Datasets:

m-a-p
/

IN-Scientific

Dataset card Data Studio Files Files and versions Community

Split (1)

train · 20.3k rows

id string	meta dict	quality_signals dict	content_image sequence	md string
0802.1098	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 68529, "num_imgs": 0, "llama3_tokens_count": 20363 }	[]	# Model dynamics on a multigrid across multiple length and time scales A. J. Roberts Computational Engineering and Sciences Research Centre, Dept Maths and Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia. ###### Abstract Most methods for modelling dynamics posit just two time scales: a fast and a slow scale. But many applications, including many in continuum mechanics, possess a wide variety of space-time scales; often they possess a continuum of space-time scales. I discuss an approach to modelling the discretised dynamics of advection and diffusion with rigorous support for changing the resolved spatial grid scale by just a factor of two. The mapping of dynamics from a finer grid to a coarser grid is then iterated to generate a hierarchy of models across a wide range of space-time scales, all with rigorous support across the whole hierarchy. This approach empowers us with great flexibility in modelling complex dynamics over multiple scales. ###### Contents Contents * 1 Introduction * 2 Centre manifold theory supports multiscale models * 3 Coarsen linear advection-dispersion * 3.1 Computer algebra approximates the slow manifold * 3.2 The coarse scale, slow dynamics model precisely * 3.3 Coarse scale dynamics are attractive * 3.4 A normal form projects initial conditions * 3.5 Extend elements for a multigrid hierarchy * 4 Multiscale modelling iterates transformations * 4.1 Diffusion emerges from discrete dispersion * 4.2 Renormalise advection-dispersion * 4.3 Approximate the nonlinear Burgers’ dynamics * 5 Conclusion ## 1 Introduction Multiscale methods promise efficient computation and simulation of many important physical systems [3, e.g.]. Dolbow et al. [10] identify that critical applications include fuel cells, subsurface contaminant transport, protein folding, climate simulations, and general networks. Most multiscale mathematical methods for modelling disparate scales presume just two scales: small lengths and large lengths; fast times and long times; fast variations and slow variations; microscopic and macroscopic [10, 11, e.g.]. Most such methods then seek _effective_ models or properties on the large/long/slow macroscales by ‘averaging/homogenising over’ the small/fast microscales [11, 22, e.g.]. Here we explore a novel mathematical framework to modelling dynamics over many length and time scales; the framework is supported by modern dynamical systems theory. Of course most two scale modelling methods will also work over many scales. The same techniques that construct and support a slow model of rapid variations may also apply to construct and support a superslow model of the slow variations. The same techniques may then also apply to construct and support a megaslow model of the superslow variations; and so on. Crucially, in most established methods each of these constructions require a large ‘spectral gap’; that is, they require an absolutely clear separation between the micro and the macro scales; a parameter such as $\epsilon$ measures the scale separation, and the requirement for extreme scale separation is provided by theorems invoking “as $\epsilon\to 0$”. In contrast, multigrid iteration for solving linear equations transforms between length scales that are different by (usually) a factor of two [5, e.g.]; some variants of multigrid iteration use an even smaller ratio of length scales [28, e.g.]. Recently Brandt [4] proposed a method for molecular dynamics without large scale separation using so-called systematic upscaling. Analogously, here we explore modelling _dynamics_ on a hierachy of length scales that differ by a factor of two and hence the ‘spectral gap’ is finite and typically much smaller than required by popular extant methods for modelling dynamics. Section 2 rigorously supports such models with centre manifold theory [6, e.g.]. Recall that multigrid iteration, using restriction and prolongation operators, transforms between length scales differing by a factor of two [5, e.g.]. The full multigrid iteration involves iterating the restriction and prolongation transformations to cross large changes in length scales by taking many coarsening or refining transforms where each step changes the resolved length scales by a factor of two. Similarly, Section 3 starts our modelling of dynamics by exploring a transformation of dynamics from one length scale to another that is a factor of two coarser. Section 3, see (13), demonstrates that to some controlled approximation the discrete advection diffusion equation \[\frac{du_{j}}{dt}=-c\rat 12(u_{j+1}-u_{j-1})+d(u_{j+1}-2u_{j}+u_{j-1})\,,\] (1) for evolving grid values $u_{j}(t)$ on a grid of spacing $h$ (and hence with ‘advection speed’ $ch$ and ‘diffusion’ $dh^{2}$) is, on the coarser grid of spacing ${\bar{h}}=2h$ , justifiably modelled by \[\frac{d{\bar{u}}_{j}}{dt}\approx-\bar{c}\rat 12({\bar{u}}_{j+1}-{ \bar{u}}_{j-1})+\bar{d}({\bar{u}}_{j+1}-2{\bar{u}}_{j}+{\bar{u}}_{j-1})\] where \[\bar{c}=\rat 12c\qtq{and}\bar{d}=\rat 14d+\frac{c^{2}}{16d}\,,\] (2) for evolving coarse grid values ${\bar{u}}_{j}$; these coarse grid values are defined to be the fine grid values at every second point on the fine grid, ${\bar{u}}_{j}=u_{2j}$ . Intriguingly, the key to the approach is to take one step backwards in order to take two steps forward: at any level we embed the dynamics (1) in a higher dimensional problem, then analysis systematically derives the lower dimensional, macroscale model (2). The geometric approach to modelling of both centre manifold theory [6, e.g.] and normal form theory [20, e.g.] justifies the model (2) using a similar approach to that of holistic discretisation [26, e.g.]. The enhancement of the diffusion by $c^{2}/(16d)$ evident in (2) on the coarse grid comes from resolving the dynamics on the finer grid in constructing the model on the coarser grid:¹ the enhancement ensures the coarse model (2) stably models the fine grid dynamics (1); intriguingly this enhanced dispersion is _precisely_ that implicit in cyclic reduction, a multigrid method, to find an equilibrium of such advection-dispersion problems, but here derived for dynamic problems with a different theoretical base. The coarse model (2) implicitly prescribes a ‘restriction operator’ that transforms the dynamics of advection-dispersion from one grid to another with twice the spacing. [FOOTNOTE:1][ENDFOOTNOTE] [FIGURE:S1.F1][ENDFIGURE] Others also explore dynamics across space-time scales. Griebel, Oeltz & Vassilevski [15] developed space-time multigrid numerics to find optimal control of problems governed by parabolic differential equations. They base their cross-scale transformation on an algebraic multigrid. The systematic upscaling by Brandt [4] uses multigrid ideas to progressively coarsen atomic simulations of polymer folding. These approaches are largely computational whereas here we develop algebraic transformations that then are used computationally. Another major difference is that the slow manifolds constructed here provides a coarsening and interpolation, across length scales, that is specifically adapted to the dynamics of the problem rather than being imposed on the problem. Section 3.1 constructs the slow manifolds by systematically approximating exact closures provided by the fine scale dynamics. Section 4 explores iterating our transformation to model dynamics across each and every intervening length scale. For example, repeating the transformation from fine (1) to coarse (2) gives a hierarchy of models all of the form of the advection-dispersion equation (1) but with differing coefficients. At the $\ell$th level, with grid spacing $h^{(\ell)}=2^{\ell}h$ , the corresponding grid values $u_{j}^{(\ell)}$ evolve according to (1) but with coefficients $c^{(\ell)}$ and $d^{(\ell)}$ determined by the recurrence \[c^{(\ell+1)}=\rat 12c^{(\ell)}\quad\text{and}\quad d^{(\ell+1)}=\rat 14d^{( \ell)}+\frac{{c^{(\ell)}}^{2}}{16d^{(\ell)}}\,.\] (3) On successively coarser grids the coefficients thus are \[c^{(\ell)}=\frac{c}{2^{\ell}}\qtq{and}d^{(\ell)}=\frac{\|c\|}{2^{\ell+1}}\tilde{ d}^{(\ell)}\qtq{where}\tilde{d}^{(\ell+1)}=\frac{1}{2}\left(\tilde{d}^{(\ell)} +\frac{1}{\tilde{d}^{(\ell)}}\right)\,.\] (4) Observe that $\tilde{d}^{(\ell)}\to 1$ quickly as $\ell$ increases (as (4) is equivalent to Newton’s iteration to find the zeros of $\tilde{d}^{2}-1$); hence as the grid coarsens, the $\ell$th level model quickly becomes simply the upwind model \[\frac{du^{(\ell)}_{j}}{dt}\approx-c\frac{u^{(\ell)}_{j}-u^{(\ell)}_{j-1}}{2^{ \ell}}\quad\text{when }c>0\,.\] Our multigrid modelling transformation naturally recognises that advection dominates diffusion on coarse grids: the cross scale transformation, the map from fine (1) to coarse (2) as summarised by (3), not only preserves the advection speed, but also models the advection in a stable scheme that preserves non-negativity. Further, in the absence of advection, $c=0$ , the transformation (3) preserves the effective diffusion across all scales: $d^{(\ell+1)}=\rat 14d^{(\ell)}$ . These are some simple results. Section 4 explores further issues in transforming both linear and nonlinear discrete dynamics across many scales. The centre manifold and normal form [12, 8, 20, e.g.] approach established here provides a framework for dynamical modelling that links what are conventionally called multigrid [5, e.g.], wavelets [9, e.g.], multiple scales [23, e.g.], and singular perturbations [36, e.g.]. This framework applies to not only the linear dynamical systems that are the main focus of this article, but also applies to nonlinear systems [26, 18, e.g.] and to stochastic systems [7, 1, 30, 34, e.g.]. Here, because it is simplest, we focus on transforming dynamics within the same algebraic form, but in principle the methodology can support the emergence, via nonlinear interactions, of qualitatively different dynamics on macroscales (as promoted by the heterogeneous multiscale method [11, e.g.]). By rationally transforming across both space and time scales, a long term aim of this approach is to empower efficient simulation and analysis of multiscale systems at whatever level of detail is required and to a controllable error. This approach to transformation from one scale to anther may in the future illuminate complex systems simulations on both lattices and with cellular automata. ## 2 Centre manifold theory supports multiscale models [FIGURE:S2.F2][ENDFIGURE] This section establishes new theoretical support for coarsening dynamics from a fine grid to a coarse grid of twice the spacing. Suppose the fine grid has grid points $x_{j}$, spacing $h$ as shown in Figure 2, and has grid values $u_{j}(t)$ evolving in time. The figure also shows the coarse grid points ${\bar{x}}_{j}=x_{2j}$ , spacing ${\bar{h}}=2h$ , and the definition of the evolving coarse grid values² [FOOTNOTE:2][ENDFOOTNOTE] \[{\bar{u}}_{j}(t)=u_{2j}(t)\,.\] (5) Mostly, an overbar denotes variables and operators on the coarser grid, and unadorned variables are those on the finer grid. Using overdots to denote time derivatives, the question is: how do we transform the evolution $\dot{u}_{j}={\cal L}u_{j}$ , for some fine scale local operator ${\cal L}$, to a coarse evolution $\dot{\bar{u}}_{j}=\bar{{\cal L}}{\bar{u}}_{j}$ on the coarse grid? The theoretical support for multiscale modelling outlined by this section applies equally well to nonlinear dynamics: Section 4.3 briefly explores the specific nonlinear advection-dispersion of a discrete Burgers’ equation. Assume the fine spatial grid is periodic with $m$ grid points: that is, for definiteness assume the grid is periodic in space $x$ with period $mh$. For conciseness, write equations in terms of centred mean $\mu$ and difference $\delta$ operators [21, Ch. 7, e.g.] acting on the fine grid. Thus the advection-dispersion equation (1), but now including some ‘nonlinearity’ $f_{j}$ with some parameters $\vec{\epsilon}$, is \[\dot{u}_{j}=\big{\{}-c\mu\delta+d\delta^{2}\big{\}}u_{j}+f_{j}(\vec{u},\vec{ \epsilon})\,.\] (6) I give three illustrative examples of such nonlinearity: a local reaction could be prescribed by $f_{j}=\epsilon u_{j}-u_{j}^{3}$ ; a nonlinear advection by $f_{j}=u_{j}\mu\delta u_{j}/(2h)$ as in the discrete Burgers’ equation (41); whereas linear diffusion in a random medium could be encompassed by $f_{j}=\epsilon\delta(\kappa_{j}\delta u_{j})/h^{2}$ for some stochastic diffusivities $\kappa_{j}$. Centre manifold supportWe now describe how to support and construct the model on the coarse grid of the fine scale, nonlinearly modified, advection-dispersion dynamics. [FIGURE:S2.F3][ENDFIGURE] Analogous to holistic discretisation of s [26, 18, e.g.], divide the $m$-periodic fine grid into $m/2$ overlapping elements. Notionally let the $j$th coarse element stretch from $x_{2j-2}$ to $x_{2j+2}$ as shown ‘exploded’ in Figure 3. As shown, denote the evolving fine grid field in the $j$th element as the 5-tuple $\vec{v}_{j}=(v_{j,-2},v_{j,-1},v_{j,0},v_{j,1},v_{j,2})$ , so that at this stage we have just renamed the fine grid variables, $u_{2j+i}=v_{j,i}$ . Note that the elements overlap: the fine grid values $u_{2j\pm 1}=v_{j,\pm 1}=v_{j\pm 1,\mp 1}$ ; this overlap empowers us to couple the dynamics in neighbouring elements to derive consistent models as similarly derived for holistic discretisation [29].³ The interelement coupling conditions (8) determine the fine grid values $v_{j,\pm 2}$, at the extremes of each element, and so these are not extra dynamic variables. But, importantly, consider the overlapping fine grid values $v_{j,\pm 1}$ and $v_{j\pm 1,\mp 1}$ as independent dynamic variables satisfying the fine scale discrete equation (6), namely [FOOTNOTE:3][ENDFOOTNOTE] \[\dot{v}_{j,i}=\big{\{}-c\mu\delta+d\delta^{2}\big{\}}v_{j,i}+f_{j}(\vec{v}_{j} ,\vec{\epsilon})\,,\quad i=0,\pm 1\,,\] (7) where these differences and means operate over the fine grid index $i$. In essence I extend the dynamics of the $m$ fine grid variables $u_{j}(t)$ by an extra $m/2$ variables. This is the ‘one step backwards’ referred to in the Introduction: in order to rigorously support the modelling of the $m$-dimensional fine scale dynamics by $m/2$ coarse scale variables, I embed the fine scale system in the $3m/2$-dimensional dynamics of these overlapping elements. Section 3.4 shows how to choose these $m/2$ extra degrees of freedom to make forecasts from any given fine grid scale initial condition. Also analogous to holistic discretisation of s [26, 18, e.g.], couple neighbouring elements with the conditions \[v_{j,\pm 2}=\bar{\gamma}v_{j\pm 1,0}+(1-\bar{\gamma})v_{j,0}\,,\] (8) where the coupling parameter $\bar{\gamma}$ controls the interaction and information flow between elements: * when $\bar{\gamma}=1$ the elements are fully coupled and the condition (8) reduces to the statement that the extrapolation of the $j$th element field to the neighbouring coarse grid points, $v_{j,\pm 2}$, is identical to the neighbouring coarse grid values $v_{j\pm 1,0}(={\bar{u}}_{j\pm 1})$; * when $\bar{\gamma}=0$ the elements are completely isolated from each other and thus, _linearly_, the new fine grid values $v_{j,i}$ evolve quickly to be constant in each element. This equilibrium when $\bar{\gamma}=0$ , or space of equilibria depending upon the nonlinearity $f_{j}$, forms the base for the slow manifold model which when evaluated at $\bar{\gamma}=1$ gives the desired model for the fully coupled dynamics. I use the overbar in $\bar{\gamma}$ because it moderates information flow between the elements forming the coarse grid. By working to an error $\Ord{\bar{\gamma}^{n}}$ we account for interactions between the dynamics in an element and its $n-1$ neighbours on either side. Thus we transform _local_ dynamics on a fine grid to _local_ dynamics on a coarse grid as in other multiscale approaches [4, e.g.]. The size of the locality depends upon the order of error in the coupling parameter $\bar{\gamma}$. The decoupled dynamics have a useful spectral gapSet $\bar{\gamma}=0$ to decouple the elements, and neglect the nonlinearity by linearisation. Then, independently of all other elements, the linear dynamics in the $j$th element are governed by the differential-algebraic system \[\begin{bmatrix}0\\ \dot{v}_{j,-1}\\ \dot{v}_{j,0}\\ \dot{v}_{j,1}\\ 0\end{bmatrix}=\begin{bmatrix}1&0&-1&0&0\\ \rat 12c+d&-2d&-\rat 12c+d&0&0\\ 0&\rat 12c+d&-2d&-\rat 12c+d&0\\ 0&0&\rat 12c+d&-2d&-\rat 12c+d\\ 0&0&-1&0&1\end{bmatrix}\begin{bmatrix}v_{j,-2}\\ v_{j,-1}\\ v_{j,0}\\ v_{j,1}\\ v_{j,2}\end{bmatrix}\,.\] (9) Seeking solutions proportional to $e^{\lambda t}$ this set of linear s has three eigenvalues and three corresponding eigenvectors: \[\lambda=0,-2d,-4d\,;\qtq{and}\begin{bmatrix}1\\ 1\\ 1\\ 1\\ 1\end{bmatrix},\quad\begin{bmatrix}0\\ \rat 12c-d\\ 0\\ \rat 12c+d\\ 0\end{bmatrix},\quad\begin{bmatrix}1\\ -1\\ 1\\ -1\\ 1\end{bmatrix}.\] (10) From these, any zig-zag structures within an element decay exponentially quickly, and hence these decoupled dynamics results in constant solutions in each element arising on a time scale of $1/d$ . Over all the $m/2$ decoupled elements these piecewise constant solutions form an $m/2$ dimensional linear subspace of equilibria, the so-called slow subspace, in the $3m/2$ dimensional state space of the fine grid values $v_{j,0}$ and $v_{j,\pm 1}$. Centre manifold theory for deterministic systems [6, 20, e.g.] or for stochastic systems [2, 1, §8.4, e.g.] then assures us of the following three part theorem. For some domain of finite non-zero coupling parameter $\bar{\gamma}$, and if nonlinear, some neighbourhood of the origin in $(\vec{u},\vec{\epsilon})$: 1. there exists an $m/2$-dimensional, invariant _slow manifold_ ${\cal M}$ of the coupled dynamics of the discrete nonlinearly perturbed, advection-dis-persion (7) with coupling conditions (8)—with a dimension corresponding to each of the $m/2$ coarse grid elements; 2. the dynamics on the slow manifold ${\cal M}$ are approached exponentially quickly, roughly like $\exp(-2dt)$, by all initial conditions $v_{j,i}(0)$ of the fine grid values in some finite neighbourhood of ${\cal M}$—that is, the slow manifold dynamics faithfully model for long times generic solutions of the coupled dynamics; 3. we may construct the slow manifold model to some order of error in $\bar{\gamma}$, $\|\vec{\epsilon}\|$ and $\|\vec{\bar{u}}\|$ by solving the governing, nolinear, discrete advection-dispersion (6) with coupling conditions (8) to residuals of the same order. Two broad cases arise: if the nonlinearity $f_{j}=0$ whenever $v_{j,i}$ is independently constant in each element—for example the Burgers’-like nonlinearity $f_{j}=u_{j}\mu\delta u_{j}/(2h)$—then the approximation is global in the coarse grid variables $\vec{\bar{u}}$; alternatively, whenever $f_{j}\neq 0$ for $v_{j,i}$ independently constant in each element—for example the reaction $f_{j}=\epsilon u_{j}-u_{j}^{3}$—then the approximation is local to the origin in $\vec{\bar{u}}$ Like systematic upscaling [4, pp.6,9] and other multiscale methods, this approach uses equilibrium concepts. But one crucial difference is that centre manifold theory guarantees that the same separation of dynamics occurs in a _finite_ neighbourhood about equilibria and hence supports the separation of coarse scale dynamics from the fine scale occurs for nontrivial dynamics. This approach provides a systematic alternative to the heuristic Fourier or wavelet decompositions for a ‘local mode analysis’ [3, §8]: here the local modes are determined by the the dynamical system itself through the shape of the slow manifold. Finite domainAfter constructing an approximate slow manifold model, we evaluate it for coupling parameter $\bar{\gamma}=1$ to recover a coarse grid model for the fully coupled dynamics on the fine grid. Is $\bar{\gamma}=1$ in the ‘finite neighbourhood’ of theoretical support? It is for the analogous holistic discretisation of the Burgers’ [26]. Similarly, Section 3.3 demonstrates that the fully coupled case, $\bar{\gamma}=1$ , is indeed within the neighbourhood of theoretical support for the linear ($f_{j}=0$) dynamics of (7). ## 3 Coarsen linear advection-dispersion Using the theoretical support of centre manifold theory established by the previous section, this section analyses linear advection-dispersion to provide the multiscale modelling results summarised in the Introduction. ### Computer algebra approximates the slow manifold Elementary algebra readily constructs general slow manifold models [25, 18, e.g.]. We solve the fine grid, linear, discrete, advection-dispersion equation (7) with coupling conditions (8) by seeking solutions parametrised by the evolving coarse grid values ${\bar{u}}_{j}(t)$: \[v_{j,i}(t)=V_{j,i}(\vec{\bar{u}},\bar{\gamma})\qtq{suchthat}\dot{\bar{u}}_{j}= G_{j}(\vec{\bar{u}},\bar{\gamma})\,,\] (11) for some functions $V_{j,i}$ and $G_{j}$ to be determined by the iterative algorithm [32]. The base approximation is the slow subspace of equilibria: \[v_{j,i}(t)=V_{j,i}(\vec{\bar{u}},\bar{\gamma})\approx{\bar{u}}_{j}\qtq{ suchthat}\dot{\bar{u}}_{j}=G_{j}(\vec{\bar{u}}_{j},\bar{\gamma})\approx 0\,.\] Computer algebra code [32] systematically refine these slow manifold approximations. The refining iteration is based upon the residuals of the discrete equation (7) with coupling conditions (8). Centre manifold theory then assures us that the error in approximating the slow manifold model is of the same order in coupling parameter $\bar{\gamma}$ as any remaining residual. For example, to errors $\Ord{\bar{\gamma}^{2}}$, computer algebra [32] constructs the slow manifold ${\cal M}$ in the $j$th element as \[\vec{V}_{j}=\begin{bmatrix}(1-\bar{\gamma})+\bar{\gamma}\bar{E}^{-1}\hfill\\ 1-\rat 12\bar{\gamma}\bar{\mu}\bar{\delta}+\rat 18\bar{\gamma}(1+\rat{c}{d}- \rat{c^{2}}{4d^{2}})\bar{\delta}^{2}\\ 1\hfill\\ 1+\rat 12\bar{\gamma}\bar{\mu}\bar{\delta}+\rat 18\bar{\gamma}(1-\rat{c}{d}- \rat{c^{2}}{4d^{2}})\bar{\delta}^{2}\\ (1-\bar{\gamma})+\bar{\gamma}\bar{E}\hfill\end{bmatrix}{\bar{u}}_{j}+\Ord{\bar {\gamma}^{2}}\,,\] (12) in terms of the coarse grid centred difference and mean operators, $\bar{\delta}$ and $\bar{\mu}\bar{\delta}$, and shift operator $\bar{E}$ (define $\bar{E}{\bar{u}}_{j}={\bar{u}}_{j+1}$ or equivalently $\bar{E}=E^{2}$). The terms in (12) which are independent of advection $c$, for the fully coupled $\bar{\gamma}=1$ , are classic quadratic interpolation from the surrounding coarse grid values ${\bar{u}}_{j}$. The terms involving advection, flagged by $c$, arise through accounting for the dynamics of the fine grid values $u_{2j\pm 1}$ and their interaction with the surrounding grid values. Equation (12) corresponds to the multigrid prolongation operator, but here it is derived by accounting for the fine scale dynamics rather than being imposed. The evolution on the slow manifold (12) is then the coarse grid model \[\dot{\bar{u}}_{j}=\bar{\gamma}\left[-\rat 12c\bar{\mu}\bar{\delta}+\left(\rat 1 4d+\frac{c^{2}}{16d}\right)\bar{\delta}^{2}\right]{\bar{u}}_{j}+\Ord{\bar{ \gamma}^{2}}\,.\] (13) Neglecting the $\Ord{\bar{\gamma}^{2}}$ error, evaluate (13) at the physically relevant coupling $\bar{\gamma}=1$ to deduce the coarse grid model (2) discussed in the Introduction. ### The coarse scale, slow dynamics model precisely Consider further the linear advection-dispersion (7) with interelement coupling conditions (8). The previous subsection constructed an approximation to errors $\Ord{\bar{\gamma}^{2}}$; this subsection gives exact formula for all coupling $\bar{\gamma}$. Seek solutions with structure within the finite elements of the formal operator form $\vec{v}_{j}=\exp(t{\cal K}_{n})\vec{e}_{n}$ , where, generalising (10) to non-zero coupling, ${\cal K}_{n}$ is the $n$th ‘operator eigenvalue’ of the advection-dispersion (7)–(8) and $\vec{e}_{n}$ is the corresponding ‘operator eigenvector’. Elementary algebra for any coupling $\bar{\gamma}$ reveals the three operator eigenvalues are precisely \[{\cal K}_{1,3}=2d\left\{-1\pm\sqrt{1+\bar{\gamma}\left[\frac{1}{4}\left(1+ \frac{c^{2}}{4d^{2}}\right)\bar{\delta}^{2}-\frac{c}{2d}\bar{\mu}\bar{\delta} \right]}\right\}\qtq{and}{\cal K}_{2}=-2d\,.\] (14) The smallest (least negative) of these, namely ${\cal K}_{1}$, governs the longest time scales in the coupled dynamics. For example, the Taylor expansion⁴ in the coupling $\bar{\gamma}$ of operator ${\cal K}_{1}$ (the plus case above), upon using the identity $\bar{\mu}^{2}=1+\rat 14\bar{\delta}^{2}$ , agrees with the $\Ord{\bar{\gamma}^{2}}$ evolution (13), to the $\Ord{\bar{\gamma}^{3}}$ approximation (40), and to the $\Ord{\bar{\gamma}^{5}}$ approximation (31) of isotropic dynamics. That is, the coarse grid evolution operator $\bar{{\cal L}}={\cal K}_{1}$ . [FOOTNOTE:4][ENDFOOTNOTE] The coarse grid operator $\bar{{\cal L}}={\cal K}_{1}$ reproduces exactly the fine grid operator of linear advection-dispersion when the elements are fully coupled. In the fully coupled limit, $\bar{\gamma}=1$ , the three operator eigenvalues (14) reduce to \[\bar{{\cal L}}={\cal K}_{1}=d(2\bar{\mu}-2)-\rat 12c\bar{\delta}\,,\quad{\cal K }_{2}=-2d\,,\quad{\cal K}_{3}=-d(2\bar{\mu}+2)+\rat 12c\bar{\delta}\,.\] (15) Relate to the fine grid operators, via the coarse grid shift operator $\bar{E}$ and the fine grid shift operator $E(=\bar{E}^{1/2})$, by observing \[2\bar{\mu}-2=\bar{E}^{1/2}+\bar{E}^{-1/2}-2=E+E^{-1}-2=\delta^{2}\] and \[\rat 12\bar{\delta}=\rat 12\big{(}\bar{E}^{1/2}-\bar{E}^{-1/2} \big{)}=\rat 12\big{(}E-E^{-1}\big{)}=\mu\delta\,.\] Consequently, the coarse grid model \[\dot{\bar{u}}_{j}=\bar{{\cal L}}{\bar{u}}_{j}=\big{[}d(2\bar{\mu}-2)-\rat 12c \bar{\delta}\big{]}{\bar{u}}_{j}=\big{[}d\delta^{2}-c\mu\delta\big{]}{\bar{u}} _{j}\,,\] is _precisely_ the fine grid, linear, advection-dispersion equation (6), except that, having half the grid points, it does not resolve the fine scale, high wavenumber, spatial structures that the fine grid can resolve. Thus the operator $\bar{{\cal L}}$ does indeed model on the coarse grid all the coarse dynamics inherent in the fine grid advection-dispersion dynamics. It is only the approximation of $\bar{{\cal L}}$ by a truncated Taylor series, such as in the $\Ord{\bar{\gamma}^{2}}$ model (13), that induces errors in the coarse scale model of the long term dynamics of linear advection-dispersion. ### Coarse scale dynamics are attractive Consider the spectrum of the advection-dispersion dynamics implicitly described by the operator eigenvalues (14). On any regularly spaced grid, the centred mean and difference operators act on Fourier modes as \[\mu e^{ikj}=\cos(k/2)e^{ikj}\qtq{and}\delta e^{ikj}=2i\sin(k/2)e^{ikj}\] for a component of spatial wavenumber $k$ relative to the grid spacing; the wavenumber domain is $-\pi<k\leq\pi$ . Upon taking the discrete Fourier transform, the operators $\bar{\mu}$ and $\bar{\delta}$ thus transform to $\cos(\bar{k}/2)=\cos k$ and $2i\sin(\bar{k}/2)=2i\sin k$ , respectively, as the fine grid wavenumber $k=\bar{k}/2$ in terms of the coarse grid wavenumber $\bar{k}$. Thus from (14) the advection-dispersion dynamics on the fine grid elements has spectrum \[\lambda_{1,3}=2d\left\{-1\pm\sqrt{1+\bar{\gamma}\left[-\left(1+ \frac{c^{2}}{4d^{2}}\right)\sin^{2}\rat{\bar{k}}2-i\frac{c}{2d}2\sin\rat{\bar{ k}}2\cos\rat{\bar{k}}2\right]}\right\}\] \[\text{and}\quad\lambda_{2}=-2d\,,\] (16) for coarse grid wavenumbers $\|\bar{k}\|\leq\pi$ . Extensive numerical computations strongly suggest that $0\leq\Re{\sqrt{\cdot}}\leq 1$ , where $\sqrt{\cdot}$ denotes the square root in (16), for all wavenumbers $\|\bar{k}\|\leq\pi$ , for all coupling $0\leq\bar{\gamma}\leq 1$ , and for all advection relative to diffusion, $c/d$. Consequently, the numerics suggest the spectral ordering $\Re\lambda_{3}\leq\Re\lambda_{2}\leq\Re\lambda_{1}\leq 0$ is maintained across the whole relevant parameter domain. Thus, not only does the coarse grid model $\dot{u}_{j}=\bar{{\cal L}}{\bar{u}}_{j}={\cal K}_{1}{\bar{u}}_{j}$ accurately model the fine grid dynamics, the coarse grid model is the _slowest_ dynamics of the fine grid advection-dispersion. _Theorem 2 ensures an atttractive slow manifold exists in some neighbourhood of coupling $\bar{\gamma}=0$ ; the spectrum (16) demonstrates that the neighbourhood extends to include the case of fully coupled elements, $\bar{\gamma}=1$ ._ We usually cannot construct slow manifolds exactly, as done above; instead we usually approximate slow manifold by a multivariate power series. Thus the practical issue is not just whether a slow manifold exists, but how well a truncated power series approximates the slow manifold. Elementary algebra shows that a Taylor series of (16) in $\bar{\gamma}$ converges at $\bar{\gamma}=1$ provided \[\big{[}(1-{\mathfrak C}^{2})^{2}\sin^{2}k+4{\mathfrak C}^{2}\big{]}\sin^{2}k<1\,,\] where ${\mathfrak C}=c/(2d)$ measures the advection relative to the dispersion. For all parameter ${\mathfrak C}$ there is a finite range of small wavenumbers $k$ satisfying this inequality. This argument leads to the following lemma. Finite truncations of the Taylor series of the slow operator eigenvalue ${\cal K}_{1}$ provide accurate approximations of the evolution of the coarse grid variables provided the solutions vary slowly enough across the grid. ### A normal form projects initial conditions Suppose we know the fine grid values $u_{j}(0)$ at the initial time $t=0$ . This subsection addresses the question: what coarse grid values should we give to ${\bar{u}}_{j}(0)$ for the coarse grid model to make accurate long term predictions? The obvious answer is wrong [24, 8, 27, e.g.]: even though we define ${\bar{u}}_{j}(t)=u_{2j}(t)$ , we nonetheless should _not_ set the initial ${\bar{u}}_{j}(0)=u_{2j}(0)$ . The reason is that the transient dynamics of the subgrid scale dynamics modifies the appropriate initial value for ${\bar{u}}_{j}(0)$; this modification is sometimes called ‘initial slip’ in physics [14, 13, e.g.]. In this subsection, a normal form coordinate transform of the fine grid dynamics clearly displays the correct initial conditions for the coarse dynamics. In this discussion, restrict attention on initial conditions to the fully coupled case of coupling $\bar{\gamma}=1$ . This restriction simplifies by avoiding the complicating detail of having variable $\bar{\gamma}$, and it focusses on the physically relevant case of full interelement coupling. Consider the spectral decomposition of the dynamics of the fine grid of all the elements. Transform the fine grid evolution to its coarse grid ‘normal form’ of the spectral decomposition \[\vec{v}_{j}(t)=\vec{e}_{j,1}{\bar{u}}_{j}(t)+\vec{e}_{j,2}{\bar{v }}_{j}(t)+\vec{e}_{j,3}{\bar{w}}_{j}(t)\,,\] where \[\dot{\bar{u}}_{j}={\cal K}_{1}{\bar{u}}_{j}\,,\quad\dot{\bar{v}}_ {j}={\cal K}_{2}{\bar{v}}_{j}\,,\quad\dot{\bar{w}}_{j}={\cal K}_{3}{\bar{w}}_{ j}\,,\] for the operators ${\cal K}_{n}$ in (15) and for intraelement structure operators \[\vec{e}_{j,2}=\begin{bmatrix}\rat 12c-d\\ 0\\ \rat 12c+d\end{bmatrix}\qtq{and}\vec{e}_{j,n}=\begin{bmatrix}\rat 12c(\bar{E}^ {-1}-1)+d(\bar{E}^{-1}+1)\\ {\cal K}_{n}+2d\\ \rat 12c(\bar{E}-1)+d(\bar{E}+1)\end{bmatrix}\] (17) for $n=1,3$ . I do not record the two extreme components $v_{j,\pm 2}$ in these $\vec{e}_{j,n}$ as $v_{j,\pm 2}$ are identical to $v_{j\pm 1,0}$ when fully coupled, $\bar{\gamma}=1$ . Within each of the fully coupled elements, a formal expression for the complete evolution on the fine grid is thus \[\vec{v}_{j}(t)=\vec{e}_{j,1}\exp(t{\cal K}_{1}){\bar{u}}_{j}(0)+\vec{e}_{j,2} \exp(t{\cal K}_{2}){\bar{v}}_{j}(0)+\vec{e}_{j,3}\exp(t{\cal K}_{3}){\bar{w}}_ {j}(0)\,,\] (18) for some constants ${\bar{u}}_{j}(0)$, ${\bar{v}}_{j}(0)$ and ${\bar{w}}_{j}(0)$. For example, from (15), when advection $c=0$ the intraelement structure operators simplify to \[\vec{e}_{j,2}\propto\begin{bmatrix}-1\\ 0\\ 1\end{bmatrix}\qtq{and}\vec{e}_{j,n}\propto\begin{bmatrix}\rat 12(\bar{E}^{-1} +1)\\ \pm\bar{\mu}\\ \rat 12(\bar{E}+1)\end{bmatrix}\approx\begin{bmatrix}1\\ \pm 1\\ 1\end{bmatrix}\] where this last approximate equality holds for fields varying slowly enough along the grids. Thus $\vec{e}_{j,1}\approx(1,1,1)$ represents the smoothest variations within each element, whereas $\vec{e}_{j,2}\approx(-1,0,1)$ and $\vec{e}_{j,3}\approx(1,-1,1)$ represents fine grid scale fluctuations within an element.⁵ Since these fine grid scale fluctuations decay rapidly in time $t$, the long term slow dynamics on the slow manifold is just the restriction of (18) to ${\bar{v}}_{j}={\bar{w}}_{j}=0$ , namely [FOOTNOTE:5][ENDFOOTNOTE] \[\vec{v}_{j}(t)=\vec{e}_{j,1}\exp(t{\cal K}_{1}t){\bar{u}}_{j}(0)\,.\] (19) We must choose the initial condition, ${\bar{u}}_{j}(0)$, for the coarse grid values so that this evolution exponentially quickly equals the fine grid dynamics $u_{j}(t)=\exp(t{\cal L})u_{j}(0)$_from the specified initial condition_. Such a choice for the initial coarse grid value ${\bar{u}}_{j}(0)$ then realises the theoretical promise by Theorem 2-2 of long term fidelity between coarse grid model and fine grid dynamics. Elementary linear algebra determines the coarse grid values ${\bar{u}}_{j}(0)$ through evaluating the general solution (18) at time $t=0$ , \[\vec{v}_{j}(0)=\vec{e}_{j,1}{\bar{u}}_{j}(0)+\vec{e}_{j,2}{\bar{v}}_{j}(0)+ \vec{e}_{j,3}{\bar{w}}_{j}(0)\,,\] (20) and then take the inner product with the left eigenvector \[\vec{z}_{j,1}=\begin{bmatrix}\rat 12c+d\\ {\cal K}_{1}+2d\\ -\rat 12c+d\end{bmatrix},\] to deduce the following lemma. For linear advection-dispersion, the initial coarse grid values are \[{\bar{u}}_{j}(0)=\frac{\vec{z}_{j,1}\cdot\vec{v}_{j}(0)}{\vec{z}_{j,1}\cdot \vec{e}_{j,1}}\,,\] (21) in terms of specified fine element values $\vec{v}_{j}(0)$. Despite the definition that the coarse grid values ${\bar{u}}_{j}(t)=u_{2j}(t)$ , the normal form coordinate transform accounts for dynamics in fast time initial transients so that the correct initial conditions for the coarse grid model is the nonlocal and weighted projection (21). The initial condition mapping (21) relates to multigrid iteration. When advection $c=0$ \[\vec{z}_{j,1}\propto\begin{bmatrix}1\\ 2\bar{\mu}\\ 1\end{bmatrix}\approx\begin{bmatrix}1\\ 2\\ 1\end{bmatrix},\] and in the case of slowly varying grid values, this projection from the fine grid initial values $\vec{v}_{j}(0)$ to the coarse grid initial values is the classic multigrid restriction operator [5, e.g.]: namely, that the coarse grid value is the average of the nearest fine grid values with a weighting of $1:2:1$ . Uniquely prescribe fine element valuesWe have an additional complication: on the fine grid, the odd grid values $u_{2j\pm 1}$ are shared between two neighbouring elements. The grid value $u_{2j+1}$ is represented as both $v_{j,1}$ and $v_{j+1,-1}$, and both of these variables are treated as separate independent variables in the dynamics on each element. We must resolve this separation. Two independent suggestions resolve the separation with the same result. My first suggestion to avoid conflict between the values of $v_{j,1}$ and $v_{j+1,-1}$ is to require that $v_{j,1}=v_{j+1,-1}$ at the initial time. The shift operators rewrite this identity as \[Ev_{j,0}=\bar{E}E^{-1}v_{j,0}\,,\] (22) where the coarse grid shift $\bar{E}$ operates on the coarse grid, first subscript of $v_{j,i}$, whereas the fine grid shift operator $E$ operates on the fine grid, second subscript of $v_{j,i}$. For a domain with $m$ fine grid points, that is, $m/2$ coarse grid elements, the compatibility condition (22) provides an additional $m/2$ constraints to determine uniquely the $3m/2$ initial values $v_{j,i}(0)$ within the elements from the $m$ fine grid values $u_{j}(0)$. My second suggestion is to choose $v_{j,\pm 1}(0)$ so that the unphysical intermediate mode vanishes in the solution (18), that is, so that ${\bar{v}}_{j}(0)=0$ in the solution (18). Then there will be no intermediate scale dynamics $\exp(-2dt)$ and the approach to the slow manifold model will be the quickest: the only rapidly decaying mode will be the $\vec{e}_{j,3}\exp({\cal K}_{3}t){\bar{w}}_{j}(0)$ mode which, from the spectrum (16), decays more rapidly than $\exp(-2dt)$. Now relate ${\bar{v}}_{j}(0)$ directly to $\vec{v}_{j}(0)$ by multiplying (20) by the left eigenvector corresponding to ${\cal K}_{2}$ namely \[\vec{z}_{j,2}=\begin{bmatrix}\rat 12c(\bar{E}-1)-d(\bar{E}+1)\\ 0\\ \rat 12c(\bar{E}^{-1}-1)+d(\bar{E}^{-1}+1)\end{bmatrix}.\] Thus, noting $v_{j,\pm 1}=E^{\pm 1}v_{j,0}$ , \[{\bar{v}}_{j}(0) \propto \left\{\left[\rat 12c(\bar{E}-1)-d(\bar{E}+1)\right]E^{-1}+\left[ \rat 12c(\bar{E}^{-1}-1)+d(\bar{E}^{-1}+1)\right]E\right\}v_{j,0}\] \[= \left\{(\rat 12c-d)(\bar{E}E^{-1}-E)+(\rat 12c+d)(\bar{E}^{-1}E-E ^{-1})\right\}v_{j,0}\,.\] Consequently, ensure the mode $\exp(-2dt)$ does not appear at all, ${\bar{v}}_{j}(0)=0$ , by requiring $(\bar{E}E^{-1}-E)v_{j,0}=0$ which is precisely (22), and by requiring $(\bar{E}^{-1}E-E^{-1})v_{j,0}=0$ which is again (22) but just shifted to the left by the multiplication by the coarse grid shift $\bar{E}^{-1}$. Thus the condition (22) ensures that neighbouring elements agree at their common points _and_ that the slow manifold, long term model is approached quickest. Choosing the embedding to $3m/2$-dimensions to satisfy (22) at the initial time ensures that (22) is satisfied for all time in the linear, advection-dispersion dynamics on the fully coupled finite elements. ### Extend elements for a multigrid hierarchy As discussed briefly in the Introduction, we aim to transform dynamics across a wide range of space-time scales using the multigrid hierarchy illustrated in Figure 1. The Introduction used a model of $\Ord{\bar{\gamma}^{2}}$, see §3.1, to transform advection-dispersion on a fine grid to advection-dispersion _of the same form_ on a coarser grid. This transform iterates simply across all scales. However, when we seek more accuracy, say errors $\Ord{\bar{\gamma}^{n}}$ for $n>2$ , the linear advection-dispersion dynamics (6) transforms into a model of the form $d{\bar{u}}_{j}/dt=G(\bar{u}_{j-n+1},\ldots,\bar{u}_{j+n-1},\bar{\gamma})$ that involves $2(n-1)$ neighbouring coarse grid values. For example, to errors $\Ord{\bar{\gamma}^{4}}$, fine scale isotropic dispersion (equation (6) with $c=f_{j}=0$) transforms to the coarser scale dispersion [32] \[\frac{d{\bar{u}}_{j}}{dt}=d\left[\rat 14\bar{\gamma}\bar{\delta}^{2}-\rat 1{64 }\bar{\gamma}^{2}\bar{\delta}^{4}+\rat 1{512}\bar{\gamma}^{3}\bar{\delta}^{6} \right]{\bar{u}}_{j}+\Ord{\bar{\gamma}^{4}}\,,\] (23) that through $\bar{\delta}^{4}$ and $\bar{\delta}^{6}$ involves $\bar{u}_{j\pm 2}$ and $\bar{u}_{j\pm 3}$ . Consequently, to empower us to transform coarse models over a hierarchy of grids we must widen the elements defined in Figure 3 to include more fine grid points. This subsection widens the elements while maintaining the spectrum (10) ensuring the centre manifold support [6, 20, e.g.]. This subsection, as seen in equation (23), avoids the overdots for time derivatives as we invoke different time scales on each level of the hierarchy. Interestingly, it eventuates that not only do we overlap the elements, but also, in some sense, overlap the time scales. The general form of linear dynamics on a gridSuppose at some level of the multigrid hierarchy we know the discrete operator governing the evolution of grid values $u_{j}(t)$. Decompose the discrete evolution operator as the sum \[\frac{du_{j}}{dt}=\left[{\cal L}_{1}+{\cal L}_{2}+{\cal L}_{3}+\cdots+{\cal L} _{n-1}\right]u_{j}\,,\] (24) where the $k$th discrete operator ${\cal L}_{k}$ has stencil width $2k+1$ ; that is, ${\cal L}_{k}u_{j}$ only involves $u_{j-k},\ldots,u_{j+k}$ . This decomposition terminates, as written in (24), when we restrict attention, by working to errors $\Ord{\gamma^{n}}$, to operators of some maximum finite width. The decomposition is not unique as specified so far; however, as apparent in (23), a specific unique decomposition naturally arises when we generate the models to errors $\Ord{\gamma^{n}}$ in some coupling parameter $\gamma$. Thus suppose there is a natural ‘ordering’ parameter $\gamma$ such that, instead of (24), the discrete evolution equation may be written \[\frac{du_{j}}{dt}=\left[{\cal L}_{1}+\gamma{\cal L}_{2}+\gamma^{2}{\cal L}_{3} +\cdots+\gamma^{n-2}{\cal L}_{n-1}\right]u_{j}\,.\] (25) At all levels, except the very finest level $0$, this natural parameter $\gamma$ is to be the coupling parameter of the finite elements of the grid one level finer than than current level. As always, we suppose that evaluation of (25) at $\gamma=1$ gives the physically relevant model (24), whereas $\gamma=0$ provides a base for theory to support models at non-zero $\gamma$. Additionally insisting on the operator ${\cal L}_{1}$ being conservative implies ${\cal L}_{1}$ must represent advection-dispersion dynamics and implies that the $\gamma=0$ dynamics, $du_{j}/dt={\cal L}_{1}u_{j}$ , provides the same sound base for applying centre manifold theory. Note that the coupling parameter of finite elements at the current level is still $\bar{\gamma}$. That is, still couple neighbouring elements with the condition (8). Anticipating the support by centre manifold theory, derived in a couple of paragraphs, we expect to construct a coarse grid model of the dynamics (25) in the form \[\frac{d\bar{u}_{j}}{dt}=\left[\bar{\gamma}\bar{{\cal L}}_{1}+\bar{\gamma}^{2} \bar{{\cal L}}_{2}+\bar{\gamma}^{3}\bar{{\cal L}}_{3}+\cdots+\bar{\gamma}^{n-1 }\bar{{\cal L}}_{n-1}\right]\bar{u}_{j}+\Ord{\bar{\gamma}^{n},\gamma^{n-1}}\,,\] (26) for some coarse grid operators $\bar{{\cal L}}_{k}$ (implicitly a function of the artificial $\gamma$) which will be of stencil width $2k+1$ as the parameter $\bar{\gamma}$ counts the number of interelement communications. The renormalising transformation requires two extra ingredients: first remove the fine grid ordering by setting $\gamma=1$ (so operators $\bar{{\cal L}}_{k}$ are no longer a function of $\gamma$); and second introduce a coarse grid time scale $\bar{t}=t/\bar{\gamma}$ (which is the same time when $\bar{\gamma}=1$), then, upon dividing by $\bar{\gamma}$, the coarse grid dynamics become \[\frac{d\bar{u}_{j}}{d\bar{t}}=\left[\bar{{\cal L}}_{1}+\bar{\gamma}\bar{{\cal L }}_{2}+\bar{\gamma}^{2}\bar{{\cal L}}_{3}+\cdots+\bar{\gamma}^{n-2}\bar{{\cal L }}_{n-1}\right]\bar{u}_{j}+\Ord{\bar{\gamma}^{n-1}}\,.\] (27) The coarse model (27) has exactly the same form as the fine model (25). By introducing the coupling (8) across all levels of the hierarchy, and by introducing a hierarchy of times, which all collapse to the same real time when $\bar{\gamma}=1$ , and working to some order of error in coupling, models of the form (25) are transformed and renormalised across the entire multigrid hierarchy. Widen the elementsAssume we wish to construct slow manifolds to errors $\Ord{\bar{\gamma}^{n},\gamma^{n-1}}$ with the aim of using centre manifold theory to support the modelling of (25) by (27). Extend Section 2 by widening the $j$th element to extend over the interval $[x_{j-n},x_{j+n}]$ and also to possess the $(2n+1)$ fine grid variables $\vec{v}_{j}=(v_{j,-n},\ldots,v_{j,n})$ . These extra variables are not extra degrees of freedom. Let these fine grid variables evolve according to \[\frac{dv_{j,i}}{dt}={\cal L}_{1}v_{j,i}+\gamma{\cal L}_{2}v_{j,i}+\gamma^{2}{ \cal L}_{3}v_{j,i}+\cdots+\gamma^{n-2}{\cal L}_{n-1}v_{j,i}\,,\quad\|i\|<n\,,\] (28) where we adopt the unusual convention that when applied within the elements, the operator ${\cal L}_{k}v_{j,i}$ is its original definition when $\|i\|+k\leq n$ but is _zero otherwise_ (for $\|i\|+k>n$). Adopting this convention ensures that the operators on the right-hand side of (28) do not ‘poke outside’ of the $j$th element; in effect, this convention truncates the sum in (28) to remain within the $j$th element. Such truncation incurs an error $\Ord{\gamma^{n+1-\|i\|}}$ in the evolution of a variable $v_{j,i}$. However, as variable $v_{j,i}$ only affects the crucial central core variables of the element, $v_{j,i^{\prime}}$ for $\|i^{\prime}\|\leq 2$ as shown in Figure 3, via terms of $\Ord{\gamma^{\|i\|-2}}$, the net effect of this conventional truncation is an error $\Ord{\gamma^{n-1}}$ which is the same as the assumed order of error of the analysis. The coupling condition (8) closes the dynamics on these widened elements. In essence we do not have new dynamics outside of the central core of each element, instead, in effect, we simply extrapolate the dynamics to the outside of the central core. Centre manifold theory supportWhen the fine grid ‘ordering’ parameter $\gamma=0$ and interelement coupling parameter $\bar{\gamma}=0$ the dynamics on the $m/2$ elements reduces to \[\frac{dv_{j,i}}{dt}={\cal L}_{1}v_{j,i}\quad\text{for }\|i\|<n\,,\qtq{and}v_{j, \pm 2}=v_{j,0}\,.\] (29) Each element is decoupled from the others. The general conservative, linear, operator is the advection-dispersion operator, ${\cal L}_{1}=-c\bar{\mu}\bar{\delta}+d\delta^{2}$ for some constants $c$ and $d$. As for the earlier (9), for each of the extended elements there are still precisely three eigenvalues of (29), namely $\lambda=0,-2d,-4d$ . Corresponding eigenvectors are the constant $v_{j,i}\propto 1$ , the artificial $v_{j,i}\propto\sin(i\pi/2)\left[({1+{\mathfrak C}})/({1-{\mathfrak C}})\right] ^{i/2}$, and the zig-zag mode $v_{j,i}\propto(-1)^{i}$ . Consequently, centre manifold theory implies Theorem 2 also applies to the system (28) with coupling conditions (8) to ensure: firstly, that an $m/2$ dimensional slow manifold exists for the dynamics of the coupled elements; secondly, the coarse scale dynamics on the slow manifold are attractive; and thirdly, that we may construct the slow manifold to any desired error—this section assumes errors $\Ord{\gamma^{n-1},\bar{\gamma}^{n}}$. The next subsection proceeds to briefly explore the resultant models of advection and dispersion over a hierarchy of multiscale grids as supported by this theory. ## 4 Multiscale modelling iterates transformations This section explores three example applications of transforming dynamics repeatedly across the wide range of length and time scales on a multigrid hierarchy. Section 4.1 shows how continuum diffusion emerges from microscale dispersion. Section 4.2 deonstrates that the nonlinear transformation from one scale to another of linear advection-dispersion has a fixed point of a stable upwind model. Section 4.3 discusses briefly the transformation of the nonlinear Burgers’ . ### Diffusion emerges from discrete dispersion The multiscale modelling of discrete dispersion, when the advection coefficient $c=0$ , reduces to a remarkably simple linear transformation. Here we explore the exact slow manifold transformation from a fine grid to a coarser grid. Iterating this transformation proves that, in the absence of advection, the continuum diffusion equation naturally emerges very quickly on macroscales. Linear dynamics which are left-right symmetric (isotropic) can be expressed in terms of only even order central differences. Our slow manifold, multiscale modelling preserves this form. Suppose the evolution at grid level $\ell$ is governed by \[\frac{du_{j}^{(\ell)}}{dt^{(\ell)}}=\sum_{p=1}^{\infty}\gamma^{p-1}c_{p}^{( \ell)}{\delta^{(\ell)}}^{2p}u_{j}^{(\ell)}\,,\] (30) for some coefficients $c_{p}$ ; for example, the second order difference coefficient $c_{1}=d$ used earlier. In practical constructions, invoking an error $\Ord{\gamma^{n-1}}$ truncates to a finite sum this ‘in principle’ infinite sum. Computer algebra [32], supported by the theory of Section 3.5, derives the dynamics at the next coarser level of the multiscale hierarchy, namely \[\frac{du_{j}^{(\ell+1)}}{dt^{(\ell+1)}} = \big{[}\rat 14c_{1}^{(\ell)}{\delta^{(\ell+1)}}^{2}\] (31) \[{}+\bar{\gamma}\Big{(}-\rat 1{64}c_{1}^{(\ell)}+\rat 1{16}c_{2}^{ (\ell)}\Big{)}{\delta^{(\ell+1)}}^{4}\] \[{}+\bar{\gamma}^{2}\Big{(}\rat 1{512}c_{1}^{(\ell)}-\rat 1{128}c_ {2}^{(\ell)}+\rat 1{64}c_{3}^{(\ell)}\Big{)}{\delta^{(\ell+1)}}^{6}\] \[{}+\bar{\gamma}^{3}\Big{(}-\rat 5{16384}c_{1}^{(\ell)}+\rat 5{409 6}c_{2}^{(\ell)}-\rat 3{1024}c_{3}^{(\ell)}+\rat 1{256}c_{4}^{(\ell)}\Big{)}{ \delta^{(\ell+1)}}^{8}\] \[{}\big{]}u_{j}^{(\ell+1)}+\Ord{\bar{\gamma}^{4}}\,.\] That is, to model level $\ell$ dynamics at level $(\ell+1)$ the coefficients in (30) transform according to the linear transform \[\vec{c}^{(\ell+1)}={\cal T}\vec{c}^{(\ell)}\qtq{where}{\cal T}=\begin{bmatrix} \rat 14&0&0&0&\cdots\\ -\rat 1{64}&\rat 1{16}&0&0&\cdots\\ +\rat 1{512}&-\rat 1{128}&\rat 1{64}&0&\cdots\\ -\rat 5{16384}&\rat 5{4096}&-\rat 3{1034}&\rat 1{256}&\\ \vdots&\vdots&\vdots&&\ddots\end{bmatrix}.\] (32) By induction, the level $\ell$ dynamics have centred difference coefficients $\vec{c}^{(\ell)}={\cal T}^{\ell}\vec{c}^{(0)}$ . Consequently, the dynamics that emerge on macroscale grids are determined by the powers ${\cal T}^{\ell}$ for large $\ell$. Since ${\cal T}$ is triangular the powers are simple: the dominant structure for large $\ell$ corresponds to that of the leading eigenvalue $1/4$; its eigenvector gives the centred difference coefficients that emerge on the macroscale as \[\vec{c}^{(\ell)}\sim\frac{c_{1}^{(0)}}{4^{\ell}}\big{(}1,-\rat 1{12},\rat 1{90 },-\rat 1{560},\ldots)\quad\text{as }\ell\to\infty\,.\] (33) Recognise in this vector the coefficients of various powers of centred differences in a discrete representation of the continuum diffusion operator $\partial^{2}/\partial x^{2}$. That is,⁶_continuum diffusion emerges on the macroscale for all isotropic, conservative, linear, continuous time, microscale dynamics provided there is some component of $\delta^{2}$ in the microscale ($c_{1}^{(0)}\neq 0$)._ [FOOTNOTE:6][ENDFOOTNOTE] What is novel here? That continuum diffusion emerges on macroscopic scales has been well known for centuries. The novelty is the centre manifold theory framework I set up to prove this well known fact. This framework illuminates issues and empowers us to analyse much more difficult nonlinear dynamics, Section 4.3, and potentially stochastic problems. Furthermore, the framework shows that a consistent truncation in the interelement coupling parameter $\bar{\gamma}$ generates a macroscopic approximation to continuum diffusion that is of the same order of error in $\bar{\gamma}$. (For example, in the Introduction we discussed multiscale modelling with truncations to $\Ord{\bar{\gamma}^{2}}$.) This consistency follows because truncating the mapping operator ${\cal T}$, given in (32), simply truncates its spectrum _and_ truncates its eigenvectors (as ${\cal T}$ is triangular). ### Renormalise advection-dispersion Now consider advection-dispersion on a multiscale hierarchy. Although the dynamics are linear, and in contrast to the previous subsection, the transformation from one level to another in the hierarchy is nonlinear. For example, suppose the microscopic dynamics is simply the discrete $\dot{u}_{j}=[-\mu\delta+\delta^{2}]u_{j}$ . Then computer algebra [32] derives that the multigrid hierarchy of dynamic models is \[\frac{du_{j}^{(0)}}{dt^{(0)}} = [-\mu\delta+\delta^{2}]u_{j}^{(0)}\,,\] (34) \[\frac{du_{j}^{(1)}}{dt^{(1)}} = \rat 12\big{[}-\mu\delta+\rat 5{8}\delta^{2}+\gamma(-\rat 1{8} \delta^{2}+\rat 5{32}\mu\delta^{3}-\rat{41}{512}\delta^{4})\big{]}u_{j}^{(1)}+ \Ord{\gamma^{2}}\,,\] (35) \[\frac{du_{j}^{(2)}}{dt^{(2)}} =\] (36) \[{}+\Ord{\gamma^{2}}\,,\] \[\frac{du_{j}^{(3)}}{dt^{(3)}} = \rat 18\big{[}-\mu\delta+0.50015\delta^{2}\] \[{}+\gamma(-0.37515\delta^{2}+0.37527\mu\delta^{3}-0.18763\delta^{ 4})\big{]}u_{j}^{(3)}+\Ord{\gamma^{2}}\,,\] \[\vdots\] \[\frac{du_{j}^{(9)}}{dt^{(9)}} = \rat 1{2^{9}}\big{[}-\mu\delta+0.50000\delta^{2}\] (38) \[{}+\gamma(-0.49805\delta^{2}+0.49805\mu\delta^{3}-0.24902\delta^{ 4})\big{]}u_{j}^{(9)}+\Ord{\gamma^{2}}\,,\] where for simplicity I omit the level of the discrete mean and difference operators. Evidently, as the level $\ell$ increases, and upon renormalising the time scale by the factor of $2^{\ell}$ (the grid step), these models approach a fixed point corresponding to an upwind model of the advection. As is well known, advection dominates diffusion on large scales. This centre manifold supported multiscale transformation preserves the advection speed, and does so with stable upwind differencing. Now explore the general mapping of linear conservative dynamics from one level on the multigrid to another. Generalise the form (30) for isotropic dynamics to the general finite difference representation of conservative linear operators: \[\frac{du_{j}}{dt}=\sum_{p=1}^{\infty}\gamma^{p-1}\left\{\sum_{k=1}^{p}\left(c_ {p,2k-1}\mu{\delta}^{2k-1}+c_{p,2k}{\delta}^{2k}\right)\right\}u_{j}\,,\] (39) for some coefficients $c_{p,k}$ where $c_{1,1}=-c$ and $c_{1,2}=d$ as used earlier. The operator in braces $\{\,\}$, called ${\cal L}_{p}$ earlier, represents a general operator of stencil width $2p+1$. For example, working to error $\Ord{\gamma^{3},\bar{\gamma}^{2}}$, computer algebra [32] derives the model at the next level of the the multiscale hierarchy to be, in gory detail, \[\frac{d\bar{u}_{j}}{d\bar{t}} = \left\{\rat 12\big{[}c_{1,1}+c_{2,1}\big{]}\bar{\mu}\bar{\delta}+ \left[\frac{1}{4}(c_{1,2}+c_{2,2})\right.\right.\] (40) \[\left.\left.\quad{}+\frac{c_{1,1}}{16c_{1,2}}(c_{1,1}+2c_{2,1}-4c _{2,3})+\frac{c_{1,1}^{2}}{16c_{1,2}^{2}}(-c_{2,2}+4c_{2,4})\right]\bar{\delta }^{2}\right\}\bar{u}_{j}\] \[{}+\bar{\gamma}\left\{\left[\frac{c_{1,1}}{16c_{1,2}}(-c_{1,1}-2c _{2,1}+4c_{2,3})+\frac{c_{1,1}^{2}}{16c_{1,2}^{2}}(c_{2,2}-4c_{2,4})\right] \bar{\delta}^{2}\right.\] \[\left.\quad{}+\left[\frac{1}{16}(-c_{1,1}-c_{2,1}-2c_{2,3})+\frac {c_{1,1}^{2}}{64c_{1,2}^{2}}(-c_{1,1}-3c_{2,1}+3c_{2,3})\right.\right.\] \[\left.\left.\qquad{}+\frac{c_{1,1}^{3}}{32c_{1,2}^{3}}(c_{2,2}-4c _{2,4})\right]\bar{\mu}\bar{\delta}^{3}\right.\] \[\left.\quad{}+\left[\frac{1}{64}(-c_{1,2}-c_{2,2}-4c_{2,4})+\frac {3c_{1,1}}{128c_{1,2}}(-c_{1,1}-c_{2,1}+c_{2,3})\right.\right.\] \[\left.\left.\qquad{}+\frac{3c_{1,1}^{4}}{1024c_{1,2}^{4}}(c_{2,2} -4c_{2,4})\right]\bar{\delta}^{4}\right\}\bar{u}_{j}\] \[{}+\Ord{\bar{\gamma}^{2}}\,.\] This general mapping from (39) to (40) governs the particular exapmple hierarchy of models (34)–(38). Fine grid scale interactions generate the nonlinear dependence upon coefficients shown in the transformation to (40). The example hierarchy (34)–(38) shows that when we scale time by a further factor of two in each transformation, to correspond to the time scale of advection of a grid of twice the spacing, then the multiscale transformation possess a fixed point. Returning to the general transformation from (39) to (40), rescaling time by a factor of two, computer algebra finds precisely two non-trivial fixed points of the multiscale transformation: \[\frac{du_{j}^{()}}{dt^{()}} = c_{}\left\{\mp\mu\delta+\rat 12\delta^{2}+\gamma\big{[}-\rat 12 \delta^{2}\pm\rat 12\mu\delta^{3}-\rat 14\delta^{4}\big{]}\right\}u_{j}^{()}+ \Ord{\gamma^{2}}\,,\] for some speed $c_{}$ (positive) which will depend upon the precise microscopic system. These fixed points are purely upwind macroscale models of the advection and dispersion dynamics. Such stable upwind models naturally emerge from our rational transformation of dynamics based upon dynamical systems theory. ### Approximate the nonlinear Burgers’ dynamics Burgers’ partial differential equation, $\partial u/\partial t+u\,\partial u/\partial x=\partial^{2}u/\partial x^{2}$ , is frequently invoked as a benchmark problem in nonlinear spatio-temporal dynamics as it involves the important physical mechanisms of dissipative diffusion and nonlinear advection. As an example of a nonlinear application of our multiscale methodology, suppose Burgers’ is spatially discretised to \[\frac{du_{j}^{(0)}}{dt^{(0)}}=\delta^{2}u_{j}^{(0)}-\alpha u_{j}^{(0)}\mu \delta u_{j}^{(0)}\,,\] (41) where the time scale $t^{(0)}$ is chosen to make the coefficient unity for the centred difference approximation $\delta^{2}u_{j}$ of the diffusion $\partial^{2}u/\partial x^{2}$. Take equation (41) to be the microscale discrete nonlinear dynamics. The parameter $\alpha$ measures the importance of the nonlinear advection on this microscopic scale. Section 2 places the coarse grid modelling of such nonlinear discrete dynamics within the purview of centre manifold theory. For relatively small parameter $\alpha$, straightforward modifications of the computer algebra for the earlier linear dynamics [32] analyses nonlinear problems. The reason is that as long as the nonlinearity is relatively weak, small $\alpha$, the dominant mechanism in each element is the linear dissipation of $\delta^{2}$ just as for the linear dynamics. Our multiscale modelling transforms the fine grid dynamics (41) into the level one dynamics (42); applying the multiscale modelling again transforms the level one dynamics (42) into the level two dynamics (43). \[\frac{du_{j}^{(1)}}{dt^{(1)}} = \left[\frac{1}{4}+\frac{1}{16}\alpha^{2}{u_{j}^{(1)}}^{2}\right] \delta^{2}u_{j}^{(1)}-\frac{\gamma}{64}\delta^{4}u_{j}^{(1)}-\frac{\alpha}{2}u _{j}^{(1)}\delta^{2}u_{j}^{(1)}\] (42) \[{}+\frac{\alpha\gamma}{64}\left[4u_{j}^{(1)}\mu\delta^{3}u_{j}^{( 1)}+(\delta^{2}u_{j}^{(1)})\mu\delta^{3}u_{j}^{(1)}+(\delta^{4}u_{j}^{(1)})\mu \delta u_{j}^{(1)}\right]\] \[{}+\Ord{\gamma^{2}+\alpha^{3}}\,,\] \[\frac{du_{j}^{(2)}}{dt^{(2)}} = \left[\frac{1}{16}+\frac{5}{64}\alpha^{2}{u_{j}^{(2)}}^{2}\right] \delta^{2}u_{j}^{(2)}-\frac{5\gamma}{1024}\delta^{4}u_{j}^{(2)}-\frac{\alpha}{ 4}u_{j}^{(2)}\delta^{2}u_{j}^{(2)}\] (43) \[{}+\frac{5\alpha\gamma}{512}\left[4u_{j}^{(2)}\mu\delta^{3}u_{j}^ {(2)}+(\delta^{2}u_{j}^{(2)})\mu\delta^{3}u_{j}^{(2)}+(\delta^{4}u_{j}^{(2)}) \mu\delta u_{j}^{(2)}\right]\] \[{}+\Ord{\gamma^{2}+\alpha^{3}}\,.\] Evidently we could continue this transformation across many more levels of a multigrid hierarchy. The transformation from (41) to (42), to (43) is based upon small nonlinearity $\alpha$. However, as we should expect, the nonlinear advection appears to become more important at larger scales: the relative magnitude of the nonlinear enhancement to dissipation, $\alpha^{2}\big{(}u_{j}^{(\ell)}\big{)}^{2}$, increases when going from level $0$ to level $2$. After transforming over enough levels, the nonlinearity will begin to dominate the linear basis of the analysis here; at that length scale I expect the discrete dynamics to morph into a qualitatively new form, one dominated by nonlinear advection. Although these emergent dynamics cannot be captured by the transformation used here, a generalisation of the algebra to being based about a nonlinear subspace of piecewise constant solutions may be feasible. In that case, the centre manifold theory of Section 2 would still support the multiscale modelling of the strongly nonlinear dynamics. ## 5 Conclusion This article introduces a new dynamical systems approach to modelling and linking dynamics across a multigrid hierarchy. Because we recover continuum diffusion, §4.1, and upwind advection, §4.2, on macroscales we are reassured that the process of modelling from one grid to the next coarser grid is indeed sound, as claimed by centre manifold theory, §2. Further, errors do not appear to accumulate when we iterate the modelling transformation across many changes in length scales. At all lengths scales in the hierarchy of models, centre manifold theory assures us that the model on each scale is exponentially attractive, §3.3, and provides an estimate of the rate of attraction. This theoretical support applies for the finite spectral gaps on the multigrid hierarchy. The geometric picture of invariant slow manifolds also provides a rationale for providing initial conditions for the models at each length scale [24, 8, 27, e.g.]. Section 3.4 connects appropriate initial conditions with the restriction projection of multigrid solution of linear equations [5, e.g.]. In addition to providing the dynamics at all length scales in the hierarchy, this approach also provides intraelement structures realised by the dynamics of the grid values: in terms of the level $(\ell+1)$ dynamic variables, equation (19) with (17) describe the corresponding structures on the level $\ell$ grid. In some sense, equation (17) provide ‘wavelets’ for each grid scale [9, 16, e.g.]. Our modelling connects the dynamics of wavelets across a hierarchy of length scales. All the analysis herein is for dynamics in one space dimension. Just as for holistic discretisation of s [19], I expect extension to higher space dimensions will be straightforward. This article focussed on dynamics which to a first approximation could be modelled by advection-dispersion equations; extension to dynamics of necessarily higher-order, such as a discrete Kuramoto-Sivashinsky equation, could also be analogous to the approach of holsitic discretisation [17, 18]. Similarly, extension to stochastic mutliscale dynamics could be analogous to that of holistic discretisation of stochastic dynamics [31, 33, e.g.]. ## References [1] Ludwig Arnold. _Random Dynamical Systems_. Springer Monographs in Mathematics. Springer, June 2003. * [2] P. Boxler. A stochastic version of the centre manifold theorem. _Probab. Th. Rel. Fields_, 83:509–545, 1989. * [3] Achi Brandt. Multiscale scientific computation: review 2001. In T. F. Chan T. J. Barth and R. Haimes, editors, _Multiscale and Multiresolution Methods: Theory and Applications_, pages 1–96. Springer–Verlag, Heidelberg, 2001. * [4] Achi Brandt. Methods of systematic upscaling. Technical report, Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, March 2006. * [5] William L. Briggs, Van Emden Henson, and Steve F. McCormick. _A multigrid tutorial, second edition_. SIAM, 2nd edition, 2001. * [6] J. Carr. _Applications of centre manifold theory_, volume 35 of _Applied Math. Sci._ Springer–Verlag, 1981. * [7] Xu Chao and A. J. Roberts. On the low-dimensional modelling of Stratonovich stochastic differential equations. _Physica A_, 225:62–80, 1996. 10.1016/0378-4371(95)00387-8. * [8] S. M. Cox and A. J. Roberts. Initial conditions for models of dynamical systems. _Physica D_, 85:126–141, 1995. 10.1016/0167-2789(94)00201-Z. * [9] Ingrid Daubechies. _Ten lectures on wavelets_. SIAM, 1992. * [10] J. Dolbow, M. A. Khaleel, and J. Mitchell. Multiscale mathematics initiative: A roadmap. Report from the 3rd DoE workshop on multiscale mathematics. Technical report, Department of Energy, USA, http://www.sc.doe.gov/ascr/mics/amr, December 2004. * [11] Weinan E, Bjorn Engquist, Xiantao Li, Weiqing Ren, and Eric Vanden-Eijnden. The heterogeneous multiscale method: A review. Technical report, http:www.math.princeton.edu/multiscale/review.pdf, 2004. * [12] C. Elphick, G. Iooss, and E. Tirapegui. Normal form reductions for time-periodically driven differential equations. _Phys. Lett. A_, 120:459–463, 1987. * [13] U. Geigenmüller, U. M. Titulaer, and B. U. Felderhof. Systematic elimination of fast variables in linear systems. _Physica A_, 119:41–52, 1983. * [14] H. Grad. Asymptotic theory of the Boltzmann equation. _Phys. Fluids_, 6:147–181, 1963. * [15] Michael Griebel, Daniel Oeltz, and Panayot Vassilevski. Space-time approximation with sparse grids. _SIAM Journal on Scientific Computing_, 28:701–727, 2006. * [16] Stéphane Jaffard, Yves Meyer, and Robert D. Ryan. _Wavelets: Tools for Science and Technology_. SIAM, 2001. * [17] T. Mackenzie and A. J. Roberts. Holistic finite differences accurately model the dynamics of the Kuramoto–Sivashinsky equation. _ANZIAM J._, 42(E):C918–C935, 2000. http://anziamj.austms.org.au/V42/CTAC99/Mack. * [18] T. MacKenzie and A. J. Roberts. Accurately model the Kuramoto–Sivashinsky dynamics with holistic discretisation. _SIAM J. Applied Dynamical Systems_, 5(3):365–402, 2006. 10.1137/050627733 http://epubs.siam.org/SIADS/volume-05/art_62773.html. * [19] Tony MacKenzie. _Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation_. PhD thesis, University of Southern Queensland, 2005. * [20] James Murdock. _Normal forms and unfoldings for local dynamical systems_. Springer Monographs in Mathematics. Springer, 2003. * [21] National Physical Laboratory. _Modern Computing Methods_, volume 16 of _Notes on Applied Science_. Her Majesty’s Stationery Office, London, 2nd edition, 1961. * [22] G. A. Pavliotis and A. M. Stuart. _An introduction to multiscale methods_. University of Warwick and Imperial College London, 2006. http://www.maths.warwick.ac.uk/~stuart. * [23] G. A. Pavliotis and A. M. Stuart. Multiscale methods: averaging and homogenization. Technical report, http://www.ma.ic.ac.uk/~pavl, 2007. * [24] A. J. Roberts. Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems. _J. Austral. Math. Soc. B_, 31:48–75, 1989. * [25] A. J. Roberts. Low-dimensional modelling of dynamics via computer algebra. _Computer Phys. Comm._, 100:215–230, 1997. * [26] A. J. Roberts. Holistic discretisation ensures fidelity to Burgers’ equation. _Applied Numerical Modelling_, 37:371–396, 2001. * [27] A. J. Roberts. Holistic projection of initial conditions onto a finite difference approximation. _Computer Phys. Comm._, 142:316–321, 2001. * [28] A. J. Roberts. Simple and fast multigrid solution of Poisson’s equation using diagonally oriented grids. _ANZIAM J._, 43(E):E1–E36, July 2001. http://anziamj.austms.org.au/V43/E025. * [29] A. J. Roberts. A holistic finite difference approach models linear dynamics consistently. _Mathematics of Computation_, 72:247–262, 2002. http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5. * [30] A. J. Roberts. A step towards holistic discretisation of stochastic partial differential equations. In Jagoda Crawford and A. J. Roberts, editors, _Proc. of 11th Computational Techniques and Applications Conference CTAC-2003_, volume 45, pages C1–C15, December 2003. [Online] http://anziamj.austms.org.au/V45/CTAC2003/Robe [December 14, 2003]. * [31] A. J. Roberts. Resolving the multitude of microscale interactions accurately models stochastic partial differential equations. _LMS J. Computation and Maths_, 9:193–221, 2006. http://www.lms.ac.uk/jcm/9/lms2005-032. * [32] A. J. Roberts. Computer algebra models dynamics on a multigrid across multiple length and time scales. Technical report, University of Southern Queensland, http://eprints.usq.edu.au/3373/, November 2007. * [33] A. J. Roberts. Subgrid and interelement interactions affect discretisations of stochastically forced diffusion. In Wayne Read, Jay W. Larson, and A. J. Roberts, editors, _Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006_, volume 48 of _ANZIAM J._, pages C168–C187. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/36, June 2007. * [34] A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. _Physica A_, 387:12–38, 2008. * [35] G. Samaey, I. G. Kevrekidis, and D. Roose. The gap-tooth scheme for homogenization problems. _SIAM Multiscale Modeling and Simulation_, 4:278–306, 2005. 10.1137/030602046. * [36] Ferdinand Verhulst. _Methods and applications of singular perturbations: boundary layers and multiple timescales_, volume 50 of _Texts in Applied Maths_. Springer, 2005.
0911.3721	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 59632, "num_imgs": 0, "llama3_tokens_count": 20963 }	[]	# Optimal Paths on the Space–Time SINR Random Graph ###### Abstract We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on the SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. This paper studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both “positive” and “negative” results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite. Poisson point process, random graph, first passage percolation, shot-noise process, SINR F. Baccelli, B. Blaszczyszyn, O. Mirsadeghi [INRIA/ENS]François Baccelli ENS DI TREC, 45 rue d’ulm, 75230 Paris, FRANCE. [INRIA/ENS and Math. Inst. University of Wrocław]BartŁomiej BŁaszczyszyn ENS DI TREC, 45 rue d’ulm, 75230 Paris, FRANCE. [INRIA/ENS and Sharif University of Technology]Mir Omid Haji Mirsadeghi ENS DI TREC, 45 rue d’ulm, 75230 Paris, FRANCE. The work of this author is part of a joint PhD programme with a co-advising by Prof. Amir Daneshgar in Tehran 60D05, 05C8090C27, 60G55 ## 1 Introduction There is a rich literature on random graphs generated over a random point process. These graphs are often motivated by physical, biological or social networks. Many interesting large scale properties of these networks related to connectivity have been studied in terms of the percolation of the associated graphs. An early example of such a study can be found in [13] where the connectivity of large networks was defined as the supercritical phase in what is today called the continuum (Boolean) percolation model. More recently, a random SINR graph model for wireless networks was studied with the same perspective in [10; 11]. The routing, and more precisely, the speed of delivery of information in networks is another example of problems, which motivated the study of the random graphs. The main object in this context is the evaluation of the so called _time constant_, which gives the asymptotic behavior of the number of edges (hops) in the paths (optimal or produced by some particular routing protocol) joining two given nodes in function of the (Euclidean) distance between these nodes, when this distance tends to infinity. In the case of a shortest (in terms of the number of hops) path, this problem is usually called the _first passage percolation problem_ and was originally stated by Broadbent and Hammersley in [8] to study the spread of fluid in a porous medium. More recently, in [20; 7], such time constants were studied on so called _small word graphs_, motivated by routing in certain social networks, where any two given nodes are joined by an edge independently with a probability that decays as some power function with the Euclidean distance between them. The complete graph on a Poisson p.p. with “nearest neighbor” routing policy was studied in this context in [6]. The first passage percolation problem on the Poisson-Delaunay graph was considered in [24; 21]. In the case of graphs whose edges are marked by some weights, one can extend the notion of time constant by studying the sum of the edge weights. First passage percolation on the complete Poisson p.p. graph, with weights proportional to some power of the distance between the nodes was studied in [14]. The present paper focuses on the speed of delivery of information in SINR graphs. In contrast to previous studies of this subject, in particular to [10; 11], we consider graphs with _space-time_ vertexes. This new model is motivated by multihop routing protocols used in wireless ad-hoc networks. In this framework, the random point process on the plane describes the locations of the users of an ad-hoc network and the discrete time dimension corresponds to successive time slots in which these nodes exchange information (here packets). As in [2] we assume the spatial Aloha policy to decide which node transmit at a given time slot. We also assume some space-time fading model (already used e.g. in [5]) to describe the variability of the wireless channel conditions (see e.g. [23]). In this _space-time SINR graph_, a directed edge represents the feasibility of the wireless transmission between two given network nodes at a given time. More precisely, the direct transmission of a packet is succeeds between two nodes in a given time slot if the ratio of the power of the signal between these nodes to the interference and noise at the receiver is larger than a threshold at this time slot. This definition has an information theoretic basis (see e.g. [23]). It is rigorously defined below using some power path-loss model and an associated shot-noise model representing the interference. We study various problems on this random graph including the law of its in- and out-degree, the number of paths originating from (or terminating at) a typical node or its connectedness. The most important results bear on the first passage percolation problem in this graph. In the case of Poisson p.p. for the node locations, we show that the time constant is infinite. We then show that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite. These results lead to bounds on the delays in ad-hoc networks which hold for all routing algorithms. This subject, or more generally, the question of the speed of the delivery of information in large wireless ad-hoc networks currently receives a lot of attention in the engineering literature see e.g. [12; 15]. The paper is organized as follows. In Section 2 we introduce the space-time SINR graph model. The results are presented in Section 3, Most of the proofs are deferred to Section 4. Some implications on routing in ad-hoc networks are presented in Section 5. ## 2 The Model ### Probabilistic Assumptions Throughout the paper we consider a simple, stationary, independently marked (i.m.) point process (p.p.) $\widetilde{\Phi}=\{(X_{i},\mathbf{e}_{i},\mathbf{F}_{i},\mathbf{W}_{i})\}$ with finite, positive intensity $\lambda$ on $\mathbb{R}^{2}$. In this model, * $\Phi=\{X_{i}\}$ denotes the locations of the network nodes on the plane $\mathbb{R}^{2}$. The following three cases regarding the distribution of $\Phi$ will be considered: * $\Phi$ is a general (stationary, non-null, with finite intensity) p.p., * $\Phi$ is a Poisson p.p., * $\Phi=\Phi_{M}+\Phi_{G}$ is the superposition of two independent p.p.s; where $\Phi_{M}$ denotes a stationary Poisson p.p. with finite, non-null intensity $\lambda_{M}$ and $\Phi_{G}=s\mathbb{Z}^{2}+U_{G}$ a stationary, periodic p.p., whose nodes constitute a square grid with edge length $s$, randomly shifted by the vector $U_{G}$ that is uniformly distributed in $[0,s]^{2}$ (this makes $\Phi_{G}$ stationary). Note that the intensity of $\Phi_{G}$ is $\lambda_{G}=1/s^{2}$. * $\mathbf{e}_{i}=\{e_{i}(n)\}_{\in n\in\mathbb{Z}}$, where $\mathbb{Z}=\{\ldots,-1,0,1,\ldots\}$ denotes integers; the variables $\{e_{i}(n):i,n\}$ are i.i.d. (in $n$ and $i$) Bernoulli random variables (r.v.s) with ${\mathbf{P}}\{\,e=1\,\}=1-{\mathbf{P}}\{\,e=0\,\}=p$, where $e$ denotes the generic r.v. for this family. We always assume $0<p<1$. The variable $e_{i}(n)$ represents the _medium access indicator_ of node $X_{i}$ at time $n$; it says whether the node transmits (case $e=1$) or not at time $n$. * $\mathbf{W}_{i}=\{W_{i}(n)\}_{n}$; $\{W_{i}(n):i,n\}$ is a family of non-negative i.i.d. r.v.s with some arbitrary distribution. The variable $W_{i}(n)$ represents the power of the _thermal noise_ at node $X_{i}$ and at time $n$. Let $W$ denote the generic r.v. for this family. * $\mathbf{F}_{i}=\{F_{i,j}(n)\}_{j,n}$; $\{F_{i,j}(n):i,j,n\}$ is a family of non-negative i.i.d. r.v.s. The variable $F_{i,j}(n)$ represents the quality of the radio channel (also called _fading_) from node $X_{i}\in\Phi$ to node $X_{j}\in\Phi$ at time $n$. The following two cases regarding the distribution of $F$ (denoting the generic random variable for this family) will be considered: * when $F$ has some arbitrary distribution with finite mean. * when $F$ has _exponential_ distribution with mean $1/\mu$. (${}^{}$)¹ [FOOTNOTE:1][ENDFOOTNOTE] To complete the probabilistic description of the model we assume that, given $\Phi$, the random elements $\{\mathbf{e}_{i}\}_{i},\{\mathbf{W}_{i}\}_{i}$ and $\{\mathbf{F}_{i}\}_{i}$ are independent. For more on this framework, which is classical, see e.g. or [3; 4; 5]. Our stationary i.m. p.p. $\tilde{\Phi}$ is considered on some probability space with probability ${\mathbf{P}}$. We will denote by ${\mathbf{P}}^{0}$ the Palm probability with respect to $\Phi$; see (9, Ch.13). Recall that it can be interpreted as the conditional probability given $\Phi$ has a point at the origin $0$ of the plane. We will denote this point (considered under ${\mathbf{P}}^{0}$) by $X_{0}$ and call it the _typical node_. Under ${\mathbf{P}}^{0}$$\tilde{\Phi}$ is also an i.m. p.p. with marks distributed as in the original law. Moreover, in the case of Poisson p.p.s, the distribution of $\Phi$ under ${\mathbf{P}}^{0}$ is equal to the distribution of $\Phi\cup\{X_{0}=0\}$ under the stationary probability ${\mathbf{P}}$ (cf. the Slivnyak-Mecke Theorem (9, p.281)). ### SINR Marks Given the i.m.p.p. $\tilde{\Phi}$ described above, we construct another family of random variables $\{\text{SINR}_{ij}(n):i,j,n\}$, which will be interpreted as the SINR observed in the channel form $X_{i}\in\Phi$ to $X_{j}\in\Phi$ at time $n$. These variables, which have an information theoretic background, will be used to assess the success of transmissions. For defining these variables, we give ourselves some non-decreasing function $l:\mathbb{R}^{+}=\{t:t\geq 0\}\to\mathbb{R}^{+}$ that we call the _path-loss function_. A special example considered in this paper (and commonly accepted in the wireless communication context) is \[l(r)=(Ar)^{\beta}\qquad\text{with some $A>0$ and $\beta>2$}\,.\] (1) Denote by $\Phi^{1}(n)=\{X_{i}:e_{i}(n)=1\}$ the point process of _transmitters_ at time slot $n$ and by $\Phi^{0}(n)=\{X_{i}:e_{i}(n)=0\}$ that of (potential) _receivers_. For a given pair receiver $X_{j}\in\Phi^{0}$ and transmitter $X_{i}\in\Phi^{1}(n)$, we will assume that $X_{j}$ receives a signal from $X_{i}$ with power $F_{i,j}(n)/l(\|{X_{j}-X_{i}}\|)$ at time $n$. Node $X_{j}$ also receives signals from _other_ transmitters $X_{k}\in\Phi^{1}(n)$, $X_{k}\not=X_{i}$ at time $n$. The total received power is equal to \[I_{i,j}(n)=\sum_{X_{k}\in\Phi^{1}(n)\setminus\{X_{i}\}}F_{k,j}(n)/l(\|{X_{k}-X_ {j}}\|)\,.\] Let also \[I_{j}(n)=\sum_{X_{k}\in\Phi^{1}(n)\setminus\{X_{j}\}}F_{k,j}(n)/l(\|{X_{k}-X_{j }}\|)\,.\] Both $I_{i,j}(n)$ and $I_{j}(n)$ are _shot-noise_ r.v.s generated by $\Phi^{1}(n)$, the fading marks and the path-loss function. The are infinite sums of non-negative r.v.s. In order to check whether these r.v.s are a.s. finite, one can use the Campbell-Little-Mecke formula (Campbell for short; cf. (9, Prop. 13.3.II)), which implies that \[\mathbf{E}^{0}\Bigl{[}\sum_{X_{k}\in\Phi^{1}(n),\|{X_{k}}\|>\epsilon}F_{k,0}(n)/ l(\|{X_{k}}\|)\Bigr{]}=p\mathbf{E}[F]\int_{\mathbb{R}^{2}\setminus[0,\epsilon]^{ 2}}1/l(\|{x}\|)\breve{M}_{[2]}(\text{d}x)\,,\] (2) where $\breve{M}_{2}(\cdot)$ is the _reduced second order moment measure_ of $\Phi$ (cf (9, p. 238)). In what follows, we will always tacitly assume that $l(\cdot),\Phi$ are such that the integral in the right-hand side of (2) is finite for some $\epsilon\geq 0$, which implies that $I_{0}(n)$ is almost surely (a.s.) finite under ${\mathbf{P}}^{0}$ for all $n$ as well as all $I_{j}(n),I_{i,j}(n)$ under ${\mathbf{P}}$. If $\Phi$ is the homogeneous Poisson p.p., we have $\breve{M}_{[2]}(\text{d}x)=\lambda\,\text{d}x$ and it is easy see that the we have finiteness for $l(\cdot)$ given by (1) for all $\epsilon>0$. It is also relatively easy to see that it holds for the Poisson$+$Grid p.p. $\Phi=\Phi_{M}+\Phi_{G}$. The _SINR at the receiver $X_{j}\in\Phi^{0}(n)$ with respect to transmitter $X_{i}\in\Phi^{1}(n)$, at time $n$_ is defined as \[\text{SINR}_{i,j}(n)=\frac{F_{i,j}(n)/l(\|{X_{i}-X_{j}}\|)}{W_{j}(n)+I_{i,j}(n)}\,.\] (3) ### Space-Time SINR Graph Let \[\delta_{i,j}(n)=\begin{cases}\mathds{1}(\text{SINR}_{i,j}\geq T)&\text{if}\ e_ {i}(n)=1,e_{j}(n)=0,i\not=j,\\ 1&\text{if}\ i=j,\\ 0&\text{otherwise,}\end{cases}\] (4) where $T>0$ is a threshold assumed to be some given constant throughout the paper. We define the space-time SINR graph ${\mathbb{G}}$ as the _directed graph_ with the set of vertexes $\Phi\times\mathbb{Z}$ and a directed edge from $(X_{i},n)$ to $(X_{j},n+1)$ if $\delta_{i,j}(n)=1$. Let us stress an important convention in our terminology. By network node, or point, we understand a point of $\Phi$. A (graph) vertex is an element of $\Phi\times\mathbb{Z}$; i.e. it represents some network node at some time. The existence of a graph edge is to be interpreted as the possibility of a successful communication between two network nodes (those involved in the edge) at time $n$. This can be rephrased as follows. Suppose that at time $n$ the network node $X_{i}$ has a packet (containing some information). Then the set of graph neighbors of the vertex $(X_{i},n)$ describes all the nodes that can decode this packet at time $n+1$. Thus any path on the graph ${\mathbb{G}}$ represents some possible route of the packet in space and time. ## 3 Results In this section we present our results on ${\mathbb{G}}$. ### Existence of Paths All the results of this section are obtained under the general p.p. and fading assumptions of Section 2, under the assumption that the finiteness of the expression in (2) is granted. Note first that ${\mathbb{G}}$ has no isolated nodes in the usual sense. Indeed, we have always $(X_{i},n)$ connected to $(X_{i},n+1)$. We will consider directed paths on ${\mathbb{G}}$ and call them paths for short. Note that these paths are self-avoiding due to the fact that there are no loops in the time dimension. Denote by ${\cal H}_{i}^{out,k}(n)$ the number of paths of length $k$ (i.e. with $k$ edges) _originating_ from $(X_{i},n)$. Similarly, denote by ${\cal H}_{i}^{in,k}(n)$ the number of such path _terminating_ at $(X_{i},n)$. In particular ${\cal H}_{i}^{out}(n)={\cal H}_{i}^{out,1}(n)$ and ${\cal H}_{i}^{in}(n)={\cal H}_{i}^{in,1}(n)$ are respectively, the out- and in-degree of the node $(X_{i},n)$. For a general p.p. $\Phi$ and a general fading model, the in-degree ${\cal H}_{i}^{in}$ of any node of ${\mathbb{G}}$ is bounded from above by the constant $\xi=1/T+2$. Proof.: Assume there is an edge to node $(X_{j},n)$ from nodes $(X_{i_{1}},n-1),\ldots,(X_{i_{k}},n-1)$, for some $k>1$ and $i_{p}\not=j$ ($p=1,\ldots,k$). Then for all such $p$ \[\frac{F_{{i_{p}},j}}{l(\|X_{i_{p}}-X_{j}\|)}\geq\frac{T}{1+T}\left(\sum_{q=1}^{k }\frac{F_{{i_{q}},j}}{l(\|X_{i_{q}}-X_{j}\|)}\right)\,.\] When summing up all these inequalities, one gets that $Tk\leq 1+T$, that is $k\leq 1/T+1$. Considering the edge from $(X_{i},n-1)$ to $(X_{i},n)$, the in-degree of any node is bounded from above by $\xi=1/T+2$. ∎∎ Let \[h^{out,k}=\mathbf{E}^{0}[{\cal H}_{0}^{out,k}(n)]=\mathbf{E}^{0}[{\cal H}_{0}^ {out,k}(0)]\] and \[h^{in,k}=\mathbf{E}^{0}[{\cal H}_{0}^{in,k}(n)]=\mathbf{E}^{0}[{\cal H}_{0}^{ in,k}(0)]\] be the expected numbers of paths of length $k$ originating or terminating at the typical node, respectively. In particular $h^{out}=h^{out,1}$ and $h^{in}=h^{in,1}$ are the mean out- and in-degree of the typical node, respectively. For a general p.p. $\Phi$ and a general fading model \[h^{in,k}=h^{out,k}\,.\] (5) Proof.: We use the mass transport principle to get that $\mathbf{E}^{0}[{\cal H}_{0}^{out,k}(0)]=\mathbf{E}^{0}[{\cal H}_{0}^{in,k}(0)]$, which implies the desired result. Indeed, Campbell’s formula and stationarity give \[\lambda h^{out,k} = \lambda\int_{[0,1)^{2}}\mathbf{E}^{0}[{\cal H}_{0}^{out,k}(0)]\, \text{d}x\] \[= \mathbf{E}\Bigl{[}\sum_{{X_{i}}\in\Phi\cap[0,1)^{2}}{\cal H}_{i}^ {out,k}(0)\Bigr{]}\] \[= \sum_{v\in\mathbb{Z}}\mathbf{E}\Bigl{[}\sum_{X_{i}\in[0,1)^{2}} \sum_{X_{j}\in[0,1)^{2}+v}\#\,\text{of paths from $(X_{i},0)$ to $(X_{j},k)$} \Bigr{]}\] \[= \sum_{v\in\mathbb{Z}}\mathbf{E}\Bigl{[}\sum_{X_{i}\in[0,1)^{2}-v} \sum_{X_{j}\in[0,1)^{2}}\#\,\text{of paths from $(X_{i},0)$ to $(X_{j},k)$} \Bigr{]}\] \[= \lambda\int_{[0,1)^{2}}\mathbf{E}^{0}[{\cal H}_{0}^{in,k}(k)]\, \text{d}x=\lambda h^{in,k}\,,\] where $\#$ denotes the cardinality. This completes the proof. ∎∎ Here are immediate consequences of the two above lemmas. Under the assumptions of Lemma 3.1: ${\mathbb{G}}$ is locally finite (both on in- and out-degrees of all nodes are ${\mathbf{P}}$-a.s. finite). * ${\cal H}_{i}^{in,k}(n)\leq\xi^{k}$${\mathbf{P}}$-a.s for all $i,n,k$. * $h^{in,k}=h^{out,k}\leq\xi^{k}$ for all $k$. For all $X_{i},X_{j}\in\Phi$ and $n\in\mathbb{Z}$, we denote by we will call _local delay_ from $X_{i}$ to $X_{j}$ at time $n$ the quantity \[L_{i,j}(n)=\inf\{k\geq n:\delta_{i,j}(k)=1\}\] with the usual convention that $\inf\emptyset=\infty$. Note that $L_{i,j}(n)$ is the length (number of edges) of the shortest path (with the smallest number of edges) from $(X_{i},n)$ to $\{X_{j}\}\times\mathbb{Z}$ among the paths contained in the subgraph ${\mathbb{G}}\cap\{X_{i},X_{j}\}\times\mathbb{Z}$ of ${\mathbb{G}}$, which is of the form \[((X_{i},n),(X_{i},n+1)),\ldots,((X_{i},n+L_{i,j}(n)-1),(X_{i},n+L _{i,j}(n))),\] \[((X_{i},n+L_{i,j}(n)),(X_{j},n+L_{i,j}(n)+1)).\] Our next result gives a condition for the local delays to be a.s. finite. Assume a general p.p. $\Phi$ and a general fading model with $F$ having unbounded support (${\mathbf{P}}\{\,F>s\,\}>0$ for all $0<s<\infty$). Then, given $\Phi$, all local delays $L_{i,j}(n)$ are ${\mathbf{P}}$-a.s. finite geometric random variables. Proof.: Due to our assumption on the independence of marks in successive time slots, given $\Phi$, the variables $\{\delta_{i,j}(n):n\in\mathbb{Z}\}$ are (i.i.d.) Bernoulli r.v. and thus $L_{i,j}(n)$ is geometric r.v. It remains to show that ${\mathbf{P}}\{\,\delta_{i,j}(0)=1\,\|\,\Phi\,\}:=\pi_{i,j}(\Phi)>0$ for ${\mathbf{P}}$-almost all $\Phi$. For this, note that \[\pi_{i,j}(\Phi)=p(1-p){\mathbf{P}}\Bigl{\{}\,F_{i,j}(0)\geq l(\|{X_{j}-X_{i}}\|) \Bigl{(}W_{j}(0)+I_{i,j}(0)\Bigr{)}\,\Bigr{\}}\,.\] Under our general assumptions (including finiteness of the expression in (2)) $I_{i,j}(0)$ is a finite random variable ${\mathbf{P}}$-a.s. The result follows from the assumption that $0<p<1$ and the fact that $F_{i,j}(0)$ is independent of $I_{i,j}(0),W_{i,j}(0)$ and has infinite support. ∎∎ The next result directly follows from Lemma 3.1. Under the assumptions of Lemma 3.1, ${\mathbb{G}}$ is ${\mathbf{P}}$-a.s. _connected_ in the following _weak sense_: for all $X_{i},X_{j}\in\Phi$ and all $n\in\mathbb{Z}$, there exists a path from $(X_{i},n)$ to the set $\{(X_{j},n+l):l\in\mathbb{N}\}$, where $\mathbb{N}=\{1,2,\ldots\}$. We denote by $L_{i}(n)=\inf_{j\not=i}L_{i,j}$ the length of a shortest directed path from $(X_{i},n)$ to $(\{\Phi\setminus X_{i}\})\times\mathbb{Z}$. We will call $L_{i}(n)$ the _exit delay_ from $X_{i}$ at time $n$. Finally, we denote by $P_{i,j}(n)$ the length of a shortest path of ${\mathbb{G}}$ from $(X_{i},n)$ to $\{X_{j}\}\times\mathbb{Z}$. We call $P_{i,j}(n)$ the _delay_ from $X_{i}$ to $X_{j}$ at time $n$. Obviously for $i\not=j$ we have \[L_{i}(n)\leq P_{i,j}(n)\leq L_{i,j}(n)\] (6) and thus it follows immediately from Lemma 3.1 that all the three collections of delays finite r.v.s ${\mathbf{P}}$-a.s. ### Optimal Paths — Poisson p.p. Case We have seen in the previous section that under very general assumptions, all the delays are ${\mathbf{P}}$-a.s. finite random variables. In this section we show that under some natural assumptions (such as Poisson p.p. and exponential fading), the averaging over $\Phi$ may lead to _infinite_ mean values. This averaging is expressed in terms of the expectation for the typical node under the Palm probability. The proofs of the results stated in what follows are given in Section 4.1. Denote $\ell=\mathbf{E}^{0}[L_{0}(n)]=\mathbf{E}^{0}[L_{0}(0)]$. Assume $\Phi$ to be a Poisson p.p., $F$ to be exponential and the noise $W$ to be bounded away from $0$: ${\mathbf{P}}\{\,W>w\,\}=1$ for some $w>0$. Let the path-loss function be given by (1). Then ${\mathbf{P}}^{0}\{\,L_{0}(0)\geq q\,\}\geq 1/q$ for $q$ large enough. Under the assumptions of Proposition 3.2, we have: * The mean exit delay from the typical node is infinite; $\ell=\infty$. * In any given subset of plane with positive Lebesgue measure, at a given time, the expected number of points of $\Phi$ which have exit delays larger than $q$ decreases not faster than $1/q$ asymptotically for large $q$. The fact that the mean exit delay from the typical point is infinite ($\ell=\infty$) seems to be a consequence of the potential existence of arbitrarily large “voids” (disks without points of $\Phi$) around this point. Indeed, when conditioning on the existence of another point in the configuration $\Phi$, one obtains finite mean local delays. This will be shown in Proposition 3.2 below. Before stating it we need to formalise the notion of existence of _two given points $X,Y\in\mathbb{R}^{2}$ of $\Phi$_. For this, we consider $\Phi$ under the two-fold Palm probability ${\mathbf{P}}^{X,Y}$. Since our results on the matter bear only on the Poisson p.p. case, we can assume (by Slivnyak’s Theorem) the following version of the Palm probability of the Poisson p.p. $\Phi$: \[{\mathbf{P}}^{X,Y}\{\,\Phi\in\cdot\,\}={\mathbf{P}}\Bigl{\{}\,\Phi\cup\{X,Y\} \in\cdot\,\Bigl{\}}\,.\] (7) Moreover, under ${\mathbf{P}}^{X,Y}$, the marked Poisson p.p. $\tilde{\Phi}$ is obtained by an independent marking of the points of $\Phi\cup\{X,Y\}$ according to the original distribution of marks. Slightly abusing the notation, we denote by $L_{X,Y}(n)$ the local delay from $X$ to $Y$ at time $n$ when considered under ${\mathbf{P}}^{X,Y}$. Similar convention will be adopted in the notation of other types of delays under the Palm probabilities ${\mathbf{P}}^{X}$ or ${\mathbf{P}}^{X,Y}$. Assume $\Phi$ to be a Poisson p.p., $F$ to be exponential and the noise $W$ to have a general distribution. Then for all $X,Y\in\mathbb{R}^{2}$, the mean local delay from $X$ to $Y$ is finite _given the existence of these two points in $\Phi$_. More precisely, \[\mathbf{E}^{X,Y}[L_{X,Y}(0)]<\infty\,.\] (8) The next result follows immediately from (6). Under the assumptions of Proposition 3.2, \[\mathbf{E}^{X,Y}[L_{X}(0)]\leq\mathbf{E}^{X,Y}[P_{X,Y}(0)]<\mathbf{E}^{X,Y}[L_ {X,Y}(0)]<\infty\,.\] (9) The following result is our main “negative” result concerning ${\mathbb{G}}$ in the Poisson p.p. case: Under the assumptions of Proposition 3.2, we have \[\lim_{\|{X-Y}\|\to\infty}\frac{\mathbf{E}^{X,Y}[P_{X,Y}(0)]}{\|{X-Y}\|}=\infty\,.\] (10) In other words, the _expected shortest delay necessary to send a packet between two given points of the Poisson p.p. grows faster than the Euclidean distance between these two points_. ### Filling in Poisson Voids In this section we show that adding an independent periodic pattern of points to the Poisson p.p. allows one to get a linear scaling of the shortest path delay with Euclidean distance. In order to prove the _existence and finiteness_ of the associated time constant, we adopt a slightly different approach to the notion of paths on ${\mathbb{G}}$, which will allow us to exploit a subadditive ergodic theorem. The proofs of the results stated in what follows are given in Section 4.2. For $x\in\mathbb{R}^{2}$, let $X(x)$ be the point of $\Phi$ which is closest to $x$. The point $X(x)\in\Phi$ is a.s. well defined for all given $x\in\mathbb{R}^{2}$ since $\Phi$ is assumed simple and stationary p.p. For all $x,y\in\mathbb{R}^{2}$, define $P(x,y,n)=P_{X(x),X(y)}(n)$ to be the length of a shortest path of ${\mathbb{G}}$ from vertex $(X(s),n)$ to the set $\{(X(y),n+l),l\in\mathbb{N}\}$. We will call $P(x,y,n)$ the _delay_ from $x$ to $y$ at time $n$. For all triples of points $x,y,z\in\mathbb{R}^{2}$, we have \[P(x,z,n)\leq P(x,y,n)+P\Bigl{(}y,z,n+P(x,y,n)\Bigr{)}\,.\] (11) Let \[p(x,y,\Phi)=\mathbf{E}[P(x,y,0)\,\|\,\Phi]\,.\] (12) Using the strong Markov property, we get that, conditionally on $\Phi$, the law of $P(y,z,n+P(x,y,n))$ is the same as that of $P(y,z,n)$. Then, the last relation and (11) give \[p(x,z,\Phi)\leq p(x,y,\Phi)+p(y,z,\Phi)\,.\] (13) We are now in a position to use the subadditive ergodic theorem to show the existence of the time constant \[\kappa_{\mathrm{d}}=\lim_{t\to\infty}\frac{p(0,t\mathrm{d},\Phi)}{t}\,,\] where $\kappa_{\mathrm{d}}$ may depend on the _unit vector_$\mathrm{d}\in\mathbb{R}^{2}$ representing the direction in which the delay is measured. Here is the main result of this section. Consider the Poisson$+$Grid p.p. defined in Section 2.1 with exponential fading $F$ and with the path-loss function be given by (1). Then, for all unit vectors $\mathrm{d}\in\mathbb{R}^{2}$, the non-negative limit $\kappa_{\mathrm{d}}$ exists and is ${\mathbf{P}}$-a.s. _finite_. The convergence also holds in $L_{1}$. Notice that $\kappa_{\mathrm{d}}$ is not a constant. Indeed, the superposition of the p.p.s $\Phi=\Phi_{M}$ and $\Phi_{G}$ is ergodic but not mixing due to the fact that $\Phi$ is a (stationary) grid. For $\mathrm{d}$ parallel to say the horizontal axis of the grid $\Phi_{G}$, the limit $\kappa_{\mathrm{d}}$ will depend on the distance from the line $\{t\mathrm{d}:t\in\mathbb{R}\}$ to the nearest parallel (horizontal) line of the grid $\Phi_{G}$, i.e. on the shift $U_{G}$ of the grid. Here is a more precise formulation of the result. Under the assumptions of Proposition 3.3, the limit $\kappa_{\mathrm{d}}=\kappa_{\mathrm{d}}(U_{G})$ is measurable w.r.t. the shift $U_{G}$ of the grid p.p. $\Phi_{G}$ and _does not_ depend on the Poisson component $\Phi_{M}$ of the p.p. $\Phi$. Moreover, the set of vectors $\mathrm{d}$ in the unit sphere for which $\kappa_{\mathrm{d}}(U_{G})$ is not ${\mathbf{P}}$-a.s. a constant is at most countable. The last result on this case is: Under the assumptions of Proposition 3.3, suppose that $W$ is constant and strictly positive. Then $\mathbf{E}[\kappa_{\mathrm{d}}]>0$. Finally let us remark that the method used in this section cannot be used in the case of the Poisson p.p. (without the addition of the grid point process). The main problem is the lack of integrability of $p(x,y,\Phi)$ as stated in the following result. Note however, that this does _not_ imply immediately that $\kappa_{\mathrm{d}}=\infty$. Under the assumptions of Proposition 3.2$\mathbf{E}[p(x,y,\Phi)]=\infty$ for all $x$ and $y$ in $\mathbb{R}^{2}$. ## 4 Proofs Consider the shortest path from $(X_{i},n)$ to $(\Phi\setminus\{X_{i}\})\times\mathbb{Z}$. Let ${\cal T}_{i}(n)$ be the number of edges $(X_{i},k),(X_{i},k+1)$ in this path such that $e_{i}(k)=1$. These variables are the _number of trials_ before the first exit form $X_{i}$ at time $n$. Obviously \[{\cal T}_{i}(n)\leq L_{i}(n)\,.\] (14) We will also consider an auxiliary graph $\widehat{\mathbb{G}}$, called the _(space-time) Signal to Noise Ratio (SNR) graph_, defined exactly in the same manner as the SINR graph ${\mathbb{G}}$ except that the variables $\text{SINR}_{i,j}(n)$ defined in (15) are replaced by the variables \[\text{SNR}_{i,j}(n)=\frac{F_{i,j}(n)/l(\|{X_{i}-X_{j}}\|)}{W_{j}(n)}\,.\] (15) Note that this modification consists in suppressing the interference term $I_{i,j}(n)$ in the SINR condition in (4). The edges of ${\mathbb{G}}$ form a subset of the edges of $\widehat{\mathbb{G}}$ (both graph share the same vertexes), which will be denoted by \[{\mathbb{G}}\subset\widehat{\mathbb{G}}\,.\] (16) In what follows we will denote the delays, local delays, exit delays and numbers of trials related to $\widehat{\mathbb{G}}$ by $\widehat{P}_{i,j}(n),\widehat{L}_{i,j}(n),\widehat{L}_{i}(n)$ and $\widehat{\cal T}_{i}(n)$, respectively. The inclusion ${\mathbb{G}}\subset\widehat{\mathbb{G}}$ implies immediately that $\widehat{P}_{i,j}(n)\leq P_{i,j}(n)$ and the same inequalities hold for the three other families of variables mentioned above. ### Proofs of Results of Section 3.2 Proof.: (_of Proposition 3.2_) The inclusion (16) and the inequality (14) yield \[\widehat{\cal T}_{i}(n)\leq{\cal T}_{i}(n)\leq L_{i}(n)\,,\] which holds for all $i,n$. The results follow from the above inequalities and the next lemma. ∎∎ Under the assumptions of Proposition 3.2, ${\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)\geq q\,\}\geq 1/q$ for $q$ large enough. Proof.: Under ${\mathbf{P}}^{0}$, denote by $\tau_{k}$ the $k\,$th time slot in $\{0,1,\ldots\}$, such that $e_{0}(k)=1$. For all $q\geq 0$ we have \[{\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)>q\,\|\,\Phi\,\} = {\mathbf{P}}^{0}\Bigl{\{}\,\forall_{0\leq k\leq q}\forall_{0\not= X_{i}\in\Phi}\;\delta_{0,i}(\tau_{k})=0\,\Bigl{\|}\,\Phi\,\Bigr{\}}\] \[= {\mathbf{P}}^{0}\Bigl{\{}\,\forall_{0\leq k\leq q}\forall_{0\not= X_{i}\in\Phi}\;e_{i}(\tau_{k})=1\text{\;or\;}\text{SNR}_{0,i}(\tau_{k})<T\, \Big{\|}\,\Phi\,\Bigr{\}}\] and by the conditional independence of marks given $\Phi$ \[{\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)>q\,\|\,\Phi\,\} = \prod_{0\not=X_{i}\in\Phi}\Bigl{(}p+(1-p){\mathbf{P}}\{F<Tl(\|{X_{ i}}\|)W\}\Bigr{)}^{q}\] \[= \exp\Bigl{\{}q\sum_{0\not=X_{i}\in\Phi}\log\Bigl{(}p+(1-p)(1-e^{- \mu Tl(\|{X_{i}}\|)W})\Bigr{)}\Bigr{\}}\,,\] where $F,W$ are independent generic random variables representing fading and thermal noise, independent of $\Phi$, $F$ is exponential with mean $1/\mu$, Using the Laplace functional formula for $\Phi$ and the assumption that $W>w$ a.s. we have \[{\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)\geq q\,\} \geq \exp\left(-2\pi\lambda\int_{v>0}\left(1-\left(1-(1-p)e^{-w\mu l(v )T}\right)^{q}\right)v\,\text{d}v\right)\] (17) \[= \exp\left(-\pi\lambda\int_{v>0}\left(1-\left(1-f(v)\right)^{q} \right)\,\text{d}v\right)\,,\] where \[f(v):=(1-p)\exp(-Kv^{\beta/2})\quad\text{and}\quad K=w\mu TA^{\beta}\,.\] In what follows we will show that the expression in (17) is not smaller than $1/q$ for $q$ large enough. To this regard denote by $v_{q}$ the unique solution of $f(v)=\frac{1}{q}$. We have \[v_{q}=\frac{1}{A^{2}\left(\mu Tw\right)^{2/\beta}}(\log(q(1-p)))^{2/\beta}.\] It is clear that $f(v)$ tends to 0 when $v$ tends to infinity and that $v_{q}$ tends to infinity as $q$ tends to infinity. Therefore, there exists a constant $Q=Q(\mu,w,A,T)<\infty$ such that for all $q\geq Q$ and for all $v\geq v_{q}$, \[(1-f(v))\geq\exp(-f(v)).\] Hence, for all $q\geq Q$, \[\int_{v>0}\left(1-\left(1-f(v)\right)^{q}\right)\,\text{d}v \leq\] \[\leq v_{q}+\int_{v=v_{q}}^{\infty}\left(1-\exp(-qf(v)\right)\,\,\text {d}v\] \[\leq v_{q}+\int_{v_{q}}^{\infty}qf(v)\,\,\text{d}v\] \[= v_{q}+\int_{u=0}^{\infty}qf(u+v_{q})\,\,\text{d}u.\] The third inequality follows from the fact that $1-\exp(-x)\leq x$. Using now the fact that $(u+v_{q})^{\beta/2}\geq u+v_{q}^{\beta/2}$ (for $q$ large enough, say again $q\geq Q$) we get that \[\int_{u=0}^{\infty}qf(u+v_{q})\,\,\text{d}u = \int_{u=0}^{\infty}q(1-p)\exp(-K(u+v_{q})^{\beta/2}))\,\,\text{d}u\] \[\leq \int_{u=0}^{\infty}q(1-p)\exp(-Ku-Kv_{q}^{\beta/2})\,\,\text{d}u= \frac{1}{K}\,,\] since $(1-p)\exp(-Kv_{q}^{\beta/2})=1/q$. Hence for $q\geq Q$ \[\int_{v>0}\left(1-\left(1-f(v)\right)^{q}\right)\,\,\text{d}v\leq v_{q}+\frac{ \alpha}{K}.\] Also it is not difficult to see that $\beta>2$ implies \[v_{q}\leq\frac{\log q}{\pi\lambda}-\frac{1}{K}\] (18) for $q$ large enough. This implies for $q$ large enough, say again $q\geq Q$, \[\exp\left(-\pi\lambda\int_{v>0}\left(1-\left(1-f(v)\right)^{q}\right)\,\text{d }v\right)\geq\exp\Bigl{(}-\pi\lambda(v_{q}+1/K)\Bigr{)}\geq\frac{1}{q}\,,\] (19) which completes the proof. ∎∎ Proof.: (_of Proposition 3.2_). Assume without loss of generality $Y=0$ and $\|{X}\|=r$. Under ${\mathbf{P}}$, consider the p.p. $\Phi\cup\{X,0\}$ and its independent marking. Given $\Phi$, the r.v. $L_{X,0}(0)$ associated with the independently marked p.p. $\Phi\cup\{X,0\}$ has a geometric distribution with parameter \[\pi_{X,0}(\Phi)=p(1-p)\Pr\Bigl{\{}\,F\geq l(r)(W+I)\Bigr{)}\,\Bigr{\}}\,,\] where $F,W,I$ are independent r.v.s, $F,W$ are generic fading and noise variables and $I=\sum_{X_{i}\in\Phi}e_{i}(0)F_{i,0}(0)/l(\|{X_{i}}\|)$. Using the exponential distribution of $F$ and the independence, we obtain \[\pi_{X,0}(\Phi)=\mathbf{E}[e^{-\mu l(r)TW}]\;\mathbf{E}[e^{-\mu l(r)TI}\,\|\, \Phi]\,.\] The mean of the geometric r.v. is known to be $\mathbf{E}^{X,0}[L_{X,0}(0)\,\|\,\Phi]=1/\pi_{X,0}(\Phi)$. By unconditioning with respect to $\Phi$, one obtains The first factor in the above expression is obviously finite. In what follows we will evaluate the second one. By the conditional independence of marks and denoting by ${\cal L}_{eF}(\cdot)$ is the Laplace transform of $eF$, where $e,F$ are independent generic variables for $e_{i}(0)$ and $F_{i,0}(0)$ we have \[= \left(\mathbf{E}\left[\exp\Bigl{(}-\mu l(r)T\sum_{X_{i}\in\Phi}e_ {i}(0)F_{i,0}(n)/l(\|{X_{i}}\|)\Bigr{)}\,\Big{\|}\,\Phi\right]\right)^{-1}\] \[= \exp\left(\sum_{X_{i}\in\Phi}\log{\cal L}_{eF}\Bigl{(}\mu Tl(r)/l (\|{X_{i}}\|)\Bigr{)}\right)\,.\] Note that ${\cal L}_{eF}(\xi)=1-p+p{\cal L}_{F}(\xi)=1-p+p\mu/(\mu+\xi)$. Using this and the Laplace functional formula for $\Phi$, (cf. (9, Eq. 9.4.17)) we obtain \[\mathbf{E}\Bigl{[}\frac{1}{\mathbf{E}[e^{-\mu l(r)TI}\,\|\,\Phi]} \Bigr{]} = \exp\biggl{\{}2\pi p\lambda\int_{0}^{\infty}\frac{vTl(r)}{l(v)+(1 -p)Tl(r)}\,\,\text{d}v\biggr{\}}\,.\] (cf. (2) Using now the fact that for the Poisson p.p., $\breve{M}_{[2]}(\text{d}x)=\lambda\text{d}x$), it is now easy to see that for any path-loss function satisfying $\int_{\epsilon}^{\infty}v/l(v)\,\text{d}v<\infty$, the integral in the exponent of the last expression is finite. This completes the proof. ∎∎ Proof.: (_of Proposition 3.2_). Using the the inclusion (16), inequality (14) and the left-hand side of (6) and we have \[\widehat{\cal T}_{i}(n)\leq{\cal T}_{i}(n)\leq L_{i}(n)\leq P_{i,j}(n)\,.\] Thus, it is enough to show \[\lim_{\|{X-Y}\|\to\infty}\frac{\mathbf{E}^{X,Y}[\widehat{\cal T}_{X}(0)]}{\|{X-Y} \|}=\infty\,.\] Without loss of generality assume $X=0$ and $\|{Y}\|=r$. Using the same arguments as in the proof of Lemma 4.1 and the representation (7) of the Palm probability with respect to Poisson p.p., we obtain \[{\mathbf{P}}^{0,Y}\{\,\widehat{\cal T}_{0}(0)>q\,\|\,\Phi\,\}\] \[\geq \prod_{0,Y\not=X_{i}\in\Phi}\Bigl{(}p+(1-p){\mathbf{P}}\{F<Tl(\|{X _{i}}\|)W\}\Bigr{)}^{q}\;\Bigl{(}p+(1-p){\mathbf{P}}\{F<Tl(\|{Y}\|)W\}\Bigr{)}^{q}\] \[\geq \exp\left(-\pi\lambda\int_{v>0}\left(1-\left(1-f(v)\right)^{q} \right)\,\text{d}v\right)\;\alpha(r)^{q}\,,\] where $\alpha(r)=1-(1-p)e^{-w\mu A^{\alpha}Tr^{\beta}}$. Using (19), which holds for large $q$, more precisely $q>Q=Q(\mu,w,A,T)$, we obtain \[\frac{\mathbf{E}^{0,Y}[\widehat{\cal T}_{0}(0)]}{r}\geq\frac{1}{r}\sum_{q>Q} \frac{\alpha(r)^{q}}{q}\,.\] It is now easy to see that \[\lim_{r\to\infty}\frac{1}{r}\sum_{q>Q}\frac{\alpha(r)^{q}}{q}=\infty.\] ∎∎ ### Proofs of Results of Section 3.3 Denote by $B_{x}(R)$ the ball centered at $x\in\mathbb{R}^{2}$ of radius $R$. Similarly as for the delays, we extend the definition of the local delays to arbitrary pairs of points $x,y\in\mathbb{R}^{2}$ by taking $L(x,y,n)=L_{X(x),X(y)}(n)$. We first establish the following technical result: Under the assumptions of Proposition 3.3 let $X_{i},X_{j}\in\Phi\cap B_{0}(R)$ for some $R>0$, where $\Phi=\Phi_{M}+\Phi_{G_{s}}$. Then the conditional expectation of the local delay $L_{i,j}(0)$ given $\Phi$ satisfies \[\mathbf{E}[L_{i,j}(0)\,\|\,\Phi]\] \[= \frac{1}{p(1-p){\cal L}_{W}(T\mu A^{\beta}\|{X_{i}-X_{j}}\|^{\beta} )}\exp\Bigl{\{}-\hskip-5.690551pt\sum_{\Phi\ni X_{k},k\not=i,k}\log{\cal L}_ {eF^{\prime}}\Bigl{(}\frac{T\|{X_{i}-X_{j}}\|^{\beta}}{\|X_{j}-X_{k}\|^{\beta}} \Bigr{)}\Bigr{\}}\] \[\leq \frac{1}{p(1-p){\cal L}_{W}(T\mu(A2R)^{\beta})}\] \[\times e^{-\Phi_{M}(B_{0}(2R))\log(1-p)}\hskip 277.41437pt(b)\] where $C(s,\beta)<\infty$ is some constant (which depends on $s$ and $\beta$ but not on $\Phi$), $F^{\prime}$ is an exponential random variable of mean 1 and ${\cal L}_{eF^{\prime}}(\cdot)$ is the Laplace transform of $eF^{\prime}$. Proof.: We first prove the equality in (4.2). When using the independence assumptions, we have \[{\mathbf{P}}\left\{\,L_{i,j}(0)>m\mid\Phi\,\right\}\] \[= {\mathbf{P}}\left\{\,\forall_{n=1}^{m}\left(e_{j}(n)=1\;\text{or} \;\right.\right.\] \[\left.\left.e_{j}(n)=0\;\text{and}\;e_{i}(n)F_{i,j}(n )\leq Tl(\|{X_{i}-X_{j}}\|)(W_{j}(n)+I_{i,j}(n))\right)\,\right\}\] \[= \prod_{n=1}^{m}\Biggl{(}p+(1-p)\biggl{(}1-p+p\Bigl{(}1-{\cal L}_{ W}(T\mu A^{\beta}\|x-y\|^{\beta})\] \[\times\prod_{\Phi \ni X_{k},k\not=i,j}{\cal L}_{eF^{\prime}}\Bigl{(}\frac{T\|{X_{i}-X_{j}}\|^{ \beta}}{\|{X_{j}-X_{k}}\|^{\beta}}\Bigr{)}\Bigr{)}\biggr{)}\Biggr{)}\,.\] The result then follows from the evaluation of \[\mathbf{E}\left[L_{i,j}(0)\,\mid\,\Phi\right]=\sum_{m=0}^{\infty}{\mathbf{P}} \left\{\,L_{i,j}(0)>m\,\mid\,\Phi\right]\,.\] The bound $\|X_{i}-X_{j}\|\leq 2R$ used in the Laplace transform of $W$ leads to the first factor of the upper bound. We now factorize the exponential function in (4.2) as the product of three exponential functions \[\alpha := \exp\Bigl{\{}-\sum_{\Phi_{G_{s}}\ni X_{k},k\not=i,j}\Bigr{\}},\] \[\beta := \exp\Bigl{\{}-\sum_{\Phi_{M}\ni X_{k},k\not=i,j\|X_{k}\|\leq 2R} \Bigr{\}},\] \[\gamma := \exp\Bigl{\{}-\sum_{\Phi_{M}\ni X_{k},\|X_{k}\|>2R}\Bigr{\}}.\] Next we prove that the last three exponentials are upper-bounded by (a), (b) and (c) in (4.2), respectively. * We use $\|{X_{i}-X_{j}}\|\leq 2R$ and Jensen’s inequality to get \[\log{\cal L}_{eF^{\prime}}\Bigl{(}\frac{T\|{X_{i}-X_{j}}\|^{\beta}} {\|X_{j}-X_{k}\|^{\beta}}\Bigr{)} \geq \log{\cal L}_{eF^{\prime}}\Bigl{(}\frac{T(2R)^{\beta}}{\|X_{j}-X_{ k}\|^{\beta}}\Bigr{)}\] \[\geq \frac{-T(2R)^{\beta}\mathbf{E}[eF^{\prime}]}{\|X_{j}-X_{k}\|^{\beta}}\] \[= -pT(2R)^{\beta}\|X_{j}-X_{k}\|^{-\beta}\,.\] We now prove that \[\sum_{\Phi_{G_{s}}\ni X_{k}:\|X_{j}-X_{k}\|>3\sqrt{2}s}\|X_{j}-X_{k}\|^{-\beta} \leq C(s,\beta),\] for some constant $C(s,\beta)$. This follows from an upper-bounding of the value of $\|X_{j}-X_{k}\|^{-\beta}$ by the value of the integral $1/s^{2}\int(\|X_{j}-x\|-\sqrt{2}s)^{-\beta}\,\,\text{d}x$ over the square with corner points $X_{k}$, $X_{k}+(s,0)$, $X_{k}+(0,s)$ and $X_{k}+(s,s)$. In this way we obtain \[\sum_{\Phi_{G_{s}}\ni X_{k}:\|X_{j}-X_{k}\|>3\sqrt{2}s}\|X_{j}-X_{ k}\|^{-\beta} \leq \frac{1}{s^{2}}\int_{\|x-X_{j}\|>2\sqrt{2}s}^{\infty}(\|X_{j}-x\|- \sqrt{2}s)^{-\beta}\,\,\text{d}x\] \[= \frac{2\pi}{s^{2}}\int_{\sqrt{2}s}^{\infty}\frac{t+\sqrt{2}s}{t^{ \beta}}\,\,\text{d}t=:C(s,\beta)<\infty\,.\] Combining this and what precedes, we get that \[\exp\left\{-\sum_{X_{k}\in\Phi_{G_{s}},\|X_{j}-X_{k}\|>2\sqrt{s}}\log{\cal L}_{ eF^{\prime}}\Bigl{(}\frac{T\|{X_{j}-X_{i}}\|^{\beta}}{\|X_{j}-X_{k}\|^{\beta}} \Bigr{)}\right\}\leq\exp(T(2R)^{\beta}C(s,\beta)).\] We also have for all $X_{k}\in\Phi_{G_{s}}$ and in particular for $\|X_{j}-X_{k}\|\leq 3\sqrt{2}s$. Hence we obtain \[\exp\{-\sum_{X_{k}\in\Phi_{G_{s}}}(\dots)\}\leq e^{-49\log(1-p)+T(2R)^{\beta}C (s,\beta)}\,,\] where 49 upper-bounds the number of points $X_{k}\in\Phi_{G_{s}}$ such that $\|X_{j}-X_{k}\|\leq 3\sqrt{2}s$. * Using the bound $\|X_{j}-X_{i}\|\leq 2R$ and the inequality $\log{\cal L}_{eF^{\prime}}(\xi)\geq\log{\cal L}_{eF^{\prime}}(\infty)=\log(1-p)$, we obtain \[\exp\{-\sum_{\Phi_{M}\ni X_{k},k\not=i,j,\|X_{i}\|\leq 2R}(\dots)\}\leq e^{- \Phi_{M}(B_{0}(2R))\log(1-p)}\,.\] * Using the bounds $\|X_{j}-X_{i}\|\leq 2R$ and $\|X_{j}-X_{k}\|\geq\|X_{k}\|-R$ (the latter follows from the triangle inequality) and the expression ${\cal L}_{eF^{\prime}}(\xi)=1-p+\frac{p}{1+\xi}$, we obtain \[\exp\Bigl{\{}-\sum_{\Phi_{M}\ni X_{k},\|X_{k}\|>2R}(\dots)\Bigr{\}}\] \[\leq\] This completes the proof. ∎∎ We can now prove the following auxiliary result. Under the assumptions of Proposition 3.3 for all points $x,y$ of $\mathbb{R}^{2}$, \[\mathbf{E}\left[\sup_{x_{1},y_{1}\in[x,y]}p(x_{1},y_{1},\Phi)\right]<\infty\,,\] where the supremum is taken over $x_{1},y_{1}$ belonging to the interval $[x,y]\subset\mathbb{R}^{2}$. Proof.: Without loss of generality, we assume that $(x+y)/2=0$ is the origin of the plane. Let $B=B_{0}(R)$ be the ball centered at $0$ and of radius $R$ such that no modification of the points in the complement of $B$ modifies $X(z)$ for any $z\in[x,y]$ (recall that $X(z)$ is the point of $\Phi$ which is the closest from $z$). Since $\Phi=\Phi_{M}+\Phi_{G_{s}}$, with $\Phi_{G_{S}}$ the square lattice p.p. with intensity $1/s^{2}$, it suffices to take $R=\|u-v\|/2+\sqrt{2}s$. Let $B^{\prime}=B_{0}(2R)$. By the above choice of $B$ and the inequality (6) we have for all $x_{1},y_{1}\in[x,y]$ \[P(x_{1},y_{1},0)\leq\sum_{X_{i},X_{j}\in\Phi\cap B}L_{i,j}(0)\,\] and consequently \[\sup_{x_{1},y_{1}\in[x,y]}p(x_{1},y_{1},\Phi)\leq\sum_{X_{i},X_{j}\in\Phi\cap B }\mathbf{E}[L_{i,j}(0)\,\|\,\Phi]\,.\] Using the result of Lemma 4.2 we obtain \[\sup_{x_{1},y_{1}\in[x,y]}\overline{\|p^{}(x_{1},y_{1},\Phi)\|}\] \[\leq \frac{e^{-49\log(1-p)+(2R)^{\beta}pTC(s,\beta)}}{p(1-p){\cal L}_{ W}(T\mu A(2R)^{\beta})}\] \[\times\Bigl{(}\Phi_{M}(B)+\pi(R+\sqrt{2}s)^{2}/s^{2}\Bigr{)}e^{- \Phi_{M}(B^{\prime})\log(1-p)}\,,\] where $\pi(R+\sqrt{2}s)^{2}/s^{2}$ is an upper bound of the number of points of $\Phi_{G_{s}}$ in $B$. The first factor in the above upper bound is deterministic. The two other factors are random and independent due to the independence property of the Poisson p.p. The finiteness of the expectation of the last expression follows from the finiteness of the exponential moments (of any order) of the Poisson random variable $\Phi_{M}(B^{\prime})$. For the expectation of the second (exponential) factor, we use the known form of the Laplace transform of the Poisson SN to obtain the following expression \[\mathbf{E}\Bigl{[}\exp\Bigl{\{}-\sum(\dots)\Bigr{\}}\Bigr{]}=\exp\biggl{\{}2 \pi p\lambda_{M}\int_{R}^{\infty}\frac{T(2R)^{\beta}}{v^{\beta}+(1-p)T(2R)^{ \beta}}(v+R)\,\,\text{d}v\Bigr{\}}<\infty\,.\] ∎∎ Proof.: (_of Proposition 3.3_ The existence and finiteness of the limit $\kappa_{\mathrm{d}}$ follows from the subadditivity (13) and Lemma 4.2 by the continuous-parameter sub-additive ergodic theorem (see (19, Theorem 4)). ∎∎ Proof.: (_of Proposition 3.3_) First, we prove the second statement; i.e., that $\kappa_{\mathrm{d}}$ is constant for all $\mathrm{d}$ in the unit sphere off some countable subset. Note that the point process $\Phi$ is ergodic as the independent superposition of mixing Poisson p.p. $\Phi_{M}$ and ergodic grid process $\Phi_{G}$. This can be easily proved using e.g. the respective characterisations of above properties by means of Laplace transforms of p.p. (see (9, Prop. 12.3.VI)). From the ergodicity of $\Phi$ we _cannot_ conclude the desired property for any vector $\mathrm{d}$ since the limit $\kappa_{\mathrm{d}}=\kappa_{\mathrm{d}(\Phi)}$ is not necessarily invariant with respect to translations of $\Phi$ by _any_ vector $x\in\mathbb{R}^{2}$ but only $x=\alpha\mathrm{d}$ for any scalar $\alpha\in\mathbb{R}$. The announced result follows from (22, Th. 1). For the first statement, consider a product space with on which two independent p.p.s $(\Phi_{M},\Phi_{G})$ are defined. Fix some vector $\mathrm{d}$ and define the operator $T=T_{1}\times T_{2}$ on this product space as the product of two operators, which correspond to the shift in the direction $\mathrm{d}$, of $\Phi_{M}$ and $\Phi_{G}$ respectively. The $\sigma$-field invariant with respect to $T$ is the product of the respective $\sigma$-fields invariant with respect to $T_{1}$ and $T_{2}$. The latter is trivial since $\Phi_{M}$ is mixing (as a Poisson p.p.). Consequently every function of $(\Phi_{M},\Phi_{G})$ that is invariant with respect to the shift in the direction $\mathrm{d}$ of its first argument ($\Phi_{M}$) is a.s. constant. This concludes the proof that $\kappa_{\mathrm{d}}$ is constant in $\Phi_{M}$ and thus depends only on $U_{G}$. ∎ Proof.: (_of Proposition 3.3_) For a given path $\sigma=\{(X_{0},n_{0}),(X_{1},n_{0}+1),\ldots,\allowbreak(X_{k},n_{0}+k)\}$ on ${\mathbb{G}}$ denote by $\|{\sigma}\|=\sum_{i=1}^{k}\|{X_{i}-X_{i-1}}\|$ the Euclidean length of the projection of $\sigma$ on $\mathbb{R}^{2}$; let us call it Euclidean length of $\sigma$ for short and recall that the (graph) length of $\sigma$ is equal to $k$. For fixed $\epsilon>0$ and all $n\geq 1$ denote by $\Pi(n)=\Pi_{\epsilon}(n)$ _the event that there exists a path on ${\mathbb{G}}$ starting at $(X(0),0)$ that has (graph) length $n$ and Euclidean length larger than $n/\epsilon$_. Assume $\mathbf{E}[\kappa_{\mathrm{d}}]=0$. We show first that this implies that for any $\epsilon>0$, ${\mathbf{P}}^{0}$-a.s. the event $\Pi_{\epsilon}(n)$ holds for infinitely many $n$ \[{\mathbf{P}}^{0}\Bigl{\{}\,\bigcap_{n\geq 1}\bigcup_{k\geq n}\Pi_{\epsilon}(k) \,\Bigl{\}}=1\,.\] (21) Indeed, $\mathbf{E}[\kappa_{\mathrm{d}}]=0$ implies $\kappa_{\mathrm{d}}=0$${\mathbf{P}}$-a.s. and by Palm-Matthes definition of the Palm probability ${\mathbf{P}}^{0}$-a.s. as well. This means that $\mathbf{E}^{0}[P(0,t\mathrm{d},0)\,\|\,\Phi]/t\to 0$, when $t\to\infty$, which implies that \[\lim_{k}P(0,t_{k}\mathrm{d},0)/t_{k}\to 0\] (22) ${\mathbf{P}}^{0}$-a.s. for some subsequence $\{t_{k}:k\geq 1\}$, with $\lim_{k}t_{k}=\infty$. Recall that $P(0,t_{k}\mathrm{d},0)$ is the length of a shortest path from $(X(0),0)$ (with $X(0)=0$ under ${\mathbf{P}}^{0}$) to $\{(X(t_{k}\mathrm{d}),n):n\geq 0\}$. Denote one of such shortest paths by $\sigma_{k}$. By the triangle inequality its Euclidean length satisfies \[\|{\sigma_{k}}\|\geq\|{0-X(t_{k}\mathrm{d})}\|\geq t_{k}-\sqrt{2}s\,.\] (23) From (22) and (23) one concludes that for any $\epsilon>0$ and $k$ large enough the length of the path $\sigma_{k}$ is smaller than $\epsilon$ time its Euclidean length $\|{\sigma_{k}}\|$. Now, (21) follows from the fact that the length of the path $\sigma_{k}$ tends to infinity with $k$, which is a consequence of $t_{k}\to\infty$ and the local finiteness of the graph ${\mathbb{G}}$ (cf Corollary 3.1). We conclude the proof by showing that for $\epsilon$ small enough, \[\sum_{n}{\mathbf{P}}^{0}\{\Pi_{\epsilon}(n)\}<\infty\,,\] (24) which by the Borel–Cantelli lemma implies that $\Pi(n)$ holds ${\mathbf{P}}^{0}$-a.s. only for a finite number of integers $n$ and thus contradicts to (21). To this regard assume constant $W=w>0$ and let ${\cal P}_{w}^{n}$ denote the set of paths $\sigma$ of ${\mathbb{G}}$ of length $n$, originating from $(X(0)=0,0)$. Denote also by ${\cal P}_{0}^{n}$ the analogous set of paths on the graph constructed under assumption $W=0$. Note that by monotonicity, \[{\cal P}^{n}_{w}\subset{\cal P}^{n}_{0}.\] (25) By the definition \[{\mathbf{P}}^{0}\{\Pi_{\epsilon}(n)\,\|\,\Phi\,\}={\mathbf{P}}^{0}\Bigl{(} \bigcup_{\sigma}\Bigl{\{}\,\sigma\in{\cal P}_{w}^{n}\;\text{and}\;\|{\sigma}\| \geq n/\epsilon\,\Bigr{\}}\,\Big{\|}\,\Phi\Bigr{)}\,,\] (26) where the sum bears on all possible $n$-tuples $\sigma=((X_{j_{1}},1),\ldots,(X_{j_{n}},n))$, with $X_{j_{i}}\in\Phi$. From this we have \[{\mathbf{P}}^{0}(\Pi_{\epsilon}(n)\,\|\,\Phi)\] \[\leq \sum_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{W} ,\|\sigma\|\geq n/\epsilon\,\Big{\|}\,\Phi\,\Bigr{\}}\] \[= \sum_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{W} ,\|\sigma\|\geq n/\epsilon\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}{ \mathbf{P}}^{0}\{\,\sigma\in{\cal P}^{n}_{0}\,\|\,\Phi\}\] \[\leq \mathbf{E}^{0}[{\cal H}_{0}^{out,n;W=0}(0)\,\|\,\Phi]\sup_{\sigma} {\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{W},\|\sigma\|\geq n/\epsilon\, \Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}\,,\] (28) where ${\cal H}_{0}^{out,n;W=0}(0)$ denotes the number of paths of length $n$ originating from $(X_{0}=0,0)$ under the assumption $W=0$. But \[\sup_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{w} ,\|\sigma\|\geq n/\epsilon\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}\] \[\leq \sup_{\sigma=((X_{j_{1}},1),\ldots,(X_{j_{n}},n))\atop{\sum_{i=1} ^{n}\|X_{j_{i}}-X_{j_{i-1}}\|\geq n/\epsilon}}\mathbf{E}^{0}\Bigl{[}\prod_{i=1}^ {n}\delta_{j_{i-1},{j_{i}}}(i-1,w)\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0} \Bigr{]},\] where $X_{j_{0}}=0$ and $\delta_{j_{i-1},{j_{i}}}(i-1,w)=\delta_{j_{i-1},{j_{i}}}(i-1)$ is the indicator of the existence of the edge from $(X_{j_{i-1}},i-1)$ to $(X_{j_{i}},i)$ defined by (4), and where we add in the notation the dependence on the noise $W=w$. Using the conditional independence of marks, (4), (15) and lack of memory of the exponential distribution of $F$ of parameter $\mu$ we have for the path-loss function (1) \[\mathbf{E}^{0}\Bigl{[}\prod_{i=1}^{n}\delta_{j_{i-1},{j_{i}}}(i-1 ,w)\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\Bigr{]} = \prod_{i=1}^{n}\mathbf{E}^{0}\bigl{[}\delta_{j_{i-1},{j_{i}}}(i-1 ,w)\,\big{\|}\,\Phi,\delta_{j_{i-1},{j_{i}}}(i-1,0)=1\bigr{]}\] \[= \prod_{i=1}^{n}\exp\left(-\mu(A\|X_{j_{i-1}}-X_{j_{i}}\|)^{\beta}{ Tw}\right)\,.\] Hence \[\sup_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{w},\|\sigma\|\geq n /\epsilon\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}\leq\exp\left(- \mu A^{\beta}Twn\epsilon^{-\beta}\right)\,,\] where the last inequality follows from a convexity argument. Using this and (4.2), we get \[\mathbf{E}^{0}(\Pi_{\epsilon}(n)) \leq \mathbf{E}^{0}[{\cal H}_{0}^{out,n;W=0}(0)]\exp\left(-\mu A^{ \beta}Twn\epsilon^{-\beta}\right)\] \[\leq \xi^{n}\exp\left(-\mu A^{\beta}Twn\epsilon^{-\beta}\right)\] \[\leq \exp\left(n(\log(\xi)-K/\epsilon^{\beta})\right)\,,\] where in the second inequality we used the following result of Corollary 3.1 \[\mathbf{E}^{0}[{\cal H}_{0}^{out,n;W=0}(0)]=h^{out,n;W=0}=h^{in,k,W=0}\leq\xi^ {k}\,\] and where $K$ is a positive constant. This shows (24) for $\epsilon$ small enough, and thus concludes the proof. ∎∎ Proof.: (_of Corollary 3.3_) Without loss of generality assume $x=0$. We use the left inequality in (6), (14) and the inclusion (16) to obtain \[P_{X(0),X(y)}(0)\geq L_{X(0)}(0)\geq{\cal T}_{X(0)}(0)\geq\widehat{\cal T}_{X( 0)}(0)\,\] and in consequence \[p(0,y,\Phi)\geq\mathbf{E}[\widehat{\cal T}_{X(0)}(0)\,\|\,\Phi]\,.\] Using the isotropy and the strong Markov property of the Poisson p.p. \[\mathbf{E}[\widehat{\cal T}_{X(0)}(0)\,\|\,\Phi]=\mathbf{E}^{0}[\widehat{\cal T }_{0}(0)\,\|\,\Phi\|_{\overline{B}})]\,,\] $\Phi\|_{\overline{B}})$ is the restriction of $\Phi$ to the complement of the open ball $B=B_{(0,R)}(R)$, centered at $(0,R)$ of radius $R\geq 0$, where $R$ is r.v. independent of $\Phi$ and having for density \[\frac{\,\text{d}\theta}{2\pi}2\pi\lambda r\exp(-\lambda\pi r^{2})\,.\] But since we consider here the SNR graph $\widehat{\mathbb{G}}$ \[\mathbf{E}^{0}[\widehat{\cal T}_{0}(0)\,\|\,\Phi\|_{\overline{B}})]\geq\mathbf{E }^{0}[\widehat{\cal T}_{0}(0)\,\|\,\Phi]\,.\] The result follows now from Lemma 4.1. ∎∎ ## 5 SINR space-time graph and routing Let us now translate our results regarding the SINR graph into properties of routing in ad-hoc networks. Firstly, it makes sense to assume that any routing algorithm builds paths on ${\mathbb{G}}$. This takes two key phenomena into account: contention for channel (nodes have to wait for some particular time slots to transmit a packet) and collisions (lack of capture due to insufficient SINR). Our time constant gives bounds on the delays that can be attained in the ad-hoc network by any routing algorithms. Of course, realistic routing policies cannot use information about future channel conditions. In the case of Poisson p.p. there is hence no routing algorithm with a finite time constant. The existence of such an algorithm in the case of the Poisson$+$Grid p.p. remains an open question. In the Poisson p.p. case; one can ask about the exact asymptotics of the optimal delay (we know it is not linear) and of the delay realizable by some non-anticipating algorithm. Let us discuss now the relation of our results to those obtained in [12; 15]. In these papers the so called delay-tolerant networks are considered and modeled by a spatial SINR or signal-to-noise ratio (SNR) graph with no time dimension. In these models, the time constant (defined there as the asymptotic ratio of the graph distance to the Euclidean distance) is announced to be finite, even in the pure Poisson case. The reason for the different performance of these models lays in the fact that they do not take the time required for a successful transmission from a given node in the evaluation of the end-to-end delay. The heavy-tailness of this time (which follows from that of the exit time (cf. Proposition 3.2) makes the time constant infinite in the space-time Poisson scenario. The reason for the heavy-tailness of the successful transmission time is linked to the so called “RESTART” algorithm (see e.g. [17; 18; 1; 16]). In our case the spatial irregularities in the ad-hoc network play a role similar to that of the file size variability in the RESTART scenario. ## References [1] S. Asmussen, P. Fiorini, L. Lipsky, T. Rolski, and R. Sheahan. Asymptotic behavior of total times for jobs that must start over if a failure occurs. _Mathematics of Operations Research_, 33(4):932–944, 2008. * [2] F. Baccelli, B. Blaszczyszyn, and P. Mühlethaler. An Aloha protocol for multihop mobile wireless networks. In _Proceedings of the Allerton Conference_, Urbana Champaign, Illinois, November 2003. and _IEEE Transactions on Information Theory_, 52(2):421–436, 2006. * [3] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler. An aloha protocol for multihop mobile wireless networks. _IEEE Transactions on Information Theory_, 52(2):421–436, 2006. * [4] F. Baccelli, B. Błaszczyszyn, and P. Mühlethaler. Stochastic analysis of spatial and opportunistic Aloha. _IEEE JSAC, special issue on Stochastic Geometry and Random Graphs for Wireless Networks_, 27:1109–1119, 2009. * [5] F. Baccelli, B. Błaszczyszyn, and P. Mühlethaler. Time-space opportunistic routing in wireless ad hoc networks, algorithms and performance. _the Computer Journal_, 2009. to appear. * [6] F. Baccelli and C. Bordenave. The radial spanning tree of a poisson point process. _Annals of Applied Probab._, 17(1):305–359, 2007. * [7] Charles Bordenave. Navigation on a poisson point process. _Ann. Appl. Probab._, 18:708–746, 2008. * [8] Simon Broadbent and John Hammersley. Percolation processes I. crystals and mazes. _Proc. Camb. Phil. Soc_, 53:629––641, 1957. * [9] D. J. Daley and D. Vere-Jones. _An Introduction to the Theory of Point Processes, vol. II_. Springer, New York, 2008. * [10] O Dousse, F. Baccelli, and P Thiran. Impact of interferences on connectivity in ad-hoc networks. _IEEE/ACM Trans. Networking_, 13:425–543, 2005. * [11] O. Dousse, M. Franceschetti, N. Macris, R. Meester, and P. Thiran. Percolation in the signal to interference ratio graph. _Journal of Applied Probability_, 43(2):552–562, 2006. * [12] R. K. Ganti and M. Haenggi. Bounds on information propagation delay in interference-limited ALOHA networks. In _Proc. of Workshop on Spatial Stochastic Models for Wireless Networks_, 2009. * [13] E. N. Gilbert. Random plane networks. _SIAM J._, 9:533–543, 1961. * [14] C. D. Howard and C. M. Newman. Euclidean models of first-passage percolation. _Probab. Theory Relat. Fields_, 108:153–170, 1997. * [15] P. Jacquet, B. Mans, P. Mühlethaler, and G. Rodolakis. Opportunistic routing in wireless ad hoc networks: Upper bounds for the packet propagation speed. _IEEE JSAC, special issue on Stochastic Geometry and Random Graphs for Wireless Networks_, 27:1192–1202, 2009. * [16] P. R. Jelenkovic and J. Tan. Stability of finite population aloha with variable packets. Technical Report arXiv:0902.4481v2, 2009. * [17] P.R. Jelenković and J. Tan. Can retransmissions of superexponential documents cause subexponential delays? In _Proc. of INFOCOM_, Anchorage, AL, USA, 2007. IEEE. * [18] P.R. Jelenković and J. Tan. Is aloha causing power law delays? In L. Mason, T. Drwiega, and J. Yan, editors, _Managing Traffic Performance in Converged Networks_. Springer, Berlin, 2007. * [19] J.F.C. Kingman. Subadditive ergodic theory. _Annals of Probab._, 1:883–899, 1973. * [20] J.M. Kleinberg. The small-world phenomenon: an algorithmic perspective. In _Proc. 32nd Annual ACM Symposium on the Theory of Computing_, pages 163–170, 2000. * [21] L.P.R. Pimentel. The time constant and critical probabilities in percolation models. _Elect. Comm. in Probab._, 11:160–168, 2006. * [22] Ch. Pugh and M. Shub. Ergodic elements of ergodic actions. _Compositio Mathematica_, 23(1):115–122, 1971. * [23] D. Tse and P. Viswanath. _Foundamentals of Wireless Communication_. Cambridge University Press, 2005. * [24] M. Q. Vahidi-Asl and J. C. Wierman. First-passage percolation on the Voronoi tessellation and Delaunay trangulation. In M. Karośki, J. Jaworski, and A. Ruciński, editors, _Random Graphs’87; Based on Proceedings of the 3rd International Seminar on Random Graphs and Probabilistic Methods in Combinatorics, June 27 – July 3_, pages 341–359. John Wiley & sons, Chichester, 1990.
1304.2618	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 18829, "num_imgs": 0, "llama3_tokens_count": 5818 }	[]	# Lexicographic identifying codes Maximilien Gadouleau School of Engineering and Computing Sciences Durham University m.r.gadouleau@durham.ac.uk ###### Abstract An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this paper, we design an algorithm that finds an identifying code in a twin-free graph on $n$ vertices in $O(n^{3})$ binary operations, and returns a failure if the graph is not twin-free. We also determine an alternative for sparse graphs with a running time of $O(n^{2}d\log n)$ binary operations, where $d$ is the maximum degree. We also prove that these algorithms can return any identifying code with minimum cardinality, provided the vertices are correctly sorted. ## 1 Introduction Identifying codes were introduced in [1] for fault diagnosis in multiprocessor systems, and have since then found applications in location and detection problems. In general, an identifying code in a graph $G$ can be defined as follows. First, we denote the (closed) neighborhood of any vertex $v$ as $N(v)=\{v\}\cup\{w:vw\in E(G)\}$. An _identifying code_ is a subset of vertices which satisfies the following property: for any two vertices $v$ and $w$, we have $N(v)\cap C\neq N(w)\cap C\neq\emptyset$. Equivalently, it is any subset of vertices $C$ such that for all $v_{1},v_{2}\in V(G)$, $(N(v_{1})\Delta N(v_{2}))\cap C\neq\emptyset$, where $\Delta$ is the symmetric difference between two sets. A graph admits an identifying code if and only if it is _twin-free_[2], where twins are two vertices with the same neighborhood. We remark that the definitions above are commonly used for a so-called $1$-identifying code, where an $r$-identifying code is defined in terms of balls of radius $r$ around a vertex. Since any $r$-identifying code can be seen as a $1$-identifying code for a related graph, we do not lose any generality in considering $1$-identifying codes only. For a thorough survey of identifying codes, the reader is invited to [3], and an exhaustive literature bibliography on identifying codes and related topics is maintained in [4]. Since any superset of an identifying code is itself an identifying code, it is natural to search for the minimum cardinality $i(G)$ of an identifying code of a given graph $G$. Let us refer to an identifying code as _minimal_ if it has no proper subset which itself is an identifying code and as _minimum_ if it has the smallest cardinality amongst all codes. The problem of finding the minimum cardinality of an identifying code was shown to be NP-hard in [3]. Viewing this problem as an instance of the subset cover problem [5], a greedy heuristic was also designed and analyzed in [3]. Its running time is on the order of $O(n^{4})$ binary operations and has the following performance guarantees. It always finds an identifying code whose cardinality is less than $c_{1}i(G)\ln n$ for some nonnegative constant $c_{1}$; however, there are graphs for which the algorithm always returns a code with cardinality greater than $c_{2}i(G)\ln n$ for another nonnegative constant $c_{2}$. _Lexicographic codes_ were introduced in [6] and independently rediscovered in [7] to design large constant-weight codes, which are sets of binary vectors of equal Hamming weight with a prescribed minimum Hamming distance (see [8] for a detailed review of constant-weight codes and lexicographic codes). The principle is to first sort all the vectors with the same Hamming weight, and then construct the code as we run through them. Adding a codeword is done according to a simple criterion: it must be at distance at least $d$ from the code constructed so far. The performance of the algorithm depends on the order in which the vectors have been sorted; moreover, some modifications can be added, such as starting with a predetermined set of vectors. Many record-holding constant-weight codes have been designed using lexicographic codes. However, this idea is not limited to constant-weight codes, and their application to nonrestricted binary codes has led to many interesting results [9]. They have also been recently applied to the construction of codes on subspaces in [10], also yielding record-holding codes. In this paper, we investigate adapting the idea of lexicographic codes to identifying codes. The main contribution is an algorithm running in $O(n^{3})$ binary operations which returns an identifying code for a twin-free graph, and returns a failure if the graph is not twin-free. This algorithm is then adapted to sparse graphs to run in $O(n^{2}d\log n)$ binary operations. Both algorithms have the same guarantees in terms of cardinality of the output. Although we are unable to give an upper bound which does not depend on the ordering of the vertices, we show that provided the vertices are properly sorted, the algorithm returns a minimum identifying code. This is fundamentally different to the greedy approach in $O(n^{4})$. ## 2 Algorithm for general graphs ### Description and correctness Let $G$ be a graph on $n$ vertices with adjacency matrix ${\bf A}$, and let ${\bf B}={\bf I}_{n}+{\bf A}$. We denote the vertices as $v_{1},v_{2},\ldots,v_{n}$, thus $b_{i,j}=1$ if and only if $v_{i}\in N(v_{j})$; yet we shall abuse notation and identify a vertex with its index. For instance, we refer to the vertex with minimum index in the neighborhood of $v_{i}$ as $\mathtt{min1}(i)$. Also, the output of our algorithm is actually the set of indices of the vertices in the code. Before giving the pseudocode of Algorithm 1, we describe it schematically below. Its input is the matrix ${\bf B}$ of the graph. It then runs along all vertices $v_{j}$, adding a new codeword to the code $C$ if $N(v_{j})\cap C=\emptyset$ or $N(v_{j})\cap C=N(v_{k})\cap C$ for some $k<j$. While searching for a new codeword to add, the algorithm may return a failure if the graph is not twin-free, which we identify as $n+1\in C$. After the $j$-th step, the code $C$ then ‘identifies’ the first $j$ vertices, i.e. they are all covered in a distinct fashion. We keep track of the intersections $N(v_{i})\cap C$ in a matrix ${\bf X}$. After going through all vertices, the algorithm then returns an identifying code $C$ or a failure (if $n+1\in C$) if the graph is not twin-free. ``` $C\leftarrow\emptyset$, $X\leftarrow{\bf 0}_{n}$, $j\gets 1$ while $j\leq n$ and $n+1\notin C$ do $l\gets 0$ if ${\bf X}(j)={\bf 0}$ then {$v_{j}$ is not covered} $l\leftarrow\mathtt{min1}(j)$ else $k\gets 1$ while ${\bf X}(j)\neq{\bf X}(k)$ and $k<j$ do {$v_{j}$ is covered, so we search if it is identified} $k\gets k+1$ end while if $k<j$ then {$v_{j}$ is not identified} $l\leftarrow\mathtt{min2}(j,k)$ end if end if if $1\leq l\leq n$ then {A new codeword has been found} $C\gets C\cup\{l\}$ ${\bf X}^{T}(l)\leftarrow{\bf B}^{T}(l)$ end if $j\gets j+1$ end while return $C$ ``` Algorithm 1 Main algorithm for general graphs The subroutine $\mathtt{min2}(j,k)$ returns the first vertex which identifies $v_{j}$ if it exists and a failure otherwise, i.e. it determines the first vertex in lexicographic order in $N(v_{j})\Delta N(v_{k})$. If $N(v_{j})=N(v_{k})$, then it returns $n+1$. It is given in Algorithm 2. ``` $l\gets 1$ while $l\leq n$ and ${\bf B}(j,l)={\bf B}(k,l)$ do $l\gets l+1$ end while return $l$ ``` Algorithm 2$\mathtt{min2}(j,k)$ subroutine We now justify this claim in Lemma 1 below. Lemma 1: _The subroutine $\mathtt{min2}(j,k)$ returns the minimum element in $N(v_{j})\Delta N(v_{k})$ if this symmetric difference is non-empty, and a failure ($l=n+1$) otherwise._ Proof First, if $N(v_{j})=N(v_{k})$, then ${\bf B}(j,l)={\bf B}(k,l)$ for all $1\leq l\leq n$. Therefore, the while loop will only stop once $l=n+1$, and hence the subroutine returns a failure. Second, if $N(v_{j})\neq N(v_{k})$, then the minimum element in $N(v_{j})\Delta N(v_{k})$ is the smallest $l$ such that ${\bf B}(j,l)\neq{\bf B}(k,l)$. It is clear that the subroutine returns this value. $\Box$ Proposition 1: _Algorithm 1 returns an identifying code if the input graph is twin-free, and a failure ($n+1\in C$) otherwise._ Proof First of all, we prove that the algorithm returns a failure if and only if the graph is not twin-free. In the latter case, let $k$ be the smallest integer such that the set $\{i\neq k:N(v_{k})=N(v_{i})\}$ is not empty, and let $j$ be the minimum element of this set (hence $k<j$, $N(v_{k})=N(v_{j})$). It is easily shown that after the $k$-th step, $v_{k}$ is covered. On the $j$-th step, Algorithm 1 first checks if $v_{j}$ is covered. Since $v_{k}$ is covered and $N(v_{k})=N(v_{j})$, then $v_{j}$ is also covered. Algorithm 1 then finds that $k$ is the smallest integer satisfying ${\bf X}(k)={\bf X}(j)$, and hence calls the subroutine $\mathtt{min2}(j,k)$. By Lemma 1 this returns a failure, and hence the whole algorithm returns a failure. Conversely, the only case where the subroutine (and hence the algorithm) returns a failure is when there exist $k<j$ such that $N(v_{j})=N(v_{k})$, i.e. the graph is not twin-free. We now assume that the graph is twin-free, and hence we have $l\leq n$ at any step. We need to show that the output $C$ of Algorithm 1 is an identifying code. Let us denote the matrix ${\bf X}$ and the code $C$ obtained after $j$ steps as ${\bf X}^{j}$ as $C^{j}$, respectively. Note that for all $a$, ${\bf X}^{j}(a)$ reflects how the vertex $v_{a}$ is covered by $C^{j}$: $N(v_{a})\cap C^{j}=\mathrm{supp}({\bf X}(a))=\{b:{\bf X}^{j}(a,b)=1\}$. The following claim is the cornerstone of the proof. Claim: After step $j$, all ${\bf X}^{j}(i)$’s are nonzero and distinct for $1\leq i\leq j$. The proof goes by induction on $j$, and is trivial for $j=1$. Suppose it is true for $j-1$, then \[\mathrm{supp}({\bf X}^{j-1}(a))=N(v_{a})\cap C^{j-1}\subseteq N(v_{a})\cap C^{ j}=\mathrm{supp}({\bf X}^{j}(a)).\] It is hence easy to show that if ${\bf X}^{j-1}(a)\neq{\bf 0}$, then ${\bf X}^{j}(a)\neq{\bf 0}$ and if ${\bf X}^{j-1}(a)\neq{\bf X}^{j-1}(b)$, then ${\bf X}^{j}(a)\neq{\bf X}^{j}(b)$ for all $a$ and $b$. It immediately follows that the vectors ${\bf X}^{j}(i)$’s are all nonzero and distinct for $1\leq i\leq j-1$, and we only have to consider ${\bf X}^{j}(j)$. Three cases occur when the algorithm reaches step $j$. * Case I: ${\bf X}^{j-1}(j)$ is nonzero and distinct to any ${\bf X}^{j-1}(i)$ for $1\leq i\leq j-1$. Then as shown above, ${\bf X}^{j}(j)$ is nonzero and distinct to all ${\bf X}^{j}(i)$’s. * Case II: ${\bf X}^{j-1}(j)$ is nonzero and equal to ${\bf X}^{j-1}(k)$ for some $k<j$. First, we remark that $k$ is unique, as ${\bf X}^{j-1}(k)\neq{\bf X}^{j-1}(i)$ for all other $i$. The $\mathtt{min2}(k,j)$ subroutine then returns an element $v_{l}\in N(v_{j})\Delta N(v_{k})$, and hence ${\bf X}^{j}(j,l)\neq{\bf X}^{j}(k,l)$. * Case III: ${\bf X}^{j-1}(j)={\bf 0}$. Then by hypothesis ${\bf X}^{j-1}(j)\neq{\bf X}^{j-1}(i)$ for all $1\leq i\leq j-1$, and hence ${\bf X}^{j}(j)\neq{\bf X}^{j}(i)$. Also, ${\bf X}^{j}(j)$ is the unit vector ${\bf e}_{\mathtt{min1}(j)}$, which is nonzero. Therefore, for the code $C^{n}=C$ obtained after $n$ steps, $N(v_{a})\cap C$ are all nonzero and distinct for all $1\leq a\leq n$. It is hence an identifying code. $\Box$ ### Performance We now investigate the performance of Algorithm 1. We are first interested in the cardinality of its output. Clearly, this depends on the order in which the vertices are sorted. We show below that provided the order is suitable, the algorithm can find any minimal identifying code, and hence can return a minimum one. Proposition 2: _Suppose that the graph is twin-free and that $M=\{v_{1},v_{2},\ldots,v_{m}\}$ forms an identifying code. Then Algorithm 1 returns an identifying code that is a subset of $M$._ Proof We know by Proposition 1 that the algorithm returns an identifying code; we only have to prove that all codewords are in $M$. At step $j$, three cases need to be distinguished. * Case I: $v_{j}$ is covered and identified, then no codeword is added. * Case II: $v_{j}$ is covered but not identified, i.e. $(N(v_{j})\Delta N(v_{k}))\cap C^{j-1}=\emptyset$ for some $k<j$. The subroutine returns the smallest element $v_{l}$ in $N(v_{j})\Delta N(v_{k})$. Since $M$ is an identifying code, the set $(N(v_{j})\Delta N(v_{k}))\cap M$ is not empty, hence $v_{l}\in M$. * Case III: $v_{j}$ is not covered. The algorithm then selects the next codeword to be $\mathtt{min1}(j)$, which is necessarily in $M$ as $N(v_{j})\cap M\neq\emptyset$. Therefore, the algorithm only adds codewords of $M$, and hence returns a subcode of $M$. $\Box$ We remark that Algorithm 1 does not necessarily return a minimal code, as seen in Figure 1. Algorithm 1 would return the code $\{1,2,3,4,5,6\}$ while $\{2,3,4,5,6\}$ is a minimal identifying code. [FIGURE:S2.F1][ENDFIGURE] On the other hand, if $M$ is minimal, then it has no proper subset that itself is an identifying code; Algorithm 1 thus returns it. We obtain the following corollary. Corollary 1: _Provided that the vertices are sorted such that $v_{1},v_{2},\ldots,v_{m}$ form a minimal identifying code for some $1\leq m\leq n$, Algorithm 1 will return this identifying code._ Proposition 2 also implies that the probability that the output has cardinality no more than $K$ is at least the probability that the first $K$ vertices form an identifying code. Hence our algorithm returns a minimum identifying code with probability at least $\frac{1}{{n\choose i(G)}}$. Proposition 3: _The running time of Algorithm 1 is $O(n^{3})$ binary operations._ Proof Clearly, we have to run the iteration for $j$ exactly $n$ times. For each iteration, the step demanding the highest number of operations is the search for $k$. We consider at most $j-1$ values of $k$, comparing at most $n$ bits to verify whether ${\bf X}(j)\neq{\bf X}(k)$. Therefore, the running time is $O(n^{3})$. $\Box$ ## 3 Algorithm for sparse graphs For sparse graphs, it is more efficient not to work with the whole adjacency matrix, but with the neighborhood array $A\in\mathcal{P}(E)^{n}$, defined as $A(v_{i})=N(v_{i})$, where the neighborhood is sorted in increasing lexicographic order. Then, instead of adding the column of the adjacency matrix corresponding to a new codeword, we only update the code array $X(v)$ for all vertices adjacent to the new codeword. THe algorithm for sparse graphs is given in Algorithm 3; its input is the neighorhood array, and it returns an identifying code $C$ or a failure ($n+1\in C$) if the graph is not twin-free. ``` $C\leftarrow\emptyset$, $X\leftarrow\emptyset^{n}$, $j\gets 1$, $f\gets 0$ while $j\leq n$ and $n+1\notin C$ do $l\gets 0$ if $X(j)=\emptyset$ then {$v_{j}$ not covered} $l\gets A(j,1)$ else $m\gets X(j,1)$, $k\gets 1$ while $X(j)\neq X(k)$ and $k<j$ do $k\gets k+1$ end while if $k<j$ then {$v_{j}$ not identified} $l\leftarrow\mathtt{min3}(j,k)$ end if end if if $1\leq l\leq n$ then $C\gets C\cup\{l\}$ for $i$ from $1$ to $d_{l}$ do $X(A(l,i))\gets X(A(l,i))\cup\{l\}$ end for end if $j\gets j+1$ end while return $C$ ``` Algorithm 3 Main algorithm for sparse graphs Similar to the general case, the $\mathtt{min3}(j,k)$ subroutine produces the first vertex $v_{l}$ which identifies $v_{j}$ if it exists and a failure otherwise, i.e. it determines the first vertex in lexicographic order which covers either $j$ or $k$, but not both. It is given in Algorithm 4. ``` $l\gets n+1$ while $a\leq\min\{d_{j},d_{k}\}$ do if $A(j,a)\neq A(k,a)$ then $l\leftarrow\min\{A(j,a),A(k,a)\}$ end if $a\gets a+1$ end while if $l=n+1$ then if $d_{j}<d_{k}$ then $l\gets A(k,d_{j}+1)$ else if $d_{k}<d_{j}$ then $l\gets A(j,d_{k}+1)$ end if end if return $l$ ``` Algorithm 4$\mathtt{min3}(j,k)$ subroutine The same results on correctness and the possibility of returning a minimum code also hold for Algorithm 3; they are summarized below. Proposition 4: _If the graph is not twin-free, then Algorithm 3 returns a failure. Otherwise, the algorithm returns an identifying code contained in $\{v_{1},v_{2},\ldots,v_{m}\}$, where $m$ is the minimum integer such that this forms an identifying code._ _The running time of Algorithm 3 is $O(n^{2}d\log n)$ binary operations._ Proof The proof of correctness of Algorithm 3 is similar to that of Algorithm 1, and is hence omitted. We hence determine the running time of the algorithm. $\Box$ ## References * [1] M. G. Karpovsky, K. Chakrabarty, and L. B. Levitin, “A new class of codes for identification of vertices in graphs,” _IEEE Trans. Info. Theory_, vol. 44, no. 2, pp. 599–611, March 1998. * [2] I. Charon, I. Honkala, O. Hudry, and A. Lobstein, “Structural properties of twin-free graphs,” _The Electronic Journal of Combinatorics_, vol. 14, no. 1, p. R16, January 2007. * [3] M. Laifenfeld and A. Trachtenberg, “Identifying codes and covering problems,” _IEEE Trans. Info. Theory_, vol. 54, no. 9, pp. 3929–3950, September 2008. * [4] A. Lobstein. Watching systems, identifying, locating-dominating and discriminating codes in graphs. [Online]. Available: http://www.infres.enst.fr/{̃}lobstein/bibLOCDOMetID.html * [5] T. Cormen, C. Leiserson, and R. Rivest, _Introduction to Algorithms_. MIT Press, 2001. * [6] V. Levenshtein, “A class of systematic codes,” _Soviet Math. Dokl. 1_, pp. 368–371, 1960. * [7] J. H. Conway and N. J. A. Sloane, “Lexicographic codes: error-correcting codes from game theory,” _IEEE Trans. Info. Theory_, vol. 32, pp. 337–348, May 1986. * [8] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, and W. D. Smith, “A new table of constant weight codes,” _IEEE Trans. Info. Theory_, vol. 36, no. 6, pp. 1334–1380, November 1990. * [9] A. Trachtenberg, “Error-correcting codes on graphs: Lexicodes, trellises, and factor graphs,” Ph.D. dissertation, University of Illinois at Urbana-Champaign, 2000. * [10] N. Silberstein and T. Etzion, “Large constant-dimension codes and lexicodes,” in _Proc. Algebraic Combinatorics and Applications_, Thurnau, Germany, April 2010.
1311.6600	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 32960, "num_imgs": 1, "llama3_tokens_count": 10209 }	[ "content_image/1311.6600/x1.png" ]	# Optimal condition for measurement observable via error-propagation Wei Zhong${}^{1}$, Xiao-Ming Lu${}^{2}$, Xiao-Xing Jing${}^{1}$ and Xiaoguang Wang${}^{1}$ ${}^{1}$ Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China. ${}^{2}$ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore. xgwang@zimp.zju.edu.cn ###### Abstract Propagation of error is a widely used estimation tool in experiments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramér-Rao theorem in parameter estimation. By invoking the Schrödinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for single parameter estimate. By applying this condition to Greenberg-Horne-Zeilinger states, we obtain the general expression of the optimal observable for separable measurements to achieve the Heisenberg-limit precision and show that it is closely related to the parity measurement. However, Jose _et al_ [Phys. Rev. A 87, 022330 (2013)] have claimed that the Heisenberg limit may not be obtained via separable measurements. We show this claim is incorrect. pacs: 06.20.Dk, 42.50.St, 03.65.Ta, 03.67.-a, † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction An essential task in quantum parameter estimation is to suppress the fundamental bound on measurement precision imposed by quantum mechanics. Various quantum strategies have been developed to enhance the accuracy of the parameter estimation, which are closely related to some practical applications, such as the Ramsey spectroscopies, atomic clocks, and the gravitational wave detection [1, 2, 3, 4, 5, 6, 7, 8]. Two approaches in common use for high-precision measurements are the parallel protocol with correlated multi-probes [9] and multi-round protocol with a single probe [10, 11]. Most recently, some novel methods, like environment-assisted metrology [12] and enhanced metrology by quantum error correction [13, 14, 15, 16], were raised to achieve high precision in realistic experiments. Rather than engineering the sensitivity-enhanced strategies, we concentrate on the problem of how to attain the maximal sensitivity in realistic experiments. In general, a noiseless procedure of the quantum single parameter estimation can be abstractly modeled by four steps (see figure 1): (i) preparing the input state $\rho_{\rm in}$, (ii) parameterizing it under the evolution of the parameter-dependent Hamiltonian, for instance, a unitary evolution $U_{\varphi}$ with $\varphi$ the parameter to be estimated, (iii) performing measurements of the observable $\hat{\mathcal{O}}$ on the output state $\rho_{\varphi}$, (iv) finally estimating the value of the parameter from the estimator $\varphi_{\rm est}$ as a function of the outcomes of the measurements. <figure><img src="content_image/1311.6600/x1.png"><figcaption>Figure 1: The schematic representation of a general scheme of (noiseless)quantum parameter estimation is composed of four components: input state ρin,parametrization process Uφ, measurements ^O, and estimator φest. Here, weconcentrate on the part in shadow to find the optimal ^O attaining the highestsensitivity to the parameter φ in ρφ.</figcaption></figure> From estimation theory, the estimation precision is statistically measured by the units-corrected mean-square deviation of the estimator $\varphi_{\rm est}$ from the true value $\varphi$[17, 18], (1) where the brackets $\langle\,\rangle_{\rm av}$ denote statistical average and the derivative $\partial_{\varphi}\langle\varphi_{\rm{est}}\rangle\equiv\partial\langle\varphi _{\rm{est}}\rangle/\partial\varphi$ removes the local difference in the “units” of $\varphi_{\rm{est}}$ and $\varphi$. Whatever is the measurement scheme employed, the ultimate limit to the precision of the unbiased estimate is given by the quantum Cramér-Rao bound (QCRB) from below as \[(\delta\varphi)^{2}_{\rm est}\geq(\upsilon\mathcal{F}_{\varphi})^{-1},\] (2) where $\upsilon$ is the repetitions of the experiment and $\mathcal{F}_{\varphi}$ is the quantum Fisher information (QFI) (see equation (7) for definition), which measures the statistical distinguishability of the parameter in quantum states. This bound is asymptotically achieved for large $\upsilon$ under optimal measurements followed by the maximum likelihood estimator [17, 18, 19, 20]. On the other hand, it is well-known that error-propagation is a widely acceptable theory in experiments [1, 3, 9, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. With this theory, to estimate the parameter $\varphi$ is reduced to measuring the average value of a physical observable $\hat{\mathcal{O}}$. After repeating the experiment $\upsilon$ times, the real accessible precision on $\varphi$ is given by the error-propagation formula as follows [1, 3, 9, 21, 22, 23, 24], \[(\delta\varphi)^{2}_{\rm ep}:=\frac{1}{\upsilon}\frac{\langle(\Delta\hat{ \mathcal{O}})^{2}\rangle}{\,\,\|\partial_{\varphi}\langle\hat{\mathcal{O}} \rangle\|^{2}},\] (3) where $\Delta\hat{\mathcal{O}}=\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle$ and $\langle\hat{\mathcal{O}}\rangle={\rm Tr}(\rho_{\varphi}\hat{\mathcal{O}})$. Note that the two estimation errors defined in equations (1) and (3) are closely related. To show the relationship between the two kinds of the estimation errors, $(\delta\varphi)^{2}_{\mathrm{est}}$ and $(\delta\varphi)^{2}_{\mathrm{ep}}$, we introduce $\Delta\varphi_{\mathrm{est}}:=\varphi_{\mathrm{est}}-\langle\varphi_{\mathrm{ est}}\rangle_{\mathrm{av}}$. Then, it is easy to show that [17] \[(\delta\varphi)^{2}_{\mathrm{est}}=\frac{\langle(\Delta\varphi_{\rm est})^{2} \rangle_{\rm av}}{\|\partial_{\varphi}\langle\varphi_{\rm{est}}\rangle_{\rm av} \|^{2}}+\bigg{\langle}\frac{\varphi_{\rm{est}}}{\|\partial_{\varphi}\langle \varphi_{\rm{est}}\rangle_{\rm av}\|}-\varphi\bigg{\rangle}_{\rm av}^{2}.\] (4) When viewing the arithmetic mean of the measurement outcomes of $\hat{\mathcal{O}}$ over repetitions of the experiment as the estimator in the quantum setting, one has in general $(\delta\varphi)^{2}_{\rm est}\geq(\delta\varphi)^{2}_{\rm ep}\geq(\upsilon \mathcal{F}_{\varphi})^{-1}$ by noting that $\langle(\Delta\varphi_{\rm est})^{2}\rangle=\langle(\Delta\hat{\mathcal{O}})^{ 2}\rangle/\upsilon$ for sufficiently large $\upsilon$ according to the central limit theorem [34] and comparison of the two definitions of the errors given by equations (1) and (3). In such situation, $(\delta\varphi)^{2}_{\rm est}$ and $(\delta\varphi)^{2}_{\rm ep}$ have the same QCRB, and the saturation of the former implies that of the latter. The formula equation (3) indicates that the fluctuation of the observable $\hat{\mathcal{O}}$ propagates to the estimated values of the parameter $\varphi$. This means that what kinds observable $\hat{\mathcal{O}}$ employed directly affects the estimating precision of the parameter $\varphi$. The purpose of this paper is to address the question of with which kind of observable does the estimation error given by equation (3) achieve the QCRB given by equation (2). In this paper, we derive the necessary and sufficient (N&S) condition for the optimal observable saturating the QCRB for the single parameter estimation by using the Schrödinger-Robertson uncertainty relation (SRUR). We then apply this condition to GHZ states and find the general form of the optimal observable for separable measurements to achieve the Heisenberg-limit sensitivity (i.e., $1/N$). Moreover, we discuss the relation between the optimal separable observable and parity measurements. However, Jose _et al._, in a recent work [35], made a contradictory conclusion with respect to the above result. They claimed that separable measurements are impossible to go beyond the shot-noise limit (i.e., $1/{\sqrt{N}}$) for any entangled states. To clarify this issue, we revisit the method in [35] and show the causes for this inconsistency. This paper is structured as follows. In section 2, we first briefly review the single parameter estimation and obtain the N&S condition for the optimal observable. In section 3, we give an application of this condition to obtain the optimal separable observables for GHZ states to saturate the Heisenberg-limit precision. In section 4, we further elucidate the reasons for contradiction between the result given in [35] and ours. At last, a conclusion is given in section 5. ## 2 N&S condition for optimal observable in single parameter estimation We start by a brief review of quantum single parameter estimation via the general estimator. Consider a parametric family of density matrices $\rho_{\varphi}$ containing an unknown parameter $\varphi$ to be estimated. Suppose that the general quantum measurement performed on $\rho_{\varphi}$ is characterized by a positive-operator-valued measure $\hat{M}:=\{\hat{M}_{x}\}$ with $x$ the results of measurement. The value of the parameter is inferred via an estimator $\varphi_{\rm est}$, which maps the measurement outcomes to the estimated value. After repeating the experiment $\upsilon$ times, the standard estimation error $(\delta\varphi)^{2}_{\rm est}$ in equation (1) is bounded from below as \[(\delta\varphi)^{2}_{\rm est}\geq(\upsilon F_{\varphi})^{-1},\] (5) where \[F_{\varphi}:=\sum_{x}p_{\varphi}(x)[\partial_{\varphi}\ln p_{\varphi}(x)]^{2}\] (6) is the (classical) Fisher information of the measurement-induced probability distribution $p_{\varphi}(x)=\mathrm{Tr}(\rho_{\varphi}\hat{M}_{x})$. The maximization over all POVMs gives rise to the so-called QFI, which is defined by \[\mathcal{F}_{\varphi}:={\rm Tr}(\rho_{\varphi}\hat{L}_{\varphi}^{2}).\] (7) Hence, a more tighter bound of equation (5) is given by equation (2). Here $\hat{L}_{\varphi}$ is the symmetric logarithmic derivative (SLD) operator, which is a Hermitian operator determined by \[\partial_{\varphi}\rho_{\varphi}=\frac{1}{2}[\rho_{\varphi},\hat{L}_{\varphi}] _{+}\] (8) with $[\cdot\,,\,\cdot]_{+}$ denoting the anti-commutator, see reference [17]. It is remarkable that $\hat{L}_{\varphi}$ may not be uniquely determined by equation (8) when $\rho_{\varphi}$ is not of full rank [36]. However, in general the value of the parameter $\varphi$ may not be directly measured. The most general method of estimating the value of $\varphi$ in practice involves measurements corresponding to a physical observable $\hat{\mathcal{O}}$ which is generally $\varphi$-independent. In such cases, the estimation error is given by the error-propagation formula equation (3), in which the fluctuations on the observable $\hat{\mathcal{O}}$ propagate to the uncertainty in the estimation of $\varphi$. In the following, we follow Hotta and Ozawa [24] to derive the achievable lower bound of the estimation error $(\delta\varphi)^{2}_{\rm ep}$ by using the SRUR. Let us first recall the SRUR [37, 38], which states that the uncertainty of two non-commuting observables $\hat{X},\,\hat{Y}$ must obey the following inequality \[\langle(\Delta\hat{X})^{2}\rangle\langle(\Delta\hat{Y})^{2}\rangle\geq\frac{1} {4}\|\langle[\hat{X},\hat{Y}]\rangle\|^{2}+\frac{1}{4}\langle[\Delta\hat{X}, \Delta\hat{Y}]_{+}\rangle^{2},\] (9) where $[\cdot\,,\,\cdot]$ denotes the commutator. The SRUR follows from the Schwarz inequality for the Hilbert-Schmidt inner product, and naturally reduces to the Heisenberg uncertainty relation under the condition $\langle[\Delta\hat{X},\Delta\hat{Y}]_{+}\rangle=0$. By substituting $\hat{X}\,(\hat{Y})$ with $\hat{\mathcal{O}}\,(\hat{L}_{\varphi}$) and utilizing \[\mathcal{F}_{\varphi}=\langle\hat{L}_{\varphi}^{2}\rangle=\langle(\Delta\hat{L }_{\varphi})^{2}\rangle,\] (10) as a result of $\langle\hat{L}_{\varphi}\rangle=2\,\partial_{\theta}\mathrm{Tr}(\rho_{\varphi} )=0$, equation (9) becomes \[\langle(\Delta\hat{\mathcal{O}})^{2}\rangle\,\mathcal{F}_{\varphi}\geq\frac{1} {4}\|\langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]\rangle\|^{2}+\frac{1}{4} \langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rangle^{2}.\] (11) Moreover, since the observable operator $\hat{\mathcal{O}}$ is independent of $\varphi$, we have \[\langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rangle = \mathrm{Tr}([\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rho_{ \varphi})\] (12) \[= \mathrm{Tr}(\hat{\mathcal{O}}[\hat{L}_{\varphi},\rho_{\varphi}]_{ +})\] \[= 2\,\partial_{\varphi}\langle\hat{\mathcal{O}}\rangle,\] where the second equality is obtained by employing the cyclic property of the trace operation, and the third equality is due to the SLD equation (8). Provided that $\langle\hat{\mathcal{O}}\rangle$ is nonzero, combining equations (3), (11) and (12) yields \[(\delta\varphi)^{2}_{\rm ep} \geq \frac{1}{\upsilon\mathcal{F}_{\varphi}}\bigg{(}1+\frac{\|\langle[ \hat{\mathcal{O}},\hat{L}_{\varphi}]\rangle\|^{2}}{\langle[\hat{\mathcal{O}}, \hat{L}_{\varphi}]_{+}\rangle^{2}}\bigg{)}\] (13) \[= \frac{1}{\upsilon\mathcal{F}_{\varphi}}\bigg{[}1+\bigg{(}\frac{{ \rm Im}\langle\hat{\mathcal{O}}\hat{L}_{\varphi}\rangle}{{\rm Re}\langle\hat{ \mathcal{O}}\hat{L}_{\varphi}\rangle}\bigg{)}^{2}\bigg{]}\] (14) \[\geq (\upsilon\mathcal{F}_{\varphi})^{-1}.\] (15) The bound in equation (13) describes the achievable sensitivity of $\varphi$ when employing an observable $\hat{\mathcal{O}}$. The bound in equation (15) gives the highest precision for $\varphi$ for the optimal observable $\hat{\mathcal{O}}_{\rm opt}$, which coincides with the QCRB in equation (2). It is shown that the estimation error $(\delta\varphi)^{2}_{\rm ep}$ achieves the QCRB only when the two equalities in equations (13) and (15) hold simultaneously. Below, we consider the attainability of the above bounds and give the N&S condition for optimal observables. From the N&S condition for equality in the SRUR, the equality in equation (13) holds if and only if \[\Delta\hat{\mathcal{O}}\sqrt{\rho_{\varphi}}=\alpha\hat{L}_{\varphi}\sqrt{\rho _{\varphi}}\] (16) is satisfied with a nonzero complex number $\alpha$. Note that we restrict here $\alpha\neq 0$ at the request of $\langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rangle\neq 0$ in the denominator of equation (13). Furthermore, the equality in equation (15) holds if and only if \[\mathrm{Im}\langle\hat{\mathcal{O}}\hat{L}_{\varphi}\rangle=0.\] (17) This condition can be combined into the condition (16) by restricting $\alpha$ to be a nonzero real number, i.e., \[\Delta\hat{\mathcal{O}}\sqrt{\rho_{\varphi}}=\alpha\hat{L}_{\varphi}\sqrt{\rho _{\varphi}}\quad\mbox{with $\alpha\in\mathbb{R}\!\setminus\!\!\{0\}$}.\] (18) This is the of the optimal observable for density matrix $\rho_{\varphi}$. It implies that the estimation error achieves the QCRB given by the QFI for $\rho_{\varphi}$_only_ when the observable that we choose satisfies equation (18). This is the main result of the paper. For pure states $\rho_{\varphi}=\|\psi_{\varphi}\rangle\langle\psi_{\varphi}\|$, the condition (18) is equivalent to \[\Delta\hat{\mathcal{O}}\|\psi_{\varphi}\rangle=\alpha\hat{L}_{\varphi}\|\psi_{ \varphi}\rangle\quad\mbox{with $\alpha\in\mathbb{R}\!\setminus\!\!\{0\}$}.\] (19) If we assume that the parameter $\varphi$ here is imprinted via a unitary operation [9], i.e., $\rho_{\varphi}=\exp({-i\hat{G}\varphi})\,\rho_{\rm in}\exp({i\hat{G}\varphi})$ with $\hat{G}$ the generator, associating with the equality $\partial_{\varphi}\rho_{\varphi}=-i[\hat{G},\rho_{\varphi}]$, then condition (19) further reduces to \[\Delta\hat{\mathcal{O}}\|\psi_{\varphi}\rangle=-2i\alpha\Delta\hat{G}\|\psi_{ \varphi}\rangle\quad\mbox{with $\alpha\in\mathbb{R}\!\setminus\!\!\{0\}$}.\] (20) This condition was alternatively obtained in Ref. [31]. It is deserved to note that their proof is only valid in the case of unitary parametrization for pure states, and cannot be generalized to obtain the condition (18). Here, we discuss the relations between the saturation of the QCRB with respect to $(\delta\varphi)^{2}_{\rm est}$ and that with respect to $(\delta\varphi)^{2}_{\rm ep}$. Following Braunstein and Caves [17], the saturation of the QCRB with respect to the error $(\delta\varphi)^{2}_{\rm est}$ can be separated as the saturation of a classical Cramér-Rao bound (CCRB) equation (5) and finding an optimal measurement attaining the QFI. The CCRB can always be asymptotically achieved by the maximum likelihood estimator, so whether the QCRB can be asymptotically saturated is determined by whether the measurement attains the QFI. The N&S condition for the optimal measurement attaining the QFI reads [17] \[\sqrt{\hat{M}_{x}}\sqrt{\rho_{\varphi}}=u_{x}\sqrt{\hat{M}_{x}}\hat{L}_{ \varphi}\sqrt{\rho_{\varphi}},\] (21) where $\{\hat{M}_{x}\}$ denotes the POVM of the measurement and $u_{x}$ are real numbers. In the following, we show that the N&S condition (18) for the saturation of the QCRB with respect to $(\delta\varphi)^{2}_{\rm ep}$ identifies an optimal measurement attaining QFI. Let $\hat{\mathcal{O}}_{\rm opt}$ be the optimal observable satisfying Eq. (18) and $P_{x}$ the eigenprojectors of $\hat{\mathcal{O}}_{\rm opt}$ with the eigenvalues $x$. Left multiplying $P_{x}$ on both sides of Eq. (18), it is easy to see that $\{P_{x}\}$ is the optimal measurement attaining the QFI. That is to say, the projective measurement $\{P_{x}\}$ followed by the maximum likelihood estimator of the measurement outcomes saturate the QCRB with respect to the standard estimation error $(\delta\varphi)^{2}_{\rm est}$. ## 3 Optimal separable observable for GHZ states Below, we apply the N&S condition to show the general optimal observable for GHZ states. Let us specifically consider an experimentally realizable Ramsey interferometry to estimate the transition frequency $\omega$ of the two-level atoms loaded in the ion trap [1, 2]. The Hamiltonian of the system with $N$ atoms is $\hat{H}=(\omega/2)\sum_{i=1}^{N}\hat{\sigma}_{z}^{i}$ where $\hat{\sigma}_{z}^{i}$ is the Pauli matrix acting on the $i$th particle. In this setup, the measurements are limited to be performed separately on each atom. The observable operator may be described as a tensor product of Hermitian matrices $\hat{\mathcal{O}}=\hat{\mathcal{O}}_{\rm q}^{\otimes N}$ with $\hat{\mathcal{O}}_{\rm q}=a_{0}\mathbb{I}+\bm{a}\cdot\hat{\bm{\sigma}}$ dependent of four real coefficients $\{a_{0},a_{1},a_{2},a_{3}\}$, where $\mathbb{I}$ is the identity matrix of dimension $2$. Suppose that the input state is the maximally entangled states, i.e., GHZ states, which provides the Heisenberg-limit-scaling sensitivity of frequency estimation in the absence of noise [1, 9, 39]. Under the time evolution $\hat{U}=\exp{(-i\hat{H}t)}$, the output state can be represented as \[\|\psi_{\rm GHZ}(\varphi)\rangle=\frac{1}{\sqrt{2}}\big{(}\|0\rangle^{\otimes N} +e^{iN\varphi}\|1\rangle^{\otimes N}\big{)},\] (22) up to an irrelevant global phase with $\varphi=\omega t$. Here, we adopt the standard notation where $\|0\rangle$ and $\|1\rangle$ are the eigenvectors of $\sigma_{z}$ corresponding to eigenvalues $+1$ and $-1$, respectively. To determine the optimal separable observable $\hat{\mathcal{O}}$, we need to find the solutions of the coefficients $\{a_{0},a_{1},a_{2},a_{3}\}$ to satisfy equation (19). With $\hat{L}_{\varphi}=2\partial_{\varphi}(\|\psi_{\varphi}\rangle\langle\psi_{ \varphi}\|)$ for pure states, the SLD operator for the state of equation (22) is given by \[\hat{L}_{\varphi}=-iNe^{-iN\varphi}\left(\|0\rangle\langle 1\|\right)^{\otimes N }+iNe^{iN\varphi}\left(\|1\rangle\langle 0\|\right)^{\otimes N}.\] (23) We find that equation (19) is always satisfied for $a_{0}=a_{3}=0$ and arbitrary real number $a_{1},\,a_{2}$ that do not vanish simultaneously. Therefore, the general expression of the optimal separable observable is given by \[\hat{\mathcal{O}}_{\rm opt}=(a_{1}\hat{\sigma}_{x}+a_{2}\hat{\sigma}_{y})^{ \otimes N},\] (24) which is independent of the parameter $\varphi$, i.e., globally optimal in the whole range of the parameter. It is easy to check that such observables saturate the Heisenberg-limit sensitivity. Actually, according to the error-propagation formula equation (3), we have \[\delta\varphi_{\rm GHZ}=\frac{1}{\sqrt{\upsilon}}\frac{\sqrt{\langle\hat{ \mathcal{O}}_{\rm opt}^{2}\rangle-\langle\hat{\mathcal{O}}_{\rm opt}\rangle^{2 }}}{\|\partial_{\varphi}\langle\hat{\mathcal{O}}_{\rm opt}\rangle\|}=\frac{1}{ \sqrt{\upsilon}N},\] (25) as a result of \[\langle\hat{\mathcal{O}}_{\rm opt}\rangle = {\rm Re}[e^{-iN\varphi}(a_{1}+ia_{2})^{N}],\] (26) \[\langle\hat{\mathcal{O}}_{\rm opt}^{2}\rangle = (a_{1}^{2}+a_{2}^{2})^{N}.\] (27) When setting $a_{1}=1,a_{0}=a_{2}=a_{3}=0$, the optimal observable in equation (24) reduces to $\hat{\sigma}_{x}^{\otimes N}$, as given in [9]. Note that here measuring the observable $\hat{\sigma}_{x}^{\otimes N}$ fails to attain the Heisenberg limit for the cases of $\varphi=k\pi/N,\,(k\in\mathbb{Z})$ in which equation (25) becomes singular. Besides, we note that measuring the spin observable $\hat{\sigma}_{y}^{\otimes N}$ also fail in these cases when $N$ is even, and it is useful except for the cases of $\varphi=(2k+1)\pi/2N,\,(k\in\mathbb{Z})$ when $N$ is odd. We next show that the optimal observable in the form of equation (24) is closely related to the parity measurement proposed originally by Bollinger _et al_[3]. As is well known, in the standard Ramsey interferometry, there are generally two Ramsey pulses applying before and after the free evolution (with an accumulated phase $\varphi$), and measurements often take place after the second pulse [1, 3]. Here the action of the pulse is modeled by a $\pi/2$-rotation operation about the $y$ axis, i.e., $R_{y}\big{[}\frac{\pi}{2}\big{]}=\exp[-i(\frac{\pi}{2})\hat{J}_{y}]$, and the measurement observable is denoted as the operator $\hat{\mathcal{O}}_{f}$. With equation (24), one has \[\hat{\mathcal{O}}_{f}=R_{y}^{\dagger}\bigg{[}\frac{\pi}{2}\bigg{]}\,\hat{ \mathcal{O}}\,R_{y}\bigg{[}\frac{\pi}{2}\bigg{]}=(a_{1}\hat{\sigma}_{z}+a_{2} \hat{\sigma}_{y})^{\otimes N}.\] (28) When setting $a_{1}=1,a_{2}=0$, equation (28) reduces to \[\hat{\mathcal{O}}_{f}=\hat{\sigma}_{z}^{\otimes N}\equiv(-1)^{j-\hat{J}_{z}}\] (29) with $j=N/2$, which is the so-called parity measurement [3]. It is shown that only a parity measurement is necessary for the optimal estimate of the phase parameter $\varphi$ for GHZ states, and it is more experimentally feasible than the detection strategy, as discussed in [9], that applies local operations and classical communication. ## 4 Further discussions However, in a recent work [35], it was pointed out that the separable measurement (the restricted readout procedure) might not be possible to go beyond the shot-noise limit even for arbitrary entangled states. It seems that this conclusion is inconsistent with ours in the above discussion. In what follows, we clarify this issue by revisiting the method in [35] and showing the causes for this inconsistency. For simplicity, let us consider the two-qubit parametric GHZ state \[\|\psi^{(2)}_{\rm GHZ}(\varphi)\rangle=\frac{1}{\sqrt{2}}\big{(}\|0 0\rangle+e^{2i\varphi}\|11\rangle\big{)}.\] (30) Following Ref. [35], we restrict the separable measurement to be the projective measurements $\{\|+\rangle\langle+\|,\|-\rangle\langle-\|\}$ for each qubit with \[\|\pm\rangle=\frac{1}{\sqrt{2}}(\|0\rangle\pm\|1\rangle).\] (31) According to the condition of equation (21), whether the above restricted measurement presented by equation (31) is the optimal measurement saturating the QCRB can be tested by asking whether or not the operators of the form \[\hat{K} = \lambda_{++}\|++\rangle\langle++\|+\lambda_{+-}\|+-\rangle\langle+-\|\] (32) \[+\,\lambda_{-+}\|-+\rangle\langle-+\|+\lambda_{--}\|--\rangle\langle --\|\] can be the SLD operator for the state of equation (30). By domenstrating that for the state in equation (30) with $\varphi=0$, there is no solution of the SLD equation (8) for the coefficients $\{\lambda_{++},\lambda_{+-},\lambda_{-+},\lambda_{--}\}$ in equation (32), the authors in Ref. [35] claimed that the projective measurement about $\{\|++\rangle,\|+-\rangle,\|-+\rangle,\|--\rangle\}$ is not the optimal measurement for the state of equation (30). However, as we showed in the Sec. 3, $\sigma_{x}\otimes\sigma_{x}$ is an optimal observable saturating the QCRB with respect to $(\delta\varphi)^{2}_{\rm ep}$ for the states (30). Although the estimation error considered in the Ref. [35] is $(\delta\varphi)^{2}_{\rm est}$, a contradiction still arises, as the projective measurement of $\sigma_{x}\otimes\sigma_{x}$ attains the QFI of states (30) (see the end in Sec. 2) so that $\{\|++\rangle,\|+-\rangle,\|-+\rangle,\|--\rangle\}$ is the optimal measurement regarding the estimation error $(\delta\varphi)^{2}_{\rm est}$. Below, we shall show that actually for any other point except for $\varphi=k\pi/2,\,(k\in\mathbb{Z})$ in the range of the parameter, there do exist the SLD operator in form of equations (32). First, note that the SLD operator for the non-full-rank density matrices is not uniquely determined, but $\hat{L}_{\varphi}\rho_{\varphi}$ (or $\hat{L}_{\varphi}\|\psi_{\varphi}\rangle$ for pure state) is uniquely determined. Second, from equation (23), we see \[\hat{L}_{\varphi}=-2ie^{-2i\varphi}\|00\rangle\langle 11\|+2ie^{2i\varphi}\|11 \rangle\langle 00\|.\] (33) is a SLD operator for the state of equation (30). Third, since $\hat{L}_{\varphi}\|\psi_{\varphi}\rangle$ is uniquely determined, then if $\hat{K}$ is the SLD operator for $\|\psi_{\varphi}\rangle$ if and only if \[\hat{L}_{\varphi}\|\psi_{\varphi}\rangle=\hat{K}\|\psi_{\varphi}\rangle\] (34) is satisfied. Thus, substituting equations (30), (32) and (33) into equation (34), we obtain the solutions for the coefficients as \[\lambda_{++}=\lambda_{--}=-2\tan\varphi,\quad\lambda_{+-}=\lambda_{-+}=2\cot\varphi.\] (35) The above solutions are singular for $\varphi=k\pi/2,\,(k\in\mathbb{Z})$, which coincide with the results discussed below equation (27). Note that here the $\varphi=0,\,(k=0)$ case is just considered in Ref. [35]. Whilst, for a general value of the parameter except those singular points, the restricted separable measurement considered here indeed saturate the Heisenberg-limit-scaling sensitivity for the parametric state of equation (30). Moreover, it is easy to check that the same results of Eq. (35) can be obtained when restricting the separable measurement to be the projective measurements $\{\|+\rangle_{y}\langle+\|,\|-\rangle_{y}\langle-\|\}$ for each qubit with \[\|\pm\rangle_{y}=\frac{1}{\sqrt{2}}(\|0\rangle\pm i\|1\rangle)\] (36) the eigenvectors of $\sigma_{y}$. This is coincided with the result shown below equation (27) that measuring the observable $\sigma_{y}^{\otimes N}$ fails to attain Heisenberg limit for the $\varphi=k\pi/N,\,(k\in\mathbb{Z})$ cases when $N$ is even. ## 5 Conclusion We have addressed the optimization problem of measurements for achieving the ultimate sensitivity determined by the QCRB. From the propagation of error, we derive the N&S condition of the optimal observables for single parameter estimate by using the SRUR. As an application of this condition, we examine the optimal observables for GHZ states to achieve the ultimate sensitivity at the Heisenberg limit. We consider an experimentally feasible case that the observable operators are restricted to separably acting on the subsystem. We then find the general expression of the optimal separable observable by applying the N&S condition, and show that it is exactly equivalent to the parity measurement when applying a $\pi/2$ pulse operation. However, Jose _et al_ in [35] gave a contradictory conclusion with respect to ours that separable measurements are impossible to beat the shot-noise limit even for entangled states. We show that for the GHZ state case, their conclusion is established _only_ for some particular values of the parameter. Our results may be helpful for further investigation of the quantum metrology. We would like to thank Dr. Heng-Na Xiong and Dr. Qing-Shou Tan for helpful discussions. We also thank the second referee for constructive suggestions. This work is supported by the NFRPC with Grant No. 2012CB921602, the NSFC with Grants No. 11025527 and No. 10935010, and National Research Foundation and Ministry of Education, Singapore, with Grant No. WBS: R-710-000-008-271. ## References * [1] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865 (1997). * [2] R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, Phys. Rev. Lett. 111, 120401 (2013). * [3] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996). * [4] V. Bužek, R. Derka, and S. Massar, Phys. Rev. Lett. 82, 2207 (1999). * [5] The LIGO Scientic Collaboration, Nat. Phys. 7, 962 (2011). * [6] K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler and P. K. Lam, Phys. Rev. Lett. 88, 231102 (2002). * [7] H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 95, 211102 (2005). * [8] H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, Phys. Rev. D 65, 022002 (2001). * [9] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006). * [10] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Nature (London) 450, 393 (2007). * [11] W. van Dam, G. M. D’Ariano, A. Ekert, C. Macchiavello, and M. Mosca, Phys. Rev. Lett. 98, 090501 (2007). * [12] G. Goldstein, P. Cappellaro, J. R. Maze, J. S. Hodges, L. Jiang, A. S. Sørensen, and M. D. Lukin, Phys. Rev. Lett. 106, 140502 (2011). * [13] W. Dür, M. Skotiniotis, F. Fröwis, B. Kraus, Phys. Rev. Lett. 112, 080801 (2014). * [14] G. Arrad, Y. Vinkler, D. Aharonov, A. Retzker, Phys. Rev. Lett. 112, 150801 (2014). * [15] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Phys. Rev. Lett. 112, 150802 (2014). * [16] R. Ozeri, e-print arXiv:1310.3432. * [17] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994). * [18] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247, 135 (1996). * [19] C. W. Helstrom, _Quantum Detection and Estimation Theory_ (Academic, New York, 1976). * [20] A. S. Holevo, _Probabilistic and Statistical Aspects of Quantum Theory_ (North-Holland, Amsterdam, 1982). * [21] B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). * [22] T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, Phys. Rev. A 57, 4004 (1998). * [23] R. A. Campos, C. C. Gerry, and A. Benmoussa, Phys. Rev. A 68, 023810 (2003). * [24] M. Hotta and M. Ozawa, Phys. Rev. A 70, 022327 (2004). * [25] G. A. Durkin and J. P. Dowling, Phys. Rev. Lett. 99, 070801 (2007). * [26] L. Pezzé and A. Smerzi, Phys. Rev. Lett. 100. 073601 (2008). * [27] S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, Phys. Rev. Lett. 101. 040403 (2008). * [28] S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan, and C. M. Caves, Phys. Rev. A 77. 012317 (2008). * [29] P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Phys. Rev. Lett. 104. 103602 (2010). * [30] C. C. Gerry and J. Mimih, Phys. Rev. A 82, 013831 (2010). * [31] H. F. Hofmann, Phys. Rev. A 79, 033822 (2009). * [32] J. Joo, W. J. Munro, and T. P. Spiller, Phys. Rev. Lett. 107, 083601 (2011). * [33] K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, Phys. Rev. A 87, 043833 (2013). * [34] Scott Miller, Donald Childers, _Probability and Random Processes: With Applications to Signal Processing and Communications_, (Academic, Waltham, 2012), Chap. 7.4. * [35] S. Jose, N. Jaseem, and A. Shaji, Phys. Rev. A 87, 022330 (2013). * [36] A. Fujiwara, in _Geometry in Present Day Science_, edited by O. E. Barndorff-Nielsen and E. B. V. Jensen (World Scientific, Singapore, 1999), pp. 35–48. * [37] E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl. 19, 296 (1930). * [38] H. R. Robertson, Phy. Rev. 46, 794 (1934). * [39] D. M. Greenberger, M. Horne, and A. Zeilinger, Bells Theorem, Quantum Theory, and Conceptions of the Universe (Kluwer, Dordrecht, Netherlands, 1989).
1510.07423	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 86158, "num_imgs": 1, "llama3_tokens_count": 35723 }	[ "content_image/1510.07423/x1.png" ]	# Anisotropic scaling of random grain model with application to network traffic Vytautė Pilipauskaitė${}^{1,2}$ and Donatas Surgailis${}^{1}$ ${}^{1}$ Vilnius University, Institute of Mathematics and Informatics, 08663 Vilnius, Lithuania ${}^{2}$ Université de Nantes, Laboratoire de Mathématiques Jean Leray, 44322 Nantes Cedex 3, France ###### Abstract We obtain a complete description of anisotropic scaling limits of random grain model on the plane with heavy tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian and some ‘intermediate’ infinitely divisible random fields. Asymptotic form of the covariance function of the random grain model is obtained. Application to superposed network traffic is included. _Keywords:_ random grain model; anisotropic scaling; long-range dependence; Lévy sheet; fractional Brownian sheet; workload process ## 1 Introduction The present paper studies scaling limits of _random grain model_: \[X(t,s)\ :=\ \sum_{i}{\bf 1}\big{(}\big{(}(t-x_{i})/R_{i}^{p},(s- y_{i})/R_{i}^{1-p}\big{)}\in B\big{)},\quad(t,s)\in\mathbb{R}^{2},\] (1.1) where $B\subset\mathbb{R}^{2}$ (‘generic grain’) is a measurable bounded set of finite Lebesgue measure $\operatorname{leb}(B)<\infty$, $0<p<1$ is a shape parameter, $\{(x_{i},y_{i}),R_{i}\}$ is a Poisson point process on $\mathbb{R}^{2}\times\mathbb{R}_{+}$ with intensity $\mathrm{d}x\mathrm{d}yF(\mathrm{d}r)$. We assume that $F$ is a probability distribution on $\mathbb{R}_{+}$ having a density function $f$ such that \[f(r)\ \sim\ c_{f}\,r^{-1-\alpha}\quad\mbox{as }r\to\infty,\quad\exists\,1< \alpha<2,\ c_{f}>0.\] (1.2) The sum in (1.1) counts the number of uniformly scattered and randomly dilated grains $(x_{i},y_{i})+R_{i}^{P}B$ containing $(t,s)$, where $R^{P}B:=\{(R^{p}x,R^{1-p}y):(x,y)\in B\}\subset\mathbb{R}^{2}$ is the dilation of $B$ by factors $R^{p}$ and $R^{1-p}$ in the horizontal and vertical directions, respectively. The case $p=1/2$ corresponds to uniform or isotropic dilation. Note that the area $\operatorname{leb}(R^{P}B)=\operatorname{leb}(B)R$ of generic randomly dilated grain is proportional to $R$ and does not depend on $p$ and has a heavy-tailed distribution with finite mean $\mathrm{E}\operatorname{leb}(R^{P}B)<\infty$ and infinite second moment $\mathrm{E}\operatorname{leb}(R^{P}B)^{2}=\infty$ according to (1.2). Condition (1.2) also guarantees that covariance of the random grain model is not integrable: $\int_{\mathbb{R}^{2}}\|{\rm Cov}(X(0,0),X(t,s))\|\mathrm{d}t\mathrm{d}s=\infty,$ see Sec. 3, hence (1.1) is a long-range dependent (LRD) random field (RF). Examples of the grain set $B$ are the unit ball and the unit square, leading respectively to the _random ellipses model_: \[X(t,s)\ =\ \sum_{i}{\bf 1}\big{(}(t-x_{i})^{2}/R_{i}^{2p}+(s-y_{ i})^{2}/R_{i}^{2(1-p)}\leq 1\big{)}\] (1.3) and the _random rectangles model_: \[X(t,s)\ =\ \sum_{i}{\bf 1}(x_{i}<t\leq x_{i}+R_{i}^{p},\,y_{i}<s \leq y_{i}+R_{i}^{1-p}).\] (1.4) Note that the ratio of sides of a generic rectangle in (1.4) \[\frac{R^{p}}{R^{1-p}}\ =\ R^{2p-1}\ \to\ \begin{cases}0,&0<p<1/2,\\ \infty,&1/2<p<1\end{cases}\qquad\mbox{as }R\to\infty,\] implying that large rectangles are ‘elongated’ or ‘flat’ unless $p=1/2$, and resulting in a strong anisotropy of (1.4). A similar observation applies to the general random grain model in (1.1). The present paper obtains a complete description of _anisotropic_ scaling limits \[a^{-1}_{\lambda,\gamma}\int_{(0,\lambda x]\times(0,\lambda^{ \gamma}y]}(X(t,s)-\mathrm{E}X(t,s))\mathrm{d}t\mathrm{d}s\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ V_{\gamma}(x,y),\qquad(x,y)\in\mathbb{R}^{2}_{+},\quad\mbox{as } \lambda\to\infty\] (1.5) for the centered random grain model in (1.1) under assumption (1.2), where $\{(0,\lambda x]\times(0,\lambda^{\gamma}y]\subset\mathbb{R}^{2}_{+},\,\lambda>0\}$ is a family of rectangles with sides growing at possibly different rate $O(\lambda)$ and $O(\lambda^{\gamma})$ and $\gamma>0$ is _arbitrary_. In (1.5), $a_{\lambda,\gamma}\to\infty$ is a normalization. See the end of this section for all unexplained notation. <figure><img src="content_image/1510.07423/x1.png"><figcaption>Figure 1: Scaling limits of random grain model</figcaption></figure> Our main results are summarized in Fig. 1 which shows a panorama of scaling limits $V_{\gamma}$ in (1.5) as $\gamma$ changes between $0$ and $\infty$. Precise formulations pertaining to Fig. 1 and the terminology therein are given in Sec. 2. Below we explain the most important facts about this diagram. First of all note that, due to the symmetry of the random grain model in (1.1), the scaling limits in (1.5) are symmetric under simultaneous exchange $x\leftrightarrow y,\gamma\leftrightarrow 1/\gamma,p\leftrightarrow 1-p$ and a reflection transformation of $B$. This symmetry is reflected in Fig. 1, where the left region $0<\gamma\leq\gamma_{-}$ and the right region $\gamma_{+}\leq\gamma<\infty$ including the change points of the scaling limits \[\gamma_{-}:=\frac{1-p}{\alpha-(1-p)},\qquad\gamma_{+}:=\frac{\alpha}{p}-1,\] (1.6) are symmetric with respect to the above transformations. The middle region $\gamma_{-}<\gamma<\gamma_{+}$ in Fig. 1 corresponds to an $\alpha$_-stable Lévy sheet_ defined as a stochastic integral over $(0,x]\times(0,y]$ with respect to (w.r.t.) an $\alpha$-stable random measure on $\mathbb{R}^{2}_{+}$. According to Fig. 1, for $\gamma>\gamma_{+}$ the scaling limits in (1.5) exhibit a dichotomy depending on parameters $\alpha,p$, featuring a Gaussian (fractional Brownian sheet) limit for $2-p\leq\alpha<2$, and an $\alpha_{+}$-stable limit for $1<\alpha<2-p$ with stability parameter \[\alpha_{+}:=\frac{\alpha-p}{1-p}>\alpha\] (1.7) larger than the parameter $\alpha$. The terminology $\alpha_{\pm}$_-stable Lévy slide_ refers to a RF of the form $xL_{+}(y)$ or $yL_{-}(x)$ ‘sliding’ linearly to zero along one of the coordinate axes, where $L_{\pm}$ are $\alpha_{\pm}$-stable Lévy processes (see Sec. 2 for definition). Finally, the ‘intermediate Poisson’ limits in Fig. 1 at $\gamma=\gamma_{\pm}$ are not stable although infinitely divisible RFs given by stochastic integrals w.r.t. Poisson random measure on $\mathbb{R}^{2}\times\mathbb{R}_{+}$ with intensity measure $c_{f}\mathrm{d}u\mathrm{d}vr^{-1-\alpha}\mathrm{d}r$. The results of this paper are related to works [1], [3], [5], [6], [7], [10], [11], [12], [13], [14] and others, which discuss the occurrence of different scaling regimes for various classes of LRD models, particularly, heavy-tailed duration models. Isotropic scaling limits (case $\gamma=1$) of random grain and random balls models in arbitrary dimension were discussed in Kaj et al. [6] and Biermé et al. [1]. See also the monograph [9] for a nice discussion of limit behavior of heavy-tailed duration models. From an application viewpoint, probably the most interesting is the study of different scaling regimes of superposed network traffic models [10], [5], [6], [7]. In these studies, it is assumed that traffic is generated by independent sources and the problem concerns the limit distribution of the aggregated traffic as the time scale $T$ and the number of sources $M$ both tend to infinity, possibly at different rate. The present paper extends the above-mentioned work, by considering the limit behavior of the aggregated workload process: \[A_{M,K}(Tx) := \int_{0}^{Tx}W_{M,K}(t)\mathrm{d}t,\qquad\mbox{where}\] (1.8) \[W_{M,K}(t) := \sum_{i}(R^{1-p}_{i}\wedge K){\bf 1}(x_{i}<t\leq x_{i}+R^{p}_{i}, \,0<y_{i}<M),\quad t\geq 0,\] and where $\{(x_{i},y_{i}),R_{i}\}$ is the same Poisson point process as in (1.1). The quantity $W_{M,K}(t)$ in (1.8) can be interpreted as the active workload at time $t$ from sources arriving at $x_{i}$ with $0<y_{i}<M$ and transmitting at rate $R^{1-p}_{i}\wedge K$ during time interval $(x_{i},x_{i}+R^{p}].$ Thus, the transmission rate in (1.8) is a (deterministic) function $(R^{p})^{(1-p)/p}\wedge K$ of the transmission duration $R^{p}$ depending on parameter $0<p\leq 1$, with $0<K\leq\infty$ playing the role of the maximal rate bound. The limiting case $p=1$ in (1.8) corresponds to a constant rate workload from stationary M/G/$\infty$ queue. Theorems 4.1-4.3 obtain the limit distributions of the centered and properly normalized process $\{A_{M,K}(Tx),\,x\geq 0\}$ with heavy-tailed distribution of $R$ in (1.2) when the time scale $T$, the source intensity $M$ and the maximal source rate $K$ tend jointly to infinity so as $M=T^{\gamma},K=T^{\beta}$ for some $0<\gamma<\infty,0<\beta\leq\infty$. The main cases of Theorems 4.1 and 4.2 are summarized in Table 1. The workload process in (1.8) featuring a power-law dependence between transmission rate and duration is closely related to the random rectangles model in (1.4), the last fact being reflected in Table 1, where most (but not all) of the limit processes can be linked to the scaling limits in Fig. 1 and where $\gamma_{+},\alpha_{+}$ are the same as in (1.6), (1.7). Parameter region \| Limit process ---\|--- (1+γ)(1−p)<αβ≤∞ \| 1<α<2 \| α-stable Lévy process 0<αβ<(1+γ)(1−p) \| 1<α<2p \| (α/p)-stable Lévy process 1∨2p<α<2 \| Brownian motion a) Slow connection rate: 0<γ<γ+ Parameter region \| Limit process ---\|--- 0<α+β<γ+ \| 1<α<2p \| Fractional Brownian motion, H=(3−(α/p))/2 1∨2p<α<2 \| Brownian motion γ+<α+β<γ \| 1<α<2−p \| Gaussian line γ<α+β≤∞ \| α+-stable line γ+<α+β≤∞ \| 2−p<α<2 \| Fractional Brownian motion, H=(2−α+p)/2p b) Fast connection rate: γ+<γ<∞ Table 1: Limit distribution of the workload process in (1.8) with M=Tγ,K=Tβ Parameter region \| Limit process ---\|--- 0<α+β<γ+ \| 1<α<2p \| Fractional Brownian motion, H=(3−(α/p))/2 1∨2p<α<2 \| Brownian motion γ+<α+β<γ \| 1<α<2−p \| Gaussian line γ<α+β≤∞ \| α+-stable line γ+<α+β≤∞ \| 2−p<α<2 \| Fractional Brownian motion, H=(2−α+p)/2p b) Fast connection rate: γ+<γ<∞ Table 1: Limit distribution of the workload process in (1.8) with M=Tγ,K=Tβ The rest of the paper is organized as follows. Sec. 2 contains rigorous formulations (Theorems 2.1-2.5) of the asymptotic results pertaining to Fig. 1. Sec. 3 discusses LRD properties and asymptotics of the covariance function of the random grain model. Sec. 4 obtains limit distributions of the aggregated workload process in (1.8). All proofs are relegated to Sec. 5. _Notation._ In this paper, $\begin{array}[t]{c}\stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}$ and $\begin{array}[t]{c}\stackrel{{\rm fdd}}{{=}}\\ \end{array}$ denote the weak convergence and equality of finite dimensional distributions, respectively. $C$ stands for a generic positive constant which may assume different values at various locations and whose precise value has no importance. $\mathbb{R}_{+}:=(0,\infty),\mathbb{R}^{2}_{+}:=(0,\infty)^{2}$. ## 2 Scaling limits of random grain model We can rewrite the sum (1.1) as the stochastic integral \[X(t,s) = \int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}{\bf 1}\Big{(}\Big{(} \frac{t-u}{r^{p}},\frac{s-v}{r^{1-p}}\Big{)}\in B\Big{)}N(\mathrm{d}u,\mathrm{ d}v,\mathrm{d}r),\quad(t,s)\in\mathbb{R}^{2}\] (2.1) w.r.t. a Poisson random measure $N(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)$ on $\mathbb{R}^{2}\times\mathbb{R}_{+}$ with intensity measure $\mathrm{E}N(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)=\mathrm{d}u\mathrm{d}vF( \mathrm{d}r)$. The integral (2.1) is well-defined and follows a Poisson distribution with mean $\mathrm{E}X(t,s)=\operatorname{leb}(B)\int_{0}^{\infty}rF(\mathrm{d}r)$. The RF $X$ in (2.1) is stationary with finite variance and the covariance function (2.2) Let \[S_{\lambda,\gamma}(x,y) := \int_{0}^{\lambda x}\int_{0}^{\lambda^{\gamma}y}(X(t,s)-\mathrm{E }X(t,s))\mathrm{d}t\mathrm{d}s\] \[= \int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}\Big{\{}\int_{0}^{ \lambda x}\int_{0}^{\lambda^{\gamma}y}{\bf 1}\Big{(}\Big{(}\frac{t-u}{r^{p}}, \frac{s-v}{r^{1-p}}\Big{)}\in B\Big{)}\mathrm{d}t\mathrm{d}s\Big{\}}\widetilde {N}(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r),\quad(x,y)\in\mathbb{R}_{+}^{2},\] where $\widetilde{N}(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)=N(\mathrm{d}u,\mathrm{d}v, \mathrm{d}r)-\mathrm{E}N(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)$ is the centered Poisson random measure in (2.1). Recall the definition of $\gamma_{\pm}$: \[\gamma_{-}\ :=\ \frac{1-p}{\alpha-(1-p)},\qquad\gamma_{+}\ :=\ \frac{\alpha}{p }-1.\] (2.4) The subsequent Theorems 2.1-2.5 precise the limit RFs $V_{\gamma}$ and normalizations $a_{\lambda,\gamma}$ in (1.5) for all $\gamma>0$ and $\alpha\in(1,2),0<p<1$ in Fig. 1. Throughout the paper we assume that $B$ is a bounded Borel set whose boundary $\partial B$ has zero Lebesgue measure: $\operatorname{leb}(\partial B)=0$. ### Case $\gamma_{-}<\gamma<\gamma_{+}$ For $1<\alpha<2$, we introduce an $\alpha$-stable Lévy sheet \[L_{\alpha}(x,y)\ :=\ Z_{\alpha}((0,x]\times(0,y]),\quad(x,y)\in \mathbb{R}^{2}_{+}\] (2.5) as a stochastic integral w.r.t. an $\alpha$-stable random measure $Z_{\alpha}(\mathrm{d}u,\mathrm{d}v)$ on $\mathbb{R}^{2}$ with control measure $\sigma^{\alpha}\mathrm{d}u\mathrm{d}v$ and skewness parameter 1, where the constant $\sigma^{\alpha}$ is given in (5.5) below. Thus, $\mathrm{E}\exp\{{\bf i}\theta Z_{\alpha}(A)\}=\exp\{-\operatorname{leb}(A) \sigma^{\alpha}\|\theta\|^{\alpha}(1-{\bf i}\operatorname{sgn}(\theta)\tan(\pi \alpha/2))\},\,\theta\in\mathbb{R}$, for any Borel set $A\subset\mathbb{R}^{2}$ of finite Lebesgue measure $\operatorname{leb}(A)<\infty$. Note $\mathrm{E}Z_{\alpha}(A)=0$. Theorem 2.1: _Let $\gamma_{-}<\gamma<\gamma_{+}$, $1<\alpha<2$. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ L_{\alpha}(x,y)\quad\mbox{as }\lambda\to\infty,\] (2.6) _where $H(\gamma):=(1+\gamma)/\alpha$ and $L_{\alpha}$ is an $\alpha$-stable Lévy sheet defined in (2.5)._ ### Cases $\gamma>\gamma_{+},1<\alpha<2-p$ and $\gamma<\gamma_{-},1<\alpha<1+p$ For $1<\alpha<2-p$ and $1<\alpha<1+p$ introduce totally skewed stable Lévy processes $\{L_{+}(y),\,y\geq 0\}$ and $\{L_{-}(x),\,x\geq 0\}$ with respective stability indices $\alpha_{\pm}\in(1,2)$ defined as \[\alpha_{+}:=\frac{\alpha-p}{1-p},\qquad\alpha_{-}:=\frac{\alpha-1+p}{p}\] (2.7) and characteristic functions \[\mathrm{E}\exp\{{\bf i}\theta L_{\pm}(1)\} := \exp\{-\sigma^{\alpha_{\pm}}\|\theta\|^{\alpha_{\pm}}(1-{\bf i} \operatorname{sgn}(\theta)\tan(\pi\alpha_{\pm}/2))\},\quad\theta\in\mathbb{R},\] (2.8) where $\sigma^{\alpha_{+}}$ is given in (5.10) and $\sigma^{\alpha_{-}}$ can be found by symmetry, see (5.1) below. Theorem 2.2: _(i) Let $\gamma>\gamma_{+}$, $1<\alpha<2-p$. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ xL_{+}(y)\quad\mbox{as }\lambda\to\infty,\] (2.9) _where $H(\gamma):=1+\gamma/\alpha_{+}$ and $L_{+}$ is the $\alpha_{+}$-stable Lévy process defined by (2.8)._ _(ii) Let_ $0<\gamma<\gamma_{-}$_,_ $1<\alpha<1+p$_. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ yL_{-}(x)\quad\mbox{as }\lambda\to\infty,\] _where_ $H(\gamma):=\gamma+1/\alpha_{-}$ _and_ $L_{-}$ _is the_ $\alpha_{-}$_-stable Lévy process defined by (_2.8_)._ ### Cases $\gamma>\gamma_{+},\ 2-p\leq\alpha<2$ and $\gamma<\gamma_{-},\ 1+p\leq\alpha<2$ A (standard) fractional Brownian sheet (FBS) $B_{H_{1},H_{2}}$ with Hurst indices $0<H_{1},H_{2}\leq 1$ is defined as a Gaussian process with zero mean and covariance \[\mathrm{E}B_{H_{1},H_{2}}(x_{1},y_{1})B_{H_{1},H_{2}}(x_{2},y_{2}) = (1/4)(x_{1}^{2H_{1}}+x_{2}^{2H_{1}}-\|x_{1}-x_{2}\|^{2H_{1}})(y_{1} ^{2H_{2}}+y_{2}^{2H_{2}}-\|y_{1}-y_{2}\|^{2H_{2}}),\] \[(x_{i},y_{i})\in\mathbb{R}^{2}_{+},\ i= 1,2.\] The constants $\sigma_{+}$ and $\widetilde{\sigma}_{+}$ appearing in Theorems 2.3 (i) and 2.4 (i) are defined in (5.14) and (5.16), respectively. The corresponding constants $\sigma_{-}$ and $\widetilde{\sigma}_{-}$ in parts (ii) of these theorems can be found by symmetry (see (5.1)). Theorem 2.3: _(i) Let $\gamma>\gamma_{+}$, $2-p<\alpha<2$. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ \sigma_{+}B_{H_{+},1/2}(x,y)\quad\mbox{as }\lambda\to\infty,\] (2.10) _where $H(\gamma):=H_{+}\,+\,\gamma/2,\,H_{+}:=1/p\,-\,\gamma_{+}/2=(2-\alpha+p)/2p\in (1/2,1)$ and $B_{H_{+},1/2}$ is an FBS with parameters $(H_{+},1/2)$._ _(ii) Let_ $\gamma<\gamma_{-}$_,_ $1+p<\alpha<2$_. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ \sigma_{-}B_{1/2,H_{-}}(x,y)\quad\mbox{as }\lambda\to\infty,\] _where_ $H(\gamma):=\gamma H_{-}\,+\,1/2,\,H_{-}:=1/(1-p)+(1-p-\alpha)/2(1-p)\in(1/2,1)$ _and_ $B_{1/2,H_{-}}$ _is an FBS with parameters_ $(1/2,H_{-})$_._ Theorem 2.4: _(i) Let $\gamma>\gamma_{+}$, $\alpha=2-p$. Then_ \[\lambda^{-H(\gamma)}(\log\lambda)^{-1/2}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c}\stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ \widetilde{\sigma}_{+}B_{1,1/2}(x,y)\quad\mbox{as }\lambda\to\infty,\] (2.11) _where $H(\gamma):=1+\gamma/2$, $B_{1,1/2}$ is an FBS with parameters $(1,1/2).$_ _(ii) Let_ $\gamma<\gamma_{-}$_,_ $\alpha=1+p$_. Then_ \[\lambda^{-H(\gamma)}(\log\lambda)^{-1/2}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c}\stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ \widetilde{\sigma}_{-}B_{1/2,1}(x,y)\quad\mbox{as }\lambda\to\infty,\] _where_ $H(\gamma):=\gamma+1/2$ _and_ $B_{1/2,1}$ _is an FBS with parameters_ $(1/2,1)$_._ ### Cases $\gamma=\gamma_{\pm}$ Define ‘intermediate Poisson’ RFs $I_{\pm}=\{I_{\pm}(x,y),\,(x,y)\in\mathbb{R}^{2}_{+}\}$ as stochastic integrals \[I_{+}(x,y) := \int_{\mathbb{R}\times(0,y]\times\mathbb{R}_{+}}\widetilde{M}( \mathrm{d}u,\mathrm{d}v,\mathrm{d}r)\int_{(0,x]\times\mathbb{R}}{\bf 1}\Big{(} \Big{(}\frac{t-u}{r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B\Big{)}\mathrm{d}t\, \mathrm{d}s,\] (2.12) \[I_{-}(x,y) := \int_{(0,x]\times\mathbb{R}\times\mathbb{R}_{+}}\widetilde{M}( \mathrm{d}u,\mathrm{d}v,\mathrm{d}r)\int_{\mathbb{R}\times(0,y]}{\bf 1}\Big{(} \Big{(}\frac{t}{r^{p}},\frac{s-v}{r^{1-p}}\Big{)}\in B\Big{)}\mathrm{d}t\, \mathrm{d}s\] w.r.t. the centered Poisson random measure $\widetilde{M}(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)=M(\mathrm{d}u,\mathrm{d}v, \mathrm{d}r)-\mathrm{E}M(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)$ on $\mathbb{R}^{2}\times\mathbb{R}_{+}$ with intensity measure $\mathrm{E}M(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)=c_{f}\mathrm{d}u\mathrm{d}vr^ {-(1+\alpha)}\mathrm{d}r.$ Proposition 2.1: _(i) The RF_ $I_{+}$ _in (_2.12_) is well-defined for_ $1<\alpha<2$_,_ $0<p<1$ _and_ $\mathrm{E}\|I_{+}(x,y)\|^{q}<\infty$ _for any_ $0<q<\alpha_{+}\wedge 2$_. Moreover, if_ $2-p<\alpha<2$ _then_ $\mathrm{E}\|I_{+}(x,y)\|^{2}<\infty$ _and_ (2.13) _where_ $\sigma_{+},H_{+}$ _are the same as in Theorem_ 2.3 _(i)._ _(ii) The RF_ $I_{-}$ _in (_2.12_) is well-defined for_ $1<\alpha<2$_,_ $0<p<1$ _and_ $\mathrm{E}\|I_{-}(x,y)\|^{q}<\infty$ _for any_ $0<q<\alpha_{-}\wedge 2$_. Moreover,_ _if_ $1+p<\alpha<2$ _then_ $\mathrm{E}\|I_{-}(x,y)\|^{2}<\infty$ _and_ _where_ $\sigma_{-},H_{-}$ _are the same as in Theorem_ 2.3 _(ii)._ Theorem 2.5: _(i) Let $\gamma=\gamma_{+}$, $1<\alpha<2$. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ I_{+}(x,y)\quad\mbox{as }\lambda\to\infty,\] (2.14) _where $H(\gamma):=1/p$ and RF $I_{+}$ is defined in (2.12)._ _(ii) Let_ $\gamma=\gamma_{-}$_,_ $1<\alpha<2$_. Then_ \[\lambda^{-H(\gamma)}S_{\lambda,\gamma}(x,y)\ \begin{array}[t]{c} \stackrel{{\rm fdd}}{{\longrightarrow}}\\ \end{array}\ I_{-}(x,y)\quad\mbox{as }\lambda\to\infty,\] _where_ $H(\gamma):=\gamma_{-}/(1-p)$ _and RF_ $I_{-}$ _is defined in (_2.12_)._ Remark 2.1: It can be easily verified that the ‘intermediate Poisson’ RFs $I_{\pm}$ in (2.12) have stationary rectangular increments (see [13], [14] for the definition) and satisfy the operator self-similarity property in [2], viz., $\{I_{\pm}(\lambda x,\lambda^{\gamma_{\pm}}y)\}\begin{array}[t]{c}\stackrel{{ \rm fdd}}{{=}}\\ \end{array}\{\lambda^{H(\gamma_{\pm})}I_{\pm}(x,y)\}$ for any $\lambda>0$. __ Remark 2.2: The normalizing exponent $H(\gamma)\equiv H(\gamma,\alpha,p)$ in Theorems 2.1-2.5 is a jointly continuous (albeit non-analytic) function of $(\gamma,\alpha,p)\in(0,\infty)\times(1,2)\times(0,1)$. Remark 2.3: Restriction $\alpha<2$ is crucial for our results. Indeed, if $\alpha>2$ then for _any_$\gamma>0$, $p\in(0,1)$ the normalized integrals $\lambda^{-(1+\gamma)/2}S_{\lambda,\gamma}(x,y)\begin{array}[t]{c}\stackrel{{ \rm fdd}}{{\longrightarrow}}\\ \end{array}\sigma B_{1/2,1/2}(x,y)$ tend to a classical Brownian sheet with variance $\sigma^{2}=\operatorname{leb}(B)^{2}\int_{0}^{\infty}r^{2}F(\mathrm{d}r)$. We omit the proof of the last result which follows a general scheme of the proofs in Sec. 5. ## 3 LRD properties of random grain model It is well-known that scaling limits characterize the dependence structure and large-scale properties of the underlying random process. Anisotropic scaling of a stationary RF $Y$ on $\mathbb{R}^{2}$ as in (1.5) with arbitrary $\gamma>0$ results in a one-dimensional family $\{V_{\gamma},\gamma>0\}$ of scaling limits and provides a more complete ‘large-scale summary of $Y$’ compared to the usual (isotropic) scaling with fixed $\gamma=1$. [13] observed that for many LRD RFs $Y$ in $\mathbb{Z}^{2}$, there exists a _unique_ point $\gamma_{0}>0$ such that the scaling limits $V_{\gamma}\begin{array}[t]{c}\stackrel{{\rm fdd}}{{=}}\\ \end{array}V_{\pm}$ do not depend on $\gamma$ for $\gamma<\gamma_{0}$ and $\gamma>\gamma_{0}$ and $V_{+}\begin{array}[t]{c}\stackrel{{\rm fdd}}{{\neq}}\\ \end{array}V_{-}$. [13] termed this phenomenon scaling transition (at $\gamma=\gamma_{0}$). The existence of scaling transition was established for a class of aggregated nearest-neighbor autoregressive RFs [13] and a natural class of Gaussian LRD RFs [14]. It also arises under joint temporal and contemporaneous aggregation of independent LRD processes in telecommunication and economics, see [10], [5], [7], [11], [12], also ([13], Remark 2.3). The results of the present work (Fig. 1) show a more complicated picture with _two_ change-points $\gamma_{-}<\gamma_{+}$ of scaling limits which does not fit into the definition of scaling transition in [13] and suggests that this concept might be more complex and needs further studies. One of the most common definitions of LRD property pertains to stationary random processes with non-summable (non-integrable) autocovariance function. In the case of anisotropic RFs, the autocovariance function may decay at different rate in different directions, motivating a more detailed classification of LRD as in Definition 3.1 below. In this Sec. we also verify these LRD properties for the random grain model in (1.1)-(1.2) and relate them to the change of the scaling limits or the dichotomies in Fig. 1; see Remark 3.1 below. Definition 3.1: _Let $Y=\{Y(t,s),\,(t,s)\in\mathbb{R}^{2}\}$ be a stationary RF with finite variance and nonnegative covariance function $\rho_{Y}(t,s):={\rm Cov}(Y(0,0),Y(t,s))\geq 0$. We say that:_ _(i)_ $Y$ _has short-range dependence (SRD) property if_ $\int_{\mathbb{R}^{2}}\rho_{Y}(t,s)\mathrm{d}t\mathrm{d}s<\infty$_; otherwise we say that_ $Y$ _has long-range dependence (LRD) property;_ _(ii)_ $Y$ _has vertical SRD property if_ $\int_{[-Q,Q]\times\mathbb{R}}\rho_{Y}(t,s)\mathrm{d}t\mathrm{d}s<\infty$ _for any_ $0<Q<\infty$_; otherwise we say that_ $Y$ _has vertical LRD property;_ _(iii)_ $Y$ _has horizontal SRD property if_ $\int_{\mathbb{R}\times[-Q,Q]}\rho_{Y}(t,s)\mathrm{d}t\mathrm{d}s<\infty$ _for any_ $0<Q<\infty$_; otherwise we say that_ $Y$ _has horizontal LRD property._ The main result of this Sec. is Theorem 3.1 providing the asymptotics of the covariance function of the random grain model in (1.1)-(1.2) as $\|t\|+\|s\|\to\infty$ and enabling the verification of its integrability properties in Definition 3.1. Let \[w:=(\|t\|^{1/p}+\|s\|^{1/(1-p)})^{p},\quad\mbox{for }(t,s)\in\mathbb{R}^{2}.\] For $p=1/2$, $w$ is the Euclidean norm and $(w,{\rm arccos}(t/w))$ are the polar coordinates of $(t,s)\in\mathbb{R}^{2},\,s\geq 0$. Introduce a function $b(z),z\in[-1,1]$ by \[b(z) := c_{f}\int_{0}^{\infty}\operatorname{leb}\Big{(}B\cap\Big{(}B+ \big{(}z/r^{p},(1-\|z\|^{1/p})^{1-p}/r^{1-p}\big{)}\Big{)}\Big{)}r^{-\alpha} \mathrm{d}r,\] (3.1) playing the role of the ‘angular function’ in the asymptotics (3.2). For the random balls model (1.3) with $p=1/2$, $b(z)$ is a constant function independent on $z$. Theorem 3.1: _Let $1<\alpha<2$, $0<p<1$._ _(i) The function_ $b(z)$ _in (_3.1_) is bounded, continuous and strictly positive on_ $[-1,1]$_._ _(ii) The covariance function_ $\rho(t,s):={\rm Cov}(X(0,0),X(t,s))$ _in (_2.2_) has the following asymptotics:_ \[\rho(t,s)\ \sim\ b(\operatorname{sgn}(s)t/w)w^{-(\alpha-1)/p} \quad\mbox{as }\|t\|+\|s\|\to\infty.\] (3.2) Theorem 3.1 implies the following bound for covariance function $\rho(t,s)={\rm Cov}(X(0,0),X(t,s))$ of the random grain model: there exist $Q>0$ and strictly positive constants $0<C_{-}<C_{+}<\infty$ such that for any $\|t\|+\|s\|>Q$ \[C_{-}(\|t\|^{1/p}+\|s\|^{1/(1-p)})^{1-\alpha}\ \leq\ \rho(t,s)\ \leq\ C_{+}(\|t\|^{1 /p}+\|s\|^{1/(1-p)})^{1-\alpha}.\] (3.3) The bounds in (3.3) together with easy integrability properties of the function $({\|t\|^{1/p}+\|s\|^{1/(1-p)}})^{1-\alpha}$ on $\{\|t\|+\|s\|>Q\}$ imply the following corollary. Corollary 3.1: _The random grain model in (1.1)-(1.2) has:_ _(i) LRD property for any_ $1<\alpha<2,\ 0<p<1$_;_ _(ii) vertical LRD property for_ $1<\alpha\leq 2-p$ _and vertical SRD property for_ $2-p<\alpha<2$ _and any_ $0<p<1$_;_ _(iii) horizontal LRD property for_ $1<\alpha\leq 1+p$ _and horizontal SRD property for_ $1+p<\alpha<2$ _and any_ $0<p<1$_._ Remark 3.1: The above corollary indicates that the dichotomy at $\alpha=2-p$ in Fig. 1, region $\gamma>\gamma_{+}$ is related to the change from the vertical LRD to the vertical SRD property in the random grain model. Similarly, the horizontal transition from the LRD to the SRD explains the dichotomy at $\alpha=1+p$ in Fig. 1, region $\gamma<\gamma_{-}$. __ [13] introduced Type I distributional LRD property for RF $Y$ with two-dimensional ‘time’ in terms of dependence properties of rectangular increments of scaling limits $V_{\gamma},\gamma>0$. The increment of a RF $V=\{V(x,y),(x,y)\in\mathbb{R}^{2}_{+}\}$ on rectangle $K=(u,x]\times(v,y]\subset\mathbb{R}^{2}_{+}$ is defined as the double difference $V(K)=V(x,y)-V(u,y)-V(x,v)+V(u,v)$. Let $\ell\subset\mathbb{R}^{2}$ be a line, $(0,0)\in\ell$. According to ([13], Definition 2.2), a RF $V=\{V(x,y),(x,y)\in\mathbb{R}^{2}_{+}\}$ is said to have: * _independent rectangular increments in direction $\ell$_ if $V(K)$ and $V(K^{\prime})$ are independent for any two rectangles $K,K^{\prime}\subset\mathbb{R}^{2}_{+}$ which are separated by an orthogonal line $\ell^{\prime}\perp\ell$; * _invariant rectangular increments in direction $\ell$_ if $V(K)=V(K^{\prime})$ for any two rectangles $K,K^{\prime}$ such that $K^{\prime}=(x,y)+K$ for some $(x,y)\in\ell$; * _properly dependent rectangular increments_ if $V$ has neither independent nor invariant increments in arbitrary direction $\ell$. Further on, a stationary RF $Y$ on $\mathbb{Z}^{2}$ is said to have _Type I distributional LRD_ ([13], Definition 2.4) if there exists a unique point $\gamma_{0}>0$ such that its scaling limit $V_{\gamma_{0}}$ has properly dependent rectangular increments while all other scaling limits $V_{\gamma},\gamma\neq\gamma_{0}$ have either independent or invariant rectangular increments in some direction $\ell=\ell(\gamma)$. The above definition trivially extends to RF $Y$ on $\mathbb{R}^{2}$. We end this Sec. with the observation that _all scaling limits of the random grain model in (1.1)-(1.2) in Theorems 2.1-2.5 have either independent or invariant rectangular increments in direction of one or both coordinate axes._ The last fact is immediate from stochastic integral representations in (2.5), (2.12), the covariance function of FBS with Hurst indices $H_{1},H_{2}$ equal to $1$ or $1/2$ (see also ([13], Example 2.3)) and the limit RFs in (2.9) and (2.2). We conclude that the random grain model in (1.1)-(1.2) _does not have Type I distributional LRD_ in contrast to Gaussian and other classes of LRD RFs discussed in [13], [14]. The last conclusion is not surprising since similar facts about scaling limits of heavy-tailed duration models with one-dimensional time are well-known; see e.g. [8]. ## 4 Limit distributions of aggregated workload process We rewrite the accumulated workload in (1.8) as the integral \[A_{M,K}(Tx) = \int_{\mathbb{R}\times(0,M]\times\mathbb{R}_{+}}\Big{\{}(r^{1-p} \wedge K)\int_{0}^{Tx}{\bf 1}(u<t\leq u+r^{p})\mathrm{d}t\Big{\}}N(\mathrm{d}u ,\mathrm{d}v,\mathrm{d}r),\] (4.1) where $N(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)$ is the same Poisson random measure on $\mathbb{R}^{2}\times\mathbb{R}_{+}$ with intensity $\mathrm{E}N(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)=\mathrm{d}u\mathrm{d}F( \mathrm{d}r)$ as in (1.1). We assume that $F(\mathrm{d}r)$ has a density $f(r)$ satisfying (1.2) with $1<\alpha<2$ as in Sec. 2. We let $p\in(0,1]$ in (4.1) and thus the parameter may take value $p=1$ as well. We assume that $K$ and $M$ grow with $T$ in such a way that \[M=T^{\gamma},\ K=T^{\beta}\quad\mbox{for some }0<\gamma<\infty,\ 0<\beta\leq\infty.\] (4.2) We are interested in the limit distribution \[b_{T}^{-1}(A_{M,K}(Tx)-\mathrm{E}A_{M,K}(Tx))\ \begin{array}[t]{c}\stackrel{{ \rm fdd}}{{\longrightarrow}}\\ \end{array}\ {\mathcal{A}}(x)\quad\mbox{as }T\to\infty,\] (4.3) where $b_{T}\equiv b_{T,\gamma,\beta}\to\infty$ is a normalization. Recall from (1.6) and (1.7) the definitions \[\gamma_{+}\ =\ \frac{\alpha}{p}-1,\qquad\alpha_{+}\ =\ \frac{\alpha-p}{1-p}.\] For $p=1$, let $\alpha_{+}:=\infty$. By assumption (1.2), transmission durations $R_{i}^{p},i\in\mathbb{Z}$ have a heavy-tailed distribution with tail parameter $\alpha/p>1$. Following the terminology in [3], [5], [6], [10], the regions $\gamma<\gamma_{+}$, $\gamma>\gamma_{+}$ and $\gamma=\gamma_{+}$ will be respectively referred to as _slow connection rate_, _fast connection rate_ and _intermediate connection rate_. For each of these ‘regimes’, Theorems 4.1, 4.2 and 4.3 detail the limit processes and normalizations in (4.3) depending on parameters $\beta,\alpha,p$. Apart from the classical Gaussian and stable processes listed in Table 1, some ‘intermediate’ infinitely divisible processes arise. Let us introduce \[I(x)\ :=\ \int_{\mathbb{R}\times\mathbb{R}_{+}}\Big{\{}\int_{0}^ {x}1(u<t<u+r^{p})\mathrm{d}t\Big{\}}\widetilde{\cal M}(\mathrm{d}u,\mathrm{d}r ),\quad x\geq 0,\] (4.4) where $\widetilde{M}(\mathrm{d}u,\mathrm{d}r)$ is a centered Poisson random measure with intensity measure $c_{f}\mathrm{d}ur^{-(1+\alpha)}\mathrm{d}r$. The process in (4.4) essentially depends on the ratio $\alpha/p$ only and is well-defined for $1<\alpha<2p$ and $1/2<p\leq 1$. Under the ‘intermediate’ regime this process arises for many heavy-tailed duration models (see e.g. [3], [5], [7]). It was studied in detail in [4]. We introduce a ‘truncated’ version of (4.4): \[\widehat{I}(x)\ :=\ \int_{\mathbb{R}\times\mathbb{R}_{+}}\Big{\{} (r^{1-p}\wedge 1)\int_{0}^{x}{\bf 1}(u<t<u+r^{p})\mathrm{d}t\Big{\}}\widetilde {\cal M}(\mathrm{d}u,\mathrm{d}r),\quad x\geq 0\] (4.5) and its Gaussian counterpart \[\widehat{Z}(x)\ :=\ \int_{\mathbb{R}\times\mathbb{R}_{+}}\Big{\{} (r^{1-p}\wedge 1)\int_{0}^{x}{\bf 1}(u<t<u+r^{p})\mathrm{d}t\Big{\}}{\cal Z}( \mathrm{d}u,\mathrm{d}r),\quad x\geq 0,\] (4.6) where ${\cal Z}(\mathrm{d}u,\mathrm{d}r)$ is a Gaussian random measure on $\mathbb{R}\times\mathbb{R}_{+}$ with the same variance $c_{f}\mathrm{d}ur^{-(1+\alpha)}\mathrm{d}r$ as the centered Poisson random measure $\widetilde{\cal M}(\mathrm{d}u,\mathrm{d}r)$. The processes in (4.5) and (4.6) are well-defined for any $1<\alpha<2$, $0<p\leq 1$ and have the same covariance functions. The RFs defined in Sec. 2 reappear in Theorems 4.1-4.3 for the certain grain set, namely the unit square $B=\{(u,v):0<u,v<1\}\subset\mathbb{R}^{2}$. Recall that a homogeneous Lévy process $\{L(x),\,x\geq 0\}$ is completely specified by its characteristic function $\mathrm{E}\mathrm{e}^{{\bf i}\theta L(1)},\theta\in\mathbb{R}$. A (standard) fractional Brownian motion with Hurst parameter $H\in(0,1]$ is a Gaussian process $\{B_{H}(x),\,x\geq 0\}$ with zero mean and covariance function $(1/2)(x^{2H}+y^{2H}-\|x-y\|^{2H}),\,x,y\geq 0$. Theorem 4.1: (Slow connection rate.) _Let $0<\gamma<\gamma_{+}$. The convergence in (4.3) holds with the limit ${\cal A}$ and normalization $b_{T}=T^{\mathcal{H}}$ specified in (i)-(v) below._ _(i) Let_ $(1+\gamma)(1-p)<\alpha\beta\leq\infty$_. Then_ ${\mathcal{H}}:=(1+\gamma)/\alpha$ _and_ ${\cal A}:=\{L_{\alpha}(x,1),\,x\geq 0\}$ _is an_ $\alpha$_-stable Lévy process defined by (_2.5_)._ _(ii) Let_ $0<\alpha\beta<(1+\gamma)(1-p)$ _and_ $1<\alpha<2p$_. Then_ ${\mathcal{H}}:=\beta+(1+\gamma)p/\alpha$ _and_ ${\mathcal{A}}:=\{L_{\alpha/p}(x),\,x\geq 0\}$ _is an_ $(\alpha/p)$_-stable Lévy process with characteristic function given by (_5.22_)._ _(iii) Let_ $0<\alpha\beta<(1+\gamma)(1-p)$ _and_ $1\lor 2p<\alpha<2$_. Then_ ${\mathcal{H}}:=(1/2)(1+\gamma+\beta(2-\alpha)/(1-p))$ _and_ ${\mathcal{A}}:=\{\sigma_{1}B(x),\,x\geq 0\}$ _is a Brownian motion with variance_ $\sigma^{2}_{1}$ _given by (_5.23_)._ _(iv) Let_ $0<\alpha\beta<(1+\gamma)(1-p)$ _and_ $\alpha=2p$_. Then_ $b_{T}:=T^{{\mathcal{H}}}(\log T)^{1/2}$ _with_ ${\mathcal{H}}:=\beta+(1+\gamma)/2$ _and_ ${\mathcal{A}}:=\{\widehat{\sigma}_{1}B(x),\,x\geq 0\}$ _is a Brownian motion with variance_ $\widehat{\sigma}^{2}_{1}$ _given by (_5.24_)._ _(v) Let_ $\alpha\beta=(1+\gamma)(1-p)$_. Then_ ${\mathcal{H}}:=(1+\gamma)/\alpha$ _and_ ${\mathcal{A}}:=\{\widehat{L}(x),\,x\geq 0\}$ _is a Lévy process with characteristic function in (_5.25_)._ Theorem 4.2: (Fast connection rate.) _Let $\gamma_{+}<\gamma<\infty$. The convergence in (4.3) holds with the limit ${\cal A}$ and normalization $b_{T}:=T^{\mathcal{H}}$ specified in (i)-(ix) below._ _(i) Let_ $0<\alpha_{+}\beta<\gamma_{+}$ _and_ $1<\alpha<2p$_. Then_ ${\cal H}:=H+\beta+\gamma/2$ _and_ ${\cal A}:=\{\sigma_{2}B_{H}(x),\,x\geq 0\}$ _is a fractional Brownian motion with_ $H=(3-\alpha/p)/2$ _and variance_ $\sigma_{2}^{2}$ _given by (_5.26_)._ _(ii) Let_ $0<\alpha_{+}\beta<\gamma_{+}$ _and_ $1\lor 2p<\alpha<2$_. Then_ ${\cal H}$ _and_ ${\cal A}$ _are the same as in Theorem_ 4.1 _(iii)._ _(iii) Let_ $\gamma_{+}<\alpha_{+}\beta<\gamma$ _and_ $1<\alpha<2-p$_. Then_ ${\cal H}:=1+(1/2)(\gamma+\beta(2-\alpha-p)/(1-p))$ _and_ ${\cal A}:=\{xZ,\,x\geq 0\}$ _is a Gaussian line with random slope_ $Z\sim N(0,\sigma_{3}^{2})$ _and_ $\sigma_{3}^{2}$ _given in (_5.27_)._ _(iv) Let_ $\gamma<\alpha_{+}\beta\leq\infty$ _and_ $1<\alpha<2-p$_. Then_ ${\cal H}:=1+\gamma/\alpha_{+}$ _and_ ${\cal A}:=\{xL_{+}(1),\,x\geq 0\}$ _is an_ $\alpha_{+}$_-stable line with random slope_ $L_{+}(1)$ _having_ $\alpha_{+}$_-stable distribution defined by (_2.8_)._ _(v) Let_ $\gamma_{+}<\alpha_{+}\beta\leq\infty$ _and_ $2-p<\alpha<2$_. Then_ ${\cal H}:=H_{+}+\gamma/2$ _and_ ${\cal A}:=\{\sigma_{+}B_{H_{+},1/2}(x,1),\,x\geq 0\}$ _is a fractional Brownian motion with_ $H=H_{+}=(2-\alpha+p)/2p$ _and variance_ $\sigma_{+}^{2}$ _given by (_5.14_)._ _(vi) Let_ $0<\alpha_{+}\beta<\gamma_{+}$ _and_ $\alpha=2p$_. Then_ $b_{T}:=T^{\cal H}(\log T)^{1/2}$ _with_ ${\cal H}:=\beta+(1+\gamma)/2$ _and_ ${\cal A}:=\{\widehat{\sigma}_{2}B(x),\,x\geq 0\}$ _is a Brownian motion with variance_ $\widehat{\sigma}_{2}^{2}$ _in (_5.28_)._ _(vii) Let_ $\alpha_{+}\beta=\gamma_{+}$_. Then_ ${\cal H}:=(1/2)(1+\gamma+(2-\alpha)/p)$ _and_ ${\cal A}:=\{\widehat{Z}(x),\,x\geq 0\}$ _in an intermediate Gaussian process defined by (_4.6_)._ _(viii) Let_ $\alpha_{+}\beta=\gamma$ _and_ $1<\alpha<2-p$_. Then_ ${\cal H}=1+\beta$ _and_ ${\cal A}:=\{x\widehat{Z},\,x\geq 0\}$_, where a slope_ $\widehat{Z}$ _is a r.v. defined by (_5.29_)._ _(ix) If_ $\gamma_{+}<\alpha_{+}\beta\leq\infty$ _and_ $\alpha=2-p$_. Then_ $b_{T}:=T^{\cal H}(\log T)^{1/2}$_,_ ${\cal H}:=1+\gamma/2$ _and_ ${\cal A}:=\{\widetilde{\sigma}_{+}B_{1,1/2}(x,1),\,x\geq 0\}=\{x\widetilde{Z}, \,x\geq 0\}$ _is a Gaussian line with random slope_ $\widetilde{Z}\sim N(0,\widetilde{\sigma}_{+}^{2})$ _and_ $\widetilde{\sigma}_{+}^{2}$ _given by (_5.16_)._ Theorem 4.3: (Intermediate connection rate.) _Let $\gamma=\gamma_{+}$. The convergence in (4.3) holds with the limit ${\cal A}$ and normalization $b_{T}:=T^{\mathcal{H}}$ specified in (i)-(v) below._ _(i) Let_ $0<\alpha_{+}\beta<\gamma_{+}$ _and_ $1<\alpha<2p$_. Then_ ${\cal H}:=1+\beta$ _and_ ${\cal A}:=\{I(x),\,x\geq 0\}$ _is an intermediate process defined by (_4.4_)._ _(ii) Let_ $0<\alpha_{+}\beta<\gamma_{+}$ _and_ $1\lor 2p<\alpha<2$_. Then_ ${\cal H}$ _and_ ${\cal A}$ _are the same as in Theorem_ 4.1 _(iii)._ _(iii) Let_ $0<\alpha_{+}\beta<\gamma_{+}$ _and_ $\alpha=2p$_. Then_ ${\cal H}$ _and_ ${\cal A}$ _are the same as in Theorem_ 4.1 _(iv)._ _(iv) Let_ $\alpha_{+}\beta=\gamma_{+}$_. Then_ ${\cal H}:=1/p$ _and_ ${\cal A}:=\{\widehat{I}(x),\,x\geq 0\}$ _is an intermediate process defined by (_4.5_)._ _(v) Let_ $\gamma_{+}<\alpha_{+}\beta\leq\infty$_. Then_ ${\cal H}:=1/p$ _and_ ${\cal A}:=\{I_{+}(x,1),\,x\geq 0\}$ _is an intermediate process defined by (_2.12_)._ Remark 4.1: For $\gamma=\gamma_{+}$ we have $(1+\gamma)(1-p)/\alpha=\gamma_{+}/\alpha_{+}=(1-p)/p$. Note that $p=1$ implies $\gamma_{+}=\alpha-1$. In this case, Theorem 4.1 reduces to the $\alpha$-stable limit in (i), whereas Theorem 4.2 reduces to the fractional Brownian motion limit in (v) discussed in [10] and other papers. A similar dichotomy appears for $\beta$ close to zero and $1<\alpha<2p$ with the difference that $\alpha$ is now replaced by $\alpha/p$. Intuitively, it can be explained as follows. For small $\beta>0$, the workload process $W_{M,K}(t)$ in (1.8) behaves like a constant rate process $K\sum_{i}{\bf 1}(x_{i}<t\leq x_{i}+R^{p}_{i},\,0<y_{i}<M)$ with transmission lengths $R^{p}_{i}$ that are i.i.d. and follow the same distribution $\mathrm{P}(R^{p}_{i}>r)=\mathrm{P}(R_{i}>r^{1/p})\sim(c_{f}/\alpha)r^{-(\alpha /p)},r\to\infty$ with tail parameter $1<\alpha/p<2$. Therefore, for small $\beta$ our results agree with [10], including the Gaussian limit in Theorems 4.1 (iii) and 4.2 (ii) arising when the $R^{p}_{i}$’s have finite variance. __ Remark 4.2: As it follows from the proof, the random line limits in Theorem 4.2 (iv) and (iii) are caused by extremely long sessions starting in the past at times $x_{i}<0$ and lasting $R^{p}_{i}=O(T^{\gamma/\gamma_{+}}),\,\gamma_{+}<\gamma<\alpha_{+}\beta$ or $R^{p}_{i}=O(T^{\alpha_{+}\beta/\gamma_{+}}),\,\gamma_{+}<\alpha_{+}\beta<\gamma$, respectively, so that typically these sessions end at times $x_{i}+R^{p}_{i}\gg T$. ## 5 Proofs ### Proofs of Sections 2 and 3 Let \[X^{}(t,s)\ :=\ \int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}{\bf 1} \Big{(}\Big{(}\frac{t-u}{r^{1-p}},\frac{s-v}{r^{p}}\Big{)}\in B^{}\Big{)}N( \mathrm{d}u,\mathrm{d}v,\mathrm{d}r),\quad(t,s)\in\mathbb{R}^{2},\] be a ‘reflected’ version of (2.1), with $B$ replaced by $B^{}:=\{(u,v)\in\mathbb{R}^{2}:(v,u)\in B\},$$p$ replaced by $1-p$ and the same Poisson random measure $N(\mathrm{d}u,\mathrm{d}v,\mathrm{d}r)$ as in (2.1). Let $S^{}_{\lambda_{},\gamma_{}}(x,y):=\int_{0}^{\lambda_{}x}\int_{0}^{\lambda_ {}^{\gamma_{}}y}(X^{}(t,s)-\mathrm{E}X^{}(t,s))\mathrm{d}t\mathrm{d}s,(x,y )\in\mathbb{R}^{2}_{+}$ be the corresponding partial integral in (2). If $\lambda_{},\gamma_{}$ are related to $\lambda,\gamma$ as $\lambda_{}=\lambda^{\gamma},\gamma_{}=1/\gamma$ then \[S^{}_{\lambda_{},\gamma_{}}(y,x)\ \begin{array}[t]{c}\stackrel{{ \rm fdd}}{{=}}\\ \end{array}\ S_{\lambda,\gamma}(x,y)\] (5.1) holds by symmetry property of the Poisson random measure. As noted in the Introduction, relation (5.1) allows to reduce the limits of $S_{\lambda,\gamma}(x,y)$ as $\lambda\to\infty$ and $\gamma\leq\gamma_{-}$ to the limits of $S^{}_{\lambda_{},\gamma_{}}(y,x)$ as $\lambda_{}\to\infty$ and $\gamma_{}\geq\gamma_{+}:=\alpha/(1-p)-1$. As a consequence, the proofs of parts (ii) of Theorems 2.2-2.5 can be omitted since they can be deduced from parts (i) of the corresponding statements. The convergence of normalized partial integrals in (1.5) is equivalent to the convergence of characteristic functions: \[\mathrm{E}\exp\Big{\{}{\bf i}a_{\lambda,\gamma}^{-1}\sum_{i=1}^{m }\theta_{i}S_{\lambda,\gamma}(x_{i},y_{i})\Big{\}} \to \mathrm{E}\exp\Big{\{}{\bf i}\sum_{i=1}^{m}\theta_{i}V_{\gamma}(x _{i},y_{i})\Big{\}}\quad\mbox{as }\lambda\to\infty,\] (5.2) for all $m=1,2,\dots$, $(x_{i},y_{i})\in\mathbb{R}^{2}_{+}$, $\theta_{i}\in\mathbb{R}$, $i=1,\dots,m$. We restrict the proof of (5.2) to one-dimensional convergence for $m=1$, $(x,y)\in\mathbb{R}_{+}^{2}$ only. The general case of (5.2) follows analogously. We have \[W_{\lambda,\gamma}(\theta) := \log\mathrm{E}\exp\{{\bf i}\theta a^{-1}_{\lambda,\gamma}S_{ \lambda,\gamma}(x,y)\}\] (5.3) \[= \int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}\Psi\Big{(}\frac{\theta} {a_{\lambda,\gamma}}\int_{0}^{\lambda x}\int_{0}^{\lambda^{\gamma}y}{\bf 1} \Big{(}\Big{(}\frac{t-u}{r^{p}},\frac{s-v}{r^{1-p}}\Big{)}\in B\Big{)}\mathrm{ d}t\mathrm{d}s\Big{)}\mathrm{d}u\mathrm{d}vf(r)\mathrm{d}r.\] where $\Psi(z):=\mathrm{e}^{{\bf i}z}-1-{{\bf i}z}$, $z\in\mathbb{R}$. We shall use the following inequality: \[\|\Psi(z)\|\ \leq\ \min(2\|z\|,z^{2}/2),\quad z\in\mathbb{R}.\] (5.4) _Proof of Theorem 2.1._ In the integrals on the r.h.s. of (5.3) we change the variables: \[\frac{t-u}{r^{p}}\ \to\ t,\quad\frac{s-v}{r^{1-p}}\ \to\ s,\quad u\ \to\ \lambda u,\quad v\ \to\ \lambda^{\gamma}v,\quad r\ \to\ \lambda^{H(\gamma)}r.\] This yields $W_{\lambda,\gamma}(\theta)=\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r) \mathrm{d}r$, where \[f_{\lambda}(r)\ :=\ \lambda^{(1+\alpha)H(\gamma)}f(\lambda^{H(\gamma)}r)\ \to \ c_{f}\,r^{-(1+\alpha)},\qquad\lambda\to\infty\] according to (1.2), and \[g_{\lambda}(r) := \int_{\mathbb{R}^{2}}\Psi(\theta h_{\lambda}(u,v,r))\mathrm{d}u \mathrm{d}v,\] \[h_{\lambda}(u,v,r) := r\int_{B}{\bf 1}(0<u+\lambda^{-\delta_{1}}r^{p}t\leq x,\,0<v+ \lambda^{-\delta_{2}}r^{1-p}s\leq y)\mathrm{d}t\mathrm{d}s,\] where the exponents $\delta_{1}:=1-H(\gamma)p=(\gamma_{+}-\gamma)/(1+\gamma_{+})>0$, $\delta_{2}:=\gamma-H(\gamma)(1-p)=(\gamma-\gamma_{-})/(1+\gamma_{-})>0$. Clearly, \[h_{\lambda}(u,v,r)\ \to\ \operatorname{leb}(B)r\,{\bf 1}(0<u\leq x,0<v\leq y), \qquad\lambda\to\infty\] for any fixed $(u,v,r)\in\mathbb{R}^{2}\times\mathbb{R}_{+}$, $u\not\in\{0,x\},v\not\in\{0,y\}$, implying \[g_{\lambda}(r)\ \to\ xy\Psi(\theta\operatorname{leb}(B)r)\] for any $r>0$. Since $\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v=xyr\operatorname {leb}(B)$ and $h_{\lambda}(u,v,r)\leq Cr$, the dominating bound $\|g_{\lambda}(r)\|\leq C\min(r,r^{2})$ follows by (5.4). Whence and from Lemma 5.1 we conclude that \[W_{\lambda,\gamma}(\theta) \to W_{\gamma}(\theta)\ :=\ xy\,c_{f}\int_{0}^{\infty}(\mathrm{e}^{{ \bf i}\theta\operatorname{leb}(B)r}-1-{\bf i}\theta\operatorname{leb}(B)r)r^{- (1+\alpha)}\mathrm{d}r.\] It remains to verify that where \[\sigma^{\alpha}\ :=\ c_{f}\operatorname{leb}(B)^{\alpha}\cos(\pi\alpha/2) \Gamma(2-\alpha)/\alpha(1-\alpha).\] (5.5) This proves the one-dimensional convergence in (2.6) and Theorem 2.1, too. $\Box$ _Proof of Theorem 2.2._ In (5.3), change the variables as follows: \[t\ \to\ \lambda t,\quad s-v\ \to\ \lambda^{(1-p)\gamma/(\alpha-p)}s,\quad u\ \to\ \lambda^{p\gamma/(\alpha-p)}u,\quad v\ \to\ \lambda^{\gamma}v,\quad r\ \to\ \lambda^{\gamma/(\alpha-p)}r.\] (5.6) This yields $W_{\lambda,\gamma}(\theta)=\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r) \mathrm{d}r$, where \[f_{\lambda}(r)\ :=\ \lambda^{(1+\alpha)\gamma/(\alpha-p)}f(\lambda^{\gamma/( \alpha-p)}r)\ \to\ c_{f}\,r^{-(1+\alpha)},\qquad\lambda\to\infty\] (5.7) and $g_{\lambda}(r):=\int_{\mathbb{R}^{2}}\Psi(\theta h_{\lambda}(u,v,r))\mathrm{d} u\mathrm{d}v$ with \[h_{\lambda}(u,v,r) := \int_{0}^{x}\mathrm{d}t\int_{\mathbb{R}}{\bf 1}\Big{(}\Big{(} \frac{\lambda^{-\delta_{1}}t-u}{r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B\Big{)}{ \bf 1}(0<v+\lambda^{-\delta_{2}}s<y)\mathrm{d}s,\] (5.8) where $\delta_{1}:=p\gamma/(\alpha-p)-1=(\gamma-\gamma_{+})/\gamma_{+}>0$, $\delta_{2}:=\gamma(\alpha-1)/(\alpha-p)>0$. Let $B(u):=\{v\in\mathbb{R}:(u,v)\in B\}$ and write $\operatorname{leb}_{1}(A)$ for the Lebesgue measure of a set $A\subset\mathbb{R}$. By the dominated convergence theorem, \[h_{\lambda}(u,v,r)\ \to\ h(u,v,r) := x\,{\bf 1}(0<v<y)\int_{\mathbb{R}}{\bf 1}\Big{(}\Big{(}\frac{-u} {r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B\Big{)}\mathrm{d}s\] \[= x\,{\bf 1}(0<v<y)r^{1-p}\operatorname{leb}_{1}(B(-u/r^{p}))\] for any $(u,v,r)\in\mathbb{R}^{2}\times\mathbb{R}_{+},v\not\in\{0,y\}$, implying \[g_{\lambda}(r) \to g(r)\ :=\ \int_{\mathbb{R}^{2}}\Psi(\theta h(u,v,r))\mathrm{d}u \mathrm{d}v\ =\ y\,r^{p}\int_{\mathbb{R}}\Psi\big{(}\theta xr^{1-p} \operatorname{leb}_{1}(B(u))\big{)}\mathrm{d}u\] for any $r>0$. Indeed, since $B$ is bounded, for fixed $r>0$ the function $(u,v)\mapsto h_{\lambda}(u,v,r)$ has a bounded support uniformly in $\lambda\geq 1$. Therefore it is easy to verify domination criterion for the above convergence. Combining $h_{\lambda}(u,v,r)\leq Cr^{1-p}$ with $\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v=xyr\operatorname {leb}(B)$ gives $\|g_{\lambda}(r)\|\leq C\min(r,r^{2-p})$ by (5.4). Hence and by Lemma 5.1, $W_{\lambda,\gamma}(\theta)\to W_{\gamma}(\theta):=c_{f}\int_{0}^{\infty}g(r)r^ {-(1+\alpha)}\mathrm{d}r$. By change of variable, the last integral can be rewritten as \[W_{\gamma}(\theta) = c_{f}\,y\,x^{\alpha_{+}}(1-p)^{-1}\int_{\mathbb{R}}\operatorname {leb}_{1}(B(u))^{\alpha_{+}}\mathrm{d}u\int_{0}^{\infty}(\mathrm{e}^{{\bf i} \theta w}-1-{\bf i}\theta w)w^{-(1+\alpha_{+})}\mathrm{d}w\] \[=\] where \[\sigma^{\alpha_{+}}\ :=\ \frac{c_{f}\Gamma(2-\alpha_{+})\cos(\pi \alpha_{+}/2)}{(1-p)\alpha_{+}(1-\alpha_{+})}\int_{\mathbb{R}}\operatorname{ leb}_{1}(B(u))^{\alpha_{+}}\mathrm{d}u,\] (5.10) thus completing the proof of one-dimensional convergence in (2.9). Theorem 2.2 is proved. $\Box$ _Proof of Theorem 2.3._ In (5.3), change the variables as follows: \[t\ \to\ \lambda t,\quad s-v\ \to\ \lambda^{(1/p)-1}s,\quad u\ \to\ \lambda u, \quad v\ \to\ \lambda^{\gamma}v,\quad r\ \to\ \lambda^{1/p}r.\] (5.11) We get $W_{\lambda,\gamma}(\theta)=\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r) \mathrm{d}r$, where \[f_{\lambda}(r)\ :=\ \lambda^{(1+\alpha)/p}f(\lambda^{1/p}r),\quad g_{\lambda}( r)\ :=\ \int_{\mathbb{R}^{2}}\lambda^{2(H(\gamma)-1/p)}\Psi(\theta\lambda^{(1/ p)-H(\gamma)}h_{\lambda}(u,v,r))\mathrm{d}u\mathrm{d}v,\] (5.12) with \[h_{\lambda}(u,v,r) := \int_{0}^{x}\mathrm{d}t\int_{\mathbb{R}}{\bf 1}(0<v+\lambda^{- \delta}s<y){\bf 1}\Big{(}\Big{(}\frac{t-u}{r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B \Big{)}\mathrm{d}s\] (5.13) \[\to\] \[= {\bf 1}(0<v<y)\,r^{1-p}\int_{0}^{x}\operatorname{leb}_{1}(B((t-u) /r^{p}))\mathrm{d}t\ =:\ h(u,v,r),\qquad\lambda\to\infty,\] for all $(u,v,r)\in\mathbb{R}^{2}\times\mathbb{R}_{+}$, $v\not\in\{0,y\}$, since $\delta:=1+\gamma-(1/p)>0.$ Note that $2(H(\gamma)-1/p)=\gamma-\gamma_{+}>0$ and hence \[\lambda^{2(H(\gamma)-1/p)}\Psi(\theta\lambda^{(1/p)-H(\gamma)}h_{\lambda}(u,v, r))\ \to\ -(\theta^{2}/2)h^{2}(u,v,r),\qquad\lambda\to\infty.\] Next, by the dominated convergence theorem \[g_{\lambda}(r)\ \to\ g(r):=-\frac{\theta^{2}}{2}\int_{\mathbb{R}^{2}}h^{2}(u,v ,r)\mathrm{d}u\mathrm{d}v\] for any $r>0$. Using $\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v=xy\operatorname{ leb}(B)r$ and $h_{\lambda}(u,v,r)\leq C\min(r^{1-p},r)$ similarly as in the proof of Theorem 2.2 we obtain $\|g_{\lambda}(r)\|\leq C\int_{\mathbb{R}^{2}}h^{2}_{\lambda}(u,v,r)\mathrm{d}u \mathrm{d}v\leq C\min(r^{2-p},r^{2})$. Then by Lemma 5.1, \[W_{\lambda,\gamma}(\theta)\ \to\ W_{\lambda}(\theta)\ :=\ c_{f}\int_{0}^{ \infty}g(r)r^{-(1+\alpha)}\mathrm{d}r\ =\ -(\theta^{2}/2)\sigma^{2}_{+}x^{2H_{ +}}y,\] where \[\sigma^{2}_{+}\ :=\ c_{f}\int_{\mathbb{R}}\mathrm{d}u\int_{0}^{ \infty}\Big{(}\int_{0}^{1}\operatorname{leb}_{1}(B((t-u)/r^{p}))\mathrm{d}t \Big{)}^{2}r^{1-\alpha-2p}\mathrm{d}r,\] (5.14) where the last integral converges. (Indeed, since $u\mapsto\operatorname{leb}_{1}(B(u))=\int 1((u,v)\in B)\mathrm{d}v$ is a bounded function with compact support, the inner integral in (5.14) does not exceed $C(1\wedge r^{p}){\bf 1}(\|u\|<K(1+r^{p}))$ for some $C,K>0$ implying $\sigma^{2}_{+}\leq C\int_{0}^{\infty}(1\wedge r^{p})^{2}(1+r^{p})r^{1-\alpha-2 p}\mathrm{d}r<\infty$ since $2-p<\alpha<2$.) This ends the proof of one-dimensional convergence in (2.10). Theorem 2.3 is proved. $\Box$ _Proof of Theorem 2.4._ After the same change of variables as in (5.6), viz., \[t\ \to\ \lambda t,\quad s-v\ \to\ \lambda^{\gamma/2}s,\quad u\ \to\ \lambda^{p \gamma/2(1-p)}u,\quad v\ \to\ \lambda^{\gamma}v,\quad r\ \to\ \lambda^{\gamma/ 2(1-p)}r,\] we obtain $W_{\lambda,\gamma}(\theta)=\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r) \mathrm{d}r$ with $f_{\lambda}(r)$ as in (5.7) and $g_{\lambda}(r):=\int_{\mathbb{R}^{2}}\Psi(\theta(\log\lambda)^{-1/2}h_{\lambda }(u,v,r))\mathrm{d}u\mathrm{d}v$, where \[h_{\lambda}(u,v,r) := \int_{0}^{x}\mathrm{d}t\int_{\mathbb{R}}{\bf 1}\Big{(}\Big{(} \frac{\lambda^{-\delta_{1}}t-u}{r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B\Big{)}{ \bf 1}(0<v+\lambda^{-\delta_{2}}s<y)\mathrm{d}s,\] $\delta_{1}:=p\gamma/2(1-p)-1=(\gamma-\gamma_{+})/\gamma_{+}>0$, $\delta_{2}:=\gamma/2>0$ are the same as in (5.8) and \[h_{\lambda}(u,v,r)\ \to\ h(u,v,r)\ :=\ x\,{\bf 1}(0<v<y)\int_{ \mathbb{R}}{\bf 1}\Big{(}\Big{(}\frac{-u}{r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B \Big{)}\mathrm{d}s\] c.f. (5.1). Below we prove that the main contribution to the limit of $W_{\lambda,\gamma}(\theta)$ comes from the interval $\lambda^{-\delta_{1}/p}<r<1$, namely, that $W_{\lambda,\gamma}(\theta)-W^{0}_{\lambda,\gamma}(\theta)\to 0$, where \[W^{0}_{\lambda,\gamma}(\theta)\ :=\ \int_{\lambda^{-\delta_{1}/p }}^{1}g_{\lambda}(r)f_{\lambda}(r)\mathrm{d}r \sim -\frac{\theta^{2}}{2}\,\frac{c_{f}}{\log\lambda}\int_{\lambda^{- \delta_{1}/p}}^{1}\frac{\mathrm{d}r}{r^{3-p}}\int_{\mathbb{R}^{2}}h^{2}(u,v,r) \mathrm{d}u\mathrm{d}v\] \[= -\frac{\theta^{2}}{2}x^{2}yc_{f}\int_{\mathbb{R}}(\operatorname{ leb}_{1}(B(u)))^{2}\mathrm{d}u\,\frac{1}{\log\lambda}\int_{\lambda^{-\delta_{1 }/p}}^{1}r^{-1}\mathrm{d}r\] \[= -\frac{\theta^{2}}{2}\widetilde{\sigma}^{2}_{+}x^{2}y\ =:\ W_{ \gamma}(\theta),\] where \[\widetilde{\sigma}^{2}_{+}\ :=\ \frac{c_{f}(\gamma-\gamma_{+})}{2(1-p)}\int_{ \mathbb{R}}\operatorname{leb}(B\cap(B+(0,u)))\mathrm{d}u\] (5.16) and where we used the fact that $\int_{\mathbb{R}^{2}}h^{2}(u,v,r)\mathrm{d}u\mathrm{d}v=x^{2}yr^{2-p}\int_{ \mathbb{R}}\operatorname{leb}_{1}(B(u))^{2}\mathrm{d}u=x^{2}yr^{2-p}\int_{ \mathbb{R}}\operatorname{leb}(B\cap(B+(0,u)))\mathrm{d}u$. Accordingly, write $W_{\lambda,\gamma}(\theta)=W^{0}_{\lambda,\gamma}(\theta)+W^{-}_{\lambda, \gamma}(\theta)+W^{+}_{\lambda,\gamma}(\theta)$, where $W^{-}_{\lambda,\gamma}(\theta):=\int_{0}^{\lambda^{-\delta_{1}/p}}g_{\lambda}( r)f_{\lambda}(r)\mathrm{d}r$ and $W^{+}_{\lambda,\gamma}(\theta):=\int_{1}^{\infty}g_{\lambda}(r)f_{\lambda}(r) \mathrm{d}r$ are remainder terms. Indeed, using (5.4) and \[\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v\ =\ xyr \operatorname{leb}(B),\qquad h_{\lambda}(u,v,r)\ \leq\ C(\lambda^{\delta_{1}}r )\wedge r^{1-p}.\] (5.17) it follows that \[\|W^{+}_{\lambda,\gamma}(\theta)\| \leq \frac{C}{(\log\lambda)^{1/2}}\int_{1}^{\infty}\frac{\mathrm{d}r}{ r^{3-p}}\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v\ =\ O(( \log\lambda)^{-1/2})\ =\ o(1).\] Similarly, \[\|W^{-}_{\lambda,\gamma}(\theta)\| \leq \frac{C\lambda^{\delta_{1}}}{\log\lambda}\int_{0}^{\lambda^{- \delta_{1}/p}}rf_{\lambda}(r)\mathrm{d}r\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r )\mathrm{d}u\mathrm{d}v\ \leq\ \frac{C\lambda^{\delta_{1}}}{\log\lambda}\int_{ 0}^{\lambda^{-\delta_{1}/p}}r^{2}f_{\lambda}(r)\mathrm{d}r\] \[= \frac{C}{\lambda\log\lambda}\int_{0}^{\lambda^{1/p}}r^{2}f(r) \mathrm{d}r\ =\ O((\log\lambda)^{-1})\ =\ o(1).\] since $\delta_{1}=p\gamma/2(1-p)-1$. Consider the main term $W^{0}_{\lambda,\gamma}(\theta)$ in (5.1). Let $\widetilde{W}_{\lambda,\gamma}(\theta):=-\frac{\theta^{2}}{2\log\lambda}\int_{ \lambda^{-\delta_{1}/p}}^{1}f_{\lambda}(r)\mathrm{d}r\int_{\mathbb{R}^{2}}h^{2 }_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v$. Then using (5.17) and $\|\Psi(z)+z^{2}/2\|\leq\|z\|^{3}/6$ we obtain \[\|W^{0}_{\lambda,\gamma}(\theta)-\widetilde{W}_{\lambda,\gamma}( \theta)\| \leq \frac{C}{(\log\lambda)^{3/2}}\int_{\lambda^{-\delta_{1}/p}}^{1}r^ {2-2p}f_{\lambda}(r)\mathrm{d}r\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{ d}u\mathrm{d}v\] \[\leq \frac{C}{(\log\lambda)^{3/2}}\int_{\lambda^{-\delta_{1}/p}}^{1}r^ {3-2p}f_{\lambda}(r)\mathrm{d}r\] \[\leq \frac{C}{(\log\lambda)^{3/2}}\int_{0}^{1}r^{-p}\mathrm{d}r\ =\ O( (\log\lambda)^{-3/2})\ =\ o(1).\] Finally, it remains to estimate the difference $\|\widetilde{W}_{\lambda,\gamma}(\theta)-W_{\gamma}(\theta)\|\leq C(J^{\prime}_{ \lambda}+J^{\prime\prime}_{\lambda})$, where \[J^{\prime}_{\lambda} := \frac{1}{\log\lambda}\int_{\lambda^{-\delta_{1}/p}}^{1}f_{\lambda }(r)\mathrm{d}r\int_{\mathbb{R}^{2}}\|h^{2}_{\lambda}(u,v,r)-h^{2}(u,v,r)\| \mathrm{d}u\mathrm{d}v,\] \[J^{\prime\prime}_{\lambda} := \frac{1}{\log\lambda}\int_{\lambda^{-\delta_{1}/p}}^{1}r^{2-p}\|f_ {\lambda}(r)-c_{f}r^{p-3}\|\mathrm{d}r.\] Let \[\widetilde{h}_{\lambda}(u,v,r) := x\int_{\mathbb{R}}{\bf 1}\Big{(}\Big{(}\frac{-u}{r^{p}},\frac{s} {r^{1-p}}\Big{)}\in B\Big{)}{\bf 1}(0<v+\lambda^{-\delta_{2}}s<y)\mathrm{d}s.\] Then $J^{\prime}_{\lambda}\leq J^{\prime}_{\lambda 1}+J^{\prime}_{\lambda 2}$, where $J^{\prime}_{\lambda 1}:=(\log\lambda)^{-1}\int_{\lambda^{-\delta_{1}/p}}^{1}f_ {\lambda}(r)\mathrm{d}r\int_{\mathbb{R}^{2}}\|h^{2}_{\lambda}(u,v,r)-\widetilde {h}^{2}_{\lambda}(u,v,r)\|\mathrm{d}u\mathrm{d}v$ and $J^{\prime}_{\lambda 2}:=$$(\log\lambda)^{-1}\int_{\lambda^{-\delta_{1}/p}}^{1}f_{\lambda}(r)\mathrm{d}r \int_{\mathbb{R}^{2}}\|\widetilde{h}^{2}_{\lambda}(u,v,r)-h^{2}(u,v,r)\|\mathrm{ d}u\mathrm{d}v$. Using the fact that $B$ is a bounded set with $\operatorname{leb}(\partial B)=0$ we get that \[\int_{\mathbb{R}^{2}}\|h_{\lambda}(u,v,r)-\widetilde{h}_{\lambda}( u,v,r)\|\mathrm{d}u\mathrm{d}v \leq\] \[\leq r\epsilon(1/\lambda^{\delta_{1}}r^{p}),\] where $\epsilon(z),z\geq 0$ is a bounded function with $\lim_{z\to 0}\epsilon(z)=0$. We also have $h_{\lambda}(u,v,r)+\widetilde{h}_{\lambda}(u,v,r)\leq Cr^{1-p}$ as in (5.17). Using these bounds together with $f_{\lambda}(r)\leq Cr^{p-3},r>\lambda^{-\delta_{1}/p}$ we obtain \[J^{\prime}_{\lambda 1}\log\lambda\ \leq\ C\int_{\lambda^{-\delta _{1}/p}}^{1}\epsilon(1/\lambda^{\delta_{1}}r^{p})r^{-1}\mathrm{d}r\ =\ C\int_{ \lambda^{-\delta_{1}}}^{1}\epsilon(z)z^{-1}\mathrm{d}z\ =\ o(\log\lambda),\] proving $J^{\prime}_{\lambda 1}\to 0$ as $\lambda\to\infty$. In a similar way, using $\int_{\mathbb{R}^{2}}\|\widetilde{h}_{\lambda}(u,v,r)-h(u,v,r)\|\mathrm{d}u \mathrm{d}v\leq xr\int_{\mathbb{R}^{3}}{\bf 1}((-u,s)\in B)\|{\bf 1}(0<v+ \lambda^{-\delta_{2}}r^{1-p}s<y)-{\bf 1}(0<v<y)\|\mathrm{d}u\mathrm{d}v\mathrm{ d}s\leq Cr^{2-p}\lambda^{-\delta_{2}}$ we obtain $J^{\prime}_{\lambda 2}\log\lambda\leq C\lambda^{-\delta_{2}}\int_{0}^{1}r^{-p} \mathrm{d}r=O(\lambda^{-\delta_{2}})$, proving $J^{\prime}_{\lambda 2}\to 0$ and hence $J^{\prime}_{\lambda}\to 0$. Finally, $J^{\prime\prime}_{\lambda}=(\log\lambda)^{-1}\int_{\lambda^{1/p}}^{\infty}r^{2 -p}\|f(r)-c_{f}r^{p-3}\|\mathrm{d}r\to 0$ follows from (1.2). This proves the limit $\lim_{\lambda\to\infty}W_{\lambda,\gamma}(\theta)=W_{\gamma}(\theta)=-(\theta^ {2}/2)\widetilde{\sigma}^{2}_{+}x^{2}y$ for any $\theta\in\mathbb{R}$, or one-dimensional convergence in (2.11). Theorem 2.4 is proved. $\Box$ _Proof of Proposition 2.1._ We use well-known properties of Poisson stochastic integrals and inequality (3.3) in [11]. Accordingly, $I_{+}(x,y)$ is well-defined and satisfies $\mathrm{E}\|I_{+}(x,y)\|^{q}\leq 2J_{q}(x,y)\,(1\leq q\leq 2)$ provided \[J_{q}(x,y) := c_{f}\int_{0}^{\infty}r^{-(1+\alpha)}\mathrm{d}r\int_{\mathbb{R} \times(0,y]}\mathrm{d}u\mathrm{d}v\Big{\|}\int_{(0,x]\times\mathbb{R}}{\bf 1} \Big{(}\Big{(}\frac{t-u}{r^{p}},\frac{s}{r^{1-p}}\Big{)}\in B\Big{)}\mathrm{d} t\mathrm{d}s\Big{\|}^{q}\] \[= c_{f}y\int_{0}^{\infty}r^{q(1-p)-(1+\alpha)}\mathrm{d}r\int_{ \mathbb{R}}\mathrm{d}u\Big{\|}\int_{0}^{x}\operatorname{leb}_{1}\Big{(}B\Big{(} \frac{t-u}{r^{p}}\Big{)}\Big{)}\mathrm{d}t\Big{\|}^{q}\ <\ \infty.\] Split $J_{q}(x,y)=c_{f}y[\int_{0}^{1}\mathrm{d}r+\int_{1}^{\infty}]\cdots\mathrm{d}r= :c_{f}y[J^{\prime}+J^{\prime\prime}]$. Then $J^{\prime\prime}\leq C\int_{1}^{\infty}r^{q(1-p)-(1+\alpha)}\mathrm{d}r\int{ \bf 1}(\|u\|\leq Cr^{p})\mathrm{d}u\leq C\int_{1}^{\infty}r^{q(1-p)-(1+\alpha)+p }\mathrm{d}r<\infty$ provided $q<(\alpha-p)/(1-p).$ Similarly, $J^{\prime}\leq C\int_{0}^{1}r^{q(1-p)-(1+\alpha)}\mathrm{d}r\|\int{\bf 1}(\|t\| \leq Cr^{p})\mathrm{d}t\|^{q}\leq C\int_{0}^{1}r^{q(1-p)-(1+\alpha)+qp}\mathrm{ d}r<\infty$ provided $\alpha<q$. Note that $\alpha<(\alpha-p)/(1-p)\leq 2$ for $1<\alpha\leq 2-p$ and $(\alpha-p)/(1-p)>2$ for $2-p<\alpha<2$. Relation (2.13) follows from (2.3) and $J_{2}(x,y)=\sigma^{2}_{+}yx^{2H_{+}}$ by a change of variables. This proves part (i). The proof of part (ii) is analogous. $\Box$ _Proof of Theorem 2.5._ Using the change of variables as in (5.11) we get $W_{\lambda,\gamma}(\theta)=\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r) \mathrm{d}r$ with the same $f_{\lambda}(r),g_{\lambda}(r)$ as in (5.12) and $h_{\lambda}(u,v,r)$ satisfying (5.13). (Note $H(\gamma)=H(\gamma_{+})=1/p$ hence $\lambda^{H(\gamma_{+})-(1/p)}=1$ in the definition of $g_{\lambda}(r)$ in (5.12).) Particularly, $\Psi(\theta h_{\lambda}(u,v,r))\to\Psi(\theta h(u,v,r))$ for any $(u,v,r)\in\mathbb{R}^{2}\times\mathbb{R}_{+}$, $v\not\in\{0,y\}$. Then $g_{\lambda}(r)\to g(r):=\int_{\mathbb{R}^{2}}\Psi(\theta h(u,v,r))\mathrm{d}u \mathrm{d}v$ follows by the dominated convergence theorem. Using $\int_{\mathbb{R}^{2}}h_{\lambda}(u,v,r)\mathrm{d}u\mathrm{d}v=xyr\operatorname {leb}(B)$ and $h_{\lambda}(u,v,r)\leq Cr$ we obtain $\|g_{\lambda}(r)\|\leq C\min(r,r^{2})$ and hence $W_{\lambda,\gamma}(\theta)\to\int_{0}^{\infty}g(r)r^{-(1+\alpha)}\mathrm{d}r= \log\mathrm{E}\exp\{{\bf i}\theta I_{+}(x,y)\}$, proving the one-dimensional convergence in (2.14). The proof of Theorem 2.5 is complete. $\Box$ _Proof of Theorem 3.1._ (i) Write $D_{r}(x,y):=\{(u,v)\in\mathbb{R}^{2}:(u-x)^{2}+(v-y)^{2}\leq r^{2}\}$ for a ball in $\mathbb{R}^{2}$ centered at $(x,y)$ and having radius $r$. Recall that $B$ is bounded. Note that $\inf_{z\in[-1,1]}(\|z\|/r^{p}+(1-\|z\|^{1/(p-1)})^{1-p}/r^{1-p})\geq c_{0}\min(r^{ -p},r^{-(1-p)})$ for some constant $c_{0}>0$. Therefore, there exists $r_{0}>0$ such that for all $0<r<r_{0}$ the intersection in (3.1). Hence $b(z)\leq C<\infty$ uniformly in $z\in[-1,1]$. Let $(x,y)\in B\setminus\partial B$. Then $D_{2r}(x,y)\subset B$ for all $r<r_{0}$ and some $r_{0}>0$. If we translate $B$ by distance $r_{0}$ at most, the translated set still contains the ball $D_{r_{0}}(x,y)$. Since $\sup_{z\in[-1,1]}(\|z\|/r^{p}+(1-\|z\|^{1/p})^{1-p}/r^{1-p})\leq 2\max(r^{-p},r^{- (1-p)})$, there exists $r_{1}>0$ for which $\inf_{r>r_{1}}\operatorname{leb}(B_{z,r})\geq\pi r_{0}^{2},$ proving $\inf_{z\in[-1,1]}b(z)>0$. The continuity of $b(z)$ follows from the above argument and the continuity of the mapping $z\mapsto\operatorname{leb}(B_{z,r}):[-1,1]\to\mathbb{R}_{+}$, for each $r>0$. (ii) Let $s\geq 0$. In the integral (2.2) we change the variables: $u\to r^{p}u$, $v\to r^{1-p}v$, $r\to w^{1/p}r$. Then \[\rho(t,s)\ =\ w^{-(\alpha-1)/p}\int_{0}^{\infty}\operatorname{leb}(B_{t/w,r})f _{w}(r)r\mathrm{d}r,\] where $f_{w}(r):=w^{(1+\alpha)/p}f(w^{1/p}r)\to c_{f}\,r^{-(1+\alpha)}$, $w\to\infty$. Then (3.2) follows by Lemma 5.1 and the afore-mentioned properties of $\operatorname{leb}(B_{t/w,r})$. Theorem 3.1 is proved. $\Box$ In this paper we often use the following lemma which is a version of Lemma 2 in [6] or Lemma 2.4 in [1]. Lemma 5.1: _Let $F$ be a probability distribution that has a density function $f$ satisfying (1.2). Set $f_{\lambda}(r):=\lambda^{1+\alpha}f(\lambda r)$ for $\lambda\geq 1$. Assume that $g$, $g_{\lambda}$ are measurable functions on $\mathbb{R}_{+}$ such that $g_{\lambda}(r)\to g(r)$ as $\lambda\to\infty$ for all $r>0$ and such that the inequality_ \[\|g_{\lambda}(r)\|\ \leq\ C(r^{\beta_{1}}\wedge r^{\beta_{2}})\] (5.18) _holds for all $r>0$ and some $0<\beta_{1}<\alpha<\beta_{2}$, where $C$ does not depend on $r,\lambda$. Then_ \[\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r)\mathrm{d}r\ \to\ c_{f}\int_{0}^{ \infty}g(r)r^{-(1+\alpha)}\mathrm{d}r\quad\mbox{as }\lambda\to\infty.\] _Proof._ Split $\int_{0}^{\infty}g_{\lambda}(r)f_{\lambda}(r)\mathrm{d}r=(\int_{0}^{\epsilon}+ \int_{\epsilon}^{\infty})g_{\lambda}(r)f_{\lambda}(r)\mathrm{d}r=:I_{1}( \lambda)+I_{2}(\lambda)$, where $\epsilon>0$. It suffices to prove \[\lim_{\lambda\to\infty}I_{2}(\lambda)\ =\ c_{f}\int_{\epsilon}^{\infty}g(r)r^{ -(1+\alpha)}\mathrm{d}r\qquad\mbox{and}\qquad\lim_{\epsilon\to 0}\limsup_{ \lambda\to\infty}I_{1}(\lambda)\ =\ 0.\] (5.19) The first relation in (5.19) follows by the dominated convergence theorem, using (5.18) and the bound $f_{\lambda}(r)\leq Cr^{-(1+\alpha)}$ which holds for all $r>\rho/\lambda$ and a sufficiently large $\rho>0$ by virtue of (1.2). The second relation in (5.19) follows from $\|I_{1}(\lambda)\|\leq C\int_{0}^{\epsilon}r^{\beta_{2}}f_{\lambda}(r)\mathrm{d} r=C\lambda^{\alpha-\beta_{2}}\int_{0}^{\lambda\epsilon}x^{\beta_{2}}f(x) \mathrm{d}x\leq C\lambda^{\alpha-\beta_{2}}+C\lambda^{\alpha-\beta_{2}}\int_{1 }^{\lambda\epsilon}x^{\beta_{2}-(1+\alpha)}\mathrm{d}x\leq C(\lambda^{\alpha- \beta_{2}}+\epsilon^{\beta_{2}-\alpha}).$$\Box$ ### Proofs of Section 4 _Proof of Theorem 4.1._ We have \[W_{T,\gamma,\beta}(\theta) := \log\mathrm{E}\exp\big{\{}{\bf i}\theta b_{T}^{-1}\big{(}A_{M,K}( Tx)-\mathrm{E}A_{M,K}(Tx)\big{)}\big{\}}\] (5.20) \[= T^{\gamma}\int_{\mathbb{R}\times\mathbb{R}_{+}}\Psi\Big{(}\theta T ^{-\cal H}(r^{1-p}\wedge T^{\beta})\int_{0}^{Tx}{\bf 1}(u<t<u+r^{p})\mathrm{d} t\Big{)}\mathrm{d}uf(r)\mathrm{d}r,\] where $\Psi(z)=\mathrm{e}^{{\bf i}z}-1-{{\bf i}z},z\in\mathbb{R}$ as in Sec. 5.1. (i) Let $0<p<1,\delta_{1}:={\beta-(1+\gamma)(1-p)/\alpha}>0,\delta_{2}:=1-(1+\gamma)p/ \alpha=(\gamma_{+}-\gamma)p/\alpha>0$. Using the change of variables $(t-u)/r^{p}\to t$, $u\to Tu$, $r\to T^{(1+\gamma)/\alpha}r$ in (5.20), we obtain \[W_{T,\gamma,\beta}(\theta)\ =\ \int_{0}^{\infty}g_{T}(r)f_{T}(r)\mathrm{d}r,\] (5.21) where $f_{T}(r):=T^{(1+\alpha)(1+\gamma)/\alpha}f(T^{(1+\gamma)/\alpha}r)$ and \[g_{T}(r)\ :=\ \int_{\mathbb{R}}\Psi\big{(}\theta(r^{1-p}\wedge T^{\delta_{1}}) r^{p}h_{T}(u,r))\big{)}\mathrm{d}u\] and where $h_{T}(u,r):=\int_{0}^{1}{\bf 1}(0<u+T^{-\delta_{2}}r^{p}t<x)\mathrm{d}t\to{\bf 1 }(0<u<x)$ for fixed $(u,r)\in\mathbb{R}\times\mathbb{R}_{+}$, $u\not\in\{0,x\}$. Hence $g_{T}(r)\to g(r):=x\Psi(\theta r)$ follows by the dominated convergence theorem. The bound $\|g_{T}(r)\|\leq C\min(r,r^{2})$ follows from (5.4) and $\int_{\mathbb{R}}h_{T}(u,r)\mathrm{d}u=x$ with $h_{T}(u,r)\leq 1$. Finally, by Lemma 5.1, $W_{T,\gamma,\beta}(\theta)\to xc_{f}\int_{0}^{\infty}\Psi(\theta r)r^{-(1+ \alpha)}\mathrm{d}r=\log\mathrm{E}\exp\{{\bf i}\theta L_{\alpha}(x,1)\}$, proving part (i) for $0<p<1$. The case $p=1$ follows similarly. (ii) Using the same change of variables as in part (i) we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where \[g_{T}(r)\ :=\ \int_{\mathbb{R}}\Psi\big{(}\theta((T^{-\delta_{1} }r^{1-p})\wedge 1)r^{p}h_{T}(u,r)\big{)}\mathrm{d}u,\] where $\delta_{1},f_{T}(r),h_{T}(u,r)$ are the same as in (5.21) except that now $\delta_{1}<0$. Next, $g_{T}(r)\to x\Psi(\theta r^{p})$ by the dominated convergence theorem while $\|g_{T}(r)\|\leq C\min(r^{p},r^{2p})$ follows by (5.4) and $\int_{\mathbb{R}}\min(h_{T}(u,r),h^{2}_{T}(u,r))\mathrm{d}u\leq C$. Then $W_{T,\gamma,\beta}(\theta)\to W_{\gamma,\beta}(\theta):=xc_{f}\int_{0}^{\infty }\Psi(\theta r^{p})r^{-(1+\alpha)}\mathrm{d}r$ follows by Lemma 5.1. To finish the proof of part (ii) it suffices to check that (5.22) (iii) Denote $\delta_{1}:=1+\gamma-\alpha\beta/(1-p)>0,\,\delta_{2}:=1-p\beta/(1-p)>0$. Then by change of variables: $(t-u)/r^{p}\to t$, $u\to Tu$, $r\to T^{\beta/(1-p)}r$ we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where $f_{T}(r):=T^{(1+\alpha)\beta/(1-p)}f(T^{\beta/(1-p)}r)$ and \[g_{T}(r)\ :=\ \int_{\mathbb{R}}T^{\delta_{1}}\Psi\big{(}\theta T ^{-\delta_{1}/2}(r^{1-p}\wedge 1)r^{p}h_{T}(u,r)\big{)}\mathrm{d}u\] with $h_{T}(u,r):=\int_{0}^{1}{\bf 1}(0<u+T^{-\delta_{2}}r^{p}t<x)\mathrm{d}t\to{\bf 1 }(0<u<x).$ Then $g_{T}(r)\to-(\theta^{2}/2)(r^{1-p}\wedge 1)^{2}r^{2p}x$ by the dominated convergence theorem using the bounds $\|\Psi(z)\|\leq z^{2}/2$, $z\in\mathbb{R}$ and $h_{T}(u,r)\leq{\bf 1}(-r^{p}<u<x)$. Moreover, $\|g_{T}(r)\|\leq C\min(r^{2p},r^{2})$ holds in view of $\int_{\mathbb{R}}h^{2}_{T}(u,r)\mathrm{d}u\leq C$. Using Lemma 5.1 we get $W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)xc_{f}\int_{0}^{\infty}(r^{1-p} \wedge 1)^{2}r^{2p-(1+\alpha)}\mathrm{d}r=-(\theta^{2}/2)\sigma^{2}_{1}x,$ where \[\sigma^{2}_{1}\ :=\ \frac{2c_{f}(1-p)}{(2-\alpha)(\alpha-2p)}<\infty\] (5.23) since $\max(1,2p)<\alpha<2$. This proves part (iii). (iv) By the same change of variables as in part (iii), we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where \[g_{T}(r)\ :=\ \int_{\mathbb{R}}T^{\delta_{1}}\Psi\big{(}\theta T^{-\delta_{1}/ 2}(\log T)^{-1/2}(r^{1-p}\wedge 1)r^{p}h_{T}(u,r)\big{)}\mathrm{d}u\] and $f_{T}(r)$ and $\delta_{1}$, $\delta_{2}>0$ and $h_{T}(u,r):=\int_{0}^{1}{\bf 1}(0<u+T^{-\delta_{2}}r^{p}t<x)\mathrm{d}t\to{\bf 1 }(0<u<x)$ are the same as in (iii). We split $W_{T,\gamma,\beta}(\theta)=W_{T,\gamma,\beta}^{-}(\theta)+W_{T,\gamma,\beta}^{ 0}(\theta)+W_{T,\gamma,\beta}^{+}(\theta)$ and next prove that $W_{T,\gamma,\beta}^{-}(\theta):=\int_{0}^{1}g_{T}(r)f_{T}(r)\mathrm{d}r$ and $W_{T,\gamma,\beta}^{+}(\theta):=\int_{T^{\delta_{1}/2p}}^{\infty}g_{T}(r)f_{T} (r)\mathrm{d}r$ are the remainder terms, whereas \[W_{T,\gamma,\beta}^{0}(\theta)\ :=\ \int_{1}^{T^{\delta_{1}/2p}} g_{T}(r)f_{T}(r)\mathrm{d}r \sim -\frac{\theta^{2}}{2}\frac{xc_{f}}{\log T}\int_{1}^{T^{\delta_{1} /2p}}r^{2p-(1+2p)}\mathrm{d}r\] \[= -\frac{\theta^{2}}{2}\widehat{\sigma}^{2}_{1}x\ =:\ W_{\gamma, \beta}(\theta),\] where \[\widehat{\sigma}_{1}^{2}\ :=\ c_{f}\frac{\delta_{1}}{2p}\ =\ \frac{c_{f}}{2p(1 -p)}((1+\gamma)(1-p)-2p\beta).\] (5.24) By (1.2), there exists $\rho>0$ such that $f_{T}(r)\leq Cr^{-(1+2p)}$ for all $r>\rho/T^{\beta/(1-p)}$. Using this bound along with $\int_{\mathbb{R}}h_{T}(u,r)\mathrm{d}u=x$, $h_{T}(u,r)\leq 1$ and (5.4), we get \[\|W_{T,\gamma,\beta}^{-}(\theta)\| \leq \frac{C}{\log T}\int_{0}^{1}r^{2}f_{T}(r)\mathrm{d}r\ =\ O((\log T )^{-1})\ =\ o(1),\] \[\qquad\|W_{T,\gamma,\beta}^{+}(\theta)\| \leq C\frac{T^{\delta_{1}/2}}{(\log T)^{1/2}}\int_{T^{\delta_{1}/2p}} ^{\infty}r^{p-(1+2p)}\mathrm{d}r\ =\ O((\log T)^{-1/2})\ =\ o(1).\] We now consider the main term $W_{T,\gamma,\beta}^{0}(\theta)$. Let $\widetilde{W}_{T,\gamma,\beta}(\theta):=-\frac{\theta^{2}}{2\log T}\int_{1}^{T ^{\delta_{1}/2p}}r^{2p}f_{T}(r)\mathrm{d}r\int_{\mathbb{R}}h_{T}^{2}(u,r) \mathrm{d}u$. Then, by $\|\Psi(z)+z^{2}/2\|\leq\|z\|^{3}/6$, $z\in\mathbb{R},$ it follows that \[\|W_{T,\gamma,\beta}^{0}(\theta)-\widetilde{W}_{T,\gamma,\beta}( \theta)\| \leq \frac{C}{(\log T)^{3/2}T^{\delta_{1}/2}}\int_{1}^{T^{\delta_{1}/2 p}}r^{3p}f_{T}(r)\mathrm{d}r\int_{\mathbb{R}}h_{T}^{3}(u,r)\mathrm{d}u\] \[\leq \frac{C}{(\log T)^{3/2}T^{\delta_{1}/2}}\int_{1}^{T^{\delta_{1}/2 p}}r^{p-1}\mathrm{d}r\ =\ O((\log T)^{-3/2})\ =\ o(1).\] Finally, we estimate $\|\widetilde{W}_{T,\gamma,\beta}(\theta)-W_{\gamma,\beta}(\theta)\|\leq C(J^{ \prime}_{T}+J^{\prime\prime}_{T})$, where \[J^{\prime}_{T} :=\] \[J^{\prime\prime}_{T} := \frac{1}{\log T}\int_{1}^{T^{\delta_{1}/2p}}r^{2p}\|f_{T}(r)-c_{f} r^{-(1+2p)}\|\mathrm{d}r.\] Using \[\int_{\mathbb{R}}\|h_{T}^{2}(u,r)-{\bf 1}(0<u<x)\|\mathrm{d}u\ \leq\ 2\int_{0}^{ 1}\mathrm{d}t\int_{\mathbb{R}}\|{\bf 1}(0<u+T^{-\delta_{2}}r^{p}t<x)-{\bf 1}(0< u<x)\|\mathrm{d}u\ \leq\ Cr^{p}T^{-\delta_{2}},\] we obtain $J^{\prime}_{T}\leq C(\log T)^{-1}T^{-\delta_{2}}\int_{1}^{T^{\delta_{1}/2p}}r^ {p-1}\mathrm{d}r=o(1)$, since $\delta_{1}/2\leq\delta_{2}$ for $\gamma\leq\gamma_{+}$. Then $J^{\prime\prime}_{T}=o(1)$ follows from (1.2), since $\|f_{T}(r)-c_{f}r^{-(1+2p)}\|\leq\epsilon c_{f}r^{-(1+2p)}$ for all $r>\rho/T^{\beta/(1-p)}$ and some $\rho>0$ if given any $\epsilon>0$. This completes the proof of $W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)\widehat{\sigma}_{1}^{2}x=\log \mathrm{E}\exp\{{\bf i}\theta\widehat{\sigma}_{1}B(x)\}$ as $T\to\infty$ for any $\theta\in\mathbb{R}$. (v) After the same change of variables as in part (iii) we get $W_{T,\gamma,\beta}(\theta)$ in (5.21), where \[g_{T}(r)\ :=\ \int_{\mathbb{R}}\Psi\big{(}\theta(r^{1-p}\wedge 1)r^{p}h_{T}(u, r)\big{)}\mathrm{d}u\] with the same $f_{T}(r)$ and $h_{T}(u,t)\to{\bf 1}(0<u<x)$ as in (iii). By dominated convergence theorem, $g_{T}(r)\to x\Psi(\theta(r^{1-p}\wedge 1)r^{p})$, where we justify its use by (5.4), and $h_{T}(u,r)\leq{\bf 1}(-r^{p}<u<x)$. The bound $\|g_{T}(r)\|\leq C\min(r^{p},r^{2})$ follows from (5.4) and $\int_{\mathbb{R}}h_{T}(u,r)\mathrm{d}u=x$ with $h_{T}(u,r)\leq 1$. Finally, by Lemma 5.1, (5.25) The proof of Theorem 4.1 is complete. $\Box$ _Proof of Theorem 4.2_. (i) Denote $\delta_{1}:=1+\gamma-\alpha/p=\gamma-\gamma_{+}>0$ and $\delta_{2}:=(1-p)/p-\beta>0$. By changing the variables in (5.3): $t\to Tt$, $u\to Tu$, $r\to T^{1/p}r$ we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where $f_{T}(r):=T^{(1+\alpha)/p}f(T^{1/p}r)$ and \[g_{T}(r)\ :=\ \int_{\mathbb{R}}T^{\delta_{1}}\Psi\big{(}\theta T^{-\delta_{1}/ 2}((T^{\delta_{2}}r^{1-p})\wedge 1)h(u,r)\big{)}\mathrm{d}u\] with $h(u,r):=\int_{0}^{x}{\bf 1}(u<t<u+r^{p})\mathrm{d}t$. The dominated convergence $g_{T}(r)\to g(r):=-(\theta^{2}/2)\int_{\mathbb{R}}h^{2}(u,r)\mathrm{d}u$ follows by (5.4). The latter combined with $\int_{\mathbb{R}}h^{2}(u,r)\mathrm{d}u\leq C\min(1,r^{p})\int_{\mathbb{R}}h(u, r)\mathrm{d}u\leq C\min(r^{p},r^{2p})$ gives the bound $\|g_{T}(r)\|\leq C\min(r^{p},r^{2p})$. Finally, by Lemma 5.1, $W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)\sigma^{2}_{2}x^{2H}$, where \[\sigma^{2}_{2} := c_{f}\int_{\mathbb{R}\times\mathbb{R}}\Big{(}\int_{0}^{1}{\bf 1} (u<t<u+r^{p})\mathrm{d}t\Big{)}^{2}\frac{\mathrm{d}u\mathrm{d}r}{r^{1+\alpha}} \ =\ \frac{2c_{f}}{\alpha(2-\alpha/p)(3-\alpha/p)(\alpha/p-1)},\] (5.26) proving part (i). (ii) The proof is the same as of Theorem 4.1 (iii). (iii) Let $\delta_{1}:=\gamma-\alpha_{+}\beta>0,\,\delta_{2}:=\alpha_{+}\beta/\gamma_{+}- 1>0$. By change of variables: $t\to Tt$, $u\to T^{\beta p/(1-p)}u$, $r\to T^{\beta/(1-p)}r$ we get (5.21) with $f_{T}(r):=T^{(1+\alpha)\beta/(1-p)}f(T^{\beta/(1-p)}r)$ and \[g_{T}(r)\ :=\ \int_{\mathbb{R}}T^{\delta_{1}}\Psi(\theta T^{-\delta_{1}/2}(r^{ 1-p}\wedge 1)h_{T}(u,r))\mathrm{d}u,\] with $h_{T}(u,r):=\int_{0}^{x}{\bf 1}(0<(T^{-\delta_{2}}t-u)/r^{p}<1)\mathrm{d}t\to h (u,r):=x{\bf 1}(-r^{p}<u<0)$. Then (5.4) and $h^{2}_{T}(u,r)\leq x{\bf 1}(-r^{p}<u<1)$ justify the dominated convergence $g_{T}(r)\to-(\theta^{2}/2)(r^{1-p}\wedge 1)^{2}r^{p}x^{2}$. By (5.4) and $\int_{\mathbb{R}}h^{2}_{T}(u,r)\mathrm{d}u\leq C\int_{\mathbb{R}}h_{T}(u,r) \mathrm{d}u\leq Cr^{p}$, we have $\|g_{T}(r)\|\leq C\min(r^{p},r^{2-p})$. Finally, by Lemma 5.1$W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)x^{2}c_{f}\int_{0}^{\infty}(r^{1-p }\wedge 1)^{2}r^{p-(1+\alpha)}\mathrm{d}r=-(\theta^{2}/2)x^{2}\sigma^{2}_{3}$ with \[\sigma^{2}_{3}\ :=\ \frac{2c_{f}(1-p)}{(2-p-\alpha)(\alpha-p)},\] (5.27) proving part (iii). (iv) Denote $\delta_{1}:=\beta-\gamma/\alpha_{+}>0,\delta_{2}:=\gamma/\gamma_{+}-1>0$. Using the change of variables: $t\to Tt$, $u\to T^{\gamma/\gamma_{+}}u$, $r\to T^{\gamma/\gamma_{+}p}r$ we get (5.21) with $f_{T}(r):=T^{(1+\alpha)\gamma/\gamma_{+}p}f(T^{\gamma/\gamma_{+}p}r)$ and \[g_{T}(r)\ :=\ \int_{\mathbb{R}}\Psi(\theta(r^{1-p}\wedge T^{ \delta_{1}})h_{T}(u,r))\mathrm{d}u,\] where $h_{T}(u,r):=\int_{0}^{x}{\bf 1}(u<T^{-\delta_{2}}t<u+r^{p})\mathrm{d}t\to h(u, r):=x{\bf 1}(-r^{p}<u<0)$. Then $g_{T}(r)\to g(r):=\int_{\mathbb{R}}\Psi(\theta xr^{1-p}{\bf 1}(-r^{p}<u<0)) \mathrm{d}u$ and similarly as in the proof of Theorem 2.2 (ii). (v) Set $\delta_{1}:=\gamma-\gamma_{+}>0$, $\delta_{2}:=\beta-(1-p)/p>0$. After a change of variables: $t\to Tt$, $u\to Tu$, $r\to T^{1/p}r$, we get (5.21) with $f_{T}(r):=T^{(1+\alpha)/p}f(T^{1/p}r)$ and \[g_{T}(r)\ :=\ \int_{\mathbb{R}}T^{\delta_{1}}\Psi(\theta T^{-\delta_{1}/2}(r^{ 1-p}\wedge T^{\delta_{2}})h(u,r))\mathrm{d}u,\] where $h(u,r):=\int_{0}^{x}{\bf 1}(u<t<u+r^{p})\mathrm{d}t$. Then $g_{T}(r)\to g(r):=-(\theta^{2}/2)\int_{\mathbb{R}}r^{2(1-p)}h^{2}(u,r)\mathrm{ d}u$ and similarly to the proof of Theorem 2.3 (i). (iii) We follow the proof of Theorem 4.1 (iv). By the same change of variables, we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21). We split $W_{T,\gamma,\beta}(\theta)=W_{T,\gamma,\beta}^{-}(\theta)+W_{T,\gamma,\beta}^{ 0}(\theta)+W_{T,\gamma,\beta}^{+}(\theta)$ with the same $W_{T,\gamma,\beta}^{\pm}(\theta)$ being the remainder terms. Note that now $\delta_{2}<\delta_{1}/2$, since $\gamma>\gamma_{+}$. Next, we split $W_{T,\gamma,\beta}^{0}(\theta)=W^{\prime}_{T,\gamma,\beta}(\theta)+W^{\prime \prime}_{T,\gamma,\beta}(\theta)$, where \[W^{\prime}_{T,\gamma,\beta}(\theta)\ :=\ \int_{1}^{T^{\delta_{2}/p}}g_{T}(r)f_ {T}(r)\mathrm{d}r,\qquad W^{\prime\prime}_{T,\gamma,\beta}(\theta)\ :=\ \int_{ T^{\delta_{2}/p}}^{T^{\delta_{1}/2p}}g_{T}(r)f_{T}(r)\mathrm{d}r.\] Analogously to the proof of Theorem 4.1 (iv), we show the convergence $W^{\prime}_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)\widehat{\sigma}^{2}_{2}x$, where \[\widehat{\sigma}^{2}_{2}\ :=\ c_{f}\frac{\delta_{2}}{p}\ =\ c_{f}\Big{(}\frac{ 1}{p}-\frac{\beta}{1-p}\Big{)}.\] (5.28) Using (5.4) and $\int_{\mathbb{R}}h_{T}(r,u)\mathrm{d}u=x$ with $h_{T}(r,u)\leq x(T^{\delta_{2}}/r^{p})$, we get \[\|W^{\prime\prime}_{T,\gamma,\beta}(\theta)\|\ \leq\ \frac{C}{\log T}\int_{T^{ \delta_{2}/p}}^{T^{\delta_{1}/2p}}\frac{\mathrm{d}r}{r}\int_{\mathbb{R}}h_{T}^ {2}(r,u)\mathrm{d}u\ \leq\ \frac{CT^{\delta_{2}}}{\log T}\int_{T^{\delta_{2}/p }}^{T^{\delta_{1}/2p}}\frac{\mathrm{d}r}{r^{1+p}}\ =\ O((\log T)^{-1})\ =\ o(1),\] which completes the proof of $W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)\widehat{\sigma}^{2}_{2}x=\log \mathrm{E}\exp\{{\bf i}\theta\widehat{\sigma}_{2}B(x)\}$ as $T\to\infty$ for any $\theta\in\mathbb{R}$. (vii) By the same change of variables as in part (i), we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where \[g_{T}(r)\ :=\ \int_{\mathbb{R}}T^{\delta_{1}}\Psi\big{(}\theta T^{-\delta_{1}/ 2}(r^{1-p}\wedge 1)h(u,r)\big{)}\mathrm{d}u\] and where $\delta_{1}$, $h(u,r)$, $f_{T}(r)$ are the same as in (i). Then $g_{T}(r)\to-(\theta^{2}/2)\int_{\mathbb{R}}h^{2}(u,r)\mathrm{d}u$ along with $\int_{R}h^{2}(u,r)\mathrm{d}u\leq C\min(r^{p},r^{2p})$ and (5.4) imply $W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)c_{f}\int_{0}^{\infty}\int_{ \mathbb{R}}(r^{1-p}\wedge 1)^{2}h^{2}(u,r)r^{-(1+\alpha)}\mathrm{d}r\mathrm{d} u=:\log\mathrm{E}\exp\{{\bf i}\theta\widehat{Z}(x)\}$ as $T\to\infty$ for any $\theta\in\mathbb{R}$, by Lemma 5.1. (viii) By the same change of variables as in part (iii) we obtain $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where $g_{T}(r):=\int_{\mathbb{R}}\Psi(\theta(r^{1-p}\wedge 1)h_{T}(u,r))\mathrm{d}u$ with $f_{T}(r)$, $\delta_{2}=\gamma/\gamma_{+}-1>0$ and $h_{T}(u,r):=\int_{0}^{x}{\bf 1}(u<T^{-\delta_{2}}t<u+r^{p})\mathrm{d}t\to x{ \bf 1}(-r^{p}<u<0)$ the same as in (iii). Using $\int_{\mathbb{R}}h_{T}(u,r)\mathrm{d}u=xr^{p}$ and $h_{T}(u,r)\leq x$ yields $\|g_{T}(r)\|\leq C\min(r^{p},r^{2-p})$ from (5.4). Hence, by Lemma 5.1, it follows that \[W_{T,\gamma,\beta}(\theta)\ \to\ c_{f}\int_{0}^{\infty}\Psi( \theta x(r^{1-p}\wedge 1))r^{p-(1+\alpha)}\mathrm{d}r\ =:\ \log\mathrm{E}\exp \{{\bf i}\theta x\widehat{Z}\}.\] (5.29) (ix) By the same change of variables as in the proof of part (iv), we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where \[g_{T}(r)\ :=\ \int_{\mathbb{R}}\Psi\big{(}\theta(\log T)^{-1/2}(r^{1-p}\wedge T ^{\delta_{1}})h_{T}(u,r)\big{)}\mathrm{d}u\] with $\delta_{1}$, $\delta_{2}:=\gamma/\gamma_{+}-1>0$ and $h_{T}(u,r):=\int_{0}^{x}{\bf 1}(u<T^{-\delta_{2}}t<u+r^{p})\mathrm{d}t\to x{ \bf 1}(-r^{p}<u<0)=:h(u,r)$ and $f_{T}(r)$ being the same as in (iv). We split $W_{T,\gamma,\beta}(\theta)=W^{-}_{T,\gamma,\beta}(\theta)+W^{0}_{T,\gamma, \beta}(\theta)+W^{+}_{T,\gamma,\beta}(\theta)$ and next prove that $W^{-}_{T,\gamma,\beta}(\theta):=\int_{0}^{T^{-\delta_{2}/p}}g_{T}(r)f_{T}(r)$ and $W^{+}_{T,\gamma,\beta}(\theta):=\int_{1}^{\infty}g_{T}(r)f_{T}(r)\mathrm{d}r$ are the remainder terms, whereas \[W^{0}_{T,\gamma,\beta}(\theta)\ :=\ \int_{T^{-\delta_{2}/p}}^{1} g_{T}(r)f_{T}(r)\mathrm{d}r \sim -\frac{\theta^{2}}{2}\frac{c_{f}}{\log T}\int_{T^{-\delta_{2}/p}} ^{1}\frac{\mathrm{d}r}{r^{1+p}}\int_{\mathbb{R}}h^{2}(u,r)\mathrm{d}u\] \[= -\frac{\theta^{2}}{2}\widetilde{\sigma}^{2}_{+}x^{2}\ =:\ W_{ \gamma,\beta}(\theta),\] where the constant $\widetilde{\sigma}^{2}_{+}$ is given in (5.16). Using $\int_{\mathbb{R}}h_{T}(u,r)\mathrm{d}u=xr^{p}$ and $h_{T}(u,r)\leq x\wedge(T^{\delta_{2}}r^{p})$ along with (5.4), we show that \[\|W^{-}_{T,\gamma,\beta}(\theta)\| \leq \frac{CT^{\delta_{2}}}{\log T}\int_{0}^{T^{-\delta_{2}/p}}r^{2}f_ {T}(r)\ =\ \frac{C}{T\log T}\int_{0}^{T^{1/p}}r^{2}f(r)\mathrm{d}r\ =\ O((\log T )^{-1})\ =\ o(1),\] \[\|W^{+}_{T,\gamma,\beta}(\theta)\| \leq \frac{C}{(\log T)^{1/2}}\int_{1}^{\infty}rf_{T}(r)\mathrm{d}r\ = \ O((\log T)^{-1/2})\ =\ o(1).\] To deal with the main term $W^{0}_{T,\gamma,\beta}(\theta)$, set $\widetilde{W}_{T,\gamma,\beta}(\theta):=-\frac{\theta^{2}}{2\log T}\int_{T^{- \delta_{2}/p}}^{1}r^{2(1-p)}f_{T}(r)\mathrm{d}r\int_{\mathbb{R}}h_{T}^{2}(u,r) \mathrm{d}u$. From $\|\Psi(z)+z^{2}/2\|\leq\|z\|^{3}/6$, we obtain \[\|W_{T,\gamma,\beta}(\theta)-\widetilde{W}_{T,\gamma,\beta}(\theta)\| \leq \frac{C}{(\log T)^{3/2}}\int_{T^{-\delta_{2}/p}}^{1}r^{3(1-p)}f_{ T}(r)\mathrm{d}r\int_{\mathbb{R}}h_{T}^{3}(u,r)\mathrm{d}u\] \[\leq \frac{C}{(\log T)^{3/2}}\int_{T^{-\delta_{2}/p}}^{1}r^{3-2p}f_{T} (r)\mathrm{d}r\ =\ O((\log T)^{-3/2})\ =\ o(1).\] Finally, we consider $\|\widetilde{W}_{T,\gamma,\beta}(\theta)-W_{\gamma,\beta}(\theta)\|\leq C(J^{ \prime}_{T}+J^{\prime\prime}_{T})$, where \[J^{\prime}_{T} := \frac{1}{\log T}\int_{T^{-\delta_{2}/p}}^{1}r^{2(1-p)}f_{T}(r) \mathrm{d}r\int_{\mathbb{R}}\|h_{T}^{2}(u,r)-h^{2}(u,r)\|\mathrm{d}u,\] \[J^{\prime\prime}_{T} := \frac{1}{\log T}\int_{T^{-\delta_{2}/p}}^{1}r^{2-p}\|f_{T}(r)-c_{f }r^{p-3}\|\mathrm{d}r.\] Using \[\int_{\mathbb{R}}\|h^{2}_{T}(u,r)-h^{2}(u,r)\|\mathrm{d}u\ \leq\ C\int_{0}^{x} \mathrm{d}t\int_{\mathbb{R}}\|{\bf 1}(u<T^{-\delta_{2}}t<u+r^{p})-{\bf 1}(-r^{p }<u<0))\|\mathrm{d}u\ \leq\ CT^{-\delta_{2}}\] we obtain $J^{\prime}_{T}\leq C(\log T)^{-1}T^{-\delta_{2}}\int_{T^{-\delta_{2}/p}}^{1}r^ {-(1+p)}\mathrm{d}r=O((\log T)^{-1})=o(1)$. Then $J^{\prime\prime}_{T}=o(1)$ follows from (1.2), since $\|f_{T}(r)-c_{f}r^{p-3}\|\leq\epsilon c_{f}r^{p-3}$ for all $r>\rho/T^{\gamma/2(1-p)}$ and some $\rho>0$ if given any $\epsilon>0$. This finishes the proof of $W_{T,\gamma,\beta}(\theta)\to-(\theta^{2}/2)\widetilde{\sigma}^{2}_{+}x^{2}= \log\mathrm{E}\exp\{{\bf i}\theta\widetilde{\sigma}^{2}_{+}B_{1,1/2}(x,1)\}$ as $T\to\infty$ for any $\theta\in\mathbb{R}$. The proof of Theorem 4.2 is complete. $\Box$ _Proof of Theorem 4.3._ (i) By the same change of variables as in Theorem 4.2 (i), we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where \[g_{T}(r)\ :=\ \int_{\mathbb{R}}\Psi\big{(}\theta((T^{\delta_{2}}r^{1-p})\wedge 1 )h(u,r)\big{)}\mathrm{d}u\ \to\ \int_{\mathbb{R}}\Psi(\theta h(u,r))\mathrm{d} u\ =:\ g(r),\] since $\delta_{2}:=(1-p)/p-\beta=\gamma_{+}/\alpha_{+}-\beta>0$ with $h(u,r)$, $f_{T}(r)$ being the same as in Theorem 4.2 (i). Using (5.4) along with $\int_{\mathbb{R}}h(u,r)\mathrm{d}u=xr^{p}$ and $h(u,r)\leq r^{p}$, we get $\|g_{T}(r)\|\leq C\min(r^{p},r^{2p})$. Hence $W_{T,\gamma,\beta}(\theta)\to c_{f}\int_{0}^{\infty}\int_{\mathbb{R}}\Psi( \theta h(u,r))r^{-(1+\alpha)}\mathrm{d}r\mathrm{d}u=:\log\mathrm{E}\exp\{{\bf i }\theta I(x)\}$ by Lemma 5.1. (ii), (iii) The proof is the same as that of Theorem 4.1 (iii), (iv) respectively. (iv) By the same change of variables as in Theorem 4.2 (i), we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where $g(r):=\int_{\mathbb{R}}\Psi(\theta(r^{1-p}\wedge 1)h(u,r))\mathrm{d}u$ with $h(u,r)$, $f_{T}(r)$ being the same as in Theorem 4.2 (i). Then $\|g(r)\|\leq C\min(r^{p},r^{2})$ follows from (5.4). By Lemma 5.1, we get . (v) By the same change of variables as in Theorem 4.2 (v), we rewrite $W_{T,\gamma,\beta}(\theta)$ as in (5.21), where $f_{T}(r)$, $g_{T}(r)$ are the same as in Theorem 4.2 (v) except for $\delta_{1}=0$. Then $g_{T}(r)\to g(r):=\int_{\mathbb{R}}\Psi(\theta r^{1-p}h(u,r))\mathrm{d}u$ and $\|g_{T}(r)\|\leq C\min(r,r^{2})$ from (5.4) lead to by Lemma 5.1, similarly to the proof of Theorem 2.5. The proof of Theorem 4.3 is complete. $\Box$ ## Acknowledgement This research was supported by a grant (no. MIP-063/2013) from the Research Council of Lithuania. ## References * [1] Biermé, H., Estrade, A. and Kaj, I. (2010) Self-similar random fields and rescaled random balls models. J. Theoret. Probab. 23, 1110–1141. * [2] Biermé, H., Meerschaert, M.M. and Scheffler, H.P. (2007) Operator scaling stable random fields. Stochastic Process. Appl. 117, 312–332. * [3] Dombry, C. and Kaj, I. (2011) The on-off network traffic under intermediate scaling. Queueing Sys. 69, 29–44. * [4] Gaigalas, R. (2006) A Poisson bridge between fractional Brownian motion and stable Lévy motion. Stochastic Process. Appl. 116, 447–462. * [5] Gaigalas, R. and Kaj, I. (2003) Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9, 671–703. * [6] Kaj, I., Leskelä, L., Norros, I. and Schmidt, V. (2007) Scaling limits for random fields with long-range dependence. Ann. Probab. 35, 528–550. * [7] Kaj, I. and Taqqu, M.S. (2008) Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In: Vares, M.E. and Sidoravicius, V. (eds.) _An Out of Equilibrium 2._ Progress in Probability, vol. 60, pp. 383–427. Birkhäuser, Basel. * [8] Leipus, R., Paulauskas, V. and Surgailis, D. (2005) Renewal regime switching and stable limit laws. J. Econometrics 129, 299–327. * [9] Lifshits, M. (2014) _Random Processes by Example._ World Scientific, New Jersey. * [10] Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002) Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12, 23–68. * [11] Pilipauskaitė, V. and Surgailis, D. (2014) Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes. Stochastic Process. Appl. 124, 1011–1035. * [12] Pilipauskaitė, V. and Surgailis, D. (2015) Joint aggregation of random-coefficient AR(1) processes with common innovations. Statist. Probab. Lett. 101, 73–82. * [13] Puplinskaitė, D. and Surgailis, D. (2015) Aggregation of autoregressive random fields and anisotropic long-range dependence. Bernoulli DOI 10.3150/15-BEJ733. Also available at at arXiv:1303.2209v3 [math.ST]. * [14] Puplinskaitė, D. and Surgailis, D. (2015) Scaling transition for long-range dependent Gaussian random fields. Stochastic Process. Appl. 125, 2256–2271.
1812.06383	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 46015, "num_imgs": 0, "llama3_tokens_count": 19408 }	[]	# Exact normalized eigenfunctions for general deformed Hulthén potentials Richard L. Hall Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montréal, Québec, Canada H3G 1M8, richard.hall@concordia.ca Nasser Saad School of Mathematical and Computational Sciences, University of Prince Edward Island, 550 University Avenue, Charlottetown, PEI, Canada C1A 4P3, nsaad@upei.ca K. D. Sen INSA Senior Scientist, School of Chemistry, University of Hyderabad 500046, India. kds77@uohyd.ac.in ###### Abstract The exact solutions of Schrödinger’s equation with the deformed Hulthén potential $V_{q}(x)=-{\mu\,e^{-\delta\,x}}/({1-q\,e^{-\delta\,x}}),~{}\delta,\mu,q>0$ are given, along with a closed–form formula for the normalization constants of the eigenfunctions for arbitrary $q>0$. The Crum-Darboux transformation is then used to derive the corresponding exact solutions for the extended Hulthén potentials $V(x)=-{\mu\,e^{-\delta\,x}}/({1-q\,e^{-\delta\,x}})+{q\,j(j+1)\,e^{-\delta\,x} }/({1-q\,e^{-\delta\,x}})^{2},j=0,1,2,\dots.$ A general formula for the new normalization condition is also provided. Hulthén potentials, Crum-Darboux transformation, Normalization constants, generalized Kampé de Fériet functions, Generalized Hypergeometric functions. pacs: 31.15.-p 31.10.+z 36.10.Ee 36.20.Kd 03.65.Ge. ## I Introduction The Hulthén potentials [1; 2] \[V(x)=-\dfrac{\mu\,e^{-\delta\,x}}{1-e^{-\delta\,x}},\qquad x\in( 0,\infty),\] (1) where $\mu$ is a constant and $\delta>0$ is the screening parameter that determines the potential range, have important applications in nuclear and particle physics, in atomic physics, and in condensed–matter physics [3; 4; 5; 6; 7; 8]. Except for the $\ell$=0 case, where the exact energy eigenvalues are known, a variety of numerical methods have been employed in the literature to obtain the eigenvalues and eigenfunctions. The so called centrifugal term approximation [10] has been widely used in these calculations. Interesting energy level ordering involving this potential has been derived using the elementary comparison theorem of quantum mechanics which serves to provide deeper insights [11; 12] in to the various other screened Coulomb potentials. A variation of the Hulthén potential can be written as \[V_{q}(x)=-\dfrac{\mu\,e^{-\delta\,x}}{1-q\,e^{-\delta\,x}}\quad q >0,~{}x\in(\log(q)/\delta,\infty).\] (2) Although such potential variations may be found in the literature [13; 14; 15] for a variety of applications, the potential in (2) is essentially the same as the original potential (1) since a simple shift of the independent variable such as $x\to x+\log(q)/\delta$ transforms $V_{q}(x)$ back into $V(x)$ with the only difference being a new coupling constant $\mu$. In future developments, in which angular momentum will be considered, it may be advantageous to revert the viewpoint to (1) so that the potential and the centrifugal terms all have singularities at the origin. However, since $V_{q}(x)$ has been widely used in the literature, we shall adopt this form in the present work. Therefore, in atomic units, Schrödinger’s equation for our problem becomes \[-\frac{1}{2}\dfrac{d^{2}\psi(x)}{dx^{2}}-\dfrac{\mu\,e^{-\delta\,x}}{1-q\,e^{- \delta\,x}}\,\psi(x)=E\,\psi(x),\quad\int_{\log(q)/\delta}^{\infty}\|\psi(x)\|^{ 2}dx=1,\quad\psi(\log(q)/\delta)=\psi(\infty)=0.\] (3) To our knowledge, no closed-form expression for the normalization constants for arbitrary $q>0$ was given in the literature (including the classical Hulthén potential (1)). The present work provides the complete normalized solutions of Schrödinger’s equation (3). Crum-Darboux transformations are then used to generate the eigenfunctions of the equation \[-\frac{1}{2}\dfrac{d^{2}\phi(x)}{dx^{2}}+\left(\dfrac{\mu\,e^{-\delta\,x}}{(1- q\,e^{-\delta\,x})^{2}}-\dfrac{v\,e^{-\delta\,x}}{1-q\,e^{-\delta\,x}}\right) \,\phi(x)=\mathfrak{E}\,\phi(x),\quad\int_{\log(q)/\delta}^{\infty}\|\phi(x)\|^{ 2}dx=1,\quad\phi(\log(q)/\delta)=\phi(\infty)=0.\] (4) A general formula is provided for the normalization constants of the eigenfunctions $\phi(x)$ in terms of the eigenfunctions $\psi(x)$, for arbitrary $q>0$. The paper is organized as follows: in section II, we discuss the exact solutions of Schrödinger’s equation (3). In section III, we develop an analytic expression for the normalization constants in terms of the generalized hypergeometric function ${}_{3}F_{2}$. In section IV, we give a general review of the Crum-Darboux transformation, and a simplified formula for the normalization constants. These ideas are used in section V to generate the exact eigenfunctions of equation (4). The approach given here provides the logical framework [16; 17; 18; 19] behind the success of the centrifugal–term approximation $1/x^{2}\approx e^{-\delta x}/(1-e^{-\delta x})^{2}$ used to estimate the eigenvalues and eigenfunctions for the Hulthén potential ($q=1$) for $\ell\neq 0$. ## II Generalized Hulthén potential: exact solutions In this section, we show that the exact solutions of Schrödinger’s equation (3) may be expressed in terms of the Gauss hypergeometric functions as \[\psi(x)=e^{-\sqrt{-2\,E}\,x}(1-q\,e^{-\delta\,x})\,{}_{2}F_{1}\left(1+\frac{ \sqrt{-2\,E}}{\delta}-\frac{1}{\delta}\sqrt{\frac{2\mu}{q}-2E},1+\frac{\sqrt{- 2\,E}}{\delta}+\frac{1}{\delta}\sqrt{\frac{2\mu}{q}-2E};1+\frac{2\sqrt{-2\,E}} {\delta};q\,e^{-\delta\,x}\right).\] (5) Indeed, the change of variable $z=e^{-\delta\,x}$ allows equation (3) to be written as \[z^{2}\dfrac{d^{2}\psi}{dz^{2}}+z\dfrac{d\psi}{dz}+\frac{v\,z}{1-q\,z}\psi=- \varepsilon\,\psi,\quad 0<z<\frac{1}{q},\quad\psi(0)=\psi\left(1/q\right)=0,\] (6) where $v={2\mu}/\delta^{2}$ and $\varepsilon=2E/\delta^{2}.$ The differential equation (6) has two regular singular points $z=0$ and $z=1/q$ with the indicial equations $\eta^{2}+\varepsilon=0$ (or $\eta=\sqrt{-2E}/\delta$) and $s(s-1)=0$ (or $s=0,1$), respectively, in addition to an irregular singular point at $z=\infty$. Thus, the general solution of equation (6) assumes the form \[\psi(z)=z^{\eta}\left(1-qz\right)^{s}f(z).\] (7) The factor $(1-qz)^{s}$ must have $s$ a positive integer so that $\psi(1/q)=0$. Thus $s$ cannot be zero. On substituting the ansatz (7) in equation (6), it is not difficult to show that the function $f(z)$, after implementing the indicial equations, satisfies the differential equation \[z(1-q\,z)f^{\prime\prime}(z) +(-q(2\eta+2s+1)\,z+2\eta+1)f^{\prime}(z)+\left(v-q\,s\,\left(2\, \eta+s\right)\right)f(z)=0.\] (8) This is a hypergeometric differential equation with exact solutions given, in terms of the Gauss hypergeometric functions, as \[f(z) =C_{1}\,{}_{2}F_{1}\left(\eta+s-\sqrt{\eta^{2}+\frac{v}{q}},\eta+ s+\sqrt{\eta^{2}+\frac{v}{q}};1+2\,\eta;q\,z\right)\] \[+C_{2}\,z^{-2\eta}\,_{2}F_{1}\left(s-\eta-\sqrt{\eta^{2}+\frac{v} {q}},s-\eta+\sqrt{\eta^{2}+\frac{v}{q}};1-2\,\eta;q\,z\right).\] (9) However, the boundary condition $\psi(0)=0$ forces that the vanishing of the constant $C_{2}=0$. Thus, \[f(z) ={}_{2}F_{1}\left(\eta+s-\sqrt{\eta^{2}+\frac{v}{q}},\eta+s+\sqrt {\eta^{2}+\frac{v}{q}};2\,\eta+1;q\,z\right),\] (10) from which the general solution of Eq. (3) takes the form (7). For polynomial solutions $f_{n}(z)$, the termination of the hypergeometric series (10) requires $\eta+1-\sqrt{\eta^{2}+\frac{v}{q}}=-n,~{}n=0,1,2,\dots.$ that yields the following expression for the eigenvalues (using $\eta=\sqrt{-2E_{n}}/\delta$, $v=2\mu/\delta^{2}$ and $s=1$) \[E_{n} =-\frac{1}{2}\left(\frac{\mu}{q\,\delta\,(1+n)}-\frac{\delta\,(1+ n)}{2}\right)^{2},\] (11) with the exact (unnormalized) wave functions: \[\psi_{n}(x) =N_{n}\left(1-q\,e^{-\delta\,x}\right)e^{-\left(\frac{\mu}{q\, \delta\,(1+n)}-\frac{(n+1)\delta}{2}\right)\,x}\,{}_{2}F_{1}\left(-n,1+\frac{2 \mu}{q\,\delta^{2}(1+n)};\frac{2\mu}{q\,\delta^{2}\,(1+n)}-n;q\,e^{-\delta\,x}\right)\] (12) up to normalization constant $N_{n}$. Note, by using the Pfaff transformation identity \[{}_{2}F_{1}(\alpha,\beta;\gamma;z)=(1-z)^{\gamma-\alpha-\beta}{}_ {2}F_{1}(\gamma-\alpha,\gamma-\beta;\gamma;z),\] (13) we can write the exact solution (12) as \[\psi_{n}(x) =N_{n}e^{-\left(\frac{\mu}{q\,\delta\,(1+n)}-\frac{(n+1)\delta}{2 }\right)\,x}\,{}_{2}F_{1}\left(-n-1,\frac{2\mu}{q\,\delta^{2}(1+n)};\frac{2\mu }{q\,\delta^{2}\,(1+n)}-n;q\,e^{-\delta x}\right).\] (14) Thus, the number of the discrete bound-states is bounded above by the inequality \[0\leq n<-1+\frac{1}{\delta}\sqrt{\frac{2\,\mu}{q}},\qquad where \qquad 0<q<\frac{2\,\mu}{\delta^{2}}.\] (15) ## III Generalized Hulthén potential: normalization constant The normalization constant $N_{n}$ in equation (12) can be evaluated, for $2\mu>q\,\delta^{2}(m+1)(n+1)$, using the following definite integral \[I_{nm}=\int_{\log(q)/\delta}^{\infty}e^{-\frac{(2+m+n)\left(2\mu -q\,\delta^{2}(1+m)(1+n)\right)x}{2\delta(1+m)(1+n)q}} \left(1-q\,e^{-\delta x}\right)^{2}\,_{2}F_{1}\left(-n,\frac{2\mu }{\delta^{2}(n+1)q}+1;\frac{2\mu}{\delta^{2}(n+1)q}-n;qe^{-\delta\,x}\right)\] \[\times\,_{2}F_{1}\left(-m,\frac{2\mu}{\delta^{2}(m+1)q}+1;\frac{2 \mu}{\delta^{2}(m+1)q}-m;qe^{-\delta\,x}\right)dx.\] (16) To evaluate this definite integral, we use the change of variables $\tau=q\,e^{-\delta\,x}$ and note for $x=\log(q)/\delta,$ that $\tau=1/q$, while for $x=\infty,\tau=0$. Further, by the series representation of the Hypergeometric function, it follows that \[I_{nm}=\frac{1}{\delta}\sum_{i=0}^{n}\sum_{j=0}^{m}\dfrac{(-n)_{ i}\left(1+\frac{2\mu}{q\delta^{2}(1+n)}\right)_{i}}{\left(\frac{2\mu}{q\delta^ {2}(1+n)}-n\right)_{i}\,i!}\dfrac{(-m)_{j}\left(1+\frac{2\mu}{q\delta^{2}(1+m) }\right)_{j}}{\left(\frac{2\mu}{q\delta^{2}(1+m)}-m\right)_{j}\,j!}q^{i+j}\int _{0}^{1/q}(q\tau-1)^{2}\tau^{\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q}-\frac{m} {2}-\frac{n}{2}-2+i+j}d\tau.\] (17) The integral on the left-hand side can be evaluated in terms of the Gamma function to yield \[I_{nm}= \frac{2q^{1+\frac{m+n}{2}-\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q }}\Gamma\left(\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1\right)} {\delta\,\Gamma\left(2-\frac{m+n}{2}+\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q} \right)}\] \[\times\sum_{i=0}^{n}\sum_{j=0}^{m}\frac{(-n)_{i}(-m)_{j}\left( \frac{2\mu}{q\delta^{2}(n+1)}+1\right)_{i}\left(\frac{2\mu}{q\delta^{2}(m+1)}+ 1\right)_{j}\left(\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1 \right)_{i+j}}{i!\,j!\,\left(\frac{2\mu}{q\delta^{2}(n+1)}-n\right)_{i}\left( \frac{2\mu}{q\delta^{2}(m+1)}-m\right)_{j}\left(\frac{\mu(m+n+2)}{\delta^{2}(m +1)(n+1)q}-\frac{m+n}{2}+2\right)_{i+j}}\] (18) Using the Pochhammer identity $(\alpha)_{i+j}=(\alpha+i)_{j}(\alpha)_{i}\Gamma(\alpha)$, equation (III) can be written as \[I_{nm} =\frac{2q^{1+\frac{m+n}{2}-\frac{\mu(m+n+2)}{q\,\delta^{2}(m+1)(n +1)}}\Gamma\left(\frac{(m+n+2)\mu}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1 \right)}{\delta\,\Gamma\left(\frac{(m+n+2)\mu}{\delta^{2}(m+1)(n+1)q}-\frac{m+ n}{2}+2\right)}\sum_{i=0}^{n}\frac{(-n)_{i}\left(\frac{2\mu}{q\delta^{2}(n+1)} +1\right)_{i}\left(\frac{(m+n+2)\mu}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1 \right)_{i}}{i!\left(\frac{2\mu}{q\delta^{2}(n+1)}-n\right)_{i}\left(\frac{(m+ n+2)\mu}{q\,\delta^{2}(m+1)(n+1)}-\frac{m+n}{2}+2\right)_{i}}\] \[\times{}_{3}F_{2}\left(\begin{array}[]{lll}-m,&\frac{2\mu}{q\, \delta^{2}(1+m)}+1,&i-1-\frac{m+n}{2}+\frac{\mu\,(2+m+n)}{q\,\delta^{2}(1+m)(1 +n)}\\ \frac{2\mu}{q\,\delta^{2}(1+m)}-m,&i+2-\frac{m+n}{2}+\frac{\mu\,(2+m+n)}{q\, \delta^{2}(1+m)(1+n)}&\\ \end{array}\bigg{\|}1\right)\] (19) For $m=n$, it follow that \[I_{nn} =\frac{2q^{1+n-\frac{2\mu}{q\,\delta^{2}(n+1)}}\Gamma\left(\frac{ 2\mu}{\delta^{2}(n+1)q}-n-1\right)}{\delta\,\Gamma\left(\frac{2\mu}{\delta^{2} (n+1)q}-n+2\right)}\sum_{i=0}^{n}\frac{(-n)_{i}\left(\frac{2\mu}{q\delta^{2}(n +1)}+1\right)_{i}\left(\frac{2\mu}{\delta^{2}(n+1)q}-n-1\right)_{i}}{i!\left( \frac{2\mu}{q\delta^{2}(n+1)}-n\right)_{i}\left(\frac{2\mu}{q\,\delta^{2}(n+1) }-n+2\right)_{i}}\] \[\times{}_{3}F_{2}\left(\begin{array}[]{lll}-n,&\frac{2\mu}{q\, \delta^{2}(1+n)}+1,&i+\frac{2\mu}{q\,\delta^{2}(1+n)}-n-1\\ \frac{2\mu}{q\,\delta^{2}(1+n)}-n,&i+2-n+\frac{2\mu}{q\,\delta^{2}(1+n)}&\\ \end{array}\bigg{\|}1\right).\] (20) Breaking the finite sum, the equation can be written as \[I_{nn}\] \[+\frac{2q^{1+n-\frac{2\mu}{q\,\delta^{2}(n+1)}}\Gamma\left(\frac{ 2\mu}{\delta^{2}(n+1)q}-n-1\right)}{\delta\,\Gamma\left(\frac{2\mu}{\delta^{2} (n+1)q}-n+2\right)}\sum_{i=1}^{n}\frac{(-n)_{i}\left(\frac{2\mu}{q\delta^{2}(n +1)}+1\right)_{i}\left(\frac{2\mu}{\delta^{2}(n+1)q}-n-1\right)_{i}}{i!\left( \frac{2\mu}{q\delta^{2}(n+1)}-n\right)_{i}\left(\frac{2\mu}{q\,\delta^{2}(n+1) }-n+2\right)_{i}}\] \[\times{}_{3}F_{2}\left(\begin{array}[]{lll}-n,&\frac{2\mu}{q\, \delta^{2}(1+n)}+1,&i+\frac{2\mu}{q\,\delta^{2}(1+n)}-n-1\\ \frac{2\mu}{q\,\delta^{2}(1+n)}-n,&i+2-n+\frac{2\mu}{q\,\delta^{2}(1+n)}&\\ \end{array}\bigg{\|}1\right)\] (21) Using the hypergeometric identity (20, formula 7.4.4.1) \[{}_{3}F_{2}\left(\begin{array}[]{lll}a,&b,&c\\ d,&e,&\end{array}\bigg{\|}1\right) =\frac{\Gamma(d)\Gamma(d+e-a-b-c)}{\Gamma(d+e-a-b)\Gamma(d-c)}{}_ {3}F_{2}\left(\begin{array}[]{lll}e-a,&e-b,&c\\ d+e-a-b,&e,&\end{array}\bigg{\|}1\right),\] \[(Re(d+e-a-b-c)>0,~{}Re(d-c)>0),\] (22) it follows \[I_{nn}\] \[+\frac{2q^{1+n-\frac{2\mu}{q\,\delta^{2}(n+1)}}\Gamma\left(\frac{ 2\mu}{\delta^{2}(n+1)q}-n-1\right)}{\delta\,\Gamma\left(\frac{2\mu}{\delta^{2} (n+1)q}-n+2\right)}\sum_{i=1}^{n}\frac{(-n)_{i}\left(\frac{2\mu}{\delta^{2}(n+ 1)q}+1\right)_{i}\left(\frac{(2n+2)\mu}{\delta^{2}(n+1)^{2}q}-n-1\right)_{i} \Gamma\left(i-n+\frac{2\mu}{\delta^{2}(n+1)q}+2\right)}{\,i!\left(\frac{2\mu}{ q\,\delta^{2}(n+1)}-n\right)_{i}\left(\frac{(2n+2)\mu}{q\,\delta^{2}(n+1)^{2}} -n+2\right)_{i}\Gamma(i-n+1)\,\Gamma\left(\frac{2\mu}{q\,\delta^{2}(n+1)}+3 \right)}\] \[\times\,_{3}F_{2}\left(1-i,\frac{2\mu}{\delta^{2}(n+1)q}+1,\frac{ 2\mu}{q\,\delta^{2}(n+1)};\frac{2\mu}{q\,\delta^{2}(n+1)}+3,\frac{2\mu}{q\, \delta^{2}(n+1)}-n;1\right)\] However, because of the reciprocal of the Gamma function $1/\Gamma(i-n+1)$, where $1/\Gamma(-m)=0,m=0,1,2,\dots$, the finite sum survives only for $i=n$ (otherwise each term is zero) whence \[I_{nn}\] \[\times\,_{3}F_{2}\left(1-n,\frac{2\mu}{\delta^{2}(n+1)q}+1,\frac{ 2\mu}{q\,\delta^{2}(n+1)};\frac{2\mu}{q\,\delta^{2}(n+1)}+3,\frac{2\mu}{q\, \delta^{2}(n+1)}-n;1\right).\] (23) Upon using the Pochhammer identities \[(-n)_{n}=(-1)^{n}\,n!,\quad\frac{\left(\frac{2\mu}{q\,\delta^{2}(n+1)}-n-1 \right)_{n}}{\left(\frac{2\mu}{\delta^{2}(n+1)q}-n\right)_{n}}=\frac{\frac{2 \mu}{q\,\delta^{2}(n+1)}-n-1}{\frac{2\mu}{q\,\delta^{2}(n+1)}-1},\quad\frac{ \left(\frac{2\mu}{\delta^{2}(n+1)q}+1\right)_{n}}{\left(\frac{2\mu}{q\,\delta^ {2}(n+1)}-n+2\right)_{n}}=\frac{(-1)^{n}\left(\frac{2\mu}{q\,\delta^{2}(n+1)}+ 1\right)_{n}}{\left(-\frac{2\mu}{q\,\delta^{2}(n+1)}-1\right){}_{n}},\] it follows finally that \[\int_{{\log(q)/\delta}}^{\infty}\left(1-q\,e^{-\delta\,x}\right)^ {2}e^{-\left(\frac{2\mu}{\delta\,q\,(1+n)}-(1+n)\delta\right)x}\left[{}_{2}F_{ 1}\left(-n,\frac{2\mu}{q\,\delta^{2}(n+1)}+1;\frac{2\mu}{q\,\delta^{2}(n+1)}-n ;q\,e^{-\delta x}\right)\right]^{2}dx\] \[=\frac{2q^{n+1-\frac{2\mu}{q\,\delta^{2}(n+1)}}\,\Gamma\left( \frac{2\mu}{q\,\delta^{2}(n+1)}-n-1\right)}{\delta\,\Gamma\left(\frac{2\mu}{q \,\delta^{2}(n+1)}-n+2\right)}\bigg{[}\frac{q\,\delta^{2}(n+1)\left(2\mu-q\, \delta^{2}(n+1)^{2}\right)\left(\frac{2\mu}{q\,\delta^{2}(n+1)}+1\right)_{n}}{ 2\left(2\mu-q\,\delta^{2}(n+1)\right)\left(q\,\delta^{2}(n+1)+\mu\right)\left( -\frac{2\mu}{q\,\delta^{2}(n+1)}-1\right)_{n}}\] (24) Next, for the case where $m\neq n$, \[I_{nm} =\frac{2q^{1+\frac{m+n}{2}-\frac{\mu(m+n+2)}{q\,\delta^{2}(m+1)(n +1)}}\Gamma\left(\frac{(m+n+2)\mu}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1 \right)}{\delta\,\Gamma\left(\frac{(m+n+2)\mu}{\delta^{2}(m+1)(n+1)q}-\frac{m+ n}{2}+2\right)}\sum_{i=0}^{n}\frac{(-n)_{i}\left(\frac{2\mu}{q\delta^{2}(n+1)} +1\right)_{i}\left(\frac{(m+n+2)\mu}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1 \right)_{i}}{i!\left(\frac{2\mu}{q\delta^{2}(n+1)}-n\right)_{i}\left(\frac{(m+ n+2)\mu}{q\,\delta^{2}(m+1)(n+1)}-\frac{m+n}{2}+2\right)_{i}}\] \[\times{}_{3}F_{2}\left(\begin{array}[]{lll}-m,&\frac{2\mu}{q\, \delta^{2}(1+m)}+1,&i-1-\frac{m+n}{2}+\frac{\mu\,(2+m+n)}{q\,\delta^{2}(1+m)(1 +n)}\\ \frac{2\mu}{q\,\delta^{2}(1+m)}-m,&i+2-\frac{m+n}{2}+\frac{\mu\,(2+m+n)}{q\, \delta^{2}(1+m)(1+n)}&\\ \end{array}\bigg{\|}1\right).\] (25) Using the identity (20, formula 7.4.4.90) \[{}_{3}F_{2}\left(\begin{array}[]{lll}-m,&a,&b\\ a-\ell,&b-s,&\end{array}\bigg{\|}1\right)=0,\qquad{\rm if}\quad(\ell+s=1,2,3, \dots,m-1),\] (26) we see that it is enough to consider the case $n=0$ and $m\neq 0$, for every other (fixed) value of $n$, the parameter $a$ is varied by a constant factor. Thus, \[I_{0m} =\frac{2q^{1+\frac{m}{2}-\frac{\mu(m+2)}{q\,\delta^{2}(m+1)}} \Gamma\left(\frac{(m+2)\mu}{q\,\delta^{2}(m+1)}-\frac{m}{2}-1\right)}{\delta\, \Gamma\left(\frac{(m+2)\mu}{q\,\delta^{2}(m+1)}-\frac{m}{2}+2\right)}{}_{3}F_{ 2}\left(\begin{array}[]{lll}-m,&\frac{2\mu}{q\,\delta^{2}(1+m)}+1,&-1-\frac{m} {2}+\frac{\mu\,(2+m)}{q\,\delta^{2}(1+m)}\\ \frac{2\mu}{q\,\delta^{2}(1+m)}-m,&2-\frac{m}{2}+\frac{\mu\,(2+m)}{q\,\delta^{ 2}(1+m)}&\\ \end{array}\bigg{\|}1\right)\] (27) comparing the equation (27) with equation (26), we note $\ell=m+1,s=-3$, thus $m-2<m-1$, it follows that $I_{0m}=0$, noting that by a similar argument, for $n\neq 0$ and $n\neq m$, each term of the finite sum vanishes . Finally we have \[\int_{{\log(q)/\delta}}^{\infty}\left(1-q\,e^{-\delta\,x}\right)^ {2}e^{-\left(\frac{2\mu}{\delta\,q\,(1+n)}-(1+n)\delta\right)x}\left[{}_{2}F_{ 1}\left(-n,\frac{2\mu}{q\,\delta^{2}(n+1)}+1;\frac{2\mu}{q\,\delta^{2}(n+1)}-n ;q\,e^{-\delta x}\right)\right]^{2}dx=I_{nm}\delta_{nm},\] (28) where \[I_{nn} =\frac{2q^{n+1-\frac{2\mu}{q\,\delta^{2}(n+1)}}\,\Gamma\left( \frac{2\mu}{q\,\delta^{2}(n+1)}-n-1\right)}{\delta\,\Gamma\left(\frac{2\mu}{q \,\delta^{2}(n+1)}-n+2\right)}\bigg{[}\frac{q\,\delta^{2}(n+1)\left(2\mu-q\, \delta^{2}(n+1)^{2}\right)\left(\frac{2\mu}{q\,\delta^{2}(n+1)}+1\right)_{n}}{ 2\left(2\mu-q\,\delta^{2}(n+1)\right)\left(q\,\delta^{2}(n+1)+\mu\right)\left( -\frac{2\mu}{q\,\delta^{2}(n+1)}-1\right)_{n}}\] (29) ## IV Crum-Darboux transformation: Sequential transformations An important technique for generating classes of exactly-solvable quantum potentials is build on the concept of intertwining operators [21]. Two Hamiltonian operators $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ are said to be intertwined if there exist an operator $\mathfrak{L}$ so that \[\mathcal{H}_{1}\mathfrak{L}=\mathfrak{L}\,\mathcal{H}_{0}.\] (30) In this case, if $\phi_{0;n}(x)$ and $\psi_{1;n}(x)$ are eigenfunctions of the intertwined operators $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ respectively, then the two sets of eigenfunctions are related ([22], p. 63) by the operator $\mathfrak{L}$ through the relations \[\psi_{1;n}(x)=\mathfrak{L}\,\phi_{0;n}(x),\qquad\phi_{0;n}(x)=\mathfrak{L}^{ \dagger}\,\psi_{0;n}(x),\] (31) and the Hamiltonians $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ are is said to be isospectral, i.e. share the same spectrum, except for those states that are annihilated by $\mathfrak{L}$ or $\mathfrak{L}^{\dagger}$. In the context of the one-dimensional quantum mechanics, $\mathfrak{L}$ is taken to be a first-order linear differential operator $\mathfrak{L}=\partial_{x}+f(x)$ intertwines two one-dimensional Schrödinger Hamiltonians $\mathcal{H}_{0}=-\partial_{xx}+V(x)$ and $\mathcal{H}_{1}=-\partial_{xx}+\mathcal{V}(x)$ where $x\in(a,b)$ that can be finite or infinite interval. Here, $\partial_{x}$ refers to the first-derivative with respect to the variable $x$. Direct computation, using (30), yields \[\left(-2\partial_{x}f(x)+\mathcal{V}(x)-V(x)\right)\partial_{x} \phi_{n}(x) =\left(\partial_{x}V(x)+[V(x)-\mathcal{V}(x)]f(x)+\partial_{x}^{2 }f(x)\right)\phi_{n}(x).\] (32) The consistency condition of (32) then requires \[-2\partial_{x}f(x)+\mathcal{V}(x)-V(x) =0,\quad\mbox{and}\quad\partial_{x}V(x)+[V(x)-\mathcal{V}(x)]f(x) +\partial_{x}^{2}f(x)=0.\] (33) Substituting the first condition into the second one gives \[\partial_{x}\left(V(x)-f^{2}(x)+\partial_{x}f(x)\right) =0\qquad\mbox{or}\qquad V(x)-f^{2}(x)+\partial_{x}f(x)=\lambda\] (34) for some constant $\lambda$. Clearly, the function $f(x)=-\partial_{x}\phi(x)/\phi(x)=-\partial_{x}\log\phi(x)$ is a particular solution of the Riccati equation (34) provided that \[-\partial_{xx}\,\phi(x)+V(x)\phi(x) =\lambda\,\phi(x),\] (35) that is to say, provided that $\phi$ is an eigenfunction of the Hamiltonian $\mathcal{H}_{0}$ with eigenvalue $\lambda$. An important conclusion from this approach is that every _no-node_ eigenfunction, called a seed function, $\phi=\phi_{0,0}$ of $\mathcal{H}_{0}$ (regardless of the normalizability) generates a new solvable Hamiltonian $\mathcal{H}_{1}$ with the potential $\mathcal{V}$ expressed in terms of $V$ and the seed function as \[\mathcal{V}(x)=V(x)+2\dfrac{d^{2}}{dx^{2}}\log\phi_{0,0}(x).\] (36) The first-order differential intertwining operator \[\mathfrak{L}=\dfrac{d}{dx}-\dfrac{1}{\phi_{0,0}}\dfrac{d\phi_{0,0}}{dx}\] (37) is known as a Darboux transformation [23]. The following theorem summarize these results: Theorem IV.1.: _The eigenfunctions $\psi_{1,n}(x),~{}n=1,2,\dots$ of the Schrödinger equation_ \[\left(-\dfrac{d^{2}}{dx^{2}}+\mathcal{V}(x)\right)\phi_{1,n}(x)=E _{n}\,\phi_{1,n}(x)\quad\mbox{where}\quad\mathcal{V}(x)=V(x)-2\dfrac{d^{2}}{dx ^{2}}\log\phi_{0,0}(x),\quad x\in(a,b)\] (38) _are generated using_ \[\phi_{1,n}(x)=\left(\dfrac{d}{dx}-\dfrac{\phi_{0,0}^{\prime}(x)}{ \phi_{0,0}(x)}\right)\phi_{0,n}(x)=\dfrac{W(\phi_{0,0}(x),\phi_{0,n}(x))}{W( \phi_{0,0}(x))},\quad n=1,2,\dots,\] (39) _where $W(\psi_{0,0}(x))\equiv\phi_{0,0}(x)$ and $W(\phi_{0,0}(x),\phi_{0,n}(x))=\psi_{0,0}(x)\psi_{0,n}^{\prime}(x)-\psi_{0,n}( x)\psi_{0,0}^{\prime}(x)$ is the classical Wronskian. Here, $\phi_{0,n}(x),~{}n=0,1,2,\dots,$ are solutions of the Schrödinger equation_ \[-\phi_{0,n}^{\prime\prime}(x)+V(x)\phi_{0,n}(x)=E_{n}\,\phi_{0,n} (x),~{}n=0,1,2,\dots,~{}\phi_{0,n}(a)=\phi_{0,n}(b)=0,~{}\phi_{0,0}(x)\neq 0~{ }\forall x\in(a,b).\] (40) _Further, if $\phi_{0,n}(x)$ are normalized according to the condition_ \[\int_{a}^{b}{\phi_{0,n}(x)}\phi_{0,m}(x)dx=\eta_{n}\,\delta_{mn},\] (41) _where $\delta_{mn}$ is the known Kronecker delta, then_ \[\int_{a}^{b}{\phi_{1,n}(x)}\phi_{1,m}(x)dx=(E_{n}-E_{0})\,\eta_{n }\,\delta_{mn},\quad n=1,2,\dots.\] (42) The proof of the normalization relation (42) is given in the Appendix. It is evident that such uses of Darboux’s transformation can be applied to (39) using $\mathfrak{L}_{1}=\frac{d}{dx}-\frac{\phi_{1,1}^{\prime}(x)}{\phi_{1,1}}$, again to reproduce a new solvable eigenvalue problem $\mathcal{H}_{2}$ and such a procedure may be repeated an arbitrary number of times [24] as long as the consecutive (generated) Hamiltonians support the existence of (discrete) states, as illustrated by the following diagram: M. M. Crum [25] in his remarkable paper 1955 introduced an elegant approach to evaluate the eigenfunctions of the Hamiltonian $\mathcal{H}_{j}$, expressed entirely in terms of the eigenfunctions of the initial Hamiltonian $\mathcal{H}_{0}$, without any reference to the intermediate Hamiltonians: This approach can be illustrated as follows: consider the eigenfunctions $\phi_{1,n}(x),~{}n=1,2,\dots$ of the Hamiltonian $\mathcal{H}_{1}$ generated using $\phi_{0,n}(x),n=0,1,2,\dots$; the solutions of an initial Hamiltonian $\mathcal{H}_{0}=-\partial_{xx}+V(x)$; and employed in a second Darboux transformation to obtain \[\phi_{2,n}(x) =\left(\dfrac{d}{dx}-\dfrac{d}{dx}\log({\phi_{1,1}(x)})\right)\, \phi_{1,n}(x)=\dfrac{W(\phi_{0,0}(x),\phi_{0,1}(x),\phi_{0,n}(x))}{W(\phi_{0,0 }(x),\phi_{0,1}(x))},\quad n=2,3,\dots\] (43) where we perform the computations using (39) and the definition of the Wronskian. This approach, in turn, generates a new class of solvable Schrödinger equation \[-\dfrac{d^{2}}{dx^{2}}\phi_{2,n}(x) +\left(V(x)-2\dfrac{d^{2}}{dx^{2}}\bigg{(}\log\bigg{[}W(\phi_{0,0 }(x),\phi_{0,1}(x))\bigg{]}\bigg{)}\right)\phi_{2,n}(x)=E_{n}\phi_{2,n}(x), \quad n=2,3,\dots.\] (44) expressed entirely in terms of the eigenfunctions of the Hamiltonian $\mathcal{H}_{0}$ with no reference to the intermediate Hamiltonian $\mathcal{H}_{1}$. For the eigenvalue problem (44), we may now employ the third transformation \[\phi_{3,n}(x)=\left(\dfrac{d}{dx}-\dfrac{d}{dx}\log\phi_{2,2}(x) \right)\,\phi_{2,n}(x)=\dfrac{W(\phi_{0,0}(x),\phi_{0,1}(x),\psi_{0,2}(x),\phi _{0,n}(x))}{W(\phi_{0,0}(x),\phi_{0,1}(x),\phi_{0,2}(x))},\quad n=3,4,\dots\] (45) to obtain the following solvable class of Schrödinger equations \[-\dfrac{d^{2}}{dx^{2}}\phi_{3,n}(x)+\left(V(x)-2\dfrac{d^{2}}{dx^ {2}}\bigg{(}\log\bigg{[}W(\phi_{0,0}(x),\phi_{0,1}(x),\phi_{0,2}(x)\bigg{]} \bigg{)}\right)\phi_{3,n}(x)=E_{n}\phi_{3,n}(x),\quad n=3,4,\dots.\] (46) without reference to the intermediate Hamiltonians $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$. This process may be generalized to include the case of $j$-times repeated Darboux transformation, expressed entirely in terms of the eigenfunctions for the initial Hamiltonian $\mathcal{H}_{0}$ to give: _The transformed functions_ \[\psi_{j,n}(x) =\left(\dfrac{d}{dx}-\dfrac{d}{dx}\log{\psi_{j-1,j-1}(x)}\right) \,\psi_{j-1,n}(x)\] \[=\dfrac{W(\phi_{0,0}(x),\phi_{0,1}(x),\phi_{0,2}(x),\dots,\phi_{0 ,k-1}(x),\phi_{0,n}(x))}{W(\phi_{0,0}(x),\phi_{0,1}(x),\phi_{0,2}(x),\dots, \phi_{0,k-1}(x))},\quad n=k,k+1,k+2,\dots\] (47) _satisfy the Schrödinger equation_ \[-\dfrac{d^{2}}{dx^{2}}\phi_{j,n}(x)+\left(V(x)-2\dfrac{d^{2}}{dx^ {2}}\left[\log\bigg{(}W(\phi_{0,0}(x),\phi_{0,1}(x),\phi_{0,2},\dots,\phi_{0,j -1}(x)\right)\bigg{]}\right)\phi_{j,n}(x)=E_{n}\phi_{j,n}(x),\] (48) _where $n=j,j+1,j+2,\dots.$ The proof of Curm’s formulation [25] is based on the following Wronski identity:_ \[W(\phi_{0,0},\phi_{0,1},\dots,\phi_{0,j-1}) W(\phi_{0,0},\phi_{0,1},\dots,\phi_{0,j},\phi_{0,n})\] \[=W(W(\phi_{0,0},\phi_{0,1},\dots,\phi_{0,j}),W(\phi_{0,0},\phi_{0 ,1},\dots,\psi_{0,j-1},\phi_{0,n})).\] (49) _The general expression of the normalization constants is given in terms of the normalization of $\mathcal{H}_{0}$ as:_ \[\int_{a}^{b}\phi_{j,n}(x)\phi_{j,m}(x)dx=\,\mu_{n}\,\delta_{nm}\prod_{i=0}^{j- 1}(E_{n}-E_{i}),\quad where\quad\mu_{n}=\int_{a}^{b}[\phi_{0,n}(x)]^{2}dx.\] (50) ## V The Crum-Darboux transformation and the generalized Hulthén potential From section III, we may write the exact solutions of the Schrödinger equation \[-\dfrac{d^{2}\psi_{0,n}(r)}{dr^{2}}-\dfrac{v\,e^{-r}}{1-q\,e^{-r} }\,\psi_{0,n}(r)=\,\mathscr{E}_{n}\,\psi_{0,n}(r),\qquad r\in[\log(q),\infty), \quad q>0,\quad\psi_{0,n}(\log(q))=\psi_{0,n}(\infty)=0,\] (51) are simply, for $n=0,1,2,\dots$, \[\mathscr{E}_{n} =-\left(\dfrac{v}{2\,q\,(n+1)}-\dfrac{n+1}{2}\right)^{2},\] \[\psi_{0,n}(r) =N_{n}\left(1-qe^{-r}\right)e^{-\left(\frac{v}{2(n+1)q}-\frac{n+1 }{2}\right)\,r}\,_{2}F_{1}\left(-n,\frac{v}{(n+1)q}+1;\frac{v}{(n+1)q}-n;q\,e^ {-r}\right),\] (52) up to the normalization constant evaluated using the following definite integral (see equation (28) and (III)): \[N_{n}^{2} \int_{\log(q)}^{\infty}(1-q\,e^{-r})^{2}\,e^{-\left(\frac{v}{q(n+ 1)}-n-1\right)\,r}\left[\,{}_{2}F_{1}\left(-n,\frac{v}{q(1+n)}+1;\frac{v}{q(1+ n)}-n;q\,e^{-r}\right)\right]^{2}dr=1\] (53) as (54) and subject to the parameter constraints \[v>0,\quad 0<q<v,\quad 0\leq n<\sqrt{\frac{v}{q}}-1,\] where $n$ is an integer. ### First Transformation Consider the Darboux transformation \[\psi_{1,n}(r) =\left[\dfrac{d}{dr}-\dfrac{1}{\psi_{0,0}(r)}\dfrac{d\psi_{0,0}(r )}{dr}\right]\psi_{0,n}(r)\equiv\dfrac{W(\psi_{0,0}(r),\psi_{0,n}(r))}{\psi_{0 ,0}(r)}\,,\quad n=1,2,\dots\,,\] (55) where the seed function $\psi_{0,0}(r)$ given as the ground-state wave-function (V), from which a new solvable potential is obtained: \[V_{1}(r)=V(r)-2\dfrac{d^{2}}{dr^{2}}\log\psi_{0,0}(r) =-\frac{v\,e^{-r}}{1-q\,e^{-r}}+\frac{2\,q\,e^{-r}}{\left(1-q\,e^ {-r}\right)^{2}}\] (56) Thus, the Schrödinger equation with the potential $V_{1}$ is exactly solvable \[-\dfrac{d^{2}}{dr^{2}}\psi_{1,n}(r)+\left(-\frac{ve^{-r}}{1-qe^{- r}}+\frac{2\,q\,e^{-r}}{\left(1-qe^{-r}\right)^{2}}\right)\psi_{1,n}(r) =-\,\mathscr{E}_{n}\psi_{1,n}(r)\,,\quad n=1,2,\dots\,,\] (57) with (up to normalization constant) exact wave function solutions, for $n=1,2,\dots$, \[\psi_{1,n}(r) =\frac{n\left(e^{r}-q\right)}{2q(n+1)}e^{-\frac{1}{2}r\left(\frac {v}{nq+q}-n+1\right)}\left[2q(n+1)\,_{2}F_{1}\left(1-n,\frac{v}{nq+q}+1;\frac{ v}{nq+q}-n;q\,e^{-r}\right)\right.\] \[\left.+(v-(n+1)q)\,_{2}F_{1}\left(-n,\frac{v}{nq+q}+1;\frac{v}{nq +q}-n;q\,e^{-r}\right)\right],\] (58) subject to $v>0,q>0,1\leq n<\sqrt{(q+v)/q}$. Using the contiguous relation \[(\gamma-\alpha-\beta){}_{2}F_{1}(\alpha,\beta;\gamma;z)+\alpha(1- z)\,{}_{2}F_{1}(\alpha+1,\beta;\gamma;z)-(\gamma-\beta)\,{}_{2}F_{1}(\alpha, \beta-1;\gamma;z)=0,\] (59) equation (V.1) can be written in more compact form as: \[\psi_{1,n}(r) =\frac{n\,(q\,n+q+v)}{2(n+1)q}\,e^{-\frac{r}{2}\left(\frac{v}{nq+ q}-n-1\right)}\left(1-qe^{-r}\right)^{2}\,_{2}F_{1}\left(1-n,\frac{v}{nq+q}+2; \frac{v}{nq+q}-n;e^{-r}q\right),\] \[(n=1,2,\dots,\mathfrak{n}<\sqrt{(q+v)/q}).\] (60) ### Second transformation Using these exact solutions (V.1), it is possible via Crum’s approach to generate sequential transformation of the Hulthén potential with \[\psi_{2,n}(r) =\left[\dfrac{d}{dr}-\dfrac{\psi_{1,1}^{\prime}(x)}{\psi_{1;1}(r) }\right]\psi_{1,n}(r)\equiv\frac{W\left(\psi_{0,0}(r),\psi_{0,1}(r),\psi_{0,n} (r)\right)}{W\left(\psi_{0,0}(r),\psi_{0,1}(r)\right)}\,,\quad n=2,3,\dots\,,\] (61) where the seed function $\psi_{1;1}(r)$ given by the ground-state wave function (V.1) for $n=1$. Thus, a new solvable potential is obtained: \[V_{2}(r) =V_{1}(r)-2\dfrac{d^{2}}{dr^{2}}\log\big{[}\psi_{1;1}(r)\big{]}=V (r)-2\,\dfrac{d^{2}}{dr^{2}}\log W(\psi_{0,0}(r),\psi_{0,1}(r))=-\frac{v\,e^{- r}}{1-q\,e^{-r}}+\frac{6\,q\,e^{-r}}{\left(1-q\,e^{-r}\right)^{2}}\,.\] (62) This potential has exact analytic solutions of the Schrödinger equation \[-\dfrac{d^{2}}{dr^{2}}\psi_{2,n}(r)+\left(-\frac{v\,e^{-r}}{1-q\,e^{-r}}+\frac {6\,q\,e^{-r}}{\left(1-q\,e^{-r}\right)^{2}}\right)\psi_{2,n}(r)=-\,\mathscr{E }_{n}\psi_{2,n}(r)\,,\quad n=2,3,\dots\,,\] (63) with exact solutions (up to normalization constant) given for $n=2,3,\dots,$ \[\psi_{2,n}(r)=\frac{n(n-1)(v+q\,(n+1))(v+2\,q\,(n+1))}{8(1+n)^{2} q^{2}}e^{-\frac{r}{2}\left(\frac{v}{q+nq}-n-1\right)}(1-q\,e^{-r})^{2}\] \[\times\left(\,{}_{2}F_{1}\left(1-n,\frac{v}{nq+q}+2; \frac{v}{nq+q}-n;e^{-r}q\right)\right.\] \[\left.-\frac{4\,q^{2}\,(n+1)}{\left(nq+n^{2 }q-v\right)}\,e^{-r}\,_{2}F_{1}\left(2-n,\frac{v}{nq+q}+3;\frac{v}{nq+q}-n+1;e ^{-r}q\right)\right)\] (64) That can be written as \[\psi_{n;2}(r) =\frac{n(n-1)(v+(n+1)q)(v+2(n+1)q)}{8\,q^{2}(n+1)^{2}}\,(1-qe^{-r })^{3}\,e^{-\frac{r}{2}\left(\frac{v}{nq+q}-n-1\right)}\] \[\times\,_{2}F_{1}\left(2-n,\frac{v}{nq+q}+3;\frac{v}{nq+q}-n;e^{- r}q\right),\] (65) subject to $v>0,~{}0<q<v/3,~{}2\leq n<\sqrt{(q+v)/q}.$ ### The $j^{\text{th}}$ Transformation The class of generalized Hulthén potentials given by an arbitrary $j^{\text{th}}$ transformation, $j=1,2,\dots$ can be established using Crum’s approach, see Section III, to give \[V_{j}(r) \equiv-\frac{v\,e^{-r}}{1-q\,e^{-r}}-2\dfrac{d^{2}}{dr^{2}}\log W \big{(}\psi_{0;0}(r),\psi_{1;0}(r),\dots,\psi_{j-1;0}(r)\big{)},\quad j=1,2, \dots,\mathfrak{n}.\] (66) We shall show, by induction on $j$, that \[-2\dfrac{d^{2}}{dr^{2}}\log W\big{(}\psi_{0;0}(r),\psi_{1;0}(r), \dots,\psi_{j-1;0}(r)\big{)}=\dfrac{j(j+1)\,q\,e^{r}}{(e^{r}-q)^{2}}\] (67) whence \[V_{j}(r)=-\frac{ve^{-r}}{1-q\,e^{-r}}+\dfrac{j(j+1)\,q\,e^{r}}{(e^{r}-q)^{2}}, \qquad j=1,2,\cdots\] (68) For $j=1$, using $W\big{(}\psi_{0;0}(r))=\psi_{0;0}(r)$ we note for \[\psi_{0,0}(r) =e^{-r\left(\frac{v}{2q}-\frac{1}{2}\right)}\left(1-q\,e^{-r} \right),\qquad v>q>0,\] \[\dfrac{d}{dr}\log\psi_{0;0}(r) =\frac{1}{2}+\frac{q}{e^{r}-q}-\frac{v}{2q},\qquad\dfrac{d^{2}}{ dr^{2}}\log\psi_{0;0}(r)=-\frac{q\,e^{r}}{\left(e^{r}-q\right)^{2}}.\] (69) and equation (67) is true for $j=1$. Assume, equation (67) is true for $j=j^{\prime}$, that is to say \[-2\dfrac{d^{2}}{dr^{2}}\log W\big{(}\psi_{0;0}(r),\psi_{1;0}(r), \dots,\psi_{j^{\prime}-1;0}(r)\big{)}=\dfrac{j^{\prime}(j^{\prime}+1)\,q\,e^{r }}{(e^{r}-q)^{2}}\] (70) then for $j=j^{\prime}+1$, since \[-2\dfrac{d^{2}}{dr^{2}}\log W\big{(}\psi_{0;0}(r),\psi_{1;0}(r),\dots,\psi_{j^{\prime}-1;0}( r),\psi_{j^{\prime};0}(r)\big{)}\] \[= -2\dfrac{d^{2}}{dr^{2}}\log W\big{(}\psi_{0;0}(r),\psi_{1;0}(r), \dots,\psi_{j^{\prime}-1;0}(r)\big{)}-2\dfrac{d^{2}}{dr^{2}}\log\frac{W\big{(} \psi_{0;0}(r),\psi_{1;0}(r),\dots,\psi_{j^{\prime}-1;0}(r),\psi_{j^{\prime};0} (r)\big{)}}{W\big{(}\psi_{0;0}(r),\psi_{1;0}(r),\dots,\psi_{j^{\prime}-1;0}(r) \big{)}}\] \[=\dfrac{j^{\prime}(j^{\prime}+1)\,q\,e^{r}}{(e^{r}-q)^{2}}-2 \dfrac{d^{2}}{dr^{2}}\log\psi_{j^{\prime};j^{\prime}}(x)=\dfrac{j^{\prime}(j^{ \prime}+1)\,q\,e^{r}}{(e^{r}-q)^{2}}+V_{j^{\prime}+1}(r)-V_{j^{\prime}}(r)\] \[=\dfrac{v\,e^{-r}}{1-q\,e^{-r}}+V_{j^{\prime}+1}(r),\] (71) which ensures the truth of the identity (67). This potential has exact analytic solutions \[\psi_{j,n}(r) =\psi_{j-1,j}^{\prime}(r)-\dfrac{\Psi_{j-1,j-1}^{\prime}(r)}{\psi _{j-1,j-1}(r)}\psi_{j-1,n}(r)=\dfrac{W\big{(}\psi_{0;0}(r),\psi_{1;0}(r),\dots ,\psi_{j-1;0}(r),\psi_{n;0}(r)\big{)}}{W\big{(}\psi_{0;0}(r),\psi_{1;0}(r), \dots,\psi_{j-1;0}(r)\big{)}},~{}n=j,j+1,\dots\,\mathfrak{n},\] (72) where $\mathfrak{n}$ indicate the finiteness of the discrete bound states. The eigenfunctions (V.3) are the solutions of the Schrödinger equation \[-\dfrac{d^{2}\psi_{n;j}(r)}{dr^{2}}+\left(\dfrac{j(j+1)\,q\,e^{-r }}{(1-qe^{-r})^{2}}-\frac{v\,e^{-r}}{1-q\,e^{-r}}\right)\Psi_{n;j}(r)=-\, \mathscr{E}_{n}\,\Psi_{n;j}(r),\qquad r\in[\log(q),\infty)\] (73) with \[\mathscr{E}_{n} =-\left(\dfrac{v}{2\,q\,(n+1)}-\dfrac{n+1}{2}\right)^{2},\quad n= j,~{}j+1,\dots\] (74) subject to $v>0,q>0,j\leq n<\sqrt{(q+v)/q}.$ Obviously, the evaluation of the general expression for $\psi_{j,n}(r)$ using the relation (V.3) is not straightforward. However, we can find a general expression by analyzing the possible solutions of the Schrödinger equation \[-\dfrac{d^{2}\psi(r)}{dr^{2}}+\left(\dfrac{\mu\,e^{-r}}{(1-q\,e^{ -r})^{2}}-\frac{v\,e^{-r}}{1-q\,e^{-r}}\right)\psi(r)=\mathfrak{E}_{n}\,\psi(r ),\qquad r\in[\log(q),\infty)\] (75) where $\mu$ is an arbitrary constant that supports the existence of discrete bound-states. ## VI Extended Hulthén’s potential: exact solutions In this section, we analyze the exact solutions of equation (75) which will allows us to obtain a compact formula for the $j$-transformed $\psi_{j,n}(r)$ as given by (V.3). Using a similar approach to that discussed in Section II, it is not difficult to show that the change of variable $z=e^{-r}$ along with the analysis of the regular singular points implies by means of the ansatz solution \[\psi(z)=z^{\eta}\left(1-q\,z\right)^{s}f(z),\quad where\quad(\eta ^{2}+\mathfrak{E}_{n}=0,\qquad qs^{2}-qs-\mu=0).\] (76) the following second-order differential equation for $f(z)$ \[z\,(1-q\,z)\,f^{\prime\prime}(z)+\left[1+2\,\eta-q\,(1+2\eta+2s)z\right]f^{ \prime}(z)+\left[v-q\left(2\eta s+s^{2}\right)\right]f(z)=0\] (77) The exact solutions of this equation are given in terms of the Gauss hypergeometric functions as \[f(z) ={}_{2}F_{1}\left(\eta+s-\sqrt{\eta^{2}+\frac{v}{q}},\eta+s+\sqrt {\eta^{2}+\frac{v}{q}};1+2\,\eta;q\,z\right),\] (78) up to the normalization constant where $\eta=\sqrt{-\mathfrak{E}_{n}}$ and $s_{\pm}=\frac{1}{2}\pm\sqrt{\frac{\mu}{q}+\frac{1}{4}}.$ Imposing the termination condition on the hypergeometric function, to obtain polynomial solutions, implies the exact solutions of Schrodinger’s equation (75) as \[\psi(r) =e^{-\left(\frac{v}{2q(n+s_{+})}-\frac{n+s_{ +}}{2}\right)\,r}\,\left(1-q\,e^{-r}\right)^{s_{ +}}\,{}_{2}F_{1}\left(-n,s_{+}+\frac{v}{q( n+s_{+})};1-n-s_{+}+\frac{v}{q(n+s_{ +})};q\,\,e^{-r}\right),\] (79) where \[\eta=\sqrt{-\mathfrak{E}_{n}},\qquad s_{+}= \frac{1}{2}+\sqrt{\frac{\mu}{q}+\frac{1}{4}},\qquad\mathfrak{E}_{n}=-\left( \frac{v}{2\,q\,(n+s_{+})}-\frac{n+s_{+}}{2 }\right)^{2},\] (80) up to the normalization constant that, as before, can be evaluated exactly. With $\mu=q\,j\,(j+1)$, we obtain $s_{+}=j+1,j=0,1,2,\dots,$ which proves the consistency between $\mathfrak{E}_{n}$ and $\mathcal{E}_{n}$ as given by (74). Finally, we can now write, for $q<(2v)/(1+2j+j^{2}),0\leq n<-1-j+\sqrt{2v/q}$, \[\psi_{j,n}(r) =\dfrac{W\big{(}\psi_{0;0}(r),\psi_{1;0}(r),\dots,\psi_{j-1;0}(r) ,\psi_{n;0}(r)\big{)}}{W\big{(}\psi_{0;0}(r),\psi_{1;0}(r),\dots,\psi_{j-1;0}( r)\big{)}}\] \[=e^{-\left(\frac{v}{2q(n+j+1)}-\frac{n+j+1}{2}\right)\,r}\,\left( 1-q\,e^{-r}\right)^{j+1}\,{}_{2}F_{1}\left(-n,j+1+\frac{v}{q(n+j+1)};\frac{v}{ q(n+j+1)}-n-j;q\,\,e^{-r}\right),\] \[=e^{-\left(\frac{v}{2q(n+1)}-\frac{n+1}{2}\right)\,r}\,\left(1-q \,e^{-r}\right)^{j+1}\,{}_{2}F_{1}\left(j-n,j+1+\frac{v}{q(n+1)};\frac{v}{q(n+ 1)}-n;q\,\,e^{-r}\right),~{}n=j,j+1,\dots\] (81) which is total agreement with the normalization constant obtain through the identity \[\int_{\log q}^{\infty}\psi_{j,n}(r)\psi_{j,m}(r)dx=\frac{(-j)_{j}(j+2n+2)_{j}} {(2^{j}(n+1)_{j})^{2}}\left(n-\frac{v}{(j+n+1)q}+1\right)_{j}\left(n+\frac{v}{ (j+n+1)q}+1\right)_{j}\times N_{n}\,\delta_{nm}\] (82) and $N_{n}$ is given by equation (V). ## VII Conclusion General expressions for the energy eigenvalues and wave function solutions are obtained for Schrödinger’s equation with the generalized Hulthén potential. The simplified closed–form expressions for the normalization constants for arbitrary $q>0$ in terms of the generalized Hypergeometric functions ${}_{3}F_{2}$ with the unit argument are new results. These include, as a particular case, the closed–form expression for the normalization constants of the classical Hulthén potential $q=1$. It is also of interest to note that the double sum in equation (17) can be evaluated in terms of the terminating generalized Kampé de Fériet function with unit arguments to yield \[I_{nm}=\int_{\log(q)/\delta}^{\infty}e^{-\frac{(2+m+n)\left(2\mu -q\,\delta^{2}(1+m)(1+n)\right)x}{2\delta(1+m)(1+n)q}}\left(1-q\,e^{-\delta x} \right)^{2}\,_{2}F_{1}\left(-n,\frac{2\mu}{\delta^{2}(n+1)q}+1;\frac{2\mu}{ \delta^{2}(n+1)q}-n;qe^{-\delta\,x}\right)\] \[\times\,_{2}F_{1}\left(-m,\frac{2\mu}{\delta^{2}(m+1)q}+1;\frac{2 \mu}{\delta^{2}(m+1)q}-m;qe^{-\delta\,x}\right)dx\] \[=\frac{2q^{1+\frac{m+n}{2}-\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1) q}}\Gamma\left(\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1\right) }{\delta\,\Gamma\left(2-\frac{m+n}{2}+\frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q} \right)}\] \[\times F_{1:1;1}^{1:2,2}\left[\begin{array}[]{lll}\frac{\mu(m+n+2 )}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}-1:&-n,\quad\frac{2\mu}{q\delta^{2}(n+1 )}+1;&-m,\quad\frac{2\mu}{q\delta^{2}(m+1)}+1;\\ \\ \frac{\mu(m+n+2)}{\delta^{2}(m+1)(n+1)q}-\frac{m+n}{2}+2:&\frac{2\mu}{q\delta^ {2}(n+1)}-n;&\frac{2\mu}{q\delta^{2}(m+1)}-m;\end{array}\bigg{\|}1,1\right]\] (83) Thus we obtain as a byproduct a simplified expression for the terminating generalized Kampé de Fériet function. It worth mentioning some of the earlier approaches can also be used to study the deformed Hulthén potential, for example, the improved quantization rule [26]. ## VIII Acknowledgments Partial financial support of this work under Grant Nos. GP3438 and GP249507 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by us (respectively RLH and NS). ## Appendix I: Normalization relation To prove the normalization (42), we note first \[\int_{a}^{b}\psi_{n}(x,E_{n})\psi_{m}(x,E_{m})dx =\int_{a}^{b}\phi_{n}^{\prime}(x,E_{n})\phi_{m}^{\prime}(x,E_{m}) dx-\int_{a}^{b}\phi_{m}^{\prime}(x,E_{m})\phi_{n}(x,E_{n})\dfrac{d}{dx}\log{ \phi_{0}(x,\lambda_{0})}dx\] \[-\int_{a}^{b}\phi_{n}^{\prime}(x,E_{n})\,\phi_{m}(x,E_{n}))\dfrac {d}{dx}\log{\phi_{0}(x,\lambda_{0})}dx+\int_{a}^{b}\left(\dfrac{\phi_{0}^{ \prime}(x,\lambda_{0})}{\phi_{0}(x,\lambda_{0})}\right)^{2}\,\phi_{n}(x,E_{n}) \,\phi_{m}(x,E_{n})dx.\] Using integration by parts and the boundary conditions, it is not difficult to show that \[\int_{a}^{b}\phi_{n}^{\prime}(x,E_{n}) \phi_{m}^{\prime}(x,E_{m})dx=E_{n}\mu_{n}\delta_{nm}-\int_{a}^{b} \phi_{m}(x,E_{m})V(x)\phi_{n}(x,E_{n})dx\] \[\int_{a}^{b}\phi_{m}^{\prime}(x,E_{m}) \phi_{n}(x,E_{n})\dfrac{d}{dx}\log{\phi_{0}(x,E_{0})}dx=\phi_{n}( x,E_{n})\phi_{m}(x,E_{m}))\dfrac{d}{dx}\log{\phi_{0}(x,E_{0})}\bigg{\|}_{a}^{b}\] \[-\int_{a}^{b}\phi_{n}(x,E_{n})\phi_{m}^{\prime}(x,E_{m})\dfrac{d} {dx}\log{\phi_{0}(x,E_{0})}dx-\int_{a}^{b}\phi_{n}(x,E_{n})\phi_{m}(x,E_{m}) \dfrac{d^{2}}{dx^{2}}\log{\phi_{0}(x,E_{0})}dx\] \[\int_{a}^{b}\phi_{n}^{\prime}(x,\lambda_{n}) \phi_{m}(x,\lambda_{m})\dfrac{d}{dx}\log{\phi_{0}(x,\lambda_{0})} dx=\phi_{n}(x,E_{n})\phi_{m}(x,E_{m}))\dfrac{d}{dx}\log{\phi_{0}(x,E_{0})} \bigg{\|}_{a}^{b}\] \[-\int_{a}^{b}\phi_{m}(x,E_{m})\,\phi_{n}^{\prime}(x,E_{n})\dfrac{ d}{dx}\log{\phi_{0}(x,E_{0})}dx-\int_{a}^{b}\phi_{n}(x,E_{n})\phi_{m}(x,E_{m}) \dfrac{d^{2}}{dx^{2}}\log{\phi_{0}(x,E_{0})}dx\] \[\int_{a}^{b}\left(\dfrac{\phi_{0}^{\prime}(x,\lambda_{0})}{\phi_{ 0}(x,\lambda_{0})}\right)^{2} \,\phi_{n}(x,E_{n})\,\phi_{m}(x,E_{m})dx=\int_{a}^{b}\left(V(x)-E _{0}-\dfrac{d^{2}}{dx^{2}}\log\phi_{0}(x,\lambda_{0})\right)\,\phi_{n}(x, \lambda_{n})\,\phi_{m}(x,\lambda_{m})dx,\] from which the assertion (42) is proved. ## References * (1) L. Hulthén, Ark. Mat. Astron. Fys. _Über die Eigenlösungen der Schrödinger-Gleichung des Deuterons_28A (1942) 1 - 12. * (2) S. Flügge, _Practical Quantum Mechanics_, Springer-Verlag Berlin Heidelberg (1999). * (3) L. Hulthën and M. Sugawara, _Encyclopedia of Physics_ Vol. 39, edited by S. Flugge, Springer: Berlin (1957). * (4) B. Durand and L. Durand, _Duality for heavy-quark systems,_ Phys. Rev. D 23 (1981) 1092. * (5) W. Van Dijk, _Model analysis of the relationship between ${}^{3}S_{1}$ scattering length and the root-mean-square radius of the deuteron,_ Phys. Rev. C 40 (1989) 1437. * (6) J. Gruninger, _Hulthén transform functions for the excited states of two-electron atoms,_ J. Chem. Phys. 55 (1971) 3561. * (7) K. Szalcwicz and H. J. Mokhorst, _On application of 0s orbitals in SCF calculations,_ J. Chem. Phys. 75 (1981) 5785. * (8) J. Lindhard and A. Winther, _Transient fields acting on heavy ions during slowing-down in magnetized materials_, Nucl. Phys. A 166 (1971) 413. * (9) A. K. Roy _Critical parameters and spherical confinement of H atom in screened Coulomb potential_, Int. J. Quantum Chem.126 (2016) 953. See refs. $2-17$. * (10) F.J.S. Ferreira and V. B. Bezerra _Some remarks concerning the centrifugal term approximation_ , J. Math. Phys.58 (2017) 102104. * (11) R. L. Hall _Yukawa and Hulthén potentials in quantum mechanics_, J. Phys. A25 (1992) 1373. * (12) R. L. Hall _Spectral comparison theorem for the Dirac equation_, Phys. Rev. Lett. 83 (1999) 468. * (13) H. Eğrifes, D. Demirhan and F. Büyükkiliç, _Exact solutions of the Schrödinger equation for two deformed hyperbolic molecular potentials_, Phys. Scripta 60 (1999) 195-198. * (14) C. Berkdemir, A. Berkdemir, and R. Sever, _Polynomial solutions of the Schrödinger equation for the generalized Woods-Saxon potential_, Phys. Rev. C 72 (2005) 027001. * (15) H. Akcay and R. Sever, _Analytical solutions of Schrödinger equation for the diatomic molecular potentials with any angular momentum._ J. Math. Chem 50 (2012) 1973-1987. * (16) Wen-Chao Qiang, Shi-Hai Dong, _Arbitrary l-state solutions of the rotating Morse potential through the exact quantization rule method_ , Phys. Lett. A 363 (2007) 169. * (17) Shi-Hai Dong, Wen-Chao Qiang, Guo-Hua Sun and V B Bezerra, _Analytical approximations to the $\ell$-wave solutions of the Schrödinger equation with the Eckart potential_, J. Phys. A. 40 (2007) 10535. * (18) Shishan Dong, J. García-Ravelo and Shi-Hai Dong, _Analytical approximations to the $\ell$-wave solutions of the Schrödinger equation with an exponential-type potential_, Phys. Scr. 76 (2007) 393. * (19) Shi-Hai Dong, Wen-Chao Qiang and J. García-Pavelo, _Analytical approximations to the Schrödinger equation for a second Pöschl-Teller-like potential with centrifugal term_, Int. J. Mod. Phys. A 23 (2008) 1537. * (20)A. P. Prudnikov, O.I. Marichev, and Yu. Brychkov, _Integrals and series. Volume 3: More special functions,_ Gordon and Breach: New York-London (1989). * (21) O. L. De Lange and R. E. Raab, _Operator Methods in Quantum Mechanics_, Clarendon, Oxford, 1991. * (22) A. Anderson and R. Camporesi, _Intertwining operators for solving differential equations, with applications to symmetric spaces_, Comm. Math. Phys. 130 (1990) 61 - 82. * (23) G. Darbaux, _Sur une proposition relative aux équations linéaires_, Campt. Rend. Acad. de Sci.: Paris 94 (1882) 1456 - 1459. * (24) V. B. Matveev and M. A. Salle, _Darboux transformations and solitons_, Berlin, Heidelberg: Springer Verlag (1991). * (25) M. M. Crum, _Associated Sturm-Liouville Systems_, Q. J. Math 2 (1955) 121-127. * (26)Zhong-Qi Ma, A. Gonzalez-Cisneros, Bo-Wei Xu and Shi-Hai Dong, _Energy spectrum of the trigonometric Rosen–Morse potential using an improved quantization rule_, Phys. Lett. A 371 (2007) 180.
1902.06526	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23519, "num_imgs": 2, "llama3_tokens_count": 6742 }	[ "content_image/1902.06526/x1.png", "content_image/1902.06526/x2.png" ]	# Exact results for the entanglement in 1D Hubbard models with spatial constraints Ioannis Kleftogiannis ph04917@yahoo.com Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan Ilias Amanatidis Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Vladislav Popkov Department of Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Department of Physics, Bergische Universität Wuppertal 42097 Wuppertal HISKP, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany February 23, 2024 ###### Abstract We investigate the entanglement in Hubbard models of hardcore bosons in $1D$, with an additional hardcore interaction on nearest neighbouring sites. We derive analytical formulas for the bipartite entanglement entropy for any number of particles and system size, whose ratio determines the system filling. At the thermodynamic limit the entropy diverges logarithmically for all fillings, except for half-filling, with the universal prefactor $1/2$ due to partial permutational invariance. We show how maximal entanglement can be achieved by controlling the interaction range between the particles and the filling which determines the empty space in the system. Our results show how entangled quantum phases can be created and controlled, by imposing spatial constraints on states formed in many-body systems of strongly interacting particles. ## 1 Introduction Entanglement is a key concept in understanding how quantum orders manifest in systems with many interacting particles[1; 2; 3], such as the well known example of topological order[4; 5; 6; 7; 8; 9; 10; 11; 12]. Essentially the degree of entanglement can be used as a measure for the strength of the quantum correlations in a many-body system. In the last decades enormous advancement has been achieved in coming up with ways to quantify the quantum orders based on entanglement measures. Such well known examples are the entanglement entropy[1; 13; 14; 15] or the entanglement spectrum[8; 16; 17; 18]. These measures require splitting the system in different partitions, whose reduced density matrix can be used to calculate the entanglement entropy of each respective partition. The scaling of this entanglement entropy with the partition size, reveals important properties of the system, such as quantum criticality.[1; 13; 14; 19] Due to the large complexity of the many-body systems under investigation, which usually contain an enormous number of particles, exact/analytical solutions of these problems are rarely possible and difficult to obtain. Exact methods (Bethe Ansatz and quantum inverse scattering method) are limited to the $1D$ Fermi-Hubbard model [20] and to some extent to the Bose-Hubbard model in the low-density regime [21], while approximate and numerical approaches are used to study various aspects of the Hubbard model and its extensions, see e.g. a recent review [22]. Among other extensions, a Hubbard model with additional integrability-breaking nearest-neighbor interactions was studied recently, showing an intriguing new phase of a quantum disentangled liquid [23]. In our paper we focus on purely nearest-neighbor interaction effects and derive exact/analytical results for the entanglement in 1D Hubbard models of hardcore bosons[15; 24; 25], with additional spatial constraints imposed by the nearest-neighbor interactions. Many-body/Fock states manifest as the ground states of these Hubbard models, as the particles organize in different spatial configurations[26]. In this paper we provide analytical solutions for the density matrix and the entanglement entropy for superpositions of such states, for any number of particles and system size. We study the bipartite entanglement and show how it varies for different system fillings. At the thermodynamic limit we find that the entropy diverges logarithmically for all fillings except for half-filling, with a universal prefactor 1/2. In addition, we show how the maximal entanglement can achieved by varying the filling. In overall, our results provide a way to tune the entanglement in Hubbard models with strong interactions, based on the empty space in the system and the interaction range between the particles. ## 2 Model The states studied in this paper can be obtained by considering the ground state of a $1D$ Hubbard-like Hamiltonian with nearest-neighbor interactions [26], in the limit of large interaction $U$, when the hopping part can be neglected, i.e. \[H_{U}=U\sum_{i=1}^{M-1}n_{i}n_{i+1},\] (1) where $n_{i}=c_{i}^{\dagger}c_{i}$ is the particle number operator, with $c_{i}^{\dagger},c_{i}$ being the creation and annihilation operators for spin-less hardcore bosons on site $i$. The ground state of this system filled by $N<M/2$ hardcore bosons (hereafter also called particles) has a large degeneracy, since every spatial configuration of the particles respecting the hard-core restriction on sites which are nearest-neighbours (apart from the on-site hardcore restriction), has the lowest energy (see figure 1 for an illustration). We use the name NN for these states. The NN states are separated from the first excited states by an energy gap, equal to the strength of the nearest-neighbor interaction U. A superposition of all possible NN states with equal amplitudes, has the form \[\left\|\Psi\right\rangle\equiv\left\|M,N\right\rangle=\frac{1}{\sqrt{d(M,N)}} \sum_{P}^{d(M,N)}\left\|1010100\ldots\right\rangle\] (2) \[d(M,N)=\binom{M-N+1}{N},\] (3) where $d(M,N)$ is the number of ways to distribute the $N$ particles on $M$ sites, assuming at least one hole between all the particles, due to the nearest-neighbor interaction. An appearance of the binomial coefficients Eq. 3 in the sum Eq. 2 signalizes a presence of permutation group symmetry in the problem. Quantifying the impact of the hardcore constraint on nearest-neighbor sites(spatial constraint) on the entanglement entropy of a block is one of our objectives. Note that the states of type Eq. (2) also arise in Hubbard models with dynamic restrictions not allowing cluster formation of the particles. For example, the unique groundstate of the following Hubbard-like Hamiltonian \[H =U\sum_{i=1}^{M-1}n_{i}n_{i+1}\] (4) \[+t\sum_{i=1}^{M-1}[(1-n_{i-1})c_{i}^{+}c_{i+1}(1-n_{i+2})+h.c.],\] with $U>0,t>0$ contains contribution from all the NN states with different amplitudes, while at the limit $t\to 0$, the state (2) becomes one of degenerate groundstates of (4). ## 3 Bipartite entanglement <figure><img src="content_image/1902.06526/x1.png"><figcaption>Figure 1: a) The possible NN states with nearest-neighbor interaction for N=3particles distributed in M=7 sites corresponding to filling f=3/7. The reduceddensity matrix of a partition containing m=4 sites, can be written in a blockdiagonal form. Each block corresponds to a sector according to the number ofparticles it contains, as shown on the right. b) NN states with N=4 and M=6for filling f>1/2. By applying particle-hole exchange, the NN states transformto those corresponding to a system with N=2 and M=6, for f<1/2. This way, byignoring the two empty edge sites we can calculate the entanglement entropyfor f>1/2 by knowing the one for f<1/2.</figcaption></figure> In this section we derive analytical results for the reduced density matrix and the entanglement entropy for partitioned superpositions of the NN states described by Eq. 2. Due to partial permutational symmetry enjoyed by the global pure state of the system, the eigenvalues of the reduced density matrix can be obtained after splitting the system of size $M$ in two parts containing $m$ and $M-m$ sites respectively, and tracing out the degrees of freedom of the $m$ sites. The tracing out procedure can be characterized via a permutation group analysis. As a result of the analysis, the number of non-zero eigenvalues of the reduced density matrix grows not exponentially, but linearly with the subsystem size $m$. Origin of all the nonzero eigenvalues have been identified, as belonging to sectors with different symmetry and particle number and it was understood how to obtain them analytically, via a recursion procedure. An example of this procedure is shown schematically in figure 1(a) for a small system. The full analytic answer has been obtained for arbitrary $N,M$. Also, the respective thermodynamic limit has been analyzed. After tracing out the $m$ sites, and using recurrently a well known formula \[\binom{F+1}{N} =\binom{F}{N-1}+\binom{F}{N},\] (5) after some algebra we obtain that the reduced density matrix is split into blocks \[\rho_{M-m} =\sum_{k=0}^{m/2}A_{k0}\left\|0_{k}\right\rangle\left\langle 0_{k} \right\|+A_{k1}\left\|1_{k}\right\rangle\left\langle 1_{k}\right\|\] (6) \[\left\|0_{k}\right\rangle =\left\|M-m,N-k\right\rangle\] (7) \[\left\|1_{k}\right\rangle =\left\|M-m-1,N-k\right\rangle\otimes\left\|1,0\right\rangle\] (8) \[A_{k0}d(M,N) =\binom{m-k}{k}\binom{M-N+1-m+k}{N-k}\] (9) \[A_{k1}d(M,N) =\binom{m-k}{k-1}\binom{M-N-m+k}{N-k}\] (10) \[\left\langle 1_{k}\|1_{k}\right\rangle =\left\langle 0_{k}\|0_{k}\right\rangle=1.\] (11) Note that the property $Tr\rho_{M-m}=1$ is guaranteed by \[\sum_{k=0}^{m/2}(A_{k0}+A_{k1}) =d(M,N).\] (12) Now, the states $\left\|0_{k}\right\rangle,\left\|\alpha_{k^{\prime}}\right\rangle$ are orthogonal for $k\neq k^{\prime}$ but they are not orthogonal for $k^{\prime}=k$. The overlap between $\left\|0_{k}\right\rangle,\left\|1_{k}\right\rangle$ can readily be found from the combinatorial arguments to be \[\eta_{k}=\left\langle 0_{k}\|1_{k}\right\rangle =\sqrt{\frac{\binom{M-N-m+k}{N-k}}{\binom{M-N+1-m+k}{N-k}}}.\] (13) Each block with $N-k$ particles thus contains two eigenvalues $\lambda_{k},\mu_{k}$, which can be found by diagonalizing the $2\times 2$ block in (6). It is then straightforward to obtain the relations \[\lambda_{k}+\mu_{k} =A_{k0}+A_{k1}=b\] (14) \[\lambda_{k}\mu_{k} =A_{k0}A_{k1}(1-\eta_{k}^{2})=c.\] (15) In terms of the above notations we have \[\lambda_{k} =\frac{b}{2}+\frac{1}{2}\sqrt{b^{2}-4c}\] (16) \[\mu_{k} =\frac{b}{2}-\frac{1}{2}\sqrt{b^{2}-4c}.\] (17) The set $\lambda_{k}$ and $\mu_{k}$ for all $0\leq k\leq m/2$ gives the exact spectrum of the reduced density matrix for arbitrary system parameters. ### Thermodynamic limit First, consider the limit \[N\gg m\gg 1,\ \ \ N/M=f<1/2.\] (18) In this limit, analogically to [13], and denoting \[p =\frac{f}{1-f}\] (19) \[q =1-p=\frac{1-2f}{1-f}\] (20) \[n =m-k\] (21) we obtain \[A_{k0} \approx\frac{1}{\sqrt{2\pi npq}}e^{-{\frac{(k-np)^{2}}{2npq}}}\] (22) \[A_{k1} \approx\frac{1}{\sqrt{2\pi npq}}e^{-{\frac{(k-1-np)^{2}}{2npq}}} \frac{N-k+1}{N-M+1-n}\] (23) valid for $npq\gg 1$. After some algebra, denoting \[x =\frac{1-k/m}{1-f}\] (24) we obtain \[A_{k0}\equiv A_{k0}(x) =(1-f)\frac{1}{m(1-f)}g(A,x)\] (25) \[A_{k1}\equiv A_{k1}(x) =f\frac{1}{m(1-f)}g\left(A,x-\frac{\kappa(f)}{m}\right)\] (26) \[g(A,x) =\sqrt{\frac{A}{x\pi}}e^{-A\frac{(x-1)^{2}}{x}}\] (27) \[\int_{0}^{\infty}g(A,x)dx =1\] (28) \[\sum_{k}\ldots \approx m(1-f)\int_{0}^{\infty}\ldots dx\] (29) \[A =m\frac{M}{M-m}\frac{1-f}{2f(1-2f)}.\] (30) Note that the last formula is valid for comparable $M\gg 1,m\gg 1,$$m/M=const$, and $\kappa(f)$ is of order $1$. Finally, in the zero non-vanishing order of $1/m$, the term $\kappa(f)/m\ll 1$ in Eq. 26 can be neglected, and we obtain the final formula for the eigenvalues of the reduced density matrix (RDM) of the form \[\lambda_{k}\equiv\lambda(x) =\frac{C_{0}}{m(1-f)}g(A,x)\] (31) \[\mu_{k}\equiv\mu(x) =\frac{C_{1}}{m(1-f)}g(A,x)\] (32) \[C_{0},C_{1} =\frac{1}{2}\pm\frac{\sqrt{1-4f^{2}}}{2}.\] (33) It can be proved easily that \[m(1-f)\int_{0}^{\infty}(\lambda(x)+\mu(x))dx =1.\] (34) Finally, we can find the von Neumann entropy (VNE) of the RDM, $S=-\mathop{\mathrm{tr}}\limits\rho\log\rho$, $\rho$ being the reduced density matrix \[S(f,m,M) =-\sum_{k}(\lambda_{k}\log\lambda_{k}+\mu_{k}\log\mu_{k})\approx\] (35) \[-m(1-f) \int_{0}^{\infty}\left(\lambda(x)\log\lambda(x)+\mu(x)\log\mu(x) \right)dx.\] (36) Performing the calculations, we obtain \[S(f,m,M) =Q_{0}(\frac{m}{M},f)+\frac{1}{2}\log m\] (37) \[Q_{0} =-\sum_{\alpha=0,1}C_{\alpha}\log C_{\alpha}+\log\frac{(1-f)\sqrt {\pi e}}{\sqrt{A/m}}.\] (38) Thus we have the same logarithmic growth of the entanglement entropy of Von Neumann, $\frac{1}{2}\log m$ as in fully permutational states of the Heisenberg ferromagnet at isotropic point, see [13; 27], which is apparently due to partial underlying permutational symmetry of the initial pure state. The prefactor $1/2$ in $\frac{1}{2}\log m$ is thus simply the value of effective local spin as discussed in [27]. A comparison between the exact results of Eq. 16-17 and the thermodynamic limit Eq. 37-38 is shown on figure 2b. <figure><img src="content_image/1902.06526/x2.png"><figcaption>Figure 2: a) The scaling of the bipartite entanglement entropy S with thenumber of particles N, for different fillings. A logarithmic divergence at thethermodynamic limit can be observed in all cases, apart for the half-filledcase f=1/2. b)Scaling of von Neumann entropy S with the system size M. Anexcellent agreement can be seen between the exact results(points) obtained byEq. 16-17 and the curves obtained in the thermodynamic limit via Eq. 37-38. c)Comparison with the entanglement entropy of a Heisenberg spin chain. Thedifference between the entropies of the respective systems is plotted versusthe filling using Eq. 39 and a symmetry property S(f)=S(1−f) established insec. 3.2. d) The entropy versus f for a chain with M=10000 and m=5000 usingEq. 38. Maximum entanglement is obtained at f≃0.305 and f≃0.695. For f=1/2minimum entanglement with S=log(2) is achieved. An excellent agreement can beseen between the exact results(points) and the thermodynamic limitapproximation(curve).</figcaption></figure> Note that in the form (38) an arbitrary base of logarithm can be considered. In particular, comparing (38) with the VNE computed for the ground state of the isotropic Heisenberg ferromagnet, denoted below as $S_{Heis}$, see [13], we obtain \[S_{Heis}-S =\sum_{\alpha=0,1}C_{\alpha}\log C_{\alpha}-\frac{1}{2}\log(1-2f).\] (39) As further analysis shows, $S_{Heis}>S$ for all nonzero fillings $f$. This has the following interpretation: the ground state of the isotropic Heisenberg ferromagnet is fully permutational invariant state with no constraints except the hard-core constraint: the minimal distance between two particles is equal to $1$: two particles can be at neighbouring sites. The wave function Eq. 2 has an additional constraint of a minimal distance between particles being equal to $2$. This additional constraint lowers the symmetry of the problem, and respectively the entanglement becomes smaller. The difference $S_{Heis}-S$, shown in figure 2c, quantifies this excess of entanglement in a state with full permutational symmetry. The difference $S_{Heis}-S$ increases with the filling $f$, reaching a maximum at $f=1/2$, since the effect of the additional constraint with increasing number of particles $fM=N$ becomes more and more pronounced. Our approach of controlling the entanglement via spatial constraints in hardcore bosonic systems, could be applied also to other systems that obey similar rules. One example would be spinless fermions on a chain, since also in this case only one particle is the maximum occupation number per site. The corresponding state described by Eq. 2 should contain Fock states which are antisymmetric under exchange of two fermions, this being one of the differences with hard-core bosons, which obey the symmetry principle instead. Nevertheless, similar entanglement properties should be observed to the hardcore bosonic system, as long as the fermionic system lies in the strong interaction regime, where the fermions behave as localized(point) particles. In general, the entanglement properties of the ground state are fully determined by the microstructure inside the Fock states in equation 2 along with their superposition amplitudes, irrespectively of the type of particles. ### Entanglement for $f>1/2$ The analysis we presented so far is valid for fillings $f<1/2$, as we have considered a wavefunction of the form Eq. 2, which has at least one hole/empty site between all the particles (minimal distance $2$ between the particles). This analysis can be easily generalized to the states for $f>1/2$ which will contain clusters of particles and a fixed number of particle pairs. These $f>1/2$ states can be transformed to states with $N\to M-N$ and $M\to M-2$, i.e. to those with $f<1/2$. This can be seen by taking the states for $f>1/2$ (note that all configurations for $f>1/2$ have edge sites filled), exchanging particles with holes and tracing out the edge sites $1$ and $M$, see figure 1b for an example. Therefore, the system of $M$ sites, $N$ particles, corresponding to $f=N/M>1/2$ is mapped onto a system of $M^{\prime}=M-2$ sites, $N^{\prime}=M-N$ particles, with filling factor $f^{\prime}=(M-N)/(M-2)\leq 1/2$. Note that for odd system size $M$ and $N=(M+1)/2$, only one NN state $\left\|1010\ldots 101\right\rangle$ contributes to the superposition Eq. 2, leading to $S=0$ for all $m$. The eigenvalues of the reduced density matrix for $f=N/M>1/2$ are given by substituting $M\to M-2$ and $N\to M-N$ in Eq. 16-17. In the thermodynamic limit, the configurations with fillings $f>1/2$ are mapped onto configurations with fillings $f^{\prime}=1-f$, leading to logarithmic behavior of the entropy for all $f\neq 1/2$. ### Entanglement control Another point of interest is the dependence of the entanglement strength on the filling. In figure 2d, we plot $S$ versus the filling using Eq. 38(curve) and compare with the exact result using Eq. 16-17(points). The case $M$ odd and $N=(M+1)/2$ which gives S=0, is not present in figure 2d, since the system size $M$ is even. The entropy is symmetric around $f=1/2$ where $S=log(2)$, due to the $f\to 1-f$ symmetry, as we have analyzed in the previous section. The entropy obtains a maximum value at $f\simeq 0.305$ and $f\simeq 0.695$ leading to a maximally entangled quantum phase, irrespectively of the partitioning as long as both $fm,fM$ are large (the asymptotic value $f\simeq 0.305884$ is obtained in the limit $m,M\rightarrow\infty$). The maximization of the entropy at this filling is a consequence of the spatial restrictions due to the nearest-neighbor interaction, that impose the constraint of a minimal distance $2$ between the particles. Changing this minimal distance by controlling the interaction range, for example by adding a second nearest instead of nearest-neighbor interaction term in the Hamiltonian, will lead to different fillings where the maximum entanglement occurs. Consequently the entanglement strength in superpositions of states like the NN ones considered in the paper, can be tuned by the system’s filling and the range of interaction between the particles. ## 4 Summary and Conclusions We have studied analytically the entanglement properties of NN states, which appear as the ground states of Hubbard chains of hardcore bosons, with strong nearest-neighbor interactions i.e. 1D Hubbard models with spatial constraints. We have derived exact expressions for the entanglement entropy and the reduced density matrix for partitioned superpositions of the NN states. We have done that for any number of particles and system size, whose ratio determines the system filling. We show that the bipartite entanglement entropy diverges logarithmically for all fillings, apart from half-filling, as in the critical phases of XY spin chains. We present a detailed analysis of the mechanism that creates the entanglement and make a comparison with the entanglement of spin chains. The entanglement entropy obtains a maximum value for specific fillings, revealing a maximally entangled quantum phase. In overall, the conditions under which this phase occurs are determined by the spatial restrictions imposed by the empty space in the system and interaction range between the particles. In conclusion, we show analytically how the entanglement can be tuned in Hubbard models with strong nearest-neighbor interactions, by controlling the empty space in the system and the nature of the interactions between the particles, which impose spatial restrictions on their self-organization. We hope that our results motivate further investigations on the mechanisms that allow controllable entanglement in many-body systems, to reveal novel quantum phases of matter and help with their potential application in the rapidly evolving field of quantum information technology. ## Acknowledgements IK acknowledges resources and financial support provided by the National Center for Theoretical Sciences of R.O.C. Taiwan. IA acknowledges support from the Center for Theoretical Physics of Complex Systems in Daejeon Korea under the project IBS-R024-D1. VP acknowledges financial support from Deutsche Forschungsgemeinschaft through DFG projects KO 4771/3-1 and KL 645/20-1 and support from ERC grant 694544 OMNES, and thanks Center for Theoretical Physics of Complex Systems in Daejeon Korea for a hospitality during his stay, during which this project has initiated. ## References * (1) Vidal G, Latorre J I, E. Rico and Kitaev A 2003 Phys. Rev. Lett. 90 227902 * (2) Amico L, Fazio R, Osterloh A and Vedral V 2008 Rev. Mod. Phys. 80 517 * (3) Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys. 81 865 * (4) Chen X, Gu Z-C and Wen X-G 2010 Phys. Rev. B 82 155138. * (5) Kitaev A and Preskill J 2006 Phys. Rev. Lett. 96 110404 * (6) Kitaev A Y 2003 Ann. Phys. (NY) 303 * (7) Levin M and Wen X-G 2006 Phys. Rev. Lett. 96 110405 * (8)Li H and Haldane F D M 2008 Phys. Rev. Lett. 101 010504 * (9)Haldane F D M 1983a Phys. Lett. A 93 464 * (10)Christopher N V, Sun K, Rigol M and Galitski V 2010 Phys. Rev. B 82, 115125 * (11) Isakov S V, Hastings M B and Melko R G 2011 Nature Physics volume 7, pages 772-775 * (12) Affleck I, Kennedy T, Lieb E H and Tasaki H 1987 Phys. Rev. Lett. 59, 799 * (13) Popkov V and Salerno M 2005 Phys. Rev. A 71, 012301 * (14)Hamma A Ionicioiu R and Zanardi P 2005 Phys. Rev. A 71, 022315 * (15)Wang Y, Gulden T and Kamenev A 2016 Phys. Rev. B 95, 075401 * (16) Alba V, Haque M and Luchli A M 2013 Phys. Rev. Lett. 110, 260403 * (17)Calabrese P and Lefevre A 2008 Phys. Rev. A 78, 032329 * (18)Pollmann F, Turner A M, Berg E and Oshikawa M 2010 Phys. Rev. B 81, 064439 * (19) Eisert J, Cramer M and Plenio M B 2010 Rev. Mod. Phys. 82, 277 * (20) F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, V. E. Korepin, The one-dimensional Hubbard model (Cambridge University Press, Cambridge, UK, 2005) * (21) Krauth W 1991 Phys. Rev. B 44, 9772 * (22) Carmelo J M P and Sacramento P D 2018 Physics Reports 749 1 * (23) Garrison J R, Mishmash R V, Fisher Matthew P A 2017 Phys. Rev. B 95, 054204 * (24) Hen I and Rigol M 2009 Phys. Rev. B 80, 134508 * (25)Nishimoto S, Ejima S and Fehske H 2013 Phys. Rev. B 87, 045116 * (26) Kleftogiannis I, Amanatidis I 2017 arXiv:1707.07840 * (27) Popkov V, Salerno M and Schütz G M 2005 Phys. Rev. E 72, 032327
1602.00825	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 54956, "num_imgs": 21, "llama3_tokens_count": 20136 }	[ "content_image/1602.00825/x1.png", "content_image/1602.00825/x2.png", "content_image/1602.00825/x8.png", "content_image/1602.00825/x10.png", "content_image/1602.00825/x11.png", "content_image/1602.00825/x12.png", "content_image/1602.00825/x13.png", "content_image/1602.00825/x19.png", "content_image/1602.00825/x25.png", "content_image/1602.00825/x31.png", "content_image/1602.00825/x37.png", "content_image/1602.00825/x43.png", "content_image/1602.00825/x49.png", "content_image/1602.00825/x55.png", "content_image/1602.00825/x61.png", "content_image/1602.00825/x67.png", "content_image/1602.00825/x73.png", "content_image/1602.00825/x79.png", "content_image/1602.00825/x85.png", "content_image/1602.00825/x91.png", "content_image/1602.00825/x97.png" ]	# Nonclassical Particle Transport in 1-D Random Periodic Media R. Vasques ${}^{\dagger,}$¹ , K. Krycki ${}^{\ddagger}$, R. N. Slaybaugh ${}^{\dagger}$ ${}^{\dagger}$_University of California, Berkeley Department of Nuclear Engineering 4155 Etcheverry Hall, Berkeley, CA 94720-1730_ ${}^{\ddagger}$_Aachen Institute for Nuclear Training GmbH Jesuitenstraße 4, 52062 Aachen, Germany_ [FOOTNOTE:1][ENDFOOTNOTE] ###### Abstract We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path-length $s$), and models particle transport taking place in homogenized random media in which a particle’s distance-to-collision is not exponentially distributed. To solve the nonclassical equation one needs to know the $s$-dependent ensemble-averaged total cross section, $\Sigma_{t}(\mu,s)$, or its corresponding path-length distribution function, $p(\mu,s)$. We consider a 1-D spatially periodic system consisting of alternating solid and void layers, randomly placed in the $x$-axis. We obtain an analytical expression for $p(\mu,s)$ and use this result to compute the corresponding $\Sigma_{t}(\mu,s)$. Then, we proceed to numerically solve the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions $\mu=\pm 1$. To assess the accuracy of these solutions, we produce “benchmark" results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely-used atomic mix model in most problems. Keywords: nonclassical transport, random media, atomic mix ## I Introduction The classical theory of linear particle transport defines the total cross section $\Sigma_{t}$ as independent of the path-length $s$ (the distance traveled by the particle since its previous interaction) and of the direction of flight ${\bf\Omega}$. This definition leads to an exponential probability density function for a particle’s distance-to-collision: \[p(s)=\Sigma_{t}e^{-\Sigma_{t}s}.\] (1) However, a nonexponential attenuation law for the particle flux arises in certain inhomogeneous media in which the scattering centers are spatially correlated. This “nonclassical" behavior occurs in certain important applications, such as neutron transport in Pebble Bed Reactors (in which a nonexponential $p(s)$ arises due to the pebble arrangement within the core) and photon transport in atmospheric clouds (in which the locations of the water droplets in the cloud seem to be correlated in ways that measurably affect the radiative transfer within the cloud). An approach to this type of nonclassical transport problem was recently proposed [1, 2], with the assumption that the positions of the scattering centers are correlated but independent of direction ${\bf\Omega}$. Existence and uniqueness of solutions are rigorously discussed in [3]. This nonclassical theory was extended in [4] to include angular-dependent path-length distributions in order to investigate anisotropic diffusion of neutrons in 3-D PBR cores. A similar kinetic equation with path-length as an independent variable has been rigorously derived for the periodic Lorentz gas in a series of papers by Golse et al. (cf. [5] for a review), and by Marklof & Strömbergsson (cf. [6, 7]). Furthermore, related work has been performed by Grosjean in [8]; it considers a generalization of neutron transport that includes arbitrary path-length distributions, and presents a derivation of diffusion solutions for infinite isotropic point and plane source problems. Assuming monoenergetic transport and isotropic scattering, the nonclassical linear Boltzmann equation with angular-dependent path-length distributions and isotropic source is writen as \[\frac{\partial\psi}{\partial s}({\bm{x}},{\bf\Omega},s)+ {\bf\Omega}\cdot{\bf\nabla}\psi({\bm{x}},{\bf\Omega},s)+\Sigma_{t }({\bf\Omega},s)\psi({\bm{x}},{\bf\Omega},s)\] (2) \[=\frac{\delta(s)}{4\pi}\left[c\int_{4\pi}\int_{0}^{\infty}\Sigma_ {t}({\bf\Omega}^{\prime},s^{\prime})\psi({\bm{x}},{\bf\Omega}^{\prime},s^{ \prime})ds^{\prime}d\Omega^{\prime}+Q({\bm{x}})\right]\,,\] where ${\bm{x}}=(x,y,z)=$ position, ${\bf\Omega}=(\Omega_{x},\Omega_{y},\Omega_{z})=$ direction of flight (with $\|{\bf\Omega}\|=1$), $\psi$ is the nonclassical angular flux, $c$ is the scattering ratio (such that the scattering cross section $\Sigma_{s}=c\Sigma_{t}$), and $Q$ is the source. Here, the nonclassical angular-dependent ensemble-averaged total cross section $\Sigma_{t}({\bf\Omega},s)$ is defined as \[\Sigma_{t}({\bf\Omega},s)ds=\begin{array}[]{l}\text{ the probability (ensemble-averaged over all physical}\\ \text{ realizations) that a particle, scattered or born at any}\\ \text{ point ${\bm{x}}$ and traveling in the direction ${\bf\Omega}$, will experience}\\ \text{ a collision between ${\bm{x}}+s{\bf\Omega}$ and ${\bm{x}}+(s+ds){\bf \Omega}$.}\end{array}\] (3) The underlying path-length distribution and the above nonclassical cross section are related [4] by \[p({\bf\Omega},s)=\Sigma_{t}({\bf\Omega},s)\exp\left(-\int_{0}^{s }\Sigma_{t}({\bf\Omega},s^{\prime})ds^{\prime}\right).\] (4) It has been shown that, if $p(s)$ is independent of ${\bf\Omega}$, Eq.2 can be converted to an integral equation for the scalar flux that is identical to the integral equation that can be constructed for certain diffusion-based approximations [9, 10]. Moreover, if the path-length distribution function is an exponential as given in Eq.1, Eq.2 reduces to the classical linear Boltzmann equation equationparentequation \[{\bf\Omega}\cdot{{\bf\nabla}}\Psi({\bm{x}},{\bf\Omega})+\Sigma_{t }\Psi({\bm{x}},{\bf\Omega})=\frac{1}{4\pi}\left[\int_{4\pi}\Sigma_{s}\Psi({\bm {x}},{\bf\Omega}^{\prime})d\Omega^{\prime}+Q({\bm{x}})\right]\] (5a) for the classical angular flux \[\Psi({\bm{x}},{\bf\Omega})=\int_{0}^{\infty}\psi({\bm{x}},{\bf \Omega},s)ds.\] (5b) Numerical results have been provided for the asymptotic diffusion limit of this nonclassical theory [2, 11, 12, 13], and for moment models of the nonclassical equation in the diffusive regime [14]. However, very few results have been presented for the nonclassical _transport_ equation. This is because one must know $\Sigma_{t}({\bf\Omega},s)$, or $\Sigma_{t}(s)$ in the case of angular-independent path lengths, in order to solve Eq.2. In this paper we investigate the accuracy of the 1-D nonclassical transport equation. We consider a 1-D random periodic system: a spatially periodic system consisting of alternating layers, randomly placed on the $x$-axis. This means that we only know which material is present at any given point $x$ in a probabilistic sense. The 1-D version of Eq.2 is written as \[\frac{\partial\psi}{\partial s}(x,\mu,s)+\mu\frac{\partial\psi}{ \partial x}(x,\mu,s) +\Sigma_{t}(\mu,s)\psi(x,\mu,s)\] (6) \[=\frac{\delta(s)}{2}\left[c\int_{-1}^{1}\int_{0}^{\infty}\Sigma_{ t}(\mu^{\prime},s^{\prime})\psi(x,\mu^{\prime},s^{\prime})ds^{\prime}d\mu^{ \prime}+Q(x)\right]\,.\] This system was chosen because we can obtain an analytical expression for the distribution function $p(\mu,s)$ of a particle’s distance-to-collision in the direction $\mu$. Then, using the identity [4] \[\Sigma_{t}(\mu,s)=\frac{p(\mu,s)}{1-\int_{0}^{s}p(\mu,s^{\prime}) ds^{\prime}},\] (7) one can obtain a solution for Eq.6. The numerical results presented in this paper consider transport in _rod geometry_, in which particles can only move in the directions $\mu=\pm 1$. Solutions are given for a total of 72 solid-void test problems. To analyze the accuracy of these results, we compare them against “benchmark" numerical results, obtained by ensemble-averaging the solutions of the transport equation over a large number of physical realizations of the random system. Furthermore, we compare the performance of the nonclassical model against the widely-used atomic mix model. This paper is an expanded version of a recent conference paper [15]. The remainder of this paper is organized as follows. In SectionII we sketch the 1-D random periodic system under consideration. In SectionIII we analytically derive the path-length distribution function for the periodic random system; explicit expressions for solid-void media are given in SectionIII.A. In SectionIV we define the parameters of the test problems and describe the benchmark, atomic mix, and nonclassical approaches to solve them. In SectionV we examine the numerical results that confirm the accuracy of the nonclassical model. We conclude with a discussion in SectionVI. ## II The 1-D Random Periodic System Let us consider a 1-D physical system similar to the one introduced in [16], consisting of alternating layers of two distinct materials (labeled 1 and 2) periodically arranged. The period is given by $\ell=\ell_{1}+\ell_{2}$, where $\ell_{i}$ represents the length of each layer of material $i\in\{1,2\}$. A sketch of the periodic system is given in Fig.1. This periodic system is _randomly placed_ in the infinite line $-\infty<x<\infty$, such that the probability $P_{i}$ of finding material $i$ in a given point $x$ is $\ell_{i}/\ell$. Therefore, the cross sections and source are stochastic functions of space; that is, if $x$ is in material $i$, then equationparentequation \[\Sigma_{t}(x) =\Sigma_{ti}\,,\] (8a) \[\Sigma_{s}(x) =c_{i}\Sigma_{ti}\,,\] (8b) \[Q(x) =Q_{i}(x)\,,\] (8c) where $\Sigma_{ti}$, $c_{i}$, and $Q_{i}$ represent the total cross section, scattering ratio, and source in material $i$. ## III The Path-length Distribution Function Given a physical realization of the 1-D system described in SectionII, let us examine a particle that is born (or scatters) at a point $x$ in a layer of material $i\in\{1,2\}$ with direction of flight $\mu\neq 0$. We define $x_{0}$ to be the horizontal distance between $x$ (the point in which the collision or birth event took place) and the next intersection between layers in the direction $\mu$. We also define: equationparentequation \[p_{A_{i}}(x_{0},\mu,s) =\begin{array}[]{l}\text{ the probability that a particle born or scattered in}\\ \text{ material $i$, at a horizontal distance $x_{0}$ of the next}\\ \text{ intersection, with direction of flight $\mu$, will travel a}\\ \text{ distance $s$ without colliding;}\end{array}\] (9a) \[p_{B_{i}}(x_{0},\mu,s)ds =\begin{array}[]{l}\text{ the probability that a particle born or scattered in}\\ \text{ material $i$, at a horizontal distance $x_{0}$ of the next}\\ \text{ intersection, with direction of flight $\mu$, will experience}\\ \text{ a collision between $s$ and $s+ds$.}\end{array}\] (9b) For $\mu\neq 0$, we can write equationparentequation \[p_{A_{i}}(x_{0},\mu,s) =\left\{\begin{array}[]{ll}e^{-\Sigma_{ti}s},&\text{if }0\leq s\| \mu\|\leq x_{0}\\ (e^{-\Sigma_{ti}x_{0}/\|\mu\|})(e^{-\Sigma_{tj}(s-x_{0}/\|\mu\|)}),&\text{if }x_{0 }<s\|\mu\|\leq x_{0}+\ell_{j}\\ (e^{-\Sigma_{ti}(s-\ell_{j}/\|\mu\|)})(e^{-\Sigma_{tj}\ell_{j}/\|\mu\|}),&\text{if }x_{0}+\ell_{j}<s\|\mu\|\leq x_{0}+\ell\\ \,\,\,\vdots&\end{array}\right.\] (10a) and \[p_{B_{i}}(x_{0},\mu,s) =\left\{\begin{array}[]{ll}\Sigma_{ti},&\text{if }0\leq s\|\mu\| \leq x_{0}\\ \Sigma_{tj},&\text{if }x_{0}<s\|\mu\|\leq x_{0}+\ell_{j}\\ \Sigma_{ti},&\text{if }x_{0}+\ell_{j}<s\|\mu\|\leq x_{0}+\ell\\ \,\,\,\vdots&\end{array}\right.\,\,,\] (10b) such that equationparentequation \[p_{A_{i}}(x_{0},\mu,s) =\left\{\begin{array}[]{ll}e^{-\Sigma_{ti}s},&\text{if }0\leq s\| \mu\|\leq x_{0}\\ e^{-\Sigma_{tj}s-(\Sigma_{ti}-\Sigma_{tj})(x_{0}+n\ell_{i})/\|\mu\|},&\text{if } x_{0}+n\ell<s\|\mu\|\leq x_{0}+n\ell+\ell_{j}\\ e^{-\Sigma_{ti}s-(\Sigma_{tj}-\Sigma_{ti})(n+1)\ell_{j}/\|\mu\|},&\text{if }x_{0 }+n\ell+\ell_{j}<s\|\mu\|\leq x_{0}+(n+1)\ell\end{array}\right.\] (11a) and \[p_{B_{i}}(x_{0},\mu,s)\] (11b) Here, $n=0,1,2,...$; $i,j\in\{1,2\}$; $i\neq j$; and $\ell=\ell_{i}+\ell_{j}$. It is clear that \[p_{C_{i}}(x_{0},\mu,s)ds =\begin{array}[]{l}\text{ the probability that a particle born or scattered in}\\ \text{ material $i$, at a horizontal distance $x_{0}$ of the next}\\ \text{ intersection, with direction of flight $\mu$, will experience}\\ \text{ its {first collision} while traveling a distance between $s$}\\ \text{ and $s+ds$}\end{array}\] (12) \[=\,\,\,p_{A_{i}}(x_{0},\mu,s)\times p_{B_{i}}(x_{0},\mu,s)ds,\] and the _ensemble-averaged_ path-length distribution function of particles born or scattered in material $i$ with direction of flight $\mu$ is given by \[p_{i}(\mu,s) =\frac{1}{\ell_{i}}\int_{0}^{\ell_{i}}p_{C_{i}}(x_{0},\mu,s)dx_{0}.\] (13) Finally, the ensemble-averaged path-length distribution function for particles born _anywhere_ in the 1-D random periodic system with direction of flight $\mu$ is given by the weighted average \[p(\mu,s) =\lambda_{1}p_{1}(\mu,s)+\lambda_{2}p_{2}(\mu,s),\] (14) where $\lambda_{i}$ is the probability that any given birth or scattering event takes place in material $i$. It is easy to see that if $\Sigma_{t1}=\Sigma_{t2}$, Eq.11 to 14 yield the exponential \[p(\mu,s)=p(s)=\Sigma_{t1}e^{-\Sigma_{t1}s},\] (15) as given in Eq.1. ### Solid-Void Medium The numerical results included in this paper are for solid-void systems. We define material 2 as the void, such that $\lambda_{2}=\Sigma_{t2}=Q_{2}=0$, $\lambda_{1}=1$, and $p(\mu,s)=p_{1}(\mu,s)$. Depending on the lengths $\ell_{i}$ of the material layers, Eq.13 yields the following expressions for $p(\mu,s)$: equationparentequation * Case 1: $\ell_{1}<\ell_{2}$ \[p(\mu,s)=\left\{\begin{array}[]{ll}\frac{\Sigma_{t1}}{\ell_{1}}( n\ell+\ell_{1}-s\|\mu\|)e^{-\Sigma_{t1}(s-n\ell_{2}/\|\mu\|)},&\text{if }n\ell\leq s \|\mu\|\leq n\ell+\ell_{1}\\ 0,&\text{if }n\ell+\ell_{1}\leq s\|\mu\|\leq n\ell+\ell_{2}\\ \frac{\Sigma_{t1}}{\ell_{1}}(s\|\mu\|-n\ell-\ell_{2})e^{-\Sigma_{t1}[s-(n+1)\ell _{2}/\|\mu\|]},&\text{if }n\ell+\ell_{2}\leq s\|\mu\|\leq(n+1)\ell\\ \end{array}\right.\] (16a) * Case 2: $\ell_{1}=\ell_{2}$ \[p(\mu,s)=\left\{\begin{array}[]{ll}\frac{\Sigma_{t1}}{\ell_{1}}( n\ell+\ell_{1}-s\|\mu\|)e^{-\Sigma_{t1}(s-n\ell_{2}/\|\mu\|)},&\text{if }n\ell\leq s \|\mu\|\leq n\ell+\ell_{1}\\ \frac{\Sigma_{t1}}{\ell_{1}}(s\|\mu\|-n\ell-\ell_{2})e^{-\Sigma_{t1}[s-(n+1)\ell _{2}/\|\mu\|]},&\text{if }n\ell+\ell_{2}\leq s\|\mu\|\leq(n+1)\ell\\ \end{array}\right.\] (16b) * Case 3: $\ell_{1}>\ell_{2}$ (16c) where $n=0,1,2,...$ . The first and second moments of $p(\mu,s)$ in Eq.16 are given by equationparentequation \[\overline{s} =\int_{0}^{\infty}sp(\mu,s)ds=\frac{\ell_{1}+\ell_{2}}{\Sigma_{t1 }\ell_{1}}\,,\] (17a) \[\overline{s^{2}}(\mu) =\int_{0}^{\infty}s^{2}p(\mu,s)ds=\frac{2\ell_{1}+4\ell_{2}}{ \Sigma_{t1}^{2}\ell_{1}}+\frac{\ell_{2}^{2}}{\Sigma_{t1}\ell_{1}\|\mu\|}\left( \frac{e^{\Sigma_{t1}\ell_{1}/\|\mu\|}+1}{e^{\Sigma_{t1}\ell_{1}/\|\mu\|}-1}\right)\,.\] (17b) We point out that the _mean free path_$\overline{s}$ does not depend on the direction $\mu$ and it is equivalent to the inverse of the volume-averaged total cross section. On the other hand, the _mean square free path_$\overline{s^{2}}$ is a function of $\|\mu\|$. Figure2 depicts examples of path-length distributions and nonclassical cross sections assuming $\Sigma_{t1}=1$ and direction of flight $\mu=\pm 1$. Figures1.vii, 1.iv and 1.i show a comparison between numerically obtained (through Monte Carlo) $p(s)$ and the analytical expressions given in Eq.16. Figures1.vii, 1.iv and 1.i show the corresponding $\Sigma_{t}(s)$ obtained with Eq.7. The “saw-tooth" behavior of $\Sigma_{t}(s)$ is consistent with the physical process and can be easily understood. For instance, in the case of $\ell_{1}=\ell_{2}=1$ (Case 2): * A particle is born or scatters in material 1. The path-length $s$ is set to 0, and $\Sigma_{t}(0)=\Sigma_{t1}=1$ * At $s=1$, the $x$-coordinate _must be_ in material 2. Thus, $\Sigma_{t}(1)=\Sigma_{t2}=0$ * At $s=2$, the $x$-coordinate _must be_ back in material 1. Thus, $\Sigma_{t}(2)=\Sigma_{t1}=1$ The exceptions would be particles born exactly at _interface points_, which form a set of measure zero. ## IV Test Problems and Models The test problems simulated in this paper consider only _rod geometry_ transport (particles can only travel in the directions $\mu=\pm 1$) taking place in a finite 1-D random periodic system with vacuum boundaries. The classical transport equation is written as equationparentequation \[\pm\frac{\partial\Psi^{\pm}}{\partial x}(x)+\Sigma_{t}(x)\Psi^{ \pm}(x)=\frac{\Sigma_{s}(x)}{2}\left[\Psi^{+}(x)+\Psi^{-}(x)\right]+\frac{Q(x) }{2}\,,\,\,\,-X\leq x\leq X,\] (18a) \[\Psi^{+}(-X)=\Psi^{-}(X)=0\,,\] (18b) where $\Psi^{\pm}(x)=\Psi(x,\mu=\pm 1)$ and the stochastic parameters $\Sigma_{t}(x)$, $\Sigma_{s}(x)$, and $Q(x)$ are given by Eq.8. We are interested in how accurately the nonclassical model predicts the ensemble-averaged scalar flux $\big{<}\Phi\big{>}$ (over all physical realizations). To this end, we compare the nonclassical results against “benchmark" results obtained by averaging the solutions of the transport equation over a large number of physical realizations of the random system. Finally, we compare the performance of the nonclassical model against the widely-known atomic mix model. We consider 2 sets of problems ($\mathcal{A}$ and $\mathcal{B}$), each divided in 3 subsets according to the choices of the lengths $\ell_{i}$ of the material layers. For each subset we present results for 12 different choices of scattering ratios ranging from purely absorbing to diffusive; namely $c_{1}\in$ {0.0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 0.95; 0.99}. We assume vacuum boundaries at $x=\pm 10$. Material 2 is defined as void, and the parameters of material 1 are given in TableI. The source $Q_{1}(x)$ is defined as \[Q_{1}(x)=\left\{\begin{array}[]{cl}q_{1},&\text{if}-0.5\leq x \leq 0.5\\ 0,&\text{otherwise}\\ \end{array}\right.\,;\] (19) that is, particles are born _near the center_ of the random system. The reason for this choice of source region can be visualized in Fig.3, in which the “wavy" pattern that arises from the periodic structure can be seen in Fig.2(a). If we allow $Q_{1}=1$ for $-X\leq x\leq X$, the solution is smoother, and the pattern is harder to identify (Fig.2(b)). ### The Benchmark Model The random quality of the 1-D system arises from its random placement in the $x$-axis. To obtain a single physical realization one can simply choose a continuous segment of two full layers (one of each material) and randomly place the coordinate $x=0$ in this segment, which also defines the boundaries $\pm X$. Given this fixed realization of the system, the cross sections and source in Eq.18 are now deterministic functions of space. We use the diamond spatial differencing scheme with mesh interval $\triangle x=2^{-7}$ to solve for the angular flux $\Psi$, obtaining the scalar flux $\Phi(x)=\Psi^{+}(x)+\Psi^{-}(x)$ (see Fig.4). This procedure is repeated for different realizations of the random system. Finally, we calculate the ensemble-averaged _benchmark_ scalar flux $\big{<}\Phi_{B}\big{>}(x)$ by averaging the resulting scalar fluxes over all physical realizations (as shown in Fig.2(a)). Clearly, the number of different realizations that can be computed is limited by the spatial discretization, with the maximum number of different realizations being $\ell/\triangle x$. For all test problems in this paper, differences in the numerical results for $\big{<}\Phi_{B}\big{>}(x)$ were negligible when increasing the number of mesh intervals and realizations. Thus, we have concluded that these benchmark results are adequately accurate for the scope of this work. ### The Atomic Mix Model The _atomic mix model_[17, 18] consists of replacing in the classical transport equation the stochastic parameters (cross sections and source) by their volume-averages. This model is known to be accurate in 1-D geometry when the material layers are optically thin. The atomic mix equation in rod geometry for the test problems in this paper is given by equationparentequation (20a) \[-X\leq x\leq X,\] \[\big{<}\Psi^{+}\big{>}(-X)=\big{<}\Psi^{-}\big{>}(X)=0\,,\] (20b) where \[\big{<}\Sigma_{t}\big{>} =P_{1}\Sigma_{t1}+P_{2}\Sigma_{t2}=\frac{\ell_{1}}{\ell}\Sigma_{t 1},\] (20c) \[\big{<}\Sigma_{s}\big{>} =P_{1}c_{1}\Sigma_{t1}+P_{2}c_{2}\Sigma_{t2}=\frac{\ell_{1}}{\ell }c_{1}\Sigma_{t1},\] (20d) \[\big{<}Q\big{>}(x) =P_{1}Q_{1}(x)+P_{2}Q_{2}(x)=\frac{\ell_{1}}{\ell}Q_{1}(x).\] (20e) We solve Eq.20 for the ensembled-averaged angular flux $\big{<}\Psi\big{>}$ using a diamond spatial differencing scheme with mesh interval $\triangle x=2^{-7}$. The ensemble-averaged _atomic mix_ scalar flux is given by $\big{<}\Phi_{AM}\big{>}(x)=\big{<}\Psi^{+}\big{>}(x)+\big{<}\Psi^{-}\big{>}(x)$. An example is depicted in Fig.5. ### The Nonclassical Model For the rod geometry test problems included in this work, we rewrite the nonclassical Eq.6 in an initial value form (cf. [4]) as equationparentequation \[\frac{\partial\psi^{\pm}}{\partial s}(x,s)\pm\frac{\partial\psi^{ \pm}}{\partial x}(x,s)+\Sigma_{t}(s)\psi^{\pm}(x,s)=0,\,\,\,-X\leq x\leq X,\, \,s>0\] (21a) (21b) \[\psi^{+}(-X,s)=\psi^{-}(X,s)=0\,,\,\,\,s\geq 0\,,\] (21c) where $\psi^{\pm}(x,s)=\psi(x,\mu=\pm 1,s)$, $\big{<}Q\big{>}(x)$ is given by Eq.20e, and the nonclassical cross section $\Sigma_{t}(s)=\Sigma(\mu=\pm 1,s)$ is given by Eqs.16 and 7 (see Fig.2). For the numerical solution of this system, we can interpret the path-length $s$ as a pseudo-time variable. We then solve Eq.21 using a finite volume method with explicit pseudo-time discretization according to [19]. Specifically, we adapt the scheme introduced in [14] for moment models of the nonclassical transport equation. This method is of first order in the pseudo-time variable $s$ and in the spatial variable $x$. We choose a uniform grid $(x_{m},s^{n})$, where $x_{m+1}=x_{m}+\Delta x$ for all $m\in\mathbb{Z}$, and $s^{n+1}=s^{n}+\Delta s$ for all $n\in\mathbb{N}_{0}$. Furthermore, we define $\psi_{m}^{n,\pm}=\psi^{\pm}(x_{m},s^{n})$, $Q_{m}=\big{<}Q\big{>}(x_{m})$, and $\Sigma_{t}^{n}=\Sigma_{t}(s^{n})$. The fully discretized system reads equationparentequation \[\frac{\psi^{n+1,\pm}_{m}-\psi^{n,\pm}_{m}}{\Delta s}\pm\frac{\psi ^{n,\pm}_{m+1}-\psi_{m-1}^{n,\pm}}{2\Delta x}-\frac{\psi^{n,\pm}_{m+1}-2\psi_{ m}^{n,\pm}+\psi_{m-1}^{n,\pm}}{2\Delta x}+\Sigma_{t}^{n}\psi^{n,\pm}_{m}=0,\] (22a) \[\psi_{m}^{0,\pm}=\frac{c}{2}\sum\limits_{n=0}^{\infty}\omega_{n} \Sigma_{t}^{n}\left(\psi_{m}^{n,+}+\psi^{n,-}_{m}\right)+\frac{Q_{m}}{2},\] (22b) for some infinite quadrature rule given by the weights $\omega_{n}$. The second order central differences arise as a numerical diffusion term, which is typical for HLL finite volume schemes. In our calculations we cut off the integration at $s_{\text{max}}=4X=40$ and use the trapezoidal rule. We use the same mesh interval $\triangle x=2^{-7}$ as for the previous models, and a CFL number $0.5$ (that is, $\triangle s=2^{-8}$). Because of the coupling of the initial value to the full solution in Eq.21, this system is solved in a source-iteration manner, where we iterate between Eqs.22b and 22a. Finally, the ensemble-averaged _nonclassical_ scalar flux is given by $\big{<}\Phi_{NC}\big{>}(x)=\int_{0}^{40}[\psi^{+}(x,s)+\psi^{-}(x,s)]ds$. An example is depicted in Fig.6. It was shown in [14] that the contraction rate for the source iteration is given by the scattering ratio $c$. The maximum number of source iterations to converge the solution in problem set $\mathcal{A}$ was 417 (problem $\mathcal{A}_{3}$ with $c_{1}=0.99$); and in problem set $\mathcal{B}$ was 251 (problem $\mathcal{B}_{3}$ with $c_{1}=0.99$). ## V Numerical Results The atomic mix model inherently approximates the path-length distribution function by the exponential $p(s)=\big{<}\Sigma_{t}\big{>}e^{-\big{<}\Sigma_{t}\big{>}s}$. The nonclassical model uses the correct $p(\mu,s)$ that was analytically obtained in Eq.16. In this section we compare the accuracy of these two models in predicting the benchmark solutions obtained for the test problem sets $\mathcal{A}$ and $\mathcal{B}$. For a better analysis of these results, we define the relative errors of the models with respect to the benchmark solutions as equationparentequation \[Err_{AM}\] (23a) \[Err_{NC}\] (23b) ### Problem Set $\mathcal{A}$ The lengths of the material 1 layers in this set are the same order as a mean free path; that is, $\ell_{1}\Sigma_{t1}=O(1)$. It has been shown [20] that, in the diffusive asymptotic limit, the diffusion coefficient of such problems is correctly estimated by the atomic mix model. For the rod geometry problems in set $\mathcal{A}$, this diffusion coefficient is given by \[D=\frac{\ell_{1}+\ell_{2}}{\Sigma_{t1}\ell_{1}}=\frac{1}{\big{<} \Sigma_{t}\big{>}}=\left\{\begin{array}[]{cl}3.0&\text{for set $\mathcal{A}_{1 }$}\\ 2.0&\text{for set $\mathcal{A}_{2}$}\\ 1.5&\text{for set $\mathcal{A}_{3}$}\\ \end{array}\right.\,.\] (24) Therefore, we expect the atomic mix predictions of the ensemble-averaged scalar flux to _improve_ as the scattering ratio increases and the system becomes more diffusive. On the other hand, the diffusion coefficient obtained by applying the same asymptotic analysis to the the nonclassical equation (see AppendixA) is given by \[D_{NC}=\frac{1}{2}\frac{\overline{s^{2}}}{\overline{s}}\approx \left\{\begin{array}[]{cl}3.0277&\text{for set $\mathcal{A}_{1}$}\\ 2.0410&\text{for set $\mathcal{A}_{2}$}\\ 1.5137&\text{for set $\mathcal{A}_{3}$}\\ \end{array}\right.\,,\] (25) where $\overline{s}$ and $\overline{s^{2}}$ are defined in Eq.17. The solution of the nonclassical transport equation has been shown to converge to the solution of the nonclassical diffusion equation in the diffusive asymptotic limit [21]. Thus, we expect the nonclassical predictions of the ensemble-averaged scalar flux to _deteriorate_ as the system becomes diffusive, underestimating the correct solution. Figure7 depicts the ensemble-averaged scalar fluxes obtained with each model for the purely absorbing case (Figs.6.vii, 6.iv and 6.i) and for the diffusive case $c_{1}=0.99$ (Figs.6.vii, 6.iv and 6.i). The benchmark solutions present a sinuous shape due to the periodic structure of the random systems. This pattern becomes less noticeable as the solid/void ratio increases, and as the system becomes more diffusive. It is important to point out that the nonclassical model is able to capture this sinuous behavior, while the atomic mix model yields a smooth curve. It is easier to analyze the accuracy of these models by examining the relative errors to the benchmark solution. Figures13, 12, 11, 10, 9 and 8 show the (absolute) percentage error of the nonclassical and atomic mix predictions of the ensemble-averaged scalar flux with respect to the benchmark solutions. The error plots confirm the theoretical predictions; atomic mix becomes more accurate as the system becomes more diffusive, while the accuracy of the nonclassical model decreases. The nonclassical model clearly outperforms atomic mix for all the problems in $\mathcal{A}_{1}$ and for most of the problems in sets $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$. The exceptions take place for the cases $c_{1}=0.95$ and $c_{1}=0.99$, in which the accuracy of the atomic mix model overtakes that of the nonclassical. TablesIV, III and II show that the nonclassical model tends to underestimate the scalar flux, while atomic mix overestimates the solution. The nonclassical model never reaches an error larger than 3.7% in estimating the solutions’ peak (at $x=0$). On the other hand, the atomic mix estimate exceeds 5% error in several problems, reaching a maximum of 8.24%. It can also be seen from the results at the boundaries that the atomic mix model generates a solution with a large tail and it greatly overestimates the outgoing flux, in some problems by several orders of magnitude. The nonclassical model, however, never reaches an error larger than 4.7%. ### Problem Set $\mathcal{B}$ Following the work presented in SectionIII.A, Fig.14 shows the path-length distributions and nonclassical cross sections of problem set $\mathcal{B}$. We have chosen the parameters of this set such that: * The optical thickness of each layer of material 1 is one order of magnitude larger than a mean free path: $\ell_{1}\Sigma_{t1}=10$; * The volume-averaged parameters remain the same in all problems in the set: $\big{<}\Sigma_{t}\big{>}=\big{<}q_{1}\big{>}=0.5$. The large optical thickness implies that the problems in this set are _not_ the type of problems for which the atomic mix model is known to yield the correct aymptotic diffusive limit. By fixing the volume-averaged parameters, the atomic mix model will yield exactly the same ensemble-averaged scalar flux for all problems in set $\mathcal{B}$ (which is the same as in $\mathcal{A}_{2}$). The goal is to investigate whether the nonclassical model will outperform atomic mix for the diffusive cases. Figure15 depicts the ensemble-averaged scalar fluxes obtained with each model for the purely absorbing case (Figs.14.vii, 14.iv and 14.i) and for the diffusive case $c_{1}=0.99$ (Figs.14.vii, 14.iv and 14.i). The sinuous pattern of the benchmark solution is easier to notice in set $\mathcal{B}_{3}$, with the largest solid/void ratio. As in the case in set $\mathcal{A}$, the nonclassical model is able to capture the sinuous behavior. The atomic mix model generates the same smooth solution for each choice of $c_{1}$, unable to capture the differences in the scalar flux caused by the different choices of $\ell_{i}$, $\Sigma_{ti}$, and $q_{i}$. Figures21, 20, 19, 18, 17 and 16 show the percentage error of the nonclassical and atomic mix predictions of the ensemble-averaged scalar flux with respect to the benchmark solutions _in logarithmic scale_. The changes in the accuracy of both models have a different pattern than in problem set $\mathcal{A}$. The atomic mix solutions tend to grossly overestimate the ensemble-averaged scalar flux in most of the system, with errors at $x=0$ reaching 36% as seen in TablesVII, VI and V. Once $x$ approaches the boundaries, the atomic mix model systematically underestimates the solution, with errors in the outgoing flux exceeding 50% in most test problems and reaching over 80% in the least diffusive systems. Once again, the nonclassical model underestimates the solution in diffusive systems. For most problems the nonclassical error in estimating the ensemble-averaged scalar flux at $x=0$ is less than 4%. The exceptions are the most diffusive problems, with scattering ratios $c_{1}=0.95$ and $c_{1}=0.99$. Nevertheless, even in these diffusive cases the nonclassical model greatly outperforms the atomic mix approach. ## VI Conclusion This work presents an investigation of the accuracy of the nonclassical transport theory in estimating the ensemble-averaged scalar flux in 1-D random periodic media. The analytical portion of the paper considers transport in a _slab_ consisting of alternating layers of any 2 materials. The following simplifying assumptions are made for the numerical simulations: (i) the 1-D system is a periodic arrangement of _solid and void_ layers randomly placed in the $x$-axis; and (ii) particle transport takes place in _rod geometry_. This paper is an expanded version of a recent conference paper [15], in which numerical solutions for the nonclassical transport equation were provided for the first time. A total of 72 test problems are analyzed. We show that the nonclassical theory greatly outperforms the atomic mix model in estimating the ensemble-averaged scalar flux for most problems and that it qualitatively preserves the sinuous shape of the solution. The few cases in which atomic mix is more accurate are part of a class of diffusive problems in which the atomic mix model is known to converge to the correct diffusive limit (diffusive problems in set $\mathcal{A}$). In this small subset of problems the nonclassical model converges to a diffusion solution with an unphysically large diffusion coefficient, causing the nonclassical solution to underestimate the ensemble-averaged scalar flux. However, for diffusive problems that are _not_ in the atomic mix limit (set $\mathcal{B}$), the nonclassical model is clearly superior to the atomic mix approach. This gain in accuracy comes at a cost: the path-length distribution function $p(s)$ (and its corresponding $\Sigma_{t}(s)$) must be known in order to solve the nonclassical transport equation. Despite the extra work, it is our expectation that the gain in accuracy will prove the effort worthwhile in the important nuclear system where nonclassical transport takes place, such as in Pebble Bed and Boiling Water reactor cores. In particular, the nonclassical theory represents an alternative to current methods that might yield more accurate estimates of the eigenvalue and eigenfunction in a criticality calculation. Future work includes (i) performing a thorough numerical investigation of the nonclassical theory in slab geometry to further validate our analytical results; (ii) comparing the gain in accuracy against other models and experimental data; and (iii) dropping the periodic assumption to investigate results in more realistic random media. We point out that step (iii) cannot be performed with the analytical approach to obtain the path-lengths presented in this paper. It requires either a numerical approach to estimate $p(\mu,s)$, or a (much) more complex mathematical theory. ## Acknowledgments This paper was prepared by Richard Vasques and Rachel Slaybaugh under award number NRC-HQ-84-14-G-0052 from the Nuclear Regulatory Commission. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U.S. Nuclear Regulatory Commission. APPENDIX ## Appendix A 1-D Asymptotic Analysis Following [4], we scale the parameters of Eq.6 such that $\Sigma_{t}=O(1)$, $1-c=O(\varepsilon^{2})$, $Q=O(\varepsilon^{2})$, $\partial\psi/\partial s=O(1)$, and $\mu\partial\psi/\partial x=O(\varepsilon)$, with $\varepsilon\ll 1$. In this scaling, Eq.6 becomes \[\frac{\partial\psi}{\partial s}(x,\mu,s)+\varepsilon\mu\frac{ \partial\psi}{\partial x}(x,\mu,s)+\Sigma_{t}(\mu,s)\psi(x,\mu,s)=\] (26) \[\quad\quad=\frac{\delta(s)}{2}\int_{-1}^{1}\int_{0}^{\infty}[1- \varepsilon^{2}(1-c)]\Sigma_{t}(\mu^{\prime},s^{\prime})\psi(x,\mu^{\prime},s^ {\prime})\,ds^{\prime}d\mu^{\prime}++\varepsilon^{2}\delta(s)\frac{Q(x)}{2}\,.\] Let us define $\hat{\psi}(x,\mu,s)$ such that \[\psi(x,\mu,s) \equiv\hat{\psi}(x,\mu,s)\frac{e^{-\int_{0}^{s}\Sigma_{t}(\mu,s^{ \prime})ds^{\prime}}}{\overline{s}}\,,\] (27) where $\overline{s}=\frac{1}{2}\int_{-1}^{1}\int_{0}^{\infty}sp(\mu,s)dsd\mu$. Then, using Eq.4, Eq.27 becomes the following equation for $\hat{\psi}(x,\mu,s)$: \[\frac{\partial\hat{\psi}}{\partial s}(x,\mu,s)+\varepsilon\mu \frac{\partial\hat{\psi}}{\partial x}(x,\mu,s)=\] (28) \[\quad=\frac{\delta(s)}{2}\int_{-1}^{1}\int_{0}^{\infty}[1- \varepsilon^{2}(1-c)]p(\mu^{\prime},s^{\prime})\hat{\psi}(x,\mu^{\prime},s^{ \prime})\,ds^{\prime}d\mu^{\prime}+\varepsilon^{2}\delta(s)\overline{s}\frac{Q (x)}{2}\,.\] This equation is mathematically equivalent to: equationparentequation \[\frac{\partial\hat{\psi}}{\partial s}(x,\mu,s)+\varepsilon\mu \frac{\partial\hat{\psi}}{\partial x}(x,\mu,s)=0\,,\quad s>0\,,\] (29a) \[\hat{\psi}(x,\mu,0)=\frac{1}{2}\int_{-1}^{1}[1-\varepsilon^{2}(1- c)]\int_{0}^{\infty}p(\mu^{\prime},s^{\prime})\hat{\psi}(x,\mu^{\prime},s^{ \prime})ds^{\prime}d\mu^{\prime}+\varepsilon^{2}\overline{s}\frac{Q(x)}{2}\,,\] (29b) where $\hat{\psi}(x,\mu,0)=\hat{\psi}(x,\mu,0^{+})$. Integrating Eq.29a over $0<s^{\prime}<s$ we obtain: \[\hat{\psi}(x,\mu,s) =\hat{\psi}(x,\mu,0)-\varepsilon\mu\frac{\partial}{\partial x} \int_{0}^{s}\hat{\psi}(x,\mu,s^{\prime})\,ds^{\prime}\] (30) \[=\frac{1}{2}\int_{-1}^{1}[1-\varepsilon^{2}(1-c)]\int_{0}^{\infty }p(\mu^{\prime},s^{\prime})\hat{\psi}(x,\mu^{\prime},s^{\prime})ds^{\prime}d \mu^{\prime}+\] \[+\varepsilon^{2}\overline{s}\frac{Q (x)}{2}-\varepsilon\mu\frac{\partial}{\partial x}\int_{0}^{s}\hat{\psi}(x,\mu, s^{\prime})\,ds^{\prime}\,.\] Introducing into this equation the ansatz \[\hat{\psi}(x,\mu,s)=\sum_{n=0}^{\infty}\varepsilon^{n}\hat{\psi}_{n}(x,\mu,s)\] (31) and equating the coefficients of different powers of $\varepsilon$, we obtain for $n\geq 0$: \[\hat{\psi}_{n}(x,\mu,s) =\frac{1}{2}\int_{-1}^{1}\int_{0}^{\infty}p(\mu^{\prime},s^{ \prime})\hat{\psi}_{n}(x,\mu^{\prime},s^{\prime})ds^{\prime}d\mu^{\prime}-\mu \frac{\partial}{\partial x}\int_{0}^{s}\hat{\psi}_{n-1}(x,\mu,s^{\prime})\,ds^ {\prime}\] (32) \[\quad\quad-\frac{1-c}{2}\int_{-1}^{1}\int_{0}^{\infty}p(\mu^{ \prime},s^{\prime})\hat{\psi}_{n-2}(x,\mu^{\prime},s^{\prime})ds^{\prime}d\mu^ {\prime}+\delta_{n,2}\overline{s}\frac{Q(x)}{2}\,,\] with $\hat{\psi}_{-1}=\hat{\psi}_{-2}=0$. Equation32 with $n=0$ has the general solution \[\hat{\psi}_{0}(x,\mu,s)=\frac{\hat{\phi}_{0}(x)}{2}\,,\] (33) where $\hat{\phi}_{0}(x)$ is undetermined at this point. For $n=1$, Eq.32 has a particular solution of the form: \[\hat{\psi}^{part}_{1}(x,\mu,s)=-\frac{s\mu}{2}\frac{d\hat{\phi}_{0}}{dx}(x)\,,\] (34) and its general solution is given by \[\hat{\psi}_{1}(x,\mu,s)=\frac{1}{2}\left[\hat{\phi}_{1}(x)-s\mu\frac{d\hat{ \phi}_{0}}{dx}(x)\right]\,,\] (35) where $\hat{\phi}_{1}(x)$ is undetermined. Equation32 with $n=2$ has a solvability condition, which is obtained by operating on it by $\int_{-1}^{1}\int_{0}^{\infty}p(\mu,s)(\cdot)dsd\mu$; the solvability condition yields \[0=\frac{1}{2}\int_{-1}^{1}\int_{0}^{\infty}p(\mu,s) \left(\frac{(s\mu)^{2}}{2}\frac{d^{2}\hat{\phi}_{0}}{dx^{2}}(x) \right)dsd\mu\] (36) \[-\frac{1-c}{2}\int_{-1}^{1}\int_{0}^{\infty}p(\mu,s)\hat{\phi}_{0 }(x)\,dsd\mu+\overline{s}Q(x)\,.\] Thus, using the fact that $\int_{0}^{\infty}p(\mu,s)ds=1$, we can rewrite Eq.36 as: equationparentequation \[-D_{NC}\frac{d^{2}\hat{\phi}_{0}}{dx^{2}}(x)+\frac{1-c}{\overline {s}}\hat{\phi}_{0}(x)=Q(x)\,,\] (37a) where \[D_{NC}\] is the nonclassical diffusion coefficient given by \[D_{NC}=\frac{1}{4\overline{s}}\int_{-1}^{1}\mu^{2}\int_{0}^{ \infty}s^{2}p(\mu,s)dsd\mu\,.\] (37b) Therefore, the solution $\psi(x,\mu,s)$ of Eq.28 satisfies \[\psi(x,\mu,s)=\frac{\hat{\phi}_{0}(x)}{2}\frac{e^{-\int_{0}^{s}\Sigma_{t}(\mu, s^{\prime})ds^{\prime}}}{\overline{s}}+O(\varepsilon)\,,\] (38) where $\hat{\phi}_{0}(x)$ satisfies Eq.37. The classical angular flux can be obtained to leading order by integrating Eq.38 over $0<s<\infty$. For transport in _rod geometry_, Eq.37b yields \[D_{NC}=\frac{1}{2}\frac{\overline{s^{2}}}{\overline{s}},\] (39) where $\overline{s^{2}}=\int_{0}^{\infty}s^{2}p(s)ds$. ## References * [1] E. LARSEN, “A Generalized Boltzmann Equation for Non-Classical Particle Transport,” in _Proc. International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications_, Monterey, CA, Apr. 15-19, 2007. * [2] E. LARSEN and R. VASQUES, “A Generalized Linear Boltzmann Equation for Non-Classical Particle Transport,” _J. Quant. Spectrosc. Radiat. Transfer_, 112, 619 (2011). * [3] M. FRANK and T. GOUDON, “On a generalized Boltzmann equation for non-classical particle transport,” _Kin. Rel. Models_, 3, 395 (2010). * [4] R. VASQUES and E. LARSEN, “Non-classical particle transport with angular-dependent path-length distributions. I: Theory,” _Ann. Nucl. Energy_, 70, 292 (2014). * [5] F. GOLSE, “Recent Results on the Periodic Lorentz Gas,” in _Nonlinear Partial Differential Equations_, edited by X. CABRÉ and J. SOLER, pages 39–99, Springer Basel, New York, NY, 2012. * [6] J. MARKLOF and A. STROMBERGSSON, “The Boltzmann-Grad limit of the periodic Lorentz gas,” _Ann. Nucl. Math._, 174, 225 (2011). * [7] J. MARKLOF and A. STROMBERGSSON, “Generalized linear Boltzmann equations for particle transport in polycrystals,” _Appl. Math. Res. Express AMRX_, 2, 274 (2015). * [8] C. GROSJEAN, _The Exact Mathematical Theory of Multiple Scattering of Particles in an In- finite Medium_, Verh. Vlaamsche Akad. Wet. Lett. Schoone Kunsten Belgie 36, Paleis der Academien, Brussels (1951). * [9] M. FRANK, K. KRYCKI, E. LARSEN, and R. VASQUES, “The nonclassical Boltzmann equation and diffusion-based approximations to the Boltzmann equation,” _SIAM J. Appl. Math._, 75, 1329 (2015). * [10] R. VASQUES, “The nonclassical diffusion approximation to the nonclassical linear Boltzmann equation,” _Appl. Math. Lett._, 53, 63 (2016). * [11] R. VASQUES and E. LARSEN, “Anisotropic Diffusion in Model 2-D Pebble-Bed Reactor Cores,” in _Proc. International Conference on Mathematics, Computational Methods & Reactor Physics_, Saratoga Springs, NY, May 03-07, 2009. * [12] R. VASQUES, “Estimating Anisotropic Diffusion of Neutrons Near the Boundary of a Pebble Bed Random System,” in _Proc. International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering_, Sun Valley, ID, May 05-09, 2013. * [13] R. VASQUES and E. LARSEN, “Non-classical particle transport with angular-dependent path-length distributions. II: Application to pebble bed reactor cores,” _Ann. Nucl. Energy_, 70, 301 (2014). * [14] K. KRYCKI, R. TURPAULT, M. FRANK, and C. BERTHON, “Asymptotic preserving numerical schemes for a non-classical radiation transport model for atmospheric clouds,” _Math. Meth. Appl. Sci._, 36, 2101 (2013). * [15] R. VASQUES and K. KRYCKI, “On the accuracy of the non-classical transport equation in 1-D random periodic media,” in _Joint International Conference on Mathematics and Computation, Supercomputing in Nuclear Applications and the Monte Carlo Method_, Nashville, TN, Apr. 19-23, 2015. * [16] O. ZUCHUAT, R. SANCHEZ, I. ZMIJAREVIC, and F. MALVAGI, “Transport in Renewal Statistical Media: Benchmarking and Comparison with Models,” _J. Quant. Spectrosc. Radiat. Transfer_, 51, 689 (1994). * [17] G. POMRANING, _Linear Kinetic Theory and Particle Transport in Stochastic Mixtures_, World Scientific Press, Singapore (1991). * [18] L. DUMAS and F. GOLSE, “Homogenization of Transport Equations,” _SIAM J. Appl. Math._, 60, 1447 (2000). * [19] A. HARTEN, P. LAX, and B. VAN LEER, “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,” _SIAM Review_, 25, 35 (1983). * [20] E. LARSEN, R. VASQUES, and M. VILHENA, “Particle transport in the 1-D diffusive atomic mix limit,” in _Proc. Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications Meeting_, Avignon, France, Sep. 12-15, 2005. * [21] R. VASQUES, R. SLAYBAUGH, and K. KRYCKI, “Nonclassical Particle Transport in the 1-D Diffusive Limit,” _arXiv:1601.02495 [nucl-th]_ (2016). Set \| ℓ1 \| ℓ2 \| Σt1 \| q1 \| Set \| ℓ1 \| ℓ2 \| Σt1 \| q1 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- A1 \| 0.5 \| 1.0 \| 1.0 \| 1.0 \| B1 \| 20/3 \| 40/3 \| 1.5 \| 1.5 A2 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| B2 \| 10 \| 10 \| 1.0 \| 1.0 A3 \| 1.0 \| 0.5 \| 1.0 \| 1.0 \| B3 \| 40/3 \| 20/3 \| 0.75 \| 0.75 Table I: Parameters of test problems \| c \| ⟨ϕB⟩ \| ⟨ϕAM⟩ \| ⟨ϕNC⟩ \| ErrAM \| ErrNC ---\|---\|---\|---\|---\|---\|--- \| 0.0 \| 0.1420 \| 0.1537 \| 0.1421 \| 0.0824 \| 0.0006 \| 0.1 \| 0.1509 \| 0.1628 \| 0.1509 \| 0.0787 \| 0.0002 \| 0.2 \| 0.1614 \| 0.1734 \| 0.1613 \| 0.0747 \| -0.0001 \| 0.3 \| 0.1740 \| 0.1862 \| 0.1738 \| 0.0706 \| -0.0006 \| 0.4 \| 0.1895 \| 0.2021 \| 0.1893 \| 0.0662 \| -0.0012 x=0 \| 0.5 \| 0.2094 \| 0.2223 \| 0.2091 \| 0.0616 \| -0.0019 \| 0.6 \| 0.2360 \| 0.2493 \| 0.2353 \| 0.0567 \| -0.0026 \| 0.7 \| 0.2735 \| 0.2876 \| 0.2725 \| 0.0515 \| -0.0036 \| 0.8 \| 0.3316 \| 0.3469 \| 0.3300 \| 0.0462 \| -0.0048 \| 0.9 \| 0.4360 \| 0.4541 \| 0.4333 \| 0.0413 \| -0.0063 \| 0.95 \| 0.5287 \| 0.5496 \| 0.5249 \| 0.0397 \| -0.0072 \| 0.99 \| 0.6472 \| 0.6728 \| 0.6421 \| 0.0395 \| \- 0.0079 \| 0.0 \| 0.0063 \| 0.0071 \| 0.0061 \| 0.1294 \| -0.0326 \| 0.1 \| 0.0076 \| 0.0085 \| 0.0074 \| 0.1128 \| -0.0313 \| 0.2 \| 0.0093 \| 0.0103 \| 0.0091 \| 0.0972 \| -0.0301 \| 0.3 \| 0.0116 \| 0.0126 \| 0.0113 \| 0.0826 \| -0.0289 \| 0.4 \| 0.0148 \| 0.0158 \| 0.0143 \| 0.0693 \| -0.0278 x=10 \| 0.5 \| 0.0191 \| 0.0202 \| 0.0186 \| 0.0571 \| -0.0267 \| 0.6 \| 0.0255 \| 0.0267 \| 0.0248 \| 0.0464 \| -0.0256 \| 0.7 \| 0.0354 \| 0.0367 \| 0.0345 \| 0.0371 \| -0.0244 \| 0.8 \| 0.0520 \| 0.0535 \| 0.0508 \| 0.0297 \| -0.0231 \| 0.9 \| 0.0841 \| 0.0863 \| 0.0823 \| 0.0251 \| -0.0216 \| 0.95 \| 0.1141 \| 0.1169 \| 0.1117 \| 0.0246 \| -0.0206 \| 0.99 \| 0.1533 \| 0.1573 \| 0.1503 \| 0.0259 \| -0.0196 Table II: Ensemble-averaged scalar fluxes for problem set A1 \| c \| ⟨ϕB⟩ \| ⟨ϕAM⟩ \| ⟨ϕNC⟩ \| ErrAM \| ErrNC ---\|---\|---\|---\|---\|---\|--- \| 0.0 \| 0.2049 \| 0.2213 \| 0.2048 \| 0.0798 \| -0.0006 \| 0.1 \| 0.2181 \| 0.2347 \| 0.2179 \| 0.0760 \| -0.0009 \| 0.2 \| 0.2337 \| 0.2506 \| 0.2334 \| 0.0720 \| -0.0013 \| 0.3 \| 0.2527 \| 0.2698 \| 0.2522 \| 0.0677 \| -0.0019 \| 0.4 \| 0.2762 \| 0.2936 \| 0.2755 \| 0.0631 \| -0.0026 x=0 \| 0.5 \| 0.3065 \| 0.3243 \| 0.3054 \| 0.0582 \| -0.0035 \| 0.6 \| 0.3475 \| 0.3658 \| 0.3458 \| 0.0527 \| -0.0049 \| 0.7 \| 0.4072 \| 0.4263 \| 0.4045 \| 0.0467 \| -0.0069 \| 0.8 \| 0.5054 \| 0.5255 \| 0.5003 \| 0.0398 \| -0.0100 \| 0.9 \| 0.7067 \| 0.7291 \| 0.6950 \| 0.0316 \| -0.0165 \| 0.95 \| 0.9254 \| 0.9502 \| 0.9035 \| 0.0267 \| -0.0237 \| 0.99 \| 1.2915 \| 1.3204 \| 1.2451 \| 0.0223 \| -0.0359 \| 0.0 \| 0.0017 \| 0.0033 \| 0.0018 \| 0.9112 \| 0.0057 \| 0.1 \| 0.0023 \| 0.0040 \| 0.0023 \| 0.7419 \| 0.0058 \| 0.2 \| 0.0031 \| 0.0049 \| 0.0031 \| 0.5953 \| 0.0063 \| 0.3 \| 0.0043 \| 0.0063 \| 0.0043 \| 0.4695 \| 0.0070 \| 0.4 \| 0.0060 \| 0.0082 \| 0.0060 \| 0.3628 \| 0.0075 x=10 \| 0.5 \| 0.0087 \| 0.0111 \| 0.0088 \| 0.2733 \| 0.0077 \| 0.6 \| 0.0132 \| 0.0158 \| 0.0133 \| 0.1993 \| 0.0072 \| 0.7 \| 0.0211 \| 0.0241 \| 0.0212 \| 0.1390 \| 0.0055 \| 0.8 \| 0.0371 \| 0.0405 \| 0.0372 \| 0.0910 \| 0.0016 \| 0.9 \| 0.0769 \| 0.0811 \| 0.0764 \| 0.0536 \| -0.0070 \| 0.95 \| 0.1257 \| 0.1305 \| 0.1236 \| 0.0384 \| -0.0162 \| 0.99 \| 0.2126 \| 0.2185 \| 0.2061 \| 0.0278 \| -0.0305 Table III: Ensemble-averaged scalar fluxes for problem set A2 \| c \| ⟨ϕB⟩ \| ⟨ϕAM⟩ \| ⟨ϕNC⟩ \| ErrAM \| ErrNC ---\|---\|---\|---\|---\|---\|--- \| 0.0 \| 0.2732 \| 0.2835 \| 0.2730 \| 0.0376 \| -0.0007 \| 0.1 \| 0.2908 \| 0.3012 \| 0.2905 \| 0.0359 \| -0.0010 \| 0.2 \| 0.3117 \| 0.3223 \| 0.3112 \| 0.0341 \| -0.0013 \| 0.3 \| 0.3369 \| 0.3477 \| 0.3363 \| 0.321 \| -0.0018 \| 0.4 \| 0.3683 \| 0.3793 \| 0.3674 \| 0.0300 \| -0.0023 x=0 \| 0.5 \| 0.4087 \| 0.4201 \| 0.4075 \| 0.0277 \| -0.0031 \| 0.6 \| 0.4637 \| 0.4754 \| 0.4618 \| 0.0252 \| -0.0041 \| 0.7 \| 0.5442 \| 0.5564 \| 0.5412 \| 0.0224 \| -0.0056 \| 0.8 \| 0.6788 \| 0.6919 \| 0.6733 \| 0.0192 \| -0.0081 \| 0.9 \| 0.9715 \| 0.9868 \| 0.9582 \| 0.0157 \| -0.0137 \| 0.95 \| 1.3295 \| 1.3481 \| 1.3018 \| 0.0140 \| -0.0209 \| 0.99 \| 2.0777 \| 2.1055 \| 2.0011 \| 0.0134 \| -0.0369 \| 0.0 \| 0.0004 \| 0.0026 \| 0.0004 \| 4.8188 \| -0.0070 \| 0.1 \| 0.0006 \| 0.0030 \| 0.0006 \| 3.6072 \| -0.0073 \| 0.2 \| 0.0009 \| 0.0034 \| 0.0009 \| 2.6478 \| -0.0073 \| 0.3 \| 0.0014 \| 0.0041 \| 0.0014 \| 1.8989 \| -0.0071 \| 0.4 \| 0.0022 \| 0.0052 \| 0.0022 \| 1.3238 \| -0.0070 x=10 \| 0.5 \| 0.0036 \| 0.0068 \| 0.0036 \| 0.8906 \| -0.0070 \| 0.6 \| 0.0062 \| 0.0097 \| 0.0061 \| 0.5718 \| -0.0076 \| 0.7 \| 0.0114 \| 0.0153 \| 0.0113 \| 0.3438 \| -0.0090 \| 0.8 \| 0.0237 \| 0.0282 \| 0.0235 \| 0.1868 \| -0.0121 \| 0.9 \| 0.0618 \| 0.0670 \| 0.0605 \| 0.0847 \| -0.0196 \| 0.95 \| 0.1198 \| 0.1259 \| 0.1164 \| 0.0502 \| -0.0287 \| 0.99 \| 0.2570 \| 0.2647 \| 0.2449 \| 0.0302 \| -0.0469 Table IV: Ensemble-averaged scalar fluxes for problem set A3 \| c \| ⟨ϕB⟩ \| ⟨ϕAM⟩ \| ⟨ϕNC⟩ \| ErrAM \| ErrNC ---\|---\|---\|---\|---\|---\|--- \| 0.0 \| 0.1776 \| 0.2213 \| 0.1768 \| 0.2459 \| -0.0045 \| 0.1 \| 0.1896 \| 0.2347 \| 0.1892 \| 0.2379 \| -0.0018 \| 0.2 \| 0.2037 \| 0.2506 \| 0.2039 \| 0.2302 \| 0.0009 \| 0.3 \| 0.2206 \| 0.2698 \| 0.2214 \| 0.2229 \| 0.0036 \| 0.4 \| 0.2414 \| 0.2936 \| 0.2329 \| 0.2163 \| 0.0063 x=0 \| 0.5 \| 0.2678 \| 0.3243 \| 0.2700 \| 0.2111 \| 0.0085 \| 0.6 \| 0.3027 \| 0.3658 \| 0.3056 \| 0.2085 \| 0.0098 \| 0.7 \| 0.3520 \| 0.4263 \| 0.3550 \| 0.2108 \| 0.0084 \| 0.8 \| 0.4294 \| 0.5255 \| 0.4293 \| 0.2237 \| -0.0002 \| 0.9 \| 0.5772 \| 0.7291 \| 0.5585 \| 0.2630 \| -0.0325 \| 0.95 \| 0.7271 \| 0.9502 \| 0.6710 \| 0.3067 \| -0.0771 \| 0.99 \| 0.9650 \| 1.3204 \| 0.8160 \| 0.3683 \| -0.1545 \| 0.0 \| 0.0250 \| 0.0033 \| 0.0248 \| -0.8667 \| -0.0075 \| 0.1 \| 0.0270 \| 0.0040 \| 0.0273 \| -0.8515 \| 0.0079 \| 0.2 \| 0.0295 \| 0.0049 \| 0.0302 \| -0.8324 \| 0.0248 \| 0.3 \| 0.0325 \| 0.0063 \| 0.0339 \| -0.8078 \| 0.0436 \| 0.4 \| 0.0364 \| 0.0082 \| 0.0387 \| -0.7756 \| 0.0643 x=10 \| 0.5 \| 0.0414 \| 0.0111 \| 0.0450 \| -0.7325 \| 0.0871 \| 0.6 \| 0.0483 \| 0.0158 \| 0.0537 \| -0.6734 \| 0.1114 \| 0.7 \| 0.0587 \| 0.0241 \| 0.0666 \| -0.5899 \| 0.1355 \| 0.8 \| 0.0760 \| 0.0405 \| 0.0877 \| -0.4677 \| 0.1532 \| 0.9 \| 0.1126 \| 0.0811 \| 0.1284 \| -0.2799 \| 0.1408 \| 0.95 \| 0.1529 \| 0.1305 \| 0.1676 \| -0.1463 \| 0.0966 \| 0.99 \| 0.2209 \| 0.2185 \| 0.2226 \| -0.0106 \| 0.0077 Table V: Ensemble-averaged scalar fluxes for problem set B1 \| c \| ⟨ϕB⟩ \| ⟨ϕAM⟩ \| ⟨ϕNC⟩ \| ErrAM \| ErrNC ---\|---\|---\|---\|---\|---\|--- \| 0.0 \| 0.1975 \| 0.2213 \| 0.1972 \| 0.1200 \| -0.0018 \| 0.1 \| 0.2100 \| 0.2347 \| 0.2101 \| 0.1177 \| 0.0004 \| 0.2 \| 0.2245 \| 0.2506 \| 0.2252 \| 0.1159 \| 0.0028 \| 0.3 \| 0.2420 \| 0.2698 \| 0.2433 \| 0.1148 \| 0.0054 \| 0.4 \| 0.2634 \| 0.2936 \| 0.2655 \| 0.1148 \| 0.0080 x=0 \| 0.5 \| 0.2904 \| 0.3243 \| 0.2934 \| 0.1168 \| 0.0105 \| 0.6 \| 0.3261 \| 0.3658 \| 0.3301 \| 0.1218 \| 0.0125 \| 0.7 \| 0.3763 \| 0.4263 \| 0.3812 \| 0.1327 \| 0.0129 \| 0.8 \| 0.4548 \| 0.5255 \| 0.4588 \| 0.1553 \| 0.0088 \| 0.9 \| 0.6042 \| 0.7291 \| 0.5972 \| 0.2066 \| -0.0116 \| 0.95 \| 0.7553 \| 0.9502 \| 0.7233 \| 0.2581 \| -0.0423 \| 0.99 \| 0.9946 \| 1.3204 \| 0.8961 \| 0.3276 \| -0.0990 \| 0.0 \| 0.0246 \| 0.0033 \| 0.0243 \| -0.8646 \| -0.0106 \| 0.1 \| 0.0267 \| 0.0040 \| 0.0265 \| -0.8494 \| -0.0048 \| 0.2 \| 0.0291 \| 0.0049 \| 0.0292 \| -0.8302 \| 0.0020 \| 0.3 \| 0.0322 \| 0.0063 \| 0.0325 \| -0.8055 \| 0.0098 \| 0.4 \| 0.0360 \| 0.0082 \| 0.0367 \| -0.7733 \| 0.0188 x=10 \| 0.5 \| 0.0410 \| 0.0111 \| 0.0422 \| -0.7302 \| 0.0293 \| 0.6 \| 0.0480 \| 0.0158 \| 0.0500 \| -0.6711 \| 0.0413 \| 0.7 \| 0.0584 \| 0.0241 \| 0.0615 \| -0.5877 \| 0.0545 \| 0.8 \| 0.0758 \| 0.0405 \| 0.0808 \| -0.4658 \| 0.0670 \| 0.9 \| 0.1124 \| 0.0811 \| 0.1201 \| -0.2786 \| 0.0685 \| 0.95 \| 0.1527 \| 0.1305 \| 0.1604 \| -0.1455 \| 0.0500 \| 0.99 \| 0.2208 \| 0.2185 \| 0.2211 \| -0.0105 \| 0.0010 Table VI: Ensemble-averaged scalar fluxes for problem set B2 \| c \| ⟨ϕB⟩ \| ⟨ϕAM⟩ \| ⟨ϕNC⟩ \| ErrAM \| ErrNC ---\|---\|---\|---\|---\|---\|--- \| 0.0 \| 0.2089 \| 0.2213 \| 0.2088 \| 0.0593 \| -0.0002 \| 0.1 \| 0.2221 \| 0.2347 \| 0.2220 \| 0.0566 \| -0.0004 \| 0.2 \| 0.2378 \| 0.2506 \| 0.2375 \| 0.0539 \| -0.0009 \| 0.3 \| 0.2566 \| 0.2698 \| 0.2562 \| 0.0512 \| -0.0017 \| 0.4 \| 0.2800 \| 0.2936 \| 0.2791 \| 0.0487 \| -0.0031 x=0 \| 0.5 \| 0.3098 \| 0.3243 \| 0.3082 \| 0.0467 \| -0.0053 \| 0.6 \| 0.3498 \| 0.3658 \| 0.3468 \| 0.0457 \| -0.0086 \| 0.7 \| 0.4071 \| 0.4263 \| 0.4015 \| 0.0471 \| -0.0137 \| 0.8 \| 0.4985 \| 0.5255 \| 0.4875 \| 0.0542 \| -0.0220 \| 0.9 \| 0.6770 \| 0.7291 \| 0.6511 \| 0.0769 \| -0.0382 \| 0.95 \| 0.8612 \| 0.9502 \| 0.8132 \| 0.1033 \| -0.0558 \| 0.99 \| 1.1570 \| 1.3204 \| 1.0563 \| 0.1412 \| -0.0870 \| 0.0 \| 0.0073 \| 0.0033 \| 0.0073 \| -0.5438 \| 0.0027 \| 0.1 \| 0.0086 \| 0.0040 \| 0.0086 \| -0.5355 \| -0.0034 \| 0.2 \| 0.0104 \| 0.0049 \| 0.0103 \| -0.5225 \| -0.0085 \| 0.3 \| 0.0126 \| 0.0063 \| 0.0124 \| -0.5037 \| -0.0121 \| 0.4 \| 0.0156 \| 0.0082 \| 0.0154 \| -0.4776 \| -0.0139 x=10 \| 0.5 \| 0.0198 \| 0.0111 \| 0.0196 \| -0.4422 \| -0.0133 \| 0.6 \| 0.0261 \| 0.0158 \| 0.0258 \| -0.3947 \| -0.0095 \| 0.7 \| 0.0360 \| 0.0241 \| 0.0359 \| -0.3313 \| -0.0016 \| 0.8 \| 0.0537 \| 0.0405 \| 0.0543 \| -0.2463 \| 0.0108 \| 0.9 \| 0.0933 \| 0.0811 \| 0.0955 \| -0.1314 \| 0.0229 \| 0.95 \| 0.1386 \| 0.1305 \| 0.1413 \| -0.0586 \| 0.0191 \| 0.99 \| 0.2166 \| 0.2185 \| 0.2151 \| 0.0089 \| -0.0070 Table VII: Ensemble-averaged scalar fluxes for problem set B3 <figure><img src="content_image/1602.00825/x1.png"><figcaption>Figure 1: A sketch of the periodic medium</figcaption></figure> <figure><img src="content_image/1602.00825/x2.png"><figcaption>.i Case 1: ℓ1=0.5, ℓ2=1.0</figcaption></figure> <figure><img src="content_image/1602.00825/x8.png"><figcaption>(a) Source Q1 given by Eq. 19</figcaption></figure> <figure><img src="content_image/1602.00825/x10.png"><figcaption>Figure 4: Scalar flux in a fixed realization of problem set A2 with c1=0.5</figcaption></figure> <figure><img src="content_image/1602.00825/x11.png"><figcaption>Figure 5: Atomic mix scalar flux for problem set A2 with c1=0.5</figcaption></figure> <figure><img src="content_image/1602.00825/x12.png"><figcaption>Figure 6: Nonclassical scalar flux for problem set A2 with c1=0.5</figcaption></figure> <figure><img src="content_image/1602.00825/x13.png"><figcaption>.i Problem set A1 with c1=00</figcaption></figure> <figure><img src="content_image/1602.00825/x19.png"><figcaption>(a) c1=0.0</figcaption></figure> <figure><img src="content_image/1602.00825/x25.png"><figcaption>(a) c1=0.6</figcaption></figure> <figure><img src="content_image/1602.00825/x31.png"><figcaption>(a) c1=0.0</figcaption></figure> <figure><img src="content_image/1602.00825/x37.png"><figcaption>(a) c1=0.6</figcaption></figure> <figure><img src="content_image/1602.00825/x43.png"><figcaption>(a) c1=0.0</figcaption></figure> <figure><img src="content_image/1602.00825/x49.png"><figcaption>(a) c1=0.6</figcaption></figure> <figure><img src="content_image/1602.00825/x55.png"><figcaption>.i Set B1: ℓ1=20/3, ℓ2=40/3</figcaption></figure> <figure><img src="content_image/1602.00825/x61.png"><figcaption>.i Problem set B1 with c1=00</figcaption></figure> <figure><img src="content_image/1602.00825/x67.png"><figcaption>(a) c1=0.0</figcaption></figure> <figure><img src="content_image/1602.00825/x73.png"><figcaption>(a) c1=0.6</figcaption></figure> <figure><img src="content_image/1602.00825/x79.png"><figcaption>(a) c1=0.0</figcaption></figure> <figure><img src="content_image/1602.00825/x85.png"><figcaption>(a) c1=0.6</figcaption></figure> <figure><img src="content_image/1602.00825/x91.png"><figcaption>(a) c1=0.0</figcaption></figure> <figure><img src="content_image/1602.00825/x97.png"><figcaption>(a) c1=0.6</figcaption></figure>
1009.0666	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 17134, "num_imgs": 9, "llama3_tokens_count": 3996 }	[ "content_image/1009.0666/x1.png", "content_image/1009.0666/x2.png", "content_image/1009.0666/x3.png", "content_image/1009.0666/x4.png", "content_image/1009.0666/x5.png", "content_image/1009.0666/x6.png", "content_image/1009.0666/x7.png", "content_image/1009.0666/x8.png", "content_image/1009.0666/x9.png" ]	# The First Station of the Long Wavelength Array University of New Mexico E-mail: Steven W. Ellingson Virginia Polytechnic Institute and State University E-mail: ellingson@vt.edu Gregory B. Taylor, Joseph Craig, Ylva Pihlström, Lee J Rickard University of New Mexico E-mail: , , , gbtaylor@unm.edu joecraig@unm.edu ylva@unm.edu lrickard@unm.edu Tracy E. Clarke, Namir E. Kassim U.S. Naval Research Laboratory E-mail: , tracy.clarke@nrl.navy.mil Namir.Kassim@nrl.navy.mil Aaron Cohen U.S. Naval Research Laboratory and Johns Hopkins Applied Physics Laboratory E-mail: acohen00@gmail.com ###### Abstract: The Long Wavelength Array (LWA) will be a new multi-purpose radio telescope operating in the frequency range 10–88 MHz. Upon completion, LWA will consist of 53 phased array “stations” distributed over a region about 400 km in diameter in the state of New Mexico. Each station will consist of 256 pairs of dipole-type antennas whose signals are formed into beams, with outputs transported to a central location for high-resolution aperture synthesis imaging. The resulting image sensitivity is estimated to be a few mJy ($5\sigma$, 8 MHz, 2 polarizations, 1 hr, zenith) in 20–80 MHz; with resolution and field of view of ($8^{\prime\prime}$,$8^{\circ}$) and ($2^{\prime\prime}$,$2^{\circ}$) at 20 MHz and 80 MHz, respectively. All 256 dipole antennas are in place for the first station of the LWA (called LWA-1), and commissioning activities are well underway. The station is located near the core of the EVLA, and is expected to be fully operational in early 2011. ## 1 The Long Wavelength Array We are living in an era of resurgent interest in astronomy at long wavelengths, as witnessed by the development of new low frequency telescopes around the world, including LOFAR, MWA, PAPER, all discussed in this volume, and the LWA. The LWA is designed for both long-wavelength astrophysics and ionospheric science. Science to be addressed by the LWA includes cosmic evolution, the acceleration of relativistic particles, physics of the interstellar and intergalactic media, solar science and space weather, and “discovery science”; that is, the search for previously unknown sources and phenomena [1]. Specific objectives for LWA are spelled out in [3]. Upon completion, LWA will consist of 53 electronically-steered phased array “stations,” each consisting of 256 pairs of dipole-like antennas, operating with Galactic noise-limited sensitivity over the frequency range 20–80 MHz, with reduced sensitivity over 10–88 MHz. The stations will be distributed over the state of New Mexico, (Figure 1), with maximum baselines of up to 400 km, yielding resolution of $8^{\prime\prime}$ and $2^{\prime\prime}$ at 20 MHz and 80 MHz respectively. Beams formed by the stations will be transmitted to a central location and correlated to form images using aperture synthesis techniques. The full array is expected to reach mJy-class sensitivity. Stations will also be capable of operating as independent radio telescopes. <figure><img src="content_image/1009.0666/x1.png"><figcaption>Figure 1: Planned LWA station locations across the state of New Mexico. The“Y” of the EVLA is shown in blue.</figcaption></figure> ## 2 LWA-1, the First Station of the LWA The first station of the LWA, called LWA-1, is being constructed near the core of the EVLA. Its 256 crossed dipole antennas (shown in Figure 2) were completed in November 2009. The elements are distributed in a $\sim$100-m aperture, producing beam FOV of $8\deg$ and $2\deg$ at 20 MHz and 80 MHz. Every element is digitized to allow independent pointings of beams, and all-sky snapshot imaging. Using 256 dual-polarization antennas results of spacings of 3 x Nyquist at 80 MHz. The pseudorandom distribution of the antennas mitigates against aliasing (Figure 3). The minimum separation between antennas of 5 m allows easy access for maintenance, and also reduces beam desensitization due to sky noise correlation [4]. Various ionospheric, solar, and especially Galactic science goals require the ability to observe towards declinations which appear low in the southern sky from New Mexico. To compensate for the elevation-plane widening of the beam for these observations, the station footprint has been made somewhat elliptical with a NS:EW axial ratio of 1.1. <figure><img src="content_image/1009.0666/x2.png"><figcaption>Figure 2: The first station of the LWA, as seen from the EVLA antenna assemblybuilding. The equipment shelter is seen to the left.</figcaption></figure> <figure><img src="content_image/1009.0666/x3.png"><figcaption>Figure 3: Diagram of antenna locations and cabling in LWA-1. The circleindicates antennas which participated in early tests.</figcaption></figure> ### RF Signal Path, and System Status The signal from every antenna is processed by a dedicated direct-sampling receiver consisting of an analog receiver (ARX) and an analog-to-digital converter (A/D) which samples 196 million samples per second (MSPS). Beams are formed using a time-domain delay-and-sum architecture, which allows the entire 10–88 MHz passband associated with each antenna to be processed as single wideband data stream. Eventually beams from the stations will be correlated in a central location, but for individual stations it is also possible to record the beam data for later processing. To facilitate commissioning activities, diagnostics, and certain types of science observations requiring all-sky field-of-view (FOV), the station electronics will also have the capability to coherently capture and record the output of all A/Ds, where each A/D corresponds to one antenna. This will occur in two modes: the “transient buffer – wideband” (TBW) allows the raw output of the A/Ds to be collected continuously, but only for $\sim 100$ ms at a time. The “transient buffer – narrowband” (TBN), in contrast, allows a single tuning of $\sim 100$ kHz bandwidth to be recorded indefinitely. To accommodate the various uncertainties in the RFI environment, we have developed an ARX which can be electronically reconfigured between three modes: A full-bandwidth (10-88 MHz) uniform-gain mode, a full-bandwidth dual-gain mode in which frequencies below about 40 MHz can be attenuated using a “shelf filter,” and a 28–54 MHz mode, which serves as a last line of defense in the case where RFI above and/or below this range is persistently linearity-limiting. A sample spectrum, showing the strong RFI at the top and bottom ends of the spectrum, is shown in Figure 4. The noise is dominated by that from the sky, as can be seen by comparison of the sky signal with that of a terminated load. <figure><img src="content_image/1009.0666/x4.png"><figcaption>Figure 4: A spectrum from the summed 7-element beamformer after 10 seconds ofintegration with a spectral resolution of 6 kHz. The receiver noisetemperature is about 250 K and roughly constant across the LWA frequency range(black line). The signal from the sky is shown in blue. The turnover in thesky signal near 33 MHz is a result of the decreasing efficiency of the LWAdipoles.</figcaption></figure> The installation of the dipole antennas for LWA-1 is complete. At the time of writing (July 2010), cabling is on track for completion in August 2010. The system is currently operating with an interim 16-antenna analog beamformer, using the Eight-meter-wavelength Transient Array (ETA) “S60” digital receiver, and data capture system. In Figure 5 we plot the total power received for a single dipole and for 7 dipoles summed together over the course of three days. Some discrepant values are the result of thunderstorm activity on the afternoon of April 15. <figure><img src="content_image/1009.0666/x5.png"><figcaption>Figure 5: Plot showing total power for a single dipole (thin black line),integrating for 2 seconds every 5 minutes and for the sum of 7 dipoles(crosses) integrating in a similar fashion on three consecutive days. Thediscrepant high points are most likely due to thunderstorm activity. Thefrequency employed was 72.25 MHz with a 3 MHz bandwidth.</figcaption></figure> ### LWA-1 Sensitivity At low frequencies, Galactic noise can be a significant or dominant contribution to the total noise. This, combined with mutual coupling between antennas, makes it difficult to predict the sensitivity of these instruments. Ellingson (2010)[6] describes a system model and procedure for estimating the system equivalent flux density (SEFD) – a useful and meaningful metric of the sensitivity of a radio telescope – that accounts for these issues. The method is applied to LWA-1, and it is shown that the correlation of Galactic noise between antennas significantly desensitizes the array for beam pointings that are not close to the zenith. It is also shown that considerable improvement is possible using beamforming coefficients that are designed to optimize signal-to-noise ratio under these conditions (see Figure 6). The receiver noise is about 250 K, but has little influence on the SEFDs which range between 3,000 and 100,000 Jy over the frequency and elevation range plotted. <figure><img src="content_image/1009.0666/x6.png"><figcaption>Figure 6: Calculated SEFD of LWA-1 as a function of degrees away from thezenith. For each frequency, the upper (dotted) curve is the result for simplebeamforming, and the lower (solid) curve is the result for optimalbeamforming. From [6].</figcaption></figure> ### Radio Frequency Interference and Deep Integrations Radio Frequency Interference (RFI) is always an issue when working at low frequencies. We have sought to minimize internally-generated RFI by use of careful shielding of the station electronics. For externally-generated RFI we have chosen modes and gain settings based on a detailed study of RFI at the EVLA, combined with a study of A/D capabilities, leading to the conclusion that an A/D of about 200 MSPS with 8-bit sampling was probably sufficient when combined with an ARX having the capabilities described above. We currently favor a sampling rate $F_{s}=196$ MSPS, as this results in the highly desirable situation that the 88-108 MHz FM broadcast band aliases onto itself, which greatly reduces anti-alias filtering requirements. During construction of the first station, the RFI situation has actually improved with the conversion from analog to digital television (Fig. 7). At the time of the conversion in July 2009, all of the television stations in New Mexico elected to relocate their digital transmissions to frequencies above the LWA band. <figure><img src="content_image/1009.0666/x7.png"><figcaption>Figure 7: RFI environment at LWA-1. The left panels show an indicativespectrum and waterfall plot before the switch to digital TV. The right panelsshow data taken after the strong analog TV stations have vacated the spectrum.The spectrum is Galactic noise dominated. The diurnal variations seen in thewaterfall plots are predominantly due to the culmination of the GalacticCenter, and day-night variation of HF noise.</figcaption></figure> Strong RFI can cause a host of problems including compression in the receivers, and aliasing. Dealing with strong RFI requires the appropriate design choices, especially in the analog receiver, as described previously. Weak RFI can be just as damaging if it masks the faint cosmic signals that one is searching for. While station electronics are still being completed, deep integrations with a portion of the array show that the noise behaves radiometrically beyond an hour of data collecting (Fig. 8). These results were obtained at night, with 3 MHz bandwidth, centered near 74 MHz. About 20% of the frequency band was excised due to the presence of low-level RFI in this test. Deep integrations have paused while the array is being finished, so the duration of the deepest integrations may well be longer that the hour+ limit indicated by current tests. This is indeed promising for the prospects of deep imaging. <figure><img src="content_image/1009.0666/x8.png"><figcaption>Figure 8: Deep integration noise behavior. The noise continues to drop as 1/√tfor integrations as long as an hour. These drift scan data were taken atnight, with 2.45 MHz bandpass (lower panel), centered at 72.25 MHz. About 2600channels are shown. Data were corrected for diurnal total power variation.(The step at about 100 sec is possibly due to HVAC turning on.) Totalintegration times where noise continues to drop radiometrically have exceededan hour, beyond the time indicated here.</figcaption></figure> ### Digital Signal Processing The signal from every antenna is processed by a dedicated direct-sampling receiver consisting of an analog receiver (ARX) and an analog-to-digital converter (A/D) as described in §2.1. Beams are formed using a time-domain delay-and-sum architecture, which allows the entire 10–88 MHz passband associated with each antenna to be processed as single wideband data stream. Delays are implemented in two stages: A coarse delay is implemented using a first-in first-out (FIFO) buffer operating on the A/D output samples, followed by a finite impulse response (FIR) filter, which is also used to introduce corrections for polarization and other frequency-dependent effects. The raw linear polarizations are transformed into calibrated standard orthogonal circular polarizations, and The signals are then added to the signals from other antennas processed similarly. Four dual-polarization beams of bandwidth 78 MHz, each capable of fully-independent pointing over the visible sky, will be constructed in this fashion. The beams will be available for various “backends” implemented at the station level, such as data recorders, wideband spectrometers, and pulsar machines. For interferometric imaging, two “tunings” will be extracted from any frequency in the 78 MHz-wide passband, having bandwidth selectable between 400 kHz and 20 MHz divided into 4096 spectral channels. This is the output to the LWA correlator. ### Prototype All Sky Imager The Prototype All Sky Imager (PASI) will consist of a software correlator, and a near real time imager, both operating on an IBM cluster already purchased for these tasks. The PASI will be a “back-end” instrument designed specifically for the first station of the LWA. Together, this equipment will allow us to image 75% of the sky every 24 hours, with an instantaneous field-of-view of over 120${}^{\circ}$$\times$ 120${}^{\circ}$. This will provide unparalleled resolution and sensitivity all-sky imaging in a largely unexplored frequency band. Expected sources to be imaged include the Sun, planets, flaring stars, active galaxies, quasars, magnetars, black holes, and gamma-ray bursts. The discovery of entirely new classes of objects are possible and will be followed up by observations with the LWA and other ground (e.g., EVLA) and space-based (e.g., Chandra) facilities. ## 3 Timeline and Future Plans Currently the first LWA station is using analog beamformers and the ETA “S60” digital receiver to commission the station. During summer 2010 this system will be increased to make use of up to 32 antennas. At the same time, the first digital processing board will be installed with the capability of handling up to 20 antennas. This will allow for commissioning of the transient buffer modes (TBW and TBN) initially, and later the beam outputs. The installation of the complete analog and digital electronics will continue throughout the fall resulting in a fully operational station in early 2011. The first observing programs have already been selected and some of them have even begun taking data. Beyond the first LWA station we are actively seeking funding for additional stations. The land and infrastructure for a second LWA station (labeled NA in Figure 9) is in place, 19 km north of LWA1. The land for a third station at Horse Mountain (HM) is also leased, but has not yet been developed. Leases for two other sites (marked MA and HS in Figure 9) are pending. Basic research in radio astronomy at the Naval Research Laboratory is supported by 6.1 base funding. <figure><img src="content_image/1009.0666/x9.png"><figcaption>Figure 9: Plot showing planned stations, including LWA-2 (North Arm) and LWA-3(Horse Mountain).</figcaption></figure> ## References * [1] N.E. Kassim et al., “The Long Wavelength Array,” in [2], pp. 392-398. * [2] N. E. Kassim, M. R. Perez, W. Junor, and P. A. Henning (eds.), _Clark Lake to the Long Wavelength Array: Bill Erickson’s Radio Science_, ASP Conf. Ser., Volume 345, 2005. * [3] T.E. Clarke, “Scientific Requirements for the Long Wavelength Array,” Ver. 2.3, Memo 117 in [5], Nov 19, 2007. * [4] Ellingson, S.W., Clarke, T.E., Cohen, A., Craig, J., Kassim, N.E., Pihlström, Y.M., Rickard, L.J., & Taylor, G.B.,“The Long Wavelength Array”, _Proc IEEE,_ 97, 1421, 2009. * [5] Long Wavelength Array Memo Series, [on-line] http://www.phys.unm.edu/$\sim$lwa/memos. * [6] S.W. Ellingson, “Sensitivity of Antenna Arrays for Long-Wavelength Radio Astronomy,” _IEEE, Trans. Ant. & Prop._, in press [Memo 166 in [5]]
1912.09110	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 21698, "num_imgs": 1, "llama3_tokens_count": 7023 }	[ "content_image/1912.09110/x1.png" ]	# Mixed QCD-EW two-loop corrections to Drell-Yan production PRISMA${}^{+}$ Cluster of Excellence, Institut für Kernphysik, Johannes Gutenberg Universität, 55116 Mainz, Germany E-mail: Andreas von Manteuffel Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA E-mail: vmante@msu.edu Robert M. Schabinger Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA E-mail: schabing@msu.edu Hubert Spiesberger PRISMA${}^{+}$ Cluster of Excellence, Institut für Physik, Johannes Gutenberg Universität, 55116 Mainz, Germany E-mail: spiesber@uni-mainz.de ###### Abstract: The Drell-Yan production of charged lepton pairs is one of the key processes measured at hadron colliders. The QCD corrections to the cross section are known to order $\alpha_{s}^{2}$ and electroweak corrections are known to order $\alpha$. The next important step for a better theoretical understanding is the complete calculation of the mixed QCD-EW corrections of order $\alpha_{s}\alpha$. In my talk, I report on the first complete calculation of the virtual two-loop corrections of order $\alpha\alpha_{s}$ to the lepton-pair production cross section. The calculation is carried out analytically using tensor reduction, integration-by-parts relations and the method of differential equations. We validate a previous calculation of the subset of mixed QCD-QED corrections and show how the jet and soft functions of that reference can be used to subtract the infrared divergences of the complete mixed QCD-electroweak virtual corrections. ## 1 Introduction The Drell-Yan process [1] is one of the key processes at the Large Hadron Collider at CERN. The measurement of its cross section can be used to determine the masses of the $W^{\pm}$ and $Z$ bosons, the weak mixing angle, as well as parton distribution functions. In order to have an accurate understanding and better prediction of the process from the theory side, higher-order corrections must be included. Known corrections to the cross section include up to now: next-to-next-to-leading order (NNLO) Quantum Chromodynamic (QCD) corrections [2, 3, 4, 5], NNLO Quantum Electrodynamics (QED) corrections [6, 7] and NLO electroweak (EW) corrections [8, 9, 10]. Since these corrections turn out to be important, it is natural to also consider the two-loop mixed QCD-EW corrections. At present, results are known for the mixed QCD-QED corrections [11] and in the approximation of on-shell $Z$ production [12, 13, 14, 15]. In this talk at RADCOR2019, I report on the first calculation of all virtual mixed QCD-EW two-loop corrections. We calculated the master integrals contributing to this process for the first time in the physical region of phase space in terms of multiple polylogarithms and demonstrated that this is possible in the presence of algebraic letters involving unrationalizable square roots [16]. For the amplitude we find exactly the same IR structure as in the case of QCD-QED, such that soft- and jet-functions from that calculation can also be used in the extension to the full EW sector. ## 2 Calculation of the amplitude In this talk, I focus on the mixed QCD-EW two-loop corrections to charged leptons in quark-antiquark annihilation, \[q\bar{q}\to l^{+}l^{-},\] (1) where $l$ is a massless electron or muon. For the calculation of the amplitude, we use the program QGRAF[17] to generate all contributing diagrams, Form[18] to apply Feynman rules and do symbolic manipulations, and Reduze 2[19, 20, 21, 22] to generate integration-by-parts (IBP) relations. The bare amplitude is written as a sum of a set of master integrals with rational functions in the kinematic variables and the space-time dimension $d$ as coefficients. We use dimensional regularization with $d=4-2\epsilon$ for the regularization of infrared (IR) and ultraviolet (UV) singularities. All rational functions are written in a unique way using partial fractioning. In Fig. 1, we show a flow chart of the calculation process. <figure><img src="content_image/1912.09110/x1.png"><figcaption>Figure 1: Calculation flow</figcaption></figure> To evaluate the master integrals, we use the method of differential equations. Using a specific choice for the basis of master integrals, the differential equation can be cast in the so-called canonical form, in which the dependence on $\epsilon$ decouples from the dependence of the kinematic invariants, in such a way that [23, 24, 25, 26]: \[\mathrm{d}\mathbf{m}_{i}=\epsilon\;\sum_{j,k}\mathrm{d}\ln(l_{k})\big{(}A^{(k) }\big{)}_{ij}\,\mathbf{m}_{j},\] (2) where $\mathbf{m}_{i}$ is a master integral, $\big{(}A^{(k)}\big{)}_{ij}$ is an element of a rational matrix and only the $l_{k}$ (the so-called letters of the differential equation) depend on the kinematic invariants. The master integrals relevant here have been solved in the Euclidean region of the phase space in Ref. [27] and for the case of one internal mass in the physical region of phase space in Ref. [28]. Here, we aim at completing the analytical calculation of all integrals in terms of functions, which permit a fast and robust numerical evaluation in the physical region of phase space. ## 3 Integrating root valued symbols For the integrals involving two internal masses we find that some of the letters of the differential equation involve unrationalizable square roots. In the literature, such cases were up to now always solved in terms of Chen iterated integrals, in which one integration has to be performed numerically. In [16], we were able to integrate the differential equation for the first time in terms of generalized polylogarithms in that region of phase space, rendering the numerical evaluation of the amplitude accessible and efficient for practical applications. The square roots enter the differential equation the first time as leading singularities in the following three integrals: \[\epsilon^{2}sr_{1}\vbox{\hbox{\includegraphics[scale=.34]{int9}}},\qquad \epsilon^{3}r_{2}\vbox{\hbox{\includegraphics[scale=.4]{int27}}},\qquad \epsilon^{4}r_{3}\vbox{\hbox{\includegraphics[scale=.4]{int32}}},\] (3) where \[r_{1} =\sqrt{s(s-4m^{2})},\qquad r_{2}=\sqrt{-st(4m^{2}(t+m^{2})-st)},\] \[r_{3} =\sqrt{s(t^{2}(s-4m^{2})+sm^{2}(m^{2}-2t))}.\] (4) Following Ref. [27], one can define the dimensionless parameters $w$ and $z$ through: \[s =-m^{2}\frac{(1-w)^{2}}{w},\qquad t=-m^{2}\frac{w(1+z)^{2}}{z(1+w )^{2}}\] such that two square roots become rational: \[r_{1} =\frac{-m^{2}(1-w)(1+w)}{w},\qquad r_{2}=\frac{-m^{4}(1-w)(1-z)(1 +z)}{z(1+w)}.\] (5) However, the third square root becomes: \[r_{3}=\frac{m^{4}(1-w)}{wz(1+w)}r,\] (6) where $r$ is a new square root in $w$-$z$-space: \[r=\sqrt{(1+w^{2}z^{2})(w+z)^{2}+2wz(w-z)^{2}+4wz^{2}(1+w^{2})}.\] (7) It can be shown that no parametrization exists which makes $r$ rational [29]. Therefore, to integrate the differential equation new methods are needed. Using the method of Ref. [30], one can try to match the symbol defined by the differential equation to a space of functions. In Ref. [30], this is discussed for rational functions. The idea is to construct suitable ${\rm Li}$ function arguments, such that the functional basis contains no spurious letters. By considering the symbol of ${\rm Li}$ functions \[S\big{(}{\rm Li}_{n}(f)\big{)}=-(1-f)\otimes\underbrace{f\otimes...\otimes f}_ {(n-1)~{}\text{times}},\] (8) it is clear, that we can achieve the absence of spurious letters by requiring both $f$ and $1-f$ to factorize over the alphabet. A similar strategy can be applied to multiple polylogarithms of several arguments. In the presence of algebraic letters, we use a heuristic test for factorization. Specifically, we want to test if a given expression $g$ factorizes over the alphabet, i.e. can be written as a power-product of the letters: \[g=c^{a_{0}}l_{1}^{a_{1}}l_{2}^{a_{2}}\cdots,\] (9) where $c$ and $a_{n}$ are rational numbers. This implies \[\ln(g)-a_{0}\ln(c)-a_{1}\ln(l_{1})-a_{2}\ln(l_{2})-\ldots=0\,.\] (10) Eq. (10) can then be tested numerically to find the required relations. However, there are several problems with this approach: * There is no unique factorization in the case of algebraic letters; e.g. if one has letters $\sqrt{x}$ and $\sqrt{y}$, it is unclear, if we need to factor $x-y=(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})$. * One needs to consider non-integer powers of letters, e.g. $\sqrt{l}$, $l^{1/4},...$. * Often one finds very complicated letters and relations. * There is no obvious way how to choose letters. These problems can be tackled based an interesting observation: we found that, for each $l=q_{1}+q_{2}r$, where $q_{1}$ and $q_{2}$ are rational functions, $\bar{l}=q_{1}-q_{2}r$ is always factorizable over the alphabet, such that one can trade $l$ for $\bar{l}$ without changing the singularity structure. This has a very useful consequence: one can try to get a much simplified version of the alphabet, by making the ansatz \[l=q+r,\quad\bar{l}=q-r\] (11) and requiring that $l\bar{l}$ factorizes over the rational part of the alphabet. In this way, one can construct a set of simpler letters and construct the algebraic part of the alphabet in a systematic way. To demonstrate the power of this approach, we show the starting point for the alphabet in the Drell-Yan case. The rational part of the alphabet is given by \[\mathcal{L}_{r}=\{ 1-w,-w,1+w,1-w+w^{2},1-z,-z,1+z\] \[1-wz,1+w^{2}z,-z-w^{2},z-w\}.\] (12) Before simplification, the algebraic part reads \[\mathcal{L}_{a}=\{ r,-(1-w)(z-w)(1-w\,z)+r\,(1+w),-(1-w)\left(\vphantom{w^{2}}4w\,z +(w+z)(1+w\,z)\right)\] \[-r\,(1+w),r^{2}-2w\,z^{2}(1-w)^{2}+r\,(w+z)(1+w\,z),\] \[r^{2}(1-z)^{2}+2z^{2}(z+w^{2})(1+w^{2}z)+r\,(1-z)(1+z)\left( \vphantom{w^{2}}2w\,z-(w+z)(1+w\,z)\right)\}\] (13) and the highest degree of a letter is therefore $8$. Using the ideas described above, we find a simplified version of the algebraic part, \[\tilde{\mathcal{L}}_{a}\] (14) where the highest degree in $w$ and $z$ is now only $3$. Note, that the factors of $1/2$ in Eq. (14) are chosen to avoid having to introduce 2 as an auxiliary letter. With the new representation of the alphabet, we were able to integrate the differential equation in the physical region of phase space, using only ${\rm Li}_{3}$ and ${\rm Li}_{21}$ functions for weight $3$ and ${\rm Li}_{4}$, ${\rm Li}_{22}$ and ${\rm Li}_{31}$ functions for weight 4. We were able to choose a representation in which all functions are manifestly real-valued above the two-mass threshold. ## 4 UV renormalization and IR structure ### The case of QCD-QED In Ref. [11] the authors calculated the mixed QCD-QED corrections to lepton-pair production. The UV renormalization in this case is trivial, since it only affects the vacuum polarization diagrams and does not mix with IR divergences. To subtract IR divergences, the authors calculated jet and soft functions, which are given by \[\mathcal{J}^{(1,0)} =-\left(\frac{1}{2\epsilon^{2}}+\frac{3}{4\epsilon}\right)(Q_{q}^ {2}+Q_{l}^{2}),\qquad\mathcal{J}^{(0,1)}=-\left(\frac{1}{2\epsilon^{2}}+\frac{ 3}{4\epsilon}\right)C_{F},\] \[\mathcal{J}^{(1,1)} =\left(\frac{1}{4\epsilon^{4}}+\frac{3}{4\epsilon^{3}}+\frac{9}{1 6\epsilon^{2}}\right)C_{F}(Q_{q}^{2}+Q_{l}^{2})-\frac{1}{2\epsilon}\left(\frac {3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)C_{F}Q_{q}^{2},\] \[\mathcal{S}^{(1,0)} =-\frac{1}{2\epsilon}\left[(Q_{q}^{2}+Q_{l}^{2})\ln\left(\frac{ \mu^{2}}{-s}\right)+2Q_{q}Q_{l}\left(\ln\left(\frac{\mu^{2}}{-t}\right)-\ln \left(\frac{\mu^{2}}{-u}\right)\right)\right],\] \[\mathcal{S}^{(0,1)} =-\frac{1}{2\epsilon}C_{F}\ln\left(\frac{\mu^{2}}{-s}\right),\] \[\mathcal{S}^{(1,1)}\] (15) where $Q_{q}$ is the charge of the quark, $Q_{l}$ the charge of the lepton and $C_{F}$ is the color factor of $SU(N_{c})$. One can then extract the finite hard scattering amplitude $\mathcal{H}$ from the bare amplitude $\mathcal{M}$ by: \[\mathcal{M}=\mathcal{H}^{(1,0)}+\left(\frac{\alpha}{\pi}\right) \left[\mathcal{J}^{(1,0)}\mathcal{H}^{(1,0)}+\mathcal{S}^{(1,0)} \mathcal{H}^{(1,0)}+\mathcal{H}^{(2,0)}\right]\] \[+\left(\frac{\alpha_{s}}{\pi}\right) \left[\mathcal{J}^{(0,1)}\mathcal{H}^{(1,0)}+\mathcal{S}^{(0,1)} \mathcal{H}^{(1,0)}+\mathcal{H}^{(1,1)}\right]\] \[+\left(\frac{\alpha_{s}}{\pi}\right)\left(\frac{\alpha}{\pi}\right) \left[\left(\mathcal{J}^{(1,1)}+\mathcal{J}^{(0,1)}\mathcal{J}^{( 1,0)}+\mathcal{J}^{(1,0)}\mathcal{J}^{(0,1)}+\mathcal{S}^{(1,1)}\right) \mathcal{H}^{(1,0)}\right.\] \[\left.+\right(\mathcal{J}^{(1,0)}+\mathcal{S}^{(1,0)}\left) \mathcal{H}^{(1,1)}+\right(\mathcal{J}^{(0,1)}+\mathcal{S}^{(0,1)}\left) \mathcal{H}^{(2,0)}+\mathcal{H}^{(2,1)}\right].\] (16) Using these definitions, we were able to reproduce the result of this reference. In the calculation one can define $4$ gauge invariant subsets by considering the charges $Q_{l}$ and $Q_{q}$ and the quantum number $C_{F}$. As an example, consider the contributions proportional to $Q_{q}^{2}Q_{l}^{2}$. Diagramatically, the finite contribution to the hard scattering matrix element is then defined by: \[\vbox{\hbox{\includegraphics[scale=.6]{VertexTopC}}}-\left( \mathcal{J}_{Q_{q}^{2}}^{(1,0)}+\mathcal{S}_{Q_{q}^{2}}^{(1,0)}\right)\left( \vbox{\hbox{\includegraphics[scale=.6]{Vertex1LG}}}-\left(\mathcal{J}_{C_{F}}^ {(0,1)}+\mathcal{S}_{C_{F}}^{(0,1)}\right)\vbox{\hbox{\includegraphics[scale=. 4]{Tree}}}\right)\] \[-\left(\mathcal{J}_{C_{F}}^{(0,1)}+\mathcal{S}_{C_{F}}^{(0,1)} \right)\left(\vbox{\hbox{\includegraphics[scale=.6]{Vertex1LA}}}-\left( \mathcal{J}_{Q_{q}^{2}}^{(1,0)}+\mathcal{S}_{Q_{q}^{2}}^{(1,0)}\right)\vbox{ \hbox{\includegraphics[scale=.6]{Tree}}}\right)\] \[-\left(\mathcal{J}_{C_{F}Q_{q}^{2}}^{(1,1)}+\mathcal{S}_{C_{F}Q_{ q}^{2}}^{(1,1)}\right)\vbox{\hbox{\includegraphics[scale=.6]{Tree}}}=\text{ finite}.\] (17) Note, that in Eq. (17) each diagram is meant as a sum over all diagrams of the corresponding gauge class, e.g. the first diagram is a representative of all two-loop diagrams proportional to $C_{F}Q_{q}^{2}Q_{l}^{2}$. The subscripts of $\mathcal{J}$ and $\mathcal{S}$ are to be understood as instructions to select only specific subsets of the jet- and soft-functions, proportional to $Q_{q}^{2}$, $C_{F}$, or $C_{F}Q_{q}^{2}$. ### Extension to the full EW sector To extend the calculation to the EW sector, one encounters several complications. The master integrals become much more difficult due to the additional mass scale of the gauge bosons. Furthermore, one has to deal with $\gamma_{5}$ in dimensional regularization and a non-trivial overlap of IR and UV divergences. The method to calculate the new master integrals with an additional mass scale was already discussed in Sec. 3. For $\gamma_{5}$ we used Kreimer’s scheme [31, 32], in which one gives up the cyclicity of the trace in order to maintain an anticommuting $\gamma_{5}$. Since traces over $\gamma_{5}$ matrices lead to four-dimensional Levi-Civita tensors which are contracted with $d$ dimensional loop momenta, we employ a Passarino-Veltman tensor decomposition. For a general $R_{\xi}$ gauge we decompose tensor integrals with up to rank $10$, which we achieve using finite field methods [33, 34] for the involved linear algebra. For the UV renormalization we calculate the wave function counterterms for all particles in the on-shell scheme. Note, that the only genuine two-loop counter terms come from fermion self-energies and terms proportional to $n_{f}$, the latter of which we ignore here. We renormalize the strong and electroweak couplings and the particle masses in the $\overline{MS}$ scheme, but keep the setup flexible enough to facilitate a convenient transition to other possible renormalization schemes. After subtraction of all UV divergences one is left with IR divergences only. Since these originate from the photon or the gluon, one expects that the same jet and soft functions can be used as in the case of QCD-QED, cf. Eq. (15). As an example, consider the gluon plus Z corrections to the $q\overline{q}\gamma$ vertex. In this case we encounter one- and two-loop counterterms for the wave-function renormalization: \[\vbox{\hbox{\includegraphics[scale=.6]{Vertex2L1m}}}+\vbox{\hbox{ \includegraphics[scale=.6]{Vertex1LxCT}}}+\vbox{\hbox{\includegraphics[scale=. 6]{Vertex2LCT}}}-\left(\mathcal{J}_{C_{F}}^{(0,1)}+\mathcal{S}_{C_{F}}^{(0,1)} \right)\times\] \[\left(\vbox{\hbox{\includegraphics[scale=.6]{Vertex1L1m}}}+\vbox{ \hbox{\includegraphics[scale=.6]{Vertex1LCT}}}\right)=\text{finite}.\] (18) In Eq. (18) the thick lines denote the $Z$ boson and, as before, each diagram is a representative of a set of gauge invariant diagrams. In a similar way, we arrive at the finite remainder for the other contributions to the amplitude. ## 5 Conclusion and outlook We calculated all mixed QCD-EW virtual two-loop corrections to the Drell-Yan production of a charged lepton pair. The master integrals have symbol letters, which depend on an unrationalizable square root. For the first time, we integrated such differential equations in terms of multiple polylogarithms with algebraic arguments. Our analytic solution allows for a fast numerical evaluation suitable for phenomenological applications. We checked that the infrared structure of the two-loop amplitude matches the structure predicted by the soft and jet-functions available in the literature from the calculation for the two-loop QCD-QED corrections. Our two-loop amplitudes provide a crucial building block for future calculations of very precise cross sections and distributions for the Drell-Yan process, including the high-invariant mass region. The new methods we developed to calculate the master integrals motivate an investigation of whether other Feynman integrals with a similar analytic structure may be treated analogously. ## Acknowledgments AvM was supported in part by the National Science Foundation under Grant No. 1719863. MH was supported in part by the German Research Foundation (DFG), through the Collaborative Research Center, Project ID 204404729, SFB 1044, and the Cluster of Excellence PRISMA${}^{+}$, Project ID 39083149, EXC 2118/1. ## References * [1] S. D. Drell and T. M. Yan, Phys. Rev. Lett. 25 (1970) 316, Erratum: [Phys. Rev. Lett. 25 (1970) 902]. * [2] R. Hamberg, W. L. van Neerven and T. Matsuura, Nucl. Phys. B 359 (1991) 343, Erratum: [Nucl. Phys. B 644 (2002) 403]. * [3] R. V. Harlander and W. B. Kilgore, Phys. Rev. Lett. 88 (2002) 201801 [hep-ph/0201206]. * [4] C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. D 69 (2004) 094008 [hep-ph/0312266]. * [5] K. Melnikov and F. Petriello, Phys. Rev. Lett. 96 (2006) 231803 [hep-ph/0603182]. * [6] F. A. Berends, W. L. van Neerven and G. J. H. Burgers, Nucl. Phys. B 297 (1988) 429, Erratum: [Nucl. Phys. B 304 (1988) 921]. * [7] J. Blümlein, A. De Freitas, C. G. Raab and K. Schönwald, Phys. Lett. B 791 (2019) 206 [arXiv:1901.08018 [hep-ph]]. * [8] U. Baur, O. Brein, W. Hollik, C. Schappacher and D. Wackeroth, Phys. Rev. D 65 (2002) 033007 [hep-ph/0108274]. * [9] S. Dittmaier and M. Krämer, Phys. Rev. D 65 (2002) 073007 [hep-ph/0109062]. * [10] U. Baur and D. Wackeroth, Phys. Rev. D 70 (2004) 073015 [hep-ph/0405191]. * [11] W. B. Kilgore and C. Sturm, Phys. Rev. D 85 (2012) 033005 [arXiv:1107.4798 [hep-ph]]. * [12] S. Dittmaier, A. Huss and C. Schwinn, Nucl. Phys. B 885 (2014) 318 [arXiv:1403.3216 [hep-ph]]. * [13] S. Dittmaier, A. Huss and C. Schwinn, Nucl. Phys. B 904 (2016) 216 [arXiv:1511.08016 [hep-ph]]. * [14] M. Delto, M. Jaquier, K. Melnikov and R. Röntsch, arXiv:1909.08428 [hep-ph]. * [15] R. Bonciani, F. Buccioni, N. Rana, I. Triscari and A. Vicini, arXiv:1911.06200 [hep-ph]. * [16] M. Heller, A. von Manteuffel and R. M. Schabinger, arXiv:1907.00491 [hep-th]. * [17] P. Nogueira, J. Comput. Phys. 105 (1993) 279. * [18] B. Ruijl, T. Ueda and J. Vermaseren, arXiv:1707.06453 [hep-ph]. * [19] A. von Manteuffel and C. Studerus, arXiv:1201.4330 [hep-ph]. * [20] C. Studerus, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546 [physics.comp-ph]]. * [21] C. W. Bauer, A. Frink and R. Kreckel, J. Symb. Comput. 33 (2000) 1 [cs/0004015 [cs-sc]]. * [22] R. H. Lewis, “_Computer Algebra System_ Fermat.” http://home.bway.net/lewis/. * [23] A. V. Kotikov, In Diakonov, D. (ed.): _Subtleties in quantum field theory_ 150-174 [arXiv:1005.5029 [hep-th]]. * [24] J. M. Henn, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806 [hep-th]]. * [25] E. Remiddi and L. Tancredi, Nucl. Phys. B 925, 212 (2017) [arXiv:1709.03622 [hep-ph]]. * [26] L. Adams and S. Weinzierl, Phys. Lett. B 781 (2018) 270 [arXiv:1802.05020 [hep-ph]]. * [27] R. Bonciani, S. Di Vita, P. Mastrolia and U. Schubert, JHEP 1609 (2016) 091 [arXiv:1604.08581 [hep-ph]]. * [28] A. von Manteuffel and R. M. Schabinger, JHEP 1704 (2017) 129 [arXiv:1701.06583 [hep-ph]]. * [29] M. Besier, D. Festi, M. Harrison and B. Naskrecki, arXiv:1908.01079 [math.AG]. * [30] C. Duhr, H. Gangl and J. R. Rhodes, JHEP 1210 (2012) 075 [arXiv:1110.0458 [math-ph]]. * [31] J. G. Korner, D. Kreimer and K. Schilcher, Z. Phys. C 54, 503 (1992). * [32] D. Kreimer, Ph.D. thesis, Johannes Gutenberg Universität Mainz (1992). * [33] M. Kauers, Nucl. Phys. Proc. Suppl. 183, 245 (2008). * [34] A. von Manteuffel and R. M. Schabinger, Phys. Lett. B 744, 101 (2015) [arXiv:1406.4513 [hep-ph]].
1003.2834	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 74483, "num_imgs": 1, "llama3_tokens_count": 27261 }	[ "content_image/1003.2834/x1.png" ]	# Quartet of spin-3/2 baryons in chiral multiplet $(1,1/2)\oplus(1/2,1)$ with mirror assignment Keitaro Nagata _Research Institute for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8527 JAPAN_ _Present address : Department of Physics, The University of Tokyo,_ _Bunkyo-ku, Tokyo 113-0033 JAPAN_ March 1, 2024 We study the possible existence of chiral partners in the spin-$\frac{3}{2}$ sector of the baryon spectrum. We consider a quartet scheme where four spin-3/2 baryons, $P_{33}$, $D_{33}$, $D_{13}$ and $P_{13}$, group into higher-dimensional chiral multiplets $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ with a mirror assignment. With an effective $SU(2)_{R}\times SU(2)_{L}$ Lagrangian, we derive constraints imposed by chiral symmetry together with the mirror assignment on the masses and coupling constants of the quartet. Using the effective Lagrangian, we try to find a set of baryons suitable for the chiral quartet. It turns out that two cases reasonably agree with the mass pattern of the quartet: ($\Delta(1600)$, $\Delta(1940)$, $N(1520)$, $N(1720)$) and ($\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$). ## 1 Introduction Chiral symmetry $SU(N_{F})_{R}\times SU(N_{F})_{L}$ and its spontaneous breaking characterize the QCD vacuum, and is a key to understanding the strong interactions. Due to the spontaneous breaking of chiral symmetry (SBCS), the hadron spectrum is classified in terms of the residual symmetry $SU(N_{F})_{V}$, while the role of $SU(N_{F})_{R}\times SU(N_{F})_{L}$ in the hadron spectrum is unclear. Nevertheless, one expects that there exists a set of hadrons reflecting a nature of the original symmetry, which is referred to as chiral partners. Such examples are well-known for mesons, e.g. $(\sigma,\pi)$ and $(\rho,a_{1})$ [1, 2, 3], while not well established for baryons. As discussed in the meson’s case, finding chiral partners provides us with the understanding of the role of chiral symmetry in the hadron spectrum, and also a clue to study the restoration of chiral symmetry. Recently, the multiplet nature of the chiral group draws a renewed attention from an interest in the effective chiral restoration [4, 5, 6], which was suggested to be the cause of the observed parity doubling in high-energy region of the spectrum [7]. In the present work, we address the issue of the multiplet nature of the baryon’s chiral partners. We denote a chiral multiplet by $(I_{R},I_{L})$, where $I_{R}[I_{L}]$ is an isospin for $SU(2)_{R}[SU(2)_{L}]$. All the members of one chiral multiplet $(I_{R},I_{L})$ have a fixed spin. The correspondence of the charge algebra between $SU(N_{f})_{R}\times SU(N_{f})_{L}$ and $SU(N_{f})_{V}\times SU(N_{f})_{A}$ leads to a relation $I=I_{R}\oplus I_{L}=\|I_{R}+I_{L}\|,\cdots,\|I_{R}-I_{L}\|$. This implies that a chiral multiplet can contain various isospin states. In the presence of the SBCS, the mixing of different chiral representations happens, and a hadron with an isospin $I$ can be described as a superposition of various chiral representations containing $I$. We are here concerned with the case that a set of hadrons group into one or a few representations even in the presence of the SBCS, or the case where the configuration mixing is small. In order to find chiral partners, we need to understand the multiplet nature of the chiral group, such as the pattern of the spectrum and coupling constants of the multiplet. Because general relations for masses and axial charges that can be applied to arbitrary chiral representations are not established so far, the properties of the chiral partners are usually studied with focusing on a particular chiral representation. In the meson’s case, the properties of chiral partners have been investigated by using e.g. the NJL model [8, 9] and Weinberg sum rules [10]. The NJL model was applied to the nucleon [11, 12, 13, 14, 15] and $\Delta(1232)$ [16] by solving the Faddeev equation. We applied the NJL model with diquarks to the nucleon [17, 18, 19] and the Roper resonance [20], using an auxiliary field method. However, when we apply such microscopic approaches to a baryon with a mass larger than sum of the masses of the internal degrees of freedom, we encounter the difficulty of the confinement. Due to this difficulty, effective Lagrangian approaches that contain hadrons as degrees of freedom are often employed for the study of baryon’s chiral partners [21, 22, 23, 24, 25, 26, 27, 28]. In recent papers, we have developed a systematic method to construct an effective $SU(N_{f})_{R}\times SU(N_{F})_{L}$ Lagrangian including higher-dimensional representations [29, 30, 31, 32, 33], which we refer to as a projection method. This method is inspired by an NJL model for mesons, and partly extend it to baryons. In Ref. [29], we classified baryon fields consisting of three quarks in terms of chiral multiplets. The Pauli principle implemented by the Fierz transformation plays a crucial role in the classification. The projection method is performed as follows. First we find a chiral invariant operator involving direct products of the quark and diquark fields. This can be achieved by using an analogy between $(\sigma,\vec{\pi})$ and diquarks in chiral transformation property. Then, we project the direct products of the quark and diquark fields onto irreducible parts with the use of the Fierz identities. After the projection, three-quark fields are replaced by baryon fields. Thus we can systematically construct chiral invariant Lagrangians including higher-dimensional chiral representations, avoiding problems caused by the lack of the confinement. Although such simple effective Lagrangians have limited validity, they are useful for the present purpose to derive the pattern of the masses and coupling constants of the chiral multiplet. In Ref. [30], we have applied the projection method to a quartet scheme (QS). The QS was first proposed by Jido et. al. [34]. They used two kinds of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ and considered so-called mirror assignment [22, 23, 25], where four types of baryons, two with $I=\frac{1}{2}$ and the other two with $I=\frac{3}{2}$, are included in the multiplet. They applied the QS to $J=\frac{1}{2},\frac{3}{2}$ and $\frac{5}{2}$ and studied the masses and intra-coupling constants of the quartet. They did not consider Dirac structure of the Lagrangian explicitly. Owing to the projection method, we took intro account the Dirac structure in the QS Lagrangian which enables us to include transition terms between $J=\frac{1}{2}$ and $J=\frac{3}{2}$, e.g. $N$ and $\Delta(1232)$. With the QS Lagrangian, we have derived several constraints on the masses and coupling constants, which characterize the multiplet nature of the quartet. In the present work, we develop the previous study to find a set of baryons suitable for the chiral quartet of spin-$\frac{3}{2}$ baryons. Considering $J=\frac{3}{2}$, the quartet consists of $P_{33}$, $D_{33}$, $D_{13}$, and $P_{13}$. Among various candidates for this set, we adopted a particular assignment in Ref. [30]: $\Delta(1232)$, $\Delta(1700)$, $N(1520)$, $N(1720)$. It is an important question if there is other assignment suitable for the quartet. One interesting assignment is a set ($\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$). Glozman mentioned the possibility that the approximate degeneracy of these four baryons is a consequence of the effective chiral restoration [6]. If this is the case, there are two possibilities. The first one is that the four baryons form the chiral quartet. The second one is that two $\Delta$s belong to $(\frac{3}{2},0)\oplus(0,\frac{3}{2})$ and two $N^{}$ belong to $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$. We can study the first case using the QS Lagrangian. In order to take into account $\pi N$ interactions in the QS, it is necessary to determine the nucleon’s chiral representation. In standard linear $\sigma$ models of Gell-Mann-Levy type [21] the nucleon belongs to $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$. In the mirror models [22, 23, 24, 25, 26, 27], the nucleon is a mixture of two kinds of $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$. The mixing of $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ and $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ was studied in an algebraic approach [35, 36, 37] and field theoretical approaches [32, 33]. In non-relativistic quark models the nucleon wave-functions also correspond to the mixing of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ and $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$. In the present study, we assume the nucleon to be saturated with the fundamental representation $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ due to the following reasons. The linear $\sigma$ models qualitatively describe the chiral properties of the nucleon. For instance, the linear $\sigma$ models describe $g_{A}=1$ in qualitative agreement with $g_{A}^{\rm(exp)}=1.267\pm 0.004$. Secondly, the nucleon belongs to $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$, if the nucleon operator has spatially symmetric property [29]. This paper is organized as follows. In section 2, we define the baryon fields and derive their $SU(2)_{A}$ transformation properties. In section 3, we construct the $SU(2)_{R}\times SU(2)_{L}$ Lagrangian, such as mass terms and $\pi NR$ interactions with the use of the projection technique. Here $R$ denotes the member of the chiral quartet. Although the QS Lagrangian is not new, we generalize the formulation given in the previous study in a assignment-free manner in order make it feasible to test various assignment. With the Lagrangian, we derive several constraints on the properties of the quartet. Because the projection method is complicated, we show an alternative derivation of some of the present results, using chiral algebra in Appendix B. Numerical results are shown in section 4. Considering the masses, we find two suitable assignments ($\Delta(1600)$, $\Delta(1940)$, $N(1520)$, $N(1720)$) and ($\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$). We discuss the properties of the quartet for these cases together with the assignment ($\Delta(1232)$, $\Delta(1700)$, $N(1520)$, $N(1720)$). The final section is devoted to a summary. ## 2 Chiral Properties of Baryon Fields In this section, we consider baryon fields consisting of three quarks, which serves as a preparation for the projection method. Baryon fields consisting of three quarks in a local form are generally described as (1) where $q(x)=(u(x),\;d(x))^{T}$ is an iso-doublet quark field at location $x$, the superscript $T$ represents the transpose and the indices $a,\;b$ and $c$ represent the color. The antisymmetric tensor in color space $\epsilon_{abc}$ ensures the baryons being color singlets. From now on, we shall omit the color indices and space-time coordinates. $\Gamma_{1,2}$ describe Dirac and isospin matrices. With a suitable choice of $\Gamma_{1,2}$, a baryon field is defined so that it forms an irreducible representation of the Lorentz and isospin groups. Concerning $J=\frac{3}{2}$, there are three possible baryon fields with $I=\frac{1}{2}$; \[N_{V}^{\mu} =(\tilde{q}\gamma_{\nu}q)\Gamma^{\mu\nu}_{3/2}\gamma_{5}q,\] (2a) \[N_{A}^{\mu} =(\tilde{q}\gamma_{\nu}\gamma_{5}\tau^{i}q)\Gamma^{\mu\nu}_{3/2} \tau^{i}q,\] (2b) \[N_{T}^{\mu} =i(\tilde{q}\sigma_{\alpha\beta}\tau^{i}q)\Gamma^{\mu\alpha}_{3/2 }\gamma^{\beta}\gamma_{5}\tau^{i}q,\] (2c) and two with \[I=\frac{3}{2}\] ; \[\Delta_{A}^{\mu i} =(\tilde{q}\gamma_{\nu}\gamma_{5}\tau^{j}q)\Gamma^{\mu\nu}_{3/2}P ^{ij}_{3/2}q,\] (2d) \[\Delta_{T}^{\mu i} =i(\tilde{q}\sigma_{\alpha\beta}\tau^{j}q)\Gamma^{\mu\alpha}_{3/2 }\gamma^{\beta}\gamma_{5}P^{ij}_{3/2}q.\] (2e) where $\tilde{q}=q^{T}C(i\tau_{2})\gamma_{5}$ is a transposed quark field. Here we employ an isospurion formalism [38, 39] for an isospin-$\frac{3}{2}$ projection operator $P^{ij}_{3/2}$, which is given by $P^{ij}_{3/2}=\delta^{ij}-\frac{1}{3}\tau^{i}\tau^{j}.$ Similarly, $\Gamma^{\mu\nu}_{3/2}$ is a local spin-$\frac{3}{2}$ projection operator defined by $\Gamma^{\mu\nu}_{3/2}=g^{\mu\nu}-\frac{1}{4}\gamma^{\mu}\gamma^{\nu}$. In the present work, we consider only on-shell spin-$\frac{3}{2}$ states. In order to consider off-shell spin-$\frac{3}{2}$ baryons, we need to employ the non-local projector instead of the local one [40, 41, 42, 43]. Note that the baryon fields Eqs. (2) are not independent [44, 45, 46]. In addition, they belong to reducible chiral representations, which leads to unphysical mixings of different chiral representations [29]. The cause of the unphysical chiral mixings is the fact that Eqs. (2) are not totally anti-symmetric; they are anti-symmetric only for the interchange between the first and second quarks. Considering the Fierz transformation as the anti-symmetrization of the second and third quarks, we obtain the totally-antisymmetric baryon fields \[N_{1}^{\mu} =\frac{\sqrt{3}}{4}N_{V}^{\mu}+\frac{1}{4\sqrt{3}}N_{A}^{\mu},\] (3a) \[\Delta_{1}^{\mu i} =\frac{1}{2}\Delta_{A}^{\mu i}.\] (3b) These totally-antisymmetric combinations belong to the irreducible chiral multiplet [29]. The derivation of Eq. (3) is shown in Appendix A. With the baryon fields consisting of the quark fields, it is straightforward but tedious task to derive their $SU(2)_{A}$ transformations by using that of the quark field : $\delta_{5}^{\vec{a}}q=\frac{1}{2}i{\mbox{\boldmath${a}$}}\cdot{\mbox{\boldmath ${\tau}$}}\gamma_{5}q$ with $\vec{a}$ being the infinitesimal parameters for $SU(2)_{A}$. We obtain \[\delta_{5}^{\vec{a}}N_{1}^{\mu} =\frac{1}{2}\left(\frac{5}{3}i{\mbox{\boldmath${a}$}}\cdot{\mbox{ \boldmath${\tau}$}}\gamma_{5}N_{1}^{\mu}+\frac{4}{\sqrt{3}}i\gamma_{5}{\mbox{ \boldmath${a}$}}\cdot{\mbox{\boldmath${\Delta}$}_{1}^{\mu}}\right),\] (4a) \[\delta_{5}^{\vec{a}}\Delta_{1}^{\mu i} =\frac{1}{2}\left(\frac{4}{\sqrt{3}}i\gamma_{5}a^{j}P^{ij}_{\frac {3}{2}}N_{1}^{\mu}-\frac{2}{3}i\tau^{i}\gamma_{5}{\mbox{\boldmath${a}$}}\cdot{ \mbox{\boldmath${\Delta}$}_{1}^{\mu}}+i{\mbox{\boldmath${a}$}}\cdot{\mbox{ \boldmath${\tau}$}}\gamma_{5}\Delta_{1}^{\mu i}\right),\] (4b) which contain off-diagonal terms $\delta_{5}^{\vec{a}}N_{1}^{\mu}\sim{\Delta_{1}^{\mu i}}$ and $\delta_{5}^{\vec{a}}\Delta_{1}^{\mu i}\sim N_{1}^{\mu}$ as well as the diagonal ones. They restrict possible chiral invariant terms, similar to the case of $(\sigma,\pi)$ in the linear sigma model. For later convenience, we define diquark fields contained in the spin-$\frac{3}{2}$ baryon fields: a Lorentz vector isoscalar diquark $V^{\mu}$ ($I(J)^{P}=0(1)^{-}$), a Lorentz axial-vector isovector diquark $A^{\mu i}$ ($1(1)^{+}$) \[V^{\mu} =\tilde{q}\gamma^{\mu}q,\] (5a) \[A^{\mu i} =\tilde{q}\gamma^{\mu}\gamma_{5}\tau^{i}q.\] (5b) It is easy to check that $V^{\mu}$ and $A^{\mu i}$ correspond to $\sigma$ and $\vec{\pi}$ mesons in chiral transformation properties, which is a key of the projection method. We introduce the other set of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$: $(N_{2}^{\mu},\Delta_{2}^{\mu i})$, where they have the same spin and isospin as the original ones $(N_{1}^{\mu},\Delta_{1}^{\mu i})$, but the opposite $SU(2)_{A}$ transformation properties in sign, i.e., \[\delta_{5}^{\vec{a}}N_{2}^{\mu} =-\frac{1}{2}\left(\frac{5}{3}i{\mbox{\boldmath${a}$}}\cdot{\mbox {\boldmath${\tau}$}}\gamma_{5}N_{2}^{\mu}+\frac{4}{\sqrt{3}}i\gamma_{5}{\mbox{ \boldmath${a}$}}\cdot{\mbox{\boldmath${\Delta}$}_{2}^{\mu}}\right)\] (6a) \[\delta_{5}^{\vec{a}}\Delta_{2}^{\mu i} =-\frac{1}{2}\left(\frac{4}{\sqrt{3}}i\gamma_{5}a^{j}P^{ij}_{ \frac{3}{2}}N_{2}^{\mu}-\frac{2}{3}i\tau^{i}\gamma_{5}{\mbox{\boldmath${a}$}} \cdot{\mbox{\boldmath${\Delta}$}_{2}^{\mu}}+i{\mbox{\boldmath${a}$}}\cdot{ \mbox{\boldmath${\tau}$}}\gamma_{5}\Delta_{2}^{\mu i}\right).\] (6b) This property is referred to as the mirror assignment [25], and we refer to $(N_{1}^{\mu},\Delta_{1}^{\mu i})$ as naive, and to $(N_{2}^{\mu},\Delta_{2}^{\mu i})$ as mirror. There is a correspondence of the chiral transformation properties between the naive and mirror sets, \[(N_{1R}^{\mu},N_{1L}^{\mu},\Delta_{1R}^{\mu i},\Delta_{1L}^{\mu i })\leftrightarrow(N_{2L}^{\mu},N_{2R}^{\mu},\Delta_{2L}^{\mu i},\Delta_{2R}^{ \mu i}),\] (7) where the indices $R$ and $L$ denote the left- and right-handed projection with the projection operator $P_{R,L}=(1\pm\gamma_{5})/2$. The right-handed parts of $N_{1}^{\mu}$ and $\Delta_{1}^{\mu i}$ have the same chiral transformation properties as the left-handed parts of $N_{2}^{\mu}$ and $\Delta_{2}^{\mu i}$, and vice versa. Note that we defined $N_{2}$ and $\Delta_{2}$ by their transformation properties Eqs. (6). It is useful to define the baryon fields for $N_{2}$ and $\Delta_{2}$. It is impossible to describe them in terms of local three-quark fields. Since baryons are composite particles, there are generally various possible expressions for $N_{2}$ and $\Delta_{2}$. For example, we can describe them by using baryon operators having a derivative, \[N_{V}^{\prime\mu} ={/\hskip-6.544134pt{D}}V_{\nu}\Gamma^{\mu\nu}_{3/2}\gamma_{5}q,\] (8a) \[N_{A}^{\prime\mu} ={/\hskip-6.544134pt{D}}A_{\nu}^{i}\Gamma^{\mu\nu}_{3/2}\tau^{i}q,\] (8b) \[\Delta_{A}^{\prime\mu i} ={/\hskip-6.544134pt{D}}A_{\nu}^{j}\Gamma^{\mu\nu}_{3/2}P^{ij}_{3 /2}q,\] (8c) where $D_{\mu}$ denotes a covariant derivative. The mirror fields $N_{2}^{\mu}$ and $\Delta_{2}^{\mu}$ are obtained by the same equations as Eqs. (3) with substitution of the primed fields $(N_{V}^{\prime\mu},N_{A}^{\prime\mu},\Delta_{A}^{\prime\mu i})$ for the original fields $(N_{V}^{\mu},N_{A}^{\mu},\Delta_{A}^{\mu i})$. Although they would not be a unique possibility for the microscopic description of the mirror fields, Eqs. (8) are enough for the present purpose to construct the chiral invariant Lagrangian. ## 3 Lagrangian Now, we proceed to the construction of the $SU(2)_{R}\times SU(2)_{L}$ Lagrangian. It is straightforward to show the chiral invariance of the kinetic terms: ${\cal L}_{K}=\bar{N}_{{\rm n}\mu}(i\ {/\hskip-6.544134pt{\partial}})N_{\rm n}^ {\mu}+\bar{\Delta}_{{\rm n}\mu}^{i}(i\ {/\hskip-6.544134pt{\partial}})\Delta_{ \rm n}^{\mu i},({\rm n}=1,2).$ In order to find interaction terms for higher-dimensional chiral multiplets, it is useful to employ the projection method. ### Mass terms and $\pi RR$ terms The vector and axial-vector diquarks belong to the chiral multiplet $(\frac{1}{2},\frac{1}{2})$, and $V_{\mu}^{2}+A_{\mu}^{2}$ is a chiral scalar. The Gell-Mann-Levy type interaction for the quark $\bar{q}U_{5}q$ is also a chiral scalar, where $U_{5}=\sigma+i\gamma_{5}{\mbox{\boldmath${\tau}$}}\cdot{\mbox{\boldmath${\pi}$}}$. Obviously, the following combination of these two terms is also a chiral scalar, \[\bar{q}(V_{\mu}^{2}+A_{\mu}^{2})U_{5}q.\] (9) This term contains the direct products of the quark and diquark : $V^{\mu}q$ and $A^{\mu i}q$. They are decomposed into the irreducible parts as \[\left\{\begin{array}[]{l}V^{\mu}q=\gamma_{5}N_{V}^{\mu}+(J=\frac{ 1}{2}\mbox{terms}),\\ A^{\mu i}q=\Delta_{A}^{\mu i}+\frac{1}{3}\tau^{i}N_{A}^{\mu}+(J=\frac{1}{2} \mbox{terms}),\end{array}\right.\] (10a) \[\left\{\begin{array}[]{l}\bar{q}(V^{\mu})^{\dagger}=-\bar{N}_{V}^ {\mu}\gamma_{5}+(J=\frac{1}{2}\mbox{terms}),\\ \bar{q}(A^{\mu i})^{\dagger}=\bar{\Delta}_{A}^{\mu i}+\frac{1}{3}\bar{N}_{A}^{ \mu}\tau^{i}+(J=\frac{1}{2}\mbox{terms}),\end{array}\right.\] (10b) Substituting Eqs. (10) into the chiral invariant term (9), we obtain \[{\cal L}_{MRR}^{(1)}= g_{1}\left(\bar{\Delta}_{1\mu}^{i}U_{5}\Delta_{1}^{\mu i}-\frac{ 3}{4}\bar{N}_{1\mu}U_{5}N_{1}^{\mu}+\frac{1}{12}\bar{N}_{1\mu}\tau^{i}U_{5} \tau^{i}N_{1}^{\mu}+\frac{\sqrt{3}}{6}\left(\bar{N}_{1\mu}\tau^{i}U_{5}\Delta_ {1}^{\mu i}+({\rm H.c.})\right)\right)\] \[+(J=\frac{1}{2}\mbox{terms}),\] (11) where we omit $J=\frac{1}{2}$ terms, which contain the Gell-Mann-Levy type interaction with local nucleon operators $N_{V}=V_{\mu}\gamma^{\mu}q$ and $N_{A}=A_{\mu}^{i}\gamma^{\mu}\gamma_{5}\tau^{i}q$. The transition terms between $J=\frac{1}{2}$ and $\frac{3}{2}$ fields vanish due to $\gamma_{\mu}\Delta_{1}^{\mu i}=\gamma_{\mu}N_{1}^{\mu}=0$. The Lagrangian (11) describes several kinds of the interactions; the first three terms describe the diagonal interactions for $N_{1}^{\mu}$ and $\Delta_{1}^{\mu i}$ with $\sigma$ and $\pi$, and the fourth term describes a transition between $N_{1}^{\mu}$ and $\Delta_{1}^{\mu i}$ with $\pi$, where a $\sigma N_{1}\Delta_{1}$ coupling vanishes due to $\tau^{i}\Delta_{1}^{\mu i}=0$. The diagonal interactions with $\sigma$ generate the masses of $N_{1}^{\mu}$ and $\Delta_{1}^{\mu i}$ in the presence of the SBCS $\sigma\to\langle\sigma\rangle=f_{\pi}=92.4$ [MeV]. We obtain a mass relation $\|m_{\Delta_{1}}\|:\|m_{N_{1}}\|=2:1$. If we assign $N_{1}^{\mu}$ with $N(1520)$, which is the lowest lying state for $I(J)=\frac{1}{2}(\frac{3}{2})$, its partner $\Delta_{1}^{\mu i}$ has the mass of $2\times 1520\sim 3000$ MeV. We do not find a baryon suitable for this mass relation in the experimental data [47]. There are several directions to solve this mass problem: the inclusion of higher order terms in the Lagrangian and of higher-order diagrams, the extension of the chiral basis such as $(\frac{3}{2},0)\oplus(0,\frac{3}{2})$ and of the mirror assignment. It was shown [34] that the inclusion of the mirror assignment reasonably reproduces the masses and some properties of observed baryons. Using Eq. (7), we find a chiral invariant interaction term \[{\cal L}_{MRR}^{(2)}=g_{2}\left(\bar{\Delta}_{2\mu}^{i}U_{5}^{ \dagger}\Delta_{2}^{\mu i}-\frac{3}{4}\bar{N}_{2\mu}U_{5}^{\dagger}N_{2}^{\mu} +\frac{1}{12}\bar{N}_{2\mu}\tau^{i}U_{5}^{\dagger}\tau^{i}N_{2}^{\mu}+\frac{ \sqrt{3}}{6}\left(\bar{N}_{2\mu}\tau^{i}U_{5}^{\dagger}\Delta_{2}^{\mu i}+{\rm H .c.}\right)\right),\] (12) which is almost the same as Eq. (11). The difference appears in the signs of the terms accompanying $\pi$$(U_{5}\to U_{5}^{\dagger})$, which is a feature of the mirror assignment [25]. Considering Eqs. (4), (6) and (7), $\bar{\Delta}_{1R}\Delta_{2L}+\bar{N}_{1R}N_{2L}$ is chiral invariant, which leads to the following term, \[{\cal L}_{RR}=-m_{0}\left(\bar{\Delta}_{1\mu}^{i}\Delta_{2}^{\mu i }+\bar{N}_{1\mu}N_{2}^{\mu}+{\rm H.c.}\right),\] (13) which describes off-diagonal mass terms between $N_{1}^{\mu}$ and $N_{2}^{\mu}$ and between $\Delta_{1}^{\mu i}$ and $\Delta_{2}^{\mu i}$. The parameter $m_{0}$ describes a chiral scalar, so called mirror mass [25]. The mass terms included in ${\cal L}_{MRR}^{(1)}+{\cal L}_{MRR}^{(2)}+{\cal L}_{RR}$ are rewritten in the following matrix forms \[{\cal L}_{M}=-(\bar{\Delta}_{1\mu}^{i},\bar{\Delta}_{2\mu}^{i}) \left(\begin{array}[]{cc}-g_{1}f_{\pi}&m_{0}\\ m_{0}&-g_{2}f_{\pi}\end{array}\right)\left(\begin{array}[]{c}\Delta_{1}^{\mu i }\\ \Delta_{2}^{\mu i}\end{array}\right)-(\bar{N}_{1\mu},\bar{N}_{2\mu})\left( \begin{array}[]{cc}\frac{1}{2}g_{1}f_{\pi}&m_{0}\\ m_{0}&\frac{1}{2}g_{2}f_{\pi}\end{array}\right)\left(\begin{array}[]{c}N_{1}^{ \mu}\\ N_{2}^{\mu}\end{array}\right).\] (14) Because of the off-diagonal terms in these mass matrices, physical states and their masses are obtained through the diagonalization of the mass matrices. Note that the mass eigen-values can take both positive and negative values. A state with a negative eigen-value can be transformed into a state with a positive mass, but has opposite parity to the original state. It is carried out by multiplying a state having negative mass by $\gamma_{5}$ [25]. In the present paper, we consider the case that two states form a pair of positive and negative parity states both in $\Delta$ and $N^{}$ sectors. For the $\Delta$ part in Eq. (14), we obtain the mass eigen-values of two $\Delta$ states \[m_{\Delta^{\pm}} =\frac{1}{2}\left[\sqrt{(g_{1}-g_{2})^{2}f_{\pi}^{2}+4m_{0}^{2}} \mp(g_{1}+g_{2})f_{\pi}\right],\] (15) and the eigen-states \[\Delta_{+}^{\mu i} =\cos\theta_{\Delta}\Delta_{1}^{\mu i}+\sin\theta_{\Delta}\Delta_ {2}^{\mu i},\] (16a) \[\Delta_{-}^{\mu i} =\gamma_{5}(-\sin\theta_{\Delta}\Delta_{1}^{\mu i}+\cos\theta_{ \Delta}\Delta_{2}^{\mu i}),\] (16b) \[\tan 2\theta_{\Delta} =\frac{2m_{0}}{(g_{2}-g_{1})f_{\pi}}.\] (16c) Here we define $\Delta_{+}^{\mu i}$ and $\Delta_{-}^{\mu i}$ as positive and negative parity states, respectively, where the indices $\pm$ denote the parity. Hence $\Delta_{+}^{\mu i}$ and $\Delta_{-}^{\mu i}$ are identified with $\Delta(P_{33})$ and $\Delta(D_{33})$, respectively. Note that $\gamma_{5}$ in Eq. (16b) appears due to the parity redefinition. Similarly, for $N^{}$ part, we obtain the mass eigen-values \[m_{N^{\pm}} =\frac{1}{2}\left[\sqrt{\frac{1}{4}(g_{1}-g_{2})^{2}f_{\pi}^{2}+4 m_{0}^{2}}\pm\frac{(g_{1}+g_{2})f_{\pi}}{2}\right],\] (17) and the eigen-states \[N_{+}^{\mu} =\cos\theta_{N}N_{1}^{\mu}+\sin\theta_{N}N_{2}^{\mu},\] (18a) \[N_{-}^{\mu} =\gamma_{5}(-\sin\theta_{N}N_{1}^{\mu}+\cos\theta_{N}N_{2}^{\mu}),\] (18b) \[\tan 2\theta_{N} =\frac{4m_{0}}{(g_{1}-g_{2})f_{\pi}}.\] (18c) $N_{+}^{\mu}$ and $N_{-}^{\mu}$ are identified with $N(D_{13})$ and $N(P_{13})$, respectively. Again, $\gamma_{5}$ in Eq. (18b) appears due to the parity redefinition. The four masses $m_{\Delta^{\pm}}$ and $m_{N^{\pm}}$ are given by the three parameters $g_{1}$, $g_{2}$ and $m_{0}$, which offers constraints on the four masses [34], \[(m_{\Delta^{+}}+m_{\Delta^{-}}) \geq(m_{N^{+}}+m_{N^{-}}),\] (19a) \[m_{\Delta^{-}}-m_{\Delta^{+}} =2(m_{N^{+}}-m_{N^{-}}).\] (19b) The inequality in the first line of Eq. (19) is controlled by $m_{0}$. Thus, the mass splittings and average masses are determined by chiral symmetry and the mirror mass $m_{0}$. It is worthwhile considering the correspondence between the basis states and the physical states. Obviously, the mixing angles vanish in the absence of the mirror mass; $\theta_{N},\theta_{\Delta}\to 0$ for $m_{0}\to 0$. In this limit, the naive and mirror sectors decouple, and the physical states correspond to the basis states : $(\Delta_{+}^{\mu i},N_{+}^{\mu})\to(\Delta_{1}^{\mu i},N_{1}^{\mu})$ and $(\Delta_{-}^{\mu i},N_{-}^{\mu})\to(\Delta_{2}^{\mu i},N_{2}^{\mu})$. It should be noted that the decoupling of the two sectors does not violate chiral invariance. Contrarily, the two sectors are maximally mixed in the $m_{0}$ dominant case : $\theta_{N},\theta_{\Delta}=\pi/4$. The Lagrangians (11) and (12) contain the one-pion interaction terms between the spin-$\frac{3}{2}$ baryons ($\pi RR$) as well as the mass terms. Having the four spin-$\frac{3}{2}$ baryons, there are ten coupling constants $g_{\pi RR}$; four diagonal and six off-diagonal terms. All the ten coupling constants are functions of $g_{1},g_{2}$ and $m_{0}$, which are determined by the masses. It is straightforward to derive the $\pi RR$ coupling constants, $g_{\pi RR}$ from Eqs. (11) and (12). For $\Delta$ part, we obtain \[\Delta-\Delta \left\{\begin{array}[]{l}g_{\pi\Delta^{+}\Delta^{+}}=-(g_{1}\cos^ {2}\theta_{\Delta}-g_{2}\sin^{2}\theta_{\Delta})\\ g_{\pi\Delta^{-}\Delta^{-}}=(g_{1}\sin^{2}\theta_{\Delta}-g_{2}\cos^{2}\theta_ {\Delta})\\ g_{\pi\Delta^{+}\Delta^{-}}=(g_{1}+g_{2})\cos\theta_{\Delta}\sin\theta_{\Delta }\\ \end{array}\right.\] (20a) which are defined by \[{\cal L}=-g_{\pi\Delta_{P}\Delta_{P^{\prime}}}\bar{\Delta}_{P\mu i}(i\gamma_{5 }{\mbox{\boldmath${\tau}$}}\cdot{\mbox{\boldmath${\pi}$}})\Gamma_{5}\Delta_{P^ {\prime}}^{\mu i}.\] Here \[P\] and \[P^{\prime}\] denote parity, i.e., \[P,P^{\prime}= +\] or \[-\] , and \[\Gamma_{5}=1\] for \[P=P^{\prime}\] and \[\gamma_{5}\] for \[P\neq P^{\prime}\] . For \[N^{}\] part, we obtain \[N^{}-N^{} \left\{\begin{array}[]{l}g_{\pi N^{+}N^{+}}=\frac{5}{6}(g_{1}\cos ^{2}\theta_{N}-g_{2}\sin^{2}\theta_{N})\\ g_{\pi N^{-}N^{-}}=-\frac{5}{6}(g_{1}\sin^{2}\theta_{N}-g_{2}\cos^{2}\theta_{N })\\ g_{\pi N^{+}N^{-}}=-\frac{5}{6}(g_{1}+g_{2})\cos\theta_{N}\sin\theta_{N}\\ \end{array}\right.\] (20b) which are defined by \[{\cal L}=-g_{\pi N_{P}N_{P^{\prime}}}\bar{N}_{P\mu}(i\gamma_{5}{\mbox{ \boldmath${\tau}$}}\cdot{\mbox{\boldmath${\pi}$}})\Gamma_{5}N_{P^{\prime}}^{\mu}\] . For \[N^{}\] - \[\Delta\] transition terms, \[N^{}-\Delta \left\{\begin{array}[]{l}g_{\pi N^{+}\Delta^{+}}=-\frac{\sqrt{3}} {3}(g_{1}\cos\theta_{\Delta}\cos\theta_{N}-g_{2}\sin\theta_{\Delta}\sin\theta_ {N})\\ g_{\pi N^{+}\Delta^{-}}=\frac{\sqrt{3}}{3}(g_{2}\cos\theta_{\Delta}\sin\theta_ {N}+g_{1}\cos\theta_{N}\sin\theta_{\Delta})\\ g_{\pi N^{-}\Delta^{+}}=-\frac{\sqrt{3}}{3}(g_{1}\cos\theta_{\Delta}\sin\theta _{N}+g_{2}\cos\theta_{N}\sin\theta_{\Delta})\\ g_{\pi N^{-}\Delta^{-}}=\frac{\sqrt{3}}{3}(g_{1}\sin\theta_{\Delta}\sin\theta_ {N}-g_{2}\cos\theta_{N}\cos\theta_{\Delta})\\ \end{array}\right.\] (20c) which are defined by \[{\cal L}=-g_{\pi N_{P}\Delta_{P^{\prime}}}\bar{N}_{P\mu}(i\gamma_{5}\Gamma_{5} )\pi^{i}\Delta_{P^{\prime}}^{\mu i}\] . In order to understand the features of $g_{\pi RR}$, it is useful to consider the axial-charges, which are obtained by the Noether theorem \[\Delta-\Delta \left\{\begin{array}[]{l}g_{A}^{\Delta^{\pm}\Delta^{\pm}}=\pm\cos 2 \theta_{\Delta},\\ g_{A}^{\Delta^{+}\Delta^{-}}=-\sin 2\theta_{\Delta},\end{array}\right.\] \[N^{}-N^{} \left\{\begin{array}[]{l}g_{A}^{N^{\pm}N^{\pm}}=\pm\frac{5}{3} \cos 2\theta_{N},\\ g_{A}^{N^{+}N^{-}}=-\frac{5}{3}\sin 2\theta_{N},\end{array}\right.\] \[N^{}-\Delta \left\{\begin{array}[]{l}g_{A}^{N^{\pm}\Delta^{\pm}}=\pm\frac{4} {\sqrt{3}}\cos(\theta_{N}+\theta_{\Delta}),\\ g_{A}^{N^{\pm}\Delta^{\mp}}=\pm\frac{4}{\sqrt{3}}\sin(\theta_{N}+\theta_{ \Delta}).\end{array}\right.\] (21) In the limit $\theta_{N,\Delta}\to 0$ ($m_{0}\to 0$), the absolute values of the parity-non-changing interactions reach the maximum values: $\|g_{A}^{\Delta^{\pm}\Delta^{\pm}}\|\to 1$, $\|g_{A}^{N^{\pm}N^{\pm}}\|\to\frac{5}{3}$ and $\|g_{A}^{N^{\pm}\Delta^{\pm}}\|\to\frac{4}{\sqrt{3}}$, while the parity-changing terms vanish $g_{A}^{\Delta^{+}\Delta^{-}}=g_{A}^{N^{+}N^{-}}=g_{A}^{N^{\pm}\Delta^{\mp}}=0$. The mixing angles larger, as $m_{0}$ becomes larger. Since the naive and mirror sectors have the opposite axial-charges, the mixing of the two sectors suppresses the parity-non-changing interactions and enhance the parity-changing interactions. In the $m_{0}$-dominance, the parity-non-changing interactions vanish $g_{A}^{\Delta^{\pm}\Delta^{\pm}}=g_{A}^{N^{\pm}N^{\pm}}=g_{A}^{N^{\pm} \Delta^{\pm}}\to 0$, while the parity-changing terms reach the maximum values $\|g_{A}^{\Delta^{+}\Delta^{-}}\|=1$, $\|g_{A}^{N^{+}N^{-}}\|=\frac{5}{3}$ and $\|g_{A}^{N^{\pm}\Delta^{\mp}}\|=\frac{4}{\sqrt{3}}$. Of course, $g_{\pi RR}$ have the same features as the axial-charges due to the Goldberger-Treiman (GT) relations: \[\Delta-\Delta \left\{\begin{array}[]{l}f_{\pi}g_{\pi\Delta^{+}\Delta^{+}}=\cos 2 \theta_{\Delta}m_{\Delta^{+}},\\ f_{\pi}g_{\pi\Delta^{-}\Delta^{-}}=-\cos 2\theta_{\Delta}m_{\Delta^{-}},\\ f_{\pi}g_{\pi\Delta^{+}\Delta^{-}}=-\frac{1}{2}\sin 2\theta_{\Delta}(m_{\Delta ^{+}}-m_{\Delta^{-}}),\end{array}\right.\] \[N^{}-N^{} \left\{\begin{array}[]{l}f_{\pi}g_{\pi N^{+}N^{+}}=\frac{5}{3} \cos 2\theta_{N}m_{N^{+}},\\ f_{\pi}g_{\pi N^{-}N^{-}}=-\frac{5}{3}\cos 2\theta_{N}m_{N^{-}},\\ f_{\pi}g_{\pi N^{+}N^{-}}=-\frac{5}{6}\sin 2\theta_{N}(m_{N^{+}}-m_{N^{-}}), \end{array}\right.\] \[N^{}-\Delta \left\{\begin{array}[]{l}f_{\pi}g_{\pi N^{+}\Delta^{+}}=\frac{2}{ \sqrt{3}}\cos(\theta_{N}+\theta_{\Delta})(m_{N^{+}}+m_{\Delta^{+}}),\\ f_{\pi}g_{\pi N^{+}\Delta^{-}}=-\frac{2}{\sqrt{3}}\sin(\theta_{N}+\theta_{ \Delta})(m_{N^{+}}-m_{\Delta^{-}}),\\ f_{\pi}g_{\pi N^{-}\Delta^{+}}=-\frac{2}{\sqrt{3}}\sin(\theta_{N}+\theta_{ \Delta})(m_{N^{-}}-m_{\Delta^{+}}),\\ f_{\pi}g_{\pi N^{-}\Delta^{-}}=-\frac{2}{\sqrt{3}}\cos(\theta_{N}+\theta_{ \Delta})(m_{N^{-}}+m_{\Delta^{-}}).\end{array}\right.\] (22) ### Interaction with the nucleon Next, we construct the interactions between the nucleon $(N)$ and the chiral quartet. As we have discussed in the introduction, we assume that the nucleon belongs to $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$. With the nucleon’s chiral multiplet, we can classify the products of the chiral properties of $N\otimes\Delta$: \[N\otimes\Delta =\left[\left(\frac{1}{2},1\right)\oplus\left(1,\frac{1}{2}\right) \right]\otimes\left[\left(\frac{1}{2},0\right)\oplus\left(0,\frac{1}{2}\right)\right]\] \[=\left\{\begin{array}[]{lcl}(1,0)\oplus(0,1)&\mbox{for}&(N_{1}^{ \mu},\Delta_{1}^{\mu i}),\\ \left(\frac{1}{2},\frac{1}{2}\right)&\mbox{for}&(N_{2}^{\mu},\Delta_{2}^{\mu i }),\end{array}\right.\] (23) where we omit four-meson terms $(1,1)$ and $\left[\left(\frac{3}{2},\frac{1}{2}\right)\oplus\left(\frac{1}{2},\frac{3}{2} \right)\right]$. In the derivation of Eq. (23), it is important to take into account the chirality conservation. This classification implies that chiral invariant interactions between $N$ and $(N_{1}^{\mu},\Delta_{1}^{\mu i})$ accompany two-meson fields, while those between $N$ and $(N_{2}^{\mu},\Delta_{2}^{\mu i})$ accompany one-meson fields. We find two chiral scalars $\sigma V_{\mu}+i{\mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${A}$}}_{\mu}$ and $\bar{N}U_{5}q$. Multiplying them, we find two chiral invariant terms; $(-i)\bar{N}U_{5}[(\partial^{\mu}\sigma)V_{\mu}+i{\mbox{\boldmath${(}$}\partial ^{\mu}\pi)}\cdot{\mbox{\boldmath${A}$}_{\mu}}]q$ , $(-i)\bar{N}(\partial^{\mu}U_{5})(\sigma V_{\mu}+i{\mbox{\boldmath${\pi}$}} \cdot{\mbox{\boldmath${A}$}_{\mu}})q$. Using Eqs. (10), we obtain the chiral invariant interaction terms between $N$ and $(N^{\mu}_{1},\;\Delta^{\mu i}_{1})$ \[{\cal L}_{MNR}^{(1)} = \frac{g_{3}}{\Lambda^{2}}\left[\bar{N}O_{1\mu}^{i}\Delta_{1}^{\mu i }+\bar{N}O_{2\mu}N_{1}^{\mu}\right]+({\rm H.c.}),\] (24a) \[{\cal L}_{MNR}^{(2)} = \frac{g_{4}}{\Lambda^{2}}\left[\bar{N}O_{3\mu}^{i}\Delta_{1}^{\mu i }+\bar{N}O_{4\mu}N_{1}^{\mu}\right]+({\rm H.c.}),\] (24b) where the dimensional parameter \[\Lambda\] [mass] is introduced to keep the coupling constants \[g_{3}\] and \[g_{4}\] dimensionless. We also introduce shorthand notations \[O_{\rm n}\;({\rm n}=1,\cdots 4)\] for mesonic operators \[O_{1}^{\mu i} =U_{5}(\partial^{\mu}\pi^{i}),\] (24c) \[O_{2}^{\mu} =-\frac{\sqrt{3}}{2}U_{5}\left((\partial^{\mu}\sigma)\gamma_{5}+ \frac{1}{3}(i\partial^{\mu}{\mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${ \tau}$}})\right),\] (24d) \[O_{3}^{\mu i} =(\partial^{\mu}U_{5})(\pi^{i}),\] (24e) \[O_{4}^{\mu} =-\frac{\sqrt{3}}{2}(i\partial^{\mu}U_{5})\left(\sigma\gamma_{5}+ \frac{1}{3}i{\mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${\tau}$}}\right).\] (24f) One may think it possible to construct similar interaction terms for the mirror fields by the replacement Eq. (7). However, such terms are forbidden by chirality conservation, as is shown in Eq. (23) ¹. The mirror fields have one-meson interactions with the nucleon. It can be constructed by using the chiral invariant operators $(-i)(\sigma V_{\mu}+i{\mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${A}$}}_{ \mu})$ and $\bar{N}{/\hskip-6.544134pt{D}}q$. We obtain [FOOTNOTE:1][ENDFOOTNOTE] \[{\cal L}_{MNR}^{(3)} = \frac{g_{5}}{\Lambda}\left[\bar{N}O_{5\mu}^{i}\Delta_{2}^{\mu i}+ \bar{N}O_{6\mu}N_{2}^{\mu}\right],\] (25a) where \[O_{5}\] and \[O_{6}\] are also mesonic operators, \[O_{5}^{\mu i} =(\partial^{\mu}\pi^{i}),\] (25b) \[O_{6}^{\mu} =-\frac{\sqrt{3}}{2}(i\partial^{\mu})(\sigma\gamma_{5}+\frac{1}{3 }i{\mbox{\boldmath${\tau}$}}\cdot{\mbox{\boldmath${\pi}$}}).\] (25c) In the mass basis, ${\cal L}_{MNR}={\cal L}_{MNR}^{(1)}+{\cal L}_{MNR}^{(2)}+{\cal L}_{MNR}^{(3)}$ is rewritten as \[{\cal L}_{MNR} =\bar{N}\left[(O_{1\mu}^{i}+O_{3\mu}^{i})\cos\theta_{\Delta}+O_{5 \mu}^{i}\sin\theta_{\Delta}\right]\Delta_{+}^{\mu i}+\bar{N}\left[-(O_{1\mu}^{ i}+O_{3\mu}^{i})\sin\theta_{\Delta}+O_{5\mu}^{i}\cos\theta_{\Delta}\right] \gamma_{5}\Delta_{-}^{\mu i}\] \[+\bar{N}\left[(O_{2\mu}+O_{4\mu})\cos\theta_{N}+O_{6\mu}\sin \theta_{N}\right]N_{+}^{\mu i}+\bar{N}\left[-(O_{2\mu}+O_{4\mu})\sin\theta_{N} +O_{6\mu}\cos\theta_{N}\right]\gamma_{5}N_{-}^{\mu i},\] (26) which contains several kinds of the interaction terms, $\pi NR$, $\pi\pi NR$, $\sigma NR$ and $\sigma\sigma NR$. Among them, we consider $\pi NR$ and $\pi\pi NR$ interaction terms in order for the comparison with experiments. The $\pi N$ interactions of the chiral quartet are given by \[{\cal L}_{\pi NR} =\frac{g_{\pi N\Delta^{+}}}{\Lambda}\bar{N}(\partial_{\mu}\pi^{i} )\Delta^{+\mu i}+\frac{g_{\pi N\Delta^{-}}}{\Lambda}\bar{N}(\partial_{\mu}\pi^ {i})\gamma_{5}\Delta^{-\mu i}\] \[+\frac{g_{\pi NN^{-}}}{\Lambda}\bar{N}(\partial_{\mu}{\mbox{ \boldmath${\pi}$}}\cdot{\mbox{\boldmath${\tau}$}})\gamma_{5}N^{-\mu}+\frac{g_{ \pi NN^{+}}}{\Lambda}\bar{N}(\partial_{\mu}{\mbox{\boldmath${\pi}$}}\cdot{ \mbox{\boldmath${\tau}$}})N^{+\mu},\] (27a) where the coupling constants \[g_{\pi NN^{\pm}}\] and \[g_{\pi N\Delta^{\pm}}\] are given by \[g_{\pi N\Delta^{+}} =\frac{1}{\Lambda}(g_{5}\Lambda\sin\theta_{\Delta}+g_{3}f_{\pi} \cos\theta_{\Delta}),\] (27b) \[g_{\pi N\Delta^{-}} =\frac{1}{\Lambda}(g_{5}\Lambda\cos\theta_{\Delta}-g_{3}f_{\pi} \sin\theta_{\Delta}),\] (27c) \[g_{\pi NN^{+}} =\frac{\sqrt{3}}{6\Lambda}(g_{5}\Lambda\sin\theta_{N}+(g_{3}+3g_{ 4})f_{\pi}\cos\theta_{N}),\] (27d) \[g_{\pi NN^{-}} =\frac{\sqrt{3}}{6\Lambda}(g_{5}\Lambda\cos\theta_{N}-(g_{3}+3g_{ 4})f_{\pi}\sin\theta_{N}).\] (27e) Four $g_{\pi NR}$ are expressed in terms of three parameters $g_{3},g_{4}$ and $g_{5}$, which leads to one identity \[(\sin\theta_{\Delta}g_{\pi N\Delta^{+}}+\cos\theta_{\Delta}g_{\pi N \Delta^{-}})=2\sqrt{3}(\sin\theta_{N}g_{\pi NN^{+}}+\cos\theta_{N}g_{\pi NN^{ -}}).\] (28) Here it must be noted that the derivation of the $\pi N$ interactions is based on the assumption of the nucleon’s chiral multiplet. If the nucleon together with the negative parity resonance group into $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ with the mirror assignment, we can include three additional interactions, which spoils the constraint Eq. (28). Another possibility is that the nucleon contains $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ as well as $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$. In this case, we can include one additional interaction that have similar form to Eq. (11). With the new term, Eq. (28) becomes loose constraint and gives the ordering of the coupling constants. So, Eq. (28) is one of the most strict constraint. The point is that it is possible to improve this result without changing the masses and $\pi RR$ interactions of the quartet. We obtain two-pion interaction terms \[{\cal L}_{\pi\pi N\Delta} =\frac{g_{\pi\pi N\Delta_{+}}^{(v)}}{\Lambda}\bar{N}(\epsilon^{ abc}\pi^{a}\pi_{,\mu}^{b}\gamma_{5})\Delta_{+}^{\mu c}\;+\;\frac{g_{\pi\pi N \Delta^{+}}^{(t)}}{\Lambda}\bar{N}(\pi^{a}\pi_{,\mu}^{b}+\pi_{,\mu}^{a}\pi^{b} )(i\gamma_{5}\tau^{a})\Delta_{+}^{\mu b}\] \[+\frac{g_{\pi\pi N\Delta_{-}}^{(v)}}{\Lambda}\bar{N}(\epsilon^{ abc}\pi^{a}\pi_{,\mu}^{b})\Delta_{-}^{\mu c}\;+\;\frac{g_{\pi\pi N\Delta_{-}}^ {(t)}}{\Lambda}\bar{N}(\pi^{a}\pi_{,\mu}^{b}+\pi_{,\mu}^{a}\pi^{b})(i\tau^{a}) \Delta_{-}^{\mu b}\] (29) \[{\cal L}_{\pi\pi NN^{}} =\frac{g_{\pi\pi NN_{+}^{}}^{(s)}}{\Lambda}\bar{N}(i\gamma_{5}{ \mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${\pi}$}_{,\mu}})N_{+}^{\mu}\;+\; \frac{g_{\pi\pi NN_{+}^{}}^{(v)}}{\Lambda}\bar{N}(\epsilon^{abc}\pi^{a}\pi^{b }_{,\mu}\tau^{c})\gamma_{5}N_{+}^{\mu}\] \[+\frac{g_{\pi\pi NN_{-}^{}}^{(s)}}{\Lambda}\bar{N}(i{\mbox{ \boldmath${\pi}$}}\cdot{\mbox{\boldmath${\pi}$}_{,\mu}})N_{-}^{\mu}\;+\;\frac{ g_{\pi\pi NN_{-}^{}}^{(v)}}{\Lambda}\bar{N}(\epsilon^{abc}\pi^{a}\pi^{b}_{, \mu}\tau^{c})N_{-}^{\mu},\] (30) with \[\Delta\mbox{-sector}\left\{\begin{array}[]{ll}g_{\pi\pi N\Delta_{ +}}^{(v)}&=\frac{\cos\theta_{\Delta}}{2\Lambda}(g_{3}-g_{4}),\\ g_{\pi\pi N\Delta^{+}}^{(t)}&=\frac{\cos\theta_{\Delta}}{2\Lambda}(g_{3}+g_{4} ),\\ g_{\pi\pi N\Delta_{-}}^{(v)}&=-\frac{\sin\theta_{\Delta}}{2\Lambda}(g_{3}-g_{4 }),\\ g_{\pi\pi N\Delta_{-}}^{(t)}&=-\frac{\sin\theta_{\Delta}}{2\Lambda}(g_{3}+g_{4 }),\end{array}\right.\;\;\;N^{}\mbox{-sector}\left\{\begin{array}[]{ll}g_{\pi \pi NN_{+}^{}}^{(s)}&=+\frac{\sqrt{3}\cos\theta_{N}}{6\Lambda}(g_{3}+g_{4}), \\ g_{\pi\pi NN_{+}^{}}^{(v)}&=-\frac{\sqrt{3}\cos\theta_{N}}{6\Lambda}(g_{3}-g_ {4}),\\ g_{\pi\pi NN_{-}^{}}^{(s)}&=-\frac{\sqrt{3}\sin\theta_{N}}{6\Lambda}(g_{3}+g_ {4}),\\ g_{\pi\pi NN_{-}^{}}^{(v)}&=\frac{\sqrt{3}\sin\theta_{N}}{6\Lambda}(g_{3}-g_{ 4}),\end{array}\right.\] (31) where they are classified into three types: the symmetric (${\mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${\pi}$}_{,\mu}}$), anti-symmetric $(i\epsilon^{abc}\pi^{a}\pi^{b}_{,\mu})$ and symmetric type $(\pi^{a}\pi^{b}_{,\mu}+\pi^{a}_{,\mu}\pi^{b})$. They corresponds to an isoscalar (${\mbox{\boldmath${\pi}$}}\cdot{\mbox{\boldmath${\pi}$}_{,\mu}}$), isovector $(i\epsilon^{abc}\pi^{a}\pi^{b}_{,\mu})$ and isotensor $(\pi^{a}\pi^{b}_{,\mu}+\pi^{a}_{,\mu}\pi^{b})$. Since the two-pion coupling constants $g_{\pi\pi NR}$ contain only $g_{3}$ and $g_{4}$, their strengths are determined by the $\pi N$ coupling constants through $g_{3}=(\Lambda/f_{\pi})((g_{\pi N\Delta^{+}}-g_{\pi N\Delta^{-}})/(\cos\theta_ {\Delta}+\sin\theta_{\Delta}))$ and $g_{4}=(2\Lambda/\sqrt{3}f_{\pi})((g_{\pi NN^{}_{+}}-g_{\pi NN^{}_{-}})/(\cos \theta_{N}+\sin\theta_{N}))$. Furthermore, $g_{\pi\pi NR}$ are proportional to either $(g_{3}+g_{4})$ or $(g_{3}-g_{4})$, which provides a selection rule; either $\pi\pi$ isoscalar or isovector interaction is suppressed each for $N^{}_{\pm}$, and either the isovector or isotensor interaction is suppressed each for $\Delta_{\pm}$. Using the $SU(2)_{R}\times SU(2)_{L}$ Lagrangian, we have derived several constraints on the properties of the chiral quartet. We concentrate on the construction of the lowest-order terms and the derivation of the chiral constraints at tree level. In general, it is possible to insert chiral invariant operators such as $(\sigma^{2}+\pi^{2})^{n}$ into the chiral Lagrangians we derived. However, those terms does not change the above constraints and can be absorbed into the parameters. Regarding the $\pi RR$ interactions, it is possible to include additional interaction term with a derivative [25]. The constraint for the $\pi NR$ interactions rely on the assumption of the saturation of $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ in the nucleon. The inclusion of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ component in the nucleon causes one additional chiral invariant $\pi N$ interaction term similar to Eq. (11). In this case, four $g_{\pi NR}$ are given by four parameters. It must be noted that the inclusion of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ for the nucleon does not affect the multiplet nature of the quartet. ## 4 Results L2I2J \| Observed states ---\|--- P33 \| Δ(1232)∗∗∗∗, Δ(1600)∗∗∗, Δ(1920)∗∗∗ D33 \| Δ(1700)∗∗∗, Δ(1940)∗∗ D13 \| N(1520)∗∗∗∗, N(1700)∗∗∗, N(2080)∗∗ P13 \| N(1720)∗∗∗∗, N(1900)∗ Table 1: Observed states listed in PDG [47] corresponding to the quantum numbers of the members of the quartet. The number of the stars denotes PDG- ratings of the states. States R \| m(exp)R [MeV] \| Γ(exp)πN [MeV] \| g(exp)πN/Λ [GeV−1] ---\|---\|---\|--- Δ(1232)[P33] \| 1231-1233 (1232) \| 116-120 (118) \| 15.7-16.0 (15.8) Δ(1600)[P33] \| 1550-1700 (1600) \| 25.0-113 (68.8) \| 2.37-5.04 (3.70) Δ(1700)[D33] \| 1670-1750 (1700) \| 20.0-80.0 (50.0) \| 6.34-12.7 (9.51) Δ(1940)[D33] \| 1950-2030 (1990) \| 17.0-62.4 (39.7) \| 3.23-6.20 (4.72) N(1520)[D13] \| 1515-1525 (1520) \| 55.0-81.3 (68.1) \| 7.64-9.30 (8.46) N(1720)[P13] \| 1700-1750 (1720) \| 15.0-60.0 (37.5) \| 1.72-3.44 (2.58) Table 2: Data for masses, πN decay widths and πN coupling constants of the observed states used in the cases (1) and (2). The data are taken from PDG [47]. The values in the bracket for m(exp)R are central values of the observed masses, while those for Γ(exp)πN are the average values between minimum and maximum values. The definition of g(exp)πN is given in the main text. For Δ(1940) in the case (2), we use the data in Ref. [48]. Case (3-1) --- States R \| m(exp)R [MeV] \| Γ(exp)πN [MeV] \| g(exp)πN/Λ [GeV−1] \| Reference Δ(1920)[P33] \| 1900-1970 (1920) \| 7.50-60.0 (33.8) \| 0.825-2.33(1.58) \| PDG average [47] Δ(1940)[D33] \| 1950-2030 (1990) \| 17.0-62.4 (39.7) \| 3.23-6.20(4.72) \| Horn et. al. [48] N(2080)[D13] \| 1945-1947 (1946) \| 85.2-121 (103) \| 4.63-5.23(5.08) \| Penner et. al. [49] N(1900)[P13] \| 1855-1975 (1915) \| 2.80-19.8 (11.3) \| 0.574-1.53(1.05) \| Nikonov et. al. [50] Case (3-2) States R \| m(exp)R [MeV] \| Γ(exp)πN [MeV] \| g(exp)πN/Λ [GeV−1] \| Reference Δ(1920)[P33] \| 1900-1970 (1920) \| 7.50-60.0 (33.8) \| 0.825-2.33 (1.58) \| PDG average [47] Δ(1940)[D33] \| 1947-2167 (2057) \| 8.40-234 (121) \| 2.04-10.8 (6.40) \| Manley et. al. [51] N(2080)[D13] \| 1749-1859 (1804) \| 53.0-165 (109) \| 4.45-7.84 (6.15) \| Manley et. al. [51] N(1900)[P13] \| 1855-1975 (1915) \| 2.8.0-19.8 (11.3) \| 0.574-1.53 (1.05) \| Nikonov et. al. [50] Table 3: Data for masses, πN decay widths and πN coupling constants of the observed states used in the cases (3-1) and (3-2). See also the caption of Table 2. In this section, we proceed to numerical discussions and look for a set of baryons suitable for the QS. Possible candidates for the members of the quartet are shown in Table 1. There are six parameters in our model: $m_{0},g_{1}$, $g_{2}$, $g_{3},g_{4}$ and $g_{5}$. The dimensional parameter $\Lambda$ does not play any role in the present study, then we do not need to determine it. Since the masses $m_{\Delta_{\pm}}$ and $m_{N^{}_{\pm}}$ are the functions of $m_{0},g_{1}$, and $g_{2}$, we can determine them by minimizing $\chi^{2}_{\rm mass}=\sum_{R}(m_{R}-m_{R}^{{\rm(exp)}})^{2}/(\delta m_{R}^{\rm( exp)})^{2},(R=\Delta_{\pm}$ and $N^{}_{\pm})$. Here $m_{R}^{\rm(exp)}$ and $\delta m_{R}^{\rm(exp)}$ are the central values and errors of the observed masses, which are shown in Table 2 and 3. Considering the sates listed in Table 1, there are 36 possible assignments. Among them, we discuss four cases [Case (1)] $(\Delta(1232)$, $\Delta(1700)$, $N(1520)$, $N(1720))$, [Case (2)] $(\Delta(1600)$, $\Delta(1940)$, $N(1520)$, $N(1720))$, [Case (3-1)] and [(3-2)]$(\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$). Although the case (1) was studied in Ref. [34, 30], we reanalyze this case with the use of the different method for the determination of the parameters. As we will show, the case (2) agrees with the mass pattern of the QS with the smallest $\chi^{2}_{\rm mass}$. We also discuss ($\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$). Because of a variety in the data, we consider two cases, (3-1) and (3-2), for this assignment, using two different data sets shown in Table 3. There are three other assignments that reproduce the masses with $\chi^{2}_{\rm mass}$ less than one: ($\Delta(1600)$, $\Delta(1700)$, $N(1700)$, $N(1720)$), ($\Delta(1600)$, $\Delta(1940)$, $N(1700)$, $N(1900)$), ($\Delta(1920)$, $\Delta(1940)$, $N(1700)$, $N(1720)$). We concentrate on the above four cases in the present work. Instead of discussing all of them, we discuss the general behaviors of the QS later. Results for the masses are shown in Table 4. Table 4: Result for the masses and parameters. For the experimental data, see Table 2 and 3. For the case (1), the present result differs from the previous study [30], which is due to the difference of the method to determine the mass parameters. In Ref. [30], we adopted the minimization of a standard deviation $\sigma^{2}=\sum_{R}(m_{R}-m_{R}^{{\rm(exp)}})^{2}$, while we employ $\chi^{2}$-minimum method in the present work. These two methods differ in how $\Delta(1232)$ are included in the fitting procedure, because the error of the observed $\Delta(1232)$’s mass is much smaller than those of the other three states. We found $\chi^{2}_{\rm mass}$ amounts to 60, which is significantly large. It is favorable for the QS that the masses of the $\Delta_{\pm}$ are larger than those of $N^{}_{\pm}$, as shown in Eqs. (19). The mass of $\Delta(1232)$ is much smaller compared with other spin-$\frac{3}{2}$ baryons. This causes the significantly large discrepancy. We also found that $\chi^{2}_{\rm mass}$ becomes larger if assignments include $\Delta(1232)$ as a member of the quartet, which implies that the mass of $\Delta(1232)$ is too small for the QS. The cases (2), (3-1) and (3-2) are new in this work. The case (2) is the best assignment for the quartet with $\chi^{2}_{\rm mass}=0.0025$, which is the smallest value among $\chi^{2}_{\rm mass}$ for 36 possible assignments. For $\Delta(1940)$ in this case, we use the data by Horn et. al. [48]. We confirmed that the result for (2) is insensitive to the choice of the data for $\Delta(1940)$. The cases (3-1) and (3-2) also reproduce the masses of the quartet with $\chi^{2}_{\rm mass}$ = 0.26 and 0.045, respectively. gπRR \| Case (1) \| Case (2) \| Case (3-1) \| Case (3-2) ---\|---\|---\|---\|--- gπΔ+Δ+ \| 0 \| -8.6 \| 0 \| -8.9 gπΔ−Δ− \| 0 \| 11 \| 0 \| 9.6 gπΔ+Δ− \| 5.2 \| 1.9 \| 0.25 \| 0.81 gπN+N+ \| 0 \| 8.5 \| 0 \| 7.9 gπN−N− \| 0 \| -7.5 \| 0 \| -7.5 gπN+N− \| -4.3 \| -1.7 \| -0.21 \| -0.73 gπN+Δ+ \| 0 \| -5.0 \| 0 \| -5.0 gπN+Δ− \| 3.0 \| 3.4 \| 0.14 \| 2.3 gπΔ+N− \| -3.0 \| 0.92 \| -0.14 \| 1.2 gπN−Δ− \| 0 \| 5.3 \| 0 \| 5.1 Table 5: The one-pion coupling constants between the the members of the quartet, gπRR. The values of the parameters are shown in Table 4. Once the masses are determined, we obtain the one-pion coupling constants between two members of the quartet, which are shown in Table 5. First, we consider qualitative features of the one-pion coupling constants. It was found [34] that in the case (1) the parity-non-changing interactions vanish, while the parity-changing interactions remain to be finite. However, even for the parity-changing interactions, their strengths are smaller than a typical order of one-pion interactions e.g. $g_{\pi NN}\sim 13$ [18]. On the other hand, $g_{\pi RR}$ behaves in an opposite way in the case (2). All of the coupling constants survive in the case, where the parity-changing interactions are suppressed compared to the parity-non-changing ones. In addition, diagonal coupling constants are comparable to $g_{\pi NN}$, e.g. $g_{\pi\Delta_{-}\Delta_{-}}=11$. Interestingly, the cases (3-1) and (3-2) show different results, although they are the same assignment. This is caused by the difference of the ordering of the masses of the quartet, especially that of $\Delta(1920)$ and $N(2080)$. We turn back to this point later. Among various coupling constants, $g_{\pi\Delta(1232)\Delta(1232)}$ are investigated in several approaches. Quark models [52] and large $N_{c}$ [53] predict large values, especially, $g_{A}^{\pi\Delta\Delta}=(9/5)g_{A}$ in large $N_{c}$ which gives $g_{\pi\Delta(1232)\Delta(1232)}\sim 30$. A light-cone QCD sum rule reported half of the quark model prediction [54] but still large values compared to our result. The $g_{\pi\Delta(1232)\Delta(1232)}$ were also determined in coupled channel analysis. Krehl et. al. obtained $g_{\pi\Delta\Delta}=31$ [55], while Schneider et. al. obtained $g_{\pi\Delta\Delta}=12.5$ [56]. In the case (1), $g_{\pi\Delta(1232)\Delta(1232)}$ vanishes, which is inconsistent with these studies. Krehl et. al. and Schneider et. al. also investigated $g_{\pi\Delta(1232)N(1520)}$ and obtained $g_{\pi N(1520)\Delta(1232)}=0.95$ and 1.3, respectively. The present result $\|g_{\pi\Delta(1232)N(1520)}\|=3.0$ is qualitatively consistent with these values. πN coupling constants Theo (Exp) [GeV−1] --- \| Case (1) \| Case (2) \| Case (3-1) \| Case (3-2) gπNΔ+Λ \| 16 (15.7-16.0) \| 7.2 (2.37-5.04) \| 2.7 (0.825-2.33) \| 1.8 (0.825-2.33) gπNΔ−Λ \| 14 (6.34-12.7) \| 7.2 (3.23-6.20) \| 8.9 (3.23-6.20) \| 12 (2.04-10.8) gπNN∗−Λ \| 7.3 (7.64-9.30) \| 4.2 (7.64-9.30) \| 3.8 (4.63-5.23) \| 2.2 (4.45-7.84) gπNN∗+Λ \| 1.3 (1.72-3.44) \| -0.89 (1.72-3.44) \| -0.44 (0.574-1.53) \| 0.81(0.574-1.53) χ2πNR \| 1.5 \| 13 \| 7.1 \| 1.8 Parameters[GeV−1] \| Case (1) \| Case (2) \| Case (3-1) \| Case (3-2) g3fπΛ2 \| 1.1 \| -2.6 \| -4.4 \| -8.8 g4fπΛ2 \| -5.2 \| -2.9 \| -2.0 \| 2.1 g5Λ \| 21 \| 9.8 \| 8.2 \| 7.7 Table 6: Result for the πN coupling constants and parameters. For the experimental data, see Table 2 and 3. With regard to the $\pi N$ coupling constants $g_{\pi NR}$, we need to determine three parameters $g_{3},g_{4}$ and $g_{5}$. Since $g_{\pi NR}$ are the functions of $g_{3},g_{4}$ and $g_{5}$, we can determine them by $\chi^{2}$-minimum method with $\chi^{2}_{\pi NR}=\sum_{R}(g_{\pi NR}-g_{\pi NR}^{{\rm(exp)}})^{2}/(\delta g_{ \pi NR}^{\rm(exp)})^{2}$. Here $g_{\pi NR}^{\rm(exp)}$ and $\delta g_{\pi NR}^{\rm(exp)}$ are the average and errors of the coupling constants determined from the experimental $\pi N$ decay widths. We obtain them by using a relation $g_{\pi NR}^{\rm(exp)}/\Lambda=\sqrt{\Gamma_{\pi N}^{(\rm exp)}/\tilde{\Gamma}_ {\pi N}}$, where $\tilde{\Gamma}$ is $\pi N$ decay widths obtained by setting the coupling constant to be one, and $\Gamma_{\pi N}^{(\rm exp)}$ are the experimental values of the $\pi N$ decay widths shown in Table 2 and 3. The dimensional parameter $\Lambda$ does not play any role in the determination of the coupling constants because of the cancellation between the numerator and denominator in $\chi^{2}_{\pi NR}$. We obtain $\tilde{\Gamma}_{\pi N}$ by calculating the simplest tree diagram. Note that we can determine only absolute values of the coupling constants from the $\pi N$ decay widths. Hence, the positive sign of $g_{\pi NR}^{{\rm(exp)}}$ in Table 2 and 3 are our assumption. The result is shown in Table 6. The case (1) reproduces the reasonable values for the four $g_{\pi NR}$ with small $\chi^{2}_{\pi NR}$, which are almost within the ranges of the experimental values. In the case (2), $\chi^{2}_{\pi NR}$ value is significantly large. The discrepancy is mostly caused by the small values of the $\pi N$ decay width of $\Delta(1600)$ and $\Delta(1940)$. In the QS, it is favored that the average values of $g_{\pi NR}$ between $\Delta_{\pm}$ is larger than that between $N^{}_{\pm}$, as is shown in Eq. (28). Because of the same reason, $\chi^{2}_{\pi NR}$ is large for the case (3-1). We obtain reasonable results for the case (3-2) with small $\chi^{2}_{\pi NR}$. Our result underestimates the value of $g_{\pi NR}$ for $R=N(2080)(N^{}_{-})$, which gives $\pi N$ decay widths half of the minimum of the experimental values. ### Mass pattern and one-pion coupling constant The quartet scheme shows two different behavior for the one-pion coupling constants, as shown in Table 5. Especially, the assignment $(\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$) shows two different behavior, depending of the choice of the experimental data. Equations (21) shows that the one-pion coupling constants are controlled by the mixing angles. The cases (1) and (3-1) correspond to the maximally mixing with the angles $\theta_{N,\Delta}=45^{\circ}$, while the cases (2) and (3-2) correspond to moderate mixing. Since the mixing angles are the functions of $m_{0}$ and $(g_{1}-g_{2})f_{\pi}$ as shown in Eqs. (16) and (18), we can understand the behavior of the one-pion coupling constants, comparing $m_{0}$ with $(g_{1}-g_{2})f_{\pi}$. These parameters also determine the masses of the quartet. Therefore, we can relate the masses to the one-pion constants. <figure><img src="content_image/1003.2834/x1.png"><figcaption>Figure 1: Schematic figures for the mass pattern of the QS. (a) small m0 case.(b) m0-dominant case.</figcaption></figure> In order to understand their relation, we approximate the masses in two ways. In the small $m_{0}$ case, the masses are, up to ${\cal O}(m_{0}^{2})$, given by \[m_{\Delta^{\pm}} =2X\mp 2Y+Z,\] \[m_{N^{\pm}} =X\pm Y+2Z,\] where $X=f_{\pi}\|g_{1}-g_{2}\|/4$, $Y=(g_{1}+g_{2})f_{\pi}/4$ and $Z=4m_{0}^{2}/(f_{\pi}\|g_{1}-g_{2}\|)$. In the $m_{0}$ dominant case, they are, up to ${\cal O}(\left(f_{\pi}/m_{0}\right))$, given by \[m_{\Delta^{\pm}} =m_{0}\mp 2a,\] \[m_{N^{\pm}} =m_{0}\pm a,\] where $a=(g_{1}+g_{2})f_{\pi}/4$. The mass patterns for these cases are shown in Fig. 1. The two cases are different in the ordering of $\Delta^{+}$ and $N^{-}$. In the $m_{0}\to 0$ limit, they have mass ratio $2:1$ and $\Delta^{+}$ is heavier than $N^{-}$. Small values of $m_{0}$ do not change this ordering, which corresponds to the left panel in Fig. 1. When $m_{0}$ becomes much larger, the ordering is changed and $\Delta^{+}$ becomes lowest-state. The cases (1) and (3-1) correspond to the mass pattern shown in the right panel in Fig. 1, while the cases (2) and (3-2) correspond to the left panel. Actually, $m_{0}$ is not small in the cases (2) and (3-2), but comparable to $(g_{1}-g_{2})f_{\pi}$. However, the left panel in Fig. 1 well described the mass pattern of these cases. Using Eqs. (16) and (18), mixing angles in the small $m_{0}$ case takes moderate values and all the one-pion coupling constants survive. On the other hand in the $m_{0}$-dominant case, mixing angles are $\theta_{N,\Delta}\sim\pi/4$ and the parity-non-changing interactions vanish. Thus, the behavior of the one-pion coupling constants is related to the mass pattern of the quartet. According to this discussion, the cases (3-1) and (3-2) are different due to the ordering of $\Delta(1920)$ and $N(2080)$, although they describe the same assignments. This is the reason why the assignment $(\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$) is sensitive to the choice of the experimental data. This discussion can be applied to other assignments we do not take into account. As we have mentioned, other three assignments reproduces the masses of the quartet with $\chi^{2}_{\rm mass}$ less than one :($\Delta(1600)$, $\Delta(1700)$, $N(1700)$, $N(1720)$), ($\Delta(1600)$, $\Delta(1940)$, $N(1700)$, $N(1900)$), ($\Delta(1920)$, $\Delta(1940)$, $N(1700)$, $N(1720)$). According to the above discussions, the first and second cases correspond to maximally-mixing with the vanishing of the parity-non-changing interactions, while all the coupling constants survive in the third case. ## 5 Summary We have investigated the possibility that chiral partners exist in spin-$\frac{3}{2}$ baryon sector by considering the quartet scheme, where four spin-$\frac{3}{2}$ baryons, $P_{33}$, $D_{33}$, $D_{13}$ and $P_{13}$, form the chiral multiplets $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ with the mirror assignment. Using the $SU(2)_{R}\times SU(2)_{L}$ Lagrangian, we tried to find a set of four baryons suitable for the chiral quartet. We discussed three assignments: (1) $(\Delta(1232),\Delta(1700),N(1520),N(1720))$, (2) $(\Delta(1600),\Delta(1940),N(1520),N(1720))$, (3-1) and (3-2) $(\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$). Here we investigated $(\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$) using two data sets. For the case (1) we found that there is significant discrepancy for the masses, which implies the mass of $\Delta(1232)$ is too small for the quartet scheme. In addition, the vanishing of $g_{\pi\Delta(1232)\Delta(1232)}$ inconsistent with other theories. Considering the discrepancy for the masses and the inconsistencies of $g_{\pi\Delta(1232)\Delta(1232)}$, it seems that this case is less suitable for the quartet. For the case (2), the masses of the observed baryons agree well with the mass pattern of the QS. Among all the possible assignments, the $\chi^{2}$ value becomes the smallest in this case. Considering the masses, this case is most suitable for the quartet. Regarding the $\pi N$ interactions, this case does not reproduce reasonable results. For the assignment $(\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$), we consider two cases (3-1) and (3-2) with the use of different data sets because of the variety of the experimental data. Both cases reproduce the masses of the quartet with $\chi^{2}$ less than one. The one-pion coupling constants for this assignment are quite sensitive to the ordering of the masses of $\Delta(1920)$ and $N(2080)$. If the mass of $\Delta(1920)$ is smaller than that of $N(2080)$, only the parity-changing one-pion interactions survive. On the other hand, if the mass of $N(2080)$ is smaller, all the coupling constants are finite and the parity-non-changing interactions are larger than the parity-changing ones. Regarding the $\pi N$ interactions, we obtained reasonable results for the case (3-2). For further confirmation, experiments or lattice calculations for the one-pion coupling constants are needed. For instance, we can test the validity of the case (2) using coupling constants such as $g_{\pi N(1520)N(1520)}$, $g_{\pi N(1720)N(1720)}$ and $g_{\pi N(1520)N(1720)}$. For the further study of the assignment ($\Delta(1920)$, $\Delta(1940)$, $N(2080)$, $N(1900)$), we need information about the masses because of a variety of the data. Especially, detailed information of the masses of $\Delta(1920)$ and $N(2080)$ are needed, because the one-pion coupling constants are sensitive to the ordering of the masses of them. If the mass ordering are determined, we can test this assignment using one-pion coupling constants such as $g_{\pi\Delta(1920)\Delta(1920)}$. It is important to extend the present framework with the inclusion of higher-dimensional chiral representations for the nucleon. For the $\pi N$ interactions with the quartet, we adopted the assumption that the nucleon belongs to the fundamental chiral representation. There are other possibilities for the nucleon’s chiral representation. Hence, the disagreements for the $\pi N$ interactions may come from this assumption and can be resolved by including higher-dimensional chiral representations for the nucleon. Furthermore, it may be possible to test the nucleon’s chiral representations through the $\pi N$ interactions with the quartet, if we can confirm the QS by using the one-pion interactions for the quartet. In the present study, we employed the effective Lagrangian approach, where we truncated higher-order terms in the Lagrangian and we neglected quantum effects. With the high-lying baryons in the multiplet, we need to include various resonances in order to evaluate the quantum effects properly, which would cause additional difficulties. Rather, it is desired to reproduce and confirm the present result using different method. For instance, an algebraic method proposed by Weinberg is one of the useful method to study chiral partners. This method is based on the commutation relations derived from the superconvergence property of pion-nucleon scattering amplitudes, and can be applied to baryons [35, 36, 37]. We have already started a study along this line in Ref. [32]. ## Acknowledgments This work was partly supported by Grant-in-Aid for Scientific Research on Innovative Areas(20105003) and Grant-in-Aid for Scientific Research (B20340055). ## Appendix A Fierz Transformation We show the derivation of Eqs. (3). We define totally anti-symmetric fields as linear combinations of Eqs. (2) \[B_{N} ={\mbox{\boldmath${a}$}_{N}}\cdot{\mbox{\boldmath${\phi}$}_{N}},\] (32a) \[B_{\Delta} ={\mbox{\boldmath${a}$}_{\Delta}}\cdot{\mbox{\boldmath${\phi}$}_{ \Delta}},\] (32b) where \[\vec{\phi}_{N} =(N_{V}^{\mu},N_{A}^{\mu},N_{T}^{\mu}),\] (32c) \[\vec{\phi}_{\Delta} =(\Delta_{A}^{\mu i},\Delta_{T}^{\mu i}),\] (32d) \[\vec{a_{N}} =(a_{1}^{N},a_{2}^{N},a_{3}^{N}),\] (32e) \[\vec{a_{\Delta}} =(a_{1}^{\Delta},a_{2}^{\Delta}).\] (32f) The coefficients $\vec{a}_{N}$ and $\vec{a}_{\Delta}$ are determined by the totally anti-symmetric condition, which is implemented by the anti-symmetric condition under the interchange between the second and third quark is given by \[{\cal F}[B_{{\rm n}}]=-[B_{{\rm n}}],({\rm n}=N,\Delta),\] (33) where ${\cal F}[B]$ denotes a baryon field obtained from the Fierz transformation of $B$. Fierz transformation formula is given in Ref. [29]. This equation can be read as two kinds of the eigen-value problems : (a) for the vector space $\vec{B}_{N,\Delta}$, and (b) for the vector space $\vec{a}_{N,\Delta}$. The eigen-value problem (a) gives identities between the baryon operators \[N_{V}^{\mu} =N_{A}^{\mu},2N_{A}^{\mu}=N_{T}^{\mu},\] (34a) \[\Delta_{A}^{\mu i} =-\Delta_{T}^{\mu i},\] (34b) which reduce the number of the independent fields [29, 44, 45, 46]. The eigen-value problem (b) determines the values of the coefficients $\vec{a}_{N}$ and $\vec{a}_{\Delta}$ \[\vec{a}_{N} =(3,1,1),\] (35a) \[\vec{a}_{\Delta} =(-2,1),\] (35b) with which $B_{N}$ and $B_{\Delta}$ are totally anti-symmetric. This determine the ratio between $N_{V}^{\mu}$ and $N_{A}^{\mu}$ in $N_{1}^{\mu}$. It is convenient to replace $N_{T}^{\mu}$ by $N_{V}^{\mu}$ and $N_{A}^{\mu}$ and $\Delta_{T}^{\mu i}$ by $\Delta_{A}^{\mu i}$ with the use of Eqs. (34), which can be done without the change of chiral transformation properties of $B_{N}$ and $B_{\Delta}$. Finally, we obtain Eqs. (3). ## Appendix B Alternative derivation of chiral properties We show an alternative derivation of the chiral transformation properties of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ and the mass relation. Starting point is a standard definition of the transformation in terms of the chiral algebra between charges and fields. In general, the $SU(2)_{A}$ transformation is given by $\psi^{\prime}=\psi+ia^{i}[Q^{i}_{A},\psi]$ with generators $Q_{A}^{i},(i=1,2,3)$ and infinitesimal parameters $a^{i}$ for the $SU(2)_{A}$ transformation. We describe $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ by product of the isovector and isospinor $\psi^{i}=(\psi^{i})_{a},(a=1,2)$. For simplicity, we suppress the Lorentz indices in this section. In the left- and right-handed representation, they correspond to $\psi^{i}_{R}=(1,\frac{1}{2})$ and $\psi^{i}_{L}=(\frac{1}{2},1)$ : $\psi^{i}_{R}=(1,\frac{1}{2})$ transforms as $I=1$ under $SU(2)_{R}$ and $I=\frac{1}{2}$ under $SU(2)_{L}$, while $\psi^{i}_{L}=(\frac{1}{2},1)$ transforms $I=\frac{1}{2}$ under $SU(2)_{R}$ and $I=1$ under $SU(2)_{L}$. Note that this field $\psi^{i}$ corresponds to $\Delta_{T}^{i}$ and $N_{T}$ in Eq. (2). It is easy to check that $N_{A}$, $N_{V}$ and $\Delta_{A}$ consist of $(RL)R$, $(RL)L$, $(LR)R$ and $(LR)L$, while $N_{T}$ and $\Delta_{T}$ contain $(RR)L$ and $(LL)R$. Jido et. al. employed $(RR)L$ and $(LL)R$ for the description of $(1,\frac{1}{2})\oplus(\frac{1}{2},1)$ [34]. The chiral transformations of these fields are given by (36) where we have defined $\delta^{a}\psi^{b}=-i[Q^{a},\psi^{b}]$. Using $Q_{V}^{a}=Q_{R}^{a}+Q_{L}^{a}$ and $Q_{A}^{a}=Q_{R}^{a}-Q_{L}^{a}$, we obtain $SU(2)_{V}$ and $SU(2)_{A}$ transformation properties \[\delta^{a}_{V}\psi_{i}^{b} =\left[(\epsilon^{abc}+it^{a}\delta^{bc})\right]\psi^{c},\] (37) \[\delta^{a}_{A}\psi^{b} =\gamma_{5}(\epsilon^{abc}-it^{a}\delta^{bc})\psi_{i}^{c}.\] (38) Employing an isospurion formalism, $I=\frac{1}{2}$ and $I=\frac{3}{2}$ components are obtained by $\psi_{1/2}=\tau^{i}\psi^{i}$ and $\psi_{3/2}^{i}=P^{ij}_{3/2}\psi^{j}$. After the irreducible decomposition, we obtain \[\delta_{A}^{a}\psi_{1/2} =\frac{1}{2}i\gamma_{5}\left[\frac{5}{3}\tau^{a}\psi_{1/2}-4\psi_ {3/2}^{a}\right],\] (39a) \[\delta_{A}^{a}\psi_{3/2}^{b} =\frac{1}{2}i\gamma_{5}\left[\tau^{a}\psi_{3/2}^{b}-\frac{2}{3} \tau^{b}\psi_{3/2}^{a}-\frac{4}{3}P^{ba}_{3/2}\psi_{1/2}\right].\] (39b) Here note that the coefficients differ from Eqs. (4). This is because $\psi_{1/2}$ and $\psi_{3/2}^{a}$ describe $N_{T}$ and $\Delta_{T}^{i}$, respectively. Using Eqs. (3) and (34), we obtain $\psi_{1/2}=N_{T}=2\sqrt{3}N_{1}$ and $\psi_{3/2}=\Delta_{T}=-2\Delta_{1}$. Substituting these relations into Eqs. (39), we reproduce Eqs. (4). Considering the $I_{z}=\frac{1}{2}$ components, it is easy to show that the $SU(2)_{A}$ transformations of the $I=\frac{1}{2}$ and $\frac{3}{2}$ fields (40) where $T$ is the axial-transformation matrix Eq. (39) for $I_{z}=\frac{1}{2}$ components. We introduce the mass matrix for $(\psi_{1/2}^{I_{z}=\frac{1}{2}},\psi_{3/2}^{I_{z}=\frac{1}{2}})^{T}$ as $M=\mbox{diag}(a,b)$ with $a$ and $b$ being the masses of $\psi_{1/2}$ and $\psi_{3/2}$. We also introduce the pion interaction matrix $M_{\pi}$ for their pseudo-scalar couplings. With chiral invariance, the matrices $T$, $M$ and $M_{\pi}$ must obey \[M =\{T,M_{\pi}\},\] \[M_{\pi} =\{T,M\},\] which leads to a double-commutation relation \[M=\{T,\{T,M\}\}.\] (41) This double-commutation relation gives $a=-2b$, which reproduces the mass relation between $N_{1}^{\mu}$ and $\Delta_{1}^{\mu i}$. Note that the double commutator Eq. (41) is the necessity condition of chiral invariance. ## References * [1] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991). * [2] S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992). * [3] T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994), [hep-ph/9401310]. * [4] T. D. Cohen, Nucl. Phys. A775, 89 (2006), [hep-ph/0605206]. * [5] L. Y. Glozman, Phys. Rev. Lett. 99, 191602 (2007), [0706.3288]. * [6] L. Y. Glozman, Phys. Rept. 444, 1 (2007), [hep-ph/0701081]. * [7] R. L. Jaffe, D. Pirjol and A. Scardicchio, Phys. Rept. 435, 157 (2006), [hep-ph/0602010]. * [8] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961). * [9] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961). * [10] J. I. Kapusta and E. V. Shuryak, Phys. Rev. D49, 4694 (1994), [hep-ph/9312245]. * [11] N. Ishii, W. Bentz and K. Yazaki, Phys. Lett. B301, 165 (1993). * [12] N. Ishii, W. Bentz and K. Yazaki, Phys. Lett. B318, 26 (1993). * [13] N. Ishii, Nucl. Phys. A689, 793 (2001), [nucl-th/0004063]. * [14] H. Mineo, W. Bentz, N. Ishii and K. Yazaki, Nucl. Phys. A703, 785 (2002), [nucl-th/0201082]. * [15] W. Bentz, T. Horikawa, N. Ishii and A. W. Thomas, Nucl. Phys. A720, 95 (2003), [nucl-th/0210067]. * [16] N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A587, 617 (1995). * [17] L. J. Abu-Raddad, A. Hosaka, D. Ebert and H. Toki, Phys. Rev. C66, 025206 (2002), [nucl-th/0206002]. * [18] K. Nagata and A. Hosaka, Prog. Theor. Phys. 111, 857 (2004), [hep-ph/0312161]. * [19] K. Nagata, A. Hosaka and L. J. Abu-Raddad, Phys. Rev. C72, 035208 (2005), [hep-ph/0408312]. * [20] K. Nagata and A. Hosaka, J. Phys. G32, 777 (2006), [hep-ph/0506193]. * [21] M. Gell-Mann and M. Levy, Nuovo Cim. 16, 705 (1960). * [22] B. W. Lee, Gordon and Breach, New York,(1972). * [23] C. DeTar and T. Kunihiro, Phys. Rev. D39, 2805 (1989). * [24] D. Jido, Y. Nemoto, M. Oka and A. Hosaka, Nucl. Phys. A671, 471 (2000), [hep-ph/9805306]. * [25] D. Jido, M. Oka and A. Hosaka, Prog. Theor. Phys. 106, 873 (2001), [hep-ph/0110005]. * [26] H. Nagahiro, D. Jido and S. Hirenzaki, Phys. Rev. C80, 025205 (2009), [0811.4516]. * [27] S. Gallas, F. Giacosa and D. H. Rischke, 0907.5084. * [28] C. Sasaki and I. Mishustin, arXiv:1005.4811[hep-ph]. * [29] K. Nagata, A. Hosaka and V. Dmitrasinovic, Mod. Phys. Lett. A23, 2381 (2008), [0705.1896]. * [30] K. Nagata, A. Hosaka and V. Dmitrasinovic, Phys. Rev. Lett. 101, 092001 (2008), [0804.3185]. * [31] H.-X. Chen, V. Dmitrasinovic, A. Hosaka, K. Nagata and S.-L. Zhu, Phys. Rev. D78, 054021 (2008), [0806.1997]. * [32] K. Nagata, arXiv:0812.3982[hep-ph]. * [33] V. Dmitrasinovic, A. Hosaka and K. Nagata, Mod. Phys. Lett. A25, 233 (2010), [0912.2372]. * [34] D. Jido, T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 84, 3252 (2000), [hep-ph/9910375]. * [35] S. Weinberg, Phys. Rev. 177, 2604 (1969). * [36] S. R. Beane and U. van Kolck, J. Phys. G31, 921 (2005), [nucl-th/0212039]. * [37] A. Hosaka, D. Jido and M. Oka, arXiv:hep-ph/0112355. * [38] T. R. Hemmert, B. R. Holstein and J. Kambor, J. Phys. G24, 1831 (1998), [hep-ph/9712496]. * [39] V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, Phys. Rept. 437, 125 (2007). * [40] M. Benmerrouche, R. M. Davidson and N. C. Mukhopadhyay, Phys. Rev. C39, 2339 (1989). * [41] H. Haberzettl, arXiv:nucl-th/9812043. * [42] V. Pascalutsa and R. Timmermans, Phys. Rev. C60, 042201 (1999), [nucl-th/9905065]. * [43] V. Pascalutsa and M. Vanderhaeghen, Phys. Lett. B636, 31 (2006), [hep-ph/0511261]. * [44] B. L. Ioffe, Nucl. Phys. B188, 317 (1981). * [45] Y. Chung, H. G. Dosch, M. Kremer and D. Schall, Nucl. Phys. B197, 55 (1982). * [46] D. Espriu, P. Pascual and R. Tarrach, Nucl. Phys. B214, 285 (1983). * [47] Particle Data Group, C. Amsler _et al._, Phys. Lett. B667, 1 (2008). * [48] CB-ELSA, I. Horn _et al._, Eur. Phys. J. A38, 173 (2008), [0806.4251]. * [49] G. Penner and U. Mosel, Phys. Rev. C66, 055211 (2002), [nucl-th/0207066]. * [50] V. A. Nikonov, A. V. Anisovich, E. Klempt, A. V. Sarantsev and U. Thoma, Phys. Lett. B662, 245 (2008), [0707.3600]. * [51] D. M. Manley and E. M. Saleski, Phys. Rev. D45, 4002 (1992). * [52] G. E. Brown and W. Weise, Phys. Rept. 22, 279 (1975). * [53] N. Fettes and U. G. Meissner, Nucl. Phys. A679, 629 (2001), [hep-ph/0006299]. * [54] S.-L. Zhu, Phys. Rev. C63, 018201 (2001), [nucl-th/0009062]. * [55] O. Krehl, C. Hanhart, S. Krewald and J. Speth, Phys. Rev. C62, 025207 (2000), [nucl-th/9911080]. * [56] S. Schneider, S. Krewald and U.-G. Meissner, Eur. Phys. J. A28, 107 (2006), [nucl-th/0603040].
1901.03220	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 38890, "num_imgs": 3, "llama3_tokens_count": 13353 }	[ "content_image/1901.03220/x1.png", "content_image/1901.03220/x2.png", "content_image/1901.03220/x3.png" ]	# Topology-dependent quantum dynamics and entanglement-dependent topological pumping in superconducting qubit chains Feng Mei meifeng@sxu.edu.cn State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China Collaborative Innovation Center of Extreme Optics, Shanxi University,Taiyuan, Shanxi 030006, China Gang Chen chengang971@163.com State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China Collaborative Innovation Center of Extreme Optics, Shanxi University,Taiyuan, Shanxi 030006, China Lin Tian ltian@ucmerced.edu School of Natural Sciences, University of California, Merced, California 95343, USA Shi-Liang Zhu slzhu@nju.edu.cn National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China Suotang Jia State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China Collaborative Innovation Center of Extreme Optics, Shanxi University,Taiyuan, Shanxi 030006, China February 21, 2024 ###### Abstract We propose a protocol using a tunable Xmon qubit chain to construct generalized Su-Schrieffer-Heeger (SSH) models that support various topological phases. We study the time evolution of a single-excitation quantum state in a SSH-type qubit chain and find that such dynamics is linked to topological winding number. We also investigate the adiabatic transfer of a single-excitation quantum state in a generalized SSH-type qubit chain and show that this process can be connected with topological Chern number and be used to generate a novel entanglement-dependent topological pumping. All results have been demonstrated to be robust against qubit coupling imperfections and can be observed in a short Xmon qubit chain. Our study provides a simple method to directly measure topological invariants rooted in momentum space using quantum dynamics in real space. ## I Introduction Supercounducting circuits nowadays have been widely recognized as one of the leading quantum systems for quantum computation [1; 2; 3; 4]. The fundamental challenge in building a full-fledged superconducting quantum computer is to balance high coherence and straightforward connectivity. Remarkable experimental progresses have recently been made in this regard [5; 6; 7; 8; 9; 10; 11; 12]. In particular, Xmon qubits have been shown to possess excellent scalability simultaneously with high coherence [5]. Meanwhile, the coupling between Xmon qubits can be dynamically varied through a g-mon coupler [6]. Moreover, it is now believed that superconducting circuit can be further scaled up to several tens of qubits and show its quantum supremacy in the near term [13]. Such state-of-the-art enables superconducting qubit chains to be a promising platform for implementing large-scale quantum simulation [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45]. On the other hand, searching topological states in cold atoms as well as photonic systems has recently become a rapidly growing research field [46; 47; 48]. In the context of superconducting circuits, some topological states and effects have also been theoretically studied [49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62]. Experimentally, several progresses studying topological phenomenons recently have been made in superconducting qubits and resonators [63; 64; 65; 66; 67; 68; 69]. Specifically, topological concepts have been investigated in the parameter space of superconducting qubits [63; 64; 65; 66], including the topological Chern numbers and topological phase transitions; Topological quantum walks and Zak phases have been realized and measured in the phase space of microwave resonators [67; 68]. Considering the-state-of-art in Xmon qubits, a natural question to ask is whether we can realize topological phases rooted in the momentum space of a superconducting Xmon qubit lattice. It is also quite interesting to study how to detect topological invariants in this qubit lattice. In this paper, we present an experimental protocol to realize a generalized SSH model [70] in a superconducting Xmon qubit chain with tunable qubit couplings. This model has a variety of topological magnon bands and supports different topologcial insulator phases characterized by topological winding numbers and Chern numbers. We first investigate the quantum dynamics of a single-excitation quantum state in a SSH-type qubit chain. Interestingly, we find that the time average of the center of qubit excitation difference (CED) associated with this quantum dynamics is topology-dependent and can be linked to the topological winding number. Winding number is one of the basic topological invariants but its detection method is still lacking. Our result thus gives a new method to directly measure the topological winding number using quantum dynamics of single-excitation states in the real space. We also study the adiabatic transfer of a single-excitation quantum state in a generalized SSH-type qubit chain by slowly ramping the qubit couplings. We show that the shift of the center of qubit excitation (CE) after one periodic ramping is exactly quantized as topological Chern number. We also find that both the amplitude and the direction of such quantized shift are entanglement-dependent. This process thus creates a novel entanglement-dependent topological pumping. Compared with topological Thouless particle pumping [71; 72; 73; 74; 75], this pumping is with respect to a single-excitation quantum state and quantum entanglement plays an important role here. This pumping can also be used to directly detect the topological Chern numbers. In contrast to recent experiments on topological properties in superconducting circuits [63; 64; 65; 66; 67; 68; 69], which focus on the parameter space of the qubits and resonators, our study aims at the intrinsic topological properties associated with Bloch energy bands and rooted in momentum spaces. This paper is organized as follows. In Sec. II, we construct a generalized SSH model with a tunable Xmon qubit chain and study its topological features. In Sec. III, we study the time evolution of a single-excitation state in a SSH-type qubit chain. In Sec. IV, we investigate the adiabatic transfer a single-excitation state in a generalized SSH-type qubit chain by slowing ramping qubit couplings. In Sec. V, we give a summary for the main results presented in this paper. ## II Topological states in a tunable superconducting Xmon qubit chain <figure><img src="content_image/1901.03220/x1.png"><figcaption>Figure 1: Superconducting circuit for the one-dimensional Xmon qubit lattice.Two nearest neighbour Xmon qubits Qk and Qk+1 are inductively coupled and thecoupling strength can be tuned through the middle connected gmon coupler (CP).</figcaption></figure> The superconducting qubit chain we consider constituts an array of coupled Xmon qubits with tunable qubit couplings, as shown in Fig. 1. Suppose each Xmon qubit has two energy levels and the same transition frequency. The Hamiltonian of such Xmon qubit lattice can be descried by a spin-chain Hamiltonian \[H=\sum^{L}_{k=1}J_{k}\hat{\sigma}^{{\dagger}}_{k}\hat{\sigma}^{-}_{k+1}+\text{ H.c}.\] (1) where $\hat{\sigma}^{{\dagger}}_{k}=\|e\rangle_{k}\langle g\|$, $J_{k}$ is the coupling strength between two nearest neighbour Xmon qubits. Here we omit the constant qubit frequencies and only consider singe qubit excitation, which can be precisely prepared with current superconducting qubit technology [7]. Because the number of excitations is conserved in our model, the above Hamiltonian can be reduced to single excitation subspace. Based on the Matsubara-Matsuda transformation [76], the qubit chain can be rewritten into the following magnon Hamiltonian \[H=\sum^{L}_{k=1}(J_{k}\hat{m}^{{\dagger}}_{k}\hat{m}_{k+1}+\text{H.c}.\] (2) where the single excitation is called as magnon in a spin chain and its annihilation operator is $\hat{m}_{k}=\hat{\sigma}^{-}_{k}$. Such Xmon qubit chain recently has been experimentally realized for studying surface-code quantum error correction [77]. Motivated by recent experiment using gmon coupler to tune the Xmon qubit couplings [6], we assume the coupling strength \[J_{k}=g_{0}+g_{1}\cos(2\pi k/p+\theta),\] (3) where $g_{0}$ and $g_{1}$ are the coupling constants, $p$ is the number of qubits in one unit cell, and $\theta$ is the control parameter. For $p=2$, each unit cell contains two qubits labeled by $a$ and $b$, respectively. The resulted qubit chain can be described by the SSH model Hamiltonian \[\hat{H_{1}}=\sum_{x=1}^{N}(J_{1}\hat{a}_{x}^{{\dagger}}\hat{b}_{x}+J_{2}\hat{b }_{x}^{{\dagger}}\hat{a}_{x+1}+\text{H.c.),}\] (4) where $\hat{\alpha}_{x}^{{\dagger}}=\hat{\sigma}_{\alpha_{x}}^{+}$ ($\alpha=a,b$) is the magnon creation operator for qubit at $a_{x}$ ($b_{x}$), $J_{i}=g_{0}+(-1)^{i}g_{1}\cos\theta$ ($i=1,2$) and $N$ is the number of unit cells. To study its topological feature, we rewrite it in the momentum space as $\hat{H}=\sum_{k_{x}}\hat{m}_{k_{x}}^{{\dagger}}\hat{h}(k_{x})\hat{m}_{k_{x}}$, where $\hat{m}_{k_{x}}=(\hat{a}_{k_{x}},\hat{b}_{k_{x}})^{T}$, $\hat{a}_{k_{x}}$ and $\hat{b}_{k_{x}}$ are the momentum space operators, \[\hat{h}(k_{x})=d_{x}\hat{\tau}_{x}+d_{y}\hat{\tau}_{y},\] (5) where $d_{x}=J_{1}+J_{2}\cos(k_{x})$, $d_{y}=J_{2}\sin(k_{x})$, and $\hat{\tau}_{x}$ and $\hat{\tau}_{y}$ are the Pauli spin operators defined in the momentum space. The energy bands of the Hamiltonian (4) are characterized by topological winding number [78] \[\nu=\frac{1}{2\pi}\int dk_{x}\mathbf{n}\times\partial_{k_{x}}\mathbf{n}=\frac{ 1}{2}\left[1+\text{sgn}\left(g_{0}g_{1}\cos\theta\right)\right],\] (6) where $\mathbf{n}=(n_{x},n_{y})=(d_{x},d_{y})/\sqrt{d_{x}^{2}+d_{y}^{2}}$. Let $g_{0}g_{1}$ be a positive number. We find that \[\nu=\begin{cases}1,&\mbox{$\theta\in(-\pi/2,\pi/2)$}\\ 0\text{,}&\mbox{$\theta\in\left(\pi/2,3\pi/2\right)$ }\end{cases}\] (7) The winding number $\nu=1$ ($0$) shows that the above SSH-type qubit chain is in the topological nontrivial (trivial) insulator phase. For the case of $p>2$, a generalized SSH model can be formed, which supports $p$ magnon bands. Different from the $p=2$ case, their topological features are characterized by Chern numbers. In particular, in the $p=3$ case, each unit cell has three qubits labeled as $a$, $b$, and $c$, the Hamiltonian describing this generalized SSH-type qubit chain has the form \[\hat{H_{2}}=\sum_{x=1}^{N}(J_{1}\hat{a}_{x}^{{\dagger}}\hat{b}_{x}+J_{2}\hat{b }_{x}^{{\dagger}}\hat{c}_{x}+J_{3}\hat{c}_{x}^{{\dagger}}\hat{a}_{x+1}+\text{H .c.}),\] (8) where $\hat{\alpha}_{x}^{{\dagger}}$ ($\alpha=a,b,c$) is the magnon creation operator and $J_{s}=g_{0}+g_{1}\cos(2\pi s/3+\theta)$ ($s=1,2,3$) is the coupling strength. To explore topological features of the Hamiltonian (8), we rewrite it in the momentum space as $\hat{H}=\sum_{k_{x}}\hat{m}_{k_{x}}^{{\dagger}}\hat{h}(k_{x},\theta)\hat{m}_{k _{x}}$, with $\hat{m}_{k_{x}}=(\hat{a}_{k_{x}},\hat{b}_{k_{x}},\hat{c}_{k_{x}})^{T}$, \[\hat{h}(k_{x},\theta)=\sum_{i=1,4,5,6}h_{i}\hat{S}_{i},\] (9) where $h_{1}=J_{1}$, $h_{4}=J_{3}\cos\left(k_{x}\right)$, $h_{5}=-J_{3}\sin\left(k_{x}\right)$, $h_{6}=J_{2}$, and $\hat{S}_{i}$ being the $i$-th Gell-Mann spin operator. By combining the momentum space of quasimomentum $k_{x}$ with the control variable $\theta$, we have a synthetic two-dimensional space with parameters $\mathbf{k}=(k_{x},\theta)$. The energy spectrum in the first Brillouin zone $\{k_{x}\in(0,2\pi/3],$$\theta\in(0,2\pi]\}$ of this synthetic space has three energy bands. For the synthetic two-dimensional space, the underlying topological properties of the Hamiltonian (8) are characterized by the Chern number defined in the first Brillouin zone. Denote the Bloch function of the $n$-th magnon band as $\|u_{\mathbf{k}n}\rangle$. The Chern number for the $n$-th magnon band is defined as [79; 80] \[C_{n}=\frac{1}{2\pi}\int_{k_{x}}\int_{\theta}dk_{x}d\theta\,F_{n}(k_{x},\theta),\] (10) where $F_{n}(k_{x},\theta)=i(\|\langle\partial_{\theta}u_{\mathbf{k}n}\|\partial_{k_{x} }u_{\mathbf{k}n}\rangle-\text{c.c.})$ is the Berry curvature and c.c. refers to the complex conjugate. Using equation (10), we calculate the Chern numbers for the first (bottom), second (middle), and third (top) magnon bands. The results are \[\{C_{1},C_{2},C_{3}\}=\begin{cases}\{2,-4,2\},&\mbox{$-g_{1}/4<g_{0}<g_{1}/4$} \\ \{-1,2,1\}\text{,}&\mbox{otherwise}\end{cases}.\] (11) Equation (11) shows that the $p=3$ generalized SSH-type qubit chain supports two types of topological insulator phases separated by the transition points $g_{0}=\pm g_{1}/4$. <figure><img src="content_image/1901.03220/x2.png"><figcaption>Figure 2: (a) The schematic of the topology-dependent quantum dynamics. Thetime-dependent average of the CED ¯Pd(t) versus time is shown in (a) forθ=0.1π with ν=1 and in (c) for θ=0.9π with ν=0. In the presence of qubitcoupling imperfection, ¯Pd(t) in the above two cases is shown in (b) and (d),respectively, with the imperfect strength W=0.2g1. The other parameter isg0=g1. g1 is used as energy unit in this work.</figcaption></figure> ## III Topology-dependent quantum dynamics and winding number detection In this section, we will study the time evolution of a single-excitation quantum state in a SSH-type qubit chain (p=2). Suppose the SSH-type qubit chain is initially prepared into a single-excitation bulk state. As shown in Fig. 2(a), we choose to excite one of the middle qubits into the excited state $\|e\rangle$ and the other qubits are in the ground state $\|g\rangle$. Then the initial state of the system can be written as \[\|\psi(0)\rangle=\|gg\cdots e\cdots gg\rangle.\] (12) The quantum dynamics of such single excitation state is governed by the Hamiltonian in Eq. (4). After an evolution time $t$, the state of the system becomes \[\|\psi(t)\rangle=e^{-i\hat{H_{1}}t}\|\psi(0)\rangle.\] (13) The relation between the above quantum dynamics and the topological feature of the SSH-type qubit chain can be revealed through the CED in the qubit chain. The CED is defined as \[\hat{P}_{d}=\sum_{x=1}^{N}x(\hat{P}_{a_{x}}^{e}-\hat{P}_{b_{x}}^{e})\] (14) with $\hat{P}_{q}^{e}=\|e\rangle_{q}\langle e\|$ ($q=a_{x},b_{x}$). Then the time-dependent average of the CED associated with the above single-excitation quantum dynamics is given by \[\bar{P}_{d}(t)=\langle\psi(t)\|\hat{P}_{d}\|\psi(t)\rangle.\] (15) Furthermore, we transfer Eq. (15) into the momentum space and get \[\bar{P}_{d}(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk_{x}\langle\psi(0)\|e^{i\hat{h} (k_{x})t}i\partial_{k_{x}}\hat{\tau}_{z}e^{-i\hat{h}(k_{x})t}\|\psi(0)\rangle.\] (16) By substituting Eq. (5) into Eq. (16), we find $\bar{P}_{d}(t)$ can be connected with the topological winding number $\nu$ defined in Eq. (6), i.e., \[\begin{split}\bar{P}_{d}(t)&=\frac{\nu}{2}-\frac{1}{ 4\pi}\int dk_{x}\cos(2d_{t}t)\mathbf{n}\times\partial_{k_{x}}\mathbf{n},\end{split}\] (17) where $d_{t}=\sqrt{J_{1}^{2}+J_{2}^{2}+2J_{1}J_{2}\cos(k_{x})}$. The second oscillation term in Eq. (17) vanishes at the critical times \[t_{c}=(2s+1)\pi/4\sqrt{J_{1}^{2}+J_{2}^{2}},\] (18) where $s$ is an integer number. At these times, the topological winding number can be directly measured via CED, i.e., \[\nu=2\bar{P}_{d}(t_{c}).\] (19) In the long time limit, we can also obtain a relationship between the winding number and the time-averaged CED, i.e., \[\nu={\lim_{T\rightarrow\infty}}\frac{2}{T}\int_{0}^{T}dt\,\bar{P}_{d}(t).\] (20) One can find that the time-averaged CED is just the oscillation center of the CED varying with time, which dependents on the topology of the band structure of the qubit chain. Thus our result demonstrates that the quantum dynamics of a single-excitation state in a SSH-type qubit chain is topology-dependent, which can be employed for directly detecting the topological winding number. In Figs. 2(b) and 2(d), we have further numerically calculated $\bar{P}_{d}(t)$, when the qubit chain is in the topological nontrivial and trivial phases, with the topological winding numbers $\nu=1$ and $0$, respectively. The numerical results show that $\bar{P}_{d}(t)$ oscillates around the average values $0.5$ and $0$, respectively, which gives the topological winding numbers $\nu=1$ and $0$ according to Eq. (20). We have also calculated $\bar{P}_{d}(t)$ for different choices of qubit chain lengthes. We find that the oscillation center of $\bar{P}_{d}(t)$ in a chain of four qubits is same as the ones in longer qubit chains. It means that the signatures of topological winding number predicted in Eq. (20) can be unambiguously observed in a qubit chain with short length. As revealed in Eq. (19), the topological winding number can be directly detected by the CED at some critical time points. In Figs. 2(b) and 2(d), our numerical results show that, the oscillation curves of $\bar{P}_{d}(t)$ for different choices of qubit chain lengthes intersect at the time critical points $t_{c}$, with their values $\bar{P}_{d}(t_{c})=0.5$ and $0$, respectively. According to Eq. (19), $\bar{P}_{d}(t_{c})$ directly gives the topological winding number $\nu=1$ and $0$. In practical experiments, the qubit couplings can not be perfectly tuned to exact values due to the intrinsic imperfections in device fabrication. This imperfection can be described by the Hamiltonian $H_{d}=\sum_{x}\delta J_{1x}a^{{\dagger}}_{x}b_{x}+\delta J_{2x}b^{{\dagger}}_{ x}a_{x+1}+\text{H.c.}$, where the influence of the imperfection on tuning qubit couplings is characterized by a disorder variable $\delta J_{1x,2x}=W\delta$, with $\delta\in[-0.5,0.5]$ being a random number and $W$ being the imperfect strength. In Figs. 2(c) and 2(e), we have taken into account the influence of qubit coupling imperfection and numerically recalculated $\bar{P}_{d}(t)$ for the topological nontrivial and trivial cases, respectively. For each $\delta J_{1x,2x}$, we choose 30 samples to perform the numerical simulation. The resulted CED $\bar{P}_{d}(t)$ is obtained by averaging over the results of all samples. The results clearly show that $\bar{P}_{d}(t)$ still oscillate around $0.5$ and $0$. The critical time points where $\bar{P}_{d}(t)$ intersects at the oscillation center are also same as the ones shown in Figs. 2(a) and 2(c) without considering qubit coupling imperfection. Thus our results are quite robust to the practical imperfections in qubit coupling engineering and provide an experimentally promising method to directly detect the topological winding number. <figure><img src="content_image/1901.03220/x3.png"><figcaption>Figure 3: (a) The schematic of the entanglement-dependent topological pumping.The time-dependent average of the CE ¯Pn(t) versus time is shown in (a) and(b). The change of ¯Pn(t) during one periodic pumping δ¯Pn versus qubitcoupling imperfections is shown in (c) and (d). The ramping rate is Ω=0.39g1.The qubit number is L=18 and g0=g1.</figcaption></figure> ## IV Entanglement-dependent topological pumping and Chern number detection In this section, we will investigate adiabatic transfer of a single-excitation quantum state in a generalized SSH-type qubit chain by slowly ramping the control parameter $\theta$. For illustration, we take $p=3$ and $g_{0}=g_{1}$, under which the three magnon bands have the Chern numbers \[C_{1,3}=-1,C_{2}=2.\] (21) Let the control parameter \[\theta(t)=\Omega t+\varphi_{0},\] (22) where $\Omega$ is the ramping rate and $\varphi_{0}$ is the initial phase. The total time for one pumping period is then $T_{p}=2\pi/\Omega$. Such time-dependent coupling has recently been implemented in superconducting Xmon qubits [43]. At time $t=0$, let $\theta(0)=\varphi_{0}=\pi$. The coupling strengths are then $J_{1,2}=3g_{1}/2$ and $J_{3}=0$, i.e., the unit cells are isolated with zero inter-cell coupling. The Hamiltonian for a single isolated unit cell in single excitation space is $\hat{H}_{s}=J_{(}\hat{\sigma}_{a}^{+}\hat{\sigma}_{b}^{-}+\hat{\sigma}_{b}^{+} \hat{\sigma}_{c}^{-})+$H.c. with $J=3g_{1}/2$. The eigenstates for such single-excitation Hamiltonian are \[\|\chi_{1,3}\rangle = (\|egg\rangle\mp\sqrt{2}\|geg\rangle+\|gge\rangle)/2,\] \[\|\chi_{2}\rangle = (\|egg\rangle-\|gge\rangle)/\sqrt{2}.\] (23) The corresponding eigenenergies of the above three states are $E_{1}=-\sqrt{2}J$, $E_{2}=0$, and $E_{3}=\sqrt{2}J$, respectively. To prepare the state $\|\chi_{1}\rangle$, we firstly decouple the selected unit cell from the rest of the qubit chain by increasing or decreasing the detuning of qubits in this unit cell from that of all other qubits. From the ground state $\|ggg\rangle$, a driving pulse in the form of $\hat{V}_{3}=\Omega_{0}\cos\left(\omega_{d}t\right)\left(\hat{\sigma}_{a}^{x}- \sqrt{2}\hat{\sigma}_{b}^{x}+\hat{\sigma}_{c}^{x}\right)$ is applied with the driving frequency $\omega_{d}=\omega_{q}+\sqrt{2}J$. In the rotating frame of $\omega_{d}$, this driving field can be written as $\hat{V}_{3}^{\text{rot}}=\Omega_{0}(\hat{\sigma}_{a}^{x}-\sqrt{2}\hat{\sigma}_ {b}^{x}+\hat{\sigma}_{c}^{x})/2$. It can be shown that $\langle\chi_{1}\|\hat{V}_{3}^{\text{rot}}\|ggg\rangle=\Omega$ and $\langle\chi_{2,3}\|\hat{V}_{R}\|ggg\rangle=0$. By applying this pulse for a duration of $t_{3}=\pi/2\Omega_{0}$, the state $\|\chi_{1}\rangle$ is generated. The time duration of this operation is also on the order of nanoseconds. Similarly, the states $\|\chi_{2}\rangle$ and $\|\chi_{3}\rangle$ can be generated by applying corresponding driving pulses. As shown in Figs. 3(a), we assume the qubits in one of the middle selected unit cells are prepared in the state $\|\chi_{n}\rangle$$(n=1,2,3)$ defined in Eq. (23) and all other qubits are in their ground states. Then the initial state of the qubit chain can be written as \[\|\psi_{n}[\theta(0)]\rangle=\|ggg\cdots\chi_{n}\cdots ggg\rangle.\] (24) Note that $\|\psi_{n}\rangle$ is just the Wannier function of the $n$-th magnon band. When $\theta$ is swept from $t=0$ to $t=T_{p}$, the state in the initial unit cell experiences an adiabatic pumping and the entanglement will propagate to the other unit cells. Define the CE as \[\hat{P}=\sum_{x=1}^{N}x(\hat{P}_{a_{x}}^{e}+\hat{P}_{b_{x}}^{e}+\hat{P}_{c_{x} }^{e}).\] (25) The time-dependent average of the CE for an initial excitation $\|\chi_{n}\rangle$ ($n=1,2,3$) during the pumping is described by \[\bar{P}_{n}(t)=\langle\psi_{n}[\theta(t)]\|\hat{P}\|\psi_{n}[\theta(t)]\rangle.\] (26) We further write the above equation into momentum space and get \[\begin{split}\bar{P}_{n}(t)&=\frac{1}{2\pi}\int dk_{ x}i\langle u_{k_{x},\theta,n}\|\partial_{k_{x}}\|u_{k_{x},\theta,n}\rangle\\ &=\frac{1}{2\pi}\int dk_{x}A_{n}(k_{x},\theta),\end{split}\] (27) where the Wannier function $\|\psi_{n}\rangle$ has been rewritten in form of the Bloch wave function as $\|\psi_{n}\rangle=\frac{1}{2\pi}\int dk_{x}e^{ik_{x}r}\|u_{k_{x},\theta,n}\rangle$. Equation (27) indicates that the CE is related to the Berry connection $A_{n}(k_{x},\theta)=i\langle u_{k_{x},\theta,n}\|\partial{}_{k_{x}}\|u_{k_{x}, \theta,n}\rangle$, which depends on the choice of the gauge in the Bloch state. Let $\theta$ be changed continuously from $\theta_{i}$ to $\theta_{f}$. The shift of the CE is then \[\begin{split}\bar{P}_{n}(t_{f})-\bar{P}_{n}(t_{i})&= \frac{1}{2\pi}\int dk_{x}A_{n}(k_{x},\theta_{f})\\ &-\frac{1}{2\pi}\int dk_{x}A_{n}(k_{x},\theta_{i})\end{split}\] (28) Using the Stokes theorem, equation (28) can be rewritten in terms of $F_{n}(k_{x},\theta)$ with $F_{n}(k_{x},\theta)=\nabla\times A_{n}(k_{x},\theta)=i(\|\langle\partial_{ \theta}u_{\mathbf{k}n}\|\partial_{k_{x}}u_{\mathbf{k}n}\rangle-\text{c.c.})$. For a pumping circle of $2\pi$, i.e., $\theta_{f}=\theta_{i}+2\pi$, $\hat{H}(\theta_{i})=\hat{H}(\theta_{f})$, and the shift of the CE is given by the integral of the Berry curvature on the torus $\{k_{x}\in(0,2\pi/3]$,$\,\theta\in(0,2\pi]\}$. We thus find \[\begin{split}\bar{P}_{n}(T_{p})-\bar{P}_{n}(0)&= \frac{1}{2\pi}\int_{k_{x}}\int_{\theta}dk_{x}d\theta\nabla\times A_{n}(k_{x}, \theta)\\ &=\frac{1}{2\pi}\int_{k_{x}}\int_{\theta}dk_{x}d\theta\,F_{n}(k_{ x},\theta)\\ &=C_{n},\end{split}\] (29) which shows that the shift of the CE during one pumping circle is equal to the Chern number of the corresponding topological magnon band and is gauge invariant. In Figs. 3(b) and 3(c), we numerically calculate $\bar{P}_{1,3}(t)$ and $\bar{P}_{2}(t)$ for a chain of 18 qubits ($N=6$), where the qubits in the third unit cell are prepared in the state $\|\chi_{1,3}\rangle$ and $\|\chi_{2}\rangle$, respectively. It is found that $\bar{P}_{1,3}(t)$ is shifted to the left by one unit cell and $\bar{P}_{2}(t)$ is shifted to the right by two unit cells. Moreover, the shifts of the CE are equal to the corresponding Chern numbers $C_{1,3}=-1$ and $C_{2}=2$, respectively. Such process yields an entanglement-dependent topological pumping, where both the quantized shift number and pumping direction are entanglement-dependent. This pumping can be realized with the following parameters: $g_{1}/2\pi=100$ MHz and $\Omega=0.39g_{1}$. The total pumping time is $T_{p}=2\pi/\Omega=25.5$ ns, much longer than typical decoherence times of superconducting X-mon qubits. Similarly, the entanglement-dependent topological pumping also can be realized in the generalized SSH-type qubit chain with $p>3$. We now analyze the influence of practical imperfections in tuning qubit couplings on the above entanglement-dependent topological pumping. This imperfection can be described by the Hamiltonian $H_{d}=\sum_{x}\delta J_{1x}\hat{a}_{x}^{{\dagger}}\hat{b}_{x}+\delta J_{2x} \hat{b}_{3x}^{{\dagger}}\hat{c}_{x}+\delta J_{3}\hat{c}_{x}^{{\dagger}}\hat{a} _{x+1}+\text{H.c.}$, where $\delta J_{1x,2x,3x}=W\delta$, with $\delta\in[-0.5,0.5]$ being a random number and $W$ being the imperfect strength. In Figs. 3(d) and 3(e), we have numerically calculated \[\delta\bar{P}_{n}=\bar{P}_{n}(T_{p},W)-\bar{P}_{n}(0,W)\] (30) where $\bar{P}_{n}(t,W)$ ($n=1,2,3$) is the CE in the presence of imperfect qubit coupling. For each $\delta J_{1x,2x,3x}$, we choose 50 samples to perform the numerical simulation. The final derived $\bar{P}_{n}(t,W)$ is obtained by averaging over the results of all samples. Our numerical results show that the entangle-dependent topological pumping is robust against qubit coupling imperfections. The quantized shifts of the CE $\delta\bar{P}_{1,3}$ and $\delta\bar{P}_{2}$ have plateaus at the values $-1$ and $2$ when the imperfection strength $W\leq 0.1g_{1}$, which correspond to the topological Chern numbers $C_{1,3}=-1$ and $C=2$, respectively. ## V Conclusions In conclusion, we have proposed an experimentally feasible protocol using a tunable Xmon qubit chain to realize SSH and generalized SSH models that support a variety of topological magnon phases protected by topological winding numbers and Chern numbers. We have explicitly studied the dynamics of a single-excitation quantum state in these qubit chains and revealed new topological phenomenons, including the entanglement-dependent topological pumping. We have also found that the topological invariants can be directly measured from the dynamics of qubit excitation, which provides a simple method to directly measure topological invariants rooted in the momentum space using quantum dynamics in the real space. Our work may open a new prospect to realize various topological models and explore new topological effects in well-controllable quantum computing platforms. ## VI Acknowledgements This work is supported by the National Key R&D Program of China under Grants No. 2017YFA0304203 and No. 2016YFA0301803; the NSFC under Grants No. 11674200, No. 11604392, No. 11434007 and No. 91636218; the PCSIRT under Grant No. IRT13076; SFSSSP; OYTPSP; SSCC; and 1331KSC. L.T. is supported by the National Science Foundation (USA) under Award Numbers DMR-0956064 and PHY-1720501, the UC Multicampus-National Lab Collaborative Research and Training under Award No. LFR-17-477237, and the UC Merced Faculty Research Grants 2017. ## References * (1) M.H. Devoret and R.J. Schoelkopf, Science 339, 1169 (2013). * (2) J.Q. You and F. Nori, Physics Today 58, 42 (2005); Nature (London) 474, 589 (2011). * (3) J.M. Martinis, npj Quantum Information 1, 15005 (2015). * (4) J.M. Gambetta, J.M. Chow, and M. Steken, npj Quantum Information 3, 2 (2017). * (5) R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. OMalley, P. Roushan, J. Wenner, T. C. White, A.N. Cleland, and J.M. Martinis, Phys. Rev. Lett. 111, 080502 (2013). * (6) Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J.Y. Mutus, P.O. Malley, C.M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T.C. White, M. R. Geller, A.N. Cleland, and J.M. Martinis, Phys. Rev. Lett. 113, 220502 (2014). * (7) R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T.C. White, J. Mutus, A.G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P.O. Malley, P. Roushan, A. Vainsencher, J. Wenner, A.N. Korotkov, A.N. Cleland, and J.M. Martinis, Nature (London) 508, 500 (2014). * (8) J.M. Chow, J.M. Gambetta, E. Magesan, D.W. Abraham, A.W. Cross, B.R. Johnson, N.A. Masluk, C.A. Ryan, J.A. Smolin, S.J. Srinivasan and M. Steffen, Nat. Commun. 5, 4015 (2014). * (9) A.D. Corcoles, E. Magesan, S.J. Srinivasan, A.W. Cross, M. Steffen, J.M. Gambetta and J.M. Chow, Nat. Commun. 6, 6979 (2015). * (10) M. Takita, A.D. Corcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J.M. Chow, and J.M. Gambetta, Phys. Rev. Lett. 117, 210505 (2016). * (11) T. Brecht, W. Pfaff, C. Wang, Y.W. Chu, L. Frunzio, M.H. Devoret and R.J. Schoelkopf, npj Quantum Information 2, 16002 (2016). * (12) D.L. Underwood, W.E. Shanks, A.C.Y. Li, L. Ateshian, J. Koch, and A.A. Houck, Phys. Rev. X 6, 021004 (2016). * (13) S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy in near-term devices, arXiv: 1608.00263 (2016). * (14) I. Buluta and F. Nori, Science 326, 108 (2009). * (15) I.M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153 (2014). * (16) A.A. Houck, H.E. Türeci, and J. Koch, Nat. Phys, 8, 292 (2012). * (17) K. Le Hur, L. Henriet, A. Petrescu, K. Plekhanov, G. Roux, M. Schiro, C. R. Physique 17, 808 (2016). * (18) M. Hartmann, F. Brandão, and M. Plenio, Laser Photonics Rev. 2, 527 (2008). * (19) A. Tomadin, and R. Fazio, J. Opt. Soc. Am. B 27, 130 (2010). * (20) D. Rossini and R. Fazio, Phys. Rev. Lett. 99, 186401 (2007). * (21) U.L. Heras, A. Mezzacapo, L. Lamata, S. Filipp, A. Wallraff and E. Solano, Phys. Rev. Lett. 112, 200501 (2014). * (22) A. Mezzacapo, E. Rico, C. Sabin, I.L. Egusquiza, L. Lamata, and E. Solano, Phys. Rev. Lett. 115, 240502 (2015). * (23) J.J. García-Ripoll, E. Solano, and M.A. Martin-Delgado, Phys. Rev. B 77, 024522 (2008). * (24) A. Kurcz, A. Bermudez, and J.J. García-Ripoll, Phys. Rev. Lett. 112, 180405 (2014). * (25) L. Tian, Phys. Rev. Lett. 105, 167001 (2010). * (26) F. Mei, V. M. Stojanovic, I. Siddiqi, L. Tian, Phys. Rev. B 88, 224502 (2013). * (27) V. M. Stojanovic, M. Vanevic, E. Demler, L. Tian, Phys. Rev. B 89, 144508 (2014). * (28) D. Marcos, P. Rabl, E. Rico, and P. Zoller, Phys. Rev. Lett. 111, 110504 (2013). * (29) O. Viehmann, J.V. Delft, and F. Marquardt, Phys. Rev. Lett. 110, 030601 (2013). * (30) S Ashhab, New. J. Phys. 16, 113006 (2014). * (31) E. Kapit, Phys. Rev. A. 87, 062336 (2013); 92, 012302 (2015) * (32) Y.D. Wang, F. Xue, Z. Song and C.P. Sun, Phys. Rev. B. 76, 174519 (2007). * (33) G. Chen, Z.D. Chen, and J.Q. Liang, Phys. Rev. A 76, 055803 (2007). * (34) L.H. Du, X.X. Zhou, Y.J. Han, G.C. Guo, Z.W. Zhou, Phys. Rev. A 86, 032302 (2012). * (35) D.G. Ferguson, A. A. Houck, and J. Koch, Phys. Rev. X. 3, 011003 (2013). * (36) A. Baksic and C. Ciuti, Phys. Rev. Lett. 112, 173601 (2014). * (37) M. Biondi, E.P.L. van Nieuwenburg, G. Blatter, S.D. Huber, and S. Schmidt, Phys. Rev. Lett. 115, 143601 (2015). * (38) J. Ghosh, B.C. Sanders, New J. Phys. 18, 033015 (2016). * (39) Y.P. Zhong, D. Xu, P. Wang, C. Song, Q.J. Guo, W.X. Liu, K. Xu, B.X. Xia, C.Y. Lu, S.Y. Han, J.W. Pan, H.H. Wang, Phys. Rev. Lett. 117, 110501 (2016). * (40) R. Barends, L. Lamata, J. Kelly, L. García-Álvarez, A. G. Fowler, A. Megrant, E. Jeffrey, T.C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.C. Hoi, C. Neill, P.J.J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, E. Solano, J.M. Martinis, Nat. Commun. 6, 7654 (2015). * (41) Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, A. Wallraff, Phys. Rev. X 5, 021027 (2015). * (42) S.H. Gourgy, V.V. Ramasesh, C.D. Grandi, I. Siddiqi, and S.M. Girvin, Phys. Rev. Lett. 115, 240501 (2015). * (43) P. Roushan, C. Neill, A. Megrant, Y. Chen, R. Babbush, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Je?rey, J. Kelly, E. Lucero, J. Y. Mutus, P. J. J. O Malley, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, E. Kapit, H. Neven, and J. Martinis, Nat. Phys. 13, 146 (2016). * (44) P.J.J. O Malley, R. Babbush, I.D. Kivlichan, J. Romero, J.R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A.G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J.Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T.C. White, P.V. Coveney, P.J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis, Phys. Rev. X. 6, 031007 (2016). * (45) M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, and A. A. Houck, Phys. Rev. X. 7, 011016 (2017). * (46) N. Goldman, J. C. Budich, and P. Zoller, Nat. Phys. 12, 639 (2016). * (47) L. Lu, J. D. Joannopoulos, and M. Soljacic, Nat. Photon. 8, 821 (2014). * (48) T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, I. Carusotto, arXiv: 1802.04173. * (49) J.Q. You, X.F. Shi, X.D. Hu, and F. Nori, Phys. Rev. B. 81, 014505 (2010). * (50) J. Koch, A. A. Houck, K. Le Hur, and S.M. Girvin, Phys. Rev. A 82, 043811 (2010). * (51) M. Hafezi, P. Adhikari, and J. M. Taylor, Phys. Rev. B 90, 060503(R) (2014). * (52) E. Kapit, M. Hafezi, and S. H. Simon, Phys. Rev. X. 4, 031039 (2014). * (53) D. I. Tsomokos, S. Ashhab, and F. Nori, Phys. Rev. A 82, 052311 (2010). * (54) S. Gasparinetti, P. Solinas, and J. P. Pekola, Phys. Rev. Lett. 107, 207002 (2011). * (55) A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, S. Filipp, Nature 496, 482 (2013). * (56) F. Mei, J.B. You, W. Nie, R. Fazio, S.-L. Zhu, and L. C. Kwek, Phys. Rev. A 92, 041805(R) (2015). * (57) F. Mei, Z.-Y. Xue, D.-W. Zhang, L. Tian, C. Lee, and S.-L. Zhu, Quantum Sci. Technol. 1, 015006 (2016) * (58) Y.P. Wang, W.L. Yang, Y. Hu, Z.Y. Xue, Y. Wu, npj Quantum Information 2, 16015 (2016). * (59) Z.H. Yang, Y.P. Wang, Z.Y. Xue, W.L. Yang, Y. Hu, J.H. Gao, Y. Wu, Phys. Rev. A 93, 062319 (2016). * (60) J. Tangpanitanon, V. M. Bastidas, S. Al-Assam, P. Roushan, D. Jaksch, and D. G. Angelakis, Phys. Rev. Lett. 117, 213603 (2016), * (61) X. Gu, S. Chen, and Y. X. Liu, arXiv:1711.06829v1. * (62) T. Goren, K. Plekhanov, F. Appas, and K. L. Hur, Phys. Rev. B 97, 041106(R) (2018). * (63) M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M. Sandberg, J. Gao, M. R. Vissers, D. P. Pappas, A. Polkovnikov, and K. W. Lehnert, _Phys. Rev. Lett._113, 050402 (2014). * (64) P. Roushan, C. Neill, Y Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, _Nature_ 515, 241 (2014). * (65) Z. Zhang, T. Wang, L. Xiang, J. Yao, J. Wu, and Y. Yin, Phys. Rev. A 95, 042345 (2017). * (66) X. Tan, D.-W. Zhang, Q. Liu, G. Xue, H.-F. Yu, Y.Q. Zhu, H. Yan, S.-L. Zhu, and Y. Yu, Phys. Rev. Lett. 120, 130503 (2018). * (67) V. V. Ramasesh, E. Flurin, M. Rudner, SI. iddiqi, and N. Y. Yao, _Phys. Rev. Lett._118, 130501 (2017). * (68) E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi, _Phys. Rev. X_7, 031023 (2017). * (69) O. Viyuela, A. Rivas, S. Gasparinetti, A. Wallraff, S. Filipp, and M.A. Martin-Delgado, npj Quantum Information 4, 10 (2018). * (70) W. P. Su, J. R. Schrieffer, and A. J. Heeger, _Phys. Rev. Lett._42, 1698 (1979). * (71) D. J. Thouless, _Phys. Rev. B_27, 6083 (1983). * (72) S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, _Nature Phys._12, 296 (2016). * (73) M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, _Nature Phys._ 12, 350 (2016). * (74) F. Mei, J. B. You, D. W. Zhang, X. C. Yang, R. Fazio, S. L. Zhu, and L. C. Kwek, _Phys. Rev. A_90, 063638 (2014). * (75) H. I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, _Phys. Rev. Lett._116, 200402 (2016). * (76) T. Matsubara and H. Matsuda, _Prog. Theor. Phys_. 16, 569 (1956). * (77) J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C.Hoi, C. Neill, P.J.J. OMalley1, C. Quintana, P. Roushan, A.Vainsencher, J. Wenner, A.N. Cleland, and J.M. Martinis, Nature (London) 519, 66 (2015). * (78) A. P. Schnyder, S. Ryu, A. Furusaki, A. W. W. Ludwig, New J. Phys.12, 065010 (2010). * (79) M. Z. Hasan and C. L. Kane, _Rev. Mod. Phys_. 82, 3045 (2010). * (80) X.-L. Qi and S.-C. Zhang, _Rev. Mod. Phys_. 83, 1057 (2011).
1404.5710	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 41612, "num_imgs": 0, "llama3_tokens_count": 16615 }	[]	# Möbius function of semigroup posets through Hilbert series Jonathan Chappelon * Université de Montpellier, Institut de Mathématiques et de Modélisation de Montpellier, Case Courrier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 05, France jonathan.chappelon@um2.fr Ignacio García-Marco ignacio.garcia-marco@um2.fr Luis Pedro Montejano lpmontejano@gmail.com Jorge Luis Ramírez Alfonsín jramirez@um2.fr January 29, 2015 ###### Abstract. In this paper, we investigate the Möbius function $\mu_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^{m}$. We introduce and develop a new approach to study $\mu_{\mathcal{S}}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $\mu_{\mathcal{S}}$ when $\mathcal{S}$ belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given. Key words and phrases:Möbius function, locally finite poset, semigroup, Hilbert series 2010 Mathematics Subject Classification: 20M15; 05A99; 06A07; 11A25; 20M05; 20M25 * Corresponding Author: Phone/Fax: +33-467144166. Email: jonathan.chappelon@um2.fr ## 1. Introduction The _Möbius function_ is an important concept that was introduced by Gian-Carlo Rota more than 50 years ago in [10]. It is a generalization to (_locally finite_) posets of the classical Möbius arithmetic function on the integers (given by the Möbius function of the poset obtained from the positive integers partially ordered by divisibility). We refer the reader to [10] for a large number of its applications. In this paper, we investigate the Möbius function associated to posets arising naturally from subsemigroups of $\mathbb{Z}^{m}$ as follows. Let $a_{1},\ldots,a_{n}$ be nonzero vectors in $\mathbb{Z}^{m}$ and let $\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle$ denote the semigroup generated by $a_{1},\ldots,a_{n}$, that is, \[\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle=\{x_{1}a_{1}+\cdots+x_{n}a_{n}\, \|\,x_{1},\ldots,x_{n}\in\mathbb{N}\}.\] We say that $\mathcal{S}$ is _pointed_ if $\mathcal{S}\cap(-\mathcal{S})=\{0\}$, where $-\mathcal{S}:=\{-x\,\|\,x\in\mathcal{S}\}$. Whenever $\mathcal{S}$ is pointed, $\mathcal{S}$ induces on $\mathbb{Z}^{m}$ a poset structure whose partial order $\leqslant_{\mathcal{S}}$ is defined by $x\leqslant_{\mathcal{S}}y$ if and only if $y-x\in\mathcal{S}$ for all $x$ and $y$ in $\mathbb{Z}^{m}$. This (locally finite) poset will be denoted by $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$. We denote by $\mu_{\mathcal{S}}$ the Möbius function associated to $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$. As far as we are aware, $\mu_{\mathcal{S}}$ has only been investigated when $\mathcal{S}$ is a _numerical semigroup_, i.e., when $\mathcal{S}\subset\mathbb{N}$ and $\gcd\{a_{1},\ldots,a_{n}\}=1$. Moreover, the only known results concerning $\mu_{\mathcal{S}}$ are an old theorem due to Deddens [3], which determines the value of $\mu_{\mathcal{S}}$ when $\mathcal{S}$ has exactly two generators, and a recent paper due to Chappelon and Ramírez Alfonsín [2], where the authors investigate $\mu_{\mathcal{S}}$ when $\mathcal{S}=\langle a,a+d,\ldots,a+kd\rangle$ with $a,k,d\in\mathbb{Z}^{+}$. In both papers, the authors approach the problem by a thorough study of the intrinsic properties of each semigroup. Here, we introduce and develop a new and more general method to study $\mu_{\mathcal{S}}$ by means of the _Hilbert series of the semigroup $\mathcal{S}$_. This enables us to provide formulas for $\mu_{\mathcal{S}}$ when $\mathcal{S}$ belongs to some families of semigroups. We also investigate when a locally finite poset is isomorphic to a semigroup poset. This paper is organized as follows. In the next section, after reviewing some standard notions of the Möbius function, we then interpret them for semigroup posets. In Section 3, we present two general results (Theorems 3.1 and 3.3) giving a new and general approach to study $\mu_{\mathcal{S}}$ through the _Hilbert series of the semigroup $\mathcal{S}$_. This enables us in Section 4 to provide formulas for $\mu_{\mathcal{S}}$ when $\mathcal{S}$ is a _semigroup with a unique Betti element_ and when $\mathcal{S}=\langle a_{1},a_{2},a_{3}\rangle\subset\mathbb{N}$ is a _complete intersection numerical semigroup_ (generalizing results in [2, 3]). Finally, in Section 5, we characterize those locally finite posets $\mathcal{P}$ that are isomorphic to the poset associated to a semigroup $\mathcal{S}$. In this case $\mu_{\mathcal{P}}$ can be computed by means of $\mu_{\mathcal{S}}$ (this will be illustrated with the well-known classical Möbius arithmetic function). ## 2. Möbius function associated to a semigroup poset Let $(\mathcal{P},\leqslant_{\mathcal{P}})$ be a partially ordered set, or _poset_ for short. The _strict partial order_$<_{\mathcal{P}}$ is the reduction of $\leqslant_{\mathcal{P}}$ given by $a<_{\mathcal{P}}b$ if and only if $a\leqslant_{\mathcal{P}}b$ and $a\neq b$. Let $a$ and $b$ be two elements of the poset $\mathcal{P}$. The _interval_ between $a$ and $b$ is defined by \[{\left[a,b\right]}_{\mathcal{P}}:=\left\{c\in\mathcal{P}\ \middle\|\ a\leqslant _{\mathcal{P}}c\leqslant_{\mathcal{P}}b\right\}.\] A poset is said to be _locally finite_ if every interval has finite cardinality. We only consider locally finite posets in this paper. A _chain_ of length $l\geqslant 0$ between $a$ and $b$ is a subset of ${\left[a,b\right]}_{\mathcal{P}}$ containing $a$ and $b$, of cardinality $l+1$ and totally ordered by $<_{\mathcal{P}}$, that is $\left\{a_{0},a_{1},\ldots,a_{l}\right\}\subset{\left[a,b\right]}_{\mathcal{P}}$ such that \[a=a_{0}<_{\mathcal{P}}a_{1}<_{\mathcal{P}}a_{2}<_{\mathcal{P}}\cdots<_{ \mathcal{P}}a_{l-1}<_{\mathcal{P}}a_{l}=b.\] For any nonnegative integer $l$, we denote by $c_{l}(a,b)$ the number of distinct chains between $a$ and $b$ of length $l$. This number always exists because the poset $\mathcal{P}$ is supposed to be locally finite. For instance, the number of chains $c_{2}(2,12)$, where the poset is $\mathbb{N}$ partially ordered by divisibility, is equal to $2$. Indeed, there are exactly $2$ chains of length $2$ between $2$ and $12$ in ${\left[2,12\right]}_{\mathbb{N}}=\left\{2,4,6,12\right\}$, which are $\left\{2,4,12\right\}$ and $\left\{2,6,12\right\}$. For any locally finite poset $\mathcal{P}$, the _Möbius function_$\mu_{\mathcal{P}}$ is the integer-valued function on $\mathcal{P}\times\mathcal{P}$ defined by (1) \[\mu_{\mathcal{P}}(a,b)=\sum_{l\geqslant 0}{{(-1)}^{l}c_{l}(a,b)},\] for all elements $a$ and $b$ of the poset $\mathcal{P}$. Note that this sum is always finite because, for $a$ and $b$ given, the interval ${\left[a,b\right]}_{\mathcal{P}}$ has finite cardinality. The concept of Möbius function for a locally finite poset $(\mathcal{P},\leqslant)$ was introduced by Rota in [10]. There, Rota proves the following property of the Möbius function: for all $(a,b)\in\mathcal{P}\times\mathcal{P}$, (2) \[\mu_{\mathcal{P}}(a,a)=1\quad\text{and}\quad\sum_{c\in\left[a,b\right]_{ \mathcal{P}}}\mu_{\mathcal{P}}(a,c)=0\text{, if}\ a<_{\mathcal{P}}b.\] Here, only posets associated to semigroups of $\mathbb{Z}^{m}$ are considered. We begin by summarizing some generalities on semigroups that will be useful for the understanding of this work. Let $\mathcal{S}:=\langle a_{1},\ldots,a_{n}\rangle\subset\mathbb{Z}^{m}$ denote the subsemigroup of $\mathbb{Z}^{m}$ generated by $a_{1},\ldots,a_{n}\in\mathbb{Z}^{m}$, i.e., \[\mathcal{S}:=\langle a_{1},\ldots,a_{n}\rangle=\{x_{1}a_{1}+\cdots+x_{n}a_{n} \,\|\,x_{1},\ldots,x_{n}\in\mathbb{N}\}.\] The semigroup $\mathcal{S}$ induces the binary relation $\leqslant_{\mathcal{S}}$ on $\mathbb{Z}^{m}$ given by \[x\leqslant_{\mathcal{S}}y\ \Longleftrightarrow\ y-x\in\mathcal{S}.\] It turns out that $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$ is a poset if and only if $\mathcal{S}$ is pointed. Indeed, $\leqslant_{\mathcal{S}}$ is antisymmetric if and only if $\mathcal{S}$ is pointed. Moreover, if $\mathcal{S}$ is pointed then the poset $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$ is locally finite. Let $\mu_{\mathcal{S}}$ denote the Möbius function associated to $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$. It is easy to see that $\mu_{\mathcal{S}}$ can be considered as a univariate function of $\mathbb{Z}^{m}$. Indeed, for all $x,y\in\mathbb{Z}^{m}$ and for all $l\geqslant 0$, one can observe that $c_{l}(x,y)=c_{l}(0,y-x)$. Thus, we obtain \[\mu_{\mathcal{S}}(x,y)=\mu_{\mathcal{S}}(0,y-x)\] for all $x,y\in\mathbb{Z}$. In the sequel of this paper, we shall only consider the reduced Möbius function $\mu_{\mathcal{S}}:\mathbb{Z}^{m}\longrightarrow\mathbb{Z}$ defined by \[\mu_{\mathcal{S}}(x):=\mu_{\mathcal{S}}(0,x),\text{ for all}\ x\in\mathbb{Z}^{ m}.\] Thus, the formula given by (2) may now be simplified when the locally finite poset is $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$. Proposition 2.1.: _([2, Proposition 1]) Let $\mathcal{S}$ be a pointed semigroup and let $x\in\mathbb{Z}^{m}$. Then,_ \[\sum_{b\in\mathcal{S}}\mu_{\mathcal{S}}(x-b)=\left\{\begin{array}[]{ll}1&\text {if }x=0,\\ 0&\text{otherwise}.\end{array}\right.\] Proof.: From (1), we know that $\mu_{\mathcal{S}}(b)=0$ for all $b\notin\mathcal{S}$. Since $\mathcal{S}$ is pointed, it follows that \[\sum_{b\in\mathcal{S}}\mu_{\mathcal{S}}(0-b)=\mu_{\mathcal{S}}(0)=1.\] Finally, if $x\neq 0$, then we apply (2) and we obtain that \[0=\sum_{b\in[0,x]_{\mathbb{Z}^{m}}}\mu_{\mathcal{S}}(b)=\sum_{b\in\mathcal{S} \atop x-b\in\mathcal{S}}\mu_{\mathcal{S}}(b)=\sum_{b\in\mathcal{S}\atop x-b\in \mathcal{S}}\mu_{\mathcal{S}}(x-b)=\sum_{b\in\mathcal{S}}\mu_{\mathcal{S}}(x-b).\] ∎ Proposition 2.1 will be very useful to obtain most of our results. ## 3. The Hilbert and Möbius series In this section, we present two results (Theorem 3.1 and Theorem 3.3), both relating the Hilbert series of the semigroup $\mathcal{S}$ with the Möbius function of the poset $(\mathbb{Z}^{m},\leqslant_{\mathcal{S}})$. Before proving these theorems, some basic notions on multivariate Hilbert series are quickly recalled. For a thorough study of multivariate Hilbert series, we refer the reader to [8]. Let $k$ be any field and let $\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle$ be a subsemigroup of $\mathbb{Z}^{m}$. The semigroup $\mathcal{S}$ induces a grading in the ring of polynomials $R:=k[x_{1},\ldots,x_{n}]$ by assigning ${\rm deg}_{\mathcal{S}}(x_{i}):=a_{i}\in\mathbb{Z}^{m}$, for all $i\in\{1,\ldots,n\}$. Then, the _$\mathcal{S}$-degree_ of the monomial $m:=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$ is ${\rm deg}_{\mathcal{S}}(m):=\sum_{i=1}^{n}\alpha_{i}a_{i}\in\mathbb{Z}^{m}$. A polynomial is said to be _$\mathcal{S}$-homogeneous_ if all of its monomials have the same $\mathcal{S}$-degree and an ideal is _$\mathcal{S}$-homogeneous_ if it is generated by $\mathcal{S}$-homogeneous polynomials. For all $b\in\mathbb{Z}^{m}$, we denote by $R_{b}$ the $k$-vector space generated by all $\mathcal{S}$-homogeneous polynomials of $\mathcal{S}$-degree $b$. Whenever $\mathcal{S}$ is pointed, the $k$-vector space $R_{b}$ has finite dimension, for all $b\in\mathbb{Z}^{m}$ (see [8, Proposition 4.1.19]). Let $I\subset R$ be an $\mathcal{S}$-homogeneous ideal. The _multigraded Hilbert function_ of $M:=R/I$ is \[H\!F_{M}:\mathbb{Z}^{m}\longrightarrow\mathbb{N},\] defined by $H\!F_{M}(b):={\rm dim}_{k}(R_{b})-{\rm dim}_{k}(R_{b}\cap I)$, for all $b\in\mathbb{Z}^{m}$. For every $b=(b_{1},\ldots,b_{m})\in\mathbb{Z}^{m}$, we denote by $\mathbf{t}^{b}$ the monomial $t_{1}^{b_{1}}\cdots t_{m}^{b_{m}}$ in the Laurent polynomial ring $\mathbb{Z}[t_{1},\ldots,t_{m},t_{1}^{-1},\ldots,t_{m}^{-1}]$. The _multivariate Hilbert series of $M$_ is the following formal power series in $\mathbb{Z}[[t_{1},\ldots,t_{m},t_{1}^{-1},\ldots,t_{m}^{-1}]]$: \[\mathcal{H}_{M}(\mathbf{t}):=\sum_{b\in\mathbb{Z}^{m}}H\!F_{M}(b)\,\mathbf{t}^ {b}.\] We denote by $I_{\mathcal{S}}$ the _toric ideal of $\mathcal{S}$_, i.e., the kernel of the homomorphism of $k$-algebras \[\varphi:R\longrightarrow k[t_{1},\ldots,t_{m},t_{1}^{-1},\ldots,t_{m}^{-1}]\] induced by $\varphi(x_{i})=\mathbf{t}^{a_{i}}$, for all $i\in\{1,\ldots,n\}$. It is well known that $I_{\mathcal{S}}$ is $\mathcal{S}$-homogeneous (see [11, Corollary 4.3]). Moreover, the multivariate Hilbert series of $M=R/I_{\mathcal{S}}$ with respect to the grading induced by $\mathcal{S}$ is (3) \[\mathcal{H}_{M}(\mathbf{t})=\sum_{b\in\mathcal{S}}\mathbf{t}^{b}.\] Indeed, $R_{b}=\{0\}$ and $H\!F_{M}(b)=0$ whenever $b\notin\mathcal{S}$. In addition, if $b\in\mathcal{S}$, $\varphi$ induces an isomorphism of $k$-vector spaces between $R_{b}/(R_{b}\cap I)$ and $\{\alpha\,\mathbf{t}^{b}\,\|\,\alpha\in k\}$, for all $b\in\mathcal{S}$. Hence, $H\!F_{M}(b)=1$ in this case. From now on, the multivariate Hilbert series of $R/I_{\mathcal{S}}$ is called the _Hilbert series of $\mathcal{S}$_ and is denoted by $\mathcal{H}_{\mathcal{S}}(\mathbf{t})$. Theorem 3.1.: _Let $\mathcal{S}$ be a pointed semigroup and let $c_{1},\ldots,c_{k}$ be nonzero vectors in $\mathbb{Z}^{m}$. If we set_ \[\left(1-\mathbf{t}^{c_{1}}\right)\cdots\left(1-\mathbf{t}^{c_{k}}\right)\ \mathcal{H}_{\mathcal{S}}(\mathbf{t})=\sum_{b\in\mathbb{Z}^{m}}f_{b}\,\mathbf{ t}^{b}\in\mathbb{Z}[[t_{1},\ldots,t_{m},t_{1}^{-1},\ldots,t_{m}^{-1}]],\] _then,_ \[\sum_{b\in\mathbb{Z}^{m}}f_{b}\,\mu_{\mathcal{S}}(x-b)=0\] _for all $x\notin\left\{\sum_{i\in A}c_{i}\ \middle\|\ A\subset\{1,\ldots,k\}\right\}$._ Proof.: From (3), we know that \[f_{b}=\sum_{A\subset\{1,\ldots,k\}\atop b-\sum_{i\in A}c_{i}\in\mathcal{S}}(-1 )^{\|A\|},\] for all $b\in\mathbb{Z}^{m}$. Set $\Delta:=\left\{\sum_{i\in A}c_{i}\ \middle\|\ A\subset\{1,\ldots,k\}\right\}$. By Proposition 2.1, for all $x\notin\Delta$ and $A\subset\{1,\ldots,k\}$, we have that \[\sum_{b\in\mathcal{S}}\mu_{\mathcal{S}}\left(x-\sum_{i\in A}a_{i}-b\right)=0.\] Hence, for all $x\notin\Delta$, it follows that where \[\alpha_{b}=\sum_{A\subset\{1,\ldots,k\}\atop b-\sum_{i\in A}c_{i}\in\mathcal{S }}(-1)^{\|A\|}=f_{b}.\] This completes the proof. ∎ Notice that the formula $\left(1-\mathbf{t}^{c_{1}}\right)\cdots\left(1-\mathbf{t}^{c_{k}}\right)\ \mathcal{H}_{\mathcal{S}}(\mathbf{t})=\sum_{b\in\mathbb{Z}^{m}}f_{b}\,\mathbf{ t}^{b}$ might have an infinite number of terms. Nevertheless, for every $x\in\mathbb{Z}^{m}$, the formula $\sum_{b\in\mathbb{Z}^{m}}f_{b}\,\mu_{\mathcal{S}}(x-b)=0$ only involves a finite number of nonzero summands, since $\mathcal{S}$ is pointed. The following example illustrates how to apply Theorem 3.1 to compute $\mu_{\mathcal{S}}$. Example 3.2.: Consider the semigroup $\mathcal{S}=\langle 2,3\rangle\subset\mathbb{N}$. We observe that $\mathcal{S}=\mathbb{N}\setminus\{1\}$. Hence, $\mathcal{H}_{\mathcal{S}}(t)=1+\sum_{b\geqslant 2}t^{b}\in\mathbb{Z}[[t]]$ and $t^{2}\,\mathcal{H}_{\mathcal{S}}(t)=t^{2}+\sum_{b\geqslant 4}t^{b}$. It follows that \[(1-t^{2})\,\mathcal{H}_{\mathcal{S}}(t)=1+t^{3}.\] Applying Theorem 3.1, we get that \[\mu_{\mathcal{S}}(x)+\mu_{\mathcal{S}}(x-3)=0,\] for all $x\in\mathbb{Z}\setminus\{0,2\}$. Furthermore, by direct computation, we have $\mu_{\mathcal{S}}(0)=1$, $\mu_{\mathcal{S}}(2)=-1$ and $\mu_{\mathcal{S}}(x)=0$ for all $x<0$. This leads to the formula \[\mu_{\mathcal{S}}(x)=\left\{\begin{array}[]{rl}1&\text{if }x\geqslant 0\text{ and }x\equiv 0\text{ or }5\pmod{6},\\ -1&\text{if }x\geqslant 0\text{ and }x\equiv 2\text{ or }3\pmod{6},\\ 0&\text{otherwise.}\end{array}\right.\] From now on, we consider the _Möbius series_$\mathcal{G}_{\mathcal{S}}$, i.e., the generating function of the Möbius function \[\mathcal{G}_{\mathcal{S}}(\mathbf{t}):=\sum_{b\in\mathbb{Z}^{m}}\mu_{\mathcal{ S}}(b)\,\mathbf{t}^{b}\in\mathbb{Z}[[t_{1},\ldots,t_{m},t_{1}^{-1},\ldots,t_{m }^{-1}]].\] Theorem 3.3.: _Let $\mathcal{S}$ be a pointed semigroup. Then,_ \[\mathcal{H}_{\mathcal{S}}(\mathbf{t})\cdot\mathcal{G}_{\mathcal{S}}(\mathbf{t} )=1.\] Proof.: From the definitions of $\mathcal{H}_{\mathcal{S}}(\mathbf{t})$ and $\mathcal{G}_{\mathcal{S}}(\mathbf{t})$, we obtain that \[\mathcal{H}_{\mathcal{S}}(\mathbf{t})\cdot\mathcal{G}_{\mathcal{S}}(\mathbf{t} )=\left(\sum_{b\in\mathcal{S}}\mathbf{t}^{b}\right)\left(\sum_{b\in\mathbb{Z}^ {m}}\mu_{\mathcal{S}}(b)\mathbf{t}^{b}\right)=\sum_{b\in\mathbb{Z}^{m}}\left( \sum_{c\in\mathcal{S}}\mu_{\mathcal{S}}(b-c)\right)\mathbf{t}^{b}.\] The result follows by Proposition 2.1. ∎ Theorem 3.3 states that, whenever we can explicitly compute the inverse of $\mathcal{H}_{\mathcal{S}}(\mathbf{t})$, we will be able to obtain $\mu_{\mathcal{S}}$. Let us illustrate the latter with the following Example 3.4.: Let $\{e_{1},\ldots,e_{m}\}$ denote the canonical basis of $\mathbb{N}^{m}$ and let $\mathcal{S}=\langle e_{1},\ldots,e_{m}\rangle=\mathbb{N}^{m}$. Clearly, we have that \[\mathcal{H}_{\mathbb{N}^{m}}(\mathbf{t})=\sum_{b\in\mathbb{N}^{m}}\mathbf{t}^{ b}=\frac{1}{(1-t_{1})\cdots(1-t_{m})}.\] Therefore, by Theorem 3.3, we obtain \[\mathcal{G}_{\mathbb{N}^{m}}(\mathbf{t})=\left(1-t_{1}\right)\cdots\left(1-t_{ m}\right)=\sum_{A\subset\{1,\ldots,m\}}(-1)^{\mid A\mid}\,\prod_{i\in A}\,t_{i }=\sum_{A\subset\{1,\ldots,m\}}(-1)^{\mid A\mid}\,\mathbf{t}^{\sum_{i\in A}e_{ i}}.\] So we derive the following formula for $\mu_{\mathbb{N}^{m}}$: \[\mu_{\mathbb{N}^{m}}(x)=\left\{\begin{array}[]{cl}(-1)^{\|A\|}&{\text{\ }if\ }x= \sum_{i\in A}e_{i}\text{ for some }A\subset\{1,\ldots,m\},\\ \\ 0&\text{otherwise}.\end{array}\right.\] A pointed semigroup $\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle$ is called a _complete intersection_ semigroup if its corresponding toric ideal $I_{\mathcal{S}}$ is a _complete intersection_ ideal, i.e., if $I_{\mathcal{S}}$ is generated by $n-d$$\mathcal{S}$-homogeneous polynomials, where $d$ is the dimension of the $\mathbb{Q}$-vector space spanned by $a_{1},\ldots,a_{n}$. For characterizations of complete intersection toric ideals, we refer the reader to [5]. Let $B=(b_{1},b_{2},\ldots,b_{k})$ be a $k$-tuple of nonzero vectors in $\mathbb{Z}^{m}$ such that the semigroup $\mathcal{T}:=\langle b_{1},\ldots,b_{k}\rangle$ is pointed and let $b\in\mathbb{Z}^{m}$. We denote by $d_{B}(b)$ the number of non-negative integer representations of $b$ by $b_{1},\ldots,b_{k}$, that is, the number of solutions of $b=\sum_{i=1}^{k}x_{i}b_{i}$, where $x_{i}$ is a nonnegative integer for all $i$. Since $\mathcal{T}$ is pointed, we know that $d_{B}(b)$ is finite, for all $b\in\mathbb{Z}^{m}$. Moreover, $d_{B}(0)=1$. It is well known (see, e.g., [8, Theorem 5.8.15]) that its generating function is given by \[\sum_{b\in\mathbb{Z}^{m}}d_{B}(b)\,\mathbf{t}^{b}=\frac{1}{\left(1-\mathbf{t}^ {b_{1}}\right)\left(1-\mathbf{t}^{b_{2}}\right)\cdots\left(1-\mathbf{t}^{b_{k} }\right)}.\] Corollary 3.5.: _Let $\mathcal{S}$ be a complete intersection pointed semigroup and assume that $I_{\mathcal{S}}$ is generated by $n-d$$\mathcal{S}$-homogeneous polynomials of $\mathcal{S}$-degrees $b_{1},\ldots,b_{n-d}\in\mathbb{Z}^{m}$. Then,_ \[\mu_{\mathcal{S}}(x)=\sum_{A\subset\{1,\ldots,n\}}(-1)^{\mid A\mid}\,d_{B} \left(x-\sum_{i\in A}b_{i}\right),\] _for all $x\in\mathbb{Z}^{m}$, where $B=\{b_{1},\ldots,b_{n-d}\}$._ Proof.: By [8, Page 341], we have that \[\mathcal{H}_{\mathcal{S}}(\mathbf{t})=\frac{(1-\mathbf{t}^{b_{1}})\cdots(1- \mathbf{t}^{b_{n-d}})}{(1-\mathbf{t}^{a_{1}})\cdots(1-\mathbf{t}^{a_{n}})}.\] Thus, from Theorem 3.3, we obtain \[\mathcal{G}_{\mathcal{S}}(\mathbf{t})\begin{array}[t]{l}=\frac{1} {\mathcal{H}_{\mathcal{S}}(\mathbf{t})}=\frac{(1-\mathbf{t}^{a_{1}})\cdots(1- \mathbf{t}^{a_{n}})}{{(1-\mathbf{t}^{b_{1}})\cdots(1-\mathbf{t}^{b_{n-d}})}}\\ \\ =\left(\sum_{A\subset\{1,\ldots,n\}}(-1)^{\|A\|}\,\mathbf{t}^{\sum_ {i\in A}a_{i}}\right)\left(\sum_{b\in\mathbb{Z}^{m}}d_{B}(b)\,\mathbf{t}^{b} \right)\\ \\ =\sum_{b\in\mathbb{Z}^{m}}\sum_{A\subset\{1,\ldots,n\}}(-1)^{\|A\|} \,d_{B}(b)\,\mathbf{t}^{b+\sum_{i\in A}a_{i}}\\ \\ =\sum_{b\in\mathbb{Z}^{m}}\sum_{A\subset\{1,\ldots,n\}}(-1)^{\|A\|} \,d_{B}\left(b-\sum_{i\in A}a_{i}\right)\,\mathbf{t}^{b}.\end{array}\] ∎ ## 4. Explicit formulas for the Möbius function In this section, we exploit the results of the previous section to obtain explicit formulas for $\mu_{\mathcal{S}}$ when $\mathcal{S}$ is a semigroup with a unique Betti element (Theorem 4.1) and when $\mathcal{S}$ is a complete intersection numerical semigroup generated by three elements (Theorem 4.4). The results included in this section are consequences of Corollary 3.5. However, they can also be obtained with a different proof by using Theorem 3.1. ### Semigroups with a unique Betti element. A semigroup $\mathcal{S}\subset\mathbb{N}^{m}$ is said to have a _unique Betti element_$b\in\mathbb{N}^{m}$ if its corresponding toric ideal is generated by a set of $\mathcal{S}$-homogeneous polynomials of common $\mathcal{S}$-degree $b$. García-Sánchez, Ojeda and Rosales proved [6, Corollary 10] that these semigroups are always complete intersection. Theorem 4.1.: _Let $\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle\subset\mathbb{N}^{m}$ be a semigroup with a unique Betti element $b\in\mathbb{N}^{m}$. If we denote by $d$ the dimension of the $\mathbb{Q}$-vector space generated by $a_{1},\ldots,a_{n}$, then we have_ \[\mu_{\mathcal{S}}(x)=\sum_{j=1}^{t}\,(-1)^{\mid A_{j}\mid}\,{{k_{A_{j}}+n-d-1} \choose{k_{A_{j}}}},\] _where $\{A_{1},\ldots,A_{t}\}=\left\{A\subset\{1,\ldots,n\}\ \middle\|\text{ there exists }k_{A}\in\mathbb{N}\text{ such that }x-\sum_{i\in A}a_{i}=k_{A}\,b\right\}$._ Proof.: By Corollary 3.5, for all $x\in\mathbb{Z}^{m}$, we have \[\mu_{\mathcal{S}}(x)=\sum_{A\subset\{1,\ldots,m\}}(-1)^{\mid A\mid}\,d_{B} \left(x-\sum_{i\in A}a_{i}\right),\] where $B$ is the $(n-d)$-tuple $(b,\ldots,b)$. The equality \[d_{B}(y)=\left\{\begin{array}[]{cl}\binom{k+n-d-1}{k}&\text{ if } y=kb\text{ with }k\in\mathbb{N},\\ \\ 0&\text{otherwise},\end{array}\right.\] for all $y\in\mathbb{Z}^{m}$, completes the proof. ∎ When $m=1$, i.e., when $\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle\subset\mathbb{N}$, $\mathcal{S}$ is a numerical semigroup with a unique Betti element $b\in\mathbb{N}$ if and only if there exist pairwise relatively prime integers $b_{1},\ldots,b_{n}\geqslant 2$ such that $a_{i}:=\prod_{j\neq i}b_{j}$, for all $i\in\{1,\ldots,n\}$, and $b=\prod_{i=1}^{n}b_{i}$ (see [6]). In this setting, Theorem 4.1 can be refined as follows. Corollary 4.2.: _Let $\mathcal{S}=\langle a_{1},\ldots,a_{n}\rangle\subset\mathbb{N}$ be a numerical semigroup with a unique Betti element $b\in\mathbb{N}$. Then,_ \[\mu_{\mathcal{S}}(x)=\left\{\begin{array}[]{cl}(-1)^{\mid A\mid} \,{\binom{k+n-2}{k}}&\text{if }x=\sum_{i\in A}a_{i}+kb\text{ for some }A \subset\{1,\ldots,n\},k\in\mathbb{N},\\ \\ 0&\text{otherwise.}\end{array}\right.\] Proof.: Since $d=1$, it is sufficient, by Theorem 4.1, to prove that, for every $A_{1},A_{2}\subset\{1,\ldots,n\}$, if $b$ divides $\sum_{i\in A_{1}}a_{i}-\sum_{i\in A_{2}}a_{i}$, then $A_{1}=A_{2}$. Let $b_{1},\ldots,b_{n}\geqslant 2$ such that $a_{i}=\prod_{j\neq i}b_{j}$. By [6, Example 12] we have that $I_{\mathcal{S}}=(f_{2},\ldots,f_{n})$, where $f_{i}:=x_{1}^{b_{1}}-x_{i}^{b_{i}}$ for all $i\in\{2,\ldots,n\}$. Assume that there exist $A_{1},A_{2}\subset\{1,\ldots,n\}$ such that $A_{1}\not=A_{2}$ and $\sum_{i\in A_{1}}a_{i}-\sum_{i\in A_{2}}a_{i}=kb$, for some $k\in\mathbb{N}$. Thus, the binomial $g:=\prod_{i\in A_{1}}x_{i}-x_{1}^{b_{1}k}\,\prod_{i\in A_{2}}x_{i}\not=0$ belongs to $I_{\mathcal{S}}$ and it can be written as a combination of $f_{2},\ldots,f_{n}$. However, since $x_{j}^{b_{j}}$ does not divide $\prod_{i\in A_{1}}x_{i}$ for all $j\in\{1,\ldots,n\}$, we obtain a contradiction. ∎ As a direct consequence of this result, we recover the Dedden’s result. Corollary 4.3.: _[_3_]_ _Let $a,b\in\mathbb{Z}^{+}$ be relatively prime integers and consider $\mathcal{S}:=\langle a,b\rangle$. Then,_ \[\mu_{\mathcal{S}}(x)=\left\{\begin{array}[]{rl}1&{\text{\ }if\ }x\geqslant 0\ {\text{a}nd\ }x\equiv 0{\text{\ }or\ }a+b\ ({\rm mod\ }ab),\\ -1&{\text{\ }if\ }x\geqslant 0{\text{\ }and\ }x\equiv a{\text{\ }or\ }b\ ({ \text{m}od\ }ab),\\ 0&{\text{\ }otherwise.}\end{array}\right.\] ### Three generated complete intersection numerical semigroups. We provide a semi-explicit formula for $\mu_{\mathcal{S}}$, when $\mathcal{S}$ is a complete intersection numerical semigroup minimally generated by the set $\{a_{1},a_{2},a_{3}\}$. When $\mathcal{S}=\langle a_{1},a_{2},a_{3}\rangle\subset\mathbb{N}$, Herzog proves in [7] that $\mathcal{S}$ is a complete intersection if and only if $\gcd\{a_{i},a_{j}\}\,a_{k}\in\langle a_{i},a_{j}\rangle$ with $\{i,j,k\}=\{1,2,3\}$. Suppose that $da_{1}\in\langle a_{2},a_{3}\rangle$, where $d:=\gcd\{a_{2},a_{3}\}$. For every $x\in\mathbb{Z}$, there exists a unique $\alpha(x)\in\{0,\ldots,d-1\}$ such that $\alpha(x)a_{1}\equiv x\ ({\rm mod}\ d)$. It is easy to check that, for every $x,y\in\mathbb{Z}$, (4) \[\alpha(x-y)=\left\{\begin{array}[]{cl}\alpha(x)-\alpha(y)&\text{if }\alpha(x) \geqslant\alpha(y),\\ \\ d+\alpha(x)-\alpha(y)&\text{otherwise}.\end{array}\right.\] Theorem 4.4.: _Let $\mathcal{S}=\langle a_{1},a_{2},a_{3}\rangle$ be a numerical semigroup such that $da_{1}\in\langle a_{2},a_{3}\rangle$, where $d:={\gcd}\{a_{2},a_{3}\}$. For all $x\in\mathbb{Z}$, we have that $\mu_{\mathcal{S}}(x)=0$, if $\alpha(x)\geqslant 2$, and_ \[\mu_{\mathcal{S}}(x)=(-1)^{\alpha}\,\left(d_{B}(x^{\prime})-d_{B}(x^{\prime}-a _{2})-d_{B}(x^{\prime}-a_{3})+d_{B}(x^{\prime}-a_{2}-a_{3})\right)\] _otherwise, where $x^{\prime}:=x-\alpha(x)a_{1}$ and $B:=(da_{1},a_{2}\,a_{3}/d)$._ Proof.: Suppose that $da_{1}=\gamma_{2}a_{2}+\gamma_{3}a_{3}$ with $\gamma_{2},\gamma_{3}\in\mathbb{N}$. Then, by [7, Theorem 3.10], it follows that \[I_{\mathcal{S}}=\left(x_{1}^{d}-x_{2}^{\gamma_{2}}x_{3}^{\gamma_{3}},\,x_{2}^{ a_{3}/d}-x_{3}^{a_{2}/d}\right).\] So, $I_{\mathcal{S}}$ is generated by two $\mathcal{S}$-homogeneous polynomials of $\mathcal{S}$-degrees $da_{1}$ and $a_{2}a_{3}/d$. Hence, from Corollary 3.5, we have (5) \[\mu_{\mathcal{S}}(x)=\begin{array}[t]{l}d_{B}(x)-d_{B}(x-a_{1})-d_{B}(x-a_{2}) -d_{B}(x-a_{3})+d_{B}(x-(a_{1}+a_{2}))+\\ +d_{B}(x-(a_{1}+a_{3}))+d_{B}(x-(a_{2}+a_{3}))-d_{B}(x-(a_{1}+a_{2}+a_{3})),\\ \end{array}\] for all integers $x$, where $B:=(da_{1},a_{2}a_{3}/d)$. Since $\alpha(da_{1})=\alpha(a_{2}a_{3}/d)=0$. It follows that $\alpha(y)=0$ if $y\in\langle da_{1},a_{2}a_{3}/d\rangle$. As a consequence of this, $d_{B}(y)=0$ whenever $\alpha(y)\neq 0$. Let $C:=\{0,a_{1},a_{2},a_{3},a_{1}+a_{2},a_{2}+a_{3},a_{3}+a_{1},a_{1}+a_{2}+a_{3}\}$. Notice that $\alpha(y)\in\{0,1\}$, for all $y\in C$. We distinguish three different cases upon the value of $\alpha:=\alpha(x)$, for $x\in\mathbb{Z}$. _Case 1_.: $\alpha\geqslant 2$. We deduce that $\alpha(x-y)=\alpha(x)-\alpha(y)\neq 0$ and $d_{B}(x-y)=0$, for all $y\in C$. Therefore, using (5), we obtain that $\mu_{\mathcal{S}}(x)=0$. _Case 2_.: $\alpha=1$. We deduce that $\alpha(x-y)\neq 0$ and $d_{B}(x-y)=0$ for all $y\in\{0,a_{2},a_{3},a_{2}+a_{3}\}$. Therefore, using (5), we obtain that \[\mu_{\mathcal{S}}(x)=-d_{B}(x-a_{1})+d_{B}(x-a_{1}-a_{2})+d_{B}(x-a_{1}-a_{3}) -d_{B}(x-a_{1}-a_{2}-a_{3}).\] _Case 3_.: $\alpha=0$. Since $d\geqslant 2$, we deduce that $\alpha(x-y)\neq 0$ and $d_{B}(x-y)=0$ for all $y\in\{a_{1},a_{1}+a_{2},a_{1}+a_{3},a_{1}+a_{2}+a_{3}\}$. Therefore, using (5), we obtain that \[\mu_{\mathcal{S}}(x)=d_{B}(x)-d_{B}(x-a_{2})-d_{B}(x-a_{3})+d_{B}(x-a_{2}-a_{3 }).\] This completes the proof. ∎ Theorem 4.4 yields an algorithm for computing $\mu_{\mathcal{S}}(x)$, for all $x\in\mathbb{Z}$, which relies on the computation of four values of $d_{B}(y)$, where $B=(da_{1},a_{2}a_{3}/d)$. It is worth mentioning that in [9, Section 4.4] there are several results and methods to compute these values. Also note that Theorem 4.4 generalizes [2, Theorem 3], where the authors provide a semi-explicit formula for $\mathcal{S}=\langle 2q,2q+e,2q+2e\rangle$ where $q,e\in\mathbb{Z}^{+}$ and $\gcd\{2q,2q+e,2q+2e\}=1$. Indeed, if $\mathcal{S}=\langle a,a+e,\ldots,a+ke\rangle$ with $\gcd\{a,e\}=1$ and $k\geqslant 2$, then $\mathcal{S}$ is a complete intersection if and only if $k=2$ and $a$ is even (see [1]). ## 5. When is a poset equivalent to a semigroup poset? A natural question is whether a poset $\mathcal{P}$ is _isomorphic_ to a poset associated to a semigroup $\mathcal{S}$ since, in such a case, one might be able to calculate $\mu_{\mathcal{P}}$ by computing $\mu_{\mathcal{S}}$ instead. Let us illustrate this with the following two examples in which we can easily find an appropriate order isomorphism between the poset $\mathcal{P}$ and the poset associated to the semigroup $\mathbb{N}^{m}=\langle e_{1},\ldots,e_{m}\rangle$. Example 5.1.: We consider the classical arithmetic Möbius function $\mu$. Recall that for all $a,b\in\mathbb{N}$ such that $a\mid b$, we have that (6) \[\mu(a,b)=\left\{\begin{array}[]{cl}(-1)^{r}&\text{if }b/a\text{ is a product of }r\text{ different prime numbers,}\\ 0&\text{otherwise}.\end{array}\right.\] For every $m\in\mathbb{Z}^{+}$, we denote by $p_{1},\ldots,p_{m}$ the first $m$ prime numbers and by $\mathbb{N}_{m}$ the set of integers that can be written as a product of powers of $p_{1},\ldots,p_{m}$. Then, for all $m\geqslant 1$, the map $\psi:\mathbb{N}_{m}\rightarrow\mathbb{N}^{m}$ defined as $\psi(p_{1}^{\,\alpha_{1}}\cdots p_{m}^{\,\alpha_{m}})=(\alpha_{1},\ldots, \alpha_{m})$ is an order isomorphism between $\mathbb{N}_{m}$, ordered by divisibility, and the poset $(\mathbb{N}^{m},\leqslant_{\mathbb{N}^{m}})$. Hence, for every $a,b\in\mathbb{N}$, we consider $m\in\mathbb{Z}^{+}$ such that $a,b\in\mathbb{N}_{m}$ and we recover the formula (6) by means of the Möbius function of $\mathbb{N}^{m}$ given in Example 3.4. Example 5.2.: Let $D=\{d_{1},\ldots,d_{m}\}$ be a finite set and let us consider the (locally finite) poset $\mathcal{P}$ of multisets of $D$ ordered by inclusion. For every $S,T\in\mathcal{P}$ such that $T\subset S$, it is well known that (7) \[\mu_{\mathcal{P}}(T,S)=\left\{\begin{array}[]{cl}(-1)^{\|S\setminus T\|}&\text{ if }T\subset S\text{ and }S\setminus T\text{ is a set,}\\ 0&\text{otherwise.}\end{array}\right.\] We consider the map $\psi:\mathcal{P}\rightarrow\mathbb{N}^{m}$ defined as $\psi(S)=(s_{1},\ldots,s_{m})$, where $s_{i}$ denotes the multiplicity of $d_{i}$ in $S$, for all $S\in\mathcal{P}$. We consider the order in $\mathbb{N}^{m}$ induced by the semigroup $\mathbb{N}^{m}$, i.e., $\alpha\leqslant_{\mathbb{N}^{m}}\beta$ if and only if $\beta-\alpha\in\mathbb{N}^{m}$ for all $\alpha,\beta\in\mathbb{N}^{m}$. We have that $\psi$ is an _order isomorphism_, i.e., an order preserving and order reflecting bijection. Thus, we can say that the poset of multisets of a finite set is a particular case of semigroup poset. This implies that for all $S,T\in\mathcal{P}$ such that $T\subset S$, $\mu_{\mathcal{P}}(T,S)=\mu_{\mathbb{N}^{m}}(\psi(T),\psi(S))=\mu_{\mathbb{N}^{ m}}(\psi(S)-\psi(T))$ and by Example 3.4 we retrieve the formula (7). In this section, we present a characterization of those locally finite posets $\mathcal{P}$ isomorphic to the poset associated to a semigroup $\mathcal{S}$ (Theorem 5.5). Let $(\mathcal{P},\leqslant_{\mathcal{P}})$ be a locally finite poset. For every $x\in\mathcal{P}$, we set $\mathcal{P}_{x}:=\{y\in\mathcal{P}\,\|\,x\leqslant_{\mathcal{P}}y\}$ and we consider the restricted Möbius function $\mu_{\mathcal{P}}(-,x):\mathcal{P}_{x}\rightarrow\mathbb{Z}$. It is clear that, if there exists a pointed semigroup $\mathcal{S}$ and an order isomorphism $\psi:(\mathcal{P}_{x},\leqslant_{\mathcal{P}})\longrightarrow(\mathcal{S}, \leqslant_{\mathcal{S}})$, then $\mu_{\mathcal{P}}(-,x)$ can be computed by means of the Möbius function of $(\mathcal{S},\leqslant_{\mathcal{S}})$, since $\mu_{\mathcal{P}}(y,x)=\mu_{\mathcal{S}}(\psi(y))$ for all $y\in\mathcal{P}_{x}$. The poset $\mathcal{P}_{x}$ is said to be _autoequivalent_ if and only if, for all $y\in\mathcal{P}_{x}$, there exists an order isomorphism $g_{y}:\mathcal{P}_{x}\longrightarrow\mathcal{P}_{y}$ such that $g_{y}\circ g_{z}=g_{z}\circ g_{y}$, for all $y,z\in\mathcal{P}_{x}$, and $g_{x}$ is the identity. For all $y\in\mathcal{P}_{x}$, we set $l_{1}(y):=\{z\in\mathcal{P}\,\|\,\text{ there is not }u\in\mathcal{P}$ such that $y\lneq u\lneq z\}$. Whenever $\mathcal{P}_{x}$ is autoequivalent with isomorphisms $\{g_{y}\}_{x\leqslant y}$ and $l_{1}(x)$ is a finite set of $n$ elements, we associate to $\mathcal{P}$ a subgroup $L_{\mathcal{P}}\subset\mathbb{Z}^{n}$ in the following way. Let $l_{1}(x)=\{x_{1},\ldots,x_{n}\}\subset\mathcal{P}$ and consider the map \[f:\mathbb{N}^{n}\longrightarrow\mathcal{P}\] defined as $f(0,\ldots,0)=x$, and for all $\alpha\in\mathbb{N}^{n}$ and all $i\in\{1,\ldots,n\}$, $f(\alpha+e_{i})=g_{x_{i}}(f(\alpha))$, where $\{e_{1},\ldots,e_{n}\}$ is the canonical basis of $\mathbb{N}^{m}$. In particular, $f(e_{i})=g_{x_{i}}(f(0))=g_{x_{i}}(x)=x_{i}$, for all $i\in\{1,\ldots,n\}$. Lemma 5.3.: $f$ _is well defined and is surjective._ Proof.: Suppose that $\alpha+e_{i}=\beta+e_{j}$. Then, we set $\gamma:=\alpha-e_{j}=\beta-e_{i}\in\mathbb{N}^{n}$. Thus, $f(\alpha+e_{i})=g_{x_{i}}(f(\alpha))=g_{x_{i}}(g_{x_{j}}(f(\gamma)))=g_{x_{j}} (g_{x_{i}}(f(\gamma)))=g_{x_{j}}(f(\beta))=f(\beta+e_{j})$ and $f$ is well defined. Take $y\in\mathcal{P}_{x}$. If $y=x$, then $y=f(0)$. If $y\neq x$, then there exists $z\in\mathcal{P}_{x}$ such that $y\in l_{1}(z)$. Therefore $y=g_{z}(x_{j})$ for some $j\in\{1,\ldots,n\}$. We claim that if $z=f(\alpha)$, then $y=f(\alpha+e_{j})$. Indeed, $f(\alpha+e_{j})=g_{x_{j}}(f(\alpha))=g_{x_{j}}(z)=g_{x_{j}}(g_{z}(x))=g_{z}(g_ {x_{j}}(x))=g_{z}(x_{j})=y$. ∎ Now, we set $L_{\mathcal{P}}:=\{\alpha-\beta\in\mathbb{Z}^{n}\,\|\,f(\alpha)=f(\beta)\}$. Lemma 5.4.: $L_{\mathcal{P}}$ _is a subgroup of $\mathbb{Z}^{n}$._ Proof.: If $\gamma\in L_{\mathcal{P}}$, then $-\gamma\in L_{\mathcal{P}}$. Moreover, if $\gamma_{1},\gamma_{2}\in L_{\mathcal{P}}$, then $\gamma_{1}+\gamma_{2}\in L_{\mathcal{P}}$. Indeed, take $\alpha,\alpha^{\prime},\beta,\beta^{\prime}\in\mathbb{N}^{m}$ such that $f(\alpha)=f(\alpha^{\prime})$, $\gamma_{1}=\alpha-\alpha^{\prime}$, $f(\beta)=f(\beta^{\prime})$ and $\gamma_{2}=\beta-\beta^{\prime}$. Then $f(\alpha+\beta)=f(\alpha^{\prime}+\beta)=f(\alpha^{\prime}+\beta^{\prime})$ and the lemma is proved. ∎ For every subgroup $L\subset\mathbb{Z}^{n}$ the _saturation_ of $L$ is the group \[\textrm{Sat}(L):=\left\{\gamma\in\mathbb{Z}^{n}\ \middle\|\text{ there exists } d\in\mathbb{Z}^{+}\text{ such that }d\gamma\in L\right\}.\] Theorem 5.5.: _Let $\mathcal{P}$ be a locally finite poset and let $x\in\mathcal{P}$. Then, $(\mathcal{P}_{x},\leqslant)$ is isomorphic to $(\mathcal{S},\leqslant_{\mathcal{S}})$ for some (pointed) semigroup $\mathcal{S}\subset\mathbb{Z}^{n}$ if and only if $\mathcal{P}_{x}$ is autoequivalent, $l_{1}(x)$ is finite and $L_{\mathcal{P}}={\rm Sat}(L_{\mathcal{P}})$._ Proof.: $(\Rightarrow)$ Let $\mathcal{S}\subset\mathbb{Z}^{m}$ be a (pointed) semigroup and denote by $\{a_{1},\ldots,a_{n}\}$ its unique minimal set of generators. Assume that $\psi:\mathcal{P}_{x}\rightarrow\mathcal{S}$ is an order isomorphism. Let us prove that $\mathcal{P}_{x}$ is autoequivalent, i.e., $l_{1}(x)=n$ and $L_{\mathcal{P}}={\rm Sat}(L_{\mathcal{P}})$. First, we observe that setting $x_{i}:=\psi^{-1}(a_{i})$, then $l_{1}(x)=\{x_{1},\ldots,x_{n}\}$. Now, for every $y\in\mathcal{P}_{x}$, we set \[\begin{array}[]{cccl}g_{y}:&\mathcal{P}_{x}&\longrightarrow&\mathcal{P}_{y}\\ &z&\longmapsto&\psi^{-1}(\psi(z)+\psi(y)).\end{array}\] Then it is straightforward to check that $g_{y}$ is an order isomorphism. Moreover, $g_{x}$ is the identity map in $\mathcal{P}_{x}$ and $g_{y}\circ g_{z}=g_{z}\circ g_{y}$, for all $y,z\in\mathcal{P}_{x}$. Let $f:\mathbb{N}^{m}\rightarrow\mathcal{P}_{x}$ be the map associated to $\{g_{y}\}_{y\leqslant x}$, i.e., $f(0)=x$ and if $f(\alpha)=y$, then $f(\alpha+e_{j})=g_{x_{j}}(f(\alpha))$. We claim that $\psi(f(\alpha))=\sum\alpha_{i}a_{i}\in\mathcal{S}$, for all $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}$. Indeed, $\psi(f(0))=\psi(x)=0$ and if we assume that $\psi(f(\alpha))=\sum\alpha_{i}a_{i}$ for some $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{m}$, then $\psi(f(\alpha+e_{j}))=\psi(g_{x_{j}}(\alpha))=\psi(z)+\psi(x_{j})=\sum\alpha_{ i}a_{i}+a_{j}$, as desired. Since $L_{\mathcal{P}}\subset{\rm Sat}(L_{\mathcal{P}})$ by definition, let us prove that ${\rm Sat}(L_{\mathcal{P}})\subset L_{\mathcal{P}}$. We take $\gamma\in{\rm Sat}(L_{\mathcal{P}})$, then $d\gamma\in L_{\mathcal{P}}$ for some $d\in\mathbb{Z}^{+}$. This means that there exist $\alpha,\beta\in\mathbb{N}^{n}$ such that $f(\alpha)=f(\beta)$ and $d\gamma=\alpha-\beta$. Hence, we have that $\sum\alpha_{i}a_{i}=\psi(f(\alpha))=\psi(f(\beta))=\sum\beta_{i}a_{i}$. This implies that $\sum\gamma_{i}a_{i}=1/d\ (\sum(\alpha_{i}-\beta_{i})a_{i})=0$. Thus, if we take $\alpha^{\prime},\beta^{\prime}\in\mathbb{N}^{m}$ such that $\gamma=\alpha^{\prime}-\beta^{\prime}$, then $\psi(f(\alpha^{\prime}))=\psi(f(\beta^{\prime}))$ and, whence, $f(\alpha^{\prime})=f(\beta^{\prime})$ and $\gamma\in L_{\mathcal{P}}$. $(\Leftarrow)$ Since $L_{\mathcal{P}}={\rm Sat}(L_{\mathcal{P}})$, we have that $\mathbb{Z}^{n}/L_{\mathcal{P}}$ is a torsion free group. Hence there exists a group isomorphism $\rho:\mathbb{Z}^{n}/L_{\mathcal{P}}\rightarrow\mathbb{Z}^{m}$, where $m=n-{\rm rk}(L_{\mathcal{P}})$. We let $a_{i}:=\rho(e_{i}+L_{\mathcal{P}})$ for all $i\in\{1,\ldots,n\}$ and set $\mathcal{S}:=\langle a_{1},\ldots,a_{n}\rangle\subset\mathbb{Z}^{m}$. We claim that $(\mathcal{P}_{x},\leqslant)$ and $(\mathcal{S},\leqslant_{\mathcal{S}})$ are isomorphic. More precisely, it is straightforward to check that the map \[\begin{array}[]{cccl}\psi:&\mathcal{P}_{x}&\longrightarrow&\mathcal{S}\\ &y&\longmapsto&\sum\alpha_{i}a_{i},{\text{\ }if\ }f(\alpha)=y\end{array}\] is an order isomorphism. ∎ The necessity direction of Theorem 5.5 can be stated in algebraic terms as : whenever $\mathcal{P}_{x}$ is autoequivalent and $l_{1}(x)$ is finite, the subgroup $L_{\mathcal{P}}$ defines a lattice ideal $I:=(\{\mathbf{x}^{\alpha}-\mathbf{x}^{\beta}\,\|\,\alpha-\beta\in L_{\mathcal{P }}\})$. Moreover, $\mathcal{P}_{x}$ is isomorphic to a semigroup poset $(\mathcal{S},\leqslant_{\mathcal{S}})$ if and only if the ideal $I$ itself is the toric ideal of a semigroup $\mathcal{S}$. The latter holds if and only if $I$ is prime or, equivalently, if $L_{\mathcal{P}}={\rm Sat}(L_{\mathcal{P}})$ (see [4]). ## References * [1]I. Bermejo, I. García-Marco, Complete intersections in certain affine and projective monomial curves, Bull Braz Math Soc, New Series 45(4), 2014, 1-26. * [2] J. Chappelon, J. L. Ramírez Alfonsín, On the Möbius function of the locally finite poset associated with a numerical semigroup, Semigroup Forum 87 (2013), no. 2, 313–330. * [3] J. A. Deddens, A combinatorial identity involving relatively prime integers, J. Combin. Theory Ser. A 26 (1979), no. 2, 189–192. * [4] D. Eisenbud, B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1996) 1–45. * [5] K. Fischer, W. Morris and J. Shapiro, Affine semigroup rings that are complete intersections, Proc. Amer. Math. Soc. 125 (1997), 3137–3145. * [6] P. A. García-Sánchez, I. Ojeda and J. C. Rosales, Affine semigroups having a unique Betti element, J. Algebra Appl. 12 (2013), no. 3, 1250177, 11 pp. * [7]J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970) 175–193. * [8]M. Kreuzer, L. Robbiano, _Computational Commutative Algebra 2_. Springer-Verlag Berlin Heidelberg, 2005. * [9] J. L. Ramírez Alfonsín, _The Diophantine Frobenius Problem_, Oxford Lecture Series in Mathematics and Its Applications, vol. 30. Oxford University Press, Oxford (2005). * [10] G-C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, (1964) 340–368. * [11] B. Sturmfels, _Gröbner Bases and Convex Polytopes_, University Lecture Series 8, American Mathematical Society, Providence, RI, 1996.
1701.06195	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 29812, "num_imgs": 6, "llama3_tokens_count": 8569 }	[ "content_image/1701.06195/x1.png", "content_image/1701.06195/x2.png", "content_image/1701.06195/x3.png", "content_image/1701.06195/x4.png", "content_image/1701.06195/x5.png", "content_image/1701.06195/x6.png" ]	# Electronic Properties, Screening and Efficient Carrier Transport in NaSbS${}_{2}$ Jifeng Sun David J. Singh Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211-7010, USA singhdj@missouri.edu February 21, 2024 ###### Abstract NaSbS${}_{2}$ is a semiconductor that was recently shown to have remarkable efficacy as a solar absorber indicating efficient charge collection even in defected material. We report first principles calculations of properties that show (1) an indirect gap only slightly smaller than the direct gap, which may impede recombination of photoexcited carriers, (2) highly anisotropic electronic and optical properties reflecting a layered crystal structure, (3) a pushed up valence band maximum due to repulsion from the Sb $5s$ states and (4) cross-gap hybridization between the S $p$ derived valence bands and the Sb $5p$ states. This latter feature leads to enhanced Born effective charges that can provide local screening and therefore defect tolerance. These features are discussed in relation to the performance of the compound as a semiconductor with efficient charge collection. pacs: ## I Introduction NaSbS${}_{2}$ is a little studied semiconductor that was recently shown to be remarkably effective as a solar absorber material, Rahayu _et al._ (2016) and has also been mentioned as a potential thermoelectric material based on computational screening. Bhattacharya _et al._ (2016) Remarkably, an efficiency of 3.18% was obtained in the first report using NaSbS${}_{2}$ nanparticles in a dye sensitized solar cell. Rahayu _et al._ (2016) This is comparable to the efficiency of early cells of similar type made using organometallic halide perovskites. Kojima _et al._ (2009) Those materials have since proven to be a revolutionary advance in photovoltaics, but suffer from issues with long term stability and the fact that they contain Pb, which is undesirable. Lee _et al._ (2012); Giustino and Snaith (2016) NaSbS${}_{2}$ contains only environmentally friendly low cost elements, and this very promising early experimental result, and the theoretical results below suggest that it may as well represent a revolutionary new material for solar photovoltaic applications. From a valence point of view one might regard the stoichiometry as derived from PbS by splitting of the divalent Pb site into monovalent Na and trivalent Sb. In this way one might anticipate that NaSbS${}_{2}$ would be a semiconductor and that the band gap may be higher than that of PbS if normal trends are followed due to the splitting of the cation site. However, at ambient temperature the crystal structure deviates strongly the rocksalt structure of PbS, as discussed below. More significantly, the presence of Na suggests a propensity for defects, e.g. Na off-stoichiometry. Good charge collection requires a high carrier drift length, which typically occurs in high quality defect free material, and so the high performance of NaSbS${}_{2}$ as a solar absorber is surprising. However, several soft lattice solar materials have been discovered, most notably CH${}_{3}$NH${}_{3}$PbI${}_{3}$, Lee _et al._ (2012) where defects do not seem to play the same detrimental role as in more traditional materials such as CdTe. Stranks _et al._ (2013) Here we report first principles calculations aimed at understanding the properties of this compound, especially in relation to its use as a solar absorber. ## II Structure and Methods The present calculations were performed within density functional theory (DFT). The electronic structure and optical properties were calculated using the general potential linearized augmented planewave (LAPW) method, Singh and Nordstrom (2006) as implemented in the WIEN2k code. Blaha _et al._ (2001) The total energy calculations and relaxation of the atomic coordinates was done using the Perdew, Burke and Ernzerhof (PBE) generalized gradient approximation (GGA). Perdew _et al._ (1996) For this, relativity was treated at a scalar relativistic level for the valence states. The core states were treated relativistically. LAPW sphere radii of $R$=2.2 bohr were used for all elements, along with a planewave sector basis cutoff determined by, $RK_{max}$=9 (here $R$ is the radius of the smallest LAPW sphere, i.e. 2.2 bohr, and $K_{max}$ is the planewave cutoff). Local orbitals were added to the basis for the S $s$, Na $s$ and $p$, and Sb $d$ semicore states. Spin-orbit was included for the electronic and optical properties. The band gap is important for these, and accordingly these calculations were done using the modified Becke-Johnson (mBJ) potential of Tran and Blaha. Tran and Blaha (2009) This functional gives band gaps in good accord with experiment for a wide variety of simple semiconductors and insulators and also appears to give reliable band shapes and optical properties, although at least in certain semiconductors the band masses are more similar to those obtained in standard density functional calculations than those from many body calculations. Tran and Blaha (2009); Koller _et al._ (2011); Singh (12); Kim _et al._ (2010); Singh (14) Calculation of the transport function for conductivity was done using the BoltzTraP code. Madsen and Singh (2006) Optical properties were calculated based on electric dipole transitions in the independent particle approximation as implemented in the WIEN2k code. The Born effective charges and the dielectric tensor were calculated using the density functional perturbation theory (DFPT) with the PBE functional as implemented in the VASP code Kresse and Furthmüller (1996) (note that calculation of the dielectric tensor cannot be done with the mBJ potential, since it is not an energy functional and therefore cannot be used to evaluate lattice response). The structure of NaSbS${}_{2}$ has been refined into two different monoclinic groups, $C2/c$ (# 15) Olivier-Fourcade _et al._ (1978) and $C2/m$ (# 12). Volk and Schafer (1978) In addition there is a report of a triclinic structure $P\bar{1}$ (# 2). Kanischeva _et al._ (1979) These are all centrosymmetric structures. The deviation from a cubic structure was discussed in terms of lone pair activity of Sb. Olivier-Fourcade _et al._ (1978) ## III Results and Discussion We did calculations for the three reported structures, in each case using the lattice parameters from the diffraction experiments, and then relaxing the atomic positions subject to the spacegroup symmetry. We find that the energy for the $C2/m$ structure is 0.127 eV per formula unit (f.u.) above the energy of the $P\bar{1}$ structure. The $C2/c$ structure is 0.005 eV/f.u. higher than the $P\bar{1}$ structure. We did further calculations to address the issue of the ground state. Specifically, we did full relaxations, including both lattice parameters, angles and internal coordinates. For this purpose we used the VASP code with three different density functionals, specifically the local density approximation (LDA), the PBE GGA Perdew _et al._ (1996) and the PBEsol GGA. Perdew _et al._ (2009) We further did full relaxations of both the $C2/c$ and the $P\bar{1}$ structures using VASP with LDA, PBE, and PBEsol Perdew _et al._ (2009) functionals. These functionals differ in equilibrium volumes for solids. Generally, at increased volume lattices tend to distort more strongly, as was noted in the case of PbTiO${}_{3}$, an oxide ferroelectric with a ground state particularly sensitive to volume. Wu and Cohen (2006) We find that the LDA underestimates the unit cell volume of NaSbS${}_{2}$, yielding 175 Å${}^{3}$, in comparison with the experimental volume of 192 Å${}^{3}$ at 300 K. The PBE functional yields 197 Å${}^{3}$, while PBEsol yields 184 Å${}^{3}$. The LDA predicts a monoclinic ground state, while the fully relaxed triclinic structure is $<$ 2 meV/atom lower in energy for both PBE and PBEsol. Considering the very small energy, and the limitations of DFT calculations we conclude that the ground state is monoclinic $C2/c$ or possibly $P\bar{1}$ with an extremely small triclinic distortion from this monoclinic structure. The other monoclinic structure, $C2/m$, is, however, clearly not a feasible structure. <figure><img src="content_image/1701.06195/x1.png"><figcaption>Figure 1: Monoclinic structure of NaSbS2, showing S as large red spheres, Sbas gold and Na as blue. The left panel shows the layering, while the rightpanel shows a single layer (note that there are two layers with oppositeorientation of the S-Sb-S units per cell.) The short Sb-S bonds in the S-Sb-Sunits are shown by pipes.</figcaption></figure> We calculated properties for both the $C2/c$ and $P\bar{1}$ structures, but find very little difference. For example, the band gap for the triclinic structure is 1.21 eV compared to 1.22 eV for the monoclinic. This is not surprising since the two structures are very similar, though not identical. Internal coordinates for the two structures are given in Tables 1 and 2. As seen, the bond valences Brown (1985) are close to their nominal values indicative of an ionic structure, Na${}^{+}$Sb${}^{3+}$S${}_{2}^{2-}$, although the Sb value of 2.85 is slightly smaller perhaps indicative of some degree of covalency. \| x \| y \| z \| b.v. ---\|---\|---\|---\|--- Na \| 0.0000 \| 0.7500 \| 0.1340 \| 1.06 Sb \| 0.0000 \| 0.7500 \| 0.6051 \| 2.85 S \| 0.2239 \| 0.7606 \| 0.4088 \| 2.02 Table 1: Calculated atomic positions and bond valence sums for monoclinic NaSbS2, spacegroup 15, C2/c, a=8.232 Å, b=6.836 Å, c=8.252 Å, γ=55.72∘. These lattice vectors are from experiment. “b.v.” denotes the bond-valence sum. The fractional atomic coordinates are in terms of the lattice vectors and were determined with the PBE functional. \| x \| y \| z \| b.v. ---\|---\|---\|---\|--- Na \| 0.3658 \| 0.6332 \| 0.2501 \| 1.06 Sb \| 0.8947 \| 0.1049 \| 0.2499 \| 2.84 S \| 0.1330 \| 0.3156 \| 0.7393 \| 2.02 S \| 0.3154 \| 0.1320 \| 0.2390 \| 2.02 Table 2: Calculated atomic positions and bond valence sums for triclinic NaSbS2, spacegroup 2, P¯1, a=5.825 Å, b=5.828 Å, c=6.833 Å, α=113.46∘, β=113.48∘, γ=90.07∘. The atomic positions were determined using the PBE functional and the lattice parameters are from experiment. In the following, we discuss properties of the monoclinic $C2/c$ structure for simplicity. The structure is depicted in Fig. 1. As seen, it is a layered structure, with two layers per unit cell. The layers have composition NaSbS${}_{2}$, with all atoms coplanar in the monoclinic structure and very nearly coplanar in the triclinic structure. A single layer is depicted in the right panel of Fig. 1. The layers show distinct short bonds between Sb and two neighboring S in the same layer leading to apparent S-Sb-S units. These units would lead to a strong ferroelectricity in plane with polarization along the $c$-direction, except that the two layers per cell have opposite orientation so that the polarizations of the individual sheets cancel. The formation of S-Sb-S units leads to a dimerization of the S in the layers. The near neighbor Sb-S distance is 2.486 Å, while the S-Sb-S angle is 98.6${}^{\circ}$. The chains of S-Sb-S units run along the $a$-direction. Thus the structure has anisotropic layers in the $a$-$c$ plane. In the following discussion of optical and electronic properties we use a orthogonal coordinate system where $x$ is along the $a$-axis, $z$ is along the $c$-axis and $y$ is perpendicular to these. The tensor properties show an $xy$ component due to the monoclinicity. <figure><img src="content_image/1701.06195/x2.png"><figcaption>Figure 2: Band structure (left) and carrier pockets (right). The point b is(1/2,0,-1/2) in primitive cell reciprocal lattice units and are shown in thezone in the right panel. The carrier pockets are isosurfaces 0.05 eV below thevalence band maximum (blue) and 0.05 eV above the conduction band minimum(red). Note that the gap is indirect.</figcaption></figure> The band structure near the band edges is depicted in Fig. 2, which shows the band structure along lines where the band extrema occur and a isosurface visualization of the band edges. The valence band maximum (VBM) is on a zone face, as shown, at the point denoted “b”. The conduction band minimum (CBM) is near, but not at, another zone face (L). This indirect band gap has a value, $E_{g}(ind)$=1.22 eV. The direct gap, $E_{g}(dir)$ is at “b”, and is only 0.02 eV ($\sim$250 K) larger. This structure can provide a partial explanation for observed good collection of photoexcited carriers. Specifically, while the very small difference between $E_{g}(ind)$ and $E_{g}(dir)$ is insignificant from the point of view of obtaining good optical absorption for the solar spectrum, the indirect nature of the gap will impede recombination of photoexcited carriers that relax to the band edges. This effect will be stronger at room temperature if the difference between the direct and indirect gaps is a little larger, which is possible considering uncertainties in DFT calculations. We note that 0.02 eV is a small energy and so it will be of importance to verify whether the gap is indirect and if so the magnitude of the difference between the direct and indirect gaps by experiment. <figure><img src="content_image/1701.06195/x3.png"><figcaption>Figure 3: Rigid band conductivity transport function σ/τ, calculated at 300 K,as a function of carrier concentration in electrons per formula unit. Negativevalues denote holes.</figcaption></figure> <figure><img src="content_image/1701.06195/x4.png"><figcaption>Figure 4: Optical absorption spectrum. A Lorentzian broadening of 0.025 eV wasapplied.</figcaption></figure> Fig. 3 shows the transport function $\sigma/\tau$ as obtained from the electronic structure. As seen, the transport is highly two dimensional for both the conduction and valance bands, but is more so for the conduction bands. Transport in plane is also anisotropic, with better conduction along the $a$ ($x$-direction) than along $c$. This amounts to $\sim$35% for the valence bands and $\sim$20% for the conduction bands. Finally, if the effective scattering rates, $\tau^{-1}$, are similar for electrons and holes, the in-plane mobility will be higher for electrons than for holes. The calculated optical absorption spectrum is given in Fig. 4. This spectrum was calculated in the independent particle approximation, i.e. neglecting excitonic effects. These are anticipated to be small due to the small band gap and resulting high electronic (clamped ion) dielectric constant. The spectrum is similar for both in plane polarizations but differs strongly for the yy component, which has electric field polarization perpendicular to the NaSbS${}_{2}$ sheets. Regardless of polarization, the absorption is relatively weak from the onset at the direct gap to $\sim$ 2.5 eV. This emphasizes the important role of good carrier transport to realize the reported efficacy of this material as a solar absorber. Rahayu _et al._ (2016) <figure><img src="content_image/1701.06195/x5.png"><figcaption>Figure 5: Electronic density of states and S p projection onto the LAPW sphereon a per formula unit basis.</figcaption></figure> Fig. 5 shows the calculated electronic density of states along with the S $p$ contribution. The valence bands are derived from S $p$ states, so that the compound should be regarded as nominally ionic. The top of the valence band manifold also appears to be split off to higher energy. This type of splitting is seen in some other S compounds where it arises from repulsion between a lower lying metal state and the S $p$ states. Chen _et al._ (2009); Mitzi _et al._ (2011) The result is that the top of the valence band has antibonding metal - S $p$ character, and often more dispersive bands beneficial for transport. <figure><img src="content_image/1701.06195/x6.png"><figcaption>Figure 6: Sb s and p density of states by projection onto the Sb LAPW sphere.Note that the extended Sb valence orbitals lie mainly outside the 2.2 bohrLAPW spheres, so that the plot shows a quantity proportional to butconsiderably smaller than the full Sb contributions.</figcaption></figure> This antibonding mechanism is operative here. Fig. 6 shows $s$ and $p$ projections on the Sb LAPW spheres. As seen, there is Sb $s$ character at the top of the valence bands including at the VBM. The main Sb $s$ bands are at -9.5 to -7 eV relative to the VBM, and so the VBM has S $p$ - Sb $5s$ antibonding character. Besides this Sb $5s$ character at the top of the valence bands, there is considerable Sb $5p$ character in the valence bands. The Sb $5p$ states are nominally unoccupied in this compound and form the main conduction bands. Thus this Sb $5p$ contribution in the valence bands comes from cross-gap hybridization between the occupied S $p$ states and unoccupied Sb $p$ states. Such cross-gap hybridization is a characteristic of oxide ferroelectric materials where it leads to enhanced Born effective charges and thus ferroelectricity, Cohen (1992) and is also found in phase change materials. Mukhopadhyay _et al._ (2016) It is closely connected with the concept of lone pair driven distortions. Enhanced Born effective charges have also been associated with efficient carrier transport in a number of materials. Du and Singh (27, 28); Fabini _et al._ (2016); Brandt _et al._ (2015); Lehner _et al._ (2015); Du (2014) The mechanism is enhanced local screening due to the high Born charge, which leads to defect tolerance in soft lattice materials. Du and Singh (27); Brandt _et al._ (2015) The Born effective charges were obtained as ${\bf Z}_{ij}^{\ast}$ = $\frac{\Omega}{e}\frac{\partial{\bf P}_{i}}{\partial{\bf u}_{j}}$, where $\Omega$ is the volume of the unit cell, ${\bf P}_{i}$ is the total polarization in direction $i$ and ${\bf u}_{j}$ is the displacement in direction $j$. The calculated Born effective charges of monoclinic NaSbS${}_{2}$ are shown in Table 3. It can be seen that the maximum Born effective charges are 1.47 for Na, 4.69 for Sb and -3.08 for S, respectively. These are considerably larger than the corresponding nominal charges, consistent with the expectation from the electronic structure. The dielectric tensor contains both the electronic and ionic contributions as $\epsilon_{ij}$ = $\epsilon_{\infty,ij}$ + $\epsilon_{ph,ij}$. The electronic part was obtained with ion-clamped using DFPT. Gajdoš _et al._ (2006); Baroni and Resta (1986) The ionic contribution was based on the interatomic force constants calculated using DFPT.Wu _et al._ (2005) For monoclinic NaSbS${}_{2}$, there are four non-zero components, as given in Table 4. The average value given by one third of the trace is 23.8. For comparison, ZnO, which is a good oxide semiconductor that has some defect tolerance at least for $n$-type, Look (2001); McCluskey and Jokela (2009) has a dielectric constant of 9.3. Crisler _et al._ (1968) \| xx \| xy \| xz \| yx \| yy \| yz \| zx \| zy \| zz ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Na \| 1.13 \| -0.15 \| 0.00 \| -0.15 \| 1.47 \| 0.00 \| 0.00 \| 0.00 \| 1.19 Sb \| 3.43 \| 1.55 \| 0.00 \| 1.51 \| 4.69 \| 0.00 \| 0.00 \| 0.00 \| 2.31 S \| -2.28 \| -0.70 \| 1.38 \| -0.68 \| -3.08 \| 0.57 \| 0.80 \| 0.02 \| -1.75 Table 3: Calculated Born effective charge tensors of monoclinic NaSbS2. \| xx \| xy \| yy \| zz ---\|---\|---\|---\|--- ϵ∞,ij \| 9.3 \| 1.7 \| 8.4 \| 7.4 ϵph,ij \| 10.4 \| 13.0 \| 32.7 \| 3.3 ϵij \| 19.7 \| 14.7 \| 41.1 \| 10.7 Table 4: Calculated dielectric tensors of monoclinic NaSbS2. Therefore NaSbS${}_{2}$ has enhanced Born effective charges due to the cross gap hybridization, similar to several materials that have been found to have excellent charge collection in the context of radiation detection, e.g. TlBr, BiI${}_{3}$ and Tl${}_{6}$SeI${}_{4}$. Du and Singh (27, 28); Biswas _et al._ (2012) This enhanced Born charge leads to an enhanced dielectric constant, which means enhanced screening. This provides an explanation of how a material that presumably contains high concentrations of point defects can nonetheless have efficient carrier collection in an optoelectronic application. In this regard, we note that SrTiO${}_{3}$, which is near a ferroelectric transition, and consequently has a very high dielectric constant at low temperature, also has an exceptional electron mobility that exceeds 30,000 cm${}^{2}$/Vs in high quality films. Son _et al._ (2010) ## IV Summary and Conclusions We report first principles calculations for NaSbS${}_{2}$. We find that the ground state structure is monoclinic $C$2/$c$ or possibly triclinic $P\bar{1}$ with a very small triclinic distortion. We find highly anisotropic electronic and optical properties as may be expected based on the crystal structure. The results show a quasidirect band gap, with an indirect gap slightly lower than the direct gap, which may impede carrier recombination. The calculated value of the band gap using the mBJ potential is 1.22 eV. Importantly, the electronic structure shows a substantial cross gap hybridization between S $p$ and Sb $p$ states. This results in enhanced Born effective charges and a high dielectric constant. This high dielectric constant provides screening and defect tolerance for the carrier collection. Therefore it is likely that NaSbS${}_{2}$ can be a useful optoelectronic material, not only as a solar absorber, but also in applications requiring doping. ###### Acknowledgements. This work was supported by the Department of Energy through the MAGICS Center, Award DE-SC0014607. ## References * Rahayu _et al._ (2016)S. U. Rahayu, C. L. Chou, N. Suriawong, B. A. Aragaw, S. B. Shi, and M. W. Lee, “Sodium antimony sulfide (NaSbS${}_{2}$): turning an unexpected impurity into a promising, environmentally friendly novel solar absorber material,” APL Mater. 4, 116103 (2016). * Bhattacharya _et al._ (2016)S. Bhattacharya, R. Chmielowski, G. Dennler, and G. K. H. Madsen, “Novel ternary sulfide thermoelectric materials from high throughput transport and defect calculations,” J. Mater. Chem. A 4, 11086–11093 (2016). * Kojima _et al._ (2009)A. Kojima, K. Teshima, Y. Shirai, and T. Miyasaka, “Organometal halide perovskites as visible-light sensitizers for photovoltaic cells,” J. Am. Chem. Soc. 131, 6050–6051 (2009). * Lee _et al._ (2012)M. M. Lee, J. Teuscher, T. Miyasaka, T. N. Murakami, and H. J. Snaith, “Efficient hybrid solar cells based on meso-structured organometal halide perovskites,” Science 338, 643–647 (2012). * Giustino and Snaith (2016)F. Giustino and H. J. Snaith, “Toward lead-free perovskite solar cells,” ACS Energy Lett. 1, 1233–1240 (2016). * Stranks _et al._ (2013)S. D. Stranks, G. E. Eperon, G. Grancini, C. Menelaou, M. J. P. Alcocer, T. Leijtens, L. M. Herz, A. Petrozza, and H. J. Snaith, “Electron-hole diffusion lengths exceeding 1 micrometer in an organometal trihalide perovskite absorber,” Science 342, 341–344 (2013). * Singh and Nordstrom (2006)D. J. Singh and L. Nordstrom, _Planewaves Pseudopotentials and the LAPW Method, 2nd Edition_ (Springer, Berlin, 2006). * Blaha _et al._ (2001)P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, _WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties_ (K. Schwarz, Tech. Univ. Wien, Austria, 2001). * Perdew _et al._ (1996)J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996). * Tran and Blaha (2009)F. Tran and P. Blaha, “Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential,” Phys. Rev. Lett. 102, 226401 (2009). * Koller _et al._ (2011)D. Koller, F. Tran, and P. Blaha, “Merits and limits of the modified Becke-Johnson exchange potential,” Phys. Rev. B 83, 195134 (2011). * Singh (2010a)D. J. Singh, “Electronic structure calculations with the Tran-Blaha modified Becke-Johnson density functional,” Phys. Rev. B 82, 205102 (2010a). * Kim _et al._ (2010)Y. S. Kim, M. Marsman, G. Kresse, F. Tran, and P. Blaha, “Towards efficient band structure and effective mass calculations for III-V direct band gap semiconductors,” Phys. Rev. B 82, 205212 (2010). * Singh (2010b)D. J. Singh, ‘‘Structure and optical properties of high light output halide scintillators,” Phys. Rev. B 82, 155145 (2010b). * Madsen and Singh (2006)G. K. H. Madsen and D. J. Singh, “BoltzTraP a code for calculating band-structure dependent quantities,” Comput. Phys. Commun. 175, 67–71 (2006). * Kresse and Furthmüller (1996)G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169 (1996). * Olivier-Fourcade _et al._ (1978)J. Olivier-Fourcade, E. Philippot, and M. Maurin, “Structure des composes NaSbS${}_{2}$$\alpha$ et NaSbS${}_{2}$$\beta$. etude de l’influence de la paire electronique e de i’antimonine III dans ia transition NaSbS${}_{2}$$\alpha$ - NaSbS${}_{2}$$\beta$,” Z. Anorg. Allg. Chem. 446, 159–168 (1978). * Volk and Schafer (1978)K. Volk and H. Schafer, “Die kristallstruktur von b-NaSbS${}_{2}$,” Z. Naturforsch. 33b, 827–828 (1978). * Kanischeva _et al._ (1979)A. S. Kanischeva, V. G. Kuznetsov, and V. N. Batog, “Crystal structure of a-NaSbS${}_{2}$,” J. Struct. Chem. 20, 122–125 (1979). * Perdew _et al._ (2009)J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, “Restoring the density-gradient expansion for exchange in solids and surfaces,” Phys. Rev. Lett. 102, 039902 (2009). * Wu and Cohen (2006)Z. Wu and R. E. Cohen, “More accurate generalized gradient approximation for solids,” Phys. Rev. B 73, 235116 (2006). * Brown (1985)I. D. Brown, “Bond-valence parameters obtained from a systematic analysis of the inorganic crystal structure database,” Acta Cryst. B 41, 244–247 (1985). * Chen _et al._ (2009)S. Chen, X. G. Gong, A. Walsh, and S. H. Wei, ‘‘Crystal and electronic band structure of Cu${}_{2}$ZnSnX${}_{4}$ (X=S and Se) photovoltaic absorbers: first principles insights,” Appl. Phys. Lett. 94, 041903 (2009). * Mitzi _et al._ (2011)D. B. Mitzi, O. Gunawan, T. K. Totorov, and S. Guha, “The path towards a high-performance solution-processed kesterite solar cell,” Solar Energy Materials and Solar Cells 95, 1421–1436 (2011). * Cohen (1992)R. E. Cohen, “Origin of ferroelectricity in perovskite oxides,” Nature 358, 136–138 (1992). * Mukhopadhyay _et al._ (2016)S. Mukhopadhyay, J. Sun, A. Subedi, T. Siegrist, and D. J. Singh, “Competing covalent and ionic bonding in Ge-Sb-Te phase change materials,” Sci. Rep. 6, 25981 (2016). * Du and Singh (2010a)M. H. Du and D. J. Singh, “Enhanced born charge and proximity to ferroelectricity in thallium halides,” Phys. Rev. B 81, 144114 (2010a). * Du and Singh (2010b)M. H. Du and D. J. Singh, “Enhanced born charges in III-VI, IV-VII${}_{2}$ and V-VII${}_{3}$ compounds,” Phys. Rev. B 82, 045203 (2010b). * Fabini _et al._ (2016)D. H. Fabini, T. Hogan, H. A. Evans, C. C. Stoumpos, M. G. Kanatzidis, and R. Seshadri, “Dielectric and thermodynamic signatures of low-temperature glassy dynamics in the hybrid perovskites CH${}_{3}$NH${}_{3}$PbI${}_{3}$ and HC(NH${}_{2}$)(2)PbI${}_{2}$,” J. Phys. Chem. Lett. 7, 376–381 (2016). * Brandt _et al._ (2015)R. E. Brandt, V. Stevanovic, D. S. Ginley, and T. Buonassisi, “Identifying defect-tolerant semiconductors with high minority-carrier lifetimes: beyond hybrid lead halide perovskites,” MRS Commun. 5, 265–275 (2015). * Lehner _et al._ (2015)A. J. Lehner, D. H. Fabini, H. A. Evans, C. A. Hebert, S. R. Smock, J. Hu, H. B. Wang, J. W. Zwanziger, M. L. Chabinyc, and R. Seshadri, ‘‘Crystal and electronic structures of complex bismuth iodides A(3)Bi(2)I(9) (A = K, Rb, Cs) related to perovskite: Aiding the rational design of photovoltaics,” Chem. Mater. 27, 7137–7148 (2015). * Du (2014)M. H. Du, “Efficient carrier transport in halide perovskites: theoretical perspectives,” J. Mater. Chem. A 2, 9091–9098 (2014). * Gajdoš _et al._ (2006)M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, ‘‘Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006). * Baroni and Resta (1986)S. Baroni and R. Resta, “Ab initio calculation of the macroscopic dielectric constant in silicon,” Phys. Rev. B 33, 7017 (1986). * Wu _et al._ (2005)X. Wu, D. Vanderbilt, and D. R. Hamann, “Systematic treatment of displacements, strains, and electric fields in density-functional perturbation theory,” Phys. Rev. B 72, 035105 (2005). * Look (2001)D. C. Look, “Recent advances in ZnO materials and devices,” Mat. Sci. Eng. B 80, 383 (2001). * McCluskey and Jokela (2009)M. D. McCluskey and S. J. Jokela, “Defects in ZnO,” J. Appl. Phys. 106, 071101 (2009). * Crisler _et al._ (1968)D. F. Crisler, J. J. Cupal, and A.R. Moore, “Dielectric, piezoelectric and electromechanical coupling constants of zinc oxide crystals,” Proc. IEEE 56, 225–226 (1968). * Biswas _et al._ (2012)K. Biswas, M. H. Du, and D. J. Singh, “Electronic structure and defect properties of Tl${}_{6}$SeI${}_{4}$: density functional calculations,” Phys. Rev. B 86, 144108 (2012). * Son _et al._ (2010)J. Son, P. Moetakef, B. Jalan, O. Bierwagen, N. J. Wright, R. Engel-Herbert, and S. Stemmer, “Epitaxial SrTiO${}_{3}$ films with electron mobilities exceeding 30,000 cm${}^{2}$ V${}^{-1}$s${}^{-1}$,” Nature Mater. 9, 482 (2010).
1906.00232	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 124134, "num_imgs": 7, "llama3_tokens_count": 42951 }	[ "content_image/1906.00232/x1.png", "content_image/1906.00232/x1.png", "content_image/1906.00232/x2.png", "content_image/1906.00232/x3.png", "content_image/1906.00232/x3.png", "content_image/1906.00232/x5.png", "content_image/1906.00232/x6.png" ]	# Kernel Instrumental Variable Regression Rahul Singh MIT Economics rahul.singh@mit.edu &Maneesh Sahani Gatsby Unit, UCL maneesh@gatsby.ucl.ac.uk &Arthur Gretton Gatsby Unit, UCL arthur.gretton@gmail.com ###### Abstract Instrumental variable (IV) regression is a strategy for learning causal relationships in observational data. If measurements of input $X$ and output $Y$ are confounded, the causal relationship can nonetheless be identified if an instrumental variable $Z$ is available that influences $X$ directly, but is conditionally independent of $Y$ given $X$ and the unmeasured confounder. The classic two-stage least squares algorithm (2SLS) simplifies the estimation problem by modeling all relationships as linear functions. We propose kernel instrumental variable regression (KIV), a nonparametric generalization of 2SLS, modeling relations among $X$, $Y$, and $Z$ as nonlinear functions in reproducing kernel Hilbert spaces (RKHSs). We prove the consistency of KIV under mild assumptions, and derive conditions under which convergence occurs at the minimax optimal rate for unconfounded, single-stage RKHS regression. In doing so, we obtain an efficient ratio between training sample sizes used in the algorithm’s first and second stages. In experiments, KIV outperforms state of the art alternatives for nonparametric IV regression. ## 1 Introduction Instrumental variable regression is a method in causal statistics for estimating the counterfactual effect of input $X$ on output $Y$ using observational data [60]. If measurements of $(X,Y)$ are confounded, the causal relationship–also called the structural relationship–can nonetheless be identified if an instrumental variable $Z$ is available, which is independent of $Y$ conditional on $X$ and the unmeasured confounder. Intuitively, $Z$ only influences $Y$ via $X$, identifying the counterfactual relationship of interest. Economists and epidemiologists use instrumental variables to overcome issues of strategic interaction, imperfect compliance, and selection bias. The original application is demand estimation: supply cost shifters ($Z$) only influence sales ($Y$) via price ($X$), thereby identifying counterfactual demand even though prices reflect both supply and demand market forces [68; 11]. Randomized assignment of a drug ($Z$) only influences patient health ($Y$) via actual consumption of the drug ($X$), identifying the counterfactual effect of the drug even in the scenario of imperfect compliance [3]. Draft lottery number ($Z$) only influences lifetime earnings ($Y$) via military service ($X$), identifying the counterfactual effect of military service on earnings despite selection bias in enlistment [2]. The two-stage least squares algorithm (2SLS), widely used in economics, simplifies the IV estimation problem by assuming linear relationships: in _stage 1_, perform linear regression to obtain the conditional means $\bar{x}(z):=\mathbb{E}_{X\|Z=z}(X)$; in _stage 2_, linearly regress outputs $Y$ on these conditional means. 2SLS works well when the underlying assumptions hold. In practice, the relation between $Y$ and $X$ may not be linear, nor may be the relation between $X$ and $Z$. In the present work, we introduce kernel instrumental variable regression (KIV), an easily implemented nonlinear generalization of 2SLS (Sections 3 and 4).¹ In _stage 1_ we learn a conditional mean embedding, which is the conditional expectation $\mu(z):=\mathbb{E}_{X\|Z=z}\psi(X)$ of features $\psi$ which map $X$ to a reproducing kernel Hilbert space (RKHS) [56]. For a sufficiently rich RKHS, called a characteristic RKHS, the mean embedding of a random variable is injective [57]. It follows that the conditional mean embedding characterizes the full distribution of $X$ conditioned on $Z$, and not just the conditional mean. We then implement _stage 2_ via kernel ridge regression of outputs $Y$ on these conditional mean embeddings, following the two-stage distribution regression approach described by [64; 65]. As in our work, the inputs for [64; 65] are distribution embeddings. Unlike our case, the earlier work uses unconditional embeddings computed from independent samples. [FOOTNOTE:1][ENDFOOTNOTE] As a key contribution of our work, we provide consistency guarantees for the KIV algorithm for an increasing number of training samples in stages 1 and 2 (Section 5). To establish stage 1 convergence, we note that the conditional mean embedding [56] is the solution to a regression problem [34; 35; 33], and thus equivalent to kernel dependency estimation [20; 21]. We prove that the kernel estimator of the conditional mean embedding (equivalently, the conditional expectation operator) converges in RKHS-norm, generalizing classic results by [53; 54]. We allow the conditional mean embedding RKHS to be infinite-dimensional, which presents specific challenges that we carefully address in our analysis. We also discuss previous approaches to establishing consistency in both finite-dimensional [35] and infinite-dimensional [56; 55; 31; 37; 20] settings. We embed the stage 1 rates into stage 2 to get end-to-end guarantees for the two-stage procedure, adapting [14; 64; 65]. In particular, we provide a ratio of stage 1 to stage 2 samples required for minimax optimal rates in the second stage, where the ratio depends on the difficulty of each stage. We anticipate that these proof strategies will apply generally in two-stage regression settings. ## 2 Related work Several approaches have been proposed to generalize 2SLS to the nonlinear setting, which we will compare in our experiments (Section 6). A first generalization is via basis function approximation [48], an approach called sieve IV, with uniform convergence rates in [17]. The challenge in [17] is how to define an appropriate finite dictionary of basis functions. In a second approach, [16; 23] implement stage 1 by computing the conditional distribution of the input $X$ given the instrument $Z$ using a ratio of Nadaraya-Watson density estimates. Stage 2 is then ridge regression in the space of square integrable functions. The overall algorithm has a finite sample consistency guarantee, assuming smoothness of the $(X,Z)$ joint density in stage 1 and the regression in stage 2 [23]. Unlike our bound, [23] make no claim about the optimality of the result. Importantly, stage 1 requires the solution of a statistically challenging problem: conditional density estimation. Moreover, analysis assumes the same number of training samples used in both stages. We will discuss this bound in more detail in Appendix A.2.1 (we suggest that the reader first cover Section 5). Our work also relates to kernel and IV approaches to learning dynamical systems, known in machine learning as predictive state representation models (PSRs) [12; 37; 26] and in econometrics as panel data models [1; 6]. In this setting, predictive states (expected future features given history) are updated in light of new observations. The calculation of the predictive states corresponds to stage 1 regression, and the states are updated via stage 2 regression. In the kernel case, the predictive states are expressed as conditional mean embeddings [12], as in our setting. Performance of the kernel PSR method is guaranteed by a finite sample bound (37, Theorem 2), however this bound is not minimax optimal. Whereas [37] assume an equal number of training samples in stages 1 and 2, we find that unequal numbers of training samples matter for minimax optimality. More importantly, the bound makes strong smoothness assumptions on the inputs to the stage 1 and stage 2 regression functions, rather than assuming smoothness of the regression functions as we do. We show that the smoothness assumptions on the inputs made in [37] do not hold in our setting, and we obtain stronger end-to-end bounds under more realistic conditions. We discuss the PSR bound in more detail in Appendix A.2.2. Yet another recent approach is deep IV, which uses neural networks in both stages and permits learning even for complex high-dimensional data such as images [36]. Like [23], [36] implement stage 1 by estimating a conditional density. Unlike [23], [36] use a mixture density network (9, Section 5.6), i.e. a mixture model parametrized by a neural network on the instrument $Z$. Stage 2 is neural network regression, trained using stochastic gradient descent (SGD). This presents a challenge: each step of SGD requires expectations using the stage 1 model, which are computed by drawing samples and averaging. An unbiased gradient estimate requires two independent sets of samples from the stage 1 model (36, eq. 10), though a single set of samples may be used if an upper bound on the loss is optimized (36, eq. 11). By contrast, our stage 1 outputs–conditional mean embeddings–have a closed form solution and exhibit lower variance than sample averaging from a conditional density model. No theoretical guarantee on the consistency of the neural network approach has been provided. In the econometrics literature, a few key assumptions make learning a nonparametric IV model tractable. These include the completeness condition [48]: the structural relationship between $X$ and $Y$ can be identified only if the stage 1 conditional expectation is injective. Subsequent works impose additional stability and link assumptions [10; 19; 17]: the conditional expectation of a function of $X$ given $Z$ is a smooth function of $Z$. We adapt these assumptions to our setting, replacing the completeness condition with the characteristic property [57], and replacing the stability and link assumptions with the concept of prior [54; 14]. We describe the characteristic and prior assumptions in more detail below. Extensive use of IV estimation in applied economic research has revealed a common pitfall: weak instrumental variables. A weak instrument satisfies Hypothesis 1 below, but the relationship between a weak instrument $Z$ and input $X$ is negligible; $Z$ is essentially irrelevant. In this case, IV estimation becomes highly erratic [13]. In [58], the authors formalize this phenomenon with local analysis. See [44; 61] for practical and theoretical overviews, respectively. We recommend that practitioners resist the temptation to use many weak instruments, and instead use few strong instruments such as those described in the introduction. Finally, our analysis connects early work on the RKHS with recent developments in the RKHS literature. In [46], the authors introduce the RKHS to solve known, ill-posed functional equations. In the present work, we introduce the RKHS to estimate the solution to an uncertain, ill-posed functional equation. In this sense, casting the IV problem in an RKHS framework is not only natural; it is in the original spirit of RKHS methods. For a comprehensive review of existing work and recent advances in kernel mean embedding research, we recommend [43; 32]. ## 3 Problem setting and definitions Instrumental variable: We begin by introducing our causal assumption about the instrument. This prior knowledge, described informally in the introduction, allows us to recover the counterfactual effect of $X$ on $Y$. Let $(\mathcal{X},\mathcal{B}_{\mathcal{X}})$, $(\mathcal{Y},\mathcal{B}_{\mathcal{Y}})$, and $(\mathcal{Z},\mathcal{B}_{\mathcal{Z}})$ be measurable spaces. Let $(X,Y,Z)$ be a random variable on $\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$ with distribution $\rho$. Hypothesis 1.: _Assume_ 1. $Y=h(X)+e$ _and_ $\mathbb{E}[e\|Z]=0$__ 2. $\rho(x\|z)$ _is not constant in_ $z$__ We call $h$ the _structural function_ of interest. The error term $e$ is unmeasured, confounding noise. Hypothesis 1.1, known as the exclusion restriction, was introduced by [48] to the nonparametric IV literature for its tractability. Other hypotheses are possible, although a very different approach is then needed [40]. Hypothesis 1.2, known as the relevance condition, ensures that $Z$ is actually informative. In Appendix A.1.1, we compare Hypothesis 1 with alternative formulations of the IV assumption. We make three observations. First, if $X=Z$ then Hypothesis 1 reduces to the standard regression assumption of unconfounded inputs, and $h(X)=\mathbb{E}[Y\|X]$; if $X=Z$ then prediction and counterfactual prediction coincide. The IV model is a framework that allows for causal inference in a more general variety of contexts, namely when $h(X)\neq\mathbb{E}[Y\|X]$ so that prediction and counterfactual prediction are different learning problems. Second, Hypothesis 1 will permit identification of $h$ even if inputs are confounded, i.e. $X\cancel{\raisebox{0.5pt}{\rotatebox[origin={c}]{90.0}{$\models$}}}e$. Third, this model includes the scenario in which the analyst has a combination of confounded and unconfounded inputs. For example, in demand estimation there may be confounded price $P$, unconfounded characteristics $W$, and supply cost shifter $C$ that instruments for price. Then $X=(P,W)$, $Z=(C,W)$, and the analysis remains the same. <figure><img src="content_image/1906.00232/x1.png"><figcaption>Figure 2: Sigmoid design</figcaption></figure> Hypothesis 1 provides the operator equation $\mathbb{E}[Y\|Z]=\mathbb{E}_{X\|Z}h(X)$[48]. In the language of 2SLS, the LHS is the _reduced form_, while the RHS is a composition of _stage 1_ linear compact operator $\mathbb{E}_{X\|Z}$ and _stage 2_ structural function $h$. In the language of functional analysis, the operator equation is a Fredholm integral equation of the first kind [46; 41; 48; 29]. Solving this operator equation for $h$ involves inverting a linear compact operator with infinite-dimensional domain; it is an ill-posed problem [41]. To recover a well-posed problem, we impose smoothness and Tikhonov regularization. RKHS model: We next introduce our RKHS model. Let $k_{\mathcal{X}}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ and $k_{\mathcal{Z}}:\mathcal{Z}\times\mathcal{Z}\rightarrow\mathbb{R}$ be measurable positive definite kernels corresponding to scalar-valued RKHSs $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$. Denote the feature maps \[\psi :\mathcal{X}\rightarrow\mathcal{H}_{\mathcal{X}},\enskip x\mapsto k _{\mathcal{X}}(x,\cdot)\quad\quad\phi:\mathcal{Z}\rightarrow\mathcal{H}_{ \mathcal{Z}},\enskip z\mapsto k_{\mathcal{Z}}(z,\cdot)\] Define the _conditional expectation operator_$E:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ such that $[Eh](z)=\mathbb{E}_{X\|Z=z}h(X)$. $E$ is the natural object of interest for stage 1. We define and analyze an estimator for $E$ directly. The conditional expectation operator $E$ conveys exactly the same information as another object popular in the kernel methods literature, the _conditional mean embedding_$\mu:\mathcal{Z}\rightarrow\mathcal{H}_{\mathcal{X}}$ defined by $\mu(z)=\mathbb{E}_{X\|Z=z}\psi(X)$[56]. Indeed, $\mu(z)=E^{}\phi(z)$ where $E^{}:\mathcal{H}_{\mathcal{Z}}\rightarrow\mathcal{H}_{\mathcal{X}}$ is the adjoint of $E$. Analogously, in 2SLS $\bar{x}(z)=\pi^{\prime}z$ for stage 1 linear regression parameter $\pi$. The structural function $h:\mathcal{X}\rightarrow\mathcal{Y}$ in Hypothesis 1 is the natural object of interest for stage 2. For theoretical purposes, it is convenient to estimate $h$ indirectly. The structural function $h$ conveys exactly the same information as an object we call the _structural operator_$H:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{Y}$. Indeed, $h(x)=H\psi(x)$. Analogously, in 2SLS $h(x)=\beta^{\prime}x$ for structural parameter $\beta$. We define and analyze an estimator for $H$, which in turn implies an estimator for $h$. Figure 1 summarizes the relationships among equivalent stage 1 objects $(E,\mu)$ and equivalent stage 2 objects $(H,h)$. Our RKHS model for the IV problem is of the same form as the model in [45; 46; 47] for general operator equations. We begin by choosing RKHSs for the structural function $h$ and the reduced form $\mathbb{E}[Y\|Z]$, then construct a tensor-product RKHS for the conditional expectation operator $E$. Our model differs from the RKHS model proposed by [16; 23], which directly learns the conditional expectation operator $E$ via Nadaraya-Watson density estimation. The RKHSs of [28; 16; 23] for the structural function $h$ and the reduced form $\mathbb{E}[Y\|Z]$ are defined from the right and left singular functions of $E$, respectively. They appear in the consistency argument, but not in the ridge penalty. ## 4 Learning problem and algorithm 2SLS consists of two stages that can be estimated separately. Sample splitting in this context means estimating stage 1 with $n$ randomly chosen observations and estimating stage 2 with the remaining $m$ observations. Sample splitting alleviates the finite sample bias of 2SLS when instrument $Z$ weakly influences input $X$[4]. It is the natural approach when an analyst does not have access to a single data set with $n+m$ observations of $(X,Y,Z)$ but rather two data sets: $n$ observations of $(X,Z)$, and $m$ observations of $(Y,Z)$. We employ sample splitting in KIV, with an efficient ratio of $(n,m)$ given in Theorem 4. In our presentation of the general two-stage learning problem, we denote stage 1 observations by $(x_{i},z_{i})$ and stage 2 observations by $(\tilde{y}_{i},\tilde{z}_{i})$. ### Stage 1 We transform the problem of learning $E$ into a vector-valued kernel ridge regression following [34; 33; 20], where the hypothesis space is the vector-valued RKHS $\mathcal{H}_{\Gamma}$ of operators mapping $\mathcal{H}_{\mathcal{X}}$ to $\mathcal{H}_{\mathcal{Z}}$. In Appendix A.3, we review the theory of vector-valued RKHSs as it relates to scalar-valued RKHSs and tensor product spaces. The key result is that the tensor product space of $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$ is isomorphic to $\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{\mathcal{Z}})$, the space of Hilbert-Schmidt operators from $\mathcal{H}_{\mathcal{X}}$ to $\mathcal{H}_{\mathcal{Z}}$. If we choose the vector-valued kernel $\Gamma$ with feature map $(x,z)\mapsto[\phi(z)\otimes\psi(x)](\cdot)=\phi(z)\langle\psi(x),\cdot\rangle_ {\mathcal{H}_{\mathcal{X}}}$, then $\mathcal{H}_{\Gamma}=\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{ \mathcal{Z}})$ and it shares the same norm. We now state the objective for optimizing $E\in\mathcal{H}_{\Gamma}$. The optimal $E$ minimizes the expected discrepancy \[E_{\rho} =\operatorname{\arg\!\min}\mathcal{E}_{1}(E),\quad\mathcal{E}_{1 }(E)=\mathbb{E}_{(X,Z)}\\|\psi(X)-E^{}\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] Both [33] and [20] refer to $\mathcal{E}_{1}$ as the surrogate risk. As shown in (34, Section 3.1) and [33], the surrogate risk upper bounds the natural risk for the conditional expectation, where the bound becomes tight when $\mathbb{E}_{X\|Z=(\cdot)}f(X)\in\mathcal{H}_{\mathcal{Z}},\:\forall f\in \mathcal{H}_{\mathcal{X}}$. Formally, the target operator is the constrained solution $E_{\mathcal{H}_{\Gamma}}=\operatorname{\arg\!\min}_{E\in\mathcal{H}_{\Gamma}} \mathcal{E}_{1}(E)$. We will assume $E_{\rho}\in\mathcal{H}_{\Gamma}$ so that $E_{\rho}=E_{\mathcal{H}_{\Gamma}}$. Next we impose Tikhonov regularization. The regularized target operator and its empirical analogue are given by \[E_{\lambda} =\operatorname{\arg\!\min}_{E\in\mathcal{H}_{\Gamma}}\mathcal{E} _{\lambda}(E),\quad\mathcal{E}_{\lambda}(E)=\mathcal{E}_{1}(E)+\lambda\\|E\\|^{2 }_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{\mathcal{Z}})}\] \[E^{n}_{\lambda} =\operatorname{\arg\!\min}_{E\in\mathcal{H}_{\Gamma}}\mathcal{E} _{\lambda}^{n}(E),\quad\mathcal{E}_{\lambda}^{n}(E)=\dfrac{1}{n}\sum_{i=1}^{n} \\|\psi(x_{i})-E^{}\phi(z_{i})\\|^{2}_{\mathcal{H}_{\mathcal{X}}}+\lambda\\|E\\|^ {2}_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{\mathcal{Z}})}\] Our construction of a vector-valued RKHS $\mathcal{H}_{\Gamma}$ for the conditional expectation operator $E$ permits us to estimate stage 1 by kernel ridge regression. The stage 1 estimator of KIV is at once novel in the nonparametric IV literature and fundamentally similar to 2SLS. Basis function approximation [48; 17] is perhaps the closest prior IV approach, but we use infinite dictionaries of basis functions $\psi$ and $\phi$. Compared to density estimation [16; 23; 36], kernel ridge regression is an easier problem. Alternative stage 1 estimators in the literature estimate the singular system of $E$ to ensure that the adjoint of the estimator equals the estimator of the adjoint. These estimators differ in how they estimate the singular system: empirical distribution [23], Nadaraya-Watson density [24], or B-spline wavelets [18]. The KIV stage 1 estimator has the desired property by construction; $(E_{\lambda}^{n})^{}=(E^{})_{\lambda}^{n}$. See Appendix A.3 for details. ### Stage 2 Next, we transform the problem of learning $h$ into a scalar-valued kernel ridge regression that respects the IV problem structure. In Proposition 12 of Appendix A.3, we show that under Hypothesis 3 below, \[\mathbb{E}_{X\|Z=z}h(X)=[Eh](z)=\langle h,\mu(z)\rangle_{\mathcal{H}_{\mathcal{ X}}}=H\mu(z)\] where $h\in\mathcal{H}_{\mathcal{X}}$, a scalar-valued RKHS; $E\in\mathcal{H}_{\Gamma}$, the vector-valued RKHS described above; $\mu\in\mathcal{H}_{\Xi}$, a vector-valued RKHS isometrically isomorphic to $\mathcal{H}_{\Gamma}$; and $H\in\mathcal{H}_{\Omega}$, a scalar-valued RKHS isometrically isomorphic to $\mathcal{H}_{\mathcal{X}}$. It is helpful to think of $\mu(z)$ as the embedding into $\mathcal{H}_{\mathcal{X}}$ of a distribution on $\mathcal{X}$ indexed by the conditioned value $z$. When $k_{\mathcal{X}}$ is characteristic, $\mu(z)$ uniquely embeds the conditional distribution, and $H$ is identified. The kernel $\Omega$ satisfies $k_{\mathcal{X}}(x,x^{\prime})=\Omega(\psi(x),\psi(x^{\prime}))$. This expression establishes the formal connection between our model and [64; 65]. The choice of $\Omega$ may be more general; for nonlinear examples see (65, Table 1). We now state the objective for optimizing $H\in\mathcal{H}_{\Omega}$. Hypothesis 1 provides the operator equation, which may be rewritten as the regression equation \[Y=\mathbb{E}_{X\|Z}h(X)+e_{Z}=H\mu(Z)+e_{Z},\quad\mathbb{E}[e_{Z}\|Z]=0\] The unconstrained solution is \[H_{\rho} =\operatorname{\arg\!\min}\mathcal{E}(H),\quad\mathcal{E}(H)= \mathbb{E}_{(Y,Z)}\\|Y-H\mu(Z)\\|_{\mathcal{Y}}^{2}\] The target operator is the constrained solution $H_{\mathcal{H}_{\Omega}}=\operatorname{\arg\!\min}_{H\in\mathcal{H}_{\Omega}} \mathcal{E}(H)$. We will assume $H_{\rho}\in\mathcal{H}_{\Omega}$ so that $H_{\rho}=H_{\mathcal{H}_{\Omega}}$. With regularization, \[H_{\xi} =\operatorname{\arg\!\min}_{H\in\mathcal{H}_{\Omega}}\mathcal{E} _{\xi}(H),\quad\mathcal{E}_{\xi}(H)=\mathcal{E}(H)+\xi\\|H\\|^{2}_{\mathcal{H}_{ \Omega}}\] \[H^{m}_{\xi} =\operatorname{\arg\!\min}_{H\in\mathcal{H}_{\Omega}}\mathcal{E} ^{m}_{\xi}(H),\quad\mathcal{E}^{m}_{\xi}(H)=\dfrac{1}{m}\sum_{i=1}^{m}\\|\tilde {y}_{i}-H\mu(\tilde{z}_{i})\\|_{\mathcal{Y}}^{2}+\xi\\|H\\|^{2}_{\mathcal{H}_{ \Omega}}\] The essence of the IV problem is this: we do not directly observe the conditional expectation operator $E$ (or equivalently the conditional mean embedding $\mu$) that appears in the stage 2 objective. Rather, we approximate it using the estimate from stage 1. Thus our KIV estimator is $\hat{h}^{m}_{\xi}=\hat{H}^{m}_{\xi}\psi$ where \[\hat{H}^{m}_{\xi} =\operatorname{\arg\!\min}_{H\in\mathcal{H}_{\Omega}}\hat{ \mathcal{E}}^{m}_{\xi}(H),\quad\hat{\mathcal{E}}^{m}_{\xi}(H)=\dfrac{1}{m}\sum _{i=1}^{m}\\|\tilde{y}_{i}-H\mu^{n}_{\lambda}(\tilde{z}_{i})\\|_{\mathcal{Y}}^{2 }+\xi\\|H\\|^{2}_{\mathcal{H}_{\Omega}}\] and $\mu^{n}_{\lambda}=(E_{\lambda}^{n})^{}\phi$. The transition from $H_{\rho}$ to $H^{m}_{\xi}$ represents the fact that we only have $m$ samples. The transition from $H^{m}_{\xi}$ to $\hat{H}^{m}_{\xi}$ represents the fact that we must learn not only the structural operator $H$ but also the conditional expectation operator $E$. In this sense, the IV problem is more complex than the estimation problem considered by [45; 47] in which $E$ is known. ### Algorithm We obtain a closed form expression for the KIV estimator. The apparatus introduced above is required for analysis of consistency and convergence rate. More subtly, our RKHS construction allows us to write kernel ridge regression estimators for both stage 1 and stage 2, unlike previous work. Because KIV consists of repeated kernel ridge regressions, it benefits from repeated applications of the representer theorem [66; 51]. Consequently, we have a shortcut for obtaining KIV’s closed form; see Appendix A.5.1 for the full derivation. Algorithm 1.: _Let $X$ and $Z$ be matrices of $n$ observations. Let $\tilde{y}$ and $\tilde{Z}$ be a vector and matrix of $m$ observations._ \[W =K_{XX}(K_{ZZ}+n\lambda I)^{-1}K_{Z\tilde{Z}},\quad\hat{\alpha}=( WW^{\prime}+m\xi K_{XX})^{-1}W\tilde{y},\quad\hat{h}_{\xi}^{m}(x)=(\hat{\alpha })^{\prime}K_{Xx}\] _where $K_{XX}$ and $K_{ZZ}$ are the empirical kernel matrices._ Theorems 2 and 4 below theoretically determine efficient rates for the stage 1 regularization parameter $\lambda$ and stage 2 regularization parameter $\xi$, respectively. In Appendix A.5.2, we provide a validation procedure to empirically determine values for $(\lambda,\xi)$. ## 5 Consistency ### Stage 1 Integral operators: We use integral operator notation from the kernel methods literature, adapted to the conditional expectation operator learning problem. We denote by $L^{2}(\mathcal{Z},\rho_{\mathcal{Z}})$ the space of square integrable functions from $\mathcal{Z}$ to $\mathcal{Y}$ with respect to measure $\rho_{\mathcal{Z}}$, where $\rho_{\mathcal{Z}}$ is the restriction of $\rho$ to $\mathcal{Z}$. Definition 1.: _The stage 1 (population) operators are_ \[S_{1}^{} :\mathcal{H}_{\mathcal{Z}}\hookrightarrow L^{2}(\mathcal{Z},\rho_ {\mathcal{Z}}),\enskip\ell\mapsto\langle\ell,\phi(\cdot)\rangle_{\mathcal{H}_{ \mathcal{Z}}}\quad S_{1}:L^{2}(\mathcal{Z},\rho_{\mathcal{Z}})\to \mathcal{H}_{\mathcal{Z}},\enskip\tilde{\ell}\mapsto\int\phi(z)\tilde{\ell}(z) d\rho_{\mathcal{Z}}(z)\] $T_{1}=S_{1}\circ S_{1}^{}$ is the uncentered covariance operator of (30, Theorem 1). In Appendix A.4.2, we prove that $T_{1}$ exists and has finite trace even when $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$ are infinite-dimensional. In Appendix A.4.4, we compare $T_{1}$ with other covariance operators in the kernel methods literature. Assumptions: We place assumptions on the original spaces $\mathcal{X}$ and $\mathcal{Z}$, the scalar-valued RKHSs $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$, and the probability distribution $\rho(x,z)$. We maintain these assumptions throughout the paper. Importantly, we assume that the vector-valued RKHS regression is correctly specified: the true conditional expectation operator $E_{\rho}$ lives in the vector-valued RKHS $\mathcal{H}_{\Gamma}$. In further research, we will relax this assumption. Hypothesis 2.: _Suppose that $\mathcal{X}$ and $\mathcal{Z}$ are Polish spaces, i.e. separable and completely metrizable topological spaces_ Hypothesis 3.: _Suppose that_ 1. $k_{\mathcal{X}}$ _and_ $k_{\mathcal{Z}}$ _are continuous and bounded:_ $\sup_{x\in\mathcal{X}}\\|\psi(x)\\|_{\mathcal{H}_{\mathcal{X}}}\leq Q$_,_ $\sup_{z\in\mathcal{Z}}\\|\phi(z)\\|_{\mathcal{H}_{\mathcal{Z}}}\leq\kappa$__ 2. $\psi$ _and_ $\phi$ _are measurable_ 3. $k_{\mathcal{X}}$ _is characteristic_ [57]__ Hypothesis 4.: _Suppose that $E_{\rho}\in\mathcal{H}_{\Gamma}$. Then $\mathcal{E}_{1}(E_{\rho})=\inf_{E\in\mathcal{H}_{\Gamma}}\mathcal{E}_{1}(E)$_ Hypothesis 3.3 specializes the completeness condition of [48]. Hypotheses 2-4 are sufficient to bound the sampling error of the regularized estimator $E_{\lambda}^{n}$. Bounding the approximation error requires a further assumption on the smoothness of the distribution $\rho(x,z)$. We assume $\rho(x,z)$ belongs to a class of distributions parametrized by $(\zeta_{1},c_{1})$, as generalized from (54, Theorem 2) to the space $\mathcal{H}_{\Gamma}$. Hypothesis 5.: _Fix $\zeta_{1}<\infty$. For given $c_{1}\in(1,2]$, define the prior $\mathcal{P}(\zeta_{1},c_{1})$ as the set of probability distributions $\rho$ on $\mathcal{X}\times\mathcal{Z}$ such that a range space assumption is satisfied: $\exists G_{1}\in\mathcal{H}_{\Gamma}$ s.t. $E_{\rho}=T_{1}^{\frac{c_{1}-1}{2}}\circ G_{1}$ and $\\|G_{1}\\|^{2}_{\mathcal{H}_{\Gamma}}\leq\zeta_{1}$_ We use composition symbol $\circ$ to emphasize that $G_{1}:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ and $T_{1}:\mathcal{H}_{\mathcal{Z}}\rightarrow\mathcal{H}_{\mathcal{Z}}$. We define the power of operator $T_{1}$ with respect to its eigendecomposition; see Appendix A.4.2 for formal justification. Larger $c_{1}$ corresponds to a smoother conditional expectation operator $E_{\rho}$. Proposition 24 in Appendix A.6.2 shows $E_{\rho}^{}\phi(z)=\mu(z)$, so Hypothesis 5 is an indirect smoothness condition on the conditional mean embedding $\mu$. Estimation and convergence:* The estimator has a closed form solution, as noted in (34, Section 3.1) and (35, Appendix D); [20] use it in the first stage of the structured prediction problem. We present the closed form solution in notation similar to [14] in order to elucidate how the estimator simply generalizes linear regression. This connection foreshadows our proof technique. Theorem 1.: $\forall\lambda>0$_, the solution $E^{n}_{\lambda}$ of the regularized empirical objective $\mathcal{E}^{n}_{\lambda}$ exists, is unique, and_ \[E^{n}_{\lambda}=(\mathbf{T}_{1}+\lambda)^{-1}\circ\mathbf{g}_{1},\quad\mathbf{ T}_{1}=\dfrac{1}{n}\sum_{i=1}^{n}\phi(z_{i})\otimes\phi(z_{i}),\quad\mathbf{g} _{1}=\dfrac{1}{n}\sum_{i=1}^{n}\phi(z_{i})\otimes\psi(x_{i})\] We prove an original, finite sample bound on the RKHS-norm distance of the estimator $E^{n}_{\lambda}$ from its target $E_{\rho}$. The proof is in Appendix A.7. Theorem 2.: _Assume Hypotheses 2-5. $\forall\delta\in(0,1)$, the following holds w.p. $1-\delta$:_ \[\\|E^{n}_{\lambda}-E_{\rho}\\|_{\mathcal{H}_{\Gamma}} \leq r_{E}(\delta,n,c_{1}):=\dfrac{\sqrt{\zeta_{1}}(c_{1}+1)}{4^{ \frac{1}{c_{1}+1}}}\bigg{(}\dfrac{4\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{ \Gamma}})\ln(2/\delta)}{\sqrt{n\zeta_{1}}(c_{1}-1)}\bigg{)}^{\frac{c_{1}-1}{c_ {1}+1}}\] \[\lambda =\bigg{(}\dfrac{8\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma }})\ln(2/\delta)}{\sqrt{n\zeta_{1}}(c_{1}-1)}\bigg{)}^{\frac{2}{c_{1}+1}}\] The efficient rate of $\lambda$ is $n^{\frac{-1}{c_{1}+1}}$. Note that the convergence rate of $E^{n}_{\lambda}$ is calibrated by $c_{1}$, which measures the smoothness of the conditional expectation operator $E_{\rho}$. ### Stage 2 Integral operators: We use integral operator notation from the kernel methods literature, adapted to the structural operator learning problem. We denote by $L^{2}(\mathcal{H}_{\mathcal{X}},\rho_{\mathcal{H}_{\mathcal{X}}})$ the space of square integrable functions from $\mathcal{H}_{\mathcal{X}}$ to $\mathcal{Y}$ with respect to measure $\rho_{\mathcal{H}_{\mathcal{X}}}$, where $\rho_{\mathcal{H}_{\mathcal{X}}}$ is the extension of $\rho$ to $\mathcal{H}_{\mathcal{X}}$(59, Lemma A.3.16). Note that we present stage 2 analysis for general output space $\mathcal{Y}$ as in [64; 65], though in practice we only consider $\mathcal{Y}\subset\mathbb{R}$ to simplify our two-stage RKHS model. Definition 2.: _The stage 2 (population) operators are_ \[S^{} :\mathcal{H}_{\Omega}\hookrightarrow L^{2}(\mathcal{H}_{\mathcal{ X}},\rho_{\mathcal{H}_{\mathcal{X}}}),\enskip H\mapsto\Omega^{}_{(\cdot)}H\] \[S :L^{2}(\mathcal{H}_{\mathcal{X}},\rho_{\mathcal{H}_{\mathcal{X}}} )\rightarrow\mathcal{H}_{\Omega},\enskip\tilde{H}\mapsto\int\Omega_{\mu(z)} \circ\tilde{H}\mu(z)d\rho_{\mathcal{H}_{\mathcal{X}}}(\mu(z))\] where $\Omega_{\mu(z)}:\mathcal{Y}\rightarrow\mathcal{H}_{\Omega}$ defined by $y\mapsto\Omega(\cdot,\mu(z))y$ is the point evaluator of [42; 15]. Finally define $T_{\mu(z)}=\Omega_{\mu(z)}\circ\Omega^{}_{\mu(z)}$ and covariance operator $T=S\circ S^{}$. Assumptions: We place assumptions on the original space $\mathcal{Y}$, the scalar-valued RKHS $\mathcal{H}_{\Omega}$, and the probability distribution $\rho$. Importantly, we assume that the scalar-valued RKHS regression is correctly specified: the true structural operator $H_{\rho}$ lives in the scalar-valued RKHS $\mathcal{H}_{\Omega}$. Hypothesis 6.: _Suppose that $\mathcal{Y}$ is a Polish space_ Hypothesis 7.: _Suppose that_ 1. _The_ $\{\Omega_{\mu(z)}\}$ _operator family is uniformly bounded in Hilbert-Schmidt norm:_ $\exists B$ _s.t._ $\forall\mu(z)$_,_ $\\|\Omega_{\mu(z)}\\|^{2}_{\mathcal{L}_{2}(\mathcal{Y},\mathcal{H}_{\Omega})}=Tr (\Omega_{\mu(z)}^{}\circ\Omega_{\mu(z)})\leq B$__ 2. _The_ $\{\Omega_{\mu(z)}\}$ _operator family is Hölder continuous in operator norm:_ $\exists L>0$_,_ $\iota\in(0,1]$ _s.t._ $\forall\mu(z),\mu(z^{\prime})$_,_ __ Larger $\iota$ is interpretable as smoother kernel $\Omega$. Hypothesis 8.: _Suppose that_ 1. $H_{\rho}\in\mathcal{H}_{\Omega}$_. Then_ $\mathcal{E}(H_{\rho})=\inf_{H\in\mathcal{H}_{\Omega}}\mathcal{E}(H)$__ 2. $Y$ _is bounded, i.e._ $\exists C<\infty$ _s.t._ $\\|Y\\|_{\mathcal{Y}}\leq C$ _almost surely_ The convergence rate from stage 1 together with Hypotheses 6-8 are sufficient to bound the excess error of the regularized estimator $\hat{H}_{\xi}^{m}$ in terms of familiar objects in the kernel methods literature, namely the residual, reconstruction error, and effective dimension. We further assume $\rho$ belongs to a stage 2 prior to simplify these bounds. In particular, we assume $\rho$ belongs to a class of distributions parametrized by $(\zeta,b,c)$ as defined originally in (14, Definition 1), restated below. Hypothesis 9.: _Fix $\zeta<\infty$. For given $b\in(1,\infty]$ and $c\in(1,2]$, define the prior $\mathcal{P}(\zeta,b,c)$ as the set of probability distributions $\rho$ on $\mathcal{H}_{\mathcal{X}}\times\mathcal{Y}$ such that_ 1. _A range space assumption is satisfied:_ $\exists G\in\mathcal{H}_{\Omega}$ _s.t._ $H_{\rho}=T^{\frac{c-1}{2}}G$ _and_ $\\|G\\|^{2}_{\mathcal{H}_{\Omega}}\leq\zeta$__ 2. _In the spectral decomposition_ $T=\sum_{k=1}^{\infty}\lambda_{k}e_{k}\langle\cdot,e_{k}\rangle_{\mathcal{H}_{ \Omega}}$_, where_ $\{e_{k}\}_{k=1}^{\infty}$ _is a basis of_ $Ker(T)^{\perp}$_, the eigenvalues satisfy_ $\alpha\leq k^{b}\lambda_{k}\leq\beta$ _for some_ $\alpha,\beta>0$__ We define the power of operator $T$ with respect to its eigendecomposition; see Appendix A.4.2 for formal justification. The latter condition is interpretable as polynomial decay of eigenvalues: $\lambda_{k}=\Theta(k^{-b})$. Larger $b$ means faster decay of eigenvalues of the covariance operator $T$ and hence smaller effective input dimension. Larger $c$ corresponds to a smoother structural operator $H_{\rho}$[65]. Estimation and convergence:* The estimator has a closed form solution, as shown by [64; 65] in the second stage of the distribution regression problem. We present the solution in notation similar to [14] to elucidate how the stage 1 and stage 2 estimators have the same structure. Theorem 3.: $\forall\xi>0$_, the solution $H^{m}_{\xi}$ to $\mathcal{E}_{\xi}^{m}$ and the solution $\hat{H}^{m}_{\xi}$ to $\hat{\mathcal{E}}_{\xi}^{m}$ exist, are unique, and_ \[H_{\xi}^{m} =(\mathbf{T}+\xi)^{-1}\mathbf{g},\quad\mathbf{T}=\dfrac{1}{m}\sum _{i=1}^{m}T_{\mu(\tilde{z}_{i})},\quad\mathbf{g}=\dfrac{1}{m}\sum_{i=1}^{m} \Omega_{\mu(\tilde{z}_{i})}\tilde{y}_{i}\] \[\hat{H}_{\xi}^{m} =(\hat{\mathbf{T}}+\xi)^{-1}\hat{\mathbf{g}},\quad\hat{\mathbf{T} }=\dfrac{1}{m}\sum_{i=1}^{m}T_{\mu^{n}_{\lambda}(\tilde{z}_{i})},\quad\hat{ \mathbf{g}}=\dfrac{1}{m}\sum_{i=1}^{m}\Omega_{\mu^{n}_{\lambda}(\tilde{z}_{i}) }\tilde{y}_{i}\] We now present this paper’s main theorem. In Appendix A.10, we provide a finite sample bound on the excess error of the estimator $\hat{H}^{m}_{\xi}$ with respect to its target $H_{\rho}$. Adapting arguments by [65], we demonstrate that KIV is able to achieve the minimax optimal single-stage rate derived by [14]. In other words, our two-stage estimator is able to learn the causal relationship with confounded data equally well as single-stage RKHS regression is able to learn the causal relationship with unconfounded data. Theorem 4.: _Assume Hypotheses 1-9. Choose $\lambda=n^{-\frac{1}{c_{1}+1}}$ and $n=m^{\frac{a(c_{1}+1)}{\iota(c_{1}-1)}}$ where $a>0$._ 1. _If_ $a\leq\frac{b(c+1)}{bc+1}$ _then_ $\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})=O_{p}(m^{-\frac{ac}{c+1}})$ _with_ $\xi=m^{-\frac{a}{c+1}}$__ 2. _If_ $a\geq\frac{b(c+1)}{bc+1}$ _then_ $\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})=O_{p}(m^{-\frac{bc}{bc+1}})$ _with_ $\xi=m^{-\frac{b}{bc+1}}$__ At $a=\frac{b(c+1)}{bc+1}<2$, the convergence rate $m^{-\frac{bc}{bc+1}}$ is minimax optimal while requiring the fewest observations [65]. This statistically efficient rate is calibrated by $b$, the effective input dimension, as well as $c$, the smoothness of structural operator $H_{\rho}$[14]. The efficient ratio between stage 1 and stage 2 samples is $n=m^{\frac{b(c+1)}{bc+1}\cdot\frac{(c_{1}+1)}{\iota(c_{1}-1)}}$, implying $n>m$. As far as we know, asymmetric sample splitting is a novel prescription in the IV literature; previous analyses assume $n=m$[4; 37]. ## 6 Experiments We compare the empirical performance of KIV (KernelIV) to four leading competitors: standard kernel ridge regression (KernelReg) [50], Nadaraya-Watson IV (SmoothIV) [16; 23], sieve IV (SieveIV) [48; 17], and deep IV (DeepIV) [36]. To improve the performance of sieve IV, we impose Tikhonov regularization in both stages with KIV’s tuning procedure. This adaptation exceeds the theoretical justification provided by [17]. However, it is justified by our analysis insofar as sieve IV is a special case of KIV: set feature maps $\psi,\phi$ equal to the sieve bases. <figure><img src="content_image/1906.00232/x1.png"><figcaption>Figure 2: Sigmoid design</figcaption></figure> We implement each estimator on three designs. The _linear_ design [17] involves learning counterfactual function $h(x)=4x-2$, given confounded observations of continuous variables $(X,Y)$ as well as continuous instrument $Z$. The _sigmoid_ design [17] involves learning counterfactual function $h(x)=\ln(\|16x-8\|+1)\cdot sgn(x-0.5)$ under the same regime. The _demand_ design [36] involves learning demand function $h(p,t,s)=100+(10+p)\cdot s\cdot\psi(t)-2p$ where $\psi(t)$ is the complex nonlinear function in Figure 6. An observation consists of $(Y,P,T,S,C)$ where $Y$ is sales, $P$ is price, $T$ is time of year, $S$ is customer sentiment (a discrete variable), and $C$ is a supply cost shifter. The parameter $\rho\in\{0.9,0.75,0.5,0.25,0.1\}$ calibrates the extent to which price $P$ is confounded by supply-side market forces. In KIV notation, inputs are $X=(P,T,S)$ and instruments are $Z=(C,T,S)$. For each algorithm, design, and sample size, we implement 40 simulations and calculate MSE with respect to the true structural function $h$. Figures 2, 3, and 8 visualize results. In the sigmoid design, KernelIV performs best across sample sizes. In the demand design, KernelIV performs best for sample size $n+m=1000$ and rivals DeepIV for sample size $n+m=5000$. KernelReg ignores the instrument $Z$, and it is biased away from the structural function due to confounding noise $e$. This phenomenon can have counterintuitive consequences. Figure 3 shows that in the highly nonlinear demand design, KernelReg deviates further from the structural function as sample size increases because the algorithm is further misled by confounded data. Figure 2 of [36] documents the same effect when a feedforward neural network is applied to the same data. The remaining algorithms make use of the instrument $Z$ to overcome this issue. KernelIV improves on SieveIV in the same way that kernel ridge regression improves on ridge regression: by using an infinite dictionary of implicit basis functions rather than a finite dictionary of explicit basis functions. KernelIV improves on SmoothIV by using kernel ridge regression in not only stage 2 but also stage 1, avoiding costly density estimation. Finally, it improves on DeepIV by directly learning stage 1 mean embeddings, rather than performing costly density estimation and sampling from the estimated density. Remarkably, with training sample size of only $n+m=1000$, KernelIV has essentially learned as much as it can learn from the demand design. See Appendix A.11 for representative plots, implementation details, and a robustness study. <figure><img src="content_image/1906.00232/x2.png"><figcaption>Figure 3: Demand design</figcaption></figure> ## 7 Conclusion We introduce KIV, an algorithm for learning a nonlinear, causal relationship from confounded observational data. KIV is easily implemented and minimax optimal. As a contribution to the IV literature, we show how to estimate the stage 1 conditional expectation operator–an infinite by infinite dimensional object–by kernel ridge regression. As a contribution to the kernel methods literature, we show how the RKHS is well-suited to causal inference and ill-posed inverse problems. In simulations, KIV outperforms state of the art algorithms for nonparametric IV regression. The success of KIV suggests RKHS methods may be an effective bridge between econometrics and machine learning. #### Acknowledgments We are grateful to Alberto Abadie, Anish Agarwal, Michael Arbel, Victor Chernozhukov, Geoffrey Gordon, Jason Hartford, Motonobu Kanagawa, Anna Mikusheva, Whitney Newey, Nakul Singh, Bharath Sriperumbudur, and Suhas Vijaykumar. This project was made possible by the Marshall Aid Commemoration Commission. ## References * (1) Theodore W Anderson and Cheng Hsiao. Estimation of dynamic models with error components. _Journal of the American Statistical Association_, 76(375):598–606, 1981. * (2) Joshua D Angrist. Lifetime earnings and the Vietnam era draft lottery: Evidence from Social Security administrative records. _The American Economic Review_, pages 313–336, 1990. * (3) Joshua D Angrist, Guido W Imbens, and Donald B Rubin. Identification of causal effects using instrumental variables. _Journal of the American Statistical Association_, 91(434):444–455, 1996. * (4) Joshua D Angrist and Alan B Krueger. Split-sample instrumental variables estimates of the return to schooling. _Journal of Business & Economic Statistics_, 13(2):225–235, 1995. * (5) Michael Arbel and Arthur Gretton. Kernel conditional exponential family. In _International Conference on Artificial Intelligence and Statistics_, pages 1337–1346, 2018. * (6) Manuel Arellano and Stephen Bond. Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. _The Review of Economic Studies_, 58(2):277–297, 1991. * (7) Jordan Bell. Trace class operators and Hilbert-Schmidt operators. Technical report, University of Toronto Department of Mathematics, 2016. * (8) Alain Berlinet and Christine Thomas-Agnan. _Reproducing Kernel Hilbert Spaces in Probability and Statistics_. Springer, 2011. * (9) Christopher Bishop. _Pattern Recognition and Machine Learning_. Springer, 2006. * (10) Richard Blundell, Xiaohong Chen, and Dennis Kristensen. Semi-nonparametric IV estimation of shape-invariant Engel curves. _Econometrica_, 75(6):1613–1669, 2007. * (11) Richard Blundell, Joel L Horowitz, and Matthias Parey. Measuring the price responsiveness of gasoline demand: Economic shape restrictions and nonparametric demand estimation. _Quantitative Economics_, 3(1):29–51, 2012. * (12) Byron Boots, Arthur Gretton, and Geoffrey J Gordon. Hilbert space embeddings of predictive state representations. In _Uncertainty in Artificial Intelligence_, pages 92–101, 2013. * (13) John Bound, David A Jaeger, and Regina M Baker. Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak. _Journal of the American Statistical Association_, 90(430):443–450, 1995. * (14) Andrea Caponnetto and Ernesto De Vito. Optimal rates for the regularized least-squares algorithm. _Foundations of Computational Mathematics_, 7(3):331–368, 2007. * (15) Claudio Carmeli, Ernesto De Vito, and Alessandro Toigo. Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. _Analysis and Applications_, 4(4):377–408, 2006. * (16) Marine Carrasco, Jean-Pierre Florens, and Eric Renault. Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. _Handbook of Econometrics_, 6:5633–5751, 2007. * (17) Xiaohong Chen and Timothy M Christensen. Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression. _Quantitative Economics_, 9(1):39–84, 2018. * (18) Xiaohong Chen, Lars P Hansen, and Jose Scheinkman. Shape-preserving estimation of diffusions. Technical report, University of Chicago Department of Economics, 1997. * (19) Xiaohong Chen and Demian Pouzo. Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. _Econometrica_, 80(1):277–321, 2012. * (20) Carlo Ciliberto, Lorenzo Rosasco, and Alessandro Rudi. A consistent regularization approach for structured prediction. In _Advances in Neural Information Processing Systems_, pages 4412–4420, 2016. * (21) Corinna Cortes, Mehryar Mohri, and Jason Weston. A general regression technique for learning transductions. In _International Conference on Machine Learning_, pages 153–160, 2005. * (22) Felipe Cucker and Steve Smale. On the mathematical foundations of learning. _Bulletin of the American mathematical society_, 39(1):1–49, 2002. * (23) Serge Darolles, Yanqin Fan, Jean-Pierre Florens, and Eric Renault. Nonparametric instrumental regression. _Econometrica_, 79(5):1541–1565, 2011. * (24) Serge Darolles, Jean-Pierre Florens, and Christian Gourieroux. Kernel-based nonlinear canonical analysis and time reversibility. _Journal of Econometrics_, 119(2):323–353, 2004. * (25) Ernesto De Vito and Andrea Caponnetto. Risk bounds for regularized least-squares algorithm with operator-value kernels. Technical report, MIT CSAIL, 2005. * (26) Carlton Downey, Ahmed Hefny, Byron Boots, Geoffrey J Gordon, and Boyue Li. Predictive state recurrent neural networks. In _Advances in Neural Information Processing Systems_, pages 6053–6064, 2017. * (27) Vincent Dutordoir, Hugh Salimbeni, James Hensman, and Marc Deisenroth. Gaussian process conditional density estimation. In _Advances in Neural Information Processing Systems_, pages 2385–2395, 2018. * (28) Heinz W Engl, Martin Hanke, and Andreas Neubauer. _Regularization of Inverse Problems_, volume 375. Springer Science & Business Media, 1996. * (29) Jean-Pierre Florens. Inverse problems and structural econometrics. In _Advances in Economics and Econometrics: Theory and Applications_, volume 2, pages 46–85, 2003. * (30) Kenji Fukumizu, Francis R Bach, and Michael I Jordan. Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. _Journal of Machine Learning Research_, 5(Jan):73–99, 2004. * (31) Kenji Fukumizu, Le Song, and Arthur Gretton. Kernel Bayes’ rule: Bayesian inference with positive definite kernels. _Journal of Machine Learning Research_, 14(1):3753–3783, 2013. * (32) Arthur Gretton. RKHS in machine learning: Testing statistical dependence. Technical report, UCL Gatsby Unit, 2018. * (33) Steffen Grünewälder, Arthur Gretton, and John Shawe-Taylor. Smooth operators. In _International Conference on Machine Learning_, pages 1184–1192, 2013. * (34) Steffen Grünewälder, Guy Lever, Luca Baldassarre, Sam Patterson, Arthur Gretton, and Massimiliano Pontil. Conditional mean embeddings as regressors. In _International Conference on Machine Learning_, volume 5, 2012. * (35) Steffen Grünewälder, Guy Lever, Luca Baldassarre, Massimilano Pontil, and Arthur Gretton. Modelling transition dynamics in MDPs with RKHS embeddings. In _International Conference on Machine Learning_, pages 1603–1610, 2012. * (36) Jason Hartford, Greg Lewis, Kevin Leyton-Brown, and Matt Taddy. Deep IV: A flexible approach for counterfactual prediction. In _International Conference on Machine Learning_, pages 1414–1423, 2017. * (37) Ahmed Hefny, Carlton Downey, and Geoffrey J Gordon. Supervised learning for dynamical system learning. In _Advances in Neural Information Processing Systems_, pages 1963–1971, 2015. * (38) Miguel A Hernan and James M Robins. _Causal Inference_. CRC Press, 2019. * (39) Daniel Hsu, Sham Kakade, and Tong Zhang. Tail inequalities for sums of random matrices that depend on the intrinsic dimension. _Electronic Communications in Probability_, 17(14):1–13, 2012. * (40) Guido W Imbens and Whitney K Newey. Identification and estimation of triangular simultaneous equations models without additivity. _Econometrica_, 77(5):1481–1512, 2009. * (41) Rainer Kress. _Linear Integral Equations_, volume 3. Springer, 1989. * (42) Charles A Micchelli and Massimiliano Pontil. On learning vector-valued functions. _Neural Computation_, 17(1):177–204, 2005. * (43) Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, Bernhard Schölkopf, et al. Kernel mean embedding of distributions: A review and beyond. _Foundations and Trends in Machine Learning_, 10(1-2):1–141, 2017. * (44) Michael P Murray. Avoiding invalid instruments and coping with weak instruments. _Journal of Economic Perspectives_, 20(4):111–132, 2006. * (45) M Zuhair Nashed and Grace Wahba. Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind. _Mathematics of Computation_, 28(125):69–80, 1974. * (46) M Zuhair Nashed and Grace Wahba. Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations. _SIAM Journal on Mathematical Analysis_, 5(6):974–987, 1974. * (47) M Zuhair Nashed and Grace Wahba. Regularization and approximation of linear operator equations in reproducing kernel spaces. _Bulletin of the American Mathematical Society_, 80(6):1213–1218, 1974. * (48) Whitney K Newey and James L Powell. Instrumental variable estimation of nonparametric models. _Econometrica_, 71(5):1565–1578, 2003. * (49) Judea Pearl. _Causality_. Cambridge University Press, 2009. * (50) Craig Saunders, Alexander Gammerman, and Volodya Vovk. Ridge regression learning algorithm in dual variables. In _International Conference on Machine Learning_, pages 515–521, 1998. * (51) Bernhard Schölkopf, Ralf Herbrich, and Alex J Smola. A generalized representer theorem. In _International Conference on Computational Learning Theory_, pages 416–426, 2001. * (52) Rahul Singh. Causal inference tutorial. Technical report, MIT Economics, 2019. * (53) Steve Smale and Ding-Xuan Zhou. Shannon sampling II: Connections to learning theory. _Applied and Computational Harmonic Analysis_, 19(3):285–302, 2005. * (54) Steve Smale and Ding-Xuan Zhou. Learning theory estimates via integral operators and their approximations. _Constructive Approximation_, 26(2):153–172, 2007. * (55) Le Song, Arthur Gretton, and Carlos Guestrin. Nonparametric tree graphical models. In _International Conference on Artificial Intelligence and Statistics_, pages 765–772, 2010. * (56) Le Song, Jonathan Huang, Alex Smola, and Kenji Fukumizu. Hilbert space embeddings of conditional distributions with applications to dynamical systems. In _International Conference on Machine Learning_, pages 961–968, 2009. * (57) Bharath Sriperumbudur, Kenji Fukumizu, and Gert Lanckriet. On the relation between universality, characteristic kernels and RKHS embedding of measures. In _International Conference on Artificial Intelligence and Statistics_, pages 773–780, 2010. * (58) Douglas Staiger and James H Stock. Instrumental variables regression with weak instruments. _Econometrica_, 65(3):557–586, 1997. * (59) Ingo Steinwart and Andreas Christmann. _Support Vector Machines_. Springer, 2008. * (60) James H Stock and Francesco Trebbi. Retrospectives: Who invented instrumental variable regression? _Journal of Economic Perspectives_, 17(3):177–194, 2003. * (61) James H Stock, Jonathan H Wright, and Motohiro Yogo. A survey of weak instruments and weak identification in generalized method of moments. _Journal of Business & Economic Statistics_, 20(4):518–529, 2002. * (62) Masashi Sugiyama, Ichiro Takeuchi, Taiji Suzuki, Takafumi Kanamori, Hirotaka Hachiya, and Daisuke Okanohara. Least-squares conditional density estimation. _IEICE Transactions_, 93-D(3):583–594, 2010. * (63) Dougal Sutherland. Fixing an error in Caponnetto and De Vito. Technical report, UCL Gatsby Unit, 2017. * (64) Zoltán Szabó, Arthur Gretton, Barnabás Póczos, and Bharath Sriperumbudur. Two-stage sampled learning theory on distributions. In _International Conference on Artificial Intelligence and Statistics_, pages 948–957, 2015. * (65) Zoltán Szabó, Bharath Sriperumbudur, Barnabás Póczos, and Arthur Gretton. Learning theory for distribution regression. _Journal of Machine Learning Research_, 17(152):1–40, 2016. * (66) Grace Wahba. _Spline Models for Observational Data_, volume 59. SIAM, 1990. * (67) Larry Wasserman. _All of Nonparametric Statistics_. Springer, 2006. * (68) Philip G Wright. _Tariff on Animal and Vegetable Oils_. Macmillan Company, 1928. ## Appendix A Appendix ### Instrumental variable #### a.1.1 Comparison of IV assumptions Here, we compare Hypothesis 1 with two alternative formulations of the IV assumption. <figure><img src="content_image/1906.00232/x3.png"><figcaption>(a) Linear</figcaption></figure> We refer to the first formulation in the introduction: conditional independence. This formulation consists of the following assumptions: exclusion $Z\raisebox{0.5pt}{\rotatebox[origin={c}]{90.0}{$\models$}}Y\|(X,e)$; unconfounded instrument $Z\raisebox{0.5pt}{\rotatebox[origin={c}]{90.0}{$\models$}}e$; and relevance, i.e. $\rho(x\|z)$ is not constant in $z$. The directed acyclic graph (DAG) in Figure 4 encodes these assumptions. Definition 7.4.1 of [49] provides a formal graphical criterion. The second formulation is via potential outcomes [3]. Though it is beyond the scope of this work, see (38, Chapter 7) for the relation between DAGs and potential outcomes. We use a third formulation, which belongs in the moment restriction framework for causal inference. In the moment restriction approach, we encode causal assumptions via functional form restrictions and conditional expectations set to zero. Hypothesis 1, introduced by [48], involves such statements. In particular, it imposes additive separability of confounding noise $e$, and $\mathbb{E}[e\|Z]=0$. Be imposing the former, we can relax the independences $Z\raisebox{0.5pt}{\rotatebox[origin={c}]{90.0}{$\models$}}Y\|(X,e)$ and $Z\raisebox{0.5pt}{\rotatebox[origin={c}]{90.0}{$\models$}}e$ to mean independence $\mathbb{E}[e\|Z]=0$. We recommend [52] for a comparison of the DAG, potential outcome, and moment restriction frameworks for causal inference. #### a.1.2 Linear vignette To build intuition for the IV model, we walk through a classic vignette about the linear case. We show how least squares (LS) has a different estimand than two-stage least squares (2SLS) when observations are confounded, i.e. with confounding noise. We will see that the estimand of 2SLS is the structural parameter of interest. Consider the model \[Y=\beta^{\prime}X+e,\quad\mathbb{E}[Xe]\neq 0,\quad\mathbb{E}[e\|Z]=0\] where $Y,e\in\mathbb{R}$, $X\in\mathbb{R}^{d_{x}}$, $Z\in\mathbb{R}^{d_{z}}$, and $d_{z}\geq d_{x}$. Data $(X,Y)$ are confounded but we have access to instrument $Z$. We aim to recover structural parameter $\beta$. Denote the estimands of LS and 2SLS by $\beta^{LS}$ and $\beta^{2SLS}$, respectively. For clarity, we write the variables to which expectations refer. Proposition 1.: $\beta^{LS}\neq\beta=\beta^{2SLS}$__ Proof.: $\beta^{LS}$ is the projection of $Y$ onto $X$. \[\beta^{LS}=\mathbb{E}_{X}[XX^{\prime}]^{-1}\mathbb{E}_{X,Y}[XY]=\beta+\mathbb{ E}_{X}[XX^{\prime}]^{-1}\mathbb{E}_{X,e}[Xe]\neq\beta\] where the second equality substitutes $Y=X^{\prime}\beta+e$. Define $\bar{X}(Z):=\mathbb{E}[X\|Z]$ and $\bar{Y}(Z):=\mathbb{E}[Y\|Z]$. $\beta^{2SLS}$ is the projection of $Y$ onto $\bar{X}(Z)$. \[\beta^{2SLS}=\mathbb{E}_{Z}[\bar{X}(Z)\bar{X}(Z)^{\prime}]^{-1}\mathbb{E}_{Z,Y }[\bar{X}(Z)Y]\] Finally we confirm that $\beta^{2SLS}=\beta$. Taking $\mathbb{E}[\cdot\|Z]$ of the model LHS and RHS \[\bar{Y}(Z)=\bar{X}(Z)^{\prime}\beta\implies\bar{X}(Z)\bar{Y}(Z)=\bar{X}(Z)\bar {X}(Z)^{\prime}\beta\implies\mathbb{E}_{Z}[\bar{X}(Z)\bar{Y}(Z)]=\mathbb{E}_{Z }[\bar{X}(Z)\bar{X}(Z)^{\prime}]\beta\] Appealing to the definition of conditional expectation, \[\beta=\mathbb{E}_{Z}[\bar{X}(Z)\bar{X}(Z)^{\prime}]^{-1}\mathbb{E}_{Z}[\bar{X} (Z)\bar{Y}(Z)]=\mathbb{E}_{Z}[\bar{X}(Z)\bar{X}(Z)^{\prime}]^{-1}\mathbb{E}_{Z ,Y}[\bar{X}(Z)Y]\] ∎ The final equality in the proof makes an important point: in 2SLS, one may use projected outputs $\bar{Y}(Z)$ or original outputs $Y$ in stage 2. Choice of the latter simplifies estimation and analysis. In the present work, we extend this basic model and approach. We consider inputs $\psi(X)$ instead of $X$ and instruments $\phi(Z)$ instead of $Z$. Matching symbols, the model becomes \[Y=h(X)+e=H\psi(X)+e\] where the structural operator $H$ generalizes the structural parameter $\beta$. Whereas 2SLS regresses $Y$ on $\bar{X}(Z)=\mathbb{E}[X\|Z]$, KIV regresses $Y$ on $\mu(Z)=\mathbb{E}[\psi(X)\|Z]$. ### Comparison of nonparametric IV bounds In this section, we compare KIV with alternative nonparametric IV methods that have statistical guarantees. Readers may find it helpful to familiarize themselves with our results in Section 5 before reading this section. #### a.2.1 Nadaraya-Watson IV We first give a detailed account of the bound for nonparametric two-stage IV regression in [23], which provides an explicit end-to-end rate for the combined stages 1 and 2. In this work, stage 1 requires estimates of the conditional density of the input $X$ and output $Y$ given the instrument $Z$. Stage 2 is a ridge regression performed in the relevant space of square integrable functions; the ridge penalty is not directly on RKHS norm, unlike the present work. Still, (23, Assumption A.2) requires that the structural function $h$ is an element of an RKHS defined from the right singular values of the conditional expectation operator $E$ in order to prove consistency. To facilitate comparison between [23] and the present work, we present the operator equation in both notations \[\mathbb{E}[Y\|Z]=Eh,\quad r=T\varphi\] The stage 1 rate of (23, Assumption 3) directly follows from the convergence rate for the Nadaraya-Watson conditional density estimate, expressed as a ratio of unconditional estimates. Definition 4.1 of [23] describes the density estimation kernels, which should not be confused with RKHS kernels. The rate depends on the smoothness of the density (specifically, the number of derivatives that exist), the dimension of the random variables, and the smoothness of the density estimation kernel used. The combined stage 1 and 2 result in (23, Theorem 4.1, Corollary 4.2) requires a further smoothness assumption on the stage 2 regression function $h$, as outlined in (23, Proposition 3.2). Our smoothness assumption in Hypothesis 9 plays an analogous role, though it takes a different form. There are a number of significant differences between [23] and KIV. Consider stage 1 of the learning problem. Density estimation is a more general task than computing conditional mean embeddings $\mu(z)=\mathbb{E}_{X\|Z=z}\psi(X)$, which are all that stage 2 regression requires. In particular, density estimation rapidly becomes more difficult with increasing dimension (67, Section 6.5), whereas the difficulty of learning $\mu(z)$ depends solely on the smoothness of the regression function to $\mathcal{H}_{\mathcal{X}}$; recall Hypothesis 5. Thus, when the input $X$ and instrumental variable $Z$ are in moderate to high dimensions, we expect conditional density estimation in stage 1 of [23] to suffer a drop in performance unlike kernel ridge regression in stage 1 of KIV. (As an aside, the approach to conditional density estimation that involves a ratio of Nadaraya-Watson estimates is suboptimal; better direct estimates of conditional densities exist [62; 5; 27].) Finally, there is no discussion of whether the overall rate obtained in [23] is optimal under the smoothess assumptions made. Relatedly, there is no discussion of what an efficient ratio of stage 1 to stage 2 training samples might be. By contrast, our stage 2 result has a minimax optimal guarantee accompanied by a recommended ratio of training sample sizes. #### a.2.2 Kernel PSR Next we describe a bound for two-stage IV regression derived in the context of predictive state representations (PSRs) [37]. PSRs are a means of performing filtering and smoothing for a time series of observations $o_{1},\ldots,o_{t}$. In this setting, future observations are summarized as a feature vector $\varphi_{t}:=\varphi(o_{t:t+k-1})$, and past observations as a feature vector $h_{t}:=h(o_{1:t-1})$. The predictive state is the expectation of future features given the history: $q_{t}:=\mathbb{E}[\varphi_{t}\|h_{t}]$. Features can be RKHS feature maps [12]. In this case, the predictive state is a conditional mean embedding. Given history $q_{t}$, the goal of filtering is to predict the extended future state $p_{t}:=\mathbb{E}[\xi_{t}\|h_{t}]$, where $\xi_{t}:=\xi(o_{t:t+k})$(37, eq. 2). The relation with IV regression is apparent: both $q_{t}$ and $p_{t}$ are the result of stage 1 regression, and the mapping between them is the result of stage 2 regression. Theorem 2 of [37] gives a finite sample bound for the final stage 2 result, which incorporates convergence results for stage 1 from (56, Theorem 6). There are several key differences between the [37] bound and the KIV bound. First, the [37] bound does not make full use of the structure of the conditional mean embedding regression problem [34]. Rather, [37] apply matrix concentration results from [39] to the operators used in constructing the regression function. As a consequence, the stage 2 rate is slower than the minimax optimal rate proposed in [14]. Another consequence is that the analysis in [37] requires strong assumptions about the smoothness of the input to stage 2 regression. By contrast, our regression-specific analysis requires assumptions on the smoothness of the regression function; see (54, Theorem 2) and (14, Definition 1). The proof of [37] additionally assumes that the stage 2 regression is a Hilbert-Schmidt operator, which amounts to a smoothness assumption, however this is insufficient for their bound. We now show that the input smoothness assumptions from [37] make the bound inapplicable in our case. Suppose we wish to make a counterfactual prediction $y_{\mathrm{test}}=H\gamma_{\mathrm{test}}$ for some $\gamma_{\mathrm{test}}\in\mathcal{H}_{\mathcal{X}}$. From (37, Theorem 2), the required assumption is that $\exists f_{\mathrm{test}}:\mathcal{X}\rightarrow\mathcal{H}_{\mathcal{X}}$ such that Our final goal of counterfactual prediction at a single point requires $\gamma_{\mathrm{test}}=\psi(x_{\mathrm{test}})$, which will only hold in the trivial case when $\rho(x^{\prime}\|z)\rho(z)$ represents a single point mass. In the PSR setting, the assumption is not vacuous since $\gamma_{\mathrm{test}}$ will not be the kernel at a single test point; see (37, Lemma 3). An identical issue arises in the stage 1 bound of (37, Proposition C.2), since it uses a result from (56, Theorem 6) which makes an analogous input smoothness assumption. In summary, neither bound applies in our setting. Finally, (37, Theorem 2) does not explicitly determine an efficient ratio of stage 1 and stage 2 training samples. Instead, analysis assumes an equal number of training samples in each stage. By contrast, we give an efficient ratio between training sample sizes required to obtain the minimax optimal rate in stage 2. Despite the difference in setting, we believe our approach may be used to improve the results in [37]. ### Vector-valued RKHS We briefly review the theory of vector-valued RKHS as it relates to the IV regression problem. The primary reference is the appendix of [33]. Proposition 2 (Lemma 4.33 of [59]).: _Under Hypotheses 2-3, $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$ are separable._ Proposition 3 (Theorem A.2 of [33]).: _Let $I_{\mathcal{H}_{\mathcal{Z}}}:\mathcal{H}_{\mathcal{Z}}\rightarrow\mathcal{H}_ {\mathcal{Z}}$ be the identity operator. $\Gamma(h,h^{\prime})=\langle h,h^{\prime}\rangle_{\mathcal{H}_{\mathcal{X}}}I_ {\mathcal{H}_{\mathcal{Z}}}$ is a kernel of positive type._ Proposition 4 (Proposition 2.3 of [15]).: _Consider a kernel of positive type $\Gamma:\mathcal{H}_{\mathcal{X}}\times\mathcal{H}_{\mathcal{X}}\to \mathcal{L}(\mathcal{H}_{\mathcal{Z}})$, where $\mathcal{L}(\mathcal{H}_{\mathcal{Z}})$ is the space of bounded linear operators from $\mathcal{H}_{\mathcal{Z}}$ to $\mathcal{H}_{\mathcal{Z}}$. It corresponds to a unique RKHS $\mathcal{H}_{\Gamma}$ with reproducing kernel $\Gamma$._ Proposition 5 (Theorem B.1 of [33]).: _Each $E\in\mathcal{H}_{\Gamma}$ is a bounded linear operator $E:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$._ Proposition 6.: $\mathcal{H}_{\Gamma}=\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{ \mathcal{Z}})$ _and the inner products are equal._ Proof.: (8, Theorem 13) and (34, eq. 12). ∎ Proposition 7 (Theorem B.2 of [33].).: _If $\exists E,G\in\mathcal{H}_{\Gamma}$ s.t. $\forall x\in\mathcal{X},\;E\psi(x)=G\psi(x)$ then $E=G$. Furthermore, if $\psi(x)$ is continuous in $x$ then it is sufficient that $E\psi(x)=G\psi(x)$ on a dense subset of $\mathcal{X}$._ Proposition 8 (Theorem B.3 of [33]).: $\forall E\in\mathcal{H}_{\Gamma}$_, $\exists E^{}\in\mathcal{H}_{\Gamma^{}}$ where $\mathcal{H}_{\Gamma^{}}$ is the vector-valued RKHS with reproducing kernel $\Gamma^{}(l,l^{\prime})=\langle l,l^{\prime}\rangle_{\mathcal{H}_{\mathcal{Z} }}I_{\mathcal{H}_{\mathcal{X}}}$. $\forall h\in\mathcal{H}_{\mathcal{X}}$ and $\forall\ell\in\mathcal{H}_{\mathcal{Z}}$,_ \[\langle Eh,\ell\rangle_{\mathcal{H}_{\mathcal{Z}}}=\langle h,E^{}\ell\rangle_ {\mathcal{H}_{\mathcal{X}}}\] _The operator $A\circ E=E^{}$ is an isometric isomorphism from $\mathcal{H}_{\Gamma}$ to $\mathcal{H}_{\Gamma^{}}$; $\mathcal{H}_{\Gamma}\cong\mathcal{H}_{\Gamma^{}}$ and $\\|E\\|_{\mathcal{H}_{\Gamma}}=\\|E^{}\\|_{\mathcal{H}_{\Gamma^{}}}$._ Proposition 9 (Theorem B.4 of [33].).: _The set of self-adjoint operators in $\mathcal{H}_{\Gamma}$ is a closed linear subspace._ Proposition 10 (Lemma 15 of [20]).: $\mathcal{H}_{\Gamma^{}}$ _is isometrically isomorphic to $\mathcal{H}_{\Xi}$, the vector-valued RKHS with reproducing kernel $\Xi(z,z^{\prime})=k_{\mathcal{Z}}(z,z^{\prime})I_{\mathcal{H}_{\mathcal{X}}}$. $\forall\mu\in\mathcal{H}_{\Xi}$, $\exists!E^{}\in\mathcal{H}_{\Gamma^{}}$ s.t._ \[\mu(z)=E^{}\phi(z),\quad\forall z\in\mathcal{Z}\] Proposition 11.: $\mathcal{H}_{\mathcal{X}}$ _is isometrically isomorphic to $\mathcal{H}_{\Omega}$, the scalar-valued RKHS with reproducing kernel $\Omega$ defined s.t._ \[\Omega(\psi(x),\psi(x^{\prime}))=k_{\mathcal{X}}(x,x^{\prime})\] Proof.: (65, eq. 7) and Figure 1. ∎ Proposition 12.: _Under Hypothesis 3,_ \[\mathbb{E}_{X\|Z=z}h(X)=[Eh](z)=\langle h,\mu(z)\rangle_{\mathcal{H}_{\mathcal{ X}}}=H\mu(z)\] Proof.: Hypothesis 3 implies that the feature map is Bochner integrable (59, Definition A.5.20) for the conditional distributions considered: $\forall z\in\mathcal{Z}$, $\mathbb{E}_{X\|Z=z}\\|\psi(X)\\|<\infty$. The first equality holds by definition of the conditional expectation operator $E$. The second equality follows from Bochner integrability of the feature map, since it allows us to exchange the order of expectation and dot product. \[\mathbb{E}_{X\|Z=z}h(X) =\mathbb{E}_{X\|Z=z}\langle h,\psi(X)\rangle_{\mathcal{H}_{ \mathcal{X}}}\] \[=\langle h,\mathbb{E}_{X\|Z=z}\psi(X)\rangle_{\mathcal{H}_{ \mathcal{X}}}\] \[=\langle h,\mu(z)\rangle_{\mathcal{H}_{\mathcal{X}}}\] To see the third equality, note that Riesz representation theorem implies that the inner product with a given element $h\in\mathcal{H}_{\mathcal{X}}$ is uniquely represented by a bounded linear functional $H$ on $\mathcal{H}_{\mathcal{X}}$. ∎ Proposition 13.: _Our RKHS construction implies that_ \[[E_{\rho}h](\cdot)=\mathbb{E}_{X\|Z=(\cdot)}[h(X)]\in\mathcal{H}_{\mathcal{Z}}, \quad\forall h\in\mathcal{H}_{\mathcal{X}}\] Proof.: After defining $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$, we define the conditional expectation operator $E:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ such that $[Eh](z)=\mathbb{E}_{X\|Z=z}h(X)$. By construction, $\mathbb{E}_{X\|Z=(\cdot)}f(X)\in\mathcal{H}_{\mathcal{Z}},\:\forall f\in \mathcal{H}_{\mathcal{X}}$. This is precisely the condition required for the surrogate risk $\mathcal{E}_{1}$ to coincide with the natural risk for the conditional expectation operator [34; 33]. As such, $E_{\rho}=\operatorname{\arg\!\min}\mathcal{E}_{1}(E)$ is the true conditional expectation operator. ∎ ### Covariance operator #### a.4.1 Definitions Definition 3.: $\mu^{-}:\mathcal{Z}\rightarrow\mathcal{H}_{\mathcal{X}}$ _is the function that satisfies_ \[\mu^{-}(z)=\mathbb{E}_{X\|Z=z}\psi(X),\quad\forall z\in D_{\rho\|\mathcal{Z}}\] _where $D_{\rho\|\mathcal{Z}}\subset\mathcal{Z}$ is the support of $Z$, and $\mu^{-}(z)=0$ otherwise._ Proposition 14* (Lemma 8 of [20]).: _Assume Hypotheses 2-3. $\mu^{-}\in L^{2}(\mathcal{Z},\mathcal{H}_{\mathcal{X}},\rho_{\mathcal{Z}})$, where $L^{2}(\mathcal{Z},\mathcal{H}_{\mathcal{X}},\rho_{\mathcal{Z}})$ is the space of square integrable functions from $\mathcal{Z}$ to $\mathcal{H}_{\mathcal{X}}$ with respect to measure $\rho_{\mathcal{Z}}$._ Definition 4.: _Additional stage 1 population operators are_ \[\tilde{T}_{1} =S_{1}^{}\circ S_{1}\] \[R_{1}^{} :\mathcal{H}_{\mathcal{X}}\to L^{2}(\mathcal{Z},\rho_{ \mathcal{Z}})\] \[:h\mapsto\langle h,\mu^{-}(\cdot)\rangle_{\mathcal{H}_{\mathcal{X }}}\] \[R_{1} :L^{2}(\mathcal{Z},\rho_{\mathcal{Z}})\rightarrow\mathcal{H}_{ \mathcal{X}}\] \[:\tilde{\ell}\mapsto\int\mu^{-}(z)\tilde{\ell}(z)d\rho_{\mathcal{ Z}}(z)\] \[T_{ZX} =S_{1}\circ R_{1}^{}\] $T_{ZX}$ is the uncentered cross-covariance operator of (30, Theorem 1). The formulation as $S_{1}\circ R_{1}^{}$ relates this integral operator to the integral operators in [20]. Definition 5.: _The stage 1 empirical operators are_ \[\hat{S}_{1}^{} :\mathcal{H}_{\mathcal{Z}}\rightarrow\mathbb{R}^{n}\] \[:\ell\mapsto\dfrac{1}{\sqrt{n}}\{\langle\ell,\phi(z_{i})\rangle_{ \mathcal{H}_{\mathcal{Z}}}\}_{i=1}^{n}\] \[\hat{S}_{1} :\mathbb{R}^{n}\rightarrow\mathcal{H}_{\mathcal{Z}}\] \[:\{v_{i}\}_{i=1}^{n}\mapsto\dfrac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi( z_{i})v_{i}\] \[\mathbf{T}_{1} =\hat{S}_{1}\circ\hat{S}_{1}^{}\] \[\tilde{\mathbf{T}}_{1} =\hat{S}_{1}^{}\circ\hat{S}_{1}\] \[\hat{R}_{1}^{} :\mathcal{H}_{\mathcal{X}}\rightarrow\mathbb{R}^{n}\] \[:h\mapsto\dfrac{1}{\sqrt{n}}\{\langle h,\psi(x_{i})\rangle_{ \mathcal{H}_{\mathcal{X}}}\}_{i=1}^{n}\] \[\hat{R}_{1} :\mathbb{R}^{n}\rightarrow\mathcal{H}_{\mathcal{X}}\] \[:\{v_{i}\}_{i=1}^{n}\mapsto\dfrac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi( x_{i})v_{i}\] \[\mathbf{T}_{ZX} =\hat{S}_{1}\circ\hat{R}_{1}^{}\] $\hat{S}_{1}^{}$ is the sampling operator of [54]. $\mathbf{T}_{1}$ is the scatter matrix, while $K_{ZZ}=n\tilde{\mathbf{T}}_{1}$ is the empirical kernel matrix with respect to $Z$ as in [20]. Note that $\mathbf{T}_{ZX}=\mathbf{g}_{1}$ in Theorem 1. #### a.4.2 Existence and eigendecomposition We initially abstract from the problem at hand to state useful lemmas. Recall tensor product notation: if $a,b\in\mathcal{H}_{1}$ and $c\in\mathcal{H}_{2}$ then $[c\otimes a]b=c\langle a,b\rangle_{\mathcal{H}_{1}}$. Denote by $\mathcal{L}_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$ the space of Hilbert-Schmidt operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$. Proposition 15 (eq. 3.6 of [32]).: _If $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are separable RKHSs, then_ \[\\|c\otimes a\\|_{\mathcal{L}_{2}(\mathcal{H}_{1},\mathcal{H}_{2})}=\\|a\\|_{ \mathcal{H}_{1}}\\|c\\|_{\mathcal{H}_{2}}\] _and $c\otimes a\in\mathcal{L}_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$._ Proposition 16 (eq. 3.7 of [32]).: _Assume $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are separable RKHSs. If $C\in\mathcal{L}_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$ then_ \[\langle C,c\otimes a\rangle_{\mathcal{L}_{2}(\mathcal{H}_{1},\mathcal{H}_{2})} =\langle c,Ca\rangle_{\mathcal{H}_{2}}\] In Hypothesis 2, we assume that input space $\mathcal{X}$ and instrument space $\mathcal{Z}$ are separable. In Hypothesis 3, we assume RKHSs $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$ have continuous, bounded kernels $k_{\mathcal{X}}$ and $k_{\mathcal{Z}}$ with feature maps $\psi$ and $\phi$, respectively. By Proposition 2, it follows that $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$ are separable, i.e. they have countable orthonormal bases that we now denote $\{e^{\mathcal{X}}_{i}\}_{i=1}^{\infty}$ and $\{e^{\mathcal{Z}}_{i}\}_{i=1}^{\infty}$. Denote by $\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{\mathcal{Z}})$ the space of Hilbert-Schmidt operators $E:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ with inner product $\langle E,G\rangle_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{ \mathcal{Z}})}=\sum_{i=1}^{\infty}\langle Ee^{\mathcal{X}}_{i},Ge^{\mathcal{X} }_{i}\rangle_{\mathcal{H}_{\mathcal{Z}}}$. Denote by $\mathcal{L}_{2}(\mathcal{H}_{\mathcal{Z}},\mathcal{H}_{\mathcal{Z}})$ the space of Hilbert-Schmidt operators $A:\mathcal{H}_{\mathcal{Z}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ with inner product $\langle A,B\rangle_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{Z}},\mathcal{H}_{ \mathcal{Z}})}=\sum_{i=1}^{\infty}\langle Ae^{\mathcal{Z}}_{i},Be^{\mathcal{Z} }_{i}\rangle_{\mathcal{H}_{\mathcal{Z}}}$. When it is contextually clear, we abbreviate both spaces as $\mathcal{L}_{2}$. Proposition 17.: _Assume Hypotheses 2-3. $\exists T_{ZX}\in\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{ \mathcal{Z}})$ and $\exists T_{1}\in\mathcal{L}_{2}(\mathcal{H}_{\mathcal{Z}},\mathcal{H}_{ \mathcal{Z}})$ s.t._ \[\langle T_{ZX},E\rangle_{\mathcal{L}_{2}} =\mathbb{E}\langle\phi(Z)\otimes\psi(X),E\rangle_{\mathcal{L}_{2}}\] \[\langle T_{1},A\rangle_{\mathcal{L}_{2}} =\mathbb{E}\langle\phi(Z)\otimes\phi(Z),A\rangle_{\mathcal{L}_{2}}\] Proof.: By Riesz representation theorem, $T_{ZX}$ and $T_{1}$ exist if the RHSs are bounded linear operators. Linearity follows by definition. Boundedness follows since \[\|\mathbb{E}\langle\phi(Z)\otimes\psi(X),E\rangle_{\mathcal{L}_{2}}\|\] \[\|\mathbb{E}\langle\phi(Z)\otimes\phi(Z),A\rangle_{\mathcal{L}_{2}}\|\] by Jensen, Cauchy-Schwarz, Proposition 15, and boundedness of the kernels. ∎ Proposition 18.: _Assume Hypotheses 2-3._ \[\langle\ell,T_{ZX}h\rangle_{\mathcal{H}_{\mathcal{Z}}}\] \[\langle\ell,T_{1}\ell^{\prime}\rangle_{\mathcal{H}_{\mathcal{Z}}} =\mathbb{E}[\ell(Z)\ell^{\prime}(Z)],\quad\forall\ell,\ell^{ \prime}\in\mathcal{H}_{\mathcal{Z}}\] Proof.: \[\langle\ell,T_{ZX}h\rangle_{\mathcal{H}_{\mathcal{Z}}} =\langle T_{ZX},\ell\otimes h\rangle_{\mathcal{L}_{2}}=\mathbb{E} \langle\phi(Z)\otimes\psi(X),\ell\otimes h\rangle_{\mathcal{L}_{2}}=\mathbb{E} \langle\ell,\phi(Z)\rangle_{\mathcal{H}_{\mathcal{Z}}}\langle h,\psi(X)\rangle _{\mathcal{H}_{\mathcal{X}}}\] \[\langle\ell,T_{1}\ell^{\prime}\rangle_{\mathcal{H}_{\mathcal{Z}}} =\langle T_{1},\ell\otimes\ell^{\prime}\rangle_{\mathcal{L}_{2}}= \mathbb{E}\langle\phi(Z)\otimes\phi(Z),\ell\otimes\ell^{\prime}\rangle_{ \mathcal{L}_{2}}=\mathbb{E}\langle\ell,\phi(Z)\rangle_{\mathcal{H}_{\mathcal{Z }}}\langle\ell^{\prime},\phi(Z)\rangle_{\mathcal{H}_{\mathcal{Z}}}\] by Proposition 16, Proposition 17, and Proposition 16, respectively. ∎ Proposition 19.: _Assume Hypotheses 2-3._ \[tr(T_{ZX}) \leq\kappa Q\] \[tr(T_{1}) \leq\kappa^{2}\] Proof.: \[tr(T_{ZX}) =\sum_{i=1}^{\infty}\langle e_{i}^{\mathcal{Z}},T_{ZX}e_{i}^{ \mathcal{X}}\rangle_{\mathcal{H}_{\mathcal{Z}}}\] \[=\sum_{i=1}^{\infty}\mathbb{E}\langle e_{i}^{\mathcal{Z}},\phi(Z) \rangle_{\mathcal{H}_{\mathcal{Z}}}\langle e_{i}^{\mathcal{X}},\psi(X)\rangle_ {\mathcal{H}_{\mathcal{X}}}\] \[=\mathbb{E}\sum_{i=1}^{\infty}\langle e_{i}^{\mathcal{Z}},\phi(Z) \rangle_{\mathcal{H}_{\mathcal{Z}}}\langle e_{i}^{\mathcal{X}},\psi(X)\rangle_ {\mathcal{H}_{\mathcal{X}}}\] \[=\mathbb{E}\\|\phi(Z)\\|_{\mathcal{H}_{\mathcal{Z}}}\\|\psi(X)\\|_{ \mathcal{H}_{\mathcal{X}}}\] \[\leq\kappa Q\] \[tr(T_{1}) =\sum_{i=1}^{\infty}\langle e_{i}^{\mathcal{Z}},T_{1}e_{i}^{ \mathcal{Z}}\rangle_{\mathcal{H}_{\mathcal{Z}}}\] \[=\sum_{i=1}^{\infty}\mathbb{E}\langle e_{i}^{\mathcal{Z}},\phi(Z) \rangle^{2}_{\mathcal{H}_{\mathcal{Z}}}\] \[=\mathbb{E}\sum_{i=1}^{\infty}\langle e_{i}^{\mathcal{Z}},\phi(Z) \rangle^{2}_{\mathcal{H}_{\mathcal{Z}}}\] \[=\mathbb{E}\\|\phi(Z)\\|_{\mathcal{H}_{\mathcal{Z}}}^{2}\] \[\leq\kappa^{2}\] by definition of trace, the proof of Proposition 18, monotone convergence theorem (59, Theorem A.3.5) with upper bounds $\kappa Q$ and $\kappa^{2}$, Parseval’s identity, and boundedness of the kernels. ∎ Since stage 1 covariance operator $T_{1}$ has finite trace, its eigendecomposition is well-defined. Recall that the stage 2 covariance operator $T$ consists of functions from $\mathcal{H_{\mathcal{X}}}$ to $\mathcal{Y}=\mathbb{R}$. Since these functions have finite-dimensional output, it is immediate that $T$ has finite trace and its eigendecomposition is well-defined (14, Remark 1). Definition 6.: _The powers of operators $T_{1}$ and $T$ are defined as_ \[T_{1}^{a} =\sum_{k=1}^{\infty}\nu_{k}^{a}e^{\mathcal{Z}}_{k}\langle\cdot,e^ {\mathcal{Z}}_{k}\rangle_{\mathcal{H}_{\mathcal{Z}}}\] \[T^{a} =\sum_{k=1}^{\infty}\lambda^{a}_{k}e_{k}\langle\cdot,e_{k}\rangle _{\mathcal{H}_{\Omega}}\] _where $(\{\nu_{k}\},\{e^{\mathcal{Z}}_{k}\})$ is the spectrum of $T_{1}$ and $(\{\lambda_{k}\},\{e_{k}\})$ is the spectrum of $T$._ #### a.4.3 Properties Proposition 20.: _In this operator notation,_ \[T_{1} =\int_{\mathcal{Z}}\phi(z)\otimes\phi(z)d\rho_{\mathcal{Z}}(z)\] \[T_{ZX}^{} =\int_{\mathcal{X}\times\mathcal{Z}}\psi(x)\otimes\phi(z)d\rho(x,z)\] \[T_{ZX} =\int_{\mathcal{X}\times\mathcal{Z}}\phi(z)\otimes\psi(x)d\rho(x,z)\] Proof.: (30, Appendix A.1) or (20, Proposition 13). Note that \[\phi(z)\langle\psi(x),\cdot\rangle_{\mathcal{H}_{\mathcal{X}}}=[\phi(z)\otimes \psi(x)](\cdot)\] ∎ Proposition 21.: _Under Hypotheses 2-3_ \[T_{ZX}=T_{1}\circ E_{\rho}\] Proof.: (30, Theorem 2), appealing to Proposition 13. ∎ Finally we state a property that will be useful for compositions involving covariance operators, generalizing (7, Theorem 15). Proposition 22.: _If $G\in\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{\mathcal{Z}})$ and $B\in\mathcal{L}(\mathcal{H}_{\mathcal{Z}},\mathcal{H}_{\mathcal{Z}})$ then_ \[\\|B\circ G\\|_{\mathcal{L}_{2}}\leq\\|B\\|_{\mathcal{L}}\\|G\\|_{\mathcal{L}_{2}}\] Proof.: \[\\|B\circ G\\|^{2}_{\mathcal{L}_{2}}=\sum_{i=1}^{\infty}\\|B\circ Ge_{i}^{ \mathcal{X}}\\|^{2}_{\mathcal{H}_{\mathcal{Z}}}\leq\sum_{i=1}^{\infty}\left(\\|B \\|_{\mathcal{L}}\\|Ge_{i}^{\mathcal{X}}\\|_{\mathcal{H}_{\mathcal{Z}}}\right)^{2 }=\\|B\\|_{\mathcal{L}}^{2}\\|G\\|_{\mathcal{L}_{2}}^{2}\] where $\mathcal{L}$ is the operator norm and $\mathcal{L}_{2}$ is the Hilbert-Schmidt norm, and the proof makes use of the operator norm definition. ∎ #### a.4.4 Related work Our approach allows both $\mathcal{H}_{\mathcal{X}}$ and $\mathcal{H}_{\mathcal{Z}}$ to be infinite-dimensional spaces. Prior work on conditional mean embeddings and RKHS regression has considered both finite [34; 33] and infinite [56; 55; 31; 37; 20] dimensional RKHS $\mathcal{H}_{\mathcal{X}}$. In this section, we briefly review this literature (besides the PSR case, which we covered in Section A.2.2). First, we recall results from Appendix A.3. $\mathcal{H}_{\Gamma}$ is a vector-valued RKHS consisting of operators $E:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ with kernel $\Gamma(h,h^{\prime})=\langle h,h^{\prime}\rangle_{\mathcal{H}_{\mathcal{X}}}I_ {\mathcal{H}_{\mathcal{Z}}}$. $\mathcal{H}_{\Xi}$ is a vector-valued RKHS consisting of mappings $\mu:\mathcal{Z}\rightarrow\mathcal{H}_{\mathcal{X}}$ with kernel $\Xi(z,z^{\prime})=k_{\mathcal{Z}}(z,z^{\prime})I_{\mathcal{H}_{\mathcal{X}}}$. By Propositions 8 and 10, $\mathcal{H}_{\Gamma}$ and $\mathcal{H}_{\Xi}$ are isometrically isomorphic. There is a fundamental equivalence between $E$ and $\mu$, illustrated in Figure 1: $\mu(z)=E^{}\phi(z)$. Next, we present additional notation for vector-valued RKHS $\mathcal{H}_{\Xi}$. \[\Xi_{z} :\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\Xi}\] \[:h\mapsto\Xi(\cdot,z)h=k_{\mathcal{Z}}(\cdot,z)h\] $\Xi_{z}$ is the point evaluator of [42; 15]. From this definition, \[\Xi(z,z^{\prime}) =\Xi_{z}^{}\circ\Xi_{z^{\prime}}\] \[T_{z}^{\Xi} =\Xi_{z}\circ\Xi_{z}^{}\] \[T_{1}^{\Xi} =\mathbb{E}T_{z}^{\Xi}\] and so $T_{1}^{\Xi}:\mathcal{H}_{\Xi}\rightarrow\mathcal{H}_{\Xi}$. With this notation, we can communicate the constructions and assumptions of [14; 34]. In (14, Hypothesis 1), the authors assume $\Xi_{z}$ is a Hilbert-Schmidt operator. Definition 1 of [14] goes on to define the prior with respect to operator $T_{1}^{\Xi}$. The analysis of [34] inherits this framework. Section 6 of [34] further points out that $\Xi_{z}$ is not Hilbert-Schmidt if $\mathcal{H}_{\mathcal{X}}$ is infinite-dimensional since \[\\|\Xi_{z}\\|_{\mathcal{L}_{2}}=k_{\mathcal{Z}}(z,z)\sum_{i=1}^{\infty}\langle e _{i}^{\mathcal{X}},I_{\mathcal{H}_{\mathcal{X}}}e_{i}^{\mathcal{X}}\rangle_{ \mathcal{H}_{\mathcal{X}}}=\infty\] Therefore the ‘main assumption’ (34, Table 1) is that $\mathcal{H}_{\mathcal{X}}$ is finite dimensional. The authors write, ‘It is likely that this assumption can be weakened, but this requires a deeper analysis’. In the present work, we differ in our constructions and assumptions at this juncture. We instead focus on the covariance operator $T_{1}:\mathcal{H}_{\mathcal{Z}}\rightarrow\mathcal{H}_{\mathcal{Z}}$ as defined in (30, Theorem 1), previously applied to regression with an infinite-dimensional output space in [56; 55; 31; 37; 20]. Proposition 19 shows $tr(T_{1})\leq\kappa^{2}$ under the mild assumptions in Hypotheses 2-3, so its eigendecomposition is well-defined. We place a prior with respect to $T_{1}$, and provide analysis inspired by [53; 54] rather than [14]. Specifically, in Hypothesis 4 we require that the stage 1 problem is well-specified: $E_{\rho}\in\mathcal{H}_{\Gamma}$. This requirement is stronger than the property articulated in Proposition 13. Moreover, in Hypothesis 5 we assume \[E_{\rho}=T_{1}^{\frac{c_{1}-1}{2}}\circ G_{1}\] where $G_{1}:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$, $T_{1}^{\frac{c_{1}-1}{2}}:\mathcal{H}_{\mathcal{Z}}\rightarrow\mathcal{H}_{ \mathcal{Z}}$, and $E_{\rho}:\mathcal{H}_{\mathcal{X}}\rightarrow\mathcal{H}_{\mathcal{Z}}$. By recognizing the equivalence of $E$ and $\mu$, we provide a general theory of conditional mean embedding regression in which $\mathcal{H}_{\mathcal{X}}$ is infinite. A question for further research is how to relax Hypothesis 4. A number of previous works have studied consistency of the conditional expectation operator $E$ in the infinite-dimensional setting. Theorem 1 of [55] establishes consistency in Hilbert-Schmidt norm. However, the proof requires a strong smoothness assumption: that $T_{1}^{-3/2}\circ T_{ZX}$ is Hilbert-Schmidt. Theorem 8 of [31] establishes consistency of $E^{}$ applied to embeddings of particular prior distributions, as needed to calculate a posterior by kernel Bayes’ rule. The consistency results of (20, Theorem 4, Theorem 5) for structured prediction are more relevant to our setting, and we discuss them in Appendix A.8.2 after establishing additional notation. Finally, we remark that previous work has considered infinite-dimensional feature space in a broad variety of settings, beyond conditional mean embedding. In the setting of conditional density estimation, [5] propose an infinite-dimensional natural parameter for a conditional exponential family model, with a loss function derived from the Fisher score. See (5, Lemma 1) for analysis specific to this particular loss. ### Algorithm #### a.5.1 Derivation Proof of Algorithm 1.: Rewrite the stage 1 regularized empirical objective as \[E^{n}_{\lambda} =\operatorname{\arg\!\min}_{E\in\mathcal{H}_{\Gamma}}\mathcal{E} _{\lambda}^{n}(E)\] \[\mathcal{E}_{\lambda}^{n}(E) =\dfrac{1}{n}\sum_{i=1}^{n}\\|\psi(x_{i})-E^{}\phi(z_{i})\\|^{2}_{ \mathcal{H}_{\mathcal{X}}}+\lambda\\|E\\|^{2}_{\mathcal{L}_{2}(\mathcal{H}_{ \mathcal{X}},\mathcal{H}_{\mathcal{Z}})}\] \[=\dfrac{1}{n}\\|\Psi_{X}-E^{}\Phi_{Z}\\|_{2}^{2}+\lambda\\|E\\|^{2}_ {\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{\mathcal{Z}})}\] where the $i^{th}$ column of $\Psi_{X}$ is $\psi(x_{i})$ and the $i^{th}$ column of $\Phi_{Z}$ is $\phi(z_{i})$. Hence by the standard regression formula \[(E^{n}_{\lambda})^{} =\Psi_{X}(K_{ZZ}+n\lambda I)^{-1}\Phi_{Z}^{\prime}\] \[\mu^{n}_{\lambda}(z) =(E^{n}_{\lambda})^{}\phi(z)\] \[=\Psi_{X}(K_{ZZ}+n\lambda I)^{-1}\Phi_{Z}^{\prime}\phi(z)\] \[=\Psi_{X}\gamma(z)\] \[=\sum_{i=1}^{n}\gamma_{i}(z)\psi(x_{i})\] where \[\gamma(z):=(K_{ZZ}+n\lambda I)^{-1}\Phi_{Z}^{\prime}\phi(z)=(K_{ZZ}+n\lambda I )^{-1}K_{Zz}\] Note that this expression coincides with the expression in Theorem 1 after appealing to the proof of (21, Proposition 2.1). By the representer theorem, we know that the first stage estimator $\mu_{\lambda}^{n}\in span(\{\psi(x_{i})\})$ because we are effectively regressing $\{\phi(z_{i})\}$ on $\{\psi(x_{i})\}$ to learn the conditional expectation operator [66; 51]. Indeed we have already shown \[\mu_{\lambda}^{n}(\cdot)=\sum_{j=1}^{n}\gamma_{j}(\cdot)\psi(x_{j})\] In the second stage, we are effectively regressing on $\{\tilde{y}_{i}\}$ on $\mu_{\lambda}^{n}(\tilde{z}_{i})$ to learn the structural function. By the representer theorem, then, $\hat{h}_{\xi}^{m}\in span(\{\mu_{\lambda}^{n}(\tilde{z}_{i})\})$. But $\mu_{\lambda}^{n}(\tilde{z}_{i})\in span(\{\psi(x_{i})\})$, so $\hat{h}_{\xi}^{m}\in span(\{\psi(x_{i})\})$. Thus the solution will take the form \[\hat{h}_{\xi}^{m}(\cdot)=\sum_{i=1}^{n}\alpha_{i}\psi(x_{i})\] Substituting in this functional form as well as the solution for $\mu^{n}_{\lambda}$ permits us to rewrite \[[E^{n}_{\lambda}\hat{h}_{\xi}^{m}](z) =\langle\hat{h}_{\xi}^{m},\mu^{n}_{\lambda}(z)\rangle_{\mathcal{H }_{\mathcal{X}}}\] \[=\bigg{\langle}\sum_{i=1}^{n}\alpha_{i}\psi(x_{i}),\sum_{j=1}^{n} \gamma_{j}(z)\psi(x_{j})\bigg{\rangle}_{\mathcal{H}_{\mathcal{X}}}\] \[=\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_{i}\gamma_{j}(z)k_{\mathcal{X }}(x_{i},x_{j})\] \[=\alpha^{\prime}K_{XX}\gamma(z)\] \[=\alpha^{\prime}w(z)\] where \[w(z):=K_{XX}\gamma(z)=K_{XX}(K_{ZZ}+n\lambda I)^{-1}K_{Zz}\] Note that $w$ depends on stage 1 sample matrices $X$ and $Z$ while $z$ is a test value supplied by the stage 2 sample. The regularized empirical error written in terms of dual parameter $\alpha$ is \[\hat{\mathcal{E}}^{m}_{\xi}(\alpha) =\dfrac{1}{m}\sum_{i=1}^{m}(\tilde{y}_{i}-\alpha^{\prime}w(\tilde {z}_{i}))^{2}+\xi\alpha^{\prime}K_{XX}\alpha\] \[=\dfrac{1}{m}\\|\tilde{y}-W^{\prime}\alpha\\|_{2}^{2}+\xi\alpha^{ \prime}K_{XX}\alpha\] where the $i^{th}$ column of $W$ is $w(\tilde{z}_{i})$. Note that $W=K_{XX}(K_{ZZ}+n\lambda I)^{-1}K_{Z\tilde{Z}}$. In this notation, $\tilde{y}$ and $\tilde{Z}$ are stage 2 sample vector and matrix. Hence \[\hat{\alpha} =(WW^{\prime}+m\xi K_{XX})^{-1}W\tilde{y}\] \[W =K_{XX}(K_{ZZ}+n\lambda I)^{-1}K_{Z\tilde{Z}}\] ∎ #### a.5.2 Validation Algorithm 1 takes as given the values of stage 1 and stage 2 regularization parameters $(\lambda,\xi)$. Theorems 2 and 4 theoretically determine optimal rates $\lambda=n^{\frac{-1}{c_{1}+1}}$ and $\xi=m^{-\frac{b}{bc+1}}$, respectively. For practical use, we provide a validation procedure to empirically determine values of $(\lambda,\xi)$. In some sense, the procedure implicitly estimates stage 1 prior parameter $c_{1}$ and stage 2 prior parameters $(b,c)$. The procedure is as follows. Train stage 1 estimator $\mu_{\lambda}^{n}$ on stage 1 observations $(x_{i},z_{i})$ then select stage 1 regularization parameter value $\lambda^{}$ to minimize out-of-sample loss, calculated from stage 2 observations $(\tilde{x}_{i},\tilde{z_{i}})$. Train stage 2 estimator $\hat{h}_{\xi}^{m}$ on stage 2 observations $(\tilde{y}_{i},\tilde{z}_{i})$ then select stage 2 regularization parameter value $\xi^{}$ to minimize out-of-sample loss, calculated from stage 1 observations $(y_{i},x_{i})$. Our approach assimilates the causal validation procedure of [36] with the sample splitting inherent in KIV. Algorithm 2.: _Let $(x_{i},y_{i},z_{i})$ be $n$ observations. Let $(\tilde{x}_{i},\tilde{y}_{i},\tilde{z}_{i})$ be $m$ observations._ \[\gamma_{\tilde{Z}}(\lambda) =(K_{ZZ}+n\lambda I)^{-1}K_{Z\tilde{Z}}\] \[L_{1}(\lambda) =\dfrac{1}{m}tr[K_{\tilde{X}\tilde{X}}-2K_{\tilde{X}X}\gamma_{ \tilde{Z}}(\lambda)+(\gamma_{\tilde{Z}}(\lambda))^{\prime}K_{XX}\gamma_{\tilde {Z}}(\lambda)]\] \[\lambda^{} =\operatorname{\arg\!\min}L_{1}(\lambda)\] \[L(\lambda,\xi) =\dfrac{1}{n}\sum_{i=1}^{n}\\|y_{i}-\hat{h}_{\xi}^{m}(x_{i})\\|^{2} _{\mathcal{Y}}\] \[\xi^{} =\operatorname{\arg\!\min}L(\lambda^{},\xi)\] _where $\hat{h}_{\xi}^{m}$ is calculated by Algorithm 1 with $\lambda=\lambda^{}$._ Proof of Algorithm 2.: From first principles, the stage 1 out-of-sample loss is \[L_{1}(\lambda)=\dfrac{1}{m}\sum_{i=1}^{m}\\|\psi(\tilde{x}_{i})-\mu_{\lambda}^{ n}(\tilde{z}_{i})\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] Recall from the proof of Algorithm 1 \[\mu_{\lambda}^{n}(z) =\Psi_{X}\gamma(z)\] \[\gamma(z) =(K_{ZZ}+n\lambda I)^{-1}K_{Zz}\] Therefore \[\\|\psi(\tilde{x}_{i})-\mu_{\lambda}^{n}(\tilde{z}_{i})\\|^{2}_{ \mathcal{H}_{\mathcal{X}}} =\\|\psi(\tilde{x}_{i})-\Psi_{X}\gamma(\tilde{z}_{i})\\|^{2}_{ \mathcal{H}_{\mathcal{X}}}\] \[=\langle\psi(\tilde{x}_{i})-\Psi_{X}\gamma(\tilde{z}_{i}),\psi( \tilde{x}_{i})-\Psi_{X}\gamma(\tilde{z}_{i})\rangle_{\mathcal{H}_{\mathcal{X}}}\] \[=k_{\mathcal{X}}(\tilde{x}_{i},\tilde{x}_{i})-2K_{\tilde{x}_{i}X} \gamma(\tilde{z}_{i})+(\gamma(\tilde{z}_{i}))^{\prime}K_{XX}\gamma(\tilde{z}_{ i})\] ∎ ### Stage 1: Lemmas #### a.6.1 Probability Proposition 23 (Lemma 2 of [54]).: _Let $\xi$ be a random variable taking values in a real separable Hilbert space $\mathcal{K}$. Suppose $\exists\tilde{M}$ s.t._ \[\\|\xi\\|_{\mathcal{K}} \leq\tilde{M}<\infty\quad\text{ a.s.}\] \[\sigma^{2}(\xi) :=\mathbb{E}\\|\xi\\|_{\mathcal{K}}^{2}\] _Then $\forall n\in\mathbb{N},\forall\eta\in(0,1)$,_ \[\mathbb{P}\bigg{[}\bigg{\\|}\dfrac{1}{n}\sum_{i=1}^{n}\xi_{i}-\mathbb{E}\xi \bigg{\\|}_{\mathcal{K}}\leq\dfrac{2\tilde{M}\ln(2/\eta)}{n}+\sqrt{\dfrac{2 \sigma^{2}(\xi)\ln(2/\eta)}{n}}\bigg{]}\geq 1-\eta\] #### a.6.2 Regression Proposition 24.: _Under Hypothesis 3_ \[E_{\rho}^{}\phi(z)=\mu(z)\] Proof.: For $h\in\mathcal{H}_{\mathcal{X}}$, \[\langle E_{\rho}^{}\phi(z),h\rangle_{\mathcal{H}_{\mathcal{X}}}=\langle\phi(z ),E_{\rho}h\rangle_{\mathcal{H}_{\mathcal{Z}}}=\langle\phi(z),\mathbb{E}_{X\|Z= (\cdot)}h(X)\rangle_{\mathcal{H}_{\mathcal{Z}}}=\mathbb{E}_{X\|Z=z}h(X)=\langle \mu(z),h\rangle_{\mathcal{H}_{\mathcal{X}}}\] The first equality is the definition of adjoint. The second holds by Proposition 13. The final equality is by Proposition 12. ∎ Proposition 25.: _Under Hypothesis 3_ \[\mathbb{E}\\|(E^{}-E^{}_{\rho})\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{X}}}= \mathcal{E}_{1}(E)-\mathcal{E}_{1}(E_{\rho})\] Proof.: \[\mathcal{E}_{1}(E)=\mathbb{E}\\|\psi(X)-E^{}\phi(Z)\\|^{2}_{\mathcal{H}_{ \mathcal{X}}}=\mathbb{E}\\|\psi(X)-E_{\rho}^{}\phi(Z)+E_{\rho}^{}\phi(Z)-E^{ }\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] Expanding the square we see that the cross terms are $0$ by law of iterated expectation and Proposition 24. ∎ Proposition 26.: _Under Hypotheses 3-4_ \[E_{\lambda}=\operatorname{\arg\!\min}_{E\in\mathcal{H}_{\Gamma}}\mathbb{E}\\|( E^{}-E_{\rho}^{})\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{X}}}+\lambda\\|E\\|^{2}_ {\mathcal{H}_{\Gamma}}\] Proof.: Corollary of Proposition 25. ∎ ### Stage 1: Theorems Proof of Theorem 1.: (35, Appendix D.1), substituting the empirical covariance operators; or (20, Lemma 17). ∎ To quantify the convergence rate of $\\|E^{n}_{\lambda}-E_{\rho}\\|_{\mathcal{H}_{\Gamma}}$, we decompose it into two terms: the sampling error $\\|E^{n}_{\lambda}-E_{\lambda}\\|_{\mathcal{H}_{\Gamma}}$, and the approximation error $\\|E_{\lambda}-E_{\rho}\\|_{\mathcal{H}_{\Gamma}}$. To bound the sampling error, we generalize (54, Theorem 1). Theorem 5.: _Assume Hypotheses 2-4. $\forall\delta\in(0,1)$, the following holds w.p. $1-\delta$:_ \[\\|E^{n}_{\lambda}-E_{\lambda}\\|_{\mathcal{H}_{\Gamma}}\leq\dfrac{4\kappa(Q+ \kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/\delta)}{\sqrt{n}\lambda}\] Proof.: Write \[E^{n}_{\lambda}-E_{\lambda}=\bigg{(}\mathbf{T}_{1}+\lambda I\bigg{)}^{-1}\circ \bigg{(}\mathbf{T}_{ZX}-\mathbf{T}_{1}\circ E_{\lambda}-\lambda E_{\lambda} \bigg{)}\] Observe that \[\mathbf{T}_{ZX}-\mathbf{T}_{1}\circ E_{\lambda} =\dfrac{1}{n}\sum_{i=1}^{n}\phi(z_{i})\otimes\psi(x_{i})-\dfrac{1 }{n}\sum_{i=1}^{n}[\phi(z_{i})\otimes\phi(z_{i})]\circ E_{\lambda}\] \[\lambda E_{\lambda} =T_{ZX}-T_{1}\circ E_{\lambda}=\int\phi(z)\otimes\psi(x)d\rho- \int\phi(z)\otimes\phi(z)d\rho\circ E_{\lambda}\] where the second line holds since $E_{\lambda}=(T_{1}+\lambda I)^{-1}\circ T_{ZX}$ and by appealing to Proposition 20. Write \[\xi_{i}=\phi(z_{i})\otimes\psi(x_{i})-[\phi(z_{i})\otimes\phi(z_{i})]\circ E_{ \lambda}=\phi(z_{i})\otimes[\psi(x_{i})-E_{\lambda}^{}\phi(z_{i})]\] where the second equality holds since \[\phi(z_{i})\otimes\psi(x_{i})-[\phi(z_{i})\otimes\phi(z_{i})] \circ E_{\lambda} =\phi(z_{i})\langle\psi(x_{i}),\cdot\rangle_{\mathcal{H}_{ \mathcal{X}}}-\phi(z_{i})\langle\phi(z_{i}),E_{\lambda}\cdot\rangle_{\mathcal{ H}_{\mathcal{Z}}}\] and by the definition of the adjoint operator. Thus the error bound can be rewritten as \[E^{n}_{\lambda}-E_{\lambda}=\bigg{(}\mathbf{T}_{1}+\lambda I\bigg{)}^{-1}\circ \bigg{(}\dfrac{1}{n}\sum_{i=1}^{n}\xi_{i}-\mathbb{E}\xi\bigg{)}\] Observe that \[\bigg{(}\mathbf{T}_{1}+\lambda I\bigg{)}^{-1} \in\mathcal{L}(\mathcal{H}_{\mathcal{Z}},\mathcal{H}_{\mathcal{Z}})\] \[\bigg{(}\dfrac{1}{n}\sum_{i=1}^{n}\xi_{i}-\mathbb{E}\xi\bigg{)} \in\mathcal{L}_{2}(\mathcal{H}_{\mathcal{X}},\mathcal{H}_{ \mathcal{Z}})\] where the latter is by Proposition 15. Therefore by Propositions 22 and 6, \[\\|E^{n}_{\lambda}-E_{\lambda}\\|_{\mathcal{H}_{\Gamma}} \leq\dfrac{1}{\lambda}\Delta\] \[\Delta\] Note that \[\\|\xi_{i}\\|_{\mathcal{H}_{\Gamma}} \leq\kappa Q+\kappa^{2}\\|E^{}_{\lambda}\\|_{\mathcal{L}_{2}( \mathcal{H}_{\mathcal{Z}},\mathcal{H}_{\mathcal{X}})}\] \[\sigma^{2}(\xi_{i}) =\mathbb{E}\\|\xi_{i}\\|^{2}_{\mathcal{H}_{\Gamma}}\leq\kappa^{2} \mathbb{E}\\|\psi(X)-E_{\lambda}^{}\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{X}}}= \kappa^{2}\mathcal{E}_{1}(E_{\lambda})\] By Proposition 26 with $E=0$ \[\leq\mathbb{E}\\|E_{\rho}^{}\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{ X}}}\] \[\leq\\|E_{\rho}^{}\\|^{2}_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{Z }},\mathcal{H}_{\mathcal{X}})}\mathbb{E}\\|\phi(Z)\\|_{\mathcal{H}_{\mathcal{Z}} }^{2}\] \[\leq\kappa^{2}\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}^{2}\] Hence \[\leq\kappa^{2}\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}^{2}\] \[\\|E^{}_{\lambda}\\|_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{Z}}, \mathcal{H}_{\mathcal{X}})}=\\|E_{\lambda}\\|_{\mathcal{H}_{\Gamma}} \leq\dfrac{\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}}{\sqrt{ \lambda}}\] Moreover by the definition of $E_{\rho}$ as the minimizer of $\mathcal{E}_{1}$, \[\mathcal{E}_{1}(E_{\rho})\leq\mathcal{E}_{1}(0)=\mathbb{E}\\|\psi(X)\\|^{2}_{ \mathcal{H}_{\mathcal{X}}}\leq Q^{2}\] so by Proposition 25 \[\mathcal{E}_{1}(E_{\lambda})=\mathcal{E}_{1}(E_{\rho})+\mathbb{E}\\|(E_{\lambda }^{}-E^{}_{\rho})\phi(Z)\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\leq Q^{2}+\kappa^ {2}\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}^{2}\] In summary, \[\\|\xi_{i}\\|_{\mathcal{H}_{\Gamma}} \leq\kappa Q+\kappa^{2}\dfrac{\kappa\\|E_{\rho}\\|_{\mathcal{H}_{ \Gamma}}}{\sqrt{\lambda}}=\kappa(Q+\kappa^{2}\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma }}/\sqrt{\lambda})\] \[\sigma^{2}(\xi_{i}) \leq\kappa^{2}(Q^{2}+\kappa^{2}\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma} }^{2})\] We then apply Proposition 23. With probability $1-\delta$, \[\Delta\leq\kappa(Q+\kappa^{2}\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}/\sqrt{\lambda })\dfrac{2\ln(2/\delta)}{n}+\sqrt{\kappa^{2}(Q^{2}+\kappa^{2}\\|E_{\rho}\\|_{ \mathcal{H}_{\Gamma}}^{2})\dfrac{2\ln(2/\delta)}{n}}\] There are two cases. 1. $\dfrac{\kappa}{\sqrt{n\lambda}}\leq\dfrac{1}{4\ln(2/\delta)}<1$. Because $a^{2}+b^{2}\leq(a+b)^{2}$ for $a,b\geq 0$, \[\Delta <\dfrac{2\kappa Q\ln(2/\delta)}{n}+\dfrac{2\kappa^{3}\\|E_{\rho}\\| _{\mathcal{H}_{\Gamma}}\ln(2/\delta)}{n\sqrt{\lambda}}+\kappa(Q+\kappa\\|E_{ \rho}\\|_{\mathcal{H}_{\Gamma}})\sqrt{\dfrac{2\ln(2/\delta)}{n}}\] \[=\dfrac{2\kappa Q\ln(2/\delta)}{n}+\dfrac{2\kappa^{2}\\|E_{\rho}\\| _{\mathcal{H}_{\Gamma}}\ln(2/\delta)}{\sqrt{n}}\dfrac{\kappa}{\sqrt{n\lambda}} +\dfrac{\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/\delta)}{ \sqrt{n}}\sqrt{\dfrac{2}{\ln(2/\delta)}}\] \[\leq\dfrac{2\kappa Q\ln(2/\delta)}{\sqrt{n}}+\dfrac{2\kappa^{2}\\| E_{\rho}\\|_{\mathcal{H}_{\Gamma}}\ln(2/\delta)}{\sqrt{n}}+\dfrac{2\kappa(Q+ \kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/\delta)}{\sqrt{n}}\] \[=\dfrac{4\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2 /\delta)}{\sqrt{n}}\] Then recall \[\\|E^{n}_{\lambda}-E_{\lambda}\\|_{\mathcal{H}_{\Gamma}}\leq\dfrac{1}{\lambda}\Delta\] 2. $\dfrac{\kappa}{\sqrt{n\lambda}}>\dfrac{1}{4\ln(2/\delta)}$. Observe that by the definition of $E_{\lambda}^{n}$ \[\dfrac{1}{n}\sum_{i=1}^{n}\\|\psi(x_{i})-(E_{\lambda}^{n})^{}\phi (z_{i})\\|^{2}_{\mathcal{H}_{\mathcal{X}}}+\lambda\\|E_{\lambda}^{n}\\|^{2}_{ \mathcal{H}_{\Gamma}} =\mathcal{E}_{\lambda}^{n}(E_{\lambda}^{n})\] \[\leq\mathcal{E}_{\lambda}^{n}(0)\] \[=\dfrac{1}{n}\sum_{i=1}^{n}\\|\psi(x_{i})\\|^{2}_{\mathcal{H}_{ \mathcal{X}}}\] \[\leq Q^{2}\] Hence \[\\|E_{\lambda}^{n}\\|_{\mathcal{H}_{\Gamma}}\leq\dfrac{Q}{\sqrt{ \lambda}}\] and \[\\|E_{\lambda}^{n}-E_{\lambda}\\|_{\mathcal{H}_{\Gamma}}\leq\dfrac{Q}{\sqrt{ \lambda}}+\dfrac{\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}}{\sqrt{\lambda}}= \dfrac{Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}}}{\sqrt{\lambda}}\] Finally observe that \[\dfrac{1}{4\ln(2/\delta)}<\dfrac{\kappa}{\sqrt{n\lambda}}\iff\dfrac{Q+\kappa\\| E_{\rho}\\|_{\mathcal{H}_{\Gamma}}}{\sqrt{\lambda}}<\dfrac{4\kappa(Q+\kappa\\|E_ {\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/\delta)}{\sqrt{n}\lambda}\] ∎ To bound the approximation error, we generalize (53, Theorem 4). Theorem 6.: _Assume Hypotheses 2-5._ \[\\|E_{\lambda}-E_{\rho}\\|_{\mathcal{H}_{\Gamma}}\leq\lambda^{\frac{c_{1}-1}{2}} \sqrt{\zeta_{1}}\] Proof.: First observe that \[e^{\mathcal{Z}}_{k}\langle e^{\mathcal{Z}}_{k},E_{\rho}\cdot\rangle_{\mathcal{ H}_{\mathcal{Z}}}=e^{\mathcal{Z}}_{k}\langle E^{}_{\rho}e^{\mathcal{Z}}_{k}, \cdot\rangle_{\mathcal{H}_{\mathcal{X}}}=[e^{\mathcal{Z}}_{k}\otimes E^{}_{ \rho}e^{\mathcal{Z}}_{k}](\cdot)\] By the definition of the prior, there exists a $G_{1}$ s.t. \[G_{1}=T_{1}^{\frac{1-c_{1}}{2}}\circ E_{\rho}=\sum_{k}\nu_{k}^{\frac{1-c_{1}}{ 2}}e^{\mathcal{Z}}_{k}\langle e^{\mathcal{Z}}_{k},E_{\rho}\cdot\rangle_{ \mathcal{H}_{\mathcal{Z}}}=\sum_{k}\nu_{k}^{\frac{1-c_{1}}{2}}e^{\mathcal{Z}}_ {k}\otimes[E_{\rho}^{}e^{\mathcal{Z}}_{k}]\] Hence by Proposition 6 \[\\|G_{1}\\|^{2}_{\Gamma}=\sum_{k}\nu_{k}^{1-c_{1}}\\|E_{\rho}^{}e^{\mathcal{Z}}_ {k}\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] By Proposition 21, write \[E_{\lambda}-E_{\rho} =[(T_{1}+\lambda I)^{-1}\circ T_{1}-I]\circ E_{\rho}\] \[=\sum_{k}\bigg{(}\dfrac{\nu_{k}}{\nu_{k}+\lambda}-1\bigg{)}e^{ \mathcal{Z}}_{k}\langle e^{\mathcal{Z}}_{k},E_{\rho}\cdot\rangle_{\mathcal{H}_ {\mathcal{Z}}}\] \[=\sum_{k}\bigg{(}\dfrac{\nu_{k}}{\nu_{k}+\lambda}-1\bigg{)}e^{ \mathcal{Z}}_{k}\otimes[E_{\rho}^{}e^{\mathcal{Z}}_{k}]\] Hence by Proposition 6 \[\\|E_{\lambda}-E_{\rho}\\|^{2}_{\mathcal{H}_{\Gamma}} =\sum_{k}\bigg{(}\dfrac{\nu_{k}}{\nu_{k}+\lambda}-1\bigg{)}^{2}\\| E_{\rho}^{}e^{\mathcal{Z}}_{k}\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] \[=\sum_{k}\bigg{(}\dfrac{\lambda}{\nu_{k}+\lambda}\bigg{)}^{2}\\|E_ {\rho}^{}e^{\mathcal{Z}}_{k}\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] \[=\sum_{k}\bigg{(}\dfrac{\lambda}{\nu_{k}+\lambda}\bigg{)}^{2}\\|E_ {\rho}^{}e^{\mathcal{Z}}_{k}\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\bigg{(}\dfrac{ \lambda}{\lambda}\cdot\dfrac{\nu_{k}}{\nu_{k}}\cdot\dfrac{\nu_{k}+\lambda}{\nu _{k}+\lambda}\bigg{)}^{c_{1}-1}\] \[=\lambda^{c_{1}-1}\sum_{k}\nu_{k}^{1-c_{1}}\\|E_{\rho}^{}e^{ \mathcal{Z}}_{k}\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\bigg{(}\dfrac{\lambda}{\nu_ {k}+\lambda}\bigg{)}^{3-c_{1}}\bigg{(}\dfrac{\nu_{k}}{\nu_{k}+\lambda}\bigg{)} ^{c_{1}-1}\] \[\leq\lambda^{c_{1}-1}\sum_{k}\nu_{k}^{1-c_{1}}\\|E_{\rho}^{}e^{ \mathcal{Z}}_{k}\\|^{2}_{\mathcal{H}_{\mathcal{X}}}\] \[=\lambda^{c_{1}-1}\\|G_{1}\\|^{2}_{\Gamma}\] \[\leq\lambda^{c_{1}-1}\zeta_{1}\] ∎ Theorems 5 and 6 deliver the main stage 1 result, Theorem 2, as a consequence of triangle inequality and optimizing the regularization parameter $\lambda$. Proof of Theorem 2.: By triangle inequality, \[\\|E^{n}_{\lambda}-E_{\rho}\\|_{\mathcal{H}_{\Gamma}}\leq\\|E^{n}_{\lambda}-E_{ \lambda}\\|_{\mathcal{H}_{\Gamma}}+\\|E_{\lambda}-E_{\rho}\\|_{\mathcal{H}_{ \Gamma}}\leq\dfrac{4\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/ \delta)}{\sqrt{n}\lambda}+\lambda^{\frac{c_{1}-1}{2}}\sqrt{\zeta_{1}}\] Minimize the RHS w.r.t. $\lambda$. Rewrite the objective as \[A\lambda^{-1}+B\lambda^{\frac{c_{1}-1}{2}}\] then the FOC yields \[\lambda=\bigg{(}\dfrac{2A}{B(c_{1}-1)}\bigg{)}^{\frac{2}{c_{1}+1}}=\bigg{(} \dfrac{8\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/\delta)}{ \sqrt{n\zeta_{1}}(c_{1}-1)}\bigg{)}^{\frac{2}{c_{1}+1}}=O(n^{\frac{-1}{c_{1}+1 }})\] Substituting this value of $\lambda$, the RHS becomes \[A\bigg{(}\dfrac{2A}{B(c_{1}-1)}\bigg{)}^{-\frac{2}{c_{1}+1}}+B \bigg{(}\dfrac{2A}{B(c_{1}-1)}\bigg{)}^{\frac{c_{1}-1}{c_{1}+1}}\] \[=\dfrac{B(c_{1}+1)}{4^{\frac{1}{c_{1}+1}}}\bigg{(}\dfrac{A}{B(c_{ 1}-1)}\bigg{)}^{\frac{c_{1}-1}{c_{1}+1}}\] \[=\dfrac{\sqrt{\zeta_{1}}(c_{1}+1)}{4^{\frac{1}{c_{1}+1}}}\bigg{(} \dfrac{4\kappa(Q+\kappa\\|E_{\rho}\\|_{\mathcal{H}_{\Gamma}})\ln(2/\delta)}{ \sqrt{n\zeta_{1}}(c_{1}-1)}\bigg{)}^{\frac{c_{1}-1}{c_{1}+1}}\] ∎ ### Stage 1: Corollary We present a corollary necessary to link stage 1 with stage 2. In doing so, we relate our work to conditional mean embedding regression. #### a.8.1 Bound Proposition 27.: _Assume the loss attains a minimum on $\mathcal{H}_{\Xi}$. Then the minimizer with minimal norm $\\|\cdot\\|_{\mathcal{H}_{\Xi}}$ is_ \[\mu^{-}(z) =E_{\rho}^{}\phi(z)\] \[E_{\rho}^{} =T_{ZX}^{}\circ T_{1}^{\dagger}\] \[E_{\rho} =T_{1}^{\dagger}\circ T_{ZX}\] _where $\mu^{-}(z)$ is given in Definition 3 and $T_{1}^{\dagger}$ is the pseudo-inverse of $T_{1}$._ Proof.: (20, Lemma 16). Note that the first equation recovers Proposition 24. The third equation recovers Proposition 21, which we know from (30, Theorem 2). ∎ Proposition 28* (Lemma 17 of [20]).: $\forall\lambda>0$_, the solution $\mu^{n}_{\lambda}\in\mathcal{H}_{\Xi}$ of the regularized empirical objective $\dfrac{1}{n}\sum_{i=1}^{n}\\|\psi(x_{i})-\mu(z_{i})\\|^{2}_{\mathcal{H}_{ \mathcal{X}}}+\lambda\\|\mu\\|^{2}_{\mathcal{H}_{\Xi}}$ exists, is unique, and satisfies_ \[\mu^{n}_{\lambda}(z)=(E^{n}_{\lambda})^{}\phi(z)\] Corollary 1.: $\forall\delta\in(0,1)$_, the following holds w.p. $1-\delta$: $\forall z\in\mathcal{Z}$,_ \[\\|\mu^{n}_{\lambda}(z)-\mu^{-}(z)\\|_{\mathcal{H}_{\mathcal{X}}}\leq r_{\mu}( \delta,n,c_{1}):=\kappa\cdot r_{E}(\delta,n,c_{1})\] Proof.: By Propositions 21 and 27 \[\mu^{-}(z)=(T_{1}^{\dagger}\circ T_{ZX})^{}\phi(z)=(T_{1}^{\dagger}\circ T_{1 }\circ E_{\rho})^{}\phi(z)=E_{\rho}^{}\phi(z)\] so by Proposition 28 \[\\|\mu^{n}_{\lambda}(z)-\mu^{-}(z)\\|_{\mathcal{H}_{\mathcal{X}}}=\\|[E^{n}_{ \lambda}-E_{\rho}]^{}\phi(z)\\|_{\mathcal{H}_{\mathcal{X}}}\leq\\|E^{n}_{ \lambda}-E_{\rho}\\|_{\mathcal{H}_{\Gamma}}\\|\phi(z)\\|_{\mathcal{H}_{\mathcal{X }}}\] ∎ #### a.8.2 Related work We relate $\mu$ to $E$ directly–an insight from [33]. In Theorem 2, we generalize work by [53; 54] to obtain a regression bound for $E$. In Corollary 1, we arrive at an RKHS-norm (and hence uniform) bound for conditional mean embedding $\mu$ that adapts to the smoothness of conditional expectation operator $E$, making use of Theorem 2. The uniform bound on $\mu$ is precisely what we will need in Theorem 7. Our strategy affords weaker input assumptions and tighter bounds than the stage 1 approach of [37], which uses (56, Theorem 6). See Section A.2.2 for a detailed comparison. We also make weaker assumptions than (55, Theorem 1), as detailed in Section A.4.4. Whereas Corollary 1 is a bound on RKHS-norm difference $\\|\mu_{\lambda}^{n}-\mu^{-}\\|_{\mathcal{H}_{\Xi}}$, (20, Lemma 18) contains a bound on excess risk $\mathcal{E}^{\Xi}_{1}(\mu_{\lambda}^{n})-\mathcal{E}_{1}^{\Xi}(\mu^{-})$. To facilitate comparison, we translate the latter to our notation. $\forall\lambda\leq\kappa^{2}$ and $\delta>0$, the following holds w.p. $1-\delta$: \[\mathcal{E}^{\Xi}_{1}(\mu_{\lambda}^{n})-\mathcal{E}_{1}^{\Xi}( \mu^{-}) =\\|(E_{\lambda}^{n})^{}\circ S_{1}-R_{1}\\|_{\mathcal{L}_{2}(L^{2 }(\mathcal{Z},\rho_{\mathcal{Z}}),\mathcal{H}_{\mathcal{X}})}\] \[\leq 4\dfrac{Q+\mathcal{A}_{2}^{\Xi}(\lambda)}{\sqrt{\lambda n}} \bigg{(}1+\sqrt{\dfrac{4\kappa^{2}}{\lambda\sqrt{n}}}\bigg{)}ln^{2}\dfrac{8}{ \delta}+\mathcal{A}_{1}^{\Xi}(\lambda)\] where \[\mathcal{A}_{1}^{\Xi}(\lambda) :=\lambda\\|R_{1}\circ(\tilde{T}_{1}+\lambda)^{-1}\\|_{\mathcal{L}_ {2}(L^{2}(\mathcal{Z},\rho_{\mathcal{Z}}),\mathcal{H}_{\mathcal{X}})}\] \[\mathcal{A}_{2}^{\Xi}(\lambda) :=\kappa\\|T_{ZX}^{}\circ(T_{1}+\lambda)^{-1}\\|_{\mathcal{L}_{2}( \mathcal{H}_{\mathcal{Z}},\mathcal{H}_{\mathcal{X}})}\] Interestingly, the proof of (20, Lemma 18) does not require Hypothesis 5, and it uses different techniques. In future work, we will leverage this result in the KIV setting and compare the consequent rates. ### Stage 2: Lemmas #### a.9.1 Probability Proposition 29* (Proposition 4 of [25]).: _Let $\xi$ be a random variable taking values in a real separable Hilbert space $\mathcal{K}$. Suppose $\exists L,\sigma>0$ s.t._ \[\\|\xi\\|_{\mathcal{K}} \leq L/2\text{ a.s}\] \[\mathbb{E}\\|\xi\\|_{\mathcal{K}}^{2} \leq\sigma^{2}\] _Then $\forall m\in\mathbb{N},\forall\eta\in(0,1)$,_ \[\mathbb{P}\bigg{[}\bigg{\\|}\dfrac{1}{m}\sum_{i=1}^{m}\xi_{i}-\mathbb{E}\xi \bigg{\\|}_{\mathcal{K}}\leq 2\bigg{(}\dfrac{L}{m}+\dfrac{\sigma}{\sqrt{m}} \bigg{)}\ln(2/\eta)\bigg{]}\geq 1-\eta\] #### a.9.2 Regression Proposition 30 (Lemma A.3.16 of [59]).: _The solution to the unconstrained structural operator regression problem is well-defined and satisfies_ \[H_{\rho}\mu(z)=\int_{\mathcal{Y}}yd\rho(y\|\mu(z))\] #### a.9.3 Bounds Definition 7.: _The residual $\mathcal{A}(\xi)$, reconstruction error $\mathcal{B}(\xi)$, and effective dimension $\mathcal{N}(\xi)$ are_ \[\mathcal{A}(\xi) =\\|\sqrt{T}(H_{\xi}-H_{\rho})\\|^{2}_{\mathcal{H}_{\Omega}}\] \[\mathcal{B}(\xi) =\\|H_{\xi}-H_{\rho}\\|^{2}_{\mathcal{H}_{\Omega}}\] \[\mathcal{N}(\xi) =Tr[(T+\xi)^{-1}\circ T]\] Proposition 31.: _If $\rho\in\mathcal{P}(\zeta,b,c)$ then_ \[\mathcal{A}(\xi) \leq\zeta\xi^{c}\] \[\mathcal{B}(\xi) \leq\zeta\xi^{c-1}\] \[\mathcal{N}(\xi) \leq\beta^{1/b}\dfrac{\pi/b}{sin(\pi/b)}\xi^{-1/b}\] Proof.: The bounds for $\mathcal{A}(\xi)$ and $\mathcal{B}(\xi)$ follow from (14, Proposition 3) and the definition of a prior. The bound for $\mathcal{N}(\xi)$ is from [63]. ∎ Proposition 32 (Theorem 2 of [65]).: _The excess error of the stage 2 estimator can be bounded by 5 terms._ \[\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})\leq 5[S_{-1}+S_{0}+ \mathcal{A}(\xi)+S_{1}+S_{2}]\] _where_ \[S_{-1} =\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}(\hat{\mathbf{g}}- \mathbf{g})\\|^{2}_{\mathcal{H}_{\Omega}}\] \[S_{0} =\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}\circ(\mathbf{T}-\hat{ \mathbf{T}})H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}}\] \[S_{1} =\\|\sqrt{T}\circ(\mathbf{T}+\xi)^{-1}(\mathbf{g}-\mathbf{T}H_{ \rho})\\|^{2}_{\mathcal{H}_{\Omega}}\] \[S_{2} =\\|\sqrt{T}\circ(\mathbf{T}+\xi)^{-1}\circ(T-\mathbf{T})(H_{\xi}- H_{\rho})\\|^{2}_{\mathcal{H}_{\Omega}}\] Definition 8.: _Fix $\eta\in(0,1)$ and define the following constants_ \[C_{\eta} =96\ln^{2}(6/\eta)\] \[M =2(C+\\|H_{\rho}\\|_{\mathcal{H}_{\Omega}}\sqrt{B})\] \[\Sigma =\dfrac{M}{2}\] The choice of $C_{\eta}$ reflects a correction by [63] to [14]. The choices of $(M,\Sigma)$ are as in (65, Theorem 2). Proposition 33.: _If $m\geq\dfrac{2C_{\eta}B\mathcal{N}(\xi)}{\xi}$ and $\xi\leq\\|T\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}$ then w.p. $1-\eta/3$_ \[\Theta(\xi):=\\|(T-\mathbf{T})\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{ \Omega})}\leq 1/2\] Proof.: Step 2.1 of (14, Theorem 4). ∎ Proposition 34.: _If $m\geq\dfrac{2C_{\eta}B\mathcal{N}(\xi)}{\xi}$, $\xi\leq\\|T\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}$, and Hypotheses 7-8 hold then w.p. $1-2\eta/3$_ \[S_{1} \leq 32\ln^{2}(6/\eta)\bigg{[}\dfrac{BM^{2}}{m^{2}\xi}+\dfrac{ \Sigma^{2}\mathcal{N}(\xi)}{m}\bigg{]}\] \[S_{2} \leq 8\ln^{2}(6/\eta)\bigg{[}\dfrac{4B^{2}\mathcal{B}(\xi)}{m^{2} \xi}+\dfrac{B\mathcal{A}(\xi)}{m\xi}\bigg{]}\] Proof.: Steps 2 and 3 of (14, Theorem 4), appealing to Propositions 29 and 33. ∎ Proposition 35.: $S_{-1}$ _and $S_{0}$ may be bounded by_ \[S_{-1} \leq\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}\\|^{2}_{\mathcal{L} (\mathcal{H}_{\Omega})}\\|\hat{\mathbf{g}}-\mathbf{g}\\|^{2}_{\mathcal{H}_{ \Omega}}\] \[S_{0} \leq\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}\\|^{2}_{\mathcal{L} (\mathcal{H}_{\Omega})}\\|\mathbf{T}-\hat{\mathbf{T}}\\|^{2}_{\mathcal{L}( \mathcal{H}_{\Omega})}\\|H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}}\] Proof.: Definition of $\\|\cdot\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}$. ∎ Proposition 36 (Supplement 9.1 of [65]).: _Suppose Hypotheses 7-8 hold. If $m\geq\dfrac{2C_{\eta}B\mathcal{N}(\xi)}{\xi}$ and $\xi\leq\\|T\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}$, then_ \[\\|H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}}\] \[\leq 6\bigg{(}\dfrac{16}{\xi}\ln^{2}(6/\eta)\bigg{[}\dfrac{M^{2}B }{m^{2}\xi}+\dfrac{\Sigma^{2}\mathcal{N}(\xi)}{m}\bigg{]}+\dfrac{4}{\xi^{2}} \ln^{2}(6/\eta)\bigg{[}\dfrac{4B^{2}\mathcal{B}(\xi)}{m^{2}}+\dfrac{B\mathcal{ A}(\xi)}{m}\bigg{]}+\mathcal{B}(\xi)+\\|H_{\rho}\\|^{2}_{\mathcal{H}_{\Omega}} \bigg{)}\] Proposition 37 (Supplement 7.1.1 and 7.1.2 of [65]).: _If $\\|\mu^{n}_{\lambda}(z)-\mu^{-}(z)\\|_{\mathcal{H}_{\mathcal{X}}}\leq r_{\mu}= \kappa\cdot r_{E}$ w.p. $1-\delta$ and Hypotheses 7-8 hold then w.p. $1-\delta$_ \[\\|\hat{\mathbf{g}}-\mathbf{g}\\|^{2}_{\mathcal{H}_{\Omega}} \leq L^{2}C^{2}r_{\mu}^{2\iota}\] \[\\|\mathbf{T}-\hat{\mathbf{T}}\\|^{2}_{\mathcal{L}(\mathcal{H}_{ \Omega})} \leq 4BL^{2}r_{\mu}^{2\iota}\] Proposition 38.: \[\\|\sqrt{T}\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}\leq\frac{1}{ 2\sqrt{\xi}}\] Proof.: (14, Step 2.1) and (64, Supplement A.1.11) use this spectral result, which we provide for completeness. Observe that \[\\|\sqrt{T}\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}=\sup_{ \lambda^{\prime}\in\{\lambda_{k}\}}\frac{\sqrt{\lambda^{\prime}}}{\lambda^{ \prime}+\xi}\] where $\{\lambda\}_{k}$ are the eigenvalues of $T$. By arithmetic-geometric mean inequality, \[\sqrt{\lambda^{\prime}\xi}\leq\frac{\lambda^{\prime}+\xi}{2}\iff\frac{\sqrt{ \lambda^{\prime}}}{\lambda^{\prime}+\xi}\leq\frac{1}{2\sqrt{\xi}}\] ∎ Proposition 39.: _If $\\|\mu^{n}_{\lambda}(z)-\mu^{-}(z)\\|_{\mathcal{H}_{\mathcal{X}}}\leq r_{\mu}$ w.p. $1-\delta$, $m\geq\max\bigg{\{}\dfrac{2C_{\eta}B\mathcal{N}(\xi)}{\xi},\bar{m}(\delta,c_{1} )\bigg{\}}$, $\xi\leq\\|T\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}$, and Hypotheses 7-8 hold then w.p. $1-\eta/3-\delta$_ \[\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega} )}\leq\dfrac{2}{\sqrt{\xi}}\] Proof.: (64, Supplement A.1.11) provides the following bound. \[\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega} )}\leq\\|\sqrt{T}\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}\sum_{k =0}^{\infty}\\|(T-\hat{\mathbf{T}})\circ(T+\xi)^{-1}\\|^{k}_{\mathcal{L}( \mathcal{H}_{\Omega})}\] Examine the RHS. By Proposition 38 \[\\|\sqrt{T}\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}\leq\frac{1}{ 2\sqrt{\xi}}\] By a telescoping argument in (64, Supplement A.1.11) \[\\|(T-\hat{\mathbf{T}})\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})} \leq\Theta(\xi)+\\|(\mathbf{T}-\hat{\mathbf{T}})\circ(T+\xi)^{-1}\\|_{\mathcal{L }(\mathcal{H}_{\Omega})}\] Proposition 33 bounds the first term w.p. $1-\eta/3$. Examine the second term. \[\\|(\mathbf{T}-\hat{\mathbf{T}})\circ(T+\xi)^{-1}\\|_{\mathcal{L}( \mathcal{H}_{\Omega})} \leq\\|\mathbf{T}-\hat{\mathbf{T}}\\|_{\mathcal{L}(\mathcal{H}_{ \Omega})}\\|(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}\] \[\leq 2\sqrt{B}Lr^{\iota}_{\mu}\cdot\dfrac{1}{\xi}\] \[\leq 1/4\] where the second inequality is by Proposition 37 and the third inequality reflects a choice of $m$ sufficiently large. In particular, by Corollary 1 it is sufficient that \[m\geq\bar{m}(\delta,c_{1}):=\bigg{[}\dfrac{\sqrt{\zeta_{1}}(c_{1}+1)}{4^{\frac {1}{c_{1}+1}}}\kappa\bigg{(}\dfrac{8\sqrt{B}L}{\xi}\bigg{)}^{1/\iota}\bigg{]}^ {2\frac{c_{1}+1}{c_{1}-1}}\bigg{[}\dfrac{4\kappa(Q+\kappa\\|E_{\rho}\\|_{ \mathcal{H}_{\Gamma}})\ln(2/\delta)}{\sqrt{\zeta_{1}}(c_{1}-1)}\bigg{]}^{2}\] Then \[\\|(T-\hat{\mathbf{T}})\circ(T+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega})} \leq 1/2+1/4=3/4\] and hence \[\\|\sqrt{T}\circ(\hat{\mathbf{T}}+\xi)^{-1}\\|_{\mathcal{L}(\mathcal{H}_{\Omega} )}\leq\dfrac{1}{2\sqrt{\xi}}\cdot\dfrac{1}{1-3/4}=\dfrac{2}{\sqrt{\xi}}\] ∎ ### Stage 2: Theorems Proof of Theorem 3.: (65, eq. 13, 14) provide the closed form solution. Existence and uniqueness follow from (22, Proposition 8). ∎ To quantity the convergence rate of $\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})$, we modify the central results of [65], replacing their first stage convergence argument with our own derived above. Theorem 7.: _Assume Hypotheses 1-9. If $m$ is large enough and $\xi\leq\\|T\\|_{\mathcal{L}(\mathcal{H}_{\Omega})}$ then $\forall\delta\in(0,1)$ and $\forall\eta\in(0,1)$, the following holds w.p. $1-\eta-\delta$:_ \[\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})\leq r_{H}( \delta,n,c_{1};\eta,m,b,c):=5\bigg{\{}\dfrac{4}{\xi}\cdot L^{2}C^{2}(\kappa \cdot r_{E})^{2\iota}+\dfrac{4}{\xi}\cdot 4BL^{2}(\kappa\cdot r_{E})^{2\iota}\] \[\quad\cdot 6\bigg{(}\dfrac{16}{\xi}\ln^{2}(6/\eta)\bigg{[}\dfrac{ M^{2}B}{m^{2}\xi}+\dfrac{\Sigma^{2}}{m}\beta^{1/b}\dfrac{\pi/b}{sin(\pi/b)}\xi ^{-1/b}\bigg{]}\] \[\quad+\dfrac{4}{\xi^{2}}\ln^{2}(6/\eta)\bigg{[}\dfrac{4B^{2}\zeta \xi^{c-1}}{m^{2}}+\dfrac{B\zeta\xi^{c}}{m}\bigg{]}+\zeta\xi^{c-1}+\\|H_{\rho}\\| ^{2}_{\mathcal{H}_{\Omega}}\bigg{)}\] Note that the convergence rate is calibrated by $c_{1}$, the smoothness of the conditional expectation operator $E_{\rho}$; $c$, the smoothness of the structural operator $H_{\rho}$; and $b$, the effective input dimension. Proof.: By Propositions 31 to 39, \[\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho}) \leq 5[S_{-1}+S_{0}+\mathcal{A}(\xi)+S_{1}+S_{2}]\] \[S_{-1} \leq\dfrac{4}{\xi}\cdot L^{2}C^{2}r^{2\iota}_{\mu}\] \[S_{0} \leq\dfrac{4}{\xi}\cdot 4BL^{2}r^{2\iota}_{\mu}\cdot\\|H_{\xi}^{m} \\|^{2}_{\mathcal{H}_{\Omega}}\] \[\\|H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}} \leq 6\bigg{(}\dfrac{16}{\xi}\ln^{2}(6/\eta)\bigg{[}\dfrac{M^{2}B }{m^{2}\xi}+\dfrac{\Sigma^{2}}{m}\beta^{1/b}\dfrac{\pi/b}{sin(\pi/b)}\xi^{-1/b }\bigg{]}\] \[\quad+\dfrac{4}{\xi^{2}}\ln^{2}(6/\eta)\bigg{[}\dfrac{4B^{2}\zeta \xi^{c-1}}{m^{2}}+\dfrac{B\zeta\xi^{c}}{m}\bigg{]}+\zeta\xi^{c-1}+\\|H_{\rho}\\| ^{2}_{\mathcal{H}_{\Omega}}\bigg{)}\] \[\mathcal{A}(\xi) \leq\zeta\xi^{c}\] \[S_{1} \leq 32\ln^{2}(6/\eta)\bigg{[}\dfrac{BM^{2}}{m^{2}\xi}+\dfrac{ \Sigma^{2}}{m}\beta^{1/b}\dfrac{\pi/b}{sin(\pi/b)}\xi^{-1/b}\bigg{]}\] \[S_{2} \leq 8\ln^{2}(6/\eta)\bigg{[}\dfrac{4B^{2}\zeta\xi^{c-1}}{m^{2} \xi}+\dfrac{B\zeta\xi^{c}}{m\xi}\bigg{]}\] Finally use Corollary 1 to write $r_{\mu}=\kappa\cdot r_{E}$. ∎ Proof of Theorem 4.: Ignoring constants in Theorem 7 yields \[S_{-1} =O\bigg{(}\dfrac{r^{2\iota}_{\mu}}{\xi}\bigg{)}\] \[S_{0} =O\bigg{(}\dfrac{r^{2\iota}_{\mu}}{\xi}\cdot\\|H_{\xi}^{m}\\|^{2}_{ \mathcal{H}_{\Omega}}\bigg{)}\] \[\\|H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}} =O\bigg{(}\dfrac{1}{m^{2}\xi^{2}}+\dfrac{1}{m\xi^{1+1/b}}+\dfrac{ 1}{m^{2}\xi^{3-c}}+\dfrac{1}{m\xi^{2-c}}+\xi^{c-1}+1\bigg{)}\] \[\mathcal{A}(\xi) =O(\xi^{c})\] \[S_{1} =O\bigg{(}\dfrac{1}{m^{2}\xi}+\dfrac{1}{m\xi^{1/b}}\bigg{)}\] \[S_{2} =O\bigg{(}\dfrac{1}{m^{2}\xi^{2-c}}+\dfrac{\xi^{c-1}}{m}\bigg{)}\] The last term in the bound on $\\|H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}}$ implies that the bounding terms of $S_{0}$ dominate those of $S_{-1}$. Within the terms bounding $\\|H^{m}_{\xi}\\|^{2}_{\mathcal{H}_{\Omega}}$, observe that $\frac{1}{m^{2}\xi^{2}}$ dominates $\frac{1}{m^{2}\xi^{3-c}}$; $\frac{1}{m\xi^{1+1/b}}$ dominates $\frac{1}{m\xi^{2-c}}$; and $1$ dominates $\xi^{c-1}$. These statements follow from the restrictions $b>1$ and $c\in(1,2]$ in the definition of a prior as well as $\xi\to 0$. Likewise, the terms bounding $S_{1}$ dominate the terms bounding $S_{2}$. In summary, we arrive at a statement analogous to (65, eq. 19). \[\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})=O\bigg{(} \dfrac{r^{2\iota}_{\mu}}{\xi}\cdot\bigg{[}\dfrac{1}{m^{2}\xi^{2}}+\dfrac{1}{m \xi^{1+1/b}}+1\bigg{]}+\xi^{c}+\dfrac{1}{m^{2}\xi}+\dfrac{1}{m\xi^{1/b}}\bigg{)}\] \[\quad\text{s.t. }m\xi^{1+1/b}\geq 1,r^{2\iota}_{\mu}\leq\xi^{2}\] By Corollary 1, Theorem 2, and the choices of $\lambda$ and $n$ in the statement of Theorem 4 \[r^{2\iota}_{\mu}=O\bigg{(}[(n^{-\frac{1}{2}})^{\frac{c_{1}-1}{c_{1}+1}}]^{2 \iota}\bigg{)}=O(m^{-a})\] With this substitution, we arrive at a statement analogous to (65, eq. 20). \[\mathcal{E}(\hat{H}_{\xi}^{m})-\mathcal{E}(H_{\rho})=O\bigg{(} \dfrac{1}{m^{2+a}\xi^{3}}+\dfrac{1}{m^{1+a}\xi^{2+1/b}}+\dfrac{1}{m^{a}\xi}+ \xi^{c}+\dfrac{1}{m^{2}\xi}+\dfrac{1}{m\xi^{1/b}}\bigg{)}\] \[\quad\text{s.t. }m\xi^{1+1/b}\geq 1,m^{a}\xi^{2}\geq 1\] The final result is (65, Theorem 5). ∎ ### Experiments #### a.11.1 Designs <figure><img src="content_image/1906.00232/x3.png"><figcaption>(a) Linear</figcaption></figure> The linear and sigmoid simulation designs are from [17], adapted from [48]. One simulation consists of a sample of $n+m\in\{1000,5000,10000\}$ observations. A given observation is generated from the IV model \[Y =h(X)+e,\quad\mathbb{E}[e\|Z]=0\] where $Y$ is the output, $X$ is the input, $Z$ is the instrument, and $e$ is confounding noise. In particular, for the linear design \[h(x) =4x-2\] while for the sigmoid design \[h(x) =\ln(\|16x-8\|+1)\cdot sgn(x-0.5)\] Data are sampled as \[\begin{pmatrix}e\\ V\\ W\end{pmatrix} \overset{i.i.d.}{\sim}N\left(\begin{pmatrix}0\\ 0\\ 0\end{pmatrix},\begin{pmatrix}1&\frac{1}{2}&0\\ \frac{1}{2}&1&0\\ 0&0&1\end{pmatrix}\right)\] \[X =\Phi\left(\frac{W+V}{\sqrt{2}}\right)\] \[Z =\Phi(W)\] We visualize 1 simulation, consisting of $n+m=1000$ observations, in Figure 5. The blue curve illustrates the structural function $h$. Grey dots depict noisy observations. The noise $e$ has positively sloped bias relative to the structural function $h$. From observations of $(Y,X,Z)$, we estimate $\hat{h}$ by several methods. For each estimated $\hat{h}$, we measure out-of-sample error as the mean square error of $\hat{h}$ versus true $h$ applied to 1000 evenly spaced values $x\in[0,1]$. We report $log_{10}(MSE)$. The demand simulation design is from [36]. One simulation consists of a sample of $n+m\in\{1000,5000,10000\}$ observations. A given observation is generated from the IV model \[Y =h(X)+e,\quad\mathbb{E}[e\|Z]=0\] where $Y$ is the output, $X=(P,T,S)$ are inputs, and $Z=(C,T,S)$ are instruments. Recall that $Y$ is sales, $P$ is the endogenous input instrumented by supply cost-shifter $C$, and $(T,S)$ are exogenous inputs interpretable as time of year and customer sentiment. While $(P,T,C)$ are continuous random variables, $S$ is discrete–a novel feature of this design. $e$ is confounding noise. \[h(p,t,s)=100+(10+p)s\psi(t)-2p\] \[\psi(t)=2\left[\frac{(t-5)^{4}}{600}+\exp\left(-4(t-5)^{2}\right) +\frac{t}{10}-2\right]\] <figure><img src="content_image/1906.00232/x5.png"><figcaption>Figure 6: Demand nonlinearity ψ(t)</figcaption></figure> Data are sampled as \[S \overset{i.i.d.}{\sim}Unif\{1,...,7\}\] \[T \overset{i.i.d.}{\sim}Unif[0,10]\] \[\begin{pmatrix}C\\ V\end{pmatrix} \overset{i.i.d.}{\sim}N\left(\begin{pmatrix}0\\ 0\end{pmatrix},\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\right)\] \[e \overset{i.i.d.}{\sim}N(\rho V,1-\rho^{2})\] \[P =25+(C+3)\psi(T)+V\] From observations of $(Y,P,T,S,C)$, we estimate $\hat{h}$ by several methods. For each estimated $\hat{h}$, we measure out-of-sample error as the mean square error of $\hat{h}$ versus true $h$ applied to 2800 values of $(p,t,s)$. Specifically, we consider 20 evenly spaced values of $p\in[2.5,14.5]$, 20 evenly spaced values of $t\in[0,10]$, and all 7 values $s\in\{1,...,7\}$. We report $log_{10}(MSE)$. #### a.11.2 Algorithms <figure><img src="content_image/1906.00232/x6.png"><figcaption>(c) ``</figcaption></figure>
1104.3436	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 79513, "num_imgs": 11, "llama3_tokens_count": 23981 }	[ "content_image/1104.3436/x1.png", "content_image/1104.3436/x2.png", "content_image/1104.3436/x3.png", "content_image/1104.3436/x4.png", "content_image/1104.3436/x5.png", "content_image/1104.3436/x6.png", "content_image/1104.3436/x7.png", "content_image/1104.3436/x8.png", "content_image/1104.3436/x9.png", "content_image/1104.3436/x10.png", "content_image/1104.3436/x11.png" ]	# Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping David Bolinlabel=e1]bolin@maths.lth.se [ Finn Lindgrenlabel=e2]finn@maths.lth.se [ Lund University Mathematical Statistics Centre for Mathematical Sciences Lund University, Box 118 SE-22100 Lund Sweden e1 E-mail: e2 2 20107 20102 20107 20102 20107 2010 ###### Abstract A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is computationally efficient, applicable to data on general smooth manifolds, and includes both the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions. Nonstationary covariance models are obtained by spatially varying the parameters in the SPDEs, and the model parameters are estimated using direct numerical optimization, which is more efficient than standard Markov Chain Monte Carlo procedures. The model class is used to estimate daily ozone maps using a large data set of spatially irregular global total column ozone data. 10.1214/10-AOAS383 5 1 2011 523 550 exampleExample Spatial models generated by nested SPDEs and Nested SPDEs Matérn covariances nonstationary covariances total column ozone data. ## 1 Introduction Building models for spatial environmental data is a challenging problem that has received much attention over the past years. Nonstationary covariance models are often needed since the traditional stationary assumption is too restrictive for capturing the covariance structure in many problems. Also, many environmental data sets today contain massive amounts of measurements, which makes computational efficiency another increasingly important model property. One such data set, which will be analyzed in this work, is the the Total Ozone Mapping Spectrometer (TOMS) atmospheric ozone data [McPeters et al. (1996)]. The data was collected by a TOMS instrument onboard the the near-polar, Sun-synchronous orbiting satellite Nimbus-7, launched by NASA on October 24, 1978. During the sunlit portions of the satellite’s orbit, the instrument collected data in scans perpendicular to the orbital plane. A new scan was started every eight seconds as the spacecraft moved from south to north. A number of recent papers in the statistical literature [Cressie and Johannesson (2008), Jun and Stein (2008), Stein (2007)] have studied the data, and it requires nonstationary covariance structures as well as efficient computational techniques due to the large number of observations. A covariance model that is popular in environmental statistics is the Matérn family of covariance functions [Matérn (1960)]. The Matérn covariance function has a shape parameter, $\nu$, a scale parameter, $\kappa$, and a variance¹ parameter, $\phi^{2}$, and can be parametrized as [FOOTNOTE:1][ENDFOOTNOTE] \[C(\mathbf{h})=\frac{2^{1-\nu}\phi^{2}}{(4\pi)^{{d/2}}\Gamma(\nu+ {d/2})\kappa^{2\nu}}(\kappa\\|\mathbf{h}\\|)^{\nu}K_{\nu}(\kappa\\|\mathbf{h}\\|), \qquad\mathbf{h}\in\mathbb{R}^{d},\] (1) where $K_{\nu}$ is a modified Bessel function of the second kind of order $\nu>0$. One drawback with defining the model directly through a covariance function, such as (1), is that it makes nonstationary extensions difficult. Another drawback is that, unless the covariance function has compact support, the computational complexity for calculating the Kriging predictor based on $n$ measurements is $\mathcal{O}(n^{3})$. This makes the Matérn covariance model computationally infeasible for many environmental data sets. Recently, Lindgren, Rue and Lindström (2010) derived a method for explicit, and computationally efficient, Markov representations of the Matérn covariance family. The method uses the fact that a random process on $\mathbb{R}^{d}$ with a Matérn covariance function is a solution to the stochastic partial differential equation (SPDE) \[(\kappa^{2}-\Delta)^{{\alpha/2}}X(\mathbf{s})=\phi\mathcal{W}(\mathbf{s}),\] (2) where $\mathcal{W}(\mathbf{s})$ is Gaussian white noise, $\Delta$ is the Laplace operator, and $\alpha=\nu+d/2$ [Whittle (1963)]. Instead of defining Matérn fields through the covariance functions (1), Lindgren, Rue and Lindström (2010) used the solution to the SPDE (2) as a definition. This definition is valid not only on $\mathbb{R}^{d}$ but also on general smooth manifolds, such as the sphere, and facilitates nonstationary extensions by allowing the SPDE parameters $\kappa^{2}$ and $\phi$ to vary with space. The Markov representations were obtained by considering approximate stochastic weak solutions to the SPDE; see Section 3 for details. In this paper we extend the work by Lindgren, Rue and Lindström (2010) and construct a new flexible class of spatial models by considering a generalization of (2). This model class contains a wide family of covariance functions, including both the Matérn family and oscillating covariance functions, and it maintains all desirable properties of the Markov approximated Matérn model, such as computational efficiency, easy nonstationary extensions and applicability to data on general smooth manifolds. The model class is introduced in Section 2, with derivations of some basic properties, examples of covariance functions that can be obtained from these models and a discussion on nonstationary extensions. Section 3 gives a review of the Hilbert space approximation technique and shows how it can be extended to give computationally efficient representations also for this new model class. In Section 4 a numerical parameter estimation procedure for the nested SPDE models is presented, and the section concludes with a discussion on computational complexity for parameter estimation and Kriging prediction. In Section 5 the model class is used to analyze the TOMS ozone data. In particular, all measurements available from October 1st, 1988 in the spatially and temporally irregular “Level 2” version of the data set are used. This data set contains approximately 180,000 measurements, and the nonstationary version of the model class is used to construct estimates of the ozone field for that particular day. Finally, Section 6 contains some concluding remarks and suggestions for further work. ## 2 Stationary nested SPDE models A limitation with the Matérn covariance family is that it does not contain any covariance functions with negative values, such as oscillating covariance functions. One way of constructing a larger class of stochastic fields is to consider a generalization of the SPDE (2): \[\mathcal{L}_{1}X(\mathbf{s})=\mathcal{L}_{2}\mathcal{W}(\mathbf{s}),\] (3) for some linear operators $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$. If $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are commutative operators, (3) is equivalent to the following system of nested SPDEs: \[\mathcal{L}_{1}X_{0}(\mathbf{s}) = \mathcal{W}(\mathbf{s}),\] (4) \[X(\mathbf{s}) = \mathcal{L}_{2}X_{0}(\mathbf{s}).\] This representation gives us an interpretation of the consequence of the additional differential operator $\mathcal{L}_{2}$: $X(\mathbf{s})$ is simply $\mathcal{L}_{2}$ applied to the solution one would get to (3) if $\mathcal{L}_{2}$ was the identity operator. Equation (3) generates a large class of random fields, even if the operators $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are restricted to operators closely related to (2). One of the simplest extensions of the Matérn model is to let $\mathcal{L}_{1}$ be the same as in (2) and use $\mathcal{L}_{2}=(b+\mathbf{B}^{\top}\nabla)$, where $\nabla$ is the gradient, $b\in\mathbb{R}$, and $\mathbf{B}\in\mathbb{R}^{d}$. The equation then is \[(\kappa^{2}-\Delta)^{{\alpha/2}}X(\mathbf{s})=(b+\mathbf{B}^{\top}\nabla) \mathcal{W}(\mathbf{s}),\] (5) and $X(\mathbf{s})$ is a weighted sum of a Matérn field and its directional derivative in the direction determined by the vector $\mathbf{B}$. This model is closely related to the models introduced in Jun and Stein (2007) and Jun and Stein (2008), and the connection is discussed later in Section 5. To get a larger class of models, the orders of the operators $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ can be increased, and to get a class of stochastic fields that is easy to work with, the operators are written as products, where each factor in the product is equal to one of the operators in (5). Thus, let \[\mathcal{L}_{1}=\prod_{i=1}^{n_{1}}(\kappa_{i}^{2}-\Delta)^{{\alpha_{i}/2}}\] (6) for $\alpha_{i}\in\mathbb{N}$ and $\kappa_{i}^{2}>0$, and use \[\mathcal{L}_{2}=\prod_{i=1}^{n_{2}}(b_{i}+\mathbf{B}_{i}^{\top}\nabla)\] (7) for $b_{i}\in\mathbb{R}$ and $\mathbf{B}_{i}\in\mathbb{R}^{d}$. Hence, the SPDE generating the class of nested SPDE models is \[\Biggl{(}\prod_{i=1}^{n_{1}}(\kappa^{2}-\Delta)^{{\alpha_{i}/2}}\Biggr{)}X( \mathbf{s})=\Biggl{(}\prod_{i=1}^{n_{2}}(b_{i}+\mathbf{B}_{i}^{\top}\nabla) \Biggr{)}\mathcal{W}(\mathbf{s}).\] (8) There are several alternative equations one might consider; one could, for example, let $\mathcal{L}_{2}$ be on the same form as $\mathcal{L}_{1}$, or allow for anisotropic operators on the form $(1-\nabla^{\top}\mathbf{A}\nabla)$ for some positive definite matrix $\mathbf{A}$. However, to limit our scope, we will from now on only consider model (8). ### Properties in $\mathbb{R}^{d}$ In this section some basic properties of random fields generated by (8), when $\mathbf{s}\in\mathbb{R}^{d}$, are given. First note that all Matérn fields with shape parameters satisfying $\nu+d/2\in\mathbb{N}$ are contained in the class of stochastic fields generated by (8) since $(\kappa^{2}-\Delta)^{{\alpha/2}}$ can be written on the form (6) for these values of $\nu$. Also note that the order of the operator $\mathcal{L}_{2}$ cannot be larger than the order of $\mathcal{L}_{1}$ if $X(\mathbf{s})$ should be at least as “well behaved” as white noise; hence, one must have $\sum_{i=1}^{n_{1}}\alpha_{i}\geq n_{2}$. The smoothness of $X(\mathbf{s})$ is determined by the difference of the orders of the operators $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$. In order to make a precise statement about this, the spectral density of $X(\mathbf{s})$ is needed. Proposition 2.1: _The spectral density for $X(\mathbf{s})$ defined by (8) is given by_ \[S(\mathbf{k})=\frac{\phi^{2}}{(2\pi)^{d}}\frac{\prod_{j=1}^{n_{2 }}(b_{j}^{2}+\mathbf{k}^{\top}\mathbf{B}_{j}\mathbf{B}_{j}^{\top}\mathbf{k})}{ \prod_{j=1}^{n_{1}}(\kappa_{j}^{2}+\\|\mathbf{k}\\|^{2})^{\alpha_{j}}}.\] This proposition is easily proved using linear filtering theory [see, for example, Yaglom (1987)]. Given the spectral density of $X(\mathbf{s})$, the following proposition regarding the sample function regularity can be proved using Theorem 3.4.3 in Adler (1981). Proposition 2.2: $X(\mathbf{s})$ _defined by (8) has almost surely continuous sample functions if $2\sum_{i=1}^{n_{1}}\alpha_{i}-2n_{2}>d$._ Because the stochastic field $X(\mathbf{s})$ is generated by the SPDE (8), the following corollary regarding sample path differentiability is also easily proved using the fact that the directional derivative of $X(\mathbf{s})$ is in the class of nested SPDE models. Corollary 2.3: _Given that $2\sum_{i=1}^{n_{1}}\alpha_{i}-2n_{2}-d>m$, the $m$th order directional derivative of $X(\mathbf{s})$, $(\mathbf{B}^{\top}\nabla)^{m}X(\mathbf{s})$, has almost surely continuous sample functions._ Hence, as $2\sum_{i=1}^{n_{1}}\alpha_{i}-2n_{2}$ increases, the sample paths become smoother, and eventually become differentiable, twice differentiable, and so on. One could also give a more precise characterization of the sample path regularity using the notion of Hölder continuity. This is (more or less) straightforward using properties of index-$\beta$ random fields [Adler (1981)], but outside the scope of this article. A closed-form expression for the covariance function is not that interesting since none of the methods that are later presented for parameter estimation, spatial prediction or model validation require an expression for the covariance function; however, if one were to use some technique that requires the covariance function, it can be derived. An expression for the general case is quite complicated, and will not be presented here. Instead we present a recipe for calculating the covariance function for given parameters of the SPDE, with explicit results for a few examples. To calculate the covariance function of $X(\mathbf{s})$, first calculate the covariance function, $C_{X_{0}}(\mathbf{h})$, of $X_{0}(\mathbf{s})$, given by (4). Given this covariance function, the covariance function for $X(\mathbf{s})$ is obtained as \[C(\mathbf{h})=\Biggl{(}\prod_{i=1}^{n_{2}}(b_{i}-\nabla^{\top} \mathbf{B}_{i}\mathbf{B}_{i}^{\top}\nabla)\Biggr{)}C_{X_{0}}(\mathbf{h}).\] The motivation for this expression is again directly from linear filter theory, and the $d$-dimensional equivalent of the formula for the covariance function for a differentiated stochastic process, $r_{X^{\prime}}(\tau)=-r_{X}^{\prime\prime}(\tau)$. To get an expression for $C_{X_{0}}(\mathbf{h})$, first use Proposition 2.1 with $\mathcal{L}_{2}=I$ to get the spectral density of $X_{0}(\mathbf{s})$. Using a partial fraction decomposition, the spectral density can be written as \[S_{X_{0}}(\mathbf{k})=\frac{\phi^{2}}{(2\pi)^{d}}\sum_{i=1}^{n}\sum_{j=1}^{ \alpha_{i}}\frac{p_{i,j}}{(\kappa_{i}^{2}+\\|\mathbf{k}\\|^{2})^{j}},\] (9) where $p_{i,j}$ is a real constant which can be found using the Heaviside cover-up method [see, for example, Thomas and Finney (1995), page 523]. Now, by taking the inverse Fourier transform of (9), the covariance function for $X_{0}(\mathbf{s})$ is \[C_{X_{0}}(\mathbf{h})=\sum_{i=1}^{n}\sum_{j=1}^{\alpha_{i}}p_{i, j}C_{\kappa_{i}}^{j}(\mathbf{h}),\] where $C_{\kappa}^{\nu}(\mathbf{h})$ denotes a Matérn covariance function with shape parameter $\nu$, scale parameter $\kappa$ and variance parameter $\phi^{2}$. The final step is to use that the derivative of a Matérn covariance function can be expressed using a Matérn covariance with another shape parameter. More precisely, one has \[\frac{\partial}{\partial h_{i}}C_{\kappa}^{\nu}(\mathbf{h})=- \frac{h_{i}}{2\nu}C_{\kappa}^{\nu-1}(\mathbf{h}),\] where $h_{i}$ denotes the $i$th component of the vector $\mathbf{h}$. Using these calculations, one can obtain the covariance function for any field given by (8). We conclude this section by showing the covariance function for some simple cases in $\mathbb{R}^{2}$. The covariance functions for these examples are shown in Figure 1, and realizations of Gaussian processes with these covariance functions are shown in Figure 2. <figure><img src="content_image/1104.3436/x1.png"><figcaption>Figure 1: Covariance functions of random fields obtained from model (8) withparameters from Example 2.1 (top left), Example 2.1 (top middle and right),Example 2.1 (bottom left and middle) and Example 2.1 (bottom right).</figcaption></figure> <figure><img src="content_image/1104.3436/x2.png"><figcaption>Figure 2: Realizations of random fields obtained from model (8) with differentparameters. The realization in each panel corresponds to a stochastic fieldwith the covariance function shown in the corresponding panel in Figure 1.</figcaption></figure> With $\mathcal{L}_{1}=(\kappa^{2}-\Delta)^{{\alpha/2}}$ and $\mathcal{L}_{2}$ as the identity operator, the standard Matérn covariance function (1) is obtained, shown in the top left panel of Figure 1. The simplest nested SPDE model (5) has the covariance function \[C(\mathbf{h})=bC_{\kappa}^{\nu}(\mathbf{h})+\frac{\mathbf{B}^{ \top}\mathbf{B}}{2\nu}C_{\kappa}^{\nu-1}(\mathbf{h})-\frac{\mathbf{h}^{\top} \mathbf{B}\mathbf{B}^{\top}\mathbf{h}}{4\nu(\nu-1)}C_{\kappa}^{\nu-2}(\mathbf{ h}).\] A stochastic field with this covariance function is obtained as a weighted sum of a Matérn field $X_{0}(\mathbf{s})$ and its directional derivative in the direction of $\mathbf{B}$. The field therefore has a Matérn-like behavior in the direction perpendicular to $\mathbf{B}$ and an oscillating behavior in the direction of $\mathbf{B}$. In the upper middle panel of Figure 1, this covariance function is shown for $\mathbf{B}=(1,0)^{\top}$, $\nu=3$, and $b=5$. In the upper right panel of Figure 1, it is shown for $\mathbf{B}=(1,0)^{\top}$, $\nu=3$, and $b=0$. The number of zero crossings of the covariance function in the direction of $\mathbf{B}$ is at most $n_{2}$. In the previous example we had $n_{2}=1$, and to obtain a more oscillating covariance function, the order of $\mathcal{L}_{2}$ can be increased by one: \[(\kappa^{2}-\Delta)^{{\alpha/2}}X(\mathbf{s})=(b_{1}+\mathbf{B}_{ 1}^{\top}\nabla)(b_{2}+\mathbf{B}_{2}^{\top}\nabla)\mathcal{W}(\mathbf{s}).\] This model has the covariance function \[C(\mathbf{h}) = b_{1}b_{2}C_{\kappa}^{\nu}(\mathbf{h})+\frac{b_{2}\mathbf{B}_{1} ^{\top}\mathbf{B}_{1}+b_{1}\mathbf{B}_{2}^{\top}\mathbf{B}_{2}}{2\nu}C_{\kappa }^{\nu-1}(\mathbf{h})\] \[{}+\frac{2(\mathbf{B}_{2}^{\top}\mathbf{B}_{1})^{2}+\mathbf{B}_{1 }^{\top}\mathbf{B}_{1}\mathbf{B}_{2}^{\top}\mathbf{B}_{2}-\mathbf{h}^{\top}(b_ {1}\mathbf{B}_{2}\mathbf{B}_{2}^{\top}+b_{2}\mathbf{B}_{1}\mathbf{B}_{1}^{\top })\mathbf{h}}{2^{2}\nu(\nu-1)}C_{\kappa}^{\nu-2}(\mathbf{h})\] In the bottom left panel of Figure 1 this covariance function is shown for $\nu=5$, $b_{1}=b_{2}=0$ and $\mathbf{B}_{1}=\mathbf{B}_{2}=(1,0)^{\top}$. With these parameters, the covariance function is similar to the covariance function in the previous example, but with one more zero crossing in the direction of $\mathbf{B}$. For this specific choice of parameters, the expression for the covariance function can be simplified to \[C(\mathbf{h})=3\gamma_{2}C_{\kappa}^{\nu-2}(\mathbf{h})-6\gamma_ {3}h_{1}^{2}C_{\kappa}^{\nu-3}(\mathbf{h})+\gamma_{4}h_{1}^{4}C_{\kappa}^{\nu- 4}(\mathbf{h}),\] where $\gamma_{k}=(2^{k}\Pi_{i=0}^{k-1}(\nu-k))^{-1}$. In the bottom middle panel of Figure 1 the covariance function is shown for $\nu=5$, $b_{1}=b_{2}=0$, $\mathbf{B}_{1}=(1,0)^{\top}$, and $\mathbf{B}_{2}=(0,1)^{\top}$. Thus, the field $X_{0}(\mathbf{s})$ is differentiated in two different directions, and the covariance function for $X(\mathbf{s})$ therefore is oscillating in two directions. For these parameters, the covariance function can be written as \[C(\mathbf{h})=\gamma_{2}C_{\kappa}^{\nu-2}(\mathbf{h})-\gamma_{3 }\mathbf{h}^{\top}\mathbf{h}C_{\kappa}^{\nu-3}(\mathbf{h})+\gamma_{4}h_{1}h_{2 }C_{\kappa}^{\nu-4}(\mathbf{h}).\] The bottom right panel of Figure 1 shows a covariance function for the nested SPDE As in the previous examples, the covariance function for a stochastic field generated by this SPDE can be calculated and written on the form \[C(\mathbf{h})=\sum_{k=0}^{8}\gamma_{k}f_{k}(\mathbf{h})C_{\kappa }^{\nu-k}(\mathbf{h}),\] where $f_{k}(\mathbf{h}),k=0,\ldots,8,$ are functions depending on $\mathbf{h}$ and the parameters in the SPDE. Without any restrictions on the parameters, it is a rather tedious exercise to calculate the functions $f_{k}(\mathbf{h})$, and we therefore only show them for the specific set of parameters that are used in Figure 1: $\nu=7$, $b_{1}=b_{2}=0$, $\mathbf{B}_{1}=(1,0)^{\top}$ and $\mathbf{B}_{2}=(0,1)^{\top}$. In this case $f_{0}(\mathbf{h})=f_{1}(\mathbf{h})=f_{2}(\mathbf{h})=0$, and the covariance function is \[C(\mathbf{h}) = 9\gamma_{4}C_{\kappa}^{\nu-4}(\mathbf{h})-18\gamma_{5}\mathbf{h} ^{\top}\mathbf{h}C_{\kappa}^{\nu-5}(\mathbf{h})+3\gamma_{6}(h_{1}^{4}+h_{2}^{4 }+12h_{1}^{2}h_{2}^{2})C_{\kappa}^{\nu-6}(\mathbf{h})\] \[{}-6\gamma_{7}h_{1}^{2}h_{2}^{2}\mathbf{h}^{\top}\mathbf{h}C_{ \kappa}^{\nu-7}(\mathbf{h})+\gamma_{8}h_{1}^{4}h_{2}^{4}C_{\kappa}^{\nu-8}( \mathbf{h}).\] ### Nonstationary nested SPDE models Nonstationarity can be introduced in the nested SPDE models by allowing the parameters $\kappa_{i}$, $b_{i}$ and $\mathbf{B}_{i}$ to be spatially varying: \[\Biggl{(}\prod_{i=1}^{n_{1}}\bigl{(}\kappa_{i}^{2}(\mathbf{s})- \Delta\bigr{)}^{{\alpha_{i}/2}}\Biggr{)}X_{0}(\mathbf{s}) = \mathcal{W}(\mathbf{s}),\] (10) \[X(\mathbf{s}) = \Biggl{(}\prod_{i=1}^{n_{2}}\bigl{(}b_{i}(\mathbf{s})+\mathbf{B}_ {i}(\mathbf{s})^{\top}\nabla\bigr{)}\Biggr{)}X_{0}(\mathbf{s}).\] If the parameters are spatially varying, the two operators are no longer commutative, and the solution to (10) is not necessarily equal to the solution of \[\Biggl{(}\prod_{i=1}^{n_{1}}\bigl{(}\kappa_{i}^{2}(\mathbf{s})-\Delta\bigr{)}^ {{\alpha_{i}/2}}\Biggr{)}X(\mathbf{s})=\Biggl{(}\prod_{i=1}^{n_{2}}\bigl{(}b_{ i}(\mathbf{s})+\mathbf{B}_{i}(\mathbf{s})^{\top}\nabla\bigr{)}\Biggr{)} \mathcal{W}(\mathbf{s}).\] (11) For nonstationary models, we will from now on only study the system of nested SPDEs (10), though it should be noted that the methods presented in the next sections can be applied to (11) as well. One could potentially use an approach where the spatially varying parameters also are modeled as stochastic fields, but to be able to estimate the parameters efficiently, it is easier to assume that each parameter can be written as a weighted sum of some known regression functions. In Section 5 this approach is used for a nested SPDE model on the sphere. In this case, one needs a regression basis $\{\psi_{j}(\mathbf{s})\}$ for the vector fields $\mathbf{B}_{i}(\mathbf{s})$ on the sphere. Explicit expressions for such a basis are given in the Appendix. ## 3 Computationally efficient representations In the previous section covariance functions for some examples of nested SPDE models were derived. Given the covariance function, standard spatial statistics techniques can be used for parameter estimation, spatial prediction and model simulation. Many of these techniques are, however, computationally infeasible for large data sets. Thus, in order to use the model for large environmental data sets, such as the ozone data studied in Section 5, a more computationally efficient representation of the model class is needed. In this section the Hilbert space approximation technique by Lindgren, Rue and Lindström (2010) is used to derive such a representation. The key idea in Lindgren, Rue and Lindström (2010) is to approximate the solution to the SPDE $\mathcal{L}_{1}X_{0}(\mathbf{s})=\mathcal{W}(\mathbf{s})$ in some approximation space spanned by basis functions $\varphi_{1}(\mathbf{s}),\ldots,\varphi_{n}(\mathbf{s})$. The method is most efficient if these basis functions have compact support, so, from now on, it is assumed that $\{\varphi_{i}\}$ are local basis functions. The weak solution of the SPDE with respect to the approximation space can be written as $\tilde{x}(\mathbf{s})=\sum_{i=1}^{n}w_{i}\varphi_{i}(\mathbf{s})$, where the stochastic weights $\{w_{i}\}_{i=1}^{n}$ are chosen such that the weak formulation of the SPDE is satisfied: \[[\langle{\varphi_{i}},{\mathcal{L}_{1}\tilde{x}}\rangle_{\Omega}]_{i=1,\ldots, n}\stackrel{{ D}}{{=}}[\langle{\varphi_{i}},{\mathcal{W}}\rangle_{ \Omega}]_{i=1,\ldots,n}.\] (12) Here $\stackrel{{ D}}{{=}}$ denotes equality in distribution, $\Omega$ is the manifold on which $\mathbf{s}$ is defined, and $\langle{f},{g}\rangle_{\Omega}=\int_{\Omega}f(\mathbf{s})g(\mathbf{s})\, \mathrm{d}\mathbf{s}$ is the scalar product on $\Omega$. As an illustrative example, consider the first fundamental case $\mathcal{L}_{1}=\kappa^{2}-\Delta$. One has \[\langle{\varphi_{i}},{\mathcal{L}_{1}\tilde{x}}\rangle_{\Omega}= \sum_{j=1}^{n}w_{j}\langle{\varphi_{i}},{\mathcal{L}_{1}\varphi_{j}}\rangle_{ \Omega},\] so by introducing a matrix $\mathbf{K}$, with elements $\mathbf{K}_{i,j}=\langle{\varphi_{i}},{\mathcal{L}_{1}\varphi_{j}}\rangle_{\Omega}$, and the vector $\mathbf{w}=(w_{1},\ldots,w_{n})^{\top}$, the left-hand side of (12) can be written as $\mathbf{K}\mathbf{w}$. Since \[\langle{\varphi_{i}},{\mathcal{L}_{1}\varphi_{j}}\rangle_{\Omega} = \kappa^{2}\langle{\varphi_{i}},{\varphi_{j}}\rangle_{\Omega}- \langle{\varphi_{i}},{\Delta\varphi_{j}}\rangle_{\Omega}\] \[= \kappa^{2}\langle{\varphi_{i}},{\varphi_{j}}\rangle_{\Omega}+ \langle{\nabla\varphi_{i}},{\nabla\varphi_{j}}\rangle_{\Omega},\] the matrix $\mathbf{K}$ can be written as $\mathbf{K}=\kappa^{2}\mathbf{C}+\mathbf{G}$, where $\mathbf{C}_{i,j}=\langle{\varphi_{i}},{\varphi_{j}}\rangle_{\Omega}$ and $\mathbf{G}_{i,j}=\langle{\nabla\varphi_{i}},{\nabla\varphi_{j}}\rangle_{\Omega}$. The right-hand side of (12) can be shown to be Gaussian with mean zero and covariance $\mathbf{C}$. For the Hilbert space approximations, it is natural to work with the canonical representation, $\mathbf{x}\sim\mathsf{N}_{C}(\mathbf{b},\mathbf{Q})$, of the Gaussian distribution. Here, the precision matrix $\mathbf{Q}$ is the inverse of the covariance matrix, and the vector $\mathbf{b}$ is connected to the mean, $\bolds{\mu}$, of the Gaussian distribution through the relation $\bolds{\mu}=\mathbf{Q}^{-1}\mathbf{b}$. Thus, if $\mathbf{K}$ is invertible, one has \[\mathbf{K}\mathbf{w}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{C}^{-1} )\quad\Longleftrightarrow\quad\mathbf{w}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{ K}\mathbf{C}^{-1}\mathbf{K}).\] For the second fundamental case, $\mathcal{L}_{1}=(\kappa^{2}-\Delta)^{1/2}$, Lindgren, Rue and Lindström (2010) show that $\mathbf{w}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{K})$. Given these two fundamental cases, the weak solution to $\mathcal{L}_{1}X_{0}(\mathbf{s})=\mathcal{W}(\mathbf{s})$, for any operator on the form (6), can be obtained recursively. If, for example, $\mathcal{L}_{1}=(\kappa^{2}-\Delta)^{2}$, the solution is obtained by solving $(\kappa^{2}-\Delta)X_{0}(\mathbf{s})=\tilde{x}(\mathbf{s})$, where $\tilde{x}$ is the weak solution to the first fundamental case. The iterative way of constructing solutions can be extended to calculate weak solutions to (8) as well. Let $\tilde{x}_{0}=\sum_{i=1}^{n}w_{i}^{0}\varphi_{i}(\mathbf{s})$ be a weak solution to $\mathcal{L}_{1}X_{0}(\mathbf{s})=\mathcal{W}(\mathbf{s})$, and let $\mathbf{Q}_{X_{0}}$ denote the precision for the weights $\mathbf{w}_{0}=(w_{1}^{0},\ldots,w_{n}^{0})^{\top}$. Substituting $X_{0}$ with $\tilde{x}_{0}$ in the second equation of (3), the weak formulation of the equation is \[[\langle{\varphi_{i}},{\tilde{x}}\rangle_{\Omega}]_{i=1,\ldots,n} \stackrel{{ D}}{{=}} [\langle{\varphi_{i}},{\mathcal{L}_{2}\tilde{x}_{0}}\rangle_{ \Omega}]_{i=1,\ldots,n}\] \[= \Biggl{[}\sum_{j=1}^{n}w_{j}^{0}\langle{\varphi_{i}},{\mathcal{L} _{2}\varphi_{j}}\rangle_{\Omega}\Biggr{]}_{i=1,\ldots,n}.\] First consider the case of an order-one operator $\mathcal{L}_{2}=b_{1}+\mathbf{B}_{1}^{\top}\nabla$. By introducing the matrix $\mathbf{H}_{1}$ with elements $\mathbf{H}_{1i,j}=\langle{\varphi_{i}},{\mathcal{L}_{2}\varphi_{j}}\rangle_{\Omega}$, the right-hand side of (3) can be written as $\mathbf{H}_{1}\mathbf{w}_{0}$. Introducing the vector $\mathbf{w}=(w_{1},\ldots,w_{n})^{\top}$, the left-hand side of (3) can be written as $\mathbf{C}\mathbf{w}$, and one has \[\mathbf{w}=\mathbf{C}^{-1}\mathbf{H}_{1}\mathbf{w}_{0}\quad \Longrightarrow\quad\mathbf{w}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{C}\mathbf{ H}_{1}^{-\top}\mathbf{Q}_{X_{0}}\mathbf{H}_{1}^{-1}\mathbf{C}).\] Now, if $\mathcal{L}_{2}$ is on the form (7), the procedure can be used recursively, in the same way as when producing higher order Matérn fields. For example, if the solution is obtained by solving $X(\mathbf{s})=(b_{2}+\mathbf{B}_{2}^{\top}\nabla)\tilde{x}(\mathbf{s})$, where $\tilde{x}$ is the weak solution to the previous example. Thus, when $\mathcal{L}_{2}$ is on the form (7), one has where each factor $\mathbf{H}_{i}$ corresponds to the $\mathbf{H}$-matrix obtained in the $i$th step in the recursion. ### Nonstationary fields As mentioned in Lindgren, Rue and Lindström (2010), the Hilbert space approximation technique can also be used for nonstationary models, and the technique extends to the nested SPDE models as well. One again begins by finding the weak solution of the first part of the system, $\mathcal{L}_{1}(\mathbf{s})X_{0}(\mathbf{s})=\mathcal{W}(\mathbf{s})$. The iterative procedure is used for obtaining approximations of high-order operators, so the fundamental step is to find the weak solution to the equation when $\mathcal{L}_{1}=(\kappa^{2}(\mathbf{s})-\Delta)$. Consider the weak formulation \[\bigl{[}\bigl{\langle}{\varphi_{i}},{\bigl{(}\kappa^{2}(\mathbf{s})-\Delta \bigr{)}\tilde{x}}\bigr{\rangle}_{\Omega}\bigr{]}_{i=1,\ldots,n}\stackrel{{ D}}{{=}}[\langle{\varphi_{i}},{\mathcal{W}}\rangle_{\Omega}]_{i=1 ,\ldots,n},\vspace{2pt}\] (14) and note that the right-hand side of the equation is the same as in the stationary case, $\mathsf{N}_{C}(\mathbf{0},\mathbf{C}^{-1})$. Now, using that \[\bigl{\langle}{\varphi_{i}},{\bigl{(}\kappa^{2}(\mathbf{s})- \Delta\bigr{)}\tilde{x}}\bigr{\rangle}_{\Omega} = \langle{\varphi_{i}},{\kappa^{2}(\mathbf{s})\tilde{x}}\rangle{}_{ \Omega}-\langle{\varphi_{i}},{\Delta\tilde{x}}\rangle_{\Omega}\] \[= \langle{\varphi_{i}},{\kappa^{2}(\mathbf{s})\tilde{x}}\rangle_{ \Omega}+\langle{\nabla\varphi_{i}},{\nabla\tilde{x}}\rangle_{\Omega},\] the left-hand side of (14) can be written as $(\tilde{\mathbf{C}}+\mathbf{G})\mathbf{w}_{0}$, where $\mathbf{G}$ and $\mathbf{w}_{0}$ are the same as in the stationary case and $\tilde{\mathbf{C}}$ is a matrix with elements \[\tilde{\mathbf{C}}_{i,j} = \langle{\varphi_{i}},{\kappa^{2}(\mathbf{s})\varphi_{j}}\rangle_{ \Omega}=\int_{\Omega}\kappa^{2}(\mathbf{s})\varphi_{i}(\mathbf{s})\varphi_{j}( \mathbf{s})\,\mathrm{d}\mathbf{s}\] \[\approx \kappa^{2}(\mathbf{s}_{j})\int_{\Omega}\varphi_{i}(\mathbf{s}) \varphi_{j}(\mathbf{s})\,\mathrm{d}\mathbf{s}=\kappa^{2}(\mathbf{s}_{j}) \mathbf{C}_{i,j}.\] Since $\{\varphi_{i}\}$ is assumed to be a local basis, such as B-spline wavelets or some other functions with compact support, the locations $\mathbf{s}_{j}$ can, for example, be chosen as the centers of the basis functions $\varphi_{j}(\mathbf{s})$. The error in the approximation of $\tilde{\mathbf{C}}$ is then small if $\kappa^{2}(\mathbf{s})$ varies slowly compared to the spacing of the basis functions $\varphi_{j}$. From equation (3.1), one has $\tilde{\mathbf{C}}~{}=~{}\mathbf{C}\bolds{\kappa}$, where $\bolds{\kappa}$ is a diagonal matrix with elements $\bolds{\kappa}_{j,j}=\kappa^{2}(\mathbf{s}_{j})$. Finally, with $\mathbf{K}=\bolds{\kappa}\mathbf{C}+\mathbf{G}$, one has \[\mathbf{K}\mathbf{w}_{0}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{C}^ {-1})\quad\Longrightarrow\quad\mathbf{w}_{0}\sim\mathsf{N}_{C}(\mathbf{0}, \mathbf{K}\mathbf{C}^{-1}\mathbf{K}).\] Now given the weak solution, $\tilde{x}_{0}$, to $\mathcal{L}_{1}(\mathbf{s})X_{0}(\mathbf{s})=\mathcal{W}(\mathbf{s})$, substitute $X_{0}$ with $\tilde{x}_{0}$ in the second equation of (4) and consider the weak formulation of the equation. Since the solution to the full operator again can be found recursively, only the fundamental case $\mathcal{L}_{2}=b(\mathbf{s})+\mathbf{B}(\mathbf{s})^{\top}\nabla$ is considered. The weak formulation is the same as (3), and one has \[\langle{\varphi_{i}},{\tilde{x}}\rangle_{\Omega} \stackrel{{ D}}{{=}} \langle{\varphi_{i}},{\mathcal{L}_{2}\tilde{x}_{0}}\rangle_{ \Omega}=\bigl{\langle}{\varphi_{i}},{\bigl{(}b(\mathbf{s})+\mathbf{B}(\mathbf{ s})^{\top}\nabla\bigr{)}\tilde{x}_{0}}\bigr{\rangle}_{\Omega}\] \[= \langle{\varphi_{i}},{b(\mathbf{s})\tilde{x}_{0}}\rangle_{\Omega} +\langle{\varphi_{i}},{\mathbf{B}(\mathbf{s})^{\top}\nabla\tilde{x}_{0}} \rangle_{\Omega}.\] Thus, the right-hand side of (3) can be written as $(\hat{\mathbf{C}}+\hat{\mathbf{H}})\mathbf{w}_{0}$, where \[\hat{\mathbf{C}}_{i,j} = \langle{\varphi_{i}},{b(\mathbf{s})\varphi_{j}}\rangle_{\Omega}= \int_{\Omega}b(\mathbf{s})\varphi_{i}(\mathbf{s})\varphi_{j}(\mathbf{s})\, \mathrm{d}\mathbf{s}\approx b(\mathbf{s}_{j})\mathbf{C}_{i,j},\] \[\hat{\mathbf{H}}_{i,j} = \langle{\varphi_{i}},{\mathbf{B}(\mathbf{s})^{\top}\nabla\varphi_ {j}}\rangle_{\Omega}=\int_{\Omega}\varphi_{i}(\mathbf{s})\mathbf{B}(\mathbf{s} )^{\top}\nabla\varphi_{j}(\mathbf{s})\,\mathrm{d}\mathbf{s}\] \[\approx \mathbf{B}(\tilde{\mathbf{s}}_{j})^{\top}\int_{\Omega}\varphi_{i} (\mathbf{s})\nabla\varphi_{j}(\mathbf{s})\,\mathrm{d}\mathbf{s}.\] Here, similar approximations as in equation (3.1) are used, so the expressions are accurate if the coefficients vary slowly compared to the spacing of the basis functions $\varphi_{j}$. The left-hand side of (3) can again be written as $\mathbf{C}\mathbf{w}$, so with $\mathbf{H}_{1}=\hat{\mathbf{C}}+\hat{\mathbf{H}}$, one has $\mathbf{w}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{C}\mathbf{H}_{1}^{-\top} \mathbf{Q}_{X_{0}}\mathbf{H}_{1}^{-1}\mathbf{C})$. ### Practical considerations The integrals that must be calculated to get explicit expressions for the matrices $\mathbf{C}$, $\mathbf{G}$ and $\mathbf{H}$ are In Section 5 a basis of piecewise linear functions induced by a triangulation of the Earth is used; see Figure 4. In this case, $\varphi_{i}(\mathbf{s})$ is a linear function on each triangle, and $\nabla\varphi_{i}(\mathbf{s})$ is constant on each triangle. The integrals, therefore, have simple analytic expressions in this case, and more generally for all piecewise linear bases induced by triangulated 2-manifolds. Bases induced by triangulations have many desirable properties, such as the simple analytic expression for the integrals and compact support. They are, however, not orthogonal, which causes $\mathbf{C}^{-1}$ to be dense. The weights $\mathbf{w}$, therefore, have a dense precision matrix, unless $\mathbf{C}^{-1}$ is approximated with some sparse matrix. This issue is addressed in Lindgren, Rue and Lindström (2010) by lowering the integration order of $\langle{\varphi_{i}},{\varphi_{j}}\rangle$, which results in an approximate, diagonal $\mathbf{C}$ matrix, $\bar{\mathbf{C}}$, with diagonal elements $\bar{\mathbf{C}}_{ii}=\sum_{k=1}^{n}\mathbf{C}_{ik}$. Bolin and Lindgren (2009) perform numerical studies on how this approximation affects the resulting covariance function of the process, and it is shown that the error is small if the approximation is used for piecewise linear bases. We will, therefore, from now on use the approximate $\mathbf{C}$ matrix in all places where $\mathbf{C}$ is used. A natural question is how many basis functions one should use in order to get a good approximation of the solution. The answer will depend on the chosen basis, and, more importantly, on the specific parameters of the SPDE model. Bolin and Lindgren (2009) study the approximation error in the Matérn case in $\mathbb{R}$ and $\mathbb{R}^{2}$ for different bases, and in this case the spacing of the basis functions compared to the range of the covariance function for $X(\mathbf{s})$ determines the approximation error: For a process with long range, fewer basis functions have to be used than for a process with short range to obtain the same approximation error. For more complicated, possibly nonstationary, nested SPDE models, there is no easy answer to how the number of basis functions should be chosen. Increasing the number of basis functions will decrease the approximation error but increase the computational complexity for the approximate model, so there is a trade-off between accuracy and computational cost. However, as long as the parameters vary slowly compared to the spacing of the basis functions, the approximation error will likely be much smaller than the error obtained from using a model that does not fit the data perfectly and from estimating the parameters from the data. Thus, for practical applications, the error in covariance induced by the Hilbert space approximation technique will likely not matter much. A more important consequence for practical applications when the piecewise linear basis is used is that the Kriging estimation of the field between two nodes in the triangulation is a linear interpolation of the values at the nodes. Thus, variations on a scale smaller than the spacing between the basis functions will not be captured correctly in the Kriging prediction. For practical applications, it is therefore often best to choose the number of basis functions depending on the scale one is interested in the Kriging prediction on. For the ozone data in Section 5, the goal is to estimate daily maps of global ozone. As we are not interested in modeling small scale variations, we choose the number of basis functions so that the mean distance between basis functions is about 258 km. For this basis, the smallest distance between two basis functions is 222 km, and the largest distance is about 342 km. Estimating the model parameters using different numbers of basis functions will give different estimates, as the parameters are estimated to maximize the likelihood for the approximate model instead of the exact SPDE. An example of this can be seen in Figure 3 where the estimates of the covariance parameters for model F’ (see Section 5 for a model description) for the ozone data are shown for varying numbers of basis functions. Instead of showing the actual parameter estimates, the figure shows the differences between the estimates and the estimate when using the basis shown in Figure 4, which has 9002 basis functions. Increasing the number of basis functions further, the estimates will finally converge to the estimates one would get using the exact SPDE representation. The curve that has not converged corresponds to the dominating parameter in the vector field. Together with $\kappa$, this parameter controls the correlation range of the ozone field. <figure><img src="content_image/1104.3436/x3.png"><figcaption>Figure 3: Parameter estimates for the covariance parameters in model F′ forthe ozone data as functions of the number of basis functions in the Hilbertspace approximations.</figcaption></figure> ## 4 Parameter estimation In this section a parameter estimation procedure for the nested SPDE models is presented. One alternative would be to use a Metropolis–Hastings algorithm, which is easy to implement, but computationally inefficient. A better alternative is to use direct numerical optimization to estimate the parameters. Let $Y(\mathbf{s})$ be an observation of the latent field, $X(\mathbf{s})$, given by (8) or (10), under mean zero Gaussian measurement noise, $\mathcal{E}(\mathbf{s})$, with variance $\sigma^{2}$: \[Y(\mathbf{s})=X(\mathbf{s})+\mathcal{E}(\mathbf{s}).\] (16) Using the approximation procedure from Section 3, and assuming a regression model for the latent field’s mean value function, $\mu(\mathbf{s})$, the measurement equation can then be written as \[\mathbf{Y}=\mathbf{M}\bolds{\mu}+\bolds{\Phi}\mathbf{w}+\bolds{ \varepsilon},\] where $\mathbf{M}$ is a matrix with the regression basis functions evaluated at the measurement locations, and $\bolds{\mu}$ is a vector containing the regression coefficients that have to be estimated. The matrix $\bolds{\Phi}$ contains the basis functions for the Hilbert space approximation procedure evaluated at the measurement locations, and $\mathbf{w}$ is the vector with the stochastic weights. In Section 3 it was shown that the vector $\mathbf{w}$ is Gaussian with mean zero and covariance matrix $\mathbf{H}\mathbf{Q}_{X_{0}}^{-1}\mathbf{H}^{\top}$. Both $\mathbf{Q}_{X_{0}}$ and $\mathbf{H}$ are sparse matrices, but neither the covariance matrix nor the precision matrix for $\mathbf{w}$ is sparse. Thus, it would seem as if one had to work with a dense covariance matrix, which would make maximum likelihood parameter estimation computationally infeasible for large data sets. However, because of the product form of the covariance matrix, one has that $\mathbf{w}=\mathbf{H}\mathbf{w}_{0}$, where $\mathbf{w}_{0}\sim\mathsf{N}_{C}(\mathbf{0},\mathbf{Q}_{X_{0}})$. Hence, the observation equation can be rewritten as \[\mathbf{Y}=\mathbf{M}\bolds{\mu}+\bolds{\Phi}\mathbf{H}\mathbf{w}_{0}+\bolds{ \varepsilon}.\] (17) Interpreting $\bolds{\Lambda}=\bolds{\Phi}\mathbf{H}$ as an observation matrix that depends on some of the parameters in the model, $\mathbf{Y}-\mathbf{M}\bolds{\mu}$ can now be seen as noisy observations of $\mathbf{w}_{0}$, which has a sparse precision matrix. The advantage with using (17) is that one then is in the setting of having observations of a latent Gaussian Markov random field, which facilitates the usage of sparse matrix techniques in the parameter estimation. Let $\bolds{\psi}$ denote all parameters in the model except for $\bolds{\mu}$. Assuming that $\bolds{\mu}$ and $\bolds{\psi}$ are a priori independent, the posterior density can be written as \[\pi(\mathbf{w}_{0},\bolds{\mu},\bolds{\psi}\|\mathbf{Y})\propto\pi (\mathbf{Y}\|\mathbf{w}_{0},\sigma^{2})\pi(\mathbf{w}_{0}\|\bolds{\mu},\bolds{ \psi})\pi(\bolds{\mu})\pi(\bolds{\psi}).\] Using a Gaussian prior distribution with mean $\bolds{\mu}$ and precision $\mathbf{Q}_{\mu}$ for the mean parameters, the posterior distribution can be reformulated as \[\pi(\mathbf{w}_{0},\bolds{\mu},\bolds{\psi}\|\mathbf{Y})\propto\pi(\mathbf{w}_{ 0}\|\bolds{\mu},\bolds{\psi},\mathbf{Y})\pi(\bolds{\mu}\|\bolds{\psi},\mathbf{Y} )\pi(\bolds{\psi}\|\mathbf{Y}),\] (18) where $\mathbf{w}_{0}\|\bolds{\mu},\bolds{\psi},\mathbf{Y}\sim\mathsf{N}_{C}(\mathbf{b },\hat{\mathbf{Q}})$, $\bolds{\mu}\|\bolds{\psi},\mathbf{Y}\sim\mathsf{N}_{C}(\mathbf{b}_{\mu},\hat{ \mathbf{Q}}_{\mu})$, and \[\mathbf{b} = \frac{1}{\sigma^{2}}\bolds{\Lambda}^{\top}(\mathbf{Y}-\mathbf{M} \bolds{\mu}),\qquad\mathbf{b}_{\mu}=\mathbf{Q}_{\mu}\mathbf{m}_{\mu}+\frac{ \mathbf{M}^{\top}\mathbf{Y}}{\sigma^{2}}-\frac{\mathbf{M}^{\top}\bolds{\Lambda }\hat{\mathbf{Q}}^{-1}\bolds{\Lambda}^{\top}\mathbf{Y}}{\sigma^{4}},\] \[\hat{\mathbf{Q}} = \mathbf{Q}_{w_{0}}+\frac{1}{\sigma^{2}}\bolds{\Lambda}^{\top} \bolds{\Lambda},\qquad\hat{\mathbf{Q}}_{\mu}=\mathbf{Q}_{\mu}+\frac{\mathbf{M} ^{\top}\mathbf{M}}{\sigma^{2}}-\frac{\mathbf{M}^{\top}\bolds{\Lambda}\hat{ \mathbf{Q}}^{-1}\bolds{\Lambda}^{\top}\mathbf{M}}{\sigma^{4}}.\] The calculations are omitted here since these expressions are calculated similarly to the posterior reformulation in Lindström and Lindgren (2008), which gives more computational details. Finally, the marginal posterior density $\pi(\bolds{\psi}\|\mathbf{Y})$ can be shown to be \[\pi(\bolds{\psi}\|\mathbf{Y})\propto\frac{\|\mathbf{Q}_{w_{0}}\|^{{1 /2}}\pi(\bolds{\psi})}{\|\hat{\mathbf{Q}}\|^{{1/2}}\|\hat{\mathbf{Q}}_{\mu}\|^{{1/ 2}}\|\sigma\mathbf{I}\|}\exp\biggl{(}\frac{1}{2\sigma^{2}}\mathbf{Y}^{\top} \biggl{(}\frac{\bolds{\Lambda}\hat{\mathbf{Q}}^{-1}\bolds{\Lambda}^{\top}}{ \sigma^{2}}-\mathbf{I}\biggr{)}\mathbf{Y}+\frac{\mathbf{b}_{\mu}^{\top}\hat{ \mathbf{Q}}_{\mu}^{-1}\mathbf{b}_{\mu}}{2}\biggr{)}.\] By rewriting the posterior as (18), it can be integrated with respect to $\mathbf{w}_{0}$ and $\bolds{\mu}$, and instead of optimizing the full posterior with respect to $\mathbf{w}_{0}$, $\bolds{\mu}$ and $\bolds{\psi}$, only the marginal posterior $\pi(\bolds{\psi}\|\mathbf{Y})$ has to be optimized with respect to $\bolds{\psi}$. This is a lower dimensional optimization problem, which substantially decreases the computational complexity. Given the optimum, $\bolds{\psi}_{\mathrm{opt}}=\operatorname{argmax}_{\bolds{\psi}}\pi(\bolds{ \psi}\|\mathbf{Y})$, $\bolds{\mu}_{\mathrm{opt}}$ is then given by $\bolds{\mu}_{\mathrm{opt}}=\hat{\mathbf{Q}}_{\mu}^{-1}\mathbf{b}_{\mu}$. In practice, the numerical optimization is carried out on $\log\pi(\bolds{\psi}\|\mathbf{Y})$. ### Estimating the parameter uncertainty There are several ways one could estimate the uncertainty in the parameter estimates obtained by the parameter estimation procedure above. The simplest estimate of the uncertainty is obtained by numerically estimating the Hessian of the marginal posterior evaluated at the estimated parameters. The diagonal elements of the inverse of the Hessian can then be seen as estimates of the variance for the parameter estimates. Another method for obtaining more reliable uncertainty estimates is to use a Metropolis–Hastings based MCMC algorithm with proposal kernel similar to the one used in Lindström and Lindgren (2008). A quite efficient algorithm is obtained by using random walk proposals for the parameters, where the correlation matrix for the proposal distribution is taken as a rescaled version of the inverse of the Hessian matrix [Gelman, Roberts and Gilks (1996)]. Finally, a third method for estimating the uncertainties is to use the INLA framework [Rue, Martino and Chopin (2009)], available as an R package (http://www.r-inla.org/). In settings with latent Gaussian Markov random fields, integrated nested Laplace approximations (INLA) provide close approximations to posterior densities for a fraction of the cost of MCMC. For models with Gaussian data, the calculated densities are for practical purposes exact. In the current implementation of the INLA package, handling the full nested SPDE structure is cumbersome, so further enhancements are needed before one can take full advantage of the INLA method for these models. ### Computational complexity In this section some details on the computational complexity for the parameter estimation and Kriging estimation are given. The most widely used method for spatial prediction is linear Kriging. In the Bayesian setting, the Kriging predictor simply is the posterior expectation of the latent field $X$ given data and the estimated parameters. This expectation can be written as \[\mathsf{E}(X\|\bolds{\psi},\bolds{\mu},\mathbf{Y})=\mathbf{M} \bolds{\mu}+\bolds{\Phi}\mathbf{H}\mathsf{E}(\mathbf{w}_{0})=\mathbf{M}\bolds{ \mu}+\bolds{\Phi}\mathbf{H}\hat{\mathbf{Q}}^{-1}\mathbf{b}.\] The computationally demanding part of this expression is to calculate $\hat{\mathbf{Q}}^{-1}\mathbf{b}$. Since the $n\times n$ matrix $\mathbf{Q}$ is positive definite, this is most efficiently done using Cholesky factorization, forward substitution and back substitution: Calculate the Cholesky triangle $\mathbf{L}$ such that $\hat{\mathbf{Q}}=\mathbf{L}\mathbf{L}^{\top}$, and given $\mathbf{L}$, solve the linear system $\mathbf{L}\mathbf{x}=\mathbf{b}$. Finally, given $\mathbf{x}$, solve $\mathbf{L}^{\top}\mathbf{y}=\mathbf{x}$, where now $\mathbf{y}$ satisfies $\mathbf{y}=\hat{\mathbf{Q}}^{-1}\mathbf{b}$. Solving the forward substitution and back substitution are much less computationally demanding than calculating the Cholesky triangle. Hence, the computational cost for calculating the Kriging prediction is determined by the cost for calculating $\mathbf{L}$. The computational complexity for the parameter estimation is determined by the optimization method that is used and the computational complexity for evaluating the marginal log-posterior $\log\pi(\bolds{\psi}\|\mathbf{Y})$. The most computationally demanding terms in $\log\pi(\bolds{\psi}\|\mathbf{Y})$ are the two log-determinants $\log\|\mathbf{Q}_{w_{0}}\|$ and $\log\|\hat{\mathbf{Q}}\|$ and the quadratic form $\mathbf{Y}^{\top}\bolds{\Lambda}\hat{\mathbf{Q}}^{-1}\bolds{\Lambda}\mathbf{Y}$, which are also most efficiently calculated using Cholesky factorization. Given the Cholesky triangle $\mathbf{L}$, the quadratic form can be obtained as $\mathbf{x}^{\top}\mathbf{x}$, where $\mathbf{x}$ is the solution to $\mathbf{L}\mathbf{x}=\bolds{\Lambda}\mathbf{Y}$, and the log-determinant $\log\|\hat{\mathbf{Q}}\|$ is simply the sum²$2\sum_{i=1}^{n}\log\mathbf{L}_{ii}$. Thus, the computational cost for one evaluation of the marginal posterior is also determined by the cost for calculating $\mathbf{L}$. Because of the sparsity structure of $\hat{\mathbf{Q}}$, this computational cost is $\mathcal{O}(n)$, $\mathcal{O}(n^{3/2})$ and $\mathcal{O}(n^{2})$ for problems in one, two and three dimensions respectively [see Rue and Held (2005) for more details]. [FOOTNOTE:2][ENDFOOTNOTE] The computational complexity for the parameter estimation is highly dependent on the optimization method. If a Broyden–Fletcher–Goldfarb-Shanno (BFGS) procedure is used without an analytic expression for the gradients, the marginal posterior has to be evaluated $p$ times for each step in the optimization, where $p$ is the number of covariance parameters in the model. Thus, if $p$ is large and the initial value for the optimization is chosen far from the optimal value, many thousand evaluations of the marginal posterior may be needed in the optimization. ## 5 Application: Ozone data On October 24, 1978, NASA launched the near-polar, Sun-synchronous orbiting satellite Nimbus-7. The satellite carried a TOMS instrument with the purpose of obtaining high-resolution global maps of atmospheric ozone [McPeters et al. (1996)]. The instrument measured backscattered solar ultraviolet radiation at 35 sample points along a line perpendicular to the orbital plane at 3-degree intervals from 51 degrees on the right side of spacecraft to 51 degrees on the left. A new scan was started every eight seconds, and as the measurements required sunlight, the measurements were made during the sunlit portions of the orbit as the spacecraft moved from south to north. The data measured by the satellite has been calibrated and preprocessed into a “Level 2” data set of spatially and temporally irregular Total Column Ozone (TCO) measurements following the satellite orbit. There is also a daily “Level 3” data set with values processed into a regular latitude-longitude grid. Both Level 2 and Level 3 data have been analyzed in recent papers in the statistical literature [Cressie and Johannesson (2008), Jun and Stein (2008), Stein (2007)]. In what follows, the nested SPDE models are used to obtain statistical estimates of a daily ozone map using a part of the Level 2 data. In particular, all data available for October 1st, 1988 is used, which is the same data set that was used by Cressie and Johannesson (2008). ### Statistical model The measurement model (16) is used for the ozone data. That is, the measurements, $Y(\mathbf{s})$, are assumed to be observations of a latent field of TCO ozone, $X(\mathbf{s})$, under Gaussian measurement noise $\mathcal{E}(\mathbf{s})$ with a constant variance $\sigma^{2}$. We let $X(\mathbf{s})$ have some mean value function, $\mu(\mathbf{s})$, and let the covariance structure be determined by a nested SPDE model. Inspired by Jun and Stein (2008), who proposed using differentiated Matérn fields for modeling TCO ozone, we use the simplest nested SPDE model. Thus, $Z(\mathbf{s})=X(\mathbf{s})-\mu(\mathbf{s})$ is generated by the system \[\bigl{(}\kappa^{2}(\mathbf{s})-\Delta\bigr{)}Z_{0}(\mathbf{s}) = \mathcal{W}(\mathbf{s})\] \[Z(\mathbf{s}) = \bigl{(}b(\mathbf{s})+\mathbf{B}(\mathbf{s})^{\top}\nabla\bigr{)} Z_{0}(\mathbf{s}),\] where $\mathcal{W}(\mathbf{s})$ is Gaussian white noise on the sphere. If $\kappa(\mathbf{s})$ is assumed to be constant, the ozone is modeled as a Gaussian field with a covariance structure that is obtained by applying the differential operator $(b(\mathbf{s})+\mathbf{B}(\mathbf{s})^{\top}\nabla)$ to a stationary Matérn field, which is similar to the model by Jun and Stein (2008). If, on the other hand, $\kappa$ is spatially varying, the range of the Matérn-like covariance function can vary with location. As in Stein (2007) and Jun and Stein (2008), the mean can be modeled using a regression basis of spherical harmonics; however, since the data set only contains measurements from one specific day, it is not possible to identify which part of the variation in the data that comes from a varying mean and which part that can be explained by the variance–covariance structure of the latent field. To avoid this identifiability problem, $\mu(\mathbf{s})$ is assumed to be unknown but constant. The parameter $\kappa^{2}(\mathbf{s})$ has to be positive, and for identifiability reasons, we also require $b(\mathbf{s})$ to be positive. We, therefore, let $\log\kappa^{2}(\mathbf{s})=\sum_{k,m}\kappa_{k,m}Y_{k,m}(\mathbf{s})$ and $\log b(\mathbf{s})=\sum_{k,m}b_{k,m}Y_{k,m}(\mathbf{s})$, where $Y_{k,m}$ is the spherical harmonic of order $k$ and mode $m$. Finally, the vector field $\mathbf{B}(\mathbf{s})$ is modeled using the vector spherical harmonics basis functions $\Upsilon_{k,m}^{1}$ and $\Upsilon_{k,m}^{2}$, presented in Appendix: \[\mathbf{B}(\mathbf{s})=\sum_{k,m}\bigl{(}B_{k,m}^{1}\Upsilon_{k,m }^{1}(\mathbf{s})+B_{k,m}^{2}\Upsilon_{k,m}^{2}(\mathbf{s})\bigr{)}.\] \| A \| B \| C \| D \| E \| F \| G \| H \| I \| J \| K \| L \| M ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- κ2(s) \| 0 \| 1 \| 00 \| 01 \| 02 \| 00 \| 03 \| 02 \| 00 \| 04 \| 03 \| 00 \| 04 b(s) \| 0 \| 1 \| 01 \| 01 \| 02 \| 02 \| 03 \| 02 \| 03 \| 04 \| 03 \| 04 \| 04 B(s) \| 0 \| 0 \| 01 \| 01 \| 00 \| 02 \| 00 \| 02 \| 03 \| 00 \| 03 \| 04 \| 04 Total \| 2 \| 8 \| 11 \| 14 \| 18 \| 26 \| 32 \| 34 \| 47 \| 50 \| 62 \| 75 \| 98 \tabnotetext []Notes: The actual number of basis functions for κ2(s) and b(s) are given by (ord+1)2, and for B(s), the actual number is 2(ord+1)2−2, where ord is the maximal order indicated in the table. Table 1: Maximal orders of the spherical harmonics used in the bases for the different parameters and total number of covariance parameters in the different models for X(s) To choose the number of basis functions for the parameters $\kappa^{2}(\mathbf{s})$, $b(\mathbf{s})$ and $\mathbf{B}(\mathbf{s})$, some model selection technique has to be used. Model selection for this model class is difficult since the models can have both nonstationary mean value functions and nonstationary covariance structures. This makes standard variogram techniques inadequate in general, and we instead base the model selection on Akaike’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC) [Hastie, Tibshirani and Friedman (2001)], which are suitable model selection tools for the nested SPDE models since the likelihood for the data can be evaluated efficiently. We estimate 13 models with different numbers of covariance parameters, presented in Table 1. The simplest model is a stationary Matérn model, with four parameters to estimate, and the most complicated model has $100$ parameters to estimate, including the mean and the measurement noise variance. There are three different types of models in Table 1: In the first type (models B, E, G and J), $\kappa^{2}$ and $b$ are spatially varying and the vector field $\mathbf{B}$ is assumed to be zero. In the second type (models C, F, I and L), $b$ and $\mathbf{B}$ are spatially varying and $\kappa^{2}$ is assumed to be constant. Finally, in the third type (model D, H, K and M), all parameters are spatially varying. A basis of 9002 piecewise linear functions induced by a triangulation of the Earth (see Figure 4) is used in the approximation procedure from Section 3 to get efficient representations of each model, and the parameters are estimated using the procedure from Section 4. The computational cost for the parameter estimation only depends on the number of basis functions in the Hilbert space approximation, and not on the number of data points, which makes inference efficient even for this large data set. <figure><img src="content_image/1104.3436/x4.png"><figcaption>Figure 4: The left part shows the triangulation of the Earth used to definethe piecewise linear basis functions in the Hilbert space approximation forozone data. Each basis function is one at a node in the triangulation, anddecreases linearly to zero at the neighboring nodes. The right part of thefigure shows one of these functions.</figcaption></figure> <figure><img src="content_image/1104.3436/x5.png"><figcaption>Figure 5: AIC (squares) and BIC (circles) for the models A–M (solid lines) andthe axially symmetric models A′–M′ (dashed lines), scaled by a factor 10−5.Note that the major improvement in AIC and BIC occurs when the orders of thebasis functions are increased from one to two, and that the model type withspatially varying b and B seems to be most appropriate for this data. Alsonote that the axially symmetric model F′ is surprisingly good considering thatit only has 8 covariance parameters.</figcaption></figure> AIC and BIC for each of the fitted models can be seen in Figure 5. The figure contains one panel for each of the three model types and one panel where AIC and BIC are shown for all models at once. The major improvement in AIC and BIC occurs when the orders of the basis functions are increased from one to two. For the first model type, with spatially varying $\kappa^{2}$ and $b$, the figure indicates that the results could be improved by increasing the orders of the basis functions further. However, for a given order of the basis functions, the other two model types have much lower AIC and BIC. Also, by comparing AIC and BIC for the second and third model types, one finds that there is not much gain in letting $\kappa^{2}$ be spatially varying. We therefore conclude that a model with spatially varying $b$ and $\mathbf{B}$ is most appropriate for this data. <figure><img src="content_image/1104.3436/x6.png"><figcaption>Figure 6: Estimated variance-scaling parameter, b(s), and the the norm of thevectors in the estimated vector field B(s) for model F. Note that theestimates are fairly constant with respect to longitude, which indicates thatthe latent field could be axially symmetric.</figcaption></figure> The estimated parameters $b(\mathbf{s})$ and the length of the vectors $\mathbf{B}(\mathbf{s})$ for model F are shown in Figure 6. One thing to note in this figure is that the two parameters are fairly constant with respect to longitude, which indicates that the latent field could be axially symmetric, an assumption that was made by both Stein (2007) and Jun and Stein (2008). If the latent field indeed was axially symmetric, one would only need the basis functions that are constant with respect to longitude in the parameter bases. Since there is only one axially symmetric spherical harmonic for each order, this assumption drastically reduces the number of parameters for the models in Table 1. Let A${}^{\prime}$–M${}^{\prime}$ denote the axially symmetric versions of models A–M. For these models, the number of basis functions for both $\kappa^{2}(\mathbf{s})$ and $b(\mathbf{s})$ is $\mathit{ord}+1$, and the number of basis functions for $\mathbf{B}(\mathbf{s})$ is $2(\mathit{ord}+1)-2$, where $\mathit{ord}$ is the maximal order indicated in Table 1. The dashed lines in Figure 5 show AIC and BIC calculated for these models. Among the axially symmetric models, model F${}^{\prime}$ is surprisingly good considering that it only has $8$ covariance parameters. <figure><img src="content_image/1104.3436/x7.png"><figcaption>Figure 7: Kriging estimate of TCO ozone in Dobson units using model F′.</figcaption></figure> <figure><img src="content_image/1104.3436/x8.png"><figcaption>Figure 8: Standard error in Dobson units for the Kriging estimate. The colorbar in the left part of the figure has been truncated at 6 Dobson units. Thebehavior near the north pole can be seen in the right part of the figure.</figcaption></figure> The Kriging estimate and its standard error for model F${}^{\prime}$ are shown in Figures 7 and 8 respectively. The oscillating behavior near the equator for the standard error is explained by the fact that the satellite tracks are furthest apart there, which results in sparser measurements between the different tracks. Because the measurements are collected using backscattered sunlight, the variance close to the north pole is high, as there are no measurements there. As seen in Figure 9, there is not much spatial correlation in the residuals $\hat{\mathbf{X}}-\mathbf{Y}$, which indicates a good model fit. In Figure 10, estimates of the local mean and variance of the residuals are shown. The mean is fairly constant across the globe, but there is a slight tendency for higher variance closer to the poles. This is due to the fact that the data really is space–time data, as the measurements are collected during a 24-hour period. Since the different satellite tracks are closest near the poles, the temporal variation of the data is most prominent here, and especially near the international date line where data is collected both at the first satellite track of the day and at the last track, 24 hours later. The area with high residual variance is one of those places where measurements are taken both at the beginning and the end of the time period, and where the ozone concentration has changed during the time period between the measurements. One could include this effect by allowing the variance of the measurement noise to be spatially varying; however, one should really use a spatio-temporal model for the data to correctly account for the effect, which is outside the scope of this article. \| \boldsκ \| \boldsσ \| \boldsb1 \| \boldsb2 \| \boldsb3 \| \boldsB1 \| \boldsB2 \| \boldsB3 \| \boldsB4 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Yf \| 0.74 \| 25.60 \| 5.85 \| 0.045 \| 0.34 \| 1.05 \| 2.59 \| −6.84 \| −0.84 Yl \| 0.73 \| 25.56 \| 5.82 \| 0.033 \| 0.34 \| 0.90 \| 2.38 \| −7.01 \| −0.82 Y \| 0.67 \| 34.09 \| 5.75 \| 0.054 \| 0.36 \| 0.70 \| 2.48 \| −7.10 \| −0.68 Table 2: Estimates of the covariance parameters in model F′ using all data but the first track (Yf), all data but the last track (Yl), and all data (Y) <figure><img src="content_image/1104.3436/x9.png"><figcaption>Figure 9: Estimated covariance function for the Kriging residuals using modelF′.</figcaption></figure> <figure><img src="content_image/1104.3436/x10.png"><figcaption>Figure 10: Estimates of the local mean (left) and standard deviation (right)for the Kriging residuals using model F′. The mean is fairly constant acrossthe globe, whereas the standard deviation is higher close to the poles and atthe international date line because of the temporal structure in the data.</figcaption></figure> <figure><img src="content_image/1104.3436/x11.png"><figcaption>Figure 11: The ratio between the kriging estimates using model F′ and model M(left), and the ratio between the corresponding kriging standard errors(right). Note that there is not much difference between the Kriging estimates,whereas there is a clear difference between the corresponding standard errors.</figcaption></figure> To see how much the temporal structure near the international date line influences the model fit, the parameters in model F${}^{\prime}$ are re-estimated without using the first satellite track of the day and without using the last track of the day. The estimated parameters can be seen in Table 2 and, as expected, the estimate of the measurement noise variance is much lower when not using all date line data. The estimates of the covariance parameters for the latent field also change somewhat, but the large scale structure of the nonstationarity is preserved. To study how sensitive the Kriging estimates are to the model choice, the ratio between the Kriging estimates for the simple model F${}^{\prime}$ and the large model M, and the ratio between the corresponding Kriging standard errors, are shown in Figure 11. There is not much difference between the two Kriging estimates, whereas there is a clear difference between the corresponding standard errors. Thus, if one only is interested in the Kriging estimate, it does not matter much which model is used, but if one also is interested in the standard error of the estimate, the model choice greatly influences the results. ### Discussion Before the nested SPDE models were used on the ozone data, several tests were performed on simulated data to verify that the model parameters in fact could be estimated using the estimation procedure in Section 4. These tests showed that the estimation procedure is robust given that the initial values for the parameters are not chosen too far from the true values. However, for nonstationary models with many covariance parameters, it is not easy to choose the initial values. To reduce this problem, the optimization is done in several steps. A stationary Matérn model (model A) is estimated to get initial values for $\kappa_{0,0}$, $b_{0,0}$ and $\sigma^{2}$. To estimate model B, all parameters are set to zero initially, except for the parameters that were estimated in model A. Another layer of spherical harmonics is added to the bases for $\kappa^{2}(\mathbf{s})$ and $b(\mathbf{s})$ for estimating model E using the model B parameters as initial values. This step-wise procedure of adding layers of spherical harmonics to the bases is then repeated to estimate the larger models. Numerical studies showed that this optimization procedure is quite robust even for large models; however, as in most other numerical optimization problems, there are no guarantees that the true optimal values have been found for all models for the ozone data. The application of the nested SPDE models to ozone data was inspired by Jun and Stein (2008), who proposed using differentiated Matérn fields for modeling TCO ozone, and we conclude this section with some remarks on the similarities and differences between the nested SPDEs and their models. The most general model in Jun and Stein (2008) is on the form \[X(\mathbf{s}) = P_{1}(l_{2})X_{0}(\mathbf{s})+\biggl{(}P_{2}(l_{2})\frac{ \partial}{\partial l_{2}}+P_{3}(l_{2})\frac{\partial}{\partial l_{1}}\biggr{)} X_{1}(\mathbf{s})\] \[{}+P_{4}(l_{2})\frac{\partial}{\partial l_{1}}X_{2}(\mathbf{s}),\] where $X_{i},i=0,1,2$, are i.i.d. Matérn fields in $\mathbb{R}^{3}$, $P_{i},i=1,2,3,4$, are nonrandom functions depending on latitude, $l_{1}$ denoted longitude and $l_{2}$ denoted latitude. This model is similar to the model used here, but there are some important differences. First of all, (5.2) contains a sum of three independent fields, which we cannot represent since the approximation procedure in Section 3 in this case loses its computational benefits. To get a model more similar to the nested SPDE model, one would have to let $P_{4}(l_{2})\equiv 0$, and $X_{0}(\mathbf{s})=X_{1}(\mathbf{s})$. Using $X_{0}=X_{1}$ or $X_{0}$ and $X_{1}$ as i.i.d. copies of a Matérn field gives different covariance functions, and without testing both cases it is hard to determine what is more appropriate for ozone data. Another important conceptual difference is how the methods deal with the spherical topology. The Matérn fields in Jun and Stein (2008) are stochastic fields on $\mathbb{R}^{3}$, evaluated on the embedded sphere, which is equivalent to using chordal distance as the metric in a regular Matérn covariance function. One might instead attempt to evaluate the covariance function using the arc-length distance, which is a more natural metric on the sphere. However, Theorem 2 from Gneiting (1998) shows that for Matérn covariances with $\nu\geq 1$, this procedure does not generate positive definite covariance functions. This means that the arc-length method cannot be used for any differentiable Matérn fields. On the other hand, the nested SPDEs are directly defined on the sphere, and therefore inherently use the arc-length distance. There is, in theory, no difference between writing the directional derivative of $X(\mathbf{s})$ as $(P_{2}(l_{2})\frac{\partial}{\partial l_{2}}+P_{3}(l_{2})\frac{\partial}{ \partial l_{1}})X_{1}(\mathbf{s})$ or $\mathbf{B}(\mathbf{s})^{\top}\nabla X(\mathbf{s})$, but the latter is easier to work with in practice. If a vector field basis is used to model $\mathbf{B}(s)$, the process will not have any singularities as long as the basis functions are nonsingular, which is the case for the basis used in this paper. If, on the other hand, $P_{2}(l_{2})$ and $P_{3}(l_{2})$ are modeled separately, the process will be singular at the poles unless certain restrictions on the two functions are met. This fact is indeed noted by Jun and Stein (2008), but the authors do not seem to take the restrictions into account in the parameter estimation, which causes all their estimated models to have singularities at the poles. Finally, the nested SPDE models are computationally efficient also for spatially irregular data, which allowed us to work with the TOMS Level 2 data instead of the gridded Level 3 data. ## 6 Concluding remarks There is a need for computationally efficient stochastic models for environmental data. Lindgren, Rue and Lindström (2010) introduced an efficient procedure for obtaining Markov approximations of, possibly nonstationary, Matérn fields by considering Hilbert space approximations of the SPDE \[\bigl{(}\kappa(\mathbf{s})^{2}-\Delta\bigr{)}^{{\alpha/2}}X( \mathbf{s})=\phi(\mathbf{s})\mathcal{W}(\mathbf{s}).\] In this work, the class of nonstationary nested SPDE models generated by (10) was introduced, and it was shown how the approximation methods in Lindgren, Rue and Lindström (2010) can be extended to this larger class of models. This model class contains a wider family of covariance models, including both Matérn-like covariance functions and various oscillating covariance functions. Because of the additional differential operator $\mathcal{L}_{2}$, the Hilbert space approximations for the nested SPDE models do not have the Markov structure the model in Lindgren, Rue and Lindström (2010) has, but all computational benefits from the Markov properties are preserved for the nested SPDE models using the procedure in Section 4. This allows us to fit complicated models with over 100 parameters to data sets with several hundred thousand measurements using only a standard personal computer. By choosing $\mathcal{L}_{2}=b+\mathbf{B}^{\top}\nabla$, one obtains a model similar to what Jun and Stein (2008) used to analyze TOMS Level 3 ozone data, and we used this restricted nested SPDE model to analyze the global spatially irregular TOMS Level 2 data. This application illustrates the ability to use the model class to produce nonstationary covariance models on general smooth manifolds which efficiently can be used to study large spatially irregular data sets. The most important next step in this work is to make a spatio-temporal extension of the model class. This would allow us to produce not only spatial but also spatio-temporal ozone models and increase the applicability of the model class to other environmental modeling problems where time dependence is a necessary model component. ## Appendix: Vector spherical harmonics When using the nonstationary model (10) in practice, we assume that the parameters in the model can be expressed in terms of some basis functions. If working on the sphere, spherical harmonics is a convenient basis for the parameters taking values in $\mathbb{R}$. On real form, the spherical harmonic $Y_{k,m}(\mathbf{s})$ of order $k\in\mathbb{N}_{0}$ and mode $m=-k,\ldots,k$ is defined as \[\ Y_{k,m}(\mathbf{s})=\sqrt{\frac{2k+1}{4\pi}\cdot\frac{(k-\|m\|)!} {(k+\|m\|)!}}\cdot\cases{\sqrt{2}\sin(ml_{1})P_{k,-m}(\sin l_{2}),&\quad$-k\leq m <0,$\cr P_{k,0}(\sin l_{2}),&\quad$m=0,$\cr\sqrt{2}\cos(ml_{1})P_{k,m}(\sin l_ {2}),&\quad$0<m\leq k,$}\] where $l_{2}$ is the latitude, $l_{1}$ is the longitude, and $P_{k,m}(\cdot)$ are associated Legendre functions. We, however, also need a basis for the vector fields $\mathbf{B}_{i}(\mathbf{s})$, determining the direction and magnitude of differentiation. Since the vector fields in each point on the sphere must lie in the tangent space of $\mathbb{S}^{2}$, the basis functions also must satisfy this. A basis with this property is obtained by using a subset of the vector spherical harmonics [Hill (1954)]. For each spherical harmonic $Y_{k,m}(\mathbf{s})$, $k>0$, define the two vector spherical harmonics \[\Upsilon_{k,m}^{1}(\mathbf{s}) = \nabla_{\mathbb{S}^{2}}Y_{k,m}(\mathbf{s}),\] \[\Upsilon_{k,m}^{2}(\mathbf{s}) = \nabla_{\mathbb{S}^{2}}Y_{k,m(\mathbf{s})}\times\mathbf{s}.\] Here $\times$ denotes the cross product in $\mathbb{R}^{3}$ and $\nabla_{\mathbb{S}^{2}}$ is the gradient on $\mathbb{S}^{2}$. By defining the basis in this way, all basis functions in $\Upsilon^{1}=\{\Upsilon_{k,m}^{1}\}$ and $\Upsilon^{2}=\{\Upsilon_{k,m}^{2}\}$ will obviously lie in the tangent space of $\mathbb{S}^{2}$. It is also easy to see that the basis is orthogonal in the sense that for any $k,l>0$, $-k\leq m\leq k$, and $-l\leq n\leq l$, one has \[\langle{\Upsilon_{k,m}^{1}},{\Upsilon_{l,n}^{2}}\rangle_{\mathbb{ S}^{2}} = 0,\] \[\langle{\Upsilon_{k,m}^{1}},{\Upsilon_{l,n}^{1}}\rangle_{\mathbb{ S}^{2}} = k(k+1)\delta_{k-l}\delta_{m-n},\] \[\langle{\Upsilon_{k,m}^{2}},{\Upsilon_{l,n}^{2}}\rangle_{\mathbb{ S}^{2}} = k(k+1)\delta_{k-l}\delta_{m-n}.\] These are indeed desirable properties for a vector field basis, but for the basis to be of any use in practice, a method for calculating the basis functions explicitly is needed. Such explicit expressions are given in the following proposition. Proposition .1: _With $\mathbf{s}=(x,y,z)^{\top}$, $\Upsilon_{k,m}^{1}(\mathbf{s})$ and $\Upsilon_{k,m}^{2}(\mathbf{s})$ can be written as_ \[\Upsilon_{k,m}^{1}(\mathbf{s}) = \frac{1}{1-z^{2}}\left[\matrix{-myY_{k,-m}(\mathbf{s})-c_{k,m}xzY _{k-1,m}(\mathbf{s})+kxz^{2}Y_{k,m}(\mathbf{s})\cr mxY_{k,-m}(\mathbf{s})-c_{k ,m}yzY_{k-1,m}(\mathbf{s})+kyz^{2}Y_{k,m}(\mathbf{s})\cr c_{k,m}(1-z^{2})Y_{k- 1,m}(\mathbf{s})-(1-z^{2})kzY_{k,m}(\mathbf{s})}\right],\] \[\Upsilon_{k,m}^{2}(\mathbf{s}) = \frac{1}{1-z^{2}}\left[\matrix{kzyY_{k,m}(\mathbf{s})-c_{k,m}yY_{ k-1,m}(\mathbf{s})+mzxY_{k,-m}(\mathbf{s})\cr-kxzY_{k,m}(\mathbf{s})+c_{k,m}xY _{k-1,m}(\mathbf{s})+myzY_{k,-m}(\mathbf{s})\cr-m(1-z^{2})Y_{k,-m}(\mathbf{s}) }\right],\] _where_ \[c_{k,m}=\sqrt{\frac{(2k+1)(k^{2}-\|m\|^{2})}{2k-1}}.\] One has that $\nabla_{\mathbb{S}^{2}}Y_{k,m}=P_{\mathbb{S}^{2}}(\nabla_{\mathbb{R}^{3}}Y_{k, m})$, that is, the gradient on $\mathbb{S}^{2}$ can be obtained by first calculating the gradient in $\mathbb{R}^{3}$ and then projecting the result onto $\mathbb{S}^{2}$. If $c_{k}^{m}$ denotes the normalization constant for the spherical harmonic $Y_{k,m}(\mathbf{s})$, and the recursive relation \[(1-z^{2})\frac{\partial}{\partial z}P_{k,m}(z)=kzP_{k,m}(z)-(k+m) P_{k-1,m}(z)\] is used, one has that \[\frac{\partial}{\partial z}Y_{k,m}(\mathbf{s})=\frac{1}{1-z^{2}} \biggl{(}kzY_{k,m}(\mathbf{s})-(k+\|m\|)\frac{c_{k}^{m}}{c_{k-1}^{m}}Y_{k-1,m}( \mathbf{s})\biggr{)}.\] Now, using that $\tan(l_{1})=x^{-1}y$, one has \[\frac{\partial l_{1}}{\partial x} = -\cos^{2}(l_{1})\frac{y}{x^{2}}=-\frac{y}{1-z^{2}},\] \[\frac{\partial l_{1}}{\partial y} = \cos^{2}(l_{1})\frac{1}{x}=\frac{x}{1-z^{2}},\] where the last equalities hold on $\mathbb{S}^{2}$. Using these relations gives \[\frac{\partial}{\partial x}Y_{k,m}(\mathbf{s})=-\frac{my}{1-z^{2} }Y_{k,-m}(\mathbf{s}),\qquad\frac{\partial}{\partial y}Y_{k,m}(\mathbf{s})= \frac{mx}{1-z^{2}}Y_{k,-m}(\mathbf{s}).\] Thus, with \[c_{k,m}\mathrel{\triangleq}(k+\|m\|)\frac{c_{k}^{m}}{c_{k-1}^{m}}= \sqrt{\frac{(2k+1)(k^{2}-\|m\|^{2})}{2k-1}},\] one has that Finally, the desired result is obtained by calculating \[\Upsilon_{k,m}^{1} = \nabla_{\mathbb{S}^{2}}Y_{k,m}=\mathbf{P}_{\mathbb{S}^{2}}\nabla_ {\mathbb{R}^{3}}Y_{k,m},\] \[\Upsilon_{k,m}^{2} = \Upsilon_{k,m}^{1}\times\mathbf{s}=\mathbf{S}_{\times}\Upsilon_{k ,m}^{1},\] where ## References * Adler (1981)[author] Adler, R. J.R. J. (1981). The Geometry of Random Fields. Wiley, New York. 0611857 * Bolin and Lindgren (2009)[author] Bolin, D.D. Lindgren, F.F. (2009). Wavelet Markov approximations as efficient alternatives to tapering and convolution fields (submitted). Preprints in Math. Sci. 2009:13, Lund Univ. * Cressie and Johannesson (2008)[author] Cressie, NoelN. Johannesson, GardarG. (2008). Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 209–226. 2412639 * Gelman, Roberts and Gilks (1996)[author] Gelman, A.A., Roberts, G. O.G. O. Gilks, W. R.W. R. (1996). Efficient Metropolis jumping rules. Bayesian Stat. 5 599–607. 1425429 * Gneiting (1998)[author] Gneiting, TT. (1998). Simple tests for the validity of correlation function models on the circle. Statist. Probab. Lett. 39 119–122. 1652540 * Hastie, Tibshirani and Friedman (2001)[author] Hastie, T.T., Tibshirani, R.R. Friedman, J. H.J. H. (2001). The Elements of Statistical Learning. Springer, New York. 1851606 * Hill (1954)[author] Hill, E. L.E. L. (1954). The theory of vector spherical harmonics. Amer. J. Phys. 22 211–214. 0061226 * Jun and Stein (2007)[author] Jun, MikyoungM. Stein, Michael L.M. L. (2007). An approach to producing space–time covariance functions on spheres. Technometrics 49 468–479. 2394558 * Jun and Stein (2008)[author] Jun, MikyoungM. Stein, Michael L.M. L. (2008). Nonstationary covariance models for global data. Ann. Appl. Statist. 2 1271–1289. * Lindgren, Rue and Lindström (2010)[author] Lindgren, FinnF., Rue, HávardH. Lindström, JohanJ. (2010). An explicit link between Gaussian fields and Gaussian Markov random fields, The SPDE approach (submitted). Preprints in Math. Sci. 2010:3, Lund Univ. * Lindström and Lindgren (2008)[author] Lindström, JohanJ. Lindgren, FinnF. (2008). A Gaussian Markov random field model for total yearly precipitation over the African Sahel. Preprints in Math. Sci. 2008:8, Lund Univ. * Matérn (1960)[author] Matérn, B.B. (1960). Spatial Variation. Meddelanden från statens skogsforskningsinstitut, Stockholm. * McPeters et al. (1996)[author] McPeters, R. D.R. D., Bhartia, P. K.P. K., Krueger, A. J.A. J., Herman, J. R.J. R., Schlesinger, B.B., Wellemeyer, C. G.C. G., Seftor, C. J.C. J., Jaross, G.G., Taylor, S. L.S. L., Swissler, T.T., Torres, O.O., Labow, G.G., Byerly, W.W. Cebula, R. P.R. P. (1996). Nimbus-7 Total Ozone Mapping Spectrometer (TOMS) data products user’s guide. NASA Reference Publication 1384. * Rue and Held (2005)[author] Rue, H.H. Held, L.L. (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC, Boca Raton, FL. 2130347 * Rue, Martino and Chopin (2009)[author] Rue, HH., Martino, SS. Chopin, NN. (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested laplace approximations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 71 319–392. * Stein (2007)[author] Stein, Michael L.M. L. (2007). Spatial variation of total column ozone on a global scale. Ann. Appl. Statist. 1 191–210. 2393847 * Thomas and Finney (1995)[author] Thomas, G. B.G. B. Finney, R. L.R. L. (1995). Calculus and Analytic Geometry, 9 ed. Addison Wesley, New York. * Whittle (1963)[author] Whittle, P.P. (1963). Stochastic processes in several dimensions. Bull. Inst. Internat. Statist. 40 974–994. 0173287 * Yaglom (1987)[author] Yaglom, A. M.A. M. (1987). Correlation Theory of Stationary and Related Random Functions 1. Springer, New York.
1804.03209	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35720, "num_imgs": 0, "llama3_tokens_count": 7585 }	[]	# Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition Pete Warden Google Brain Mountain View, California petewarden@google.com April 2018 ## 1 Abstract Describes an audio dataset[1] of spoken words designed to help train and evaluate keyword spotting systems. Discusses why this task is an interesting challenge, and why it requires a specialized dataset that’s different from conventional datasets used for automatic speech recognition of full sentences. Suggests a methodology for reproducible and comparable accuracy metrics for this task. Describes how the data was collected and verified, what it contains, previous versions[2] and properties. Concludes by reporting baseline results of models trained on this dataset. ## 2 Introduction Speech recognition research has traditionally required the resources of large organizations such as universities or corporations to pursue. People working in those organizations usually have free access to either academic datasets through agreements with groups like the Linguistic Data Consortium[3], or to proprietary commercial data. As speech technology has matured, the number of people who want to train and evaluate recognition models has grown beyond these traditional groups, but the availability of datasets hasn’t widened. As the example of ImageNet[4] and similar collections in computer vision has shown, broadening access to datasets encourages collaborations across groups and enables apples-for-apples comparisons between different approaches, helping the whole field move forward. The Speech Commands dataset is an attempt to build a standard training and evaluation dataset for a class of simple speech recognition tasks. Its primary goal is to provide a way to build and test small models that detect when a single word is spoken, from a set of ten or fewer target words, with as few false positives as possible from background noise or unrelated speech. This task is often known as keyword spotting. To reach a wider audience of researchers and developers, this dataset has been released under the Creative Commons BY 4.0 license[5]. This enables it to easily be incorporated in tutorials and other scripts where it can be downloaded and used without any user intervention required (for example to register on a website or email an administrator for permission). This license is also well known in commercial settings, and so can usually be dealt with quickly by legal teams where approval is required. ## 3 Related Work Mozilla’s Common Voice dataset[6] has over 500 hours from 20,000 different people, and is available under the Creative Commons Zero license (similar to public domain). This licensing makes it very easy to build on top of. It is aligned by sentence, and was created by volunteers reading requested phrases through a web application. LibriSpeech[7] is a collection of 1,000 hours of read English speech, released under a Creative Commons BY 4.0 license, and stored using the open source FLAC encoder, which is widely supported. Its labels are aligned at the sentence level only, thus lacking word-level alignment information. This makes it more suitable for full automatic speech recognition than keyword spotting. TIDIGITS[8] contains 25,000 digit sequences spoken by 300 different speakers, recorded in a quiet room by paid contributors. The dataset is only available under a commercial license from the Language Data Consortium, and is stored in the NIST SPHERE file format, which proved hard to decode using modern software. Our initial experiments on keyword spotting were performed using this dataset. CHiME-5[9] has 50 hours of speech recorded in people’s homes, stored as 16 KHz WAV files, and available under a restricted license. It’s aligned at the sentence level. ## 4 Motivations Many voice interfaces rely on keyword spotting to start interactions. For example you might say "Hey Google" or "Hey Siri"[10] to begin a query or command for your phone. Once the device knows that you want to interact, it’s possible to send the audio to a web service to run a model that’s only limited by commercial considerations, since it can run on a server whose resources are controlled by the cloud provider. The initial detection of the start of an interaction is impractical to run as a cloud-based service though, since it would require sending audio data over the web from all devices all the time. This would be very costly to maintain, and would increase the privacy risks of the technology. Instead, most voice interfaces run a recognition module locally on the phone or other device. This listens continuously to audio input from microphones, and rather than sending the data over the internet to a server, they run models that listen for the desired trigger phrases. Once a likely trigger is heard, the transfer of the audio to a web service begins. Because the local model is running on hardware that’s not under the web service provider’s control, there are hard resource constraints that the on-device model has to respect. The most obvious of these is that the mobile processors typically present have total compute capabilities that are much lower than most servers, so to run in near real-time for an interactive response, on-device models must require fewer calculations than their cloud equivalents. More subtly, mobile devices have limited battery lives and anything that is running continuously needs to be very energy efficient or users will find their device is drained too quickly. This consideration doesn’t apply to plugged-in home devices, but those do have thermal constraints on how much heat they can dissipate that restrict the amount of energy available to local models, and are encouraged by programs like EnergyStar to reduce their overall power usage as much as possible. A final consideration is that users expect a fast response from their devices, and network latency can be highly variable depending on the environment, so some initial acknowledgement that a command was received is important for a good experience, even if the full server response is delayed. These constraints mean that the task of keyword spotting is quite different to the kind of speech recognition that’s performed on a server once an interaction has been spotted: * Keyword spotting models must be smaller and involved less compute. * They need to run in a very energy-efficient way. * Most of their input will be silence or background noise, not speech, so false positives on those must be minimized. * Most of the input that is speech will be unrelated to the voice interface, so the model should be unlikely to trigger on arbitrary speech. * The important unit of recognition is a single word or short phrase, not an entire sentence. These differences mean that the training and evaluation process between on-device keyword spotting and general speech recognition models is quite different. There are some promising datasets to support general speech tasks, such as Mozilla’s Common Voice, but they aren’t easily adaptable to keyword spotting. This Speech Commands dataset aims to meet the special needs around building and testing on-device models, to enable model authors to demonstrate the accuracy of their architectures using metrics that are comparable to other models, and give a simple way for teams to reproduce baseline models by training on identical data. The hope is that this will speed up progress and collaboration, and improve the overall quality of models that are available. A second important audience is hardware manufacturers. By using a publicly-available task that closely reflects product requirements, chip vendors can demonstrate the accuracy and energy usage of their offerings in a way that’s easily comparable for potential purchasers. This increased transparency should result in hardware that better meets product requirements over time. The models should also provide clear specifications that hardware engineers can use to optimize their chips, and potentially suggest model changes that make it easier to provide efficient implementations. This kind of co-design between machine learning and hardware can be a virtuous circle, increasing the flow of useful information between the domains in a way that helps both sides. ## 5 Collection ### Requirements I made the decision to focus on capturing audio that reflected the on-device trigger phrase task described above. This meant that the use of studio-captured samples seemed unrealistic, since that audio would lack background noise, would be captured with high-quality microphones, and in a formal setting. Successful models would need to cope with noisy environments, poor quality recording equipment, and people talking in a natural, chatty way. To reflect this, all utterances were captured through phone or laptop microphones, wherever users happened to be. The one exception was that I asked them to avoid recording themselves whenever there were background conversations happening for privacy reasons, so I asked them to be in a room alone with the door closed. I also decided to focus on English. This was for pragmatic reasons, to limit the scope of the gathering process and make it easier for native speakers to perform quality control on the gathered data. I hope that transfer learning and other techniques will still make this dataset useful for other languages though, and I open-sourced the collection application to allow others to easily gather similar data in other languages. I did want to gather as wide a variety of accents as possible however, since we’re familiar from experience with the bias towards American English in many voice interfaces. Another goal was to record as many different people as I could. Keyword-spotting models are much more useful if they’re speaker-independent, since the process of personalizing a model to an individual requires an intrusive user interface experience. With this in mind, the recording process had to be quick and easy to use, to reduce the number of people who would fail to complete it. I also wanted to avoid recording any personally-identifiable information from contributors, since any such data requires handling with extreme care for privacy reasons. This meant that I wouldn’t ask for any attributes like gender or ethnicity, wouldn’t require a sign-in through a user ID that could link to personal data, and would need users to agree to a data-usage agreement before contributing. To simplify the training and evaluation process, I decided to restrict all utterances to a standard duration of one second. This excludes longer words, but the usual targets for keyword recognition are short so this didn’t seem to be too restrictive. I also decided to record only single words spoken in isolation, rather than as part of a sentence, since this more closely resembles the trigger word task we’re targeting. It also makes labeling much easier, since alignment is not as crucial. ### Word Choice I wanted to have a limited vocabulary to make sure the capture process was lightweight, but still have enough variety for models trained on the data to potentially be useful for some applications. I also wanted the dataset to be usable in comparable ways to common proprietary collections like TIDIGITS. This led me to pick twenty common words as the core of our vocabulary. These included the digits zero to nine, and in version one, ten words that would be useful as commands in IoT or robotics applications; "Yes", "No", "Up", "Down", "Left", "Right", "On", "Off", "Stop", and "Go". In version 2 of the dataset, I added four more command words; “Backward”, “Forward”, “Follow”, and “Learn”. One of the most challenging problems for keyword recognition is ignoring speech that doesn’t contain triggers, so I also needed a set of words that could act as tests of that ability in the dataset. Some of these, such as “Tree”, were picked because they sound similar to target words and would be good tests of a model’s discernment. Others were chosen arbitrarily as short words that covered a lot of different phonemes. The final list was "Bed", "Bird", "Cat", "Dog", "Happy", "House", "Marvin", "Sheila", "Tree", and "Wow". ### Implementation To meet all these requirements, I created an open-source web-based application that recorded utterances using the WebAudioAPI[11]. This API is supported on desktop browsers like Firefox and Chrome, and on Android mobile devices. It’s not available on iOS, which was considered to be unfortunate but there were no alternatives that were more attractive. I also looked into building native mobile applications for iOS and Android, but I found that users were reluctant to install them, for privacy and security reasons. The web experience requires users to grant permission to the website to access the microphone, but that seemed a lot more acceptable, based on the increased response rate. The initial test of the application was hosted at an appspot.com subdomain, but it was pointed out that teaching users to give microphone permissions to domains that were easy for malicious actors to create was a bad idea. To address this, the final home of the application was moved to: ``` https://aiyprojects.withgoogle.com/open_speech_recording ``` This is a known domain that’s controlled by Google, and so it should be much harder to create confusing spoofs of. The initial page that a new user sees when navigating to the application explains what the project is doing, and asks them to explicitly and formally agree to participating in the study. This process was designed to ensure that the resulting utterances could be freely redistributed as part of an open dataset, and that users had a clear understanding of what the application was doing. When a user clicks on “I Agree”, a session cookie is added to record their agreement. The recording portion of the application will only be shown if this session cookie is found, and all upload accesses are guarded by cross-site request forgery tokens, to ensure that only audio recorded from the application can be uploaded, and that utterances are from users who have agreed to the terms. The recording page asks users to press a “Record” button when they’re ready, and then displays a random word from the list described above. The word is displayed for 1.5 seconds while audio is recorded, and then another randomly-chosen word is shown after a one-second pause. Each audio clip is added to a list that’s stored locally on the client’s machine, and they remain there until the user has finished recording all words and has a chance to review them. The random ordering of words was chosen to avoid pronunciation changes that might be caused by repetition of the same word multiple times. Core words are shown five times each in total, whereas auxiliary words only appear once. There are 135 utterances collected overall, which takes around six minutes in total to run through completely. The user can pause and restart at any point. Once the recording process is complete, the user is asked to review all of the clips, and if they’re happy with them, upload them. This then invokes a web API which uploads the audio to the server application, which saves them into a cloud storage bucket. The WebAudioAPI returns the audio data in OGG-compressed format, and this is what gets stored in the resulting files. The session ID is used as the prefix of each file name, and then the requested word is followed by a unique instance ID for the recording. This session ID has been randomly generated, and is not tied to an account or any other demographic information, since none has been generated. It does serve as a speaker identifier for utterances however. To ensure there’s a good distribution of different speakers, once a user has gone through this process once a cookie is added to the application that ensures they can’t access the recording page again. To gather volunteers for this process, I used appeals on social media to share the link and the aims of the project. I also experimented with using paid crowdsourcing for some of the utterances, though the majority of the dataset comes from the open site. ### Quality Control The gathered audio utterances were of variable quality, and so I needed criteria to accept or reject submissions. The informal guideline I used was that if a human listener couldn’t tell what word was being spoken, or it sounded like an incorrect word, then the clip should be rejected. To accomplish this, I used several layers of review. To remove clips that were extremely short or quiet, I took advantage of the nature of the OGG compression format. Compressed clips that contained very little audio would be very small in size, so a good heuristic was that any files that were smaller than 5 KB were unlikely to be correct. To implement this rule, I used the following Linux shell command: ``` find ${BASEDIR}/oggs -iname ".ogg" -size -5k -delete ``` With that complete, I then converted the OGG files into uncompressed WAV files containing PCM sample data at 16KHz, since this is any easier format for further processing: ``` find ${BASEDIR}/oggs -iname ".ogg" -print0 \| xargs -0 basename -s .ogg \| xargs -I {} ffmpeg -i ${BASEDIR}/oggs/{}.ogg -ar 16000 ${BASEDIR}/wavs/{}.wav ``` Samples from other sources came as varying sample-rate WAV files, so they were also resampled to 16 KHz WAV files using a similar ffmpeg command. ### Extract Loudest Section From manual inspection of the results, there were still large numbers of utterances that were too quiet or completely silent. The alignment of the spoken words within the 1.5 second file was quite arbitrary too, depending on the speed of the user’s response to the word displayed. To solve both these problems, I created a simple audio processing tool called Extract Loudest Section to examine the overall volume of the clips. As a first stage, I summed the absolute differences of all the samples from zero (using a scale where -32768 in the 16-bit sample data was -1.0 as a floating-point number, and +32767 was 1.0), and looked at the mean average of that value to estimate the overall volume of the utterance. From experimentation, anything below 0.004 on this metric was likely to be to quiet to be intelligible, and so all of those clips were removed. To approximate the correct alignment, the tool then extracted the one-second clip that contained the highest overall volume. This tended to center the spoken word in the middle of the trimmed clip, assuming that the utterance was the loudest part of the recording. To run these processes, the following commands were called: ``` git clone https://github.com/petewarden/extract_loudest_section tmp/extract_loudest_section cd tmp/extract_loudest_section make cd ../.. mkdir -p ${BASEDIR}/trimmed_wavs /tmp/extract_loudest_section/gen/bin/extract_loudest_section ${BASEDIR}’/wavs/.wav’ ${BASEDIR}/trimmed_wavs/ ``` ### Manual Review These automatic processes caught technical problems with quiet or silent recordings, but there were still some utterances that were of incorrect words or were unintelligible for other reasons. To filter these out I turned to commercial crowdsourcing. The task asked workers to type in the word they heard from each clip, and gave a list of the expected words as examples. Each clip was only evaluated by a single worker, and any clips that had responses that didn’t match their expected labels were removed from the dataset. ### Release Process The recorded utterances were moved into folders, with one for each word. The original 16-digit hexadecimal speaker ID numbers from the web application’s file names were hashed into 8-digit hexadecimal IDs. Speaker IDs from other sources (like the paid crowdsourcing sites) were also hashed into the same format. This was to ensure that any connection to worker IDs or other personally-identifiable information was removed. The hash function used is stable though, so in future releases the IDs for existing files should remain the same, even as more speakers are added. ### Background Noise A key requirement for keyword spotting in real products is distinguishing between audio that contains speech, and clips that contain none. To help train and test this capability, I added several minute-long 16 KHz WAV files of various kinds of background noise. Several of these were recorded directly from noisy environments, for example near running water or machinery. Others were generated mathematically using these commands in Python: ``` scipy.io.wavfile.write(’/tmp/white_noise.wav’, 16000, np.array(((acoustics.generator.noise(1600060, color=’white’))/3) * 32767).astype(np.int16)) scipy.io.wavfile.write(’/tmp/pink_noise.wav’, 16000, np.array(((acoustics.generator.noise(1600060, color=’pink’))/3) 32767).astype(np.int16)) ``` To distinguish these files from word utterances, they were placed in a specially-named "_background_noise_" folder, in the root of the archive. ## 6 Properties The final dataset consisted of 105,829 utterances of 35 words, broken into the categories and frequencies shown in Table 1. [FIGURE:S6.F1][ENDFIGURE] Each utterance is stored as a one-second (or less) WAVE format file, with the sample data encoded as linear 16-bit single-channel PCM values, at a 16 KHz rate. There are 2,618 speakers recorded, each with a unique eight-digit hexadecimal identifier assigned as described above. The uncompressed files take up approximately 3.8 GB on disk, and can be stored as a 2.7GB gzip-compressed tar archive. ## 7 Evaluation One of this dataset’s primary goals is to enable meaningful comparisons between different models’ results, so it’s important to suggest some precise testing protocols. As a starting point, it’s useful to specify exactly which utterances can be used for training, and which must be reserved for testing, to avoid overfitting. The dataset download includes a text file called validation_list.txt, which contains a list of files that are expected to be used for validating results during training, and so can be used frequently to help adjust hyperparameters and make other model changes. The testing_list.txt file contains the names of audio clips that should only be used for measuring the results of trained models, not for training or validation. The set that a file belongs to is chosen using a hash function on its name. This is to ensure that files remain in the same set across releases, even as the total number changes, so avoid set cross-contamination when trying old models on the more recent test data. The Python implementation of the set assignment algorithm is given in the TensorFlow tutorial code[12] that is a companion to the dataset. ### Top-One Error The simplest metric to judge a trained model against is how many utterances it can correctly identify. In principle this can be calculated by running the model against all the files in the testing set, and comparing the reported against the expected label for each. Unlike image classification tasks like ImageNet, it’s not obvious how to weight all of the different categories. For example, I want a model to indicate when no speech is present, and separately to indicate when it thinks a word has been spoken that’s not one it recognizes. These “open world” categories need to be weighted according to their expected occurrence in a real application to produce a realistic metric that reflects the perceived quality of the results in a product. The standard chosen for the TensorFlow speech commands example code is to look for the ten words "Yes", "No", "Up", "Down", "Left", "Right", "On", "Off", "Stop", and "Go", and have one additional special label for “Unknown Word”, and another for “Silence” (no speech detected). The testing is then done by providing equal numbers of examples for each of the twelve categories, which means each class accounts for approximately 8.3% of the total. The "Unknown Word" category contains words randomly sampled from classes that are part of the target set. The "Silence" category has one-second clips extracted randomly from the background noise audio files. I’ve uploaded a standard set of test files[13] to make it easier to reproduce this metric. If you want to calculate the canonical Top-One error for a model, run inference on each audio clip, and compare the top predicted class against the ground truth label encoded in its containing subfolder name. The proportion of correct predictions will give you the Top-One error. There’s also a similar collection of test files[14] available for version one of the dataset. The example training code that accompanies the dataset[15] provides results of 88.2% on this metric for the highest-quality model when fully trained. This translates into a model that qualitatively gives a reasonable, but far from perfect response, so it’s expected that this will serve as a baseline to be exceeded by more sophisticated architectures. ### Streaming Error Metrics Top-One captures a single dimension of the perceived quality of the results, but doesn’t reveal much about other aspects of its performance in a real application. For example, models in products receive a continuous stream of audio data and don’t know when words start and end, whereas the inputs to Top One evaluations are aligned to the beginning of utterances. The equal weighting of each category in the overall score also doesn’t reflect the distribution of trigger words and silence in typical environments. To measure some of these more complex properties of models, I test them against continuous streams of audio and score them on multiple metrics. Here’s what the baseline model trained with V2 data produces: ``` 49.0% matched, 46.0% correctly, 3.0% wrongly, 0.0% false positives ``` To produce this result, I ran the following bash script against the 10 minute streaming test audio clip and ground truth labels: ``` bazel run tensorflow/examples/speech_commands:freeze -- --start_checkpoint=/tmp/speech_commands_train/conv.ckpt-18000 --output_file=/tmp/v2_frozen_graph.pb bazel run tensorflow/examples/speech_commands:test_streaming_accuracy -- --graph=/tmp/v2_frozen_graph.pb --wav=/tmp/speech_commands_train/streaming_test.wav --labels=/tmp/speech_commands_train/conv_labels.txt --ground_truth=/tmp/speech_commands_train/streaming_test_labels.txt ``` * Matched-percentage represents how many words were correctly identified, within a given time tolerance. * Wrong-percentage shows how many words were correctly distinguished as speech rather than background noise, but were given the wrong class label. * False-positive percentage is the number of words detected that were in parts of the audio where no speech was actually present. An algorithm for calculating these values given an audio file and a text file listing ground truth labels is implemented in TensorFlow as test_streaming_accuracy.cc[16]. Performing successfully on these metrics requires more than basic template recognition of audio clips. There has to be at least a very crude set of rules to suppress repeated recognitions of the same word in short time frames, so default logic for this is implemented in recognize_commands.cc[17]. This allows a simple template-style recognition model to be used directly to generate these statistics. One of the other configurable features of the accuracy test is the time tolerance for how close to the ground truth’s time a recognition result must be to count as a match. The default for this is set to 750ms, since that seems to match with requirements for some of the applications that are supported. To make reproducing and comparing results easier, I’ve made available a one-hour audio file[18] containing a mix of utterances at random times and noise, together with a text file marking the times and ground truth labels of each utterance. This was generated using the script included in the TensorFlow tutorial, and can be used to compare different models performance on streaming applications. ### Historical Evaluations Version 1 of the dataset[2] was released August 3rd 2017, and contained 64,727 utterances from 1,881 speakers. Training the default convolution model from the TensorFlow tutorial (based on Convolutional Neural Networks for Small-footprint Keyword Spotting[19]) using the V1 training data gave a Top-One score of 85.4%, when evaluated against the test set from V1. Training the same model against version 2 of the dataset[1], documented in this paper, produces a model that scores 88.2% Top-One on the training set extracted from the V2 data. A model trained on V2 data, but evaluated against the V1 test set gives 89.7% Top-One, which indicates that the V2 training data is responsible for a substantial improvement in accuracy over V1. The full set of results are shown in Table 2. [FIGURE:S7.F2][ENDFIGURE] These figures were produced using the checkpoints produced by the following training commands: ``` python tensorflow/examples/speech_commands/train.py --data_url=http://download.tensorflow.org/data/speech_commands_v0.01.tar.gz python tensorflow/examples/speech_commands/train.py --data_url=http://download.tensorflow.org/data/speech_commands_v0.02.tar.gz ``` The results of these commands are available as pretrained checkpoints[20]. The evaluations were performed by running variations on the following command line (with the v1/v2’s substituted as appropriate): ``` python tensorflow/examples/speech_commands/train.py --data_url=http://download.tensorflow.org/data/speech_commands_v0.0{1,1}.tar.gz --start_checkpoint=${HOME}/speech_commands_checkpoints/conv-v{1,2}.ckpt-18000 ``` ### Applications The TensorFlow tutorial gives a variety of baseline models, but one of the goals of the dataset is to enable the creation and comparison of a wide range of models on a lot of different platforms, and version one of has enabled some interesting applications. CMSIS-NN[21] covers a new optimized implementation of neural network operations for ARM microcontrollers, and uses Speech Commands to train and evaluate the results. Listening to the World[22] demonstrates how combining the dataset and UrbanSounds[23] can improve the noise tolerance of recognition models. Did you Hear That[24] uses the dataset to test adversarial attacks on voice interfaces. Deep Residual Learning for Small Footprint Keyword Spotting[25] shows how approaches learned from ResNet can produce more efficient and accurate models. Raw Waveform-based Audio Classification[26] investigates alternatives to traditional feature extraction for speech and music models. Keyword Spotting Through Image Recognition[27] looks at the effect virtual adversarial training on the keyword task. ## 8 Conclusion The Speech Commands dataset has shown to be useful for training and evaluating a variety of models, and the second version shows improved results on equivalent test data, compared to the original. ## 9 Acknowledgements Massive thanks are due to everyone who donated recordings to this data set, I’m very grateful. I also couldn’t have put this together without the help and support of Billy Rutledge, Rajat Monga, Raziel Alvarez, Brad Krueger, Barbara Petit, Gursheesh Kour, Robert Munro, Kirsten Gokay, David Klein, Lukas Biewald, and all the AIY and TensorFlow teams. ## References * [1] (2018) Speech commands dataset version 2. [Online]. Available: http://download.tensorflow.org/data/speech_commands_v0.02.tar.gz * [2] (2017) Speech commands dataset version 1. [Online]. Available: http://download.tensorflow.org/data/speech_commands_v0.01.tar.gz * [3] (2018) Linguistic data consortium. [Online]. Available: https://www.ldc.upenn.edu/ * [4] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “ImageNet: A Large-Scale Hierarchical Image Database,” in _CVPR09_, 2009. * [5] (2018) Creative commons international attribution international 4.0 license. [Online]. Available: https://creativecommons.org/licenses/by/4.0/ * [6] (2017) Mozilla common voice. [Online]. Available: https://voice.mozilla.org/en * [7] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Librispeech: an ASR corpus based on public domain audio books,” in _Proceedings of the International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2015. * [8] R. G. Leonard and G. R. Doddington. (1992) A speaker-independent connected-digit database. [Online]. Available: https://catalog.ldc.upenn.edu/docs/LDC93S10/tidigits.readme.html * [9] (2018) The 5th chime speech separation and recognition challenge. [Online]. Available: http://spandh.dcs.shef.ac.uk/chime_challenge/data.html * [10] (2017) Hey siri: An on-device dnn-powered voice trigger for apple’s personal assistant. [Online]. Available: https://machinelearning.apple.com/2017/10/01/hey-siri.html * [11] (2015) Web audio api. [Online]. Available: https://developer.mozilla.org/en-US/docs/Web/API/Web_Audio_API * [12] (2018) Implementation of set assignment algorithm. [Online]. Available: https://github.com/tensorflow/tensorflow/blob/master/tensorflow/examples/speech_commands/input_data.py#L61 * [13] (2018) Speech commands dataset test set version 2. [Online]. Available: http://download.tensorflow.org/data/speech_commands_test_set_v0.02.tar.gz * [14] (2017) Speech commands dataset test set version 1. [Online]. Available: http://download.tensorflow.org/data/speech_commands_test_set_v0.01.tar.gz * [15] (2017) Tensorflow audio recognition tutorial. [Online]. Available: https://www.tensorflow.org/tutorials/audio_recognition * [16] (2018) test_streaming_accuracy.cc source file. [Online]. Available: https://github.com/tensorflow/tensorflow/blob/master/tensorflow/examples/speech_commands/test_streaming_accuracy.cc * [17] (2018) recognize_commands.cc source file. [Online]. Available: https://github.com/tensorflow/tensorflow/blob/master/tensorflow/examples/speech_commands/recognize_commands.cc * [18] (2018) Speech commands dataset streaming test version 2. [Online]. Available: http://download.tensorflow.org/data/speech_commands_streaming_test_v0.02.tar.gz * [19] T. N. Sainath and C. Parada, “Convolutional Neural Networks for Small-Footprint Keyword Spotting,” in _Sixteenth Annual Conference of the International Speech Communication Association_, 2015. [Online]. Available: https://www.isca-speech.org/archive/interspeech_2015/papers/i15_1478.pdf * [20] (2018) Speech commands tutorial checkpoints. [Online]. Available: https://storage.googleapis.com/download.tensorflow.org/models/speech_commands_checkpoints.tar.gz * [21] L. Lai, N. Suda, and V. Chandra, “CMSIS-NN: Efficient Neural Network Kernels for Arm Cortex-M CPUs,” _ArXiv e-prints_, Jan. 2018. * [22] B. McMahan and D. Rao, “Listening to the World Improves Speech Command Recognition,” _ArXiv e-prints_, Oct. 2017. * [23] J. Salamon, C. Jacoby, and J. P. Bello, “A dataset and taxonomy for urban sound research,” in _Proceedings of the 22Nd ACM International Conference on Multimedia_, ser. MM ’14. New York, NY, USA: ACM, 2014, pp. 1041–1044. [Online]. Available: http://doi.acm.org/10.1145/2647868.2655045 * [24] M. Alzantot, B. Balaji, and M. Srivastava, “Did you hear that? Adversarial Examples Against Automatic Speech Recognition,” _ArXiv e-prints_, Jan. 2018. * [25] R. Tang and J. Lin, “Deep Residual Learning for Small-Footprint Keyword Spotting,” _ArXiv e-prints_, Oct. 2017. * [26] J. Lee, T. Kim, J. Park, and J. Nam, “Raw Waveform-based Audio Classification Using Sample-level CNN Architectures,” _ArXiv e-prints_, Dec. 2017. * [27] S. Krishna Gouda, S. Kanetkar, D. Harrison, and M. K. Warmuth, “Speech Recognition: Keyword Spotting Through Image Recognition,” _ArXiv e-prints_, Mar. 2018.
1608.05293	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 200126, "num_imgs": 17, "llama3_tokens_count": 59393 }	[ "content_image/1608.05293/x1.png", "content_image/1608.05293/x2.png", "content_image/1608.05293/x6.png", "content_image/1608.05293/x7.png", "content_image/1608.05293/x8.png", "content_image/1608.05293/x10.png", "content_image/1608.05293/x11.png", "content_image/1608.05293/x12.png", "content_image/1608.05293/x14.png", "content_image/1608.05293/x16.png", "content_image/1608.05293/x18.png", "content_image/1608.05293/x19.png", "content_image/1608.05293/x20.png", "content_image/1608.05293/x21.png", "content_image/1608.05293/x24.png", "content_image/1608.05293/x32.png", "content_image/1608.05293/x34.png" ]	# CGC factorization for forward particle production in proton-nucleus collisions at next-to-leading order E. Iancu, b A.H. Mueller, c and D.N. Triantafyllopoulos edmond.iancu@cea.fr amh@phys.columbia.edu trianta@ectstar.eu Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, FranceDepartment of Physics, Columbia University, New York, NY 10027, USAEuropean Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT) and Fondazione Bruno Kessler, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy ###### Abstract Within the Color Glass Condensate effective theory, we reconsider the next-to-leading order (NLO) calculation of the single inclusive particle production at forward rapidities in proton-nucleus collisions at high energy. Focusing on quark production for definiteness, we establish a new factorization scheme, perturbatively correct through NLO, in which there is no ‘rapidity subtraction’. That is, the NLO correction to the impact factor is not explicitly separated from the high-energy evolution. Our construction exploits the skeleton structure of the (NLO) Balitsky-Kovchegov equation, in which the first step of the evolution is explicitly singled out. The NLO impact factor is included by computing this first emission with the exact kinematics for the emitted gluon, rather than by using the eikonal approximation. This particular calculation has already been presented in the literature [1; 2], but the reorganization of the perturbation theory that we propose is new. As compared to the proposal in [1; 2], our scheme is free of the fine-tuning inherent in the rapidity subtraction, which might be the origin of the negativity of the NLO cross-section observed in previous studies. Keywords:Perturbative QCD, High-Energy Evolution, Color Glass Condensate, Proton-Nucleus Collisions ## 1 Introduction Using perturbative QCD, we would like to study particle production in high-energy proton-nucleus ($pA$) collisions in the kinematical regime where the produced particle is _semi-hard to hard_ (meaning that its transverse momenta can be larger than the nuclear saturation momentum $Q_{s}$, but not _much_ larger) and it propagates at _forward rapidity_ in the proton fragmentation region (that is, it makes a very small angle w.r.t. the collision axis) [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. What is special about this kinematics is that the scattering probes the small-$x$ part of the nuclear wavefunction, but the large-$x$ part of the proton wavefunction, so it acts as a clean probe of the nuclear gluon distribution in the interesting regime where one expects large gluon occupation numbers and strong non-linear phenomena, like gluon saturation. This probe is ‘clean’ since the large-$x$ part of the proton wavefunction is very dilute and hence well described by the standard QCD parton picture and the associated collinear factorization. Accordingly, the overall process can be depicted as follows: a collinear parton from the proton undergoes multiple scattering off the dense gluon system in the nuclear target and hence acquires some transverse momentum $k_{\perp}$, before eventually fragmenting into the hadrons that are measured in the final state. The above physical picture naturally lends itself to a _hybrid factorization_ scheme [5; 11] for the calculation of the single-inclusive hadron multiplicity, which combines the _collinear factorization_ for the parton distribution of the incoming proton and also for the fragmentation of the produced quark or gluon [16], with the _CGC factorization_ for the high-energy scattering between the collinear parton and the nucleus. The ‘CGC’ refers to the Color Glass Condensate effective theory, which is the appropriate pQCD framework to address the problem of high-energy scattering in the presence of high gluon densities [17; 18; 19; 20]. This is essentially a theory for the gauge-invariant correlations of Wilson lines and their evolution with increasing energy. A _Wilson line_ (a unitary matrix in the color group SU$(N_{c})$) is the $S$-matrix of an energetic parton which undergoes multiple scattering off a strong color field representing the gluon distribution of the target. The CGC factorization¹ for ‘dilute-dense’ scattering associates one such a Wilson line to each of the partons partaking in the collision, separately in the direct amplitude and the complex conjugate amplitude. Cross-sections are obtained by averaging over all the configurations of the color fields in the target, a procedure which generates the Wilson-line correlators aforementioned. In the simplest case, that is for single-inclusive particle production at leading order, this correlator involves the trace of the product of two Wilson lines², which can be identified with the elastic $S$-matrix of a _color dipole_ which scatters off the nuclear target. [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] Whereas this hybrid factorization may indeed look natural, in view of the underlying physical picture, its foundation in pQCD is not obvious, nor easy to establish. In order to make sense beyond tree-level, this scheme must be consistent with the QCD radiative corrections and notably with the collinear and high-energy evolutions. As we shall shortly explain, this issue is already non-trivial at leading-order (LO) and it becomes even more so at next-to-leading order (NLO) and beyond. The LO version of the hybrid factorization for single-inclusive hadron production [5; 11] includes the LO DGLAP evolution for the parton distribution in the proton and for the parton fragmentation in the final state. It furthermore includes the LO B-JIMWLK evolution of the dipole $S$-matrix. The B-JIMWLK (from Balitsky, Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner) equations [23; 24; 25; 26; 27; 28; 29] form an infinite hierarchy of coupled equations which describes the non-linear evolution of the $n$-point correlations of the Wilson lines. (Operators with different number of Wilson lines couple under the evolution due to multiple scattering.) This hierarchy drastically simplifies in the limit of a large number of colors $N_{c}\gg 1$, in which expectation values of gauge-invariant operators factorize from each other. In that limit, the evolution of the dipole $S$-matrix is governed by a closed non-linear equation, known as the Balitsky-Kovchegov (BK) equation [23; 30]. A first subtle point, which arises already at LO, refers to the relation between the cross-section for parton-nucleus scattering on one hand, and the dipole scattering amplitude on the other hand. In general, cross-sections and amplitudes are different quantities (e.g. they have different analytic properties) and it is only due to the high-energy approximations — notably, due to the fact that the high-energy amplitudes are purely absorptive — that such an identification becomes possible in the problem at hand. Yet, the consistency between this relation and the high-energy evolution is far from being trivial (see the discussion in [31]). So far, this has been demonstrated up to next-to-leading order [4; 31] and there is no obvious reason why it should remain true in higher orders. With this in mind, we can address the calculation of single-inclusive hadron production in $pA$ collisions at NLO. The NLO version of the DGLAP equation is known since long (see e.g. the textbook [16] for a pedagogical discussion). Recently, the BK equation and the full B-JIMWLK hierarchy have been promoted to NLO accuracy as well [32; 33; 34]. By itself, the NLO approximation turns out to be unstable [35; 36; 37], due to the presence of large NLO corrections enhanced by transverse logarithms. A similar difficulty was already encountered for the NLO version of the BFKL equation (the linearized version of the BK equation valid when the scattering is week; see e.g. the textbook [20]). As in that case [38; 39; 40; 41; 42; 43], resummation schemes have been devised also for the non-linear, BK and B-JIMWLK, equations [44; 37; 45; 46], to restore the convergence of perturbation theory. In particular, the collinearly-improved BK equation [37; 45], which resums to all orders the double-collinear logarithms together with a subset of the single-collinear logarithms and with the running coupling corrections, appears to be a convenient tool for the phenomenology [45; 47]. Moreover, the full NLO BK equation with collinear improvement has recently been shown to be stable and tractable via numerical methods [48]. Besides the NLO evolution, a calculation of the particle production to NLO accuracy must include an equally accurate version of the _impact factor_. The ‘impact factor’ refers to the partonic subprocess and for the present purposes can be simply defined as the cross-section for parton-nucleus scattering in the absence of any QCD evolution. For more clarity, from now on, we shall assume that the parton from the proton which participates in the collision is a quark. At LO, the impact factor is simply the cross-section for the scattering between a bare quark and a nucleus or, equivalently, the $S$-matrix for a bare dipole. At NLO, the wavefunction of the incoming quark (or dipole) may contain an additional gluon, to be referred to as the ‘primary gluon’ in what follows. This gluon can be released in the final state (‘real correction’), or not (‘virtual correction’), but in any case its emission modifies the cross-section (or the dipole amplitude) w.r.t. to its LO value. Since the kinematics of this primary gluon is integrated over, the corresponding correction to the impact factor is truly a one-loop effect. So far, this correction has been computed via two different approaches, [1; 2] and respectively [49; 50], with results which are quite difficult to compare with each other, but _a priori_ look different at NLO accuracy. In what follows, we shall mostly refer to the NLO calculation in Refs. [1; 2]. This is better suited for our new developments in this paper and this is also the context in which emerged the problem of the negativity of the cross-section [51], which attracted our interest on this topic. But our general philosophy for attacking this problem is perhaps closer in spirit to that in [50], in that it involves no subtraction for the ‘rapidity divergence’ (see below). When performing the one-loop integration alluded to above, one should keep in mind that there are regions in phase-space that have already been included at LO, at least approximately, via the collinear and the high-energy evolutions. In the absence of physical cutoffs, these regions would generate logarithmic divergences. The physical cutoffs are truly needed when solving the evolution equations (they define the boundaries of the corresponding phase-space), but they can often be avoided when computing the NLO correction to the impact factor. Namely, one can directly subtract the would-be divergences by using a suitable ‘renormalization prescription’, tuned to match the resummation performed by the LO evolution equations. This is the strategy followed by the authors of Refs. [1; 2]. Specifically, Refs. [1; 2] used dimensional regularization plus minimal subtraction to ‘remove’ the collinear divergences. This is a rather standard procedure in the context of the collinear factorization and relies on the fact that the collinear divergences can be factorized from the transverse integrations, as they refer to the renormalization of the _integrated_ parton distributions and fragmentation functions. Refs. [1; 2] furthermore proposed a ‘plus’ prescription in order to subtract the ‘rapidity divergence’, i.e. the would-be divergence³ at $x\to 0$, where $x$ is the longitudinal momentum fraction of the primary gluon w.r.t. the incoming quark. This prescription is quite common in the context of the $k_{\perp}$-factorization at next-to-leading order (see e.g. [52] and references therein) and its ‘non-linear’ extension to the CGC factorization may look natural. Recall however that, underlying this prescription, there is the strong assumption that cross-sections in perturbative QCD at high-energy can be factorized in rapidity. This assumption is highly non-trivial (and still unproven in the general case and beyond LO accuracy) because the perturbative corrections are truly non-local in rapidity. Accordingly, it is _a priori_ not obvious that the small-$x$ divergence can be factorized from the high-energy evolution of the various scattering operators — the ‘dipole $S$-matrices’ which describe the eikonal scattering between the quark-gluon projectile and the target. This being said, we shall explicitly demonstrate in this paper that, via an appropriate reorganization of the perturbative corrections, one can indeed obtain such a factorized expression, valid to NLO, which involves the ‘plus’ prescription and agrees with the proposal in Refs. [1; 2]. However, we shall also argue that the manipulations associated with this reorganization — namely, with the subtraction of the high-energy evolution from the NLO impact factor — involve a considerable amount of fine-tuning, which may be dangerous in practice. [FOOTNOTE:3][ENDFOOTNOTE] This fine-tuning might be at the origin of the negativity problem observed in explicit numerical calculations based on the factorization scheme in [1; 2]: the cross-section for single-inclusive particle production suddenly turns negative for transverse momenta slightly larger than the target saturation momentum $Q_{s}$[51; 53]. Various proposals to circumvent this problem, by either introducing a cutoff in the rapidity subtraction scheme [54; 55; 56], or via a more careful implementation of the kinematics [53; 50; 57], have managed to alleviate the problem (by pushing it to somewhat larger values of the transverse momentum), but without offering a fully satisfactory solution, at either conceptual or practical level (see also the discussion in the recent review paper [58]). Whereas high-energy approximations are expected to become less accurate at sufficiently large transverse momenta $k_{\perp}\gg Q_{s}$, we find it surprising that they fail already in the transition region towards saturation ($k_{\perp}\gtrsim Q_{s}$) — a region whose description is in fact the main focus of the CGC effective theory [17; 18; 19; 20]. This negativity problem encouraged us to reconsider the overall calculation from a new perspective and thus propose a new factorization scheme for the high-energy aspects of the problem. In presenting this proposal below, we shall restrict ourselves to quark production, that is, we shall ignore other partonic channels and also the fragmentation of the quark into hadrons in the final state. Also, we shall omit all the NLO corrections associated with the collinear resummation, i.e. the finite terms which remain after subtracting the collinear divergence. These terms can be unambiguously distinguished from those referring to the high-energy factorization [56] and can be simply added to our final results. Our main observations and new results can be summarized as follows: (i) In our opinion, the negativity problem is most likely related to the severe fine-tuning inherent in the rapidity subtraction. The ‘fine-tuning’ refers to the delicate balance between the NLO corrections which are included in the evolution of the dipole $S$-matrix and those which are subtracted via the ‘plus’ prescription in order to construct the NLO correction to the impact factor. As we shall explain in detail in Sect. 3.5, this subtraction amounts to a reorganization of the perturbation theory which exploits the integral representation for the solution to the BK equation. Any approximation in solving this equation, as well as the subsequent approximations which are in practice needed to derive the ‘plus’ prescription, will lead to an imbalance between the large, ‘added’ and ‘subtracted’, contributions and thus possibly to unphysical results. (ii) The ‘plus’ prescription is actually not needed: the would-be ‘rapidity divergence’ is truly cut off by physical mechanisms, namely by energy conservation for the ‘real’ corrections and by probability conservation for the ‘virtual’ ones. The role of the energy conservation in constraining the longitudinal phase-space for the primary emission is in fact well appreciated [53; 50]. This constraint has been used to cut off the soft divergence in Ref. [50] and to alleviate the negativity problem in the context of the ‘plus’ prescription in Ref. [57]. The corresponding constraint on the ‘virtual’ corrections has not been discussed to our knowledge, so we shall devote an appendix to an explicit NLO calculation which demonstrates this. Specifically, in App. A we show that the ‘virtual’ corrections with very short lifetimes — outside the physical range for the ‘real’ corrections — mutually cancel each other. This result is in fact natural: the ‘real’ and ‘virtual’ corrections must have the same support in longitudinal phase-space, since they should combine with each other to ensure probability conservation. (iii) To NLO accuracy, the calculation of the single-inclusive forward particle production in dilute-dense collisions can be given a different factorization, cf. Eq. (3.2), in which the small-$x$ logarithm associated with the primary gluon emission is not included in the high-energy evolution, but is implicitly kept within the impact factor⁴. The latter includes the incoming quark and its (not necessarily soft) primary gluon, which together scatter off the gluon distribution in the nucleus and thus measure its high-energy evolution. This is the same picture as for the CGC calculation of di-hadron production [63; 64; 65; 66], except that the kinematics of the primary gluon is now integrated over and one must add the ‘virtual’ corrections. [FOOTNOTE:4][ENDFOOTNOTE] (iv) In the limit where the primary gluon is soft and treated in the eikonal approximation, our general formula in Eq. (3.2) reduces to the integral representation (33) of the solution to rcBK. This is indeed the correct result for the quark multiplicity at LO. Vice-versa, our formula may be viewed as a generalization of the LO BK evolution in which the very first gluon emission (and that emission only) is treated beyond the eikonal approximation. In view of that, we expect our result for the cross-section to be positive semi-definite, albeit we have not been able to prove this explicitly. (v) Our general formula (3.2) is probably too cumbersome to be used in practice, due to the complicated structure of the transverse and longitudinal integrations, which are entangled with each other. Fortunately though, the problem can be considerably simplified in the interesting regime where the transverse momentum $k_{\perp}$ of the produced quark is sufficiently hard, $k_{\perp}\gtrsim Q_{s}$. In that regime, the primary gluon is relatively hard as well, with a transverse momentum $p_{\perp}\sim k_{\perp}$ (see the discussion in Sect. 2.4). This allows us to replace $p_{\perp}\sim k_{\perp}$ within the rapidity variables in Eq. (3.2) and thus deduce a much simpler result, Eq. (3.3), which is of the same degree of difficulty as the formulae used within previous numerical simulations [67; 51; 53; 57; 56], while at the same time avoiding the rapidity subtraction and the associated fine-tuning. (vi) To ensure the desired NLO accuracy of the overall scheme, the high-energy evolution of the color dipoles must be computed to NLO as well. In Sect. 3.4 we complete our factorization scheme by specifying the NLO corrections associated with the high-energy evolution. As we also explain there, the inclusion of the NLO evolution in the problem at hand is _a priori_ problematic, for two reasons: (a) the evolution of a dense wavefunction, like a nucleus, is not known beyond leading-order, and (b) the strict NLO approximation is expected to be unstable, due to large corrections enhanced by collinear logarithms. We provide solutions to these problems by relating the target evolution to that of the dilute projectile, which is indeed known to NLO accuracy [32], including the all-order resummation of the collinear logarithms [37; 45]. This is furthermore discussed in Appendices B and C. (vii) To make contact with the formalism in [1; 2], we consider in Sect. 3.5 the decomposition of our general result (3.3) between NLO dipole evolution and NLO corrections to the impact factor. This decomposition, shown in schematic notations in Eq. (45), relies in an essential way on the fact that the dipole $S$-matrix obeys a specific evolution equation — either the LO BK equation with running-coupling (rcBK) [68; 69; 70], or the NLO BK equation [32] with collinear improvement [37; 45], depending upon the desired accuracy. Indeed, the integral version of the evolution equation is used to reshuffle the largest contribution to the cross-section, that associated with the LO evolution. Eq. (45) is very similar, but not fully identical, to the factorization scheme proposed in [1; 2], which is schematically shown in Eq. (46). As explained in Sect. 3.5, the differences between Eqs. (45) and (46) are irrelevant to NLO accuracy, but they can be nevertheless important in practice, as they introduce an imbalance between the terms included in the dipole evolution and those subtracted via the ‘plus’ prescription. As already mentioned at point (i), we believe that this imbalance is responsible for the problem of the negativity of the cross-section. To summarize, all the potential difficulties with the subtraction method can be avoided by computing the cross-section directly from our formula (3.3), which involves no subtraction at all. This formula can be evaluated with a suitable approximation for the high-energy evolution, like rcBK or the more elaborated approximations described in Sect. 3.4 and in Appendix B. This paper includes two major sections, devoted to the LO and the NLO calculations respectively, and three appendices. Sect. 2 starts with a discussion of the kinematics and of the importance of the choice of a Lorentz frame for building a physical picture. Such a picture is first developed in the target infinite momentum frame (in Sects. 2.1 and 2.2), then extended to a mixed frame, where one of the evolution gluons (the ‘primary gluon’) is viewed as an emission by the incoming quark, whereas all the subsequent ones are included in the evolution of the nuclear target (in Sect. 2.3). In Sect. 2.4, we discuss the di-jet configurations which control the final state in the regime where the produced quark has a large transverse momentum $k_{\perp}\gg Q_{s}$. The first subsection of Sect. 3 summarizes the result for the NLO impact factor obtained in [1; 2] and also extends that result by specifying the rapidity variables for the evolution of the various dipole $S$-matrices. This discussion motivates our main result in this paper, that is, the NLO factorization displayed in Eq. (3.2). This general but rather cumbersome expression is rendered more tractable and also more explicit in Sects. 3.3, 3.4 and in Appendix B, where we simplify the kinematics via approximations appropriate at large $k_{\perp}\gtrsim Q_{s}$ and we replace the unknown NLO evolution of the nuclear target by that of the dilute quark-gluon projectile (with collinear improvement). In Sect. 3.5, we isolate LO from NLO contributions, as described at point (vii) above. Sect. 4 contains our conclusions. In Appendix A we demonstrate the mutual cancellation of the ‘virtual’ fluctuations whose lifetime is shorter than the longitudinal extent of the target. Finally, Appendices B and C give more details on the NLO evolution of color dipoles. ## 2 The leading order calculation In this section, we shall briefly review the leading-order (LO) calculation of single-inclusive quark production in high-energy proton-nucleus ($pA$) at forward rapidities (i.e. in the proton fragmentation region). This calculation relies on a hybrid factorization scheme [11] which involves collinear factorization at the level of the proton wavefunction together with the dipole picture for the scattering between a collinear quark from the proton and the nuclear gluon distribution. ### General picture and kinematics To LO in perturbative QCD and in a suitable Lorentz frame, the forward production of a quark in $pA$ collisions proceeds via the _transverse momentum broadening_ of one of the quarks from the incoming proton: the quark, which was originally collinear with the proton, acquires a transverse momentum $\bm{k}$ via scattering off the small-$x$ gluons in the nuclear wavefunction and thus emerges at a small angle $\theta\simeq k_{\perp}/k^{+}$ w.r.t. the collision axis. The typical situation is such that the quark undergoes multiple soft scattering and thus accumulates a transverse momentum of order $Q_{s}$ — the target saturation momentum at the longitudinal resolution probed by the scattering (see below). But the $k_{\perp}$-distribution of the produced quark also features a power-like tail at high momenta $k_{\perp}\gg Q_{s}$, which is the result of a single, relatively hard, Coulomb scattering off the color sources in the target. The physical picture actually depends upon the choice of a Lorentz frame. The picture that we have just described only holds in a ‘target infinite momentum frame’, where the nuclear target carries most of the total energy, so the high-energy evolution via the successive emissions of soft gluons is fully encoded in the nuclear gluon distribution. On the other hand, the picture would be different in a frame where the projectile proton carries most of the total energy; in that case, the wavefunction of the incoming quark is highly evolved, in the sense that it contains many soft gluons, which can be put on-shell by their scattering off the (un-evolved) nucleus. The final transverse momentum $\bm{k}$ acquired by the quark is then the result of the recoil from this induced gluon radiation. To transform these pictures into actual calculations, we need to better specify the kinematics. We work in a Lorentz frame where the proton is a right mover, with longitudinal momentum $Q^{+}$, while the nuclear target is a left mover, with longitudinal momentum $P^{-}$ per nucleon. The high-energy regime corresponds to the situation where the center-of-mass energy $\sqrt{s}$, with $s=2Q^{+}P^{-}$, is much larger than any of the transverse momentum (or virtuality) scales in the problem; in particular, $s\gg k_{\perp}^{2}$ and $s\gg Q_{s}^{2}$. For our purposes, the longitudinal momentum $k^{+}$ of the produced quark is most conveniently parametrized in terms of the boost-invariant ratio $x_{p}\equiv k^{+}/Q^{+}\leq 1$ (the quark longitudinal momentum fraction w.r.t. the incoming proton). Let us start by choosing a _target infinite-momentum frame_, where the scattering involves a bare quark from the proton and the highly-evolved gluon distribution of the nucleus. Prior to the collision, the quark has only a ‘plus’ momentum $q^{+}_{0}$. After the scattering, which can involve one or several gluon exchanges with color sources from the target, the quark emerges with the same longitudinal momentum, $k^{+}=q^{+}_{0}$ (since gluons from the target have negligible ‘plus’ momenta), but it acquires a transverse momentum $\bm{k}$ and also a ‘minus’ component $k^{-}$, which is needed for the produced quark to be on-shell: $k^{-}=k_{\perp}^{2}/2k^{+}$. This condition fixes the total longitudinal momentum fraction carried by the gluons from the target that were involved in the collision⁵ : [FOOTNOTE:5][ENDFOOTNOTE] \[X_{g}=\,\frac{k^{-}}{P^{-}}\,=\,\frac{k_{\perp}^{2}}{2k^{+}P^{-}}\,=\,\frac{k_ {\perp}^{2}}{x_{p}s}\,\equiv\,\frac{k_{\perp}^{2}}{\hat{s}}\,.\] (1) To relate to the experimental situation, it is customary to express the longitudinal fractions $x_{p}$ and $X_{g}$ in terms of the rapidity $\eta\equiv(1/2)\ln(k^{+}/k^{-})$ of the produced quark in the center-of-mass frame (where $Q^{+}=P^{-}=\sqrt{s/2}$). Using $x_{p}=k^{+}/Q^{+}$ and $k^{+}=(k_{\perp}/\sqrt{2}){\rm e}^{\eta}$, one finds \[x_{p}=\frac{k_{\perp}}{\sqrt{s}}\,{\rm e}^{\eta}\,,\qquad X_{g}=\frac{k_{\perp }}{\sqrt{s}}\,{\rm e}^{-\eta}\,.\] (2) The _forward kinematics_ corresponds to the situation where $\eta$ is positive and large. Then Eq. (2) makes it clear that $X_{g}\ll x_{p}<1$, thus confirming that forward particle production explores the small-$X_{g}$ part of the nuclear wavefunction, as anticipated in the Introduction. ### Dipole picture To LO in the CGC effective theory, the ‘quark multiplicity’ (i.e. the distribution of the produced quarks in transverse momentum $\bm{k}$ and COM rapidity $\eta$) is computed as follows \[\frac{{\rm d}N^{pA\to qX}}{{\rm d}^{2}\bm{k}\,{\rm d}\eta}\bigg{\|}_{{\rm LO}}=\,\frac{1}{(2\pi)^{2}}\,x_{p}q(x_{p})\,{\mathcal{S}}( \bm{k},X_{g})\,,\] (3) where the kinematic variables $\bm{k}$, $\eta$, $x_{p}$, and $X_{g}$ have already been introduced, $x_{p}q(x_{p})$ is the quark distribution in the proton for a collinear quark with longitudinal momentum fraction $x_{p}$, and $\mathcal{S}(\bm{k},X_{g})$ is the Fourier transform of the elastic $S$–matrix for the scattering between a color dipole in the fundamental representation and the nucleus: \[{\mathcal{S}}(\bm{k},X_{g})=\int{\rm d}^{2}\bm{r}\,{\rm e}^{-{\rm i}\bm{k} \cdot\bm{r}}{S}(\bm{r},X_{g})\,.\] (4) The ‘color dipole’ is a quark-antiquark pair in a color-singlet state. In the present context, this appears as merely a mathematical representation for the cross-section for the scattering between the produced quark and the nucleus: the ‘quark’ component of the dipole is the colliding quark viewed in the direct amplitude (DA) and the ‘antiquark’ is the same physical quark, but viewed in the complex conjugate amplitude (CCA). To the accuracy of interest, the dipole-nucleus scattering can be computed in the eikonal approximation, i.e. the transverse coordinates of the quark ($\bm{x}$) and the antiquark ($\bm{y}$) can be treated as fixed during the collision. Then the only effect of the collision are color rotations of the two fermions, as described by Wilson lines extending along their trajectories: \[S(\bm{x},\bm{y};X)\,\equiv\,\frac{1}{N_{c}}\left\langle{\rm tr}\big{[}U(\bm{x} )U^{\dagger}(\bm{y})\big{]}\right\rangle_{X}.\] (5) Here, $U(\bm{x})$ and $U^{\dagger}(\bm{y})$ are Wilson lines in the fundamental representation, e.g., \[U(\bm{x})={\rm P}\exp\left[{\rm i}g\int{\rm d}x^{+}A^{-}_{a}(x^{ +},\bm{x})t^{a}\right],\] (6) and $A^{-}_{a}(x)$ is (the relevant component of) the color field representing the gluons from the target with longitudinal momentum fraction $X\equiv q^{-}/P^{-}$. In general, this field is strong (corresponding to large gluon occupation numbers) and the path-ordered phase in Eq. (6) resums multiple scattering to all orders. In the Fourier transform in Eq. (4), the transverse momentum $\bm{k}$ of the produced quark is conjugated to the dipole size $\bm{r}\equiv\bm{x}-\bm{y}$. Both the l.h.s. and the r.h.s. of Eq. (4) depend upon the impact parameter $\bm{b}\equiv(\bm{x}+\bm{y})/2$, but this dependence is unessential for what follows and will be omitted: for our purposes, the target can be treated as quasi-homogeneous in the transverse plane. The brackets in the r.h.s. of Eq. (5) denote the target average over the color field $A^{-}_{a}(x)$, as computed with the CGC weight functional [17; 18; 19]. By using the JIMWLK equation for the latter, or directly the Balitsky equations for the color dipole operator, one finds an equation for the evolution of the dipole $S$-matrix with decreasing $X$. In general, this is just the first equation from an infinite hierarchy, but the situation simplifies in the limit of a large number of colors $N_{c}\gg 1$, where the dipole $S$-matrix obeys a closed, non-linear, equation, known as the Balitsky-Kovchegov (BK) equation [23; 30]. This equation will be later needed, so let us display it here: \[\frac{\partial}{\partial Y}\,S(\bm{x},\bm{y};Y)=\frac{\bar{\alpha }_{s}}{2\pi}\,\int{\rm d}^{2}\bm{z}\,\frac{(\bm{x}-\bm{y})^{2}}{(\bm{x}-\bm{z} )^{2}(\bm{z}-\bm{y})^{2}}\,\Big{[}S(\bm{x},\bm{z};Y)S(\bm{z},\bm{y};Y)-S(\bm{x },\bm{y};Y)\Big{]}\,,\] (7) where $\bar{\alpha}_{s}\equiv\alpha_{s}N_{c}/\pi$ and $Y\equiv\ln(1/X)$. The integration variable $\bm{z}$ in Eq. (7) represents the transverse coordinate of a soft gluon with longitudinal fraction $X=q^{-}/P^{-}\ll 1$, which is emitted by ‘fast’ color sources from the target (valence quarks and gluons from the previous generations, with momentum fractions $X^{\prime}\gg X$) and absorbed by the projectile dipole. Eq. (7) must be integrated from some lower value $Y_{0}\equiv\ln(1/X_{0})$, where one can use a low-energy model for the nuclear gluon distribution, up to $Y_{g}\equiv\ln(1/X_{g})=\ln(\hat{s}/k_{\perp}^{2})$, where we compute the quark production. Typically, $X_{0}\sim 1\gg X_{g}$. For instance, if one uses a valence-quark model for the nucleus, like the McLerran-Venugopalan (MV) model [71; 72], then $X_{0}$ must satisfy $\bar{\alpha}_{s}\ln(1/X_{0})\ll 1$ and the whole gluon distribution is built up via evolution. In the above discussion, we have privileged the viewpoint of _target evolution_, that is, we have described Eq. (7) as the result of a change in the gluon distribution of the target. The complementary point of view, that of _projectile_ evolution, will be useful too for what follows and will be introduced in the next subsection. Also, we implicitly assumed that the relation (3) between the quark transverse momentum broadening and the dipole $S$-matrix remains valid in the presence of the high-energy evolution; that is, both sides of Eq. (3) evolve in exactly the same way with increasing energy (i.e. with increasing $\eta$, or decreasing $X_{g}$). This is known to be true, at least, up to NLO accuracy, as demonstrated in Ref. [4] for the LO BK evolution and in Ref. [31] for the NLO one. However, Eq. (3) is not complete beyond leading order: to this ‘dipole’ piece, one must add the ‘corrections to the impact factor’, that is, the contributions from partonic configurations which do not reduce to either the dipole, or its high-energy evolution. Such corrections will represent a main topic of the NLO discussion in Sect. 3. ### Target versus projectile evolution Previously, we have insisted that the physical picture of the high-energy evolution depends upon the choice of a frame, but as a matter of facts the BK equation (7) holds exactly as written in _any_ frame that is obtained from the COM frame via a boost. This includes the infinite momentum frame of the nucleus that we have considered so far, but also the corresponding frame for the projectile, where the incoming proton carries most of the total energy and the high-energy evolution of interest refers to the emission of soft gluons in the wavefunction of the colliding dipole (or quark). By ‘soft gluons’ in this case, one means gluons which carry small fractions $x\equiv p^{+}/q^{+}_{0}\ll 1$ of the longitudinal momentum $q^{+}_{0}$ of the parent quark. This ‘boost invariance’ of the LO BK equation is not an automatic consequence of the underlying Lorentz symmetry of the problem — after all, the respective evolution variables are different: $Y=\ln(1/X)$ for the target evolution and $y\equiv\ln(1/x)$ for that of the projectile. Rather, it reflects approximations specific to the LLA at hand, whose effect is to identify these two variables $Y$ and $y$ to the accuracy of interest (up to a change of sign). In other terms, at LLA, the fact of decreasing $Y$ is indeed equivalent with increasing $y$, meaning that the evolution can be progressively transferred from the projectile to the target, and back. This point will play an important role in our subsequent discussion of the NLO contribution to particle production. In preparation for that, let us briefly remind here the kinematical assumptions underlying the LLA and thus expose their limitations. To that aim, we consider the situation where only one of the soft gluons has been emitted by the quark, while all the other ones belong to the wavefunction of the target (see Fig. 1). The gluon that has been singled out in this way is the one to be closest in rapidity ($y$) to the incoming quark; we shall refer to it as the _primary gluon_ and write its longitudinal and transverse momenta as $p^{+}=xq^{+}_{0}$ and respectively $\bm{p}$. Consider a ‘real’ graph in which the primary gluon, albeit unmeasured, is released in the final state⁶. Then longitudinal momentum conservation implies $k^{+}=(1\!-\!x)q^{+}_{0}=x_{p}Q^{+}$ and therefore $q_{0}^{+}/Q^{+}=x_{p}/(1\!-\!x)$. [Recall that $x_{p}$ is defined as the boost-invariant ratio $x_{p}\equiv k^{+}/Q^{+}$, which in the COM frame takes the form shown in Eq. (2).] [FOOTNOTE:6][ENDFOOTNOTE] Both the quark and the primary gluon must be on mass-shell in the final state. Then, light-cone energy conservation implies that the scattering off the nuclear target must transfer a total ‘minus’ component $q^{-}=X(x,p_{\perp})P^{-}$ with (recall that $\hat{s}=x_{p}s$) \[X(x,p_{\perp})\,=\,\frac{1}{P^{-}}\left(\frac{k_{\perp}^{2}}{2k^{+}}+\frac{p_{ \perp}^{2}}{2p^{+}}\right)=\frac{1}{\hat{s}}\left[k_{\perp}^{2}+\frac{1\!-\!x} {x}\,{p_{\perp}^{2}}\right]\,.\] (8) The LLA essentially relies on the two following kinematical assumptions: (i) The longitudinal fraction of the emitted gluon is small: $x\ll 1$. This allows one to simplify the calculation, notably by computing the quark-gluon vertex in Fig. 1 in the eikonal approximation. (ii) The transverse momenta of the successive emissions are parametrically of the same order: $k_{\perp}\sim p_{\perp}$ or, more precisely, $\ln(1/x)\gg\|\ln(k_{\perp}^{2}/p_{\perp}^{2})\|$. This condition is necessary to simplify the energy denominators (by neglecting $k^{-}\equiv{k_{\perp}^{2}}/{2k^{+}}$ compared to $p^{-}\equiv{p_{\perp}^{2}}/{2p^{+}}$) in the study of soft successive emissions and thus obtain the LO BK (or BFKL) equation. <figure><img src="content_image/1608.05293/x1.png"><figcaption>Figure 1: An illustration of the LO high-energy evolution of the ‘cross-section for quark production’ S(k,Xg) (the Fourier transform of the dipoleS–matrix). The ‘primary gluon’ (the first gluon emission by the quark, whichcarries a longitudinal momentum fraction x≪1 is) viewed as a part of thewavefunction of the incoming quark (a right mover). The subsequent emissionsare rather associated with the evolution of the gluon distribution in thenuclear target (a left mover), within the range X(x)<X<X0, with X(x)=Xg/x andX0∼1. Alternatively, and equivalently at LLA, they can be associated with theevolution of the wavefunction of the primary quark-gluon pair (a right mover)within the ‘plus’ momentum range xg<x′<x.</figcaption></figure> Under these assumptions, the second term in the r.h.s. of Eq. (8) (the light-cone energy of the primary gluon) dominates over the first one and, moreover, one can ignore the difference between $p_{\perp}$ and $k_{\perp}$ when computing the evolution variables $Y=\ln(1/X)$ and $y=\ln(1/x)$. The above argument can be immediately extended to an arbitrary separation of the LO high-energy evolution between the quark projectile and the nuclear target: successive emissions in the projectile are strongly ordered in $x$ but have comparable transverse momenta, hence both energy conservation and the energy denominators are controlled by the last emitted gluon — the one with the smallest value of $x$ and a transverse momentum of order $k_{\perp}$. Accordingly, instead of Eq. (8), one can use the following, simpler, relation, \[X(x)\,=\,\frac{k_{\perp}^{2}}{x\hat{s}}\,=\,\frac{X_{g}}{x}\,,\] (9) (or $Y=Y_{g}-y$) in order to connect the LO evolution of the projectile to that of the target. The differences between (8) and (9) become however important starting with NLO, as we shall see. The above discussion can be summarized by the following integral representation of the solution to the BK equation, illustrated in Fig. 1, in which the total evolution is explicitly split between exactly one soft gluon ($x\ll 1$) in the quark wavefunction and an arbitrary number of soft gluons ($X\ll 1$) in the wavefunction of the target: \[S\big{(}\bm{x},\bm{y};X_{g}\big{)}=S_{0}(\bm{x},\bm{y})+\frac{ \bar{\alpha}_{s}}{2\pi} \int_{x_{g}}^{1}\frac{{\rm d}x}{x}\int{\rm d}^{2}\bm{z}\,\frac{( \bm{x}-\bm{y})^{2}}{(\bm{x}-\bm{z})^{2}(\bm{z}-\bm{y})^{2}}\] \[\,\Big{[}S\big{(}\bm{x},\bm{z};X(x)\big{)}S\big{(}\bm{z},\bm{y};X (x)\big{)}-S\big{(}\bm{x},\bm{y};X(x)\big{)}\Big{]}\,.\] (10) In this equation, $S_{0}$ is the initial condition at $X_{0}\simeq 1$ and the lower limit $x_{g}\equiv k_{\perp}^{2}/\hat{s}$ for the integral over $x$ corresponds via (9) to $X=1$, i.e. to the situation where the soft gluon from the projectile probes bare nucleons from the target. Eq. (2.3) can also be viewed as purely target evolution provided one changes the integration variable from $x$ to $X\equiv X(x)$. Then it becomes obvious that this is the same as Eq. (7) integrated over $Y$, from $Y_{0}=0$ up to $Y_{g}$. <figure><img src="content_image/1608.05293/x2.png"><figcaption>Figure 2: Typical diagrams contributing to the high-energy evolution of thequark production at leading order. (a) Real diagram, in which all possibleinteractions of the gluon with the target cancel one another. (b) Real diagramin which the gluon in the CCA is emitted before the collision with theshockwave, while in the DA is emitted after the collision. (c,d) Virtualdiagrams in which the gluon interacts, or not, with the shockwave. Onlytransverse momenta are shown in all the graphs.</figcaption></figure> In Fig. 2, we present some Feynman graphs which contribute to the integral term in Eq. (2.3). We do not use the dipole picture, rather we show graphs which enter the cross-section for quark production (so, in particular, we use the transverse momentum representation). The nuclear target evolved up to $X(x)$ is represented as a shockwave (recall that we are still in a frame where the target is ultrarelativistic). There are two types of graphs: ‘real’, where the primary gluon appears in the final state — it is emitted in the direct amplitude (DA) and reabsorbed in the complex conjugate amplitude (CCA) — and ‘virtual’, where the gluon is both emitted and reabsorbed on the same side of the cut (either in the DA, or in the CCA). It is instructive to notice how such graphs are generated from the BK equation (2.3): decomposing the dipole kernel there as \[{\mathcal{M}}_{\bm{x}\bm{y}\bm{z}}\equiv\frac{(\bm{x}-\bm{y})^{2}}{(\bm{x}-\bm {z})^{2}(\bm{y}-\bm{z})^{2}}=\frac{1}{(\bm{x}-\bm{z})^{2}}+\frac{1}{(\bm{y}- \bm{z})^{2}}-2\,\frac{x^{i}-z^{i}}{(\bm{x}-\bm{z})^{2}}\frac{y^{i}-z^{i}}{(\bm {y}-\bm{z})^{2}}\,,\] (11) one can check that the ‘virtual’ terms are generated by the first 2 terms in the r.h.s. of Eq. (11), whereas the ‘real’ terms come from the third one. Note that, for the particular ‘real’ term where the gluon crosses the shockwave twice, cf. Fig. 2.a, the gluon interactions cancel between the DA and the CCA, by unitarity, hence the respective contribution to the r.h.s. of Eq. (2.3) involves only the scattering of the original quark (as described by the dipole $S$-matrix $S\big{(}\bm{x},\bm{y};X(x)\big{)}$). Given the ‘boost-invariance’ of the (LO) BK equation alluded to above, one may wonder what can be the utility of dividing the evolution between target and projectile, as we did above. As a matter of facts, there are several advantages for doing that. First, one should keep in mind that the laboratory frame for ‘dilute-dense’ (d+Au or p+Pb) collisions at RHIC and the LHC coincides with the COM frame (at RHIC), or is close to it (at the LHC). Hence the picture of the high-energy evolution which is directly visible in the experiments is that of an evolution shared by the two incoming hadrons. Second, we shall shortly argue that the first gluon emission by the incoming quark (the ‘primary gluon’) plays in fact a special role, at least for relatively large $k_{\perp}\gtrsim Q_{s}$. Because of that, it is preferable (and even compulsory, starting with NLO) to view this gluon as a part of the quark evolution, like in Eq. (2.3). Still beyond LO, it is conceptually simpler to associate the high-energy evolution with the target wavefunction. As we shall see, the complete result for the quark multiplicity to NLO, to be presented in Sect. 3, can be viewed as a natural generalization of Eq. (2.3). ### Hard transverse momentum and di-jet events In this subsection, we shall discuss the physical picture of forward quark production in the IMF of the projectile, or, more generally, in any ‘mixed’ frame, like that illustrated in Fig. 1, where the quark wavefunction contains at least one soft gluon. We would like to show that, in any such a frame, the tail of the quark distribution at relatively high $k_{\perp}$ comes from the recoil in the emission of the _primary gluon_ (the first gluon emitted by the quark). That is, a forward quark with large transverse momentum $k_{\perp}\gtrsim Q_{s}$ is produced in a _di-jet event_ where the quark is accompanied by a recoil gluon and the two particles propagate back-to-back in the transverse plane. This point is important in that it will affect the NLO calculation of the quark production at relatively high $k_{\perp}$, where the negativity problem in the cross-section has been observed. At a first sight, the prominence of the di-jet configuration at large $k_{\perp}$ might look rather obvious, as an immediate consequence of transverse momentum conservation at the emission vertex. But the situation is a bit more subtle, since a large transverse momentum can also be transferred by the target, via a sufficiently hard scattering. As a matter of facts, in the classical approximation at low energy (i.e. in the absence of any evolution), a power-like tail $\propto 1/k_{\perp}^{4}$ in the quark distribution at high $k_{\perp}$ is generated via Coulomb scattering (see below). The same physical picture would also hold at high energy (within the limits of the LLA), but only in the target IMF, where there is no gluon emission by the quark. But in a frame where the quark itself is allowed to radiate, the phase-space for high-energy evolution at high $k_{\perp}$ favors configurations where the momentum $\ell_{\perp}$ transferred from the target to the projectile (the quark together with its small-$x$ radiation) is relatively low, $\ell_{\perp}\ll k_{\perp}$. Because of that, the only way to produce a quark with very large $k_{\perp}$ is via a di-jet event, as anticipated. Consider first the semi-classical approximation (no evolution), that we shall treat within the MV model. Since we are interested in a relatively hard quark with $k_{\perp}\gg Q_{s}$, we can limit ourselves to the single-scattering approximation, as obtained by expanding the Wilson lines in Eq. (5) up to second order in the target color fields $A^{-}$ (see Fig. 3). Writing $S=1-T$, one finds the dipole scattering amplitude in the 2-gluon exchange approximation as (recall that $\bm{r}=\bm{x}-\bm{y}$) \[T_{0}(\bm{r}) =\frac{g^{2}}{2N_{c}}\left\langle\big{(}A^{-}_{a}(\bm{x})-A^{-}_{ a}(\bm{y})\big{)}^{2}\right\rangle\] \[=g^{2}C_{F}\!\int\frac{{\rm d}^{2}\bm{q}}{(2\pi)^{2}}\frac{\mu^{2 }}{q_{\perp}^{4}}\big{[}1-{\rm e}^{i\bm{q}\cdot\bm{r}}\big{]}\simeq\frac{ \alpha_{s}C_{F}}{4}\,r^{2}\mu^{2}\ln\frac{1}{r^{2}\Lambda^{2}}\,,\] (12) where in the second line we have used the MV model expression for the 2-point correlator of the color fields in a dense nucleus (with atomic number $A$ and transverse area $\pi R_{A}^{2}$), namely \[\int{\rm d}x^{+}{\rm d}y^{+}\left\langle A^{-}_{a}(x^{+},\bm{q}) \,A^{-}_{b}(y^{+},\bm{\ell})\right\rangle_{{\rm MV}}=(2\pi)^ {2}\delta^{(2)}(\bm{q}+\bm{\ell})\delta_{ab}\,\frac{\mu^{2}}{\bm{q}^{4}}\,, \qquad\mu^{2}=\frac{g^{2}C_{F}AN_{c}}{(N_{c}^{2}-1)\pi R_{A}^{2}}\,.\] (13) The quantity $\mu^{2}$ represents the color charge squared of the $AN_{c}$ valence quarks (treated as uncorrelated color sources) per unit transverse area. The variable $\bm{q}$ that is integrated over in Eq. (2.4) is the transverse momentum transferred from the target to the dipole and $\Lambda$ is an infrared cutoff (say, the confinement scale). The unit term within the square brackets corresponds to the case where the two exchanged gluons are attached to a same quark leg within the dipole, while the exponential ${\rm e}^{i\bm{q}\cdot\bm{r}}$ refers to attachments to both legs (see Fig. 3). For relatively small dipole sizes $r\ll 1/\Lambda$, the integral over $q_{\perp}$ develops a transverse logarithm which can be isolated by expanding out the exponential to second order. This yields the final result shown in Eq. (2.4). When this result becomes of $\mathcal{O}{(1)}$, multiple scattering becomes important and the above approximation breaks down. This condition defines the target saturation momentum $Q_{0}$ at low energy: $T_{0}(r)\sim 1$ for $r\simeq 1/Q_{0}$. <figure><img src="content_image/1608.05293/x6.png"><figcaption>Figure 3: Diagrams for the elastic scattering of the dipole in the singlescattering approximation, or 2 gluon exchange. The blob at the bottom of thediagram refers to the average over the color fields in the target, whicheffectively generates the gluon distribution on the resolution scale r of thedipole projectile.</figcaption></figure> This simple calculation makes it clear that the scattering of a small dipole ($1/r\gg Q_{0}$) is controlled by relatively soft gluon exchanges ($\Lambda\ll q_{\perp}\ll 1/r$) with the target. Let us similarly compute the quark production, for a quark with large transverse momentum $k_{\perp}\gg Q_{0}$. When taking the Fourier transform of $T_{0}(\bm{r})$, the unit term within the square brackets in Eq. (2.4) does not matter (this would describe an elastic scattering without net momentum transfer; see Fig. 4), whereas the exponential term there selects $\bm{q}=\bm{k}$. This is simply the expression of momentum conservation and confirms that one needs a hard (inelastic) scattering in order to produce a hight-$k_{\perp}$ quark. One thus finds⁷$\mathcal{T}_{0}(k_{\perp})=g^{2}C_{F}\mu^{2}/k_{\perp}^{4}$, which is recognized as the Rutherford cross-section for the Coulomb scattering between the quark and the nucleus; therefore, [FOOTNOTE:7][ENDFOOTNOTE] \[\frac{{\rm d}N^{pA\to qX}}{{\rm d}^{2}\bm{k}\,{\rm d}\eta}\bigg{\|}_{{\rm MV}}\simeq\,x_{p}q(x_{p})\,\frac{\alpha_{s}C_{F}\mu^{2}}{ \pi k_{\perp}^{4}}\,\qquad\mbox{for}\quad k_{\perp}\gg Q_{0}\,.\] (14) We shall now study the high-energy evolution of the above results, in the double logarithmic approximation (DLA) which is appropriate for sufficiently small dipole sizes, or large $k_{\perp}$. For the present purposes, it is convenient to work in a frame where this is viewed as _projectile_ evolution; that is, the soft gluons belong to the wavefunction of the quark and they are all right movers. <figure><img src="content_image/1608.05293/x7.png"><figcaption>Figure 4: Diagrams for quark production in the 2-gluon exchange approximation.The diagram on the left describes an elastic scattering in the DA and noscattering in the CCA; hence it contributes to quark production only for k=0.The diagram on the right describes an inelastic scattering in the DA andanother one in the CCA. The final momentum k of the produced quark istransferred by the target. Momenta are flowing from left to right both in theDA and in the CCA and from bottom to top in the exchange (red) gluons.</figcaption></figure> In transverse coordinate space, the DLA corresponds to the splitting of the original dipole $(\bm{x},\,\bm{y})$ into two daughter dipoles, $(\bm{x},\,\bm{z})$ and $(\bm{z},\,\bm{y})$, whose transverse sizes are much larger, but still small enough to undergo only single scattering: $r\ll\bar{z}\ll 1/Q_{0}$, with $\bar{z}\equiv\|\bm{x}-\bm{z}\|\simeq\|\bm{z}-\bm{y}\|$. The respective evolution equation is obtained from the general BK equation (7) by (i) linearizing w.r.t. $T=1-S$ (by itself, this step yields the BFKL equation), then (ii) approximating the dipole kernel as ${\mathcal{M}}_{\bm{x}\bm{y}\bm{z}}\simeq r^{2}/\bar{z}^{4}$, and (iii) keeping only the scattering amplitudes $T(\bm{x},\bm{z})+T(\bm{z},\bm{y})\simeq 2T(\bar{z})$ for the two daughter dipoles, whose scattering is stronger (since $T(r)\propto r^{2}$ in this physical regime). One thus finds (as before, we use $y=\ln(1/x)$ for the evolution ‘time’ of the projectile) \[\frac{\partial}{\partial y}\,T(r,y)={\bar{\alpha}_{s}}\,r^{2}\int _{r^{2}}^{1/Q_{0}^{2}}\frac{{\rm d}\bar{z}^{2}}{\bar{z}^{4}}\,T(\bar{z},y)\,.\] (15) Since $T(\bar{z})\propto\bar{z}^{2}$, the integral in the r.h.s. is clearly logarithmic. The first iteration of this equation, as obtained by evaluating its r.h.s. with the amplitude $T_{0}$ from Eq. (2.4), describes the first gluon emission by the parent dipole. The physical picture of this emission follows from the previous discussion: the original dipole with size $r$ emits a relatively soft gluon with transverse momentum $p_{\perp}\sim 1/\bar{z}$ within the range $Q_{0}\ll p_{\perp}\ll 1/r$, which then suffers an even softer scattering off the nuclear target, with transferred momentum $\Lambda\lesssim q_{\perp}\ll p_{\perp}$ (see Fig. 5 left). This picture extends to the whole gluon cascade generated by iterating Eq. (15): successive gluon emissions are strongly ordered not only in $x$ but also in transverse momenta, and the final exchange with the target is even softer. <figure><img src="content_image/1608.05293/x8.png"><figcaption>Figure 5: Left: one step in the DLA evolution of a small dipole, with sizer≪1/Qs. The daughter gluon is typical soft and thus emitted at a largedistance \|z−x\|≃\|z−y\|≫r from the parent dipole. The gluon exchange q with thenuclear target is even softer. Right: one step in the DLA evolution of thecross-section for quark production, at large transverse momentum k⊥≫Qs. Theprimary gluon is as hard as the produced quark and they are both much harderthan the gluon exchanged with the target: k⊥≃p⊥≫q⊥=\|k+p\|. The other diagramscontributing to this process at the level of the amplitude are shown in Fig.6.</figcaption></figure> We now turn to the corresponding picture in transverse momentum space, that is, to the problem of quark production (see Fig. 5 right and also Fig. 6). The momentum-space DLA equation reads \[\frac{\partial}{\partial y}\,\mathcal{T}(k_{\perp},y)=\frac{\bar{ \alpha}_{s}}{k_{\perp}^{4}}\int_{Q_{0}^{2}}^{k_{\perp}^{2}}{\rm d}q_{\perp}^{2 }\,q_{\perp}^{2}\mathcal{T}(q_{\perp},y)\,,\] (16) where the integral in the r.h.s. is indeed logarithmic, since $\mathcal{T}(q_{\perp},y)\propto 1/q_{\perp}^{4}$. Within this integral, $\mathcal{T}(q_{\perp},y)$ should be interpreted as the cross-section for a single scattering, with transferred momentum $q_{\perp}$, between partons in the quark wavefunction and the target. The factor ${\bar{\alpha}_{s}}/{k_{\perp}^{4}}$ in front of the integral does not represent anymore a $t$-channel exchange with the target, as in Eq. (14), but rather it comes from the propagator of the intermediate quark, or gluon, in the $s$-channel (see Fig. 6). Hence, the physical picture of the first emission is now as follows: the original quark with zero transverse momentum emits a gluon with momentum $\bm{p}$ and turns into a final quark with momentum $\bm{k}$, while at the same time receiving a momentum transfer $\bm{q}$ from the target (via a scattering that can occur either before, or after the splitting). Transverse momentum conservation requires $\bm{q}=\bm{k}+\bm{p}$. But the overall cross-section, as described by Eq. (16), favors _soft_ scattering, with transferred momenta $q_{\perp}\ll k_{\perp}$. Accordingly, the first emitted gluon must be hard, $p_{\perp}\simeq k_{\perp}$, to balance the momentum of the produced quark. <figure><img src="content_image/1608.05293/x10.png"><figcaption>Figure 6: The 3 diagrams which contribute to the production of a quark-gluonpair in the final state in the regime where both the quark and the gluon arerelatively hard (k⊥≃p⊥≫q⊥=\|k+p\|).</figcaption></figure> As for the subsequent gluon emissions, starting with the second one, they follow the standard DLA ordering, in both $x$ and $p_{\perp}$, as in the respective calculation in coordinate space, cf. Eq. (15). This argument too shows that, when computing particle production, it is quite natural to associate the primary gluon with the wavefunction of the produced particle, whereas the other gluons are more conveniently included in the gluon distribution of the target, as measured by the hard splitting process. While natural already at LLA, this viewpoint becomes almost unavoidable when moving to the next-to-leading order calculation, where the primary gluon is also allowed to have a large longitudinal momentum $p^{+}\sim q^{+}_{0}$. The NLO calculation will be discussed in the next section. ## 3 Next-to-leading order In order to move on to next-to-leading order (NLO) accuracy, one must relax some of the previous approximations and add new contributions which start at NLO. By inspection of the LO result (3), it is clear that one ingredient required in that sense is the NLO version of the B-JIMWLK (or BK) equations [32; 33; 34], together with their all-order ‘collinear’ resummations [44; 37; 45; 46]. This in particular means that some gluon emissions must be computed beyond the eikonal approximation: besides the effect of order $\alpha_{s}Y$, which dominates at high energy, one must also keep, for each such an emission, the ‘pure-$\alpha_{s}$’ corrections which are not enhanced by the rapidity logarithm $Y$ (but may be accompanied by transverse logarithms). So long as these NLO corrections refer to generic gluons inside the cascade, they can be absorbed into a renormalization of the kernel of the evolution equation. The same is true for the quark-antiquark loop which at NLO can be inserted within any of the gluon lines. But the NLO corrections associated with the ‘primary gluon’ (the very first emission by the leading quark) must rather be used to renormalize the ‘impact factor’, i.e. the value of the cross-section in the absence of high-energy evolution. At LO, the impact factor is the cross-section for the inelastic scattering between the leading quark and the low energy nucleus (say, as described by the MV model). Equivalently (to the accuracy of interest), it can be written as the $S$–matrix for the elastic scattering of a $q\bar{q}$ dipole. At NLO, one must add the impact factor encoding the inelastic scattering of the quark-gluon pair made with the leading quark and the primary gluon. Unlike the emission of the primary gluon, which must be computed exactly, the scattering between the quark-gluon pair and the target can still be computed in the eikonal approximation and thus related to elastic scattering amplitudes for color multipoles [63; 65; 2]. So, it may look like, in order to compute quark production at NLO, one must dress the two contributions to the impact factor aforementioned with the high-energy evolutions of the respective scattering amplitudes (themselves computed at NLO) and then add the results. But a moment of thinking reveals that the two pieces of the impact factor mix with each other under the high-energy evolution: a part of the primary gluon emission that we have explicitly included in the NLO impact factor is also included (within the limits of the eikonal approximation) as the first small-$x$ gluon in the evolution of the dipole $S$–matrix from the LO cross-section (3). This is the problem of over-counting. Previous papers in the literature [1; 2] proposed a solution to this problem, in the form of a ‘plus’ prescription which subtracts the LO evolution from the NLO impact factor. This prescription however appears to be responsible for the problem with the negativity of the cross-section discussed in the Introduction. In what follows, we shall propose a different way to organize the calculation, which avoids the over-counting without performing any subtraction. Our strategy will naturally exploit the structure of perturbation theory at high energy. As we shall see, the contribution to the cross-section which includes the NLO correction to the impact factor does also encode, completely and faithfully, the LO evolution of the dipole $S$-matrix. Hence, by computing this contribution as it stands, one can simultaneously include both effects without any ambiguity, or over-counting. On top of that, there is a NLO correction to the evolution of the color dipole; this will be clearly identified and related to recent results concerning the NLO version of the BK equation [32] and its collinear resummations [37; 45]. ### Revisiting the NLO calculation by Chirilli, Xiao, and Yuan In this subsection, we shall exhibit, discuss, and adapt to our present purposes the result of the NLO calculation of the impact factor by Chirilli, Xiao, and Yuan [1; 2]. First, we shall display their ‘bare’, or ‘unsubtracted’, result, where the soft divergence⁸ at $x\to 0$ is explicit. Then we shall briefly mention the ‘plus’ prescription advocated in Refs. [1; 2] in order to subtract the rapidity divergence. (We shall return to this point in Sect. 3.5.) Finally, we shall explain our strategy to deal with this problem, which is to use kinematical constraints like energy conservation in order to cut off the soft divergence and at the same time fix the rapidity variables for the evolution of the dipole $S$-matrices. The only subtle point here is the treatment of the virtual corrections, where the phase-space for the emission of the primary gluon is not directly constrained by the kinematics. Yet, as we shall demonstrate via explicit calculations (in Appendix A), the same lower limit on $x$ applies in that case too, albeit its emergence is now _dynamical_. [FOOTNOTE:8][ENDFOOTNOTE] <figure><img src="content_image/1608.05293/x11.png"><figcaption>Figure 7: Pictorial representation of a typical amplitude contributing to theNLO piece of the impact factor. This is a ‘real’ amplitude, in the sense thatthe primary gluon is released in the final state.</figcaption></figure> The NLO result in Refs. [1; 2] has been obtained by evaluating Feynman graphs like that illustrated in Fig. 7 in which the emission of the primary gluon is treated exactly. There is a similar graph where the gluon emission occurs after the scattering between the quark and the target. And there are of course virtual graphs, whose evaluation is somewhat subtle as just mentioned and that we shall deal with in some detail. (See Figs. 8 and 9 below for more examples of Feynman graphs.) After the scattering, both the quark and the gluon will fragment into hadrons and thus contribute to single-inclusive hadron production. There is also another channel where the original collinear parton is a gluon, which splits into a pair of gluons, or into a quark-antiquark pair, in the process of scattering. As before, we shall omit the discussion of the fragmentation process and concentrate on quark production alone (see Refs. [1; 2] for a complete discussion and also [50] for an alternative calculation, whose precise relation to the original results in [1; 2] is still unclear). That is, the primary gluon is not measured, so one needs to integrate out its kinematics — the longitudinal momentum fraction $x=p^{+}/q_{0}^{+}$ and the transverse momentum $\bm{p}$. The NLO result in Refs. [1; 2] can be conveniently written as the sum of 2 pieces⁹: [FOOTNOTE:9][ENDFOOTNOTE] (A) A piece proportional to the quark Casimir $C_{\rm F}$ which develops no logarithm at small $x$ (the respective integrand vanishes as $x\to 0$), but has collinear divergences in the transverse momentum integrations. In [1; 2], these divergences have been isolated with the help of dimensional regularization and reabsorbed into the leading-order DGLAP evolution of the quark distribution function $q(x_{p})$ (if the primary emission occurs prior to scattering) and of the quark-to-hadrons fragmentation function (if the emission occurs after the scattering). This prescription leaves a finite remainder of NLO order whose explicit evaluation poses no special problem. (B) A piece proportional to the gluon Casimir $N_{\rm c}$ which is free of collinear problems but develops a logarithm at small $x$ (the respective integral over $x$ exhibits a logarithmic divergence at $x\to 0$ in the absence of any physical regulator). The proper way to deal with this ‘rapidity divergence’ at small $x$ represents our main concern in this paper. To better focus on this problem while avoiding cumbersome notations, we shall omit the piece proportional to $C_{\rm F}$ in what follows. (This piece can be easily added to our main result shown in Eq. (3.2) below.) As for the second piece, proportional to $N_{\rm c}$, we start by displaying the original result, as presented in Refs. [1; 2] : \[\frac{{\rm d}N^{pA\to qX}}{{\rm d}^{2}\bm{k}\,{\rm d}\eta}\bigg{\|}_{{\rm NLO}}^{\rm unsub}=\frac{\alpha_{s}N_{\rm c}}{(2\pi)^{2}} \int_{0}^{1}{\rm d}\xi\,\frac{1+\xi^{2}}{1-\xi}\left\{\frac{x_{p}}{\xi}q\left( \frac{x_{p}}{\xi}\right)\mathcal{J}(\bm{k},\xi)-x_{p}q\left(x_{p}\right) \mathcal{J}_{v}(\bm{k},\xi)\right\},\] (17) where $\xi\equiv 1-x$ and the two functions $\mathcal{J}(\bm{k},\xi)$ and $\mathcal{J}_{v}(\bm{k},\xi)$ correspond to real and virtual contributions to the process illustrated in Fig. 7. They read (our present notations are slightly different from the original ones Refs. [1; 2], but follow closely the recent paper [56]) \[\mathcal{J}(\bm{k},\xi)=\!\int\frac{{\rm d}^{2}\bm{q}}{(2\pi)^{2}}\frac{2(\bm{ k}-\xi\bm{q})\cdot(\bm{k}-\bm{q})}{(\bm{k}-\xi\bm{q})^{2}(\bm{k}-\bm{q})^{2}} \mathcal{S}(\bm{q})-\int\!\frac{{\rm d}^{2}\bm{q}}{(2\pi)^{2}}\frac{{\rm d}^{2 }\bm{\ell}}{(2\pi)^{2}}\frac{2(\bm{k}-\xi\bm{q})\cdot(\bm{k}-\bm{\ell})}{(\bm{ k}-\xi\bm{q})^{2}(\bm{k}-\bm{\ell})^{2}}\mathcal{S}(\bm{q})\mathcal{S}(\bm{ \ell}),\] (18) and respectively \[\mathcal{J}_{v}(\bm{k},\xi)=\mathcal{S}(\bm{k})\left[\int\!\frac{{\rm d}^{2} \bm{q}}{(2\pi)^{2}}\frac{2(\xi\bm{k}-\bm{q})\cdot(\bm{k}-\bm{q})}{(\xi\bm{k}- \bm{q})^{2}(\bm{k}-\bm{q})^{2}}-\int\!\frac{{\rm d}^{2}\bm{q}}{(2\pi)^{2}} \frac{{\rm d}^{2}\bm{\ell}}{(2\pi)^{2}}\frac{2(\xi\bm{k}-\bm{q})\cdot(\bm{\ell }-\bm{q})}{(\xi\bm{k}-\bm{q})^{2}(\bm{\ell}-\bm{q})^{2}}\mathcal{S}(\bm{\ell}) \right].\] (19) As before, the dipole $S$-matrices like $\mathcal{S}(\bm{k})$ or $\mathcal{S}(\bm{q})$ refer to dipoles in the fundamental representation (cf. footnote 9). To simplify writing, we have considered the large $N_{\rm c}$ limit, in which the scattering of a system of two dipoles factorizes as the product of two individual dipole $S$-matrices, but this limit is not essential for what follows. The variables $\bm{q}$ and $\bm{\ell}$ which appear in the above integrations represent transverse momenta exchanged between the target and the quark-gluon pair. For what follows, it is important to understand their precise meaning and notably their relation with the transverse momentum $\bm{p}$ taken by the primary gluon. By following the derivation of these results in Refs. [1; 2], one can check that $\bm{q}=\bm{p}+\bm{k}$ whereas $\bm{\ell}$ is independent of $\bm{p}$. For more clarity, let us briefly discuss the physical interpretation of the various terms in Eqs. (18) and (19). The ‘real’ terms in Eq. (18) represent processes where the primary gluon, albeit not measured, is released in the final state (see Fig. 8). For such processes, longitudinal momentum conservation implies $\xi=k^{+}/q^{+}_{0}$. The first term in Eq. (18), which is linear in $\mathcal{S}(\bm{q})$, represents situations where the hard splitting occurs either after the collision, or prior to it, in _both_ the DA and the CCA. In these cases, the gluon either does not interact with the target at all (emissions after the collision), or the effects of its interaction cancel out from the final result, by unitarity, because the gluon is not measured (emissions before the collision). Accordingly, there is only one dipole $S$-matrix, $\mathcal{S}(\bm{q})$, which physically describes the _inelastic_ scattering of the quark. This scattering transfers a non-zero transverse momentum $\bm{q}$ to the quark; then momentum conservation implies $\bm{q}=\bm{p}+\bm{k}$, as aforementioned. <figure><img src="content_image/1608.05293/x12.png"><figcaption>Figure 8: Production of a quark with a transverse momentum k. Typical realdiagrams, i.e. diagrams in which the gluon is crossing the cut, but at thesame time is integrated (cf. Eq. (18)). The primary gluon is represented as aq¯q pair, as appropriate at large Nc. (a) Left: Diagram contributing to thereal term proportional to S(q) and which originates from S(x,y) in coordinatespace. All possible interactions of the gluon with the target cancel eachother. (b) Right: Diagram contributing to the real term proportional toS(q)S(ℓ) and which originates from S(x,z)S(z,y) in coordinate space. In bothdiagrams the target transfers momentum q=p+k to the final state. Momenta areflowing from left to right both in the DA and in the CCA and from bottom totop in the exchange (red) gluons.</figcaption></figure> <figure><img src="content_image/1608.05293/x14.png"><figcaption>Figure 9: Production of a quark with a transverse momentum k. Typical virtualdiagrams, i.e. diagrams in which the gluon is not crossing the cut (cf. Eq.(19)). (a) Left: Diagram contributing to the virtual term proportional toS(k), which originates from S(x,y) in coordinate space (b) Right: Diagramcontributing to the real term proportional to S(k)S(ℓ) and which originatesfrom S(x,z)S(z,y) in coordinate space. Momenta are flowing from left to rightboth in the DA and in the CCA and from bottom to top in the exchange (red)gluons.</figcaption></figure> The second term in Eq. (18), bilinear in the dipole $S$-matrix, corresponds to interference processes, where the primary gluon is emitted prior to scattering in the direct amplitude (DA) and after the scattering in the complex conjugate amplitude (CCA), or vice-versa. In such processes, both the quark and the gluon can participate in the collision. At large $N_{c}$, this yields 2 dipole $S$-matrices: one made with the quark in the DA and the antiquark piece of the gluon, the other one with the quark piece of the gluon and the antiquark in CCA. One of these $S$-matrices, denoted as $\mathcal{S}(\bm{\ell})$ in (18), describes the _elastic_ scattering of a _physical_ dipole — i.e. a dipole whose both fermion legs exist on the same side of the cut (either in the DA, or in the CCA). For this elastic scattering, there is no net transfer of transverse momentum; e.g., if $\mathcal{S}(\bm{\ell})$ is computed in the 2-gluon exchange approximation, then the momentum $\bm{\ell}$ transferred by the first exchanged gluon towards the dipole is subsequently taken back by the second exchanged gluon. The other dipole $S$-matrix, $\mathcal{S}(\bm{q})$, describes an inelastic scattering with net momentum transfer $\bm{q}=\bm{p}+\bm{k}$. Consider similarly the ‘virtual’ contributions encoded in (19) (see Fig. 9). In that case, the primary gluon is both emitted and reabsorbed on the same side of the cut, hence the momentum $\bm{k}$ of the produced quark fully comes via inelastic scattering (and $k^{+}=q_{0}^{+}$). In the first term in (19), the gluon fluctuation has no overlap with the target, hence the (inelastic) scattering refers to the quark alone. In the second term, the gluon can scatter too. Accordingly, this term involve 2 dipole $S$-matrices, one describing an elastic scattering ($\mathcal{S}(\bm{\ell})$), the other one an inelastic one ($\mathcal{S}(\bm{k})$). The following observations will be useful for the subsequent arguments: (i) In Eq. (17) one recognizes the full LO DGLAP quark-to-quark splitting function $P_{qq}(\xi)$, in line with the fact that the gluon emission has been treated exactly, and not in the eikonal approximation. (ii) In Eqs. (18) and (19), the splitting fraction $\xi$ is visible only in the various kernels describing the transverse momentum structure of the hard splitting, which in turn have been generated by combining the light-cone energy denominator with factors coming from the splitting vertex. (iii) The various dipole $S$-matrices in Eqs. (18) and (19) are supposed to describe scattering off the nuclear gluon distribution evolved up to the right ‘rapidity’ ($Y=\ln(1/X)$) scale, but this scale is left unspecified in the above equations. For the ‘real’ contributions at least, we know by now what is the typical longitudinal momentum fraction $X$ of the gluons from the target which are probed by this scattering: this is the value $X(x,p_{\perp})$ given by Eq. (8). Hence, the dipole $S$-matrices in Eq. (18) must be evaluated at $X\simeq X(x,p_{\perp})$, where it is understood that $\bm{p}=\bm{q}-\bm{k}$. We shall later demonstrate that $X(x,p_{\perp})$ with $\bm{p}=\bm{q}-\bm{k}$ is also the appropriate choice for the rapidity argument of $S$-matrices which enter the ‘virtual’ terms in Eq. (19). This means that, strictly speaking, one cannot factorize the $S$-matrix $\mathcal{S}(\bm{k})$ in front of the integrals in Eq. (19), in contrast to the results in [1; 2]. (iv) The integral over $\xi$ in Eq. (17) seems to develop a logarithmic singularity at $\xi=1$, meaning an infrared divergence associated with the emission of very soft ($x\to 0$) gluons. (This is the meaning of the upper label ‘unsub’ in the l.h.s. of Eq. (17).) As already mentioned, Refs. [1; 2] proposed to eliminate this divergence via the ‘plus’ prescription, defined as (for a generic function $F(\xi)$) \[\int_{0}^{1}{\rm d}\xi\,\frac{F(\xi)}{1-\xi}\,\longrightarrow\,\int_{0}^{1}{ \rm d}\xi\,\frac{F(\xi)}{(1-\xi)_{+}}\,\equiv\,\int_{0}^{1}{\rm d}\xi\,\frac{F (\xi)-F(1)}{1-\xi}\,.\] (20) After this subtraction, the result in Eqs. (17)– (19) is supposed to represent a purely NLO correction, to be added to the respective LO result in Eq. (3). We shall further discuss this particular prescription in Sect. 3.5, but already at this level it should be clear that, as a matter of facts, there is no physical singularity in Eq. (17): for the ‘real’ terms at least, the integral over $\xi$ is cut off near $\xi=1$ by energy conservation, cf. Eq. (8). Specifically, by using Eq. (8) together with the kinematical limit $X(x,p_{\perp})\leq 1$, one finds the following lower limit on $x=1-\xi$: \[x\,\gtrsim\,x_{\rm m}(p_{\perp})\equiv\,\frac{p_{\perp}^{2}}{\hat{s}}\,,\] (21) where we also used $x_{\rm m}\ll 1$. Still for the ‘real’ terms, there is also an upper limit $x\leq 1-x_{p}$, coming from the condition $x_{p}/(1-x)\leq 1$ on the longitudinal fraction of the incoming quark. This lower limit $x\gtrsim x_{\rm m}$, can be recognized as the condition that the lifetime $\Delta x^{+}\sim 2p^{+}/p_{\perp}^{2}$ of the softest primary gluon emission be at least as large as the longitudinal width $1/P^{-}$ of the target (a necessary condition for having significant scattering). This condition has been previously emphasized in [50] (the ‘Ioffe time’) and numerically implemented in [57] (where however the dependence of the various $S$-matrices upon the target rapidity $X(x,p_{\perp})$ has not been taken into account). The existence of a physical lower limit on $x$ is indeed crucial for our subsequent construction, which will not involve the ‘plus’ prescription, or any other infrared regularization of the integral over $x$. It is therefore important to demonstrate that such a limit exists also for the ‘virtual’ terms in Eq. (19), for which the previous argument on energy conservation does not apply. We shall do that in Appendix A, where we demonstrate that the same lower limit $x\gtrsim x_{\rm m}$ holds also for the ‘virtual’ terms, as a consequence of fine cancellations among the virtual gluon graphs which occur in the complementary region at $x<x_{\rm m}$. Whereas mathematically subtle, the occurrence of such cancellations has a clear physical interpretation: gluon fluctuations with $x<x_{\rm m}$ cannot interact with the target, since their lifetime is too short. Accordingly the emissions of such short-lived gluons cannot modify the $S$-matrix of the projectile. Since ‘real’ emissions with $x<x_{\rm m}$ are anyway forbidden by energy conservation, it follows that the respective ‘virtual’ graphs must cancel among themselves. The precise way how such cancellations occur is demonstrated in Appendix A (for the somewhat simpler, but general enough, situation where the target itself is a quark). (v) If one takes the limit $\xi\to 1$ (i.e. $x\to 0$) in the _transverse kernels_ in Eqs. (18) and (19) (but not necessarily also in their implicit dependence upon $x$ via the rapidity cutoff $X(x,p_{\perp})$), then the combination of these two terms which enters Eq. (17) with $\xi\to 1$ reduces to the Fourier transform of the r.h.s. of the BK equation Eq. (7). Specifically, (22) where the notation $\partial S(\bm{r};Y)/\partial Y$ is merely a shortcut for the r.h.s. of (7) with $\bm{r}=\bm{x}-\bm{y}$. The appearance of the BK equation was in fact to be expected: when $\xi\to 1$, Eqs. (17)–(19) describe the emission of a soft primary gluon by the incoming quark, in the eikonal approximation. By definition, the effect of this soft emission on the quark multiplicity is equivalent to the first step in the BK evolution of the respective LO result in Eq. (3). Note however that in general the $S$-matrices implicit in the r.h.s. of Eq. (22) are meant to be computed _beyond_ the LLA and their rapidity argument $X(x,p_{\perp})$ is a complicated function of the kinematics of the emitted gluon. ### CGC factorization at NLO After this preparation, we are now in a position to present our master formula for the single-inclusive quark multiplicity valid through next-to-leading order (i.e. which includes both the LO and the NLO contributions). The relation between this formula and the factorization scheme proposed in Refs. [1; 2] will be discussed later, in Sect. 3.5. As already mentioned, we systematically omit the NLO corrections proportional to the quark Casimir $C_{\rm F}$, which play no special role for the high-energy evolution. These corrections can be taken over from Refs. [1; 2] and simply added to our master formula, which reads \[=\frac{x_{p}q(x_{p})}{(2\pi)^{2}}\Big{[}{\mathcal{S}}_{0}(\bm{k}) +\Delta{\mathcal{S}}(\bm{k},X_{g})\Big{]}+\int\!\frac{{\rm d}^{2}\bm{p}}{(2\pi )^{2}}\int_{x_{\rm m}(p_{\perp})}^{1}\!{\rm d}x\ \frac{\bar{\alpha}_{s}}{2\pi} \,\frac{1+(1-x)^{2}}{2x}\] \[\times\left\{\frac{x_{p}}{1-x}q\left(\frac{x_{p}}{1-x}\right) \tilde{\mathcal{J}}\big{(}\bm{k},x;\bm{p},X(x,p_{\perp})\big{)}-x_{p}q\left(x_ {p}\right)\tilde{\mathcal{J}}_{v}\big{(}\bm{k},x;\bm{p},X(x,p_{\perp})\big{)} \right\}.\] (23) This formula is illustrated in Fig. 10. As before, ${\mathcal{S}}_{0}(\bm{k})$ denotes the tree-level contribution to the dipole $S$-matrix, say as given by the MV model (see e.g. [17; 18; 19; 20] for an explicit expression). The quantity $\Delta{\mathcal{S}}(\bm{k},X_{g})$ within the square brackets denotes a NLO correction to the dipole $S$-matrix, to be specified in Sect. 3.4. The last term in Eq. (3.2), which involves a double integration — over the longitudinal fraction $x=1-\xi$ and the transverse momentum $\bm{p}$ of the primary gluon — is the main term for our present purposes. It encodes the impact factor to NLO accuracy, the LO evolution of the dipole $S$-matrix and also a part of the respective NLO contribution (namely, the part which is not included in $\Delta{\mathcal{S}}(\bm{k},X_{g})$; see Sect. 3.4 for details). The new functions $\tilde{\mathcal{J}}\big{(}\bm{k},x;\bm{p},X(x,p_{\perp})\big{)}$ and $\tilde{\mathcal{J}}_{v}\big{(}\bm{k},x;\bm{p},X(x,p_{\perp})\big{)}$ are essentially the integrands in Eqs. (18) and respectively (19), in which we replaced $\xi\to 1-x$ and $\bm{q}\to\bm{p}+\bm{k}$. (As compared to (18) and (19), we now use $\bm{p}$ and $\bm{\ell}$ as integration variables; the integral over $\bm{p}$ is explicit in Eq. (3.2), while that over $\bm{\ell}$ is included in the definitions of $\tilde{\mathcal{J}}$ and $\tilde{\mathcal{J}}_{v}$.) The notation emphasizes that the various dipole $S$-matrices implicit in these functions should be evaluated at a target rapidity $Y=\ln(1/X)$ with $X=X(x,p_{\perp})$, cf. Eq. (8). The lower limit $x_{\rm m}(p_{\perp})$ is shown in Eq. (21). In the real term, it is understood that the support of the quark distribution limits the integration to $x<1-x_{p}$. <figure><img src="content_image/1608.05293/x16.png"><figcaption>Figure 10: Graphical illustration of the factorization in Eq. (3.2). Left: adiagram describing multiple scattering in the MV model; this is representativefor the tree-level term S0 in Eq. (3.2). Right: a diagram which exhibits theprimary quark-gluon pair (the NLO impact factor) and its multiple scatteringoff the gluon distribution of the target, itself evolved to NLO; this isrepresentative for the second term in Eq. (3.2), which encodes all theradiative corrections to the quark multiplicity through NLO.</figcaption></figure> For more clarity, let us exhibit here the function $\tilde{\mathcal{J}}\big{(}\bm{k},x;\bm{p},X(x,p_{\perp})\big{)}$ which enters the ‘real contribution (the corresponding expression for $\tilde{\mathcal{J}}_{v}$ can be similarly written): \[\tilde{\mathcal{J}}\big{(}\bm{k},x;\bm{p},X(x,p_{\perp})\big{)} =\frac{2\bm{p}\cdot\big{[}(1-x)\bm{p}-x\bm{k}\big{]}}{\bm{p}^{2} \big{[}(1-x)\bm{p}-x\bm{k}\big{]}^{2}}\,\mathcal{S}\big{(}\bm{p}+\bm{k},X(x,p_ {\perp})\big{)}\] \[+\int\!\frac{{\rm d}^{2}\bm{\ell}}{(2\pi)^{2}}\frac{2(\bm{k}-\bm{ \ell})\cdot\big{[}(1-x)\bm{p}-x\bm{k}\big{]}}{(\bm{k}-\bm{\ell})^{2}\big{[}(1- x)\bm{p}-x\bm{k}\big{]}^{2}}\,\mathcal{S}\big{(}\bm{p}+\bm{k},X(x,p_{\perp}) \big{)}\,\mathcal{S}\big{(}\bm{\ell},X(x,p_{\perp})\big{)}.\] (24) It is perhaps interesting to notice that the linear combination ${\bm{P}}\equiv(1-x)\bm{p}-x\bm{k}$ which appears in the above integrand is the momentum conjugated to the transverse separation $\bm{x}-\bm{z}$ between the quark and the primary gluon. Similarly, the total momentum ${\bm{q}}\equiv\bm{k}+\bm{p}$ is conjugated to the center-of-mass $(1-x)\bm{x}+x\bm{z}$ of the quark-gluon pair. (As in Eq. (7), $\bm{x}$ and $\bm{z}$ denote the transverse positions of the quark and the primary gluon, respectively.) To convincingly demonstrate the validity of Eq. (3.2) to the NLO accuracy of interest, one still needs to describe the correction $\Delta{\mathcal{S}}(\bm{k},X_{g})$ to the dipole $S$-matrix, which we shall do in Sect. 3.4. In the remaining part of this subsection, we shall merely check that Eq. (3) properly encodes the LO result, cf. Eq. (3), together with the NLO correction to the impact factor discussed in Sect. 3.1, without any over-counting. The LLA limit of Eq. (3.2) is obtained by making approximations appropriate at small $x$, that is, by treating the emission of the primary gluon in the eikonal approximation and by replacing $p_{\perp}\sim k_{\perp}$ in the kinematical limits and the various rapidity variables; that is, one approximates $x_{\rm m}(p_{\perp})\simeq x_{g}=k_{\perp}^{2}/\hat{s}$ and $X(x,p_{\perp})\simeq X(x)=X_{g}/x$, with $X_{g}=x_{g}$ (cf. Eq. (9)). Also, all dipoles $S$-matrices are now understood to obey the LO BK evolution, from $X_{0}\simeq 1$ down to $X(x)$. Under these assumptions, one can first commute the integrations over $\bm{p}$ and $x$ in Eq. (3.2) and then use the identity (22) to rewrite the simplified version of this equation in the following, suggestive, form \[=x_{p}q(x_{p})\,\frac{\mathcal{S}_{0}(\bm{k})}{(2\pi)^{2}}\,+\,x_ {p}q(x_{p})\,\frac{1}{(2\pi)^{2}}\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\frac{ \partial}{\partial Y}\,\mathcal{S}\big{(}\bm{k};Y=\ln 1/X(x)\big{)}\,.\] (25) The above integral runs over the longitudinal momentum fraction $x=p^{+}/k^{+}$ of the right-moving gluon, whereas the evolution of the dipole $S$-matrix has been rather performed w.r.t. the longitudinal fraction $X$ of the gluons in the target. However, to LLA, $x$ and $X$ are related via $X(x)=X_{g}/x$, so one can change the integration variable from $x$ to $Y\equiv\ln(1/X)$ and thus identify a total derivative \[\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\frac{\partial}{\partial Y}\,\mathcal{S} \big{(}\bm{k};Y=\ln 1/X(x)\big{)}=\int_{0}^{Y_{g}}{\rm d}Y\,\frac{\partial}{ \partial Y}\,\mathcal{S}(\bm{k};Y)\,=\,\mathcal{S}(\bm{k};Y_{g})-{\mathcal{S}} _{0}(\bm{k})\,.\] (26) After also adding the initial condition in Eq. (25), one recognizes the LO result (3), as anticipated. The r.h.s. of Eq. (25) is recognized as the ‘integral’ version of the LO BK equation introduced in Eq. (2.3). Hence, the full result (3.2) can be viewed as the generalization of that integral representation to NLO and to the exact kinematics for the primary gluon emission. (This will be confirmed by the discussion in Sect. 3.3.) The explicit separation of the first gluon emission from the remaining evolution, as operated by this representation, has allowed us to promote the calculation of the impact factor to NLO accuracy, while at the same time avoiding over-counting. At this point, it is important to more precisely specify the perturbative content of Eq. (3.2). As just explained, the integral term there fully encodes the LO evolution of the dipole $S$-matrix, that is, it resums corrections of the type ${(\bar{\alpha}_{s}Y_{g})^{n}}$ to all orders. It obviously encodes NLO corrections due to the fact that the emission of the primary gluon is treated exactly; that is, the integral over $x$ also generates corrections of $\mathcal{O}{(\bar{\alpha}_{s})}$ besides the dominant contribution of $\mathcal{O}{(\bar{\alpha}_{s}Y_{g})}$, which counts for the LO evolution. To ensure NLO accuracy, the evolution of the various dipole $S$-matrices must be computed to NLO as well. Indeed, the NLO BK kernel includes corrections of $\mathcal{O}{(\bar{\alpha}_{s})}$; hence, the solution to the NLO BK equation involves corrections proportional to ${\bar{\alpha}_{s}(\bar{\alpha}_{s}Y_{g})^{n}}$, which count to NLO. By a similar argument, one must include the (one-loop) running coupling corrections within the QCD coupling $\bar{\alpha}_{s}$ associated with the primary gluon vertex. This is why, in writing Eq. (3.2), we have inserted the factor $\alpha_{s}$_inside_ the double integral over $\bm{p}$ and $x$: after including the running coupling corrections, this factor will depend upon the transverse momenta $\bm{k}$ and $\bm{p}$ which enter the emission vertex and possibly also upon $x$ (via the gluon kinematics). Specifying this dependence requires a prescription, which is most conveniently formulated in the transverse coordinate representation (since this is the representation in which the BK equation is generally solved in practice). Such prescriptions will be discussed in Sect. 3.4 (see e.g. Eq. (32) there), together with the other NLO corrections to the dipole evolution. Notice however that, in order to evaluate Eq. (3.2), one also needs a prescription for the running coupling which is directly formulated in momentum space. In general such a prescription will be different from the one in coordinate space. This mismatch could have consequences for the fine-tinning issue to be discussed in Sect. 3.5. The above arguments show that, strictly speaking, the integral term in Eq. (3.2) also includes terms of NNLO, as generated by the product between the NLO correction to the impact factor and the NLO effects in the high-energy evolution, or in the running coupling. As we shall explain in Sect. 3.5, these various types of NLO effects can be disentangled from each other via a reorganization of the perturbation theory which involves a ‘rapidity subtraction’, as in Refs. [1; 2]. Yet, this procedure has some inconveniences, as we shall see (notably, it introduces the ‘fine-tuning’ issue anticipated in the Introduction). So it is important to stress here that, although going beyond a strict NLO approximation, the result (3.2) is in fact the natural outcome of perturbative QCD — that is, the direct result of evaluating Feynman graphs at the loop-order of interest, before performing additional manipulations like the ‘rapidity subtraction’. We conclude this subsection with a discussion of potential difficulties with using Eq. (3.2) in practice. All these issues will be addressed in more detail in the next two subsections, where we shall provide solutions to them, at least at the expense of further approximations. First, Eq. (3.2) looks very cumbersome, notably due the intricacy of the multiple integrations, over both transverse ($\bm{p}$, $\bm{\ell}$) and longitudinal ($x$) momenta, which are entangled with each other. It is not clear to us whether these integrations can be computed as such, not even numerically. The calculations might be further complicated by the need to compute the Fourier transform of the dipole $S$-matrix, as numerically obtained by solving the BK equation in coordinate space. Second, Eq. (3.2) is somewhat formal, in that the dipole $S$-matrices implicit there are supposed to encode the evolution of the target gluon distribution at NLO. However, the high-energy evolution of a dense nucleus has not been explicitly computed beyond LO. (All NLO calculations to date refer to the evolution of a dilute projectile, like a dipole [32; 33; 34].) Besides, a purely NLO approximation to the high-energy evolution is likely to become unstable (and thus require resummations) in the ‘collinear’ regime where the transverse momentum $k_{\perp}$ is relatively large ($k_{\perp}\gg Q_{s}$). Finally, the NLO calculation based on Eq. (3.2) is formally sensitive to the physics of the nuclear wavefunction at large values of $X$ (recall that $x\sim x_{g}$ corresponds to $X\sim 1$), which is not really under control within the present, high-energy approximations. This cannot be a serious difficulty in the physical context at hand: the NLO corrections to the impact factor are controlled by relatively hard primary gluons with $x\sim\mathcal{O}{(1)}$ and hence $X(x)\ll 1$; such a hard emission by the projectile should be well separated from the valence structure of the target at $X\sim 1$. At the end of the next subsection we shall describe an explicit procedure which implements this separation. ### Simplifying the kinematics In this subsection, we shall propose strategies to deal with some of the problems alluded to at the end of the previous subsection. First, we shall argue that one can approximate $p_{\perp}\sim k_{\perp}$ within the rapidity variables for the high-energy evolution and thus greatly simplify the structure of the transverse and longitudinal integrations in Eq. (3.2). Second, we shall discuss the prescription for the running of the coupling in the emission of the primary gluon. Third, we shall reformulate the initial condition at low energy in such a way to reduce the sensitivity to the large-$X$ region in the target wavefunction. Concerning the first point — the dependence of the kinematical constraints on the high-energy evolution upon the transverse momenta of the primary quark-gluon pair —, we note that there are two interesting physical regimes: (i)_Hard quark production and di-jet configurations:_$k_{\perp}\gg Q_{s}(X)$. This is the regime which is primarily concerned by the negativity problem discussed in Refs. [51; 53; 54; 55; 50; 57; 56]. In this case, we have already argued in Sect. 2.4 that the transverse momentum of the primary gluon and that of the produced quark must balance each other : $p_{\perp}\simeq k_{\perp}$. (ii)_Semi-hard quark production:_$k_{\perp}\gtrsim Q_{s}(X)$. This regime includes the ‘geometric scaling’ window [59], where the scattering is weak, but the $S$-matrix is still influenced by non-linear effects, via the ‘saturation boundary’ at $Q_{s}(X)$[60; 61; 62]. In this regime, all the relevant transverse momenta — the $k_{\perp}$ of the produced quark, the $p_{\perp}$ of the primary gluon, and the $q_{\perp}$ transferred by the nuclear target — take typical values of $\mathcal{O}{(Q_{s})}$, since this is the value naturally acquired via rescattering off the saturated gluon distribution of the target. We see that in both cases the quantities $p_{\perp}$ and $k_{\perp}$ cannot be very different from each other, so one can approximate $p_{\perp}\simeq k_{\perp}$ when evaluating the rapidity variables (cf. Eqs. (8) and (21)) : \[x_{\rm m}(p_{\perp})\,\to\,x_{\rm m}(k_{\perp})=\frac{k_{\perp}^{2}}{\hat{s}} \equiv x_{g}\quad\mbox{and}\quad X(x,p_{\perp})\,\to\,X(x,k_{\perp})=\frac{k_{ \perp}^{2}}{x\hat{s}}\equiv X(x)\,.\] (27) Remarkably, thanks to this approximation, we have returned to the same expressions for the rapidity variables $x_{g}$ and $X(x)$ as at LO, cf. Eq. (8). This greatly simplifies Eq. (3.2) since the transverse momentum integrations can now be performed prior to the integral over $x$. Then Eq. (3.2) takes a form closer to that in Eq. (17), namely, \[\frac{{\rm d}N^{pA\to qX}}{{\rm d}^{2}\bm{k}\,{\rm d}\eta} =\frac{x_{p}q(x_{p})}{(2\pi)^{2}}\Big{[}{\mathcal{S}}_{0}(\bm{k}) +\Delta{\mathcal{S}}(\bm{k},X_{g})\Big{]}+\frac{\bar{\alpha}_{s}(k_{\perp}^{2} )}{2\pi}\int_{x_{g}}^{1}{\rm d}x\ \frac{1+(1-x)^{2}}{2x}\] \[\qquad\qquad\times\left\{\frac{x_{p}}{1-x}q\left(\frac{x_{p}}{1-x }\right)\mathcal{J}\big{(}\bm{k},x;X(x)\big{)}-x_{p}q\left(x_{p}\right) \mathcal{J}_{v}\big{(}\bm{k},x;X(x)\big{)}\right\},\] (28) where the functions $\mathcal{J}\big{(}\bm{k},x;X(x)\big{)}$ and $\mathcal{J}_{v}\big{(}\bm{k},x;X(x)\big{)}$ have the same formal expressions as in Eqs. (18)–(19) [with $\xi=1\!-\!x$, of course], except for the fact that the rapidity variable for the evolution of the dipole $S$-matrices is now clearly identified, namely $Y=\ln\big{(}1/X(x)\big{)}$. Once again, it is understood that the integral over $x$ in the real term is restricted to $x<1-x_{p}$. In writing Eq. (3.3), we have also identified the argument of the running coupling for the primary vertex as $k_{\perp}^{2}$. This is unambiguous under the present assumptions, since $k_{\perp}\sim p_{\perp}$ is the only hard scale involved in that splitting¹⁰. Eq. (3.3) also shows that the natural ‘LO approximation’ for the high-energy evolution in the problem at hand is the LO BK equation with running coupling (rcBK). Indeed, when evaluating the second term in Eq. (3.3) within the eikonal approximation, as appropriate for $x\ll 1$, the r.h.s. of this equation becomes proportional to the integral version of rcBK; that is, this is tantamount to evaluating the LO formula (3) with the solution $\mathcal{S}_{\rm rcBK}(\bm{k},X_{g})$ to rcBK. [FOOTNOTE:10][ENDFOOTNOTE] The considerably simpler structure of Eq. (3.3) also allows us to reformulate the initial condition at low-energy, in such a way to avoid the ‘dangerous’ region at $X\sim 1$. To that aim, let us introduce a ‘better’ separation scale $X_{0}$ for the rapidity evolution of the target, which is such that the high-energy approximations are indeed justified for any $X\leq X_{0}$. (For instance the value $X_{0}=0.01$ is often used in the fits to the HERA data for deep inelastic scattering; see e.g. [73; 45].) In particular, this scale should obey $x_{g}\ll X_{0}\ll 1$. We then separate the integral over $x$ into two regions, $x_{g}<x<x_{g}/X_{0}$ and $x_{g}/X_{0}<x<1$, which in terms of $X=X_{g}/x$ correspond to $1>X>X_{0}$ and respectively $X_{0}>X>X_{g}$. (We recall that $X_{g}=k_{\perp}^{2}/\hat{s}=x_{g}$.) Within the first region, one has $x\ll 1$, hence one can replace $x\to 0$ within the transverse kernels which enter the functions $\mathcal{J}$ and $\mathcal{J}_{v}$ [e.g., $\mathcal{J}\big{(}\bm{k},x;X(x)\big{)}\to\mathcal{J}\big{(}\bm{k},0;X(x)\big{)}$] and also within the quark distribution. Under these assumptions, the sum between the ‘initial condition’ $\mathcal{S}_{0}$ in Eq. (3.3) and the part of the integral there which runs over the small-$x$ interval at $x_{g}<x<x_{g}/X_{0}$ is formally the same as the r.h.s. of the LO BK equation with running coupling (rcBK) integrated from $X=1$ down to $X_{0}$ (recall Eq. (22)). It might be tempting to interpret this sum as the solution to rcBK evaluated at $X=X_{0}$, but we shall not adopt this point of view: after all, this ‘rcBK evolution’ refers to the large-$X$ interval at $1>X>X_{0}$, where the high-energy approximations are not valid. We shall rather _replace_ the result of this fictitious ‘rcBK evolution’ with a _new_ initial condition, denoted as $\mathcal{S}(\bm{k},X_{0})$, which is formulated directly at $X_{0}$. That is, we replace Eq. (3.3) by¹¹ [FOOTNOTE:11][ENDFOOTNOTE] \[\frac{{\rm d}N^{pA\to qX}}{{\rm d}^{2}\bm{k}\,{\rm d}\eta} =\frac{x_{p}q(x_{p})}{(2\pi)^{2}}\Big{[}{\mathcal{S}}(\bm{k},X_{0 })+\Delta{\mathcal{S}}(\bm{k},X_{g})\Big{]}+\frac{\bar{\alpha}_{s}(k_{\perp}^{ 2})}{2\pi}\int_{x_{g}/X_{0}}^{1}{\rm d}x\ \frac{1+(1-x)^{2}}{2x}\] \[\qquad\qquad\times\left\{\frac{x_{p}}{1-x}q\left(\frac{x_{p}}{1-x }\right)\mathcal{J}\big{(}\bm{k},x;X(x)\big{)}-x_{p}q\left(x_{p}\right) \mathcal{J}_{v}\big{(}\bm{k},x;X(x)\big{)}\right\},\] (29) where it is now understood that all the $S$-matrices implicit in the second term in the r.h.s. are obtained by solving appropriate evolution equations with the initial condition formulated at $X=X_{0}$. The evolution equations to be used in this context will be discussed in the next subsection. ### The evolution of the color dipole beyond leading order In this subsection, we shall describe the NLO evolution of the dipole $S$-matrices to be used in conjunction with the factorization scheme in Eq. (3.2) or (3.3). In particular, we shall present an explicit expression for the NLO correction $\Delta{\mathcal{S}}$ which enters this scheme but so far has not been specified. To simplify the arguments and the notations, we shall work at the level of the kinematic approximations introduced in Sect. 3.3, that is, we shall built on top of Eq. (3.3). There are two aspects which are rather subtle and that we shall try to clarify in what follows. The first refers one to the proper inclusion of NLO corrections within $\Delta{\mathcal{S}}$ without any over-counting: this quantity must contain only those corrections to the high-energy evolution that are not already included in the integral term in Eq. (3.3). The second aspect refers to the relation between the viewpoint of _target_ evolution, that was explicitly used in our previous arguments (for instance, when specifying the rapidity arguments of the various $S$-matrices), and that of the evolution of the _projectile_ (a color dipole), for which the evolution equation is currently known at NLO accuracy [32], including the collinear improvement [44; 37; 45; 46]. Indeed, the two evolutions refer to different variables (‘evolution times’) — $x=q^{+}/q_{0}^{+}$ for the right-moving projectile and respectively $X=q^{-}/P^{-}$ for the left-moving target —, so the corresponding equations cannot be identical beyond leading order, when the differences between various transverse momenta and also the off-shell effects start to play a role. As mentioned after Eq. (3.2), the unknown quantity $\Delta{\mathcal{S}}(\bm{k},X_{g})$ represents a part of the NLO corrections to the evolution of the dipole $S$-matrix. Since these corrections are fully known by now [32], the simplest way to obtain $\Delta{\mathcal{S}}(\bm{k},X_{g})$ is by clarifying its relation to the NLO calculation in Ref. [32]. We first observe that, by definition, the full NLO result for the quark multiplicity must involve two types of NLO corrections: those associated with the impact factor and those related to the high-energy evolution of the color dipoles. Hence, if one ‘switches off’ the first type of corrections (which one can do by computing the emission of the primary gluon in the eikonal approximation), then the r.h.s. of Eq. (3.3) should be proportional to ${\mathcal{S}}(\bm{k},X_{g})$ — the dipole $S$-matrix computed at NLO accuracy and for the kinematics of interest. This argument implies \[{\mathcal{S}}(\bm{k},X_{g})={\mathcal{S}}_{0}(\bm{k})+\Delta{\mathcal{S}}(\bm{ k},X_{g})+2\pi\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\ \frac{{\rm d}x} {x}\left\{\mathcal{J}\big{(}\bm{k},0;X(x)\big{)}-\mathcal{J}_{v}\big{(}\bm{k}, 0;X(x)\big{)}\right\},\] (30) where the function $\mathcal{J}\big{(}\bm{k},0;X(x)\big{)}$ is obtained from Eq. (18) by letting $x\equiv 1-\xi\to 0$ and by evaluating the $S$-matrices there to NLO accuracy and for a rapidity argument $Y=\ln(1/X(x))$ [and similarly for the function $\mathcal{J}_{v}\big{(}\bm{k},0;X(x)\big{)}$]. Using this condition together with the NLO result for the dipole $S$-matrix which emerges from Ref. [32], it is possible to identify the quantity $\Delta{\mathcal{S}}(\bm{k},X_{g})$. By the ‘NLO result for ${\mathcal{S}}$’, we mean the integral representation for the NLO $S$-matrix, as obtained by formally solving the respective evolution equation — that is, the generalization of Eq. (2.3) to NLO. Before we proceed, it is useful to, first, clarify what we precisely mean by the ‘LO evolution’ and, second, introduce some simplified notations, which focus on the essential and help making the subsequent arguments more transparent. As already mentioned after Eq. (3.3), our LO approximation to the dipole $S$-matrix is the solution $\mathcal{S}_{\rm rcBK}(\bm{k},X)$ to the LO BK equation with running coupling (rcBK) [68; 69; 70]. This choice deserves some comment, in that it already includes a subset of the NLO (and higher-order) corrections, via the running of the coupling. But precisely because of that, rcBK offers a better starting point for a perturbative expansion than the _strict_ leading-order approximation — the LO BK equation with fixed coupling. This is related to the poor convergence of the perturbative expansion at high energy: the LO BK equation with fixed coupling is well known to predict an unrealistically fast evolution with increasing energy, meaning a too large value for the saturation exponent. Hence, for sufficiently high energies, the strict LO estimate (3) for the multiplicity becomes _exponentially larger_ (in the sense of an exponential in $Y=\ln(1/X)$) than the actual result at NLO. This problem is considerably alleviated if one instead uses rcBK as the ‘leading order’ evolution: this approximation predicts a significantly slower evolution [61; 74] and offers a reasonably good description for the phenomenology [12; 73; 75]. The all-order resummation of running coupling corrections requires a prescription. Here, we shall mention two popular such prescriptions, both built with the one-loop approximation for the running coupling and which are roughly equivalent to each other (see Ref. [45] for a recent discussion). Consider the splitting of the parent dipole $(\bm{x},\bm{y})$ into two daughter dipoles $(\bm{x},\bm{z})$ and $(\bm{z},\bm{y})$, as described by the LO BK equation (7). The _smallest dipole prescription_ consists in replacing¹²$\bar{\alpha}_{s}\to\bar{\alpha}_{s}(r_{\rm min})$, where $r_{\rm min}\equiv\min\big{\{}\|\bm{x}\!-\!\bm{y}\|,\|\bm{x}\!-\!\bm{z}\|,\|\bm{y}\! -\!\bm{z}\|\big{\}}$ and [FOOTNOTE:12][ENDFOOTNOTE] \[\bar{\alpha}_{s}(r)=\frac{1}{\bar{b}\ln\big{[}4/r\Lambda_{\rm QCD}^{2}\big{]}} \,,\qquad\bar{b}=(11N_{\rm c}-2N_{\rm f})/12N_{\rm c}\,.\] (31) The other prescription, known as _fastest apparent convergence (fac)_, amounts to $\bar{\alpha}_{s}\to\bar{\alpha}_{\rm fac}$, with \[\bar{\alpha}_{\rm fac}\equiv\left[\frac{1}{\bar{\alpha}_{s}(\|\bm{x}\!-\!\bm{y} \|)}+\frac{(\bm{x}\!-\!\bm{z})^{2}-(\bm{y}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{y}) ^{2}}\,\frac{\bar{\alpha}_{s}(\|\bm{x}\!-\!\bm{z}\|)-\bar{\alpha}_{s}(\|\bm{y}\!- \!\bm{z}\|)}{\bar{\alpha}_{s}(\|\bm{x}\!-\!\bm{z}\|)\bar{\alpha}_{s}(\|\bm{y}\!-\! \bm{z}\|)}\right]^{-1}.\] (32) This last prescription is particularly useful for what follows, in that it simplifies the expression that we shall obtain for $\Delta{\mathcal{S}}$. After a Fourier transform to momentum space, the solution to rcBK can be given the following integral representation: \[{\mathcal{S}}_{\rm rcBK}(\bm{k},X_{g})={\mathcal{S}}_{0}(\bm {k})+2\pi\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\, \big{[}\mathcal{J}_{\rm rcBK}\big{(}\bm{k},0;X(x)\big{)}- \mathcal{J}_{v,{\rm rcBK}}\big{(}\bm{k},0;X(x)\big{)}\big{]}\,,\] (33) where the factorization of the running coupling $\bar{\alpha}_{s}(k_{\perp}^{2})$ was possible because of our assumption that $k_{\perp}$ is sufficiently hard (recall the discussion after Eq. (3.3)). The functions $\mathcal{J}_{\rm rcBK}\big{(}\bm{k},0;X(x)\big{)}$ and $\mathcal{J}_{v,{\rm rcBK}}\big{(}\bm{k},0;X(x)\big{)}$ are obtained from the respective functions in Eq. (30) after replacing $\mathcal{S}\to{\mathcal{S}}_{\rm rcBK}$. <figure><img src="content_image/1608.05293/x18.png"><figcaption>Figure 11: A graphical illustration of the integral version of the LO BKequation (36). Running-coupling corrections and non-linear effects in Sdescribing multiple scattering are implicitly assumed, but not explicitlydepicted.</figcaption></figure> We now introduce more schematic notations, as anticipated. Specifically, let us ignore (just in our notations) the transverse momentum convolutions, the inessential numerical factors, and the non-linear structure of functions like $\mathcal{J}$ w.r.t. the dipole $S$-matrix. Also, in writing the cross-section, we shall omit the quark distribution function. That is, we shall rewrite Eq. (3.3) simply as \[\mathcal{N}_{{\rm LO}+{\rm NLO}}=\mathcal{ S}_{0}+\Delta{\mathcal{S}}(X_{g})+\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^ {1}\frac{{\rm d}x}{x}\,\mathcal{K}(x)\,\mathcal{S}\big{(}X(x)\big{)}\,,\] (34) where the kernel $\mathcal{K}(x)$ encodes all the momentum space variables and convolutions (but no dipole $S$-matrix) and $\mathcal{S}\big{(}X(x)\big{)}$ succinctly denotes all the factors involving the dipole $S$-matrix, which could be either linear, or bi-linear, in $\mathcal{S}$. Both ‘real’ and ‘virtual’ contributions are implicitly added in the above integral. With these new notations, Eqs. (30) and (33) become \[{\mathcal{S}}(X_{g})={\mathcal{S}}_{0}+\Delta{\mathcal{S}}(X_{g})+\bar{\alpha} _{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\mathcal{K}(0)\, \mathcal{S}\big{(}X(x)\big{)}\,,\] (35) and respectively \[{\mathcal{S}}_{\rm rcBK}(X_{g})=\mathcal{S}_{0}+\bar{\alpha} _{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\mathcal{K}(0)\, \mathcal{S}_{\rm rcBK}\big{(}X(x)\big{)}\,,\] (36) where the kernel is now evaluated at $x=0$ (or $\xi=1$), that is, in the eikonal approximation. Clearly, the compact notation $\mathcal{K}(0)$ stays for the LO BK (or dipole) kernel. In particular, the rcBK equation (36) is graphically illustrated in Fig. 11. <figure><img src="content_image/1608.05293/x19.png"><figcaption>Figure 12: A graphical illustration of the integral version of the NLO BKequation (37). The middle term with kernel K(0) describes a soft gluonemission and corresponds to the LO BK kernel, but with a running coupling. Thelast term with kernel K2(0) represents the NLO piece of the BK kernel, withthe running-coupling corrections excluded. The two s-channel gluons includedwithin K2(0) can be close in rapidity. The non-linear effects in S describingmultiple scattering are kept implicit.</figcaption></figure> These simpler notations hopefully make clear that the quantity denoted as $\Delta{\mathcal{S}}$ must encode all NLO corrections to the BK kernel except for those expressing the running of the coupling. These corrections should be computed in the large-$N_{c}$ limit, for consistency with our previous approximations. They can be inferred by inspection of the NLO version of the BK equation shown in Eq. (5) of Ref. [32]. For convenience, we display the large-$N_{c}$ version of this equation in Appendix C, where we also discuss its collinear improvement, along the lines of Refs. [37; 45]. For simplicity, we shall stick here to our schematic notations and refer to Appendix C for more explicit formulae. The integral version of the NLO BK equation reads (in momentum space and adapted to the kinematics at hand) \[{\mathcal{S}}(X_{g})={\mathcal{S}}_{0}+\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_ {g}}^{1}\frac{{\rm d}x}{x}\,\mathcal{K}(0)\,\mathcal{S}\big{(}X(x)\big{)}+\bar {\alpha}_{s}^{2}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\mathcal{K} _{2}(0)\,\mathcal{S}\big{(}X(x)\big{)}\,,\] (37) where $\mathcal{K}_{2}(0)$ is a compact notation for the NLO piece of the BK kernel alluded to above: its action on $\mathcal{S}$ generates all the NLO terms visible in the r.h.s. of Eq. (C) _except for the running coupling corrections_, which are explicitly included in the middle term of Eq. (37). More precisely, if one uses the prescription (32) for the running coupling, then in constructing $\mathcal{K}_{2}(0)$ one should omit the NLO terms in Eq. (C) which are proportional to $\bar{b}$ (but keep all the other ones). Besides, the large NLO corrections enhanced by double or single collinear logarithms require all-order resummations (‘collinear improvement’), to be shortly described (see also the discussion in Appendix C). By comparing Eqs. (35) and (37), it is now obvious that $\Delta{\mathcal{S}}(X_{g})$ must be identified with the third term in the r.h.s. of Eq. (37), that we now describe in some detail. As a general rule, the NLO corrections to the BK kernel are obtained by evaluating all the 2-loop graphs which yield a contribution of order $\alpha_{s}^{2}Y$ and hence count for a single step in the high-energy evolution (see Ref. [32] for an exhaustive list of diagrams and explicit calculations). A typical such a graph involves a sequence of two gluon emissions, whose longitudinal fractions $x_{1}$ and $x_{2}$ obey $x_{1}\sim x_{2}\ll 1$; that is, the two gluons have similar rapidities, but they are both soft compared to the projectile. The integration variable $x$ in the last term in Eq. (37) can be interpreted as $x\equiv x_{1}+x_{2}$. The other independent rapidity integration, say over the variable $u\equiv x_{2}/(x_{1}+x_{2})$ with $0<u\leq 1$, is implicitly included in the structure of the NLO kernel $\mathcal{K}_{2}(0)$. This is possible since the scattering between the 2-gluon system and the target is independent of $u$ to the accuracy of interest¹³. A rather schematic, but intuitive, graphical illustration of Eq. (37) is shown in Fig. 12. [FOOTNOTE:13][ENDFOOTNOTE] At large $N_{c}$, two successive gluon emissions from the original $q\bar{q}$ dipole can generate up to three dipoles in the fundamental representation. Accordingly, the third term in Eq. (37) involves contributions which are cubic in the dipole $S$-matrix, together with quadratic and linear terms. All these contributions are visible in Eq. (C). To the accuracy of interest, the kinematics of the primary gluon (with energy fraction $x=x_{1}+x_{2}$) can be treated in the LLA. This explains why we were able to use the same lower limit $x_{g}$ in the integral over $x$ and also the same rapidity variable $X(x)$ for the relevant $S$-matrices as in the middle term in Eq. (37), which encodes the LO evolution. <figure><img src="content_image/1608.05293/x20.png"><figcaption>Figure 13: A graphical illustration of the factorization of quark productionat NLO, as schematically encoded in Eq. (38). The primary gluon emission, withkernel K(x), is included with exact kinematics. The evolution step depicted astwo-gluon emission with kernel K2(0) succinctly represents the NLO piece ofthe BK kernel (but without the running-coupling corrections, which werealready included in the middle term). Non-linear effects in S, correspondingto multiple scattering, are implicitly included but not explicitly shown.</figcaption></figure> To conclude, the result for the quark multiplicity which is complete to NLO accuracy can be compactly, but schematically, written as \[\mathcal{N}_{{\rm LO}+{\rm NLO}}=\mathcal{ S}_{0}+\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\, \mathcal{K}(x)\,\mathcal{S}\big{(}X(x)\big{)}+\bar{\alpha}_{s}^{2}(k_{\perp}^{ 2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\mathcal{K}_{2}(0)\,\mathcal{S}\big{(}X (x)\big{)}\,,\] (38) and is illustrated in Fig. 13. To deduce a more explicit expression, one should use Eq. (3.3) together with the formula for $\Delta\mathcal{S}$ shown in Appendix C. Although the above factorization scheme is formally correct to NLO, it might be still too hard, if not impossible, to achieve a full NLO accuracy in a practical calculation. Indeed, as already mentioned, the target evolution is not known to NLO and the relation between the function $\mathcal{S}\big{(}X(x)\big{)}$ which enters the above factorization and the solution $\mathcal{S}(x)$ to the NLO BK equation for the projectile is not known at the accuracy of interest. Besides, numerical calculations might be hindered by the complexity of the NLO BK equation and of the transverse convolutions implicit in the two integral terms in Eq. (38). In view of that, we would like to propose two approximation schemes which we believe have more chances to be transposed in practice. In both schemes, the approximations refer to the dipole evolution, whereas the NLO impact factor, as represented by the kernel $\mathcal{K}(x)$ in the middle term in Eq. (38), should be treated exactly. The simplest approximation which is still physically meaningful is the LO approximation (in the sense of rcBK) to the dipole evolution. This is obtained by neglecting the third term in Eq. (38) and replacing $\mathcal{S}\simeq\mathcal{S}_{\rm rcBK}$ within the integrand of the middle term. A similar strategy has been used [51; 53; 57] in relation with the ‘plus’ prescription [1; 2] ; the respective numerical calculations, albeit very complex, turned out to be tractable [67]. The second approximation, which is more ambitious, refers to the use of the collinearly-improved version of the BK equation, as proposed in Refs. [37; 45]. We recall that, besides the running coupling corrections, this equation also resums double-collinear logarithms together with a subset of the single collinear logarithms (which includes the respective contribution at NLO). The corresponding approximation to Eq. (38) involves two aspects. On the one hand, one must relate the function $\mathcal{S}\big{(}X(x)\big{)}$, which encodes the evolution of the target, to the solution to the collinearly-improved BK equation (which refers to the evolution of the projectile). This aspect is particularly important for the middle term in Eq. (38) and will be discussed in Appendix B. On the other hand, one must use a simplified version of the NLO BK kernel $\mathcal{K}_{2}(0)$ which keeps only those corrections to the LO kernel which refer to the collinear improvement. This will be described in what follows. To that aim, it is convenient to use the coordinate representation: the last term in Eq. (38), which we recall corresponds to the piece $\Delta{\mathcal{S}}(\bm{k},X_{g})$ in Eq. (3.3), will be written as \[\Delta{\mathcal{S}}(\bm{k},X_{g})=\int{\rm d}^{2}\bm{r}\,{\rm e}^{-{\rm i}\bm{ k}\cdot\bm{r}}{\Delta S}(\bm{r},x_{\rm T})\,,\] (39) where $\bm{r}=\bm{x}-\bm{y}$ and \[{\Delta S}(\bm{x},\bm{y};x_{\rm T})\equiv\int_{ x_{\rm T}}^{1}\frac{{\rm d}x}{x}\int\frac{{\rm d}^{2}\bm{z}} {2\pi}\,\bar{\alpha}_{\rm fac}\,{\mathcal{M}}_{\bm{x}\bm{y}\bm{z}}\big{(} \mathcal{K}_{{\rm DLA}}\mathcal{K}_{{\rm SL }}-1\big{)}\Big{[}S\Big{(}\bm{x},\bm{z};\frac{x_{\rm T}}{x} \Big{)}S\Big{(}\bm{z},\bm{y};\frac{x_{\rm T}}{x}\Big{)}-S \Big{(}\bm{x},\bm{y};\frac{x_{\rm T}}{x}\Big{)}\Big{]}.\] (40) In the above equation, we recognize the dipole kernel ${\mathcal{M}}_{\bm{x}\bm{y}\bm{z}}$ and the running coupling $\bar{\alpha}_{\rm fac}$ that were introduced before, together with two multiplicative corrections to the kernel, $\mathcal{K}_{{\rm DLA}}$ and $\mathcal{K}_{{\rm SL}}$, which encode the resummations of double and respectively single collinear logarithms, as alluded to above. Physicswise, $\mathcal{K}_{{\rm DLA}}$ implements the condition of time-ordering for the successive soft gluon emissions by the projectile, whereas $\mathcal{K}_{{\rm SL}}$ resums a subset of the DGLAP logarithms (see Refs. [37; 45] for details). The new rapidity argument $x_{\rm T}$ which appears too in Eq. (40) will be later explained. Specifically, $\mathcal{K}_{{\rm DLA}}$ is defined as the function \[\mathcal{K}_{{\rm DLA}}(\rho)=\frac{{\rm J}_{1}\big{(}2\sqrt {\bar{\alpha}_{s}\rho^{2}}\big{)}}{\sqrt{\bar{\alpha}_{s}\rho^{2}}}=1-\frac{ \bar{\alpha}_{s}\rho^{2}}{2}+\frac{(\bar{\alpha}_{s}\rho^{2})^{2}}{12}+\cdots,\] (41) evaluated at $\rho^{2}={L_{\bm{x}\bm{z}r}L_{\bm{y}\bm{z}r}}$, with $L_{\bm{x}\bm{z}r}\equiv\ln[(\bm{x}-\bm{z})^{2}/r^{2}]$. If the double logarithm $L_{\bm{x}\bm{z}r}L_{\bm{y}\bm{z}r}$ is negative, then one uses its absolute value and the Bessel function ${\rm J}_{1}$ gets replaced by the modified Bessel function ${\rm I}_{1}$. Note however that if, e.g., $(\bm{x}-\bm{z})^{2}\ll r^{2}$, so that $L_{\bm{x}\bm{z}r}<0$, then $(\bm{y}-\bm{z})^{2}\simeq r^{2}$ and hence $L_{\bm{y}\bm{z}r}\simeq 0$. Accordingly, the relatively small daughter dipoles bring no significant contributions to the difference $\mathcal{K}_{{\rm DLA}}-1$. Furthermore, \[\mathcal{K}_{{\rm SL}}=\exp\left\{-\bar{\alpha}_{s}A_{1} \left\|\ln\frac{(\bm{x}\!-\!\bm{y})^{2}}{\min\{(\bm{x}\!-\!\bm{z})^{2},(\bm{y} \!-\!\bm{z})^{2}\}}\right\|\right\},\qquad A_{1}=\frac{11}{12}+\frac{N_{\rm f}} {6N_{\rm c}^{3}}\,,\] (42) where $A_{1}$ is the ‘gluonic anomalous dimension’ of the DGLAP evolution (see e.g. [40] for details). Note that the difference $\mathcal{K}_{{\rm DLA}}\mathcal{K}_{{\rm SL }}-1$ starts at $\mathcal{O}{(\bar{\alpha}_{s})}$, as expected. However, keeping only that lowest-order term in the expansion would artificially enhance the importance of the ‘collinear’ regions in phase-space — the regions where the successive gluon emissions (or dipole splittings) are strongly ordered in transverse sizes, or momenta. This is the origin of the instability of the strict NLO approximation to the high-energy evolution, as previously mentioned. Vice-versa, the all-order resummation of such corrections within the factor $\mathcal{K}_{{\rm DLA}}\mathcal{K}_{{\rm SL}}$ suppresses the contributions from the ‘collinear’ regions and thus restores the convergence of perturbation theory. This is rather obvious for the second factor $\mathcal{K}_{{\rm SL}}$, which exponentially cuts off the configurations where the daughter dipoles are either much smaller, or much larger, than the parent dipole. But this is also true for the other factor $\mathcal{K}_{{\rm DLA}}$, which, as already mentioned, becomes important only when the daughter dipoles are sufficiently large, such that $\bar{\alpha}_{s}\rho^{2}\gg 1$. In that case, the Bessel function ${\rm J}_{1}\big{(}2\sqrt{\bar{\alpha}_{s}\rho^{2}}\big{)}$ is rapidly oscillating when varying the position $\bm{z}$ of the emitted gluon, hence the integral over the regions in space where ‘$\bm{z}$ is large’ (in the sense that $\|\bm{z}-\bm{x}\|\sim\|\bm{z}-\bm{y}\|\gg r$) averages to zero. To conclude, for gluon emissions which are strongly ordered in transverse momenta, we have $\mathcal{K}_{{\rm DLA}}\mathcal{K}_{{\rm SL }}\simeq 0$ and then the overall kernel in Eq. (40) reduces to _minus_ the LO kernel $-\mathcal{K}(0)$. In turn, the latter subtracts the soft and collinear contributions to the middle term in Eq. (38), that is, it implements the collinear improvement for the LO evolution, as it should. There is one more aspect of Eq. (40) which requires a few words of explanation: the rapidity arguments of the various $S$-matrices and, related to them, the lower limit $x_{\rm T}$ for the integral over $x$. If the evolution of these $S$-matrices is computed to LO, i.e. according to rcBK, then one can identify $x_{\rm T}\simeq x_{g}=k_{\perp}^{2}/\hat{s}$ and there is no distinction between projectile and target evolutions. Albeit such an approximation would be formally justified to NLO accuracy, it is still preferable to use the collinearly-improved evolution, i.e. the BK equation with the resummed kernel ${\mathcal{M}}_{\bm{x}\bm{y}\bm{z}}\mathcal{K}_{{\rm DLA}} \mathcal{K}_{{\rm SL}}$. Indeed, this would ensure a smooth matching with the middle term in Eq. (38), where the collinearly-improved version of the evolution becomes compulsory. Since the latter has been formulated for projectile evolution alone [37; 45], one must understand what is the longitudinal phase-space for the evolution in $x=q^{+}/q_{0}^{+}$ which corresponds to the physical range for the evolution in $X=q^{-}/P^{-}$. In coordinate space and to the accuracy of interest, one can write (see e.g. [37]) \[S_{\rm T}\big{(}\bm{r},X\big{)}=S_{\rm P} \big{(}\bm{r},x_{\rm T}\big{)}\,\qquad\mbox{with}\qquad X= \frac{Q^{2}}{\hat{s}}\,\gg\,x_{\rm T}=\frac{Q^{2}_{0}}{\hat{ s}}\,,\] (43) where we have temporarily introduced the subscripts T (‘target’) and P (‘projectile’), to make the discussion more transparent and we recall that $\hat{s}=2q_{0}^{+}P^{-}$. In this equation, $Q=1/r$ is the dipole resolution scale in the transverse plane and can be also identified with the transverse momentum $k_{\perp}$ of the produced quark, via the Fourier transform (39); hence, $X\simeq X_{g}=x_{g}$. Furthermore, $Q_{0}$ is the saturation scale in the nuclear target at low energy (say, within the MV model). The interesting situation is such that $k_{\perp}\gtrsim Q_{s}(X_{g})\gg Q_{0}$ and therefore $x_{\rm T}\ll x_{g}$, as indicated too in Eq. (43). The relation (43) can be understood as follows. The scattering between the small dipole and the nuclear target probes the evolution of the latter down to values of $X$ such that the longitudinal extent $\Delta x^{+}=1/(XP^{-})$ of the softest target fluctuations is still smaller than the lifetime $2q_{0}^{+}/Q^{2}$ of the dipole projectile. This argument, which selects $X={Q^{2}}/{\hat{s}}$ as anticipated, is merely a variation of our earlier derivation of Eq. (1) in Sect. 2.1. If on the other hand the evolution is encoded in the wavefunction of the projectile, then one should allow for the small-$x$ fluctuations with transverse momenta $\bm{q}$ within the range $Q^{2}>q_{\perp}^{2}>Q_{0}^{2}$ and with large enough lifetimes $2xq_{0}^{+}/q_{\perp}^{2}\gtrsim 1/P^{-}$ (indeed, $1/P^{-}$ is the longitudinal extent of the un-evolved target). These conditions imply $x\gtrsim q_{\perp}^{2}/\hat{s}\geq Q_{0}^{2}/\hat{s}$, or $x\geq x_{\rm T}$, which explains the lower limit in the integral over $x$ in Eq. (40). One may find surprising that the rapidity interval $\Delta y=\ln(1/x_{\rm T})$ available for the evolution of the projectile is much larger than that, $\Delta Y=\ln(1/X_{g})$, allowed for the evolution of the target. But one should keep in mind that the projectile evolution with decreasing $x$ is strongly constrained by the condition of time-ordering, which limits the corresponding transverse phase-space (via the multiplicative correction $\mathcal{K}_{{\rm DLA}}$ to the kernel) and thus reduces the evolution speed. By contrast, there is no similar constraint for the evolution of the target with decreasing $X$, since in that case the physical condition of time-ordering is automatically satisfied. ### Subtracting the leading order evolution: why is this subtle The factorization scheme that we have constructed in the previous sections, cf. Eqs. (3.3) or (38), does not involve any subtraction: there is no over-counting of the relevant perturbative contributions and hence no need for a subtraction. We therefore expect that calculations based on this factorization should yield a positive result for the quark multiplicity. More explicit arguments in that sense will be presented later in this section. This represents a significant improvement over the previous proposal in Refs. [1; 2], so it is interesting to better understand the relation between these two schemes. From the discussion in Sect. 3.1, we recall that in the approach [1; 2] on explicitly subtracts the LO evolution of the dipole $S$-matrix from the NLO correction to the impact factor. Such a subtraction can also be performed within our present approach. To synthetically describe that, we shall again use the schematic notations introduced in the previous subsection. Using Eq. (37), we can express the last term in the NLO cross-section (38) (a NLO correction to the dipole evolution) as the difference between the dipole $S$-matrix at NLO and its LO evolution: \[\bar{\alpha}_{s}^{2}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\, \mathcal{K}_{2}(0)\mathcal{S}\big{(}X(x)\big{)}={\mathcal{S}}(X_{g})-\left[{ \mathcal{S}}_{0}+\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x }{x}\,\mathcal{K}(0)\,\mathcal{S}\big{(}X(x)\big{)}\right]\,.\] (44) The r.h.s. of this equation is not exactly the same as the difference $\mathcal{S}-\mathcal{S}_{\rm rcBK}$, but it is very close to it. (The expression between the square brackets would reduce to $\mathcal{S}_{\rm rcBK}$ if the $S$-matrix under the integral would be itself approximated by rcBK; recall Eq. (47).) So, clearly, Eq. (44) expresses a NLO correction which is _a priori_ small as the difference between two quantities which are individually large: each of them includes the LO contribution $\mathcal{S}_{\rm rcBK}$. Inserting Eq. (44) into Eq. (38), one deduces an alternative expression for the NLO cross-section, \[\mathcal{N}_{{\rm LO}+{\rm NLO}}=\mathcal{ S}(X_{g})\,+\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x} \,\big{[}\mathcal{K}(x)-\mathcal{K}(0)\big{]}\,\mathcal{S}\big{(}X(x)\big{)},\] (45) in which the LO evolution (as represented by the kernel $\mathcal{K}(0)$) is explicitly subtracted from the impact factor. Since the difference $\mathcal{K}(x)-\mathcal{K}(0)$ vanishes as $x\to 0$, the above integral is controlled by large values $x\sim 1$ and it does not generate a small-$x$ logarithm anymore. That is, the second term in the r.h.s. of Eq. (45) is a pure $\bar{\alpha}_{s}$ correction, that can be viewed as the NLO contribution to the impact factor. The NLO correction to the dipole evolution is now fully encoded in the first term $\mathcal{S}(X_{g})$. Indeed, to the NLO accuracy of interest, the $S$-matrix within the integral can be evaluated by using the LO approximation, $\mathcal{S}\simeq\mathcal{S}_{\rm rcBK}$. Eq. (45) is very similar to the proposal in Refs. [1; 2], which in our schematic notations reads \[\mathcal{N}_{{\rm CXY}}=\mathcal{S}(X_{g})+\bar {\alpha}_{s}(k_{\perp}^{2})\int_{0}^{1}\frac{{\rm d}x}{x}\,\big{[}\mathcal{K}( x)-\mathcal{K}(0)\big{]}\,\mathcal{S}(X_{g})\,.\] (46) The main difference w.r.t. Eq. (45) is that the $S$-matrix within the above integral over $x$ is not evaluated at the $x$-dependent rapidity argument $X(x)$, but rather at its endpoint value $X_{g}=X(1)$. In other terms, Eq. (46) is _local_ in the target rapidity $X_{g}$, as standard for the $k_{\perp}$-factorization. In spite of this difference, the results in Eqs. (46) and (45) are perturbatively equivalent to NLO accuracy. Indeed, since the integral over $x$ in (45) is controlled by $x\sim 1$, there is no loss of NLO accuracy if one replaces $\mathcal{S}\big{(}X(x)\big{)}\to\mathcal{S}(X_{g})$. This can be easily checked by using the dominant energy-behavior of the BK solution in the weak scattering regime, namely $\mathcal{S}(X)\propto 1/X^{\lambda}$ with $\lambda=\mathcal{O}{(\bar{\alpha}_{s})}$. This being said, one should keep in mind that the limiting value $\mathcal{S}(X_{g})$ is strictly larger than $\mathcal{S}\big{(}X(x)\big{)}$ for any $x<1$, because the function $\mathcal{S}(X)$ increases when decreasing $X$. Hence, albeit formally allowed to the accuracy of interest, the replacement $\mathcal{S}\big{(}X(x)\big{)}\to\mathcal{S}(X_{g})$ could still be troublesome, in that it might result in an _over-subtraction_. Indeed, as discussed in [56], the difference $\mathcal{K}(x)-\mathcal{K}(0)$ is strictly negative for sufficiently large $k_{\perp}\gtrsim Q_{s}$. Hence, the negative correction to the impact factor at large $k_{\perp}$ is over included in (46) as compared to (45), a feature which might contribute to the ‘negativity’ problem under consideration. In order to discuss this problem — the fact that the cross-section computed according to Eq. (46) becomes negative at sufficiently large (but still semi-hard) $k_{\perp}$ —, it is useful to keep in mind that previous numerical evaluations of Eq. (46) used approximate versions of the dipole $S$-matrix, like $\mathcal{S}_{\rm rcBK}$[51; 53; 57]. Hence, it would be interesting to understand why Eq. (46) with $\mathcal{S}\to\mathcal{S}_{\rm rcBK}$ can lead to a negative cross-section at large transverse momenta. To that aim, let us consider the rcBK approximation to the ‘subtracted’ version of our factorization scheme, cf. (45). This reads \[\mathcal{N}_{{\rm rcBK}}=\mathcal{S}_{{\rm rcBK }}+\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\big{[} \mathcal{K}(x)-\mathcal{K}(0)\big{]}\,\mathcal{S}_{\rm rcBK} \big{(}X(x)\big{)}\,.\] (47) Via the rcBK equation (36), this is equivalent to \[\mathcal{N}_{{\rm rcBK}}=\mathcal{S}_{0}+\bar{\alpha}_{s}(k_ {\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\mathcal{K}(x)\,\mathcal{S}_{ \rm rcBK}\big{(}X(x)\big{)},\] (48) which is of course the same as our ‘un-subtracted’ result in Eq. (34) with $\Delta\mathcal{S}=0$ and $\mathcal{S}\to\mathcal{S}_{\rm rcBK}$. We believe that the r.h.s. of Eq. (48) should be positive. Indeed, when the primary gluon emission is treated in the eikonal approximation ($\mathcal{K}(x)\to\mathcal{K}(0)$), Eq. (48) is the same as the integral representation of $\mathcal{S}_{\rm rcBK}$, which is well known to be positive definite (at least in coordinate space). With the actual kernel $\mathcal{K}(x)$, the value of the integral term should be somewhat reduced (since the correct phase-space for the primary gluon emission is smaller than its eikonal estimate), but this should remain positive, as it describes the growth of the dipole $S$-matrix via gluon emissions. Hence, the ‘subtracted’ result in Eq. (47) should be positive as well. In spite of that, we believe that explicit numerical calculations based on Eq. (47) may run into difficulties (in particular, yield a negative result) because of the high degree of ‘fine-tuning’ inherent in the subtraction method: in going from Eq. (48) to Eq. (47), we have added and subtracted the same quantity ($\mathcal{S}_{\rm rcBK}$), but we have done that in a very peculiar way: we have added the l.h.s. of Eq. (36), that is, $\mathcal{S}_{\rm rcBK}$ itself, but we have subtracted the r.h.s. of Eq. (36), that is, the integral representation of $\mathcal{S}_{\rm rcBK}$. This procedure leaves the result unchanged if and only if the function $\mathcal{S}_{\rm rcBK}$ is an exact solution to the integral equation (36). But any numerical approximation in solving rcBK, or in the Fourier transform $S(\bm{r},X)\to\mathcal{S}(\bm{k},X)$, may lead to an imbalance between the terms that have been added and respectively subtracted. For instance, such an imbalance could be introduced by the treatment of the running coupling corrections, which in general is not fully coherent between the coordinate-space and the momentum-space representations. In practice, the momentum-space version of the rcBK equation, as shown in (36), is not the _exact_ Fourier transform of the respective equation in coordinate space: the Fourier transform is not also applied to the running of the coupling. So, strictly speaking, there is some mismatch between Eqs. (47) and (48) even when using the exact solution to the rcBK equation, as obtained in coordinate space. When computing the cross-section according to Eq. (46), this mismatch can be further enhanced by the fact that the function $\mathcal{S}_{\rm rcBK}\big{(}X(x)\big{)}$ gets replaced by its maximal value $\mathcal{S}_{\rm rcBK}(X_{g})$ (cf. the discussion after Eq. (46)). From the above discussion, we see that the rcBK equation plays the role of a ‘self-consistency condition’ for rewriting the cross-section in a ‘subtracted’ form. To illustrate the importance of having ‘good’ solutions to this equation, let us consider a somewhat extreme example, where this condition is strongly violated. (A similar discussion can be found in [56].) Namely, we consider the popular approximation in which the dipole $S$-matrix is taken from the GBW model [76], which is a Gaussian: \[{\mathcal{S}}_{\rm GBW}(\bm{k},X)\,=\,\frac{4\pi}{Q_{s}^{2}} \,{\rm e}^{-\frac{k_{\perp}^{2}}{Q_{s}^{2}}}\,,\] (49) where $Q_{s}^{2}(X)=Q_{0}^{2}(X_{0}/X)^{\lambda}$ with $\lambda\simeq 0.3$. Clearly, this is a very poor approximation at high $k_{\perp}\gtrsim Q_{s}$, where it decays exponentially, in sharp contrast with the power law tail $1/k_{\perp}^{4}$ predicted by pQCD. Using this particular model within Eq. (45), one finds \[\mathcal{N}_{{\rm GBW}}=\mathcal{S}_{{\rm GBW }}(X_{g})+\bar{\alpha}_{s}\int_{x_{g}}^{1}\frac{{\rm d}x}{x}\,\big{[}\mathcal{ K}(x)-\mathcal{K}(0)\big{]}\,\mathcal{S}_{\rm GBW}\big{(}X(x )\big{)}\,.\] (50) The ‘LO’ piece $\mathcal{S}_{{\rm GBW}}$ exponentially vanishes at large $k_{\perp}\gg Q_{s}$, whereas the ‘subtracted’ piece shows a tail $\propto 1/k_{\perp}^{4}$, since obtained by iterating once the LO BK equation. So, the ‘subtracted’ piece is not only larger than the ‘LO’ one, but it is the only one to survive at large $k_{\perp}$. Hence, at $k_{\perp}\gtrsim Q_{s}$ the overall result reduces to the second, ‘NLO’, piece, which is negative in that particular region of phase-space [56], as already mentioned. This is in agreement with the numerical findings in [57; 56]. To summarize, albeit Eqs. (47) and (48) are in principle equivalent with each other, the second equation is probably safer to use in practice. A similar discussion applies to the full NLO cross-section: the ‘unsubtracted’ factorization (38) should provide a meaningful result which is positive when evaluated with either the NLO $S$-matrix $\mathcal{S}$, or with its collinearly-improved version $\mathcal{S}_{{\rm collBK}}$. On the other hand, the ‘subtracted’ version (45), which may look appealing since structurally simpler, is likely to be more tricky to use in practice. ## 4 Summary and conclusions In this paper, we have established a factorization scheme allowing the computation of single-inclusive particle production at forward rapidities in proton-nucleus collisions at next-to-leading order in pQCD, in the presence of the non-linear effects associated with the high gluon density in the nuclear target. The main difference with respect to the previous proposal in [1; 2] is that our result involves no rapidity subtraction, meaning that it is free of the fine-tuning problem inherent in any such a subtraction scheme. The ‘fine-tuning’ refers to the fact that the numerical solution to the evolution equation for the dipole $S$-matrix must be precisely known to ensure that the quantity which is included as ‘LO evolution’ is properly subtracted from the ‘NLO correction to the impact factor’. Further approximations within the subtraction scheme, which are formally allowed to NLO accuracy, or even small numerical errors in the associated calculations, may spoil such a fine cancellation and lead to unphysical results. In our opinion, this is the reason why the cross-section computed within this scheme appears to turn negative at sufficiently large transverse momenta [51; 53; 57; 56]. By contrast, our scheme is more robust and should converge to physical, positive-definite, results with considerably less numerical efforts. Our factorization scheme relies on the skeleton structure of perturbative QCD up to two-loop order, which is the relevant loop-order for the NLO calculation. The most general version of our result is exhibited in Eq. (3.2). This version however is probably too complicated to be used in practice. Fortunately, important simplifications become possible in the interesting situation where the transverse momentum $k_{\perp}$ of the produced quark is relatively large, $k_{\perp}\gtrsim Q_{s}$. In that case, the primary gluon is hard as well, $p_{\perp}\sim k_{\perp}$, and the general formula (3.2) can be then replaced by Eq. (3.3) [or (3.3)], which is considerably simpler. The latter shows the same degree of complexity, in terms of transverse integrations, like the formulae already used in practice in relation with the subtraction method. So, we are confident that Eq. (3.3) can indeed by explicitly evaluated, via the numerical tools developed in [67; 51; 53; 57; 56]. As discussed in Sect. 2.4, the fact that we can approximate $p_{\perp}\sim k_{\perp}$ is specific to the problem of particle production. This would not apply, say, to deep inelastic scattering, where the dipole evolution must be computed in transverse coordinate space. In that case, if the parent dipole is sufficiently small ($r\ll 1/Q_{s}$) — as appropriate for DIS at relatively high virtuality $Q^{2}\gg Q_{s}^{2}$ —, then the first gluon emission in the high-energy evolution is considerably _softer_ (it typically carries a transverse momentum $p_{\perp}\ll 1/r$) and the dependence upon its transverse kinematics cannot be simplified at NLO. This is visible in the NLO calculations of the DIS impact factor [77; 78; 79; 80]. To explicitly evaluate the quark multiplicity according to Eq. (3.3), one also needs a suitable approximation for the high-energy evolution of the dipole $S$-matrix. The simplest such an approximation which is still meaningful for phenomenology is the LO BK equation with running coupling (rcBK) [68; 69; 70]. A more accurate treatment of the evolution can be obtained by using the collinearly-improved BK equation [44; 37; 45; 47; 48; 46], which resums the double collinear logarithms inherent in the ‘hard-to-soft’ evolution of the projectile together with a subset of single collinear logs. The inclusion of the collinear improvement in the problem at hand is quite subtle, due to the need to properly identify the rapidity phase-space for the evolution of the dilute projectile (the incoming quark, or the pair made with this quark and the hard primary gluon). This is clarified in Sect. 3.4 and Appendix B. An important lesson emerging from our analysis is that one should not always insist in writing the result of a NLO calculation in pQCD at high-energy in a ‘$k_{\perp}$-factorized form’, which is local in rapidity. This can be best appreciated by comparing our results in Eqs. (38), (45), and (46). Eq. (38) is obtained via a direct evaluation of the relevant, one-loop and two-loop, diagrams. As such, it does not involve any rapidity subtraction at one-loop level: the would-be NLO correction to the impact factor and the first step in the high-energy evolution of the color dipole are both encoded in the middle term in Eq. (38). In Eq. (45), the two contributions that we just mentioned are explicitly separated from each other, via a rapidity subtraction, but the result is still not ‘factorized’: the $\mathcal{O}{(\alpha_{s})}$-correction to the impact factor and the dipole $S$-matrix are still entangled by the rapidity integral over $x$ (besides the transverse momentum convolutions which are implicit in our schematic notations). Finally, Eq. (46) expresses a genuine $k_{\perp}$-factorization: the r.h.s. is local in the target rapidity $X_{g}$ and the NLO impact factor is explicitly factorized (in so far as the rapidity dependence is concerned) from the dipole $S$-matrices, which encode the high-energy evolution¹⁴. These three representations for the NLO cross-section, (38), (45), and (46), are all consistent with each other to NLO accuracy. Yet, the two representations involving a rapidity subtraction, Eqs. (45) and (46), are potentially affected by the issue of fine-tuning that we have identified in this paper. By contrast, the original result in Eq. (38) is free from this problem and therefore it should provide a positive-definite estimate for the cross-section. [FOOTNOTE:14][ENDFOOTNOTE] We would like to conclude this section with some prospects for the extension of this factorization program beyond NLO. Our factorization scheme suggests an interesting pattern which is likely to survive beyond the present approximations. The distinguished feature of this pattern is the fact that the higher-order corrections to the impact factor are not explicitly separated from the relevant corrections to the high-energy evolution. To illustrate this point, let us consider quark production at NNLO. Without any claim to completeness and leaving aside the possibility of new contributions which go beyond the dipole picture, let us here indicate the generic structure that we expect in view of our previous analysis in this paper. Namely, we expect the NNLO cross-section to include the following four terms (in schematic notations, similar to those introduced in Sect. 3.4) \[\mathcal{N}_{{\rm LO}+{\rm NLO }+{\rm NNLO}}=\mathcal{S}_{0} +\bar{\alpha}_{s}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d}x}{x }\,\mathcal{K}(x)\,\mathcal{S}\big{(}X(x)\big{)}\] \[+\bar{\alpha}_{s}^{2}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d} x_{1}}{x_{1}}\int_{x_{g}}^{x_{1}}\frac{{\rm d}x_{2}}{x_{2}}\,\big{[}\mathcal{K }_{2}(x_{1},x_{2}/x_{1})-\mathcal{K}_{2}(x_{1},0)\big{]}\,\mathcal{S}\big{(}X( x_{1},x_{2})\big{)}\] \[+\bar{\alpha}_{s}^{3}(k_{\perp}^{2})\int_{x_{g}}^{1}\frac{{\rm d} x}{x}\,\mathcal{K}_{3}(0)\,\mathcal{S}\big{(}X(x)\big{)}\,,\] (51) where it is now understood that the dipole $S$-matrices evolve according to the (presently unknown) NNLO version of the BK equation. The argument of the running couplings is taken as $k_{\perp}^{2}$, for illustration, but this should be the right choice only in the limit where the transverse momentum of the produced quark is larger than any other transverse scale in the problem. Also, the lower limit $x_{g}$ on the integrals over $x$ is purely illustrative: the actual limit should depend upon the exact kinematics. The first two terms in the r.h.s. of Eq. (4) have the same structure as the respective terms in Eq. (38); in particular, the second term encodes the NLO impact factor and the LO piece of the BK kernel. The third term is a generalization of the last term in Eq. (38): it encodes the NLO piece of the BK kernel together with the NNLO correction to the impact factor. This term is generated by 2-loop graphs where the 2 emitted partons (say, gluons) can have arbitrarily energy fractions $x_{1}$ and $x_{2}$ (with $x_{2}\leq x_{1}$ for definiteness), so their emissions must be computed with exact kinematics. The sum of all such graphs is schematically represented by the kernel $\mathcal{K}_{2}(x_{1},x_{2}/x_{1})$. The subtraction of $\mathcal{K}_{2}(x_{1},0)$ is needed to avoid the over inclusion of the first two steps in the high energy evolution, that were already included in the second term, with kernel $\mathcal{K}(x)$. Finally, the fourth term in Eq. (4) represents the NNLO correction to the BK kernel, as generated by 3-loop graphs in which the 3 gluons are all soft, but close in rapidity to each other: $x_{1}\sim x_{2}\sim x_{3}\ll 1$. Clearly, in the limit where the very first emission by the projectile is soft and computed in the eikonal approximation, the r.h.s. of Eq. (4) becomes proportional to the integral representation of the NNLO BK equation. ## Acknowledgments We would like to thank Tuomas Lappi for inspiring discussions which triggered our interest on this problem. We are grateful to Bertrand Ducloué, Dmitry Ivanov, Tuomas Lappi, and Yan Zhu for useful comments on the manuscript. A.H.M. would like to thank Bowen Xiao for useful and informative discussions. D.N.T. would like to thank Guillaume Beuf for insightful remarks on the interplay between evolution and factorization at next-to-leading. A.H.M. and D.N.T. would like to acknowledge l’Institut de Physique Théorique de Saclay for hospitality during the early stages of this work. The work of E.I. is supported in part by the European Research Council under the Advanced Investigator Grant ERC-AD-267258. The work of A.H.M. is supported in part by the U.S. Department of Energy Grant # DE-FG02-92ER40699. ## Appendix A Cancellation of shortly lived virtual fluctuations As we have seen, the minus longitudinal momentum $p^{-}=p_{\perp}^{2}/2p^{+}$ of the primary gluon is constrained by light-cone energy conservation, that is, by Eq. (8). This constraint however exists only for the gluons which are crossing the cut (albeit eventually non measured), but not also for those which are emitted and reabsorbed on the same side of the cut. In principle, such ‘virtual’ gluons are allowed to have very short lifetimes $2p^{+}/p_{\perp}^{2}<1/P^{-}$, however we shall show here that their respective contributions cancel exactly. To that aim, we shall separately consider the two possible regimes in terms of the ratio $p_{\perp}/k_{\perp}$. We recall that $k_{\perp}$ is the transverse momentum of the measured quark and for the virtual diagrams it coincides with the momentum transferred by the target via scattering. (i) $p_{\perp}\gg k_{\perp}$. This is the simplest case, as the virtual fluctuation cannot be resolved by the scattering (the transverse separation $\Delta x_{\perp}\sim 1/p_{\perp}$ between the virtual gluon and its parent quark is much smaller than the transverse resolution $\sim 1/k_{\perp}$ of the exchanged gluon). Accordingly, all the diagrams shown in Fig. 14 are equally weighted by the $S$-matrix, so their sum must vanish by probability conservation. In practice, this occurs because the two self-energy graphs in figures (a) and (c) have a different sign as compared to the vertex correction in figure (b) and also an additional factor 1/2. <figure><img src="content_image/1608.05293/x21.png"><figcaption>Figure 14: The virtual diagrams when p⊥≫k⊥. In such a regime the shortly livedp-line is attached only to the projectile quark and the sum of the diagramsvanishes due to probability conservation. Transverse momenta are shown in thegraphs.</figcaption></figure> (ii) $p_{\perp}\lesssim k_{\perp}$. This case is much less trivial than (i), since now one has to also consider the diagrams in which the virtual gluon with momentum $p$ is emitted and/or absorbed by the exchanged gluon. For the calculation that follows, it is convenient to change our apporach and use Light Cone Perturbation Theory (LCPT); see e.g. [20] for an introduction. To this end, both the virtual and the ‘exchanged’ gluon are viewed as part of the projectile wavefunction and all the respective graphs are shown in Fig. 15. For simplicity, but without any loss of generality, we have taken the target to be a single quark. The four-momenta involved in the process are $q_{0}=(q_{0}^{+},0,\bm{0})$ for the incoming projectile quark, $P=(0,P^{-},\bm{0})$ for the incoming quark from the target, $\ell=(\ell^{+},\ell^{-},-\bm{k})$ for the ‘exchanged’ gluon and $p=(p^{+},p^{-},\bm{p})$ for the fluctuation. According to the rules of LCPT, all internal lines carry a positive plus longitudinal momentum and all particles are on-shell, meaning that the respective minus components are fixed by the on-shell condition; e.g. $\ell^{-}=k_{\perp}^{2}/2\ell^{+}$ and $p^{-}=p_{\perp}^{2}/2p^{+}$. Our convention in drawing Fig. 15 is that momenta flow from the left to the right (since this is the natural flow direction for the plus components). In particular, the four-momentum of the $\ell$-gluon has the opposite flow from the one in the main text, but we have set $\bm{\ell}_{\perp}=-\bm{k}_{\perp}$ so that the respective transverse momenta are still the same. The target quark must remain on-shell after the scattering, i.e. in the final state crossing the cut. This condition fixes its minus longitudinal momentum, that is, \[(P+\ell)^{2}=0\,\Rightarrow\,(P+\ell)^{-}=\frac{k_{\perp}^{2}}{2\ell^{+}}=\ell ^{-}.\] (52) Now we invoke light-cone energy conservation (between the initial and final states), which implies \[P^{-}=(P+\ell)^{-}+(q_{0}-\ell)^{-}=\ell^{-}+\frac{k_{\perp}^{2}}{2(q_{0}^{+}- \ell^{+})}\simeq\ell^{-}+\frac{k_{\perp}^{2}}{2q_{0}^{+}}.\] (53) Combining Eqs. (52) and (53) and using once more $\ell^{+}\ll q_{0}^{+}$, we see that $(P+\ell)^{-}\simeq P^{-}\simeq\ell^{-}$ and more importantly we can determine the plus component of the $\ell$-gluon which reads \[\ell^{+}=\frac{k_{\perp}^{2}}{2P^{-}}.\] (54) Looking deeply into the regime $2p^{+}/p_{\perp}^{2}\ll 1/P^{-}$, which means that the $p$-gluon is a fluctuation which has by far the shortest lifetime, and using $p_{\perp}\lesssim k_{\perp}$, we arrive at the strong ordering condition $p^{+}\ll\ell^{+}$. <figure><img src="content_image/1608.05293/x24.png"><figcaption>Figure 15: The virtual diagrams when p⊥≲k⊥. In contrast to those in Fig. 14,the shortly lived p-line can be now attached to the gluon “exchange” ℓ-line.Four-momenta are shown in the graphs and flow from the left to the right. Asin the text, q0=(q+0,0,0), P=(0,P−,0), ℓ=(ℓ+,ℓ−,−k) and p=(p+,p−,p), withℓ−=k2⊥/2ℓ+ and p−=p2⊥/2p+. Just for C1, and for our convenience in thecalculation, we let p→−p.</figcaption></figure> We are not going to give a detailed calculation for all the diagrams appearing in Fig. 15, but only deal with some representative cases. Therefore, let us start by considering the diagram B2, which is decomposed in terms of the basic tree-level graph and the loop correction as shown in Fig. 16. In LCPT, we can write this loop correction as \[{\rm B2}\|_{\rm loop}\to({\rm i}g)^{2}t^{a}t^{c}t^{a}\int \frac{{\rm d}^{2}\bm{p}\,{\rm d}p^{+}}{(2\pi)^{3}2p^{+}}\,\frac{1 }{\frac{\bm{p}^{2}}{2p^{+}}+\frac{(\bm{q}_{0}-\bm{p})^{2}}{2(q^{+}_{0}-p^{+})} }\,\frac{1}{\frac{\bm{p}^{2}}{2p^{+}}+\frac{(\bm{q}_{0}-\bm{p}+\bm{k})^{2}}{2( q_{0}^{+}-p^{+}-\ell^{+})}+\frac{\bm{k}^{2}}{2\ell^{+}}}\] \[\times\gamma\!\cdot\!\epsilon^{\lambda}(p)\,\gamma\!\cdot\!(q_{0} -p-\ell)\,\gamma^{+}\gamma\!\cdot\!(q_{0}-p)\,\gamma\!\cdot\!\epsilon^{\lambda }(p)\,\frac{1}{(2q_{0}^{+})^{2}},\] (55) where $t^{a}$ are the SU(3) generators in the fundamental representation and $\epsilon^{\lambda}(p)$ are the polarization vectors in the projectile light-cone gauge $A^{+}=0$, given by \[\epsilon^{\lambda}(p)=\Big{(}0,\epsilon^{\lambda-}=\frac{\bm{\epsilon}^{ \lambda}\!\cdot\!\bm{p}}{p^{+}},\bm{\epsilon}^{\lambda}\Big{)},\] (56) with $\bm{\epsilon}^{\lambda}$ two complex orthonormal two-dimensional vectors. Notice that since $p^{+}$ is very small, $\epsilon^{\lambda-}$ is the dominant component of the polarization vector. Using $\bm{q}_{0}=0$, $q_{0}^{+}\gg\ell^{+}\gg p^{+}$ and the condition in Eq. (54) which fixes $\ell^{+}$, one readily finds that the dominant term within both energy denominators and in the regime of interest (i.e. for $p_{\perp}\lesssim k_{\perp}$ and $2p^{+}/p_{\perp}^{2}\ll 1/P^{-}$) is the light-cone energy $\bm{p}^{2}/2p^{+}$ of the virtual gluon. This was to be expected since, as already said, the $p$-line is by far the one with the shortest lifetime. Regarding the inner products, we have $\gamma\cdot(q_{0}-p-\ell)\simeq\gamma\cdot(q_{0}-p)\simeq\gamma^{-}q_{0}^{+}$ and $\gamma\cdot\epsilon^{\lambda}(p)\simeq\gamma^{+}\bm{\epsilon}^{\lambda}\cdot \bm{p}/p^{+}$. Then, by using $t^{a}t^{c}t^{a}=-(C_{\rm F}-N_{\rm c}/2)t^{c}$ and putting aside a factor $t^{c}\gamma^{+}$ which is part of the tree-level graph, we eventually arrive at \[{\rm B2}\|_{\rm loop}=\frac{\alpha_{s}}{\pi}\left(C_{\rm F}-N_{\rm c}/2\right) \int\frac{{\rm d}p^{+}}{p^{+}}\frac{{\rm d}p_{\perp}^{2}}{p_{\perp}^{2}}.\] (57) One can calculate in a similar fashion the diagrams A, B1, D and E to find \[{\rm A}\|_{\rm loop}={\rm D}\|_{\rm loop}=-\frac{\alpha_{s}C_{\rm F}}{2\pi}\int \frac{{\rm d}p^{+}}{p^{+}}\frac{{\rm d}p_{\perp}^{2}}{p_{\perp}^{2}}\,,\qquad{ \rm B1}\|_{\rm loop}=-{\rm E}\|_{\rm loop}=\frac{\alpha_{s}N_{\rm c}}{2\pi}\int \frac{{\rm d}p^{+}}{p^{+}}\frac{{\rm d}p_{\perp}^{2}}{p_{\perp}^{2}}.\] (58) The factor $-1/2$ for diagrams A and D emerges from the calculation and it is the usual factor associated with the wavefunction renormalization. A similar argument applies to diagram E, however in that case we have put the factor 1/2 by hand, since half of the full result should be attached to the lower vertex of the $\ell$-line. Now let us move to the C diagrams, for which it is not hard to see that they will have the same color factor $N_{\rm c}/2$ with diagram B1. We have \[{\rm C}\|_{\rm loop}\to-({\rm i}g)^{2}\frac{N_{c}}{2}\int\frac{{ \rm d}^{2}\bm{p}\,{\rm d}p^{+}}{(2\pi)^{3}2p^{+}}\,\frac{1}{\frac{(\bm{k}+\bm{ p})^{2}}{2\ell^{+}}}\left[\frac{4n_{\alpha}n_{\beta}}{p^{+}}-\frac{\epsilon^{ \lambda}_{\alpha}(p)\epsilon^{\lambda}_{\beta}(p)}{\frac{\bm{p}^{2}}{2p^{+}}+ \frac{\bm{k}^{2}}{2\ell^{+}}}-\frac{\epsilon^{\lambda}_{\alpha}(p)\epsilon^{ \lambda}_{\beta}(p)}{\frac{\bm{p}^{2}}{2p^{+}}+\frac{(\bm{k}+\bm{p})^{2}}{2 \ell^{+}}}\right]\frac{\gamma^{\alpha}\,\gamma\!\cdot\!q_{0}\,\gamma^{+}}{2q_{ 0}^{+}}\,\frac{2\ell^{\beta}}{2\ell^{+}},\] (59) where each term in the square bracket represents the respective contribution from C1, C2 and C3. The 4-vector $n$, appearing in the instantaneous term, is such that $n\cdot\upsilon=\upsilon^{+}$ for any vector $\upsilon^{\alpha}$. In Eq. (59) we have already simplified the energy denominators (in particular we have neglected the terms suppressed by $1/q_{0}^{+}$) and kept only the terms that will contribute to the final result (notice also that in C1 we have let $\bm{p}\to-\bm{p}$ in order to combine them in an elegant way). Since $\epsilon^{\lambda-}$ is the large component of the polarization vector, we need to keep only the $--$ component of the tensor structure in the square bracket (notice that the indices here have been raised), which becomes \[\Big{[}\cdots\Big{]}^{--}=\frac{4}{p^{+}}-\frac{\frac{\bm{p}^{2}}{(p^{+})^{2}} }{\frac{\bm{p}^{2}}{2p^{+}}+\frac{\bm{k}^{2}}{2\ell^{+}}}-\frac{\frac{\bm{p}^{ 2}}{(p^{+})^{2}}}{\frac{\bm{p}^{2}}{2p^{+}}+\frac{(\bm{k}+\bm{p})^{2}}{2\ell^{ +}}}.\] (60) Here it becomes clear that we cannot make the immediate approximation to keep only the $\bm{p}^{2}/2p^{+}$ in the energy denominators (as we did for the previous class of diagrams), since the leading term cancels. It is a matter of straightforward algebra to find the dominant surviving terms: \[\Big{[}\cdots\Big{]}^{--}\simeq\frac{4}{\bm{p}^{2}}\,\left[\frac{(\bm{k}+\bm{p })^{2}}{2\ell^{+}}+\frac{\bm{k}^{2}}{2\ell^{+}}\right].\] (61) Since the transverse momentum of the exchange gluon is $\bm{k}+\bm{p}$ when attached to the upper vertex and $\bm{k}$ when attached to the lower one, only the first term in Eq. (61) should be kept for our purposes, and Eq. (59) leads to \[{\rm C}\|_{\rm loop}=\frac{\alpha_{s}N_{\rm c}}{2\pi}\int\frac{{\rm d}p^{+}}{p^ {+}}\frac{{\rm d}p_{\perp}^{2}}{p_{\perp}^{2}}.\] (62) <figure><img src="content_image/1608.05293/x32.png"><figcaption>Figure 16: (a) Tree level diagram and (b) loop correction for the diagram B2in Fig. 15. Four-momenta are shown in the graphs and flow from the left to theright.</figcaption></figure> Putting Eqs. (57), (58) and (62) together, it is obvious that \[{\rm A+B+C+D+E}=0,\] (63) which is the aforementioned cancellation of virtual corrections in the regime under consideration. Perhaps the most intuitive way to view this result is to realize that in diagrams C, D and E the shortly-lived gluon with momentum $p$ is emitted after the ‘exchanged’ gluon. The gluon in the ‘exchange’ line has a transverse velocity $\|\bm{k}+\bm{p}\|/\ell^{+}\sim k_{\perp}/\ell^{+}$, so that over the small lifetime $2p^{+}/p_{\perp}^{2}$ of the $p$-fluctuation it separates from the quark $q_{0}$ only a very small distance \[\Delta x_{\perp}\simeq\frac{k_{\perp}}{\ell^{+}}\,\frac{2p^{+}}{p_{\perp}^{2}} \sim\frac{2}{k_{\perp}}\,\frac{k_{\perp}^{2}}{2\ell^{+}}\,\frac{2p^{+}}{p_{ \perp}^{2}}=\frac{2}{k_{\perp}}\,\frac{2p^{+}P^{-}}{p_{\perp}^{2}}\ll\frac{2}{ k_{\perp}}.\] (64) To arrive at the above we have used Eq. (54) and the fact that $2p^{+}/p_{\perp}^{2}\ll 1/P^{-}$. Therefore, the system composed of the quark and the ‘exchanged’ gluon looks like a quark during the time interval defined by the emission and the reabsorption of the virtual, $p$-gluon. Indeed, one has \[\left({\rm C+D+E}\right)\|_{\rm loop}=-\frac{\alpha_{s}C_{\rm F}}{2\pi}\int \frac{{\rm d}p^{+}}{p^{+}}\frac{{\rm d}p_{\perp}^{2}}{p_{\perp}^{2}},\] (65) which can be associated with the quark wavefunction renormalization factor, more precisely with \[\sqrt{Z_{2}}-1=\sqrt{1+\left(Z_{2}-1\right)}-1\simeq\frac{1}{2}\left(Z_{2}-1 \right),\] (66) where the r.h.s. follows since $Z_{2}-1\sim\alpha_{s}\ll 1$. Similarly graph A gives another factor of $1/2(Z_{2}-1)$, while graphs B can be identified with the quark-gluon vertex renormalization factor $Z_{1}$ to order $\alpha_{s}$ : \[({\rm B_{1}+B_{2}})\|_{\rm loop}=\frac{1}{Z_{1}}-1\simeq 1-Z_{1}.\] (67) Therefore, it is instructive to rewrite the sum of the one-loop virtual corrections as follows \[\left[1+\frac{1}{2}\left(Z_{2}-1\right)\right]\left[1+\left(\frac{1}{Z_{1}}-1 \right)\right]\left[1+\frac{1}{2}\left(Z_{2}-1\right)\right]-1\simeq\sqrt{Z_{2 }}\,\frac{1}{Z_{1}}\,\sqrt{Z_{2}}-1,\] (68) so that the first factor in the square bracket can be identified with $1+{\rm A\|_{loop}}$, the second with $1+({\rm B_{1}+B_{2}})\|_{\rm loop}$ and the last with ${\rm 1+(C+D+E)\|_{loop}}$. The fact that our calculation gives $Z_{2}-1=$$-(1/Z_{1}-1)$ at order $\alpha_{s}$, making the right hand side in Eq. (68) equal to zero, can be interpreted as the condition $Z_{1}=Z_{2}$ valid in light cone gauge. At a first glance, it may seem strange to identify our calculation with an evaluation of renormalization constants when $p_{\perp}$ is not large, however one can check that the Slavnov-Taylor-Ward identities indeed require ${\rm 2A+B}=0$, as explicitly shown above. ## Appendix B Target versus projectile evolution beyond LO The NLO expressions for the quark multiplicity in Eqs. (3.3) or (3.3) involve dipole $S$-matrices like $\mathcal{S}\big{(}\bm{q},X(x)\big{)}$, which in the main text have been assumed to follow the target evolution with decreasing $X=q^{-}/P^{-}$, from $X=X_{0}$ down to $X(x)\ll 1$. The viewpoint of target evolution was indeed more convenient at a conceptual level, for developing a physical picture and also for formulating kinematical constraints like the energy conservation (8). This is however less convenient in practice, because the high-energy evolution of a dense nucleus is not known beyond LO (i.e. beyond rcBK). Yet, as we shall now explain, this unknown target evolution can be replaced for the present purposes with the evolution of the dilute projectile, which is indeed known to the accuracy of interest. More precisely, what is known is the NLO version of the BK equation in coordinate space [32; 33; 34] together with its ‘collinear improvement’ [44; 37; 45; 46]. When adapting these equations to the problem at hand, the only subtlety refers to the proper identification of the longitudinal phase-space for the evolution of the projectile — that is, the interval in $p^{+}$ which is spanned by the evolution gluons. This identification was straightforward at LO, where the ‘plus’ and ‘minus’ rapidities can be identified with each other, as discussed in Sect. 2.3, but it is less trivial beyond LO, where the exact kinematics becomes important (recall also the discussion towards the end of Sect. 3.4). When discussing projectile evolution in what follows, one should keep in mind that the relevant ‘projectile’ is the right-moving partonic pair made with the incoming quark and its primary gluon. The kinematics of that pair, in particular the 3-momentum $(p^{+}=xq_{0}^{+},\bm{p})$ of the primary gluon, must be considered as fixed for the purposes of the evolution — this matter only for the boundaries of the evolution phase-space. The evolution rather refers to the three dipoles $S$-matrices $\mathcal{S}(\bm{q})$, $\mathcal{S}(\bm{\ell})$, and $\mathcal{S}(\bm{k})$ which appear in Eqs. (18) and (19) — and hence implicitly in equations like (3.2) or (3.3) — and describe the scattering between the primary quark-gluon pair (the ‘projectile’) and the nuclear target (whose wavefunction is not evolving anymore, as we now work in the infinite momentum frame of the projectile). These dipoles, that we referred to as ‘daughter dipoles’ when discussing the primary emission in Sect. 3.1, will now act as _parent_ dipoles for the high-energy evolution. That is, the ‘evolution gluons’ will be soft gluons which belong to the wavefunctions of these three dipoles and whose longitudinal momenta $p^{+}_{i}=x_{i}q_{0}^{+}$ are necessarily smaller than the corresponding momentum $p^{+}=xq_{0}^{+}$ of the primary gluon. In what follows, we shall pick one of them, say $\mathcal{S}(\bm{q})$, and study its high-energy evolution beyond LO. Previously, we have (formally) expressed the result in terms of the evolution of the target, like $\mathcal{S}\big{(}\bm{k},X(x)\big{)}$. In what follows, we would like to compute the same quantity from the evolution of the dipole $\mathcal{S}(\bm{q})$ itself, which is known beyond NLO. Let us first recall, from the discussion in Sect. 2.3, that at LO one simply has $\mathcal{S}_{\rm T}\big{(}\bm{q},X(x)\big{)}=\mathcal{S}_{ \rm P}\big{(}\bm{q},x_{g}/x\big{)}$, where we have temporarily introduced the subscripts T (‘target’) and P (‘projectile’), to make the discussion more transparent. The rapidity argument $x_{g}/x$ of $\mathcal{S}_{\rm P}$ can be understood as follows: the evolution variable is the longitudinal fraction $z_{i}\equiv x_{i}/x$ of a generic evolution gluon w.r.t. the parent dipole. At LO, this is bounded by $x_{g}/x<z_{i}<1$, where the lower limit comes from the kinematical limit $X(x_{i})=x_{g}/x_{i}\leq 1$. In other terms, the function $\mathcal{S}_{\rm P}\big{(}\bm{q},x_{g}/x\big{)}$ encodes the probability to find a gluon with longitudinal fraction $x_{g}$ within the quark-gluon projectile (see also Fig. 1). In what follows, we shall argue that beyond LO, the above relation should be extended to (69) where $Q_{0}$ is the ‘infrared cutoff’ introduced by the initial condition at $x_{0}\sim 1$, e.g. the target saturation momentum at low energy (and the lowest transverse momentum scale in the problem). As before, $x_{\rm T}$ refers to the softest gluon in the wavefunction of the projectile which is involved in the collision, whereas $x_{\rm P}$ to the hardest gluon that can be emitted by the primary quark-gluon pair. The differences between $x_{\rm T}$ and $x_{g}$ and, respectively, between $x_{\rm P}$ and $x$, are consequences of _time-ordering_, as we shall shortly explain. It is understood that the function $\mathcal{S}_{P}$ obeys a suitable ‘beyond LO’ version of the BK equation, which includes ‘collinear improvement’ [44; 37; 45; 47; 48; 46] — that is, which includes an all-order resummation of the radiative corrections enhanced by double collinear logarithms. This equation, conveniently written as a differential equation in the _plus_ rapidity $Y\equiv\ln(x/x_{\rm T})$, must be integrated from $Y_{0}\simeq 0$ (where one can use the initial condition $\mathcal{S}_{0}$ from the MV model) up to $Y_{\rm P}\equiv\ln({x_{\rm P}}/{x_{\rm T}})$. Full NLO accuracy can be achieved by using the NLO version of the BK equation [32] amended by collinear improvement [37; 45] (the feasibility of such a calculation has been demonstrated in [48]). To justify Eq. (69), consider the evolution of the original quark-gluon pair via successive emissions of soft gluons (see Fig. 17). We focus on the ‘collinear regime’ at $k_{\perp}\gg Q_{0}$, where the collinear resummations are actually needed. Then the projectile evolution looks rather similar as at LO, in the sense that it is dominated by gluon emissions which are strongly ordered in both longitudinal ($x\gg x_{1}\gg x_{2}\dots\gg x_{\rm T}$) and transverse ($p_{\perp}\gg p_{\perp 1}\gg p_{\perp 2}\dots\gg Q_{0}$) momenta, but which beyond LO must be also ordered in _time_ : $\Delta x^{+}>\Delta x^{+}_{1}>\Delta x^{+}_{2}\dots>1/P^{-}$. In the above, $x=p^{+}/q^{+}_{0}$ and $p_{\perp}=\|\bm{p}\|$ refer to the kinematics of the primary gluon, as before, while $x_{i}=p^{+}_{i}/q^{+}_{0}$ and $p_{\perp i}=\|\bm{p}_{i}\|$, with $i=1,\,2,\,\dots n$, refer to the subsequent gluons in the cascade, which are softer. The ‘lifetime’¹⁵ of the $i$th generation, as given by the uncertainty principle or by the respective energy denominator, is roughly $\Delta x^{+}_{i}\simeq 2p^{+}_{i}/p_{\perp i}^{2}$. The ‘lifetime’ $\Delta x^{+}$ of the primary gluon can be similarly estimated as $\Delta x^{+}\simeq 2p^{+}/p_{\perp}^{2}$ or, more precisely (after using the complete energy denominator for the quark-gluon fluctuation), [FOOTNOTE:15][ENDFOOTNOTE] \[\frac{1}{\Delta x^{+}}\simeq\frac{k_{\perp}^{2}}{2(1-x)q^{+}_{0}}+\frac{p_{ \perp}^{2}}{2xq^{+}_{0}}\sim\frac{k_{\perp}^{2}}{2x(1-x)q^{+}_{0}}\,,\] (70) where in the final equality we have used $p_{\perp}\sim k_{\perp}$, in line with the arguments leading to Eq. (3.3). Notice that, since both longitudinal ($p^{+}_{i}$) and transverse ($p_{\perp i}$) momenta are simultaneously decreasing during the evolution, the condition of time-ordering, $\Delta x^{+}_{i}>\Delta x^{+}_{i+1}$, is not automatically satisfied: the LLA includes unphysical configurations for which this condition is in fact violated. This is corrected when going beyond LO, but the respective corrections are large (since they have to ‘subtract’ for double-logarithmic regions in phase-space) and hence must be resummed to all orders. This is achieved by the ‘collinear resummations’ aforementioned [44; 37; 45; 46], which essentially amount to enforcing time-ordering within the projectile evolution. <figure><img src="content_image/1608.05293/x34.png"><figcaption>Figure 17: A sequence of emissions contributing to the evolution of the quark-gluon projectile in the collinear regime. Both longitudinal and transversemomenta are strongly ordered down the cascade. The transferred momenta q⊥i arerelatively soft, q⊥i≪p⊥i, hence we also have q⊥≃p⊥1, q⊥1≃p⊥2, etc.</figcaption></figure> Specifically, for the first emission, the condition $\Delta x^{+}_{1}<\Delta x^{+}$ immediately implies $x_{1}\lesssim x_{\rm P}$ with $x_{\rm P}$ as defined in Eq. (69). (To obtain this, we have also used the fact that $p_{\perp 1}\simeq q_{\perp}\ll k_{\perp}$, cf. Fig. 17.) That is, the longitudinal momentum of the first gluon which counts for the evolution of $\mathcal{S}_{\rm P}(\bm{q})$ is not limited (from the above) by the respective momentum $p^{+}=xq_{0}^{+}$ of the primary quark, but by the generally smaller value $x_{\rm P}q_{0}^{+}$. Similarly, the condition that the lifetime $\Delta x^{+}_{n}$ of the _last_ (i.e. softest) emitted gluon be larger than the longitudinal extent $1/P^{-}$ of the target, implies \[x_{n}\,\gtrsim\,\frac{p_{\perp n}^{2}}{2q_{0}^{+}P^{-}}=(1-x)\frac{p_{\perp n} ^{2}}{\hat{s}}\,>\,(1-x)\frac{Q_{0}^{2}}{\hat{s}}=x_{\rm T}\,,\] (71) where we have also used $q_{0}^{+}/Q^{+}=x_{p}/(1-x)$ for the ‘real’ terms. These considerations confirm that the longitudinal phase-space for the high-energy evolution of the quark-gluon ‘projectile’ is given by $x_{\rm T}<x_{i}<x_{\rm P}$, in agreement with Eq. (69). Note that rapidity interval $Y_{\rm P}=\ln(x_{\rm P}/x_{\rm T})$ available for the evolution of the projectile is larger (when ${q_{\perp}}\gg{Q_{0}}$) than the corresponding interval $Y=\ln\big{(}{1}/{X(x)}\big{)}$ for the evolution of the target: \[Y_{\rm P}=\,\ln\frac{x_{\rm P}}{x_{\rm T}}\,=\,\ln\frac{x\hat{s}}{k_{\perp}^{2}}+\ln\frac{q_{\perp }^{2}}{Q_{0}^{2}}\,=Y+\ln\frac{q_{\perp}^{2}}{Q_{0}^{2}}\,.\] (72) This difference is compensated by time-ordering, which effectively reduces the phase-space for the evolution of the projectile. ## Appendix C NLO BK evolution and its collinear improvement The BK equation describes the rapidity evolution of the $S$-matrix $S_{\bm{x}\bm{y}}=1-T_{\bm{x}\bm{y}}$ for the scattering of a color dipole with transverse coordinates ($\bm{x},\bm{y}$) off a hadronic target. This equation is currently known to NLO accuracy [32]. Neglecting the terms suppressed in the multi-color limit $N_{\rm c}\gg 1$, one finds a closed equation for $S_{\bm{x}\bm{y}}$ which, in the spirit of Eq. (2.3), can be written in the integral form \[\delta S_{\bm{x}\bm{y}}(x_{\rm T})=\] \[+\frac{\bar{\alpha}_{s}^{2}}{8\pi^{2}}\int_{x_{\rm T}}^{1}\frac{{\rm d}x}{x}\int\frac{{\rm d}^{2}\bm{u}\,{\rm d }^{2}\bm{z}}{(\bm{u}\!-\!\bm{z})^{4}}\bigg{\{}-2+\frac{(\bm{x}\!-\!\bm{u})^{2} (\bm{y}\!-\!\bm{z})^{2}+(\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u})^{2}-4(\bm{x }\!-\!\bm{y})^{2}(\bm{u}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\! \bm{z})^{2}-(\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u})^{2}}\ln\frac{(\bm{x}\!- \!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u })^{2}}\] \[+\frac{(\bm{x}\!-\!\bm{y})^{2}(\bm{u}\!-\! \bm{z})^{2}}{(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}}\left[1+\frac{(\bm {x}\!-\!\bm{y})^{2}(\bm{u}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!- \!\bm{z})^{2}-(\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u})^{2}}\right]\ln\frac{( \bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{z})^{2}(\bm{y} \!-\!\bm{u})^{2}}\bigg{\}}\] \[\left[S_{\bm{x}\bm{u}}(x)S_{\bm{u}\bm{z}}( x)S_{\bm{z}\bm{y}}(x)-S_{\bm{x}\bm{u}}(x)S_{\bm{u}\bm{y}}(x)\right]\] \[+\frac{\bar{\alpha}_{s}^{2}}{8\pi^{2}}\,\frac{N_{\rm f}}{N_{\rm c }}\int_{x_{\rm T}}^{1}\frac{{\rm d}x}{x}\int\frac{{\rm d}^{2 }\bm{u}\,{\rm d}^{2}\bm{z}}{(\bm{u}\!-\!\bm{z})^{4}}\bigg{[}2-\frac{(\bm{x}\!- \!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}+(\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u} )^{2}-(\bm{x}\!-\!\bm{y})^{2}(\bm{u}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{u})^{2}( \bm{y}\!-\!\bm{z})^{2}-(\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u})^{2}}\ln\frac {(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}}{(\bm{x}\!-\!\bm{z})^{2}(\bm{y }\!-\!\bm{u})^{2}}\bigg{]}\] \[\left[S_{\bm{x}\bm{z}}(x)S_{\bm{u}\bm{y}}( x)-S_{\bm{x}\bm{u}}(x)S_{\bm{u}\bm{y}}(x)\right],\] (73) where $N_{\rm f}$ is the number of flavors, the coupling $\bar{\alpha}_{s}$ is evaluated at the renormalization scale $\mu$, while $\bar{\alpha}_{s}$ and $\bar{b}$ have been defined in the main text. Notice also that, for economy with respect to the notation used in Eq. (40) and afterwards, we have let $x\to x_{\rm T}/x$ and put the dependence on the transverse coordinates as subscripts. The $\delta$ in front $S_{\bm{x}\bm{y}}(x_{\rm T})$ stands for the change from the initial condition, while the quantity $\Delta S$ that we have used in the main text stands for the $\bar{\alpha}_{s}^{2}$ terms on the r.h.s. of the above equation, except for those which are proportional to $\bar{b}$. The term with a single integration (SI) over the transverse coordinate $\bm{z}$ keeps the same structure as the LO equation, but receives a correction of order $\mathcal{O}(\bar{\alpha}_{s}^{2})$ to the kernel. In particular, it contains the running coupling corrections proportional to $\bar{b}$. The terms of order $\mathcal{O}(\bar{\alpha}_{s}^{2})$ with a double integration (DI) over the coordinates $\bm{u}$ and $\bm{z}$ arise from partonic fluctuations involving two additional partons at the time of scattering. The first of such terms is independent of $N_{\rm f}$ and clearly represents the case where both daughter partons are gluons. The $S$-matrix structure $S_{\bm{x}\bm{u}}S_{\bm{u}\bm{z}}S_{\bm{z}\bm{y}}-S_{\bm{x}\bm{u}}S_{\bm{u}\bm{ y}}$ corresponds to a sequence of emissions in which the original dipole $(\bm{x},\bm{y})$ emits a gluon at $\bm{u}$ giving rise to the two dipoles $(\bm{x},\bm{u})$ and $(\bm{u},\bm{y})$, and then the dipole $(\bm{u},\bm{y})$ emits a gluon at $\bm{z}$ leading to the dipoles $(\bm{u},\bm{z})$ and $(\bm{z},\bm{y})$. The ‘real’ term $S_{\bm{x}\bm{u}}S_{\bm{u}\bm{z}}S_{\bm{z}\bm{y}}$ describes the situation where both gluons interact with the target, while the ‘virtual’ term $-S_{\bm{x}\bm{u}}S_{\bm{u}\bm{y}}$ stands for the the case where the gluon at $\bm{z}$ has been emitted and reabsorbed either before, or after, the scattering. This ‘virtual’ term ensures that the potential ‘ultraviolet’ singularity due to the $1/(\bm{u}-\bm{z})^{4}$ factor in the kernel is in fact harmless. The same discussion applies to the second DI term, proportional to $N_{\rm f}$, which represents the case that the additional partons at the time of scattering are a quark and an antiquark, and thus the number of dipoles involved in the scattering is just two. The issue with the NLO BK equation given in (C) is that there are terms in the kernels which can get large in certain kinematic regimes, thus invalidating the strict $\bar{\alpha}_{s}$-expansion. The first class of such terms contain the corrections proportional to $\bar{b}$ in the SI term in Eq. (C), and has already been discussed in Sect. 3.4. The scale $\mu$ should be chosen in such way, that the logarithms of transverse dipole sizes proportional to $\bar{b}$ (and only those) become innocuous in any kinematic regime. It is trivial to see that both choices suggested in Sect. 3.4 will cancel any potentially large logarithmic contribution. In fact, our _fac_ prescription, cf. Eq. (32), is by definition the one in which the sum of all terms proportional to $\bar{b}$ vanishes identically. Now we wish to discuss NLO corrections enhanced by logarithms associated with large separations in transverse sizes (or momenta) between successive emissions. These ‘collinear’ corrections become large only in the weak-scattering regime where all the dipoles are small compared to the target saturation scale $1/Q_{s}$, so that we can linearize w.r.t. to the scattering amplitude $T$. At this level, we can drop the term proportional to $N_{\rm f}/N_{\rm c}$ in Eq. (C) since it vanishes after linearization, as one can check by using the symmetry of the kernel under the interchange $\bm{u}\leftrightarrow\bm{z}$. More precisely, we consider the strongly ordered regime \[1/Q_{s}\gg\|\bm{z}-\bm{x}\|\simeq\|\bm{z}-\bm{y}\|\simeq\|\bm{z}-\bm{u}\|\gg\|\bm{u}- \bm{x}\|\simeq\|\bm{u}-\bm{y}\|\gg\|\bm{x}-\bm{y}\|,\] (74) which means that the parent dipole is the smallest one, a gluon is emitted far away at $\bm{u}$, a second one even further at $\bm{z}$, but all the possible dipole sizes remain smaller than the inverse saturation momentum. Denoting by $r$, $\bar{u}$ and $\bar{z}$ the size of the parent dipole, the size of the dipoles involving $\bm{u}$ and the size of the dipoles involving $\bm{z}$, respectively, we have $r^{2}\ll\bar{u}^{2}\ll\bar{z}^{2}$. Inspecting the SI piece in the NLO BK equation (C), it is obvious that the term which involves the double transverse logarithm (DTL) is the dominant one in the kinematic region defined in Eq. (74). In this regime we are also allowed to write $S_{\bm{x}\bm{z}}S_{\bm{z}\bm{y}}-S_{\bm{x}\bm{y}}\simeq-T_{\bm{x}\bm{z}}-T_{ \bm{z}\bm{y}}+T_{\bm{x}\bm{y}}\simeq-2T(\bar{z})$, with the second approximate equality arriving from the fact that the dipole amplitude for a small dipole is roughly proportional to the dipole size squared. Thus, the net result comes from the ‘real’ term, i.e. the one which involves the large daughter dipoles. A single transverse logarithm (STL) is hidden in the DI term. To see that, let us isolate the kernel \[\mathcal{M}_{{\rm STL}}\equiv\frac{\bar{\alpha} _{s}^{2}}{8\pi^{2}(\bm{u}\!-\!\bm{z})^{4}}\bigg{[}-2+ \,\frac{(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}+(\bm{x}\!- \!\bm{z})^{2}(\bm{y}\!-\!\bm{u})^{2}-4(\bm{x}\!-\!\bm{y})^{2}(\bm{u}\!-\!\bm{z })^{2}}{(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}-(\bm{x}\!-\!\bm{z})^{2} (\bm{y}\!-\!\bm{u})^{2}}\] \[\,\times\ln\frac{(\bm{x}\!-\!\bm{u})^{2}(\bm{y}\!-\!\bm{z})^{2}}{ (\bm{x}\!-\!\bm{z})^{2}(\bm{y}\!-\!\bm{u})^{2}}\bigg{]},\] (75) which in the collinear regime can be successively written as \[\mathcal{M}_{{\rm STL}}\simeq \,\frac{\bar{\alpha}_{s}^{2}}{8\pi^{2}\bar{z}^{4}}\bigg{[}-2+ \frac{2\bar{u}^{2}-2\bar{u}r\cos\phi-3r^{2}}{r^{2}-2\bar{u}r\cos\phi}\ln\left( 1+\frac{r^{2}-2\bar{u}r\cos\phi}{\bar{u}^{2}}\right)\bigg{]}\] \[\simeq \,-\frac{\bar{\alpha}_{s}^{2}}{\pi^{2}}\,\frac{6-\cos^{2}\phi}{12 }\,\frac{r^{2}}{\bar{u}^{2}\bar{z}^{4}}\to-\frac{11\bar{\alpha}_{s}^{2}}{24\pi ^{2}}\,\frac{r^{2}}{\bar{u}^{2}\bar{z}^{4}},\] (76) where $\phi$ is the angle between $\bm{r}$ and any of the two dipoles involving $\bm{u}$. To arrive at (C), we first set all dipole sizes which include $\bm{z}$ equal to each other, since any subleading term would be highly suppressed by inverse powers of $\bar{z}$. This simplifies significantly the expansion in the regime of interest, since the only $z$ dependence left is the one explicit in the prefactor. Then we have taken the limit $r\ll\bar{u}$ and finally we have performed an average over the angle $\phi$ between the parent dipole and those involving $\bm{u}$. As evident in Eq. (C), the would-be leading term of order $1/\bar{z}^{4}$ has cancelled and the first non-vanishing term is suppressed by the power factor $r^{2}/\bar{u}^{2}$. Now by linearizing the $S$-matrices multiplying $\mathcal{M}_{{\rm STL}}$ and realizing, as in the DTL case, that the ‘real’ term dominates, we can approximate $S_{\bm{x}\bm{u}}S_{\bm{u}\bm{z}}S_{\bm{z}\bm{y}}-S_{\bm{x}\bm{u}}S_{\bm{u}\bm{ y}}\simeq-T_{\bm{u}\bm{z}}-T_{\bm{z}\bm{y}}+T_{\bm{u}\bm{y}}\simeq-2T(\bar{z})$. Since this is independent of the intermediate dipole size $\bar{u}$, the integration over the latter, within the range limited by $r$ and $\bar{z}$, leads to the anticipated STL. Thus, putting together the LO term and the two NLO ones enhanced by the STL and the DTL, one finds that the NLO BK equation in the collinear regime (74) reduces to \[\delta T(r;x_{\rm T})=\bar{\alpha}_{s}\int_{x_{\rm T}}^{1}\frac{{\rm d}x}{x}\int_{r^{2}}^{1/Q_{s}^{2}}{\rm d} \bar{z}^{2}\,\frac{r^{2}}{\bar{z}^{4}}\left(1-\frac{1}{2}\,\bar{\alpha}_{s}\ln ^{2}\frac{\bar{z}^{2}}{r^{2}}-\frac{11}{12}\,\bar{\alpha}_{s}\ln\frac{\bar{z}^ {2}}{r^{2}}\right)T(\bar{z};x).\] (77) Clearly, for sufficiently large daughter dipoles, the NLO contributions are enhanced by large transverse logarithms and become comparable to, or larger than, the LO one. Then, the present perturbative expansion of the kernel, which is of fixed order in $\bar{\alpha}_{s}$, cannot be trusted anymore. Eventually this problematic behavior is transmitted to the solution of equations like (77) and (C) which becomes unstable, as indeed seen in [35; 36; 37]. Therefore, in order to have a meaningful evolution equation, we need to identify the physical origin of the large transverse logarithms and subsequently resum them to all orders in the coupling $\bar{\alpha}_{s}$. It should be obvious by now, that such higher order terms become more important than the pure $\bar{\alpha}_{s}^{2}$ NLO terms (that is, the $\mathcal{O}(\bar{\alpha}_{s}^{2})$ corrections not enhanced by any transverse logarithm), so that the latter will be discarded in what follows. This is the approach taken in [37; 45] leading to the collinearly improved BK evolution equation which admits a stable solution. Before giving this equation, we shall first write its collinear limit, i.e. we shall first resum the kernel appearing in Eq. (77). The origin of the DTLs is in the kinematics [44; 37]. Our main observation in [37] was that these large corrections arise from LCPT Feynman graphs in which the successive gluon emissions are not only strongly ordered in both longitudinal momenta and transverse momenta (or ‘dipole sizes’), but are also ordered in _lifetimes_ (or, equivalently, in light-cone energies [44]). The all-order resummation of the double collinear logarithms $[\bar{\alpha}_{s}\ln^{2}(\bar{z}^{2}/r^{2})]^{n}$, with $n=1,2,3,\dots$, eventually leads to a modification of the kernel by a multiplicative factor given in terms of the Bessel function ${\rm J}_{1}$ (see below). The STLs find their origin in the DGLAP dynamics [45]. Since each power of $\bar{\alpha}_{s}$ is accompanied by a collinear logarithm, it is intuitively clear that such terms must represent DGLAP corrections to the BFKL dynamics. This is further confirmed by the fact that the coefficient $-11/12$ in front of the single logarithm in Eq. (77) can be recognized as the second-order term in the small-$\omega$ expansion of the largest eigenvalue $\mathcal{P}(\omega)$ of the DGLAP anomalous dimension matrix in the large-$N_{\rm c}$ limit, more precisely \[\mathcal{P}(\omega)\simeq\int_{0}^{1}{\rm d}z\,z^{\omega}\left[P_{\rm GG}(z)+ \frac{C_{\rm F}}{N_{\rm c}}\,P_{\rm qG}(z)\right]=\frac{1}{\omega}-A_{1}+ \mathcal{O}\left(\omega\right)\quad\mbox{with}\quad A_{1}=\frac{11}{12}+\frac{ N_{\rm f}}{6N_{\rm c}^{3}},\] (78) where the second term of $A_{1}$ can be dropped when working in the large-$N_{\rm c}$ limit. Like the factor $1/\omega$ generates a small-$x$ gluon, the piece $-A_{1}$ corresponds to the most dominant sub-leading gluon emission (the term linear in $\omega$ would correspond to the next-to-most dominant one and so on). It is not hard to follow the combinatorics for an arbitrary number of such gluon emissions associated with $A_{1}$, and one finds that the resummation of these $[\bar{\alpha}_{s}\ln(\bar{z}^{2}/r^{2})]^{n}$ terms leads to the exponentiation of the NLO correction. Of course this is only a partial resummation of STLs, but it is well-defined and in the spirit of the $\omega$-expansion introduced and developed in [39; 40; 42]. Higher orders in $\omega$ would be related to next-to-most dominant gluons and beyond, and the corresponding STLs would start at higher orders in perturbation theory (NNLO or higher). Putting together the two types of resummation, one finds that the collinear kernel in Eq. (77) should be replaced by [37; 45] \[\frac{r^{2}}{\bar{z}^{4}}\,\,\frac{{\rm J}_{1}\Big{(}2\sqrt{\bar{\alpha}_{s} \ln^{2}(\bar{z}^{2}/r^{2})}\Big{)}}{\sqrt{\bar{\alpha}_{s}\ln^{2}(\bar{z}^{2}/ r^{2})}}\,\left(\frac{r^{2}}{\bar{z}^{2}}\right)^{\bar{\alpha}_{s}A_{1}}.\] (79) The last step in our construction consists of matching the above kernel with the LO BFKL/BK one in a consistent way and reinserting the virtual and non-linear terms in the respective evolution equation. Then one arrives at the local equation \[\delta S_{\bm{x}\bm{y}}(x_{\rm T})=\frac{\bar{\alpha}_{s}}{2 \pi}\int_{x_{\rm T}}^{1}\frac{{\rm d}x}{x}\int{\rm d}^{2}\bm {z}\,\frac{(\bm{x}-\bm{y})^{2}}{(\bm{x}-\bm{z})^{2}(\bm{y}-\bm{z})^{2}}\, \mathcal{K}_{\rm DLA}\mathcal{K}_{\rm SL}\left[S_{\bm{x}\bm{z}}(x)S_{\bm{z}\bm {y}}(x)-S_{\bm{x}\bm{y}}(x)\right],\] (80) where the precise form of the kernels $\mathcal{K}_{\rm DLA}$ and $\mathcal{K}_{\rm SL}$ has been given in Eqs. (41) and (42). ## References (1) G. A. Chirilli, B.-W. Xiao, and F. Yuan, “One-loop Factorization for Inclusive Hadron Production in $pA$ Collisions in the Saturation Formalism,” _Phys. Rev. Lett._108 (2012) 122301, arXiv:1112.1061 [hep-ph]. * (2) G. A. Chirilli, B.-W. Xiao, and F. Yuan, “Inclusive Hadron Productions in pA Collisions,” _Phys. Rev._D86 (2012) 054005, arXiv:1203.6139 [hep-ph]. * (3) Y. V. Kovchegov and A. H. Mueller, “Gluon production in current nucleus and nucleon - nucleus collisions in a quasiclassical approximation,” _Nucl.Phys._B529 (1998) 451–479, arXiv:hep-ph/9802440 [hep-ph]. * (4) Y. V. Kovchegov and K. Tuchin, “Inclusive gluon production in DIS at high parton density,” _Phys.Rev._D65 (2002) 074026, arXiv:hep-ph/0111362 [hep-ph]. * (5) A. Dumitru and J. Jalilian-Marian, “Forward quark jets from protons shattering the colored glass,” _Phys. Rev. Lett._89 (2002) 022301, arXiv:hep-ph/0204028 [hep-ph]. * (6) J. L. Albacete, N. Armesto, A. Kovner, C. A. Salgado, and U. A. Wiedemann, “Energy dependence of the Cronin effect from nonlinear QCD evolution,” _Phys.Rev.Lett._92 (2004) 082001, arXiv:hep-ph/0307179 [hep-ph]. * (7) D. Kharzeev, Y. V. Kovchegov, and K. Tuchin, “Cronin effect and high p(T) suppression in pA collisions,” _Phys.Rev._D68 (2003) 094013, arXiv:hep-ph/0307037 [hep-ph]. * (8) E. Iancu, K. Itakura, and D. Triantafyllopoulos, “Cronin effect and high p-perpendicular suppression in the nuclear gluon distribution at small x,” _Nucl.Phys._A742 (2004) 182–252, arXiv:hep-ph/0403103 [hep-ph]. * (9) J. P. Blaizot, F. Gelis, and R. Venugopalan, “High-energy pA collisions in the color glass condensate approach. 1. Gluon production and the Cronin effect,” _Nucl.Phys._A743 (2004) 13–56, arXiv:hep-ph/0402256 [hep-ph]. * (10) J. P. Blaizot, F. Gelis, and R. Venugopalan, “High energy p A collisions in the color glass condensate approach. II: Quark production,” _Nucl. Phys._A743 (2004) 57–91, arXiv:hep-ph/0402257. * (11) A. Dumitru, A. Hayashigaki, and J. Jalilian-Marian, “The Color glass condensate and hadron production in the forward region,” _Nucl. Phys._A765 (2006) 464–482, arXiv:hep-ph/0506308 [hep-ph]. * (12) J. L. Albacete and C. Marquet, “Single Inclusive Hadron Production at RHIC and the LHC from the Color Glass Condensate,” _Phys.Lett._B687 (2010) 174–179, arXiv:1001.1378 [hep-ph]. * (13) P. Tribedy and R. Venugopalan, “QCD saturation at the LHC: comparisons of models to p+p and A+A data and predictions for p+Pb collisions,” _Phys.Lett._B710 (2012) 125–133, arXiv:1112.2445 [hep-ph]. * (14) A. H. Rezaeian, “CGC predictions for p+A collisions at the LHC and signature of QCD saturation,” _Phys.Lett._B718 (2013) 1058–1069, arXiv:1210.2385 [hep-ph]. * (15) T. Lappi and H. Mäntysaari, “Single inclusive particle production at high energy from HERA data to proton-nucleus collisions,” _Phys.Rev._D88 (2013) 114020, arXiv:1309.6963 [hep-ph]. * (16) R. K. Ellis, W. J. Stirling, and B. R. Webber, _QCD and collider physics, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol._, vol. 8. Cambridge University Press, 1996. * (17) E. Iancu, A. Leonidov, and L. McLerran, “The Color glass condensate: An Introduction,” arXiv:hep-ph/0202270 [hep-ph]. * (18) E. Iancu and R. Venugopalan, “The color glass condensate and high energy scattering in QCD,” arXiv:hep-ph/0303204. * (19) F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, “The Color Glass Condensate,” _Ann.Rev.Nucl.Part.Sci._60 (2010) 463–489, arXiv:1002.0333 [hep-ph]. * (20) Y. V. Kovchegov and E. Levin, _Quantum chromodynamics at high energy_. Cambridge University Press, 2012. * (21) S. Catani, M. Ciafaloni, and F. Hautmann, “High-energy factorization and small x heavy flavor production,” _Nucl. Phys._B366 (1991) 135–188. * (22) S. Catani and F. Hautmann, “High-energy factorization and small x deep inelastic scattering beyond leading order,” _Nucl. Phys._B427 (1994) 475–524, arXiv:hep-ph/9405388 [hep-ph]. * (23) I. Balitsky, “Operator expansion for high-energy scattering,” _Nucl. Phys._B463 (1996) 99–160, arXiv:hep-ph/9509348. * (24) J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, “The BFKL equation from the Wilson renormalization group,” _Nucl. Phys._B504 (1997) 415–431, arXiv:hep-ph/9701284. * (25) J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, “The Wilson renormalization group for low x physics: Towards the high density regime,” _Phys.Rev._D59 (1998) 014014, arXiv:hep-ph/9706377 [hep-ph]. * (26) A. Kovner, J. G. Milhano, and H. Weigert, “Relating different approaches to nonlinear QCD evolution at finite gluon density,” _Phys. Rev._D62 (2000) 114005, arXiv:hep-ph/0004014. * (27) E. Iancu, A. Leonidov, and L. D. McLerran, “Nonlinear gluon evolution in the color glass condensate. I,” _Nucl. Phys._A692 (2001) 583–645, arXiv:hep-ph/0011241. * (28) E. Iancu, A. Leonidov, and L. D. McLerran, “The renormalization group equation for the color glass condensate,” _Phys. Lett._B510 (2001) 133–144, arXiv:hep-ph/0102009. * (29) E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, “Nonlinear gluon evolution in the color glass condensate. II,” _Nucl. Phys._A703 (2002) 489–538, arXiv:hep-ph/0109115. * (30) Y. V. Kovchegov, “Small-x F2 structure function of a nucleus including multiple pomeron exchanges,” _Phys. Rev._D60 (1999) 034008, arXiv:hep-ph/9901281. * (31) A. H. Mueller and S. Munier, “$p_{\perp}$-broadening and production processes versus dipole/quadrupole amplitudes at next-to-leading order,” _Nucl. Phys._A893 (2012) 43–86, arXiv:1206.1333 [hep-ph]. * (32) I. Balitsky and G. A. Chirilli, “Next-to-leading order evolution of color dipoles,” _Phys.Rev._D77 (2008) 014019, arXiv:0710.4330 [hep-ph]. * (33) I. Balitsky and G. A. Chirilli, “Rapidity evolution of Wilson lines at the next-to-leading order,” _Phys.Rev._D88 (2013) 111501, arXiv:1309.7644 [hep-ph]. * (34) A. Kovner, M. Lublinsky, and Y. Mulian, “Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner evolution at next to leading order,” _Phys.Rev._D89 (2014) no. 6, 061704, arXiv:1310.0378 [hep-ph]. * (35) E. Avsar, A. Stasto, D. Triantafyllopoulos, and D. Zaslavsky, “Next-to-leading and resummed BFKL evolution with saturation boundary,” _JHEP_1110 (2011) 138, arXiv:1107.1252 [hep-ph]. * (36) T. Lappi and H. Mäntysaari, “Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next to leading order,” _Phys.Rev._D91 (2015) no. 7, 074016, arXiv:1502.02400 [hep-ph]. * (37) E. Iancu, J. Madrigal, A. Mueller, G. Soyez, and D. Triantafyllopoulos, “Resumming double logarithms in the QCD evolution of color dipoles,” _Phys.Lett._B744 (2015) 293–302, arXiv:1502.05642 [hep-ph]. * (38) J. Kwiecinski, A. D. Martin, and A. Stasto, “A Unified BFKL and GLAP description of F2 data,” _Phys.Rev._D56 (1997) 3991–4006, arXiv:hep-ph/9703445 [hep-ph]. * (39) G. Salam, “A Resummation of large subleading corrections at small x,” _JHEP_9807 (1998) 019, arXiv:hep-ph/9806482 [hep-ph]. * (40) M. Ciafaloni, D. Colferai, and G. Salam, “Renormalization group improved small x equation,” _Phys.Rev._D60 (1999) 114036, arXiv:hep-ph/9905566 [hep-ph]. * (41) G. Altarelli, R. D. Ball, and S. Forte, “Resummation of singlet parton evolution at small x,” _Nucl.Phys._B575 (2000) 313–329, arXiv:hep-ph/9911273 [hep-ph]. * (42) M. Ciafaloni, D. Colferai, G. Salam, and A. Stasto, “Renormalization group improved small x Green’s function,” _Phys.Rev._D68 (2003) 114003, arXiv:hep-ph/0307188 [hep-ph]. * (43) A. Sabio Vera, “An ’All-poles’ approximation to collinear resummations in the Regge limit of perturbative QCD,” _Nucl.Phys._B722 (2005) 65–80, arXiv:hep-ph/0505128 [hep-ph]. * (44) G. Beuf, “Improving the kinematics for low-x QCD evolution equations in coordinate space,” _Phys.Rev._D89 (2014) 074039, arXiv:1401.0313 [hep-ph]. * (45) E. Iancu, J. D. Madrigal, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, “Collinearly-improved BK evolution meets the HERA data,” _Phys. Lett._B750 (2015) 643–652, arXiv:1507.03651 [hep-ph]. * (46) Y. Hatta and E. Iancu, “Collinearly improved JIMWLK evolution in Langevin form,” _JHEP_08 (2016) 083, arXiv:1606.03269 [hep-ph]. * (47) J. L. Albacete, “Resummation of double collinear logs in BK evolution versus HERA data,” arXiv:1507.07120 [hep-ph]. * (48) T. Lappi and H. Mäntysaari, “Next-to-leading order Balitsky-Kovchegov equation with resummation,” _Phys. Rev._D93 (2016) no. 9, 094004, arXiv:1601.06598 [hep-ph]. * (49) T. Altinoluk and A. Kovner, “Particle Production at High Energy and Large Transverse Momentum - ’The Hybrid Formalism’ Revisited,” _Phys. Rev._D83 (2011) 105004, arXiv:1102.5327 [hep-ph]. * (50) T. Altinoluk, N. Armesto, G. Beuf, A. Kovner, and M. Lublinsky, “Single-inclusive particle production in proton-nucleus collisions at next-to-leading order in the hybrid formalism,” _Phys. Rev._D91 (2015) no. 9, 094016, arXiv:1411.2869 [hep-ph]. * (51) A. M. Stasto, B.-W. Xiao, and D. Zaslavsky, “Towards the Test of Saturation Physics Beyond Leading Logarithm,” _Phys. Rev. Lett._112 (2014) no. 1, 012302, arXiv:1307.4057 [hep-ph]. * (52) D. Yu. Ivanov and A. Papa, “Inclusive production of a pair of hadrons separated by a large interval of rapidity in proton collisions,” _JHEP_07 (2012) 045, arXiv:1205.6068 [hep-ph]. * (53) A. M. Staśto, B.-W. Xiao, F. Yuan, and D. Zaslavsky, “Matching collinear and small $x$ factorization calculations for inclusive hadron production in $pA$ collisions,” _Phys. Rev._D90 (2014) no. 1, 014047, arXiv:1405.6311 [hep-ph]. * (54) Z.-B. Kang, I. Vitev, and H. Xing, “Next-to-leading order forward hadron production in the small-$x$ regime: rapidity factorization,” _Phys. Rev. Lett._113 (2014) 062002, arXiv:1403.5221 [hep-ph]. * (55) B.-W. Xiao and F. Yuan, “Comment on ”Next-to-leading order forward hadron production in the small-x regime: rapidity factorization” arXiv:1403.5221 by Kang et al,” arXiv:1407.6314 [hep-ph]. * (56) B. Ducloué, T. Lappi, and Y. Zhu, “Single inclusive forward hadron production at next-to-leading order,” _Phys. Rev._D93 (2016) no. 11, 114016, arXiv:1604.00225 [hep-ph]. * (57) K. Watanabe, B.-W. Xiao, F. Yuan, and D. Zaslavsky, “Implementing the exact kinematical constraint in the saturation formalism,” _Phys. Rev._D92 (2015) no. 3, 034026, arXiv:1505.05183 [hep-ph]. * (58) A. M. Stasto and D. Zaslavsky, “Saturation in inclusive production beyond leading logarithm accuracy,” arXiv:1608.02285 [hep-ph]. * (59) A. M. Stasto, K. J. Golec-Biernat, and J. Kwiecinski, “Geometric scaling for the total gamma* p cross-section in the low x region,” _Phys. Rev. Lett._86 (2001) 596–599, arXiv:hep-ph/0007192. * (60) E. Iancu, K. Itakura, and L. McLerran, “Geometric scaling above the saturation scale,” _Nucl. Phys._A708 (2002) 327–352, arXiv:hep-ph/0203137. * (61) A. Mueller and D. Triantafyllopoulos, “The Energy dependence of the saturation momentum,” _Nucl.Phys._B640 (2002) 331–350, arXiv:hep-ph/0205167 [hep-ph]. * (62) S. Munier and R. B. Peschanski, “Geometric scaling as traveling waves,” _Phys. Rev. Lett._91 (2003) 232001, arXiv:hep-ph/0309177. * (63) C. Marquet, “Forward inclusive dijet production and azimuthal correlations in pA collisions,” _Nucl. Phys._A796 (2007) 41–60, arXiv:0708.0231 [hep-ph]. * (64) J. L. Albacete and C. Marquet, “Azimuthal correlations of forward di-hadrons in d+Au collisions at RHIC in the Color Glass Condensate,” _Phys. Rev. Lett._105 (2010) 162301, arXiv:1005.4065 [hep-ph]. * (65) F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan, “Universality of Unintegrated Gluon Distributions at small x,” _Phys. Rev._D83 (2011) 105005, arXiv:1101.0715 [hep-ph]. * (66) E. Iancu and J. Laidet, “Gluon splitting in a shockwave,” _Nucl.Phys._A916 (2013) 48–78, arXiv:1305.5926 [hep-ph]. * (67) D. Zaslavsky, _Probing Hadron Structure in Proton-Nucleus Collisions_. PhD thesis, Penn State U., 2014. arXiv:1409.8259 [hep-ph]. https://inspirehep.net/record/1319344/files/arXiv:1409.8259.pdf. * (68) Y. V. Kovchegov and H. Weigert, “Quark loop contribution to BFKL evolution: Running coupling and leading-N(f) NLO intercept,” _Nucl.Phys._A789 (2007) 260–284, arXiv:hep-ph/0612071 [hep-ph]. * (69) Y. V. Kovchegov and H. Weigert, “Triumvirate of Running Couplings in Small-x Evolution,” _Nucl.Phys._A784 (2007) 188–226, arXiv:hep-ph/0609090 [hep-ph]. * (70) I. Balitsky, “Quark contribution to the small-x evolution of color dipole,” _Phys.Rev._D75 (2007) 014001, arXiv:hep-ph/0609105 [hep-ph]. * (71) L. D. McLerran and R. Venugopalan, “Computing quark and gluon distribution functions for very large nuclei,” _Phys. Rev._D49 (1994) 2233–2241, arXiv:hep-ph/9309289. * (72) L. D. McLerran and R. Venugopalan, “Gluon distribution functions for very large nuclei at small transverse momentum,” _Phys. Rev._D49 (1994) 3352–3355, arXiv:hep-ph/9311205. * (73) J. L. Albacete, N. Armesto, J. G. Milhano, P. Quiroga-Arias, and C. A. Salgado, “AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks,” _Eur.Phys.J._C71 (2011) 1705, arXiv:1012.4408 [hep-ph]. * (74) J. L. Albacete and Y. V. Kovchegov, “Solving high energy evolution equation including running coupling corrections,” _Phys.Rev._D75 (2007) 125021, arXiv:0704.0612 [hep-ph]. * (75) J. L. Albacete and C. Marquet, “Gluon saturation and initial conditions for relativistic heavy ion collisions,” _Prog.Part.Nucl.Phys._76 (2014) 1–42, arXiv:1401.4866 [hep-ph]. * (76) K. J. Golec-Biernat and M. Wusthoff, “Saturation effects in deep inelastic scattering at low $Q^{2}$ and its implications on diffraction,” _Phys. Rev._D59 (1998) 014017, arXiv:hep-ph/9807513. * (77) I. Balitsky and G. A. Chirilli, “Photon impact factor in the next-to-leading order,” _Phys. Rev._D83 (2011) 031502, arXiv:1009.4729 [hep-ph]. * (78) G. Beuf, “NLO corrections for the dipole factorization of DIS structure functions at low x,” _Phys. Rev._D85 (2012) 034039, arXiv:1112.4501 [hep-ph]. * (79) I. Balitsky and G. A. Chirilli, “Photon impact factor and $k_{T}$-factorization for DIS in the next-to-leading order,” _Phys. Rev._D87 (2013) no. 1, 014013, arXiv:1207.3844 [hep-ph]. * (80) G. Beuf, “Dipole factorization for DIS at NLO I: Loop correction to the photon to quark-antiquark light-front wave-functions,” arXiv:1606.00777 [hep-ph].
1707.04938	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 28826, "num_imgs": 7, "llama3_tokens_count": 8114 }	[ "content_image/1707.04938/x1.png", "content_image/1707.04938/x2.png", "content_image/1707.04938/x3.png", "content_image/1707.04938/x4.png", "content_image/1707.04938/x5.png", "content_image/1707.04938/x6.png", "content_image/1707.04938/x7.png" ]	###### Abstract We implement a spatially fixed mesh refinement under spherical symmetry for the characteristic formulation of General Relativity. The Courant-Friedrich-Levy (CFL) condition lets us deploy an adaptive resolution in (retarded-like) time, even for the nonlinear regime. As test cases, we replicate the main features of the gravitational critical behavior and the spacetime structure at null infinity using the Bondi mass and the News function. Additionally, we obtain the global energy conservation for an extreme situation, i.e. in the threshold of the black hole formation. In principle, the calibrated code can be used in conjunction with an ADM 3+1 code to confirm the critical behavior recently reported in the gravitational collapse of a massless scalar field in an asymptotic anti-de Sitter spacetime. For the scenarios studied, the fixed mesh refinement offers improved runtime and results comparable to code without mesh refinement. Key words: Numerical Relativity; Characteristic Formulation; Fixed Mesh Refinement; Critical Behaviour; Asymptotically AdS Spacetimes; AdS/CFT correspondence; Holographic Principle. FIXED MESH REFINEMENT IN THE CHARACTERISTIC FORMULATION OF GENERAL RELATIVITY W. Barreto¹${}^{,2}$, H. P. de Oliveira², B. Rodriguez-Mueller³ [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] ## 1 Introduction The gravitational critical behavior as originally discovered by Choptuik [1] is well understood and seems to be an ubiquitous phenomenon. It emerges in many contexts, including when the gravitational collapse of a massless scalar field takes place in asymptotically anti de Sitter (AdS) spacetimes. But there are new features recently reported by Santos-Oliván and Sopuerta [2], [3]: A series of critical points arises, branching with and without mass gap. Computationally, the calculation of the gravitational critical collapse is challenging, especially so for multiple ’cascading’ critical points. There are two ways to attack these issues: i) Using Adaptive Mesh Refinement (AMR) [4], [5], [6], [7]; ii) Following the null geodesics [8], [9]. Here we report an implementation and testing of the Fixed Mesh Refinement (FMR) approach in the characteristic formulation of Numerical General Relativity. The final purpose is to use the developed code in combination with other code, which employs Domain Decomposition and the Galerkin-Collocation method in the ADM 3+1 formulation [10]. With this Characteristic-Cauchy merging we expect to make a future independent confirmation of the multiple critical points and as well as uncover fine-grained structure in asymptotically AdS spacetimes. A massless scalar field, in the strong field limit near the formation of a black hole, under spherical symmetry and minimally coupled to gravity, displays: (i) a critical behavior of type II with a very small black hole mass; (ii) an unstable naked singularity by fine-tuning generic initial data; (iii) a power law mass scaling and shows discrete self-similarity. Critical behavior of type I is found when a massive scalar field (a Compton wavelength) is considered [11], [12]. For a review on the critical phenomena for gravitational collapse, including quantum extensions, see Ref. [34]. Recently we have been involved in calculations of the gravitational critical behavior with mass gap, evolving a scalar field kink [14]. It is interesting enough to explore and confirm the new features [3] of the gravitational collapse of a massless scalar field in asymptotically AdS spacetimes. Near the multiple critical points, the mass spectra show a power law with and without mass gap. What is the meaning of this in the AdS/CFT dictionary? The AdS spacetime is the (maximally symmetric) solution for the vacuum Einstein equations with a negative cosmological constant. The boundary of AdS spacetime plays a fundamental role in the Holographic Principle implementation. The Holographic Principle establishes the equivalence between two Universes with different dimensions obeying different physical laws [15]. Maldacena [16] reported one mathematical realization of this principle: one 5-dimensional (5-D) spacetime corresponding to a hologram at the boundary of a 4-dimensional (4-D) spacetime. A black hole in a 5-D spacetime is equivalent to thermal radiation in the 4-D Hologram; they have the same entropy, but the physical origin is different for each case [17]-[18]. No experiment can establish the difference between these two descriptions of the Universe. Thus, understanding gravitational collapse for asymptotically AdS spacetimes is important. The simplest model for this scenario is a massless scalar field minimally coupled to gravitation, the Einstein-Klein-Gordon (EKG) system. The EKG system has been useful in gravitational collapse and black hole dynamics to develop a better understanding of: discovery of the gravitational collapse critical behavior [1], modeling and simulation of a black hole binary system [19]-[21], simulations to analyze the gravitational radiation detected by LIGO [22]. The instability problem for EKG collapse in an asymptotically AdS spacetime remains open. The EKG system plays a fundamental role as a toy model to discover new phenomena, translating them to observational Physics across the Holographic Principle [23]. Mesh refinement can be static or dynamic, that is, fixed or adaptive. When an (uniform) unigrid setting is not enough to resolve the fine structure, each problem determines the type of multigrid. For instance, if a discontinuity -typically formed with a shock wave- is moving, then the AMR is the right answer; if confined to some region, then FMR should be enough. In any case, if we know in advance how to handle regions with different resolutions, then FMR paves the way for AMR [24]. Here we adopt the practical point of view that a FMR code is sufficient in its own right for this problem because we know the location of the problem spot in space. We report a characteristic code with FMR intended to be combined with both Domain Decomposition and Galerkin-Collocation. The goal of this work is to create a FMR code which preludes and enables accurate and efficient study of critical behavior for the EKG system in an asymptotic AdS spacetime [2], [3]. As far as we know, the FMR method has not been implemented in the characteristic formulation of Numerical General Relativity. Basically it consists in a recursively static domain decomposition, which requires a previous knowledge of the problem. In our case we implement the FMR only in one coordinate (radial and null) which, given the structure $R\times S^{2}$, can be easily extended to higher dimensionality. We organize the work as follows. In section 2, we write the field equations (including the cosmological constant) for the Bondi-Sachs coordinates under spherical symmetry. We briefly explain how the deal with the origin and with infinity, mentioning the numerical methods employed to solve the equations. In section 3 we revisit the well established critical behavior (without cosmological constant) as discovered by Choptuik. In section 4 we show the results and tests of the implemented FMR in the characteristic formulation. Finally, we discuss our results and conclude in section 5. ## 2 Setup ### The field equations We use the Bondi-Sachs metric [25], [26] under spherical symmetry [27] \[ds^{2}=e^{2\beta}du[(V/r)du+2dr]-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),\] (1) where $\beta$ and $V$ are functions of $u$ and $r$. Here $u$ is a timelike coordinate; in a flat spacetime $u$ is just the retarded time. Therefore, surfaces $u=$constant represent null cones open to the future; $r$ is a null coordinate ($g_{rr}=0$) such that surfaces $r=$constant are spheres; $\theta$ and $\phi$ are the usual angular coordinates. Thus, we write the Einstein-Klein-Gordon system, including the cosmological term, as: \[\beta_{,r}=2\pi r\Phi_{,r}^{2},\] (2) \[V_{,r}=e^{2\beta}(1-3r^{2}/\ell^{2}),\] (3) \[2(r\Phi)_{,ur}=r^{-1}(rV\Phi_{,r})_{,r},\] (4) where the comma represents a partial derivative with respect to that coordinate, $\Phi=\Phi(u,r)$ is a massless scalar field and $\ell$ is the AdS length scale, which is related to the cosmological constant $\Lambda$ by $\ell^{2}=-3/\Lambda$. This is an initial-boundary problem. Specifying the initial null data $\Phi(u_{0},r)$ at the initial time $u_{0}$, and using the gauge freedom $\Phi\rightarrow\Phi\,\,+$ constant, we set $\Phi(u_{0},\infty)=0$ to solve the problem. We assume that $\Phi(u_{0},r)$ is not singular at $r=0$. We make $\Lambda=0$ for the purposes of this paper, then the spacetime described by metric (1) is asymptotically flat. Note that for a non-zero cosmological constant we need to transfer the evolved initial data from an interior (characteristic) to an exterior (Cauchy) asymptotically AdS. Otherwise, we have to use the affine metric of Chesler and Yaffe [28], which let us reach asymptotically the AdS spacetime using characteristics. The resulting metric does not take an asymptotic Minkowski form in the limit $r\rightarrow\infty$ of future-null-infinity (${\mathcal{J}}^{+}$). Because $\beta(u,\infty)=H(u)$, the Bondi time $u_{B}$ for a Minkowski frame at ${\mathcal{J}}^{+}$ relates to the proper time $u$ along the central geodesic by means of \[\frac{du_{B}}{du}=e^{2H}.\] (5) The coordinates $(u_{B},r,\theta,\phi)$ constitute a standard Bondi frame whose line element is given by (1) with the replacements $V\to V_{B}=e^{-2H}V$ and $\beta\rightarrow\beta_{B}=\beta-H$. Bondi time is more convenient to explore asymptotic quantities such as the mass and news function. The central time is more convenient to deal with horizons. A horizon forms at a finite central time $u^{\mathcal{H}}$ but at an infinite Bondi time $u_{B}^{\mathcal{H}}$, with a central redshift given by (5). ### Near infinity At ${\mathcal{J}}^{+}$: $g(u_{0},\infty)=Q(u_{0})$ and $\partial_{r}^{n}g(u_{0},\infty)=0$, for $n\neq 0$, where $g=r\Phi$ and $Q(u)$ is the scalar monopole moment. Assuming the scalar field has an asymptotic expansion \[\Phi(u,r)=\frac{Q(u)}{r}+\frac{c_{NP}}{r^{2}}+O(r^{-3}),\] (6) the hypersurface equations (2) y (3) lead to \[\beta(u,r)=H(u)-\frac{\pi Q^{2}(u)}{r^{2}}+O(r^{-3}),\] (7) \[V(u,r)=e^{2H}\left(r-2M(u)+\frac{\pi Q^{2}(u)}{r}\right)+O(r^{-3}),\] (8) where $H(u)$ and $M(u)$ are integration functiones with physical interpretation, as we shall see. The expansion of the wave equation (4) implies ${c_{NP}}_{,u}=0$, where $c_{NP}$ is the Newman-Penrose constant for the scalar field. On physical grounds the Bondi mass can be defined as \[M(u)=\frac{1}{2}e^{-2H}r^{2}(V/r)_{,r}\|_{r=\infty},\] (9) for which exists a mass loss equation given by \[e^{-2H}\frac{dM}{du}=-4\pi N^{2},\] (10) where \[N(u)=e^{-2H}\frac{dQ(u)}{du},\] (11) is the News function. It can be shown that the Bondi mass and the scalar News can be written as [29], [30]: \[M=2\pi\int^{\infty}_{0}rVe^{2\beta}\Phi^{2}_{,r}dr,\] (12) and \[N=\frac{1}{2}e^{-2H}\int^{\infty}_{0}\frac{V}{r}\Phi_{,r}dr.\] (13) ### Near the center Near $r=0$ we adopt the conditions \[\beta(u,r)=O(r^{2})\,\,\mbox{;}\,\,V(u,r)=r+O(r^{3}),\] (14) such that the metric reduces to the Minkowski form (polar null) along the central world line: \[ds^{2}=du^{2}+2dudr-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).\] (15) In the general dynamical case the scalar field is free at the center. For that reason we make the expansion around $r=0$ \[\Phi(u,r)=\Phi_{0}(u)+r\Phi_{1}(u)+r^{2}\Phi_{2}(u).\] (16) In this case we get from (2)-(4) \[V=r-\frac{2\pi}{3}\Phi_{1}^{2}r^{3}+\frac{4\pi}{3}\Phi_{1}\Phi_{2}r^{4}+ \mathcal{O}(r^{5}),\] (17) \[\beta=\pi\Phi_{1}^{2}r^{2}+\frac{8\pi}{3}\Phi_{1}\Phi_{2}r^{3}+\mathcal{O}(r^{ 4}),\] (18) \[\dot{\Phi}_{0} = \Phi_{1},\] (19) \[\dot{\Phi}_{1} = \frac{3}{2}\Phi_{2},\] (20) \[\dot{\Phi}_{2} = \frac{4\pi}{9}\Phi_{1}^{3}-2\Phi_{0}^{2}\Phi_{1}.\] (21) where overdot indicates a derivative with respect to $u$. The scalar curvature \[R=8\pi e^{-2\beta}\Phi_{,r}\left(2\Phi_{,u}-\frac{V}{r}\Phi_{,r}\right).\] (22) at the center is: \[R(u,r=0)=8\pi\Phi_{1}^{2}.\] (23) ### Numerical methods To solve the field equations we use a null cone evolution algorithm for nonlinear scalar waves developed in Refs. [31], [29] (the 1D Pitt code) adapted to the present setting as reported in [30]. The characteristic formulation in Numerical General Relativity is well documented by Winicour in Ref. [32]. The algorithm is based upon the compactified radial coordinate $x=r/(R+r)$, so that ${\mathcal{J}}^{+}$ is represented by a finite grid boundary, with $x=0$ at the center and $x=1$ at ${\mathcal{J}}^{+}$. The code has been tested to be globally second order accurate. This code has been used to get global energy conservation near the critical behavior [30]. Additionally, we implement a four-level FMR, as explained in section 4, and a fourth order Runge-Kutta to solve the system of equations given by (19)-(21). ## 3 Choptuik’s solution Let $\mathcal{S}$ denote a solution of (2)-(4). Choptuik [1] focused his attention on the family of one-parameter solutions $\mathcal{S}[p]$. Each solution is generated by evolving an initially incoming massless scalar field. Each family has the property that the parameter $p$ characterizes the strength of the self-interacting scalar field. There is a parameter value $p_{weak}$ such that in the limit $p\to p_{weak}$ the spacetime is flat. At the other extreme, there is a parameter value $p_{strong}$ such that as $p\to p_{strong}$ the end state of the evolution is a black hole. Between these two extremes, a critical value $p^{}$ exists where black hole does not form nor the solution disperses. Assuming that $p_{weak}<p^{}<p_{strong}$, solutions $\mathcal{S}[p^{}<p_{weak}]$ and $\mathcal{S}[p^{}>p_{strong}]$ are subcritical and supercritical, respectively. Choptuik discovered (using AMR) that the spacetime is discretely self-similar when the initial data is fine-tuned to the critical parameter $p^{}$. Spacetime is discretely self-similar (DSS) if it admits a discrete diffeomorphism $D_{\Delta}$ which leaves the metric $g$ invariant by a scale factor \[(D^{}_{\Delta})^{n}g=e^{2n\Delta}g,\] (24) where $\Delta$ is a dimensionless real constant and $n\in{\mathcal{N}}$. Let us define \[\tau=-\ln\big{\{}\frac{u^{}-u}{u^{}}\big{\}}\] (25) and \[\rho=\frac{r}{u^{}-u}=\frac{r}{u^{}}e^{\tau},\] (26) where $u^{}$ is a real number, which we call the accumulation time in the discrete self-similar spacetime. The critical function $\zeta^{}$ (scalar field or metric) satisfies the scaling relation \[\zeta^{}(\tau,\rho)=\zeta^{}(\tau-n\Delta,\rho-n\Delta),\] (27) and does not depend on the family of initial data, which means it is universal, that is, with a numerical value of $\Delta\approx 3.4$ for any initial data. But it is important to say that it occurs in a neighborhood $r=0$, at least as Choptuik reported. What does this behavior mean? If we freeze the critical evolution at some time, examine the profile $\zeta$ in some delimited region, and continue evolving for a $\delta u$ and reexamine the solution on a scale $e^{\Delta}$ times smaller than previously, we will see the same field (metric) profiles. If we then wait for an additional time interval $\delta u/e^{\Delta}$ and “zoom in” by another factor of $e^{\Delta}$, we will see again the same profiles. Thus, a precisely critical configuration will be characterized by an infinite series of “echoes” in the field patterns (as well as another form-invariant quantity) which arise from dynamics unfolding on increasingly smaller spatiotemporal scales. For each family of initial conditions, we can find $p^{}$ (to the limit of machine precision) using a binary search predicated on whether or not a black hole forms. Another universality feature is the agreement of the profiles at late times, regardless of the initial pulse shape. To generate universality and echoing we use subcritical initial data. The critical regime may also be studied using supercritical evolutions characterized by the formation of black holes. The black hole mass obeys the power law \[M_{BH}\simeq c_{f}\|p-p^{}\|^{\gamma}\] (28) where $c_{f}$ are a family dependent constants, but $\gamma$ is a universal scaling exponent which has a numerical value of $\gamma\approx 0.37$. $\gamma$ is the same for a family of initial data. These results suggest that the black hole mass turns out to be infinitesimal in this model problem. Any detail that might appear in the specification of the initial data is washed out by the interaction between the scalar and gravitational fields. The numerical values for $\Delta$ and $\gamma$ depend on the matter fields and the geometry, but under the present context, i.e., the EKG system without cosmological constant, the values of $\Delta$ and $\gamma$ obtained by numerical experimentation are the same for any initial data. $\Delta$ is a measure of the discrete self-similarity (echoing) and $\gamma$ is related with the largest Lyapunov exponent [33]. They characterize subcritical and supercritical behavior, respectively. A combination of them $\Delta/2\gamma$ explain the oscillations for the fine structure in the power law [9]. The gravitational critical behavior has been found in many systems and is well documented in Ref. [34] (and references therein). In the present context we use these well established results to test our extended code with FMR. ## 4 Characteristic FMR The implementation of the FMR in the characteristic 1D Pitt code is very simple, and can be resumed in two steps: * Fixed radial refinement: up to the selected level, in the radial compact coordinate $x$; * Adaptive Time Step: The spatial refinement determines the minimal CFL time step, which is adapted in the evolution up to the critical behavior. <figure><img src="content_image/1707.04938/x1.png"><figcaption>Figure 1: One strategy for the FMR.</figcaption></figure> We choose each refinement just on the middle of each (sub)grid to the left, with the same number of original points. For instance, as indicated in Figure 1, if $N_{x}^{(n)}$ represents the number of grid points at the refinement Level $n$, we have $N_{x}^{(0)}=5$, $N_{x}^{(1)}=7$, $N_{x}^{(2)}=9$ and $N_{x}^{(3)}=11$; and grid sizes $\Delta x^{(0)}=1/(N_{x}^{(0)}-1)$, $\Delta x^{(n+1)}=\Delta x^{(n)}/2$, with $n=1,\,2,\,3$. Therefore the grid sizes are related in a proportion $8:4:2:1$. We can choose any other proportion or more levels to go with refinement. The selection of the refinement first point can be any interior point of the original (Level 0) grid and the refinement itself can be realized to the left ($x\to 0$) or to the right ($x\to 1$). Now, knowing Choptuik’s solution we replicate the echoing and power law. This can be considered as demanding tests of our FMR implementation. Using as initial condition \[\Phi(0,r)=\lambda r^{2}e^{-(r-r_{0})^{2}/\sigma^{2}},\] (29) <figure><img src="content_image/1707.04938/x2.png"><figcaption>Figure 2: Scalar field at r=0 as a function of τ. It is apparent theperiodicity ( one and a half cycle) in time with Δ≈3.4. This calculation took42 minutes on a 2.4 GHz Intel Core i5.</figcaption></figure> <figure><img src="content_image/1707.04938/x3.png"><figcaption>Figure 3: Scalar curvature at r=0 as a function of τ. It is apparent theperiodicity Δ/2 in time (in three cycles).</figcaption></figure> with $r_{0}=0.7$, $\sigma=0.3$ and $\lambda=0.144930560446$, we show in Figure 2 the scalar field at $r=0$ as a function of $\tau$, using a grid refined from $N_{x}^{(0)}=10,001$ using the strategy shown in Fig. 1. That is, we increase in $10,001$ the number of points to the left of the midpoint in the previous Level to get $N_{x}^{(1)}=15,001$. In this specific case from Level $0$ to Level $3$ we get $N_{x}^{(3)}=25,001$. Our estimate accumulation time under the aforementioned conditions is $u^{}=2.119620369$. Figure 3 displays the scalar curvature at $r=0$ as a function of $\tau$[6] and the same conditions of Fig. 2. The accumulation time is determined when the scalar curvature begins to decay (for the subcritical evolution closest to $\lambda^{}$). <figure><img src="content_image/1707.04938/x4.png"><figcaption>Figure 4: Mass spectrum for the supercritical case λ>λ∗; the slope is γ=0.366.</figcaption></figure> <figure><img src="content_image/1707.04938/x5.png"><figcaption>Figure 5: Natural log of the Bondi mass as a function of τB; the apparentperiodicity is Δ/2.</figcaption></figure> <figure><img src="content_image/1707.04938/x6.png"><figcaption>Figure 6: News as a function of τB; the apparent periodicity is Δ.</figcaption></figure> <figure><img src="content_image/1707.04938/x7.png"><figcaption>Figure 7: Global energy conservation near critical behavior.</figcaption></figure> Only for the sake of completeness we also replicate the power law for supercritical evolutions as shown in Fig. 4. We have used the same initial condition as in Fig. 1, with $N_{x}=10,001$ and $\lambda>\lambda^{}$. We choose $a$ and $b$ to normalize the abscissa and show the smallest black hole, following Choptuik [1]. The scaling exponent was fitted using a mean squared quadrature, resulting in $\gamma=0.366\,(1\%)$. For this calculation we do not require the FMR code; each point takes 3 minutes. For each evolution, we picked up the minimum Bondi mass just before the black hole formation. The mass spectrum corresponds to the mass scaling given by Eq. (28). The oscillations in the power law, called by authors fine structure, are obtained once the linear behavior is extracted; this can be accomplished without FMR. Thus in our context, the fine structure oscillations for the super critical case are not calculated or related to the FMR method since the FMR method is used only in the subcritical case. Another feature (our main test for the FMR code) is the asymptotic structure of the spacetime at $\mathcal{J}^{+}$, particularly the Bondi mass and the news function. Figures 5 and 6 show them as a function of $\tau_{B}$ (the conditions are the same of Fig. 2), given by (25) but using the Bondi time and the accumulation Bondi time. As an extra test, Fig. 7 shows the global energy conservation with conditions as in Fig. 2. The periodicity in the Bondi mass and the News function is as in Pürrer et al. (Ref. [9]). As expected by Eq. (12) the period of the Bondi mass has to be a half of the scalar field, and by Eq. (13) the period of the News function has to be the same as the scalar field. We pick up the $r=0$ echoing for the scalar field at null infinity for the Bondi mass and News function. Remarkably the energy conservation test, a new result, is well behaved even in the threshold of the black hole formation. ## 5 Discussion and conclusions We implemented the FMR method in the radial (compactified) coordinate for the characteristic formulation of General Relativity. Our final goal is to use the developed code in conjunction with other code which uses the ADM 3+1 formulation of General Relativity, in order to reach asymptotically an AdS spacetime. It is not possible in the present context to get an AdS boundary for a non-zero cosmological constant because the Bondi-Sachs coordinates are constructed for an asymptotically flat spacetime. Thus, in this work, we set the cosmological constant to zero. As an important test we replicate the main features of the critical behavior in the collapse of a massless scalar field under spherical symmetry. We also replicate asymptotic quantities as the Bondi mass and News function. We obtain the energy conservation even in the extreme situation near the black hole formation, as an additional test and new result itself. In Table 1 we display some indicators of the performance with and without FMR. Calculation of echoing without FMR, using $N_{x}=25,001$, shed a half of the cycle for $\lambda=0.14493045.$ This last calculation took 19 minutes on a 2.4 GHz Intel Core i5. Using $N_{x}=25,001$ distributed with our implementation of FMR takes 42 minutes. Thus, a better resolution of echoing is clear with FMR. Without FMR the resolution is not improved increasing $N_{x}$. FMR \| λ \| Nx \| Exec. time (min.) \| cicle (Δ) ---\|---\|---\|---\|--- Yes \| 0.144930688869 \| 25,001 \| 42 \| 3/2 No \| 0.14493045 \| 25,001 \| 19 \| 1 No \| 0.14493045 \| 32,001 \| 24 \| 1 Table 1: Performance with and without FMR. Pretorius and Lehner [7] did an implementation of the AMR method in double null coordinates and they did a test for the particular case of the EKG system, far away from the critical behavior. We implemented the FMR in outgoing Bondi’s coordinates and report as main test the subcritical critical behavior (echoing). Also, our results are in complete agreement with Pürrer et al. [9]. The comparison was focused mainly on the asymptotic behavior at null infinity. We use a different numerical method (the FMR) to get the echoing in the gravitational critical behavior at $\mathcal{J}^{+}$. We conclude by stressing the following: At least in the characteristic formulation, the implementation of the FMR method in 1-D (radial coordinate) can be extended to non-spherical problems, because of the spatial foliation structure $\mathcal{R}\times S^{2}$. In the 3-D case, we can use the FMR orthogonal to the inflated cube technique to run efficiently in parallel [35] for high angular definition. ## Acknowledgments The authors thank the financial support of Brazilian agencies CNPq and FAPERJ; also would like to thank Jennifer Rodriguez-Mueller for her valuable input to the paper. We thank the Referees because the presentation of our work indeed improves with their comments. ## References [1] Choptuik, M. W.: Phys. Rev. Lett. 70, 9 (1993) * [2] Santos-Oliván, D., Sopuerta, C.: Phys. Rev. Lett. 116, 041101 (2016) * [3] Santos-Oliván, D., Sopuerta, C.: Phys. Rev. D 93, 104002 (2016) * [4] Berger, M. J., Oliger, J.: J. Comput. Phys. 53, 484 (1984) * [5] Choptuik, M. W.: Frontiers in Numerical Relativity, Edited by C. R. Evans, L. S. Finn and D. W. Hobill (Cambridge University Press, 1989) * [6] Hamade, R. S., Stewart, J. M.: Class. Quantum Grav. 13, 497 (1996) * [7] Pretorius, F., Lehner, L.: J. Comput. Phys. 198, 10 (2004) * [8] Garfinkle, D.: Phys. Rev. D 51, 5558 (1995) * [9] Pürrer, M., Husa, S., Aichelburg, P. C.: Phys. Rev. D 71, 104005 (2005) * [10] de Oliveira, H. P., Pando-Zayas, L., Rodrigues, E.: Phys. Rev. Lett. 111, 051101 (2013) * [11] Brady, P.R., Chambers, C. M., Goncalves, and S. M. C. V.: Phys. Rev. D 56, R6057 (1997) * [12] Seidel, E. Suen, W.-M.: Phys. Rev. Lett. 66 1659 (1991) * [13] Gundlach, C., Mart n-Garc a, J. M.: Living Rev. Relativity 10, 5 (2007) * [14] Barreto, W., Crespo, J. A., De Oliveira, H. P., Rodrigues, E. L., Rodriguez-Mueller, B.: Phys. Rev. D 93, 064042 (2016) * [15] Bekenstein, J.: Scientific American, 17, 67 (2007) * [16] Maldacena, J.: Adv. Theor. Math. Phys. 2, 231 (1998) * [17] ’t Hooft, G.: arXiv:9310026 [qr-qc]. * [18] Susskind, L.: J. Math. Phys. 36, 6377 (1995) * [19] Pretorius, F.: Phys. Rev. Lett. 95, 121101 (2005) * [20] Campanelli, M., Lousto, C., Marronetti, P., Zlochower, J.: Phys. Rev. Lett. 96, 111101 (2006) * [21] Baker, J., Centrella, J., Choi, D., Koppitz, M., van Metter, J.: Phys. Rev. Lett. 96, 111102 (2006) * [22] Abbott, B. et al.: Phys. Rev. Lett. 116, 061102 (2016) * [23] Balasubramanian, V., Buchel, A., Green, S., Lehner, L., Liebling, S.: Phys. Rev. Lett. 113, 071601 (2014) * [24] Schnetter, E., Hawley, S. H., Hawke, I.: Class. Quant. Grav.21 1465 (2004) * [25] Bondi, H., van der Burg, M. G. J., Metzner, A. W. K.: Proc. R. Soc. A 269, 21 (1962) * [26] Sachs, R. K.: Proc. R. Soc. A 270, 103 (1962) * [27] Bondi, H.: Proc. R. Soc. A 281, 39 (1964) * [28] Chesler, P., Yaffe, L.: J. High Energ. Phys. 2014, 86 (2014) * [29] Gómez R., Winicour, J.: J. Math. Phys. 33, 1445 (1992) * [30] Barreto, W.: Phys. Rev. D 89, 084071 (2014) * [31] Gómez, R., Winicour, J., Isaacson, R.: J. Comput. Phys. 98, 11 (1992) * [32] Winicour, J.: Living Rev. Relativity 15, 2 (2012) * [33] Koike, T., Hara, T., Adachi, S.: Phys. Rev. Lett. 74, 5170 (1995) * [34] Gundlach, C., Martin-Garcia, J. M.: Living Rev. Relativity 10, 5 (2007) * [35] Gómez, R., Barreto, W., Frittelli, S.: Phys. Rev. D 76, 124029 (2007)
1802.05371	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 52363, "num_imgs": 11, "llama3_tokens_count": 13188 }	[ "content_image/1802.05371/x1.png", "content_image/1802.05371/x2.png", "content_image/1802.05371/x3.png", "content_image/1802.05371/x4.png", "content_image/1802.05371/x5.png", "content_image/1802.05371/x6.png", "content_image/1802.05371/x7.png", "content_image/1802.05371/x8.png", "content_image/1802.05371/x9.png", "content_image/1802.05371/x10.png", "content_image/1802.05371/x11.png" ]	# Input-Aware Auto-Tuning of Compute-Bound HPC Kernels Philippe Tillet Harvard Universityptillet@g.harvard.edu David Cox Harvard Universitydavidcox@fas.harvard.edu 2017 ###### Abstract. Efficient implementations of HPC applications for parallel architectures generally rely on external software packages (e.g., BLAS, LAPACK, CUDNN). While these libraries provide highly optimized routines for certain characteristics of inputs (e.g., square matrices), they generally do not retain optimal performance across the wide range of problems encountered in practice. In this paper, we present an input-aware auto-tuning framework for matrix multiplications and convolutions, ISAAC, which uses predictive modeling techniques to drive highly parameterized PTX code templates towards not only hardware-, but also application-specific kernels. Numerical experiments on the NVIDIA Maxwell and Pascal architectures show up to 3x performance gains over both cuBLAS and cuDNN after only a few hours of auto-tuning. † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] Input-Aware Auto-Tuning of Compute-Bound HPC Kernels ## 1. Introduction The growing adoption of many-core devices across HPC applications has rendered on-node performance perhaps more important than ever. However, while many practitioners have effectively been able to offload the execution of compute- or data-intensive tasks to local accelerators, the wide variety of applications and architectures available on the market has made it increasingly challenging to write code whose performance is portable. In order to develop efficient application code for these diverse architectures, many developers have relied, directly or through external libraries, on automatic source code tuning (auto-tuning). There, the performance-critical portions (kernels) of the application code are parameterized, and those parameters optimized for the architecture – and inputs – of interest [(5; 21)]. The wide adoption of this technique in fields like Linear Algebra [(19; 18; 15)] and Machine Learning [(20; 2)] has given rise to a plethora of hardware-oblivious software libraries capable of efficiently adapting virtually any underlying memory hierarchies and/or multi-threading schemes. The resulting performance gains have nonetheless remained highly input¹-sensitive, often lacking portability across the wide range of problem characteristics encountered in practice; it is for instance common for Basic Linear Algebra Subroutines (BLAS) to be used for computations involving input matrices of certain aspect ratios beyond those for which the implementation was optimized (usually square or highly rectangular). [FOOTNOTE:1][ENDFOOTNOTE] <figure><img src="content_image/1802.05371/x1.png"><figcaption>Figure 1. Overview of ISAAC</figcaption></figure> This paper aims at offering a new perspective on automatic performance tuning. We present a system, ISAAC, which does not produce a fixed set of tuning parameters _per se_, but rather a function that maps input characteristics to such parameters. We show that this function can be automatically learned from empirical benchmarking data using standard machine learning techniques (i.e., multi-layer perceptron), and propose a simple statistical method to speed-up the synthesis of a proper training dataset. An important addition of our framework is the use of a relatively low-level intermediate language (i.e., NVIDIA PTX), as opposed to higher-level alternatives typically used in similar systems (i.e., C, CUDA or OpenCL). While this is not strictly necessary and restricts our numerical experiments to NVIDIA GPUs only, this strategic choice leads to (1) better code generation, (2) faster compilation, and (3) more accurate performance models (due to simpler instruction selection heuristics). Our system is composed of four major components (see Figure 1), each of which will be described in a separate section of this paper: Section 3 describes the design and implementation of efficient code generation/parameterization techniques for matrix multiplication (GEMM) and convolution (CONV). Section 4 defines the process by which we generate training data for the input-aware predictive model presented in Section 5. Section 6 shows how this model may be used at runtime to quickly infer globally optimal kernels given any input configuration. Section 7 provides a numerical evaluation of our system on various practical problems, and shows substantial performance gains over both cuBLAS and cuDNN (up to 3x), which we analyze in Section 8. Finally, Section 9 provides concluding remarks and directions for future work. Prior to diving into the details of our system, Section 2 provides insights on existing related work. ## 2. Related Work Automatic performance tuning is a well established technique that has been effectively leveraged in a wide range of core HPC libraries (e.g., FFTW [(4)], SPIRAL [(13)], ATLAS [(19)] and OSKI [(17)]). Despite their obvious merits, these projects largely focus on delivering portable performance across architectures (ISA, memory hierarchies …) rather than input properties (matrix sizes, sparsity patterns …). This mismatch can result in the inefficient use of available hardware resources, ultimately leading to sub-optimal performance and/or energy efficiency. Input-aware auto-tuning arose recently [(10)] as a way to solve this issue, and has been since then applied to a variety of problems including poly-algorithmic selection [(3)], OpenACC loops optimization [(11)], and general-purpose GPU compilers [(12; 14)]. This surge of interest is encouraging, but has yet to win over an industry dominated by manual heuristics. It is indeed common for high-budget vendor libraries (e.g., MKL, cuBLAS) to engineer a set of several highly-optimized assembly kernels, and handcraft heuristics for runtime kernel selection. In addition to being expensive and time-consuming, this process can create portions of the input-space where the performance is poorly optimized – if at all. Our work, on the other hand, depicts a fully automated approach that not only fills such “performance holes”, but also equals vendor libraries where they perform best (e.g., LINPACK). <figure><img src="content_image/1802.05371/x2.png"><figcaption>Figure 2. Programming model for our kernel generator</figcaption></figure> ## 3. Kernel Generation In this section, we present the programming model underlying our forthcoming analysis, as well as the design and implementation of flexible source code parameterization techniques for GEMM and CONV. We introduce a set of reduction splitting parameters that improves our system’s performance for deep reductions that may arise in, for instance, covariance matrices computations. This technique – which is commonly found in communication-avoiding distributed algorithms for GEMM [(1)] – is to our knowledge too often overlooked by automatically tuned on-node software libraries. ### Programming Model Fig. 2 shows the programming model assumed by our framework. This model has been adopted by many programming languages for multi/many-core devices, including PTX, CUDA and OpenCL. Threads are arranged into a grid of 1-, 2- or 3-D blocks, where they may communicate using either synchronization barriers or shared memory. Communication between blocks is however only possible upon kernel completion through the use of global memory – typically GDDR or HBM. Each individual thread and block can be globally identified, thereby allowing different program instances to execute a given algorithm on different tiles of input data. The core idea behind auto-tuning frameworks is to parameterize the shape of these tiles, hence varying the amount of computation and resources used by each thread/block, ultimately fitting the underlying memory hierarchy and hardware threading mechanisms. ### Matrix Multiplication We now describe an input- and hardware- portable kernel parameterization for the matrix multiplication problem: \[C=AB\qquad C\in\mathbb{C}^{M\times N},~{}A\in\mathbb{C}^{M\times K},~{}B\in \mathbb{C}^{K\times N}\] Since this algorithm can be compute-bound for certain values of $(M,N,K)$, achieving high-performance requires efficient data-reuse and latency hiding. The former can be obtained via tiling and prefetching, while the latter necessitates thread level parallelism (TLP) and/or instruction level parallelism (ILP). For GPUs, TLP is implemented in hardware using a runtime scheduler: whenever a thread is stalled due to e.g., unfinished data transfers, another thread takes over the compute resources and execute another independent stream of instructions. This process actually happens at a granularity of 32 threads (i.e., a “warp”) for NVIDIA hardware. On the other hand, ILP is mostly handled in software. For the sake of energy efficiency, modern accelerators indeed outsource dependencies analysis to their respective Instruction Set Architectures (ISA): assembly programs are now often required to specify stall counts along with op-codes and operands. <figure><img src="content_image/1802.05371/x3.png"><figcaption>Figure 3. Flexible parameterization of matrix multiplication. Input and tuning parameters are respectively shown in red and blue.</figcaption></figure> It is crucial to note that all these optimization techniques exhibit trade-offs with one another. Large tile sizes, for instance, promote data-reuse but require more hardware resources, potentially undermining TLP. On the other hand, if tiles are too small, independent instructions will become rare and opportunities for ILP will be reduced. This implies that, when the tiling factor along one direction is constrained to be “small“ (due to the shape of the input matrices), it becomes necessary to mindfully increase tiling along another dimension to compensate. What it means for a tile to be “small“ or “large“, however, is a hidden property of the underlying hardware – that even experts rarely fully understand. It should be clear, now, that optimal tile sizes depend not only on the target micro-architecture but also on user-provided input parameters not necessarily known in advance. Figure 3 describes an algorithmic parameterization able to adjust these factors over a wide range of potential hardware architectures and input matrices. Each thread (resp. block) computes a tile of $M_{S}\times N_{S}$ (resp. $M_{L}\times N_{L}$) elements of $C$. In order maximize data-reuse, each work-group prefetches, into shared memory, $M_{L}\times U$ elements from $A$ and $U\times N_{L}$ elements from $B$. These tiles can be transposed in-place if necessary. The actual computations are then performed in a fully unrolled fashion – thereby producing a $K/U$_dependent_ stream of $M_{S}.N_{S}.U$ multiply-accumulate instructions each. Because $M_{S}$ and/or $N_{S}$ may be constrained to be very small in practice, it can become necesarry to create additional independent arithmetic instructions by splitting the computations along the reduction axis $K$, and accumulate the resulting partial results in a separate step. We therefore introduce three parameters $K_{S}$, $K_{L}$ and $K_{G}$ to split the reductions within respectively each thread, block and grid. Accumulation may then be performed using either registers addition, shared reductions or global atomics. A common concern for practitioners is the handling of cases where $M$ (or $N$) is not a multiples of $M_{L}$ (or $N_{L}$). Fortunately, the use of predicated instructions in PTX makes it possible to perform bounds checking much more efficiently than with input padding. We will come back to this later. ### Multi-Channel Convolution We now consider the following convolution algorithm: (1) \[O_{k,:,:,n}=\sum_{c=0}^{C}I_{c,:,:,n}\star F_{c,:,:,k}\] Where \[O \in R^{K\times P\times Q\times N}\] \[I \in R^{C\times H\times W\times N}\] \[F \in R^{C\times R\times S\times K}\] This operation is a useful generalization of the usual 2D convolution operator $\star$: instead of convolving a single image with a single filter, it convolves a set of $C$ different images with $C$ different filters and and returns the sum of all the resulting matrices. This process is repeated for $N$ sets of images and $K$ sets of filters. Due to the rise of Deep Learning over the past few years, this algorithm has become a bottleneck in many industrial and academic applications – each operating on its own specific input domain. This has created a strong demand for input-aware peak performance that even the most popular frameworks fail to fully satisfy. It is indeed common for convolution libraries (e.g., cuDNN) to provide relatively poor performance on signal processing applications where (1) degenerates to $\star$ – that is $N=C=K=1$ – or even certain standard benchmarks (e.g., DeepBench). Since every element $O_{k,p,q,n}$ of the resulting tensor is the inner product of $CRS$ element from $I$ and $F$, it is possible to reformulate multi-channel convolutions as implicit matrix-multiplication problems: tiles loaded from $I$ and $F$ are scrambled while being stored to shared memory, using an indirection table in order to alleviate integer arithmetics in the algorithm’s inner loop. It follows that we can use a parameterization similar to that exposed in Figure 3, except that tiling is performed across five dimensions (K, P, Q, N, C) rather than three. Each thread (resp. block) computes a tile of $K_{S}\times P_{S}\times Q_{S}\times N_{S}$ (resp. $K_{L}\times P_{L}\times Q_{L}\times N_{L}$) elements of $O$. For the sake of data-reuse, each work-group prefetches, into shared memory, $N_{L}\times P_{L}\times Q_{L}\times U$ elements from $I$ and $U\times K_{L}$ elements from $F$. The offsets for these load operations are obtained using the aforementioned indirection table. The actual computations are then performed in a fully unrolled fashion, and the reduction along $C$ is split using three tunable parameters: $C_{S},C_{L}$ and $C_{G}$. ## 4. Data Generation Let $\mathbb{X}$ and $\hat{\mathbb{X}}$ be respectively the space of _legal_ and _possible_ configuration for the aforementioned parameterization schemes. This distinction is necessary because some kernels can be properly compiled but not safely executed on the target device, due to the excessive usage of hardware resources such as shared memory or registers. For the GEMM algorithm described above, there are 10 tuning parameters and 6 input parameters – 3 shapes, 1 data-type and 2 transposition layouts – so $\mathbb{X}\subset\hat{\mathbb{X}}\subset\mathbb{N}^{16}$. Input-aware auto-tuning works by building a device-specific regression model $\mathcal{R}$ for the performance of any combination of legal input and tuning parameters $\mathbf{x}\in\mathbb{X}$. At runtime, the set of input parameters is fixed by the user, and $\mathcal{R}$ can be optimized over tuning parameters only. While $\mathcal{R}$ could technically be analytically approximated using deep expert knowledge, doing so could reduce its portability – let alone performance portability – across alternative and future micro-architectures. In this paper, we propose to learn $\mathcal{R}$ automatically from a large amount of benchmarking data obtained via the following statistical process. ### Generative Modeling Formally speaking, the goal of the data generation step is to produce a set of pairs $(\mathbf{x},y)$, where $\mathbf{x}\in\mathbb{X}$, and $y\in\mathbb{R}$ is a performance measurement (e.g., FLOPS, Joules, FLOPS/W…) of the kernel induced by $\mathbf{x}$ on the target hardware. When the number of parameters is small enough and the underlying resource constraints are known in advance, $\mathbb{X}$ can be pre-computed, in which case random parameter values can be trivially obtained via uniform sampling. On the other hand, when only $\hat{\mathbb{X}}$ is explicitly known, uniform sampling can be extremely wasteful (For GEMM, more than 99.9% of the resulting samples are illegal). A more tractable solution is to build a generative model $\mathcal{G}$ able to sample directly from the latent space of legal configurations $\mathbb{X}$. It is easy to imagine scenarios where $\mathcal{G}$ would be defined by a complex graphical model, but this would require a thorough analysis that is beyond the scope of this paper. Instead, our framework uses a naive technique which is simple to understand yet significantly more efficient than uniform sampling. Our generative model treats $\mathbf{x}$ as a random vector whose components $\mathbf{x}_{i}$ are independent categorical variables. In other words, we assume: \[p(\mathbf{x}\in\mathbb{X})=p(\mathbf{x}_{0})p(\mathbf{x}_{1})\cdots p(\mathbf{ x}_{N})\] The probability distribution of each parameter $\mathbf{x}_{i}$ can be approximated empirically, as the proportion of accepted values after a short period of uniform sampling. For instance, assuming that $\mathbf{x}_{1}=M_{S}$ may only take four values – say, $1,2,4,8$ – which respectively appear 5, 20, 25 and 50 times out of 100 uniformly sampled valid configurations, our framework assigns: \[p(\mathbf{x}_{1}=1)=.05\qquad p(\mathbf{x}_{1}=2)=.2\] \[p(\mathbf{x}_{1}=4)=.25\qquad p(\mathbf{x}_{1}=8)=.5\] Because we never really want any such probability to be exactly zero, we initialize each such count at a value $\alpha>0$ (our implementation uses $\alpha=100$). Formally speaking, this corresponds to assuming a Dirichlet prior distribution on $\mathbf{x}_{i}$. ### Performance Table 1 shows the proportion of invalid configuration generated by the above sampling method as compared to naive, uniform sampling, when each parameter is constrained to be a power of two between 1 and 16. \| Categorical \| Uniform ---\|---\|--- GEMM \| 20% \| 0.1% CONV \| 15% \| 0.1% Table 1. Proportion of samples accepted by our categorical generative model vs uniform sampling We see that this model, although simplistic, offers large performance improvements over uniform sampling, reducing the amount of bad samples by more than two orders of magnitude. Using this method, we were able to benchmark 50,000 valid different kernels in less than two hours. Again, we emphasize that this model is only meaningful when $\mathbb{X}$ cannot be pre-computed. <figure><img src="content_image/1802.05371/x4.png"><figcaption>Figure 4. A Multi-Layer Perceptron</figcaption></figure> ## 5. Regression Analysis Once a sufficient amount of training data $\mathcal{X}$ has been gathered using the above sampling method, our system builds a predictive model for the performance of any parameter vector $\mathbf{x}\in\mathbb{X}$. This is known as _regression analysis_. We evaluated multiple potential solutions before opting for a multi-layer perceptron (MLP), as it (1) scales best with large datasets (given enough time and resources, our dataset can be made arbitrarily large) and (2) naturally handles common nonlinearities found in performance modeling such as maximums and minimums. Furthermore, since MLP involving small feature vectors (around 20 in our case) rely on highly rectangular matrix computations, our system could itself be bootstrapped to make its own auto-tuning procedure more efficient. ### Multi-Layer Perceptron Fig. 4 shows the architecture of a basic MLP. The mapping from $\mathbf{x}$ to $y$ is organized in multiple layers of nonlinearly-activating nodes. Successive layers are fully connected, meaning that each node $i$ in one layer $L_{n}$ connects to each node $j$ in the following layer $L_{n+1}$ with a trainable weight $(W_{n})_{i,j}$. In other words, $y$ can be computed from $\mathbf{x}$ using the following algorithm: ``` Input: Weights $W_{0},W_{1},\cdots,W_{L-1}$ ; Features $\mathbf{x}$. Output: Performance prediction $\hat{y}$ $\mathbf{a}_{-1}=\mathbf{x}$; for $n\gets 0$to$L-1$ do $\mathbf{z}_{n}=W_{n}\mathbf{a}_{n-1}$; $\mathbf{a}_{n}=f_{n}(\mathbf{z}_{n})$ end for $\hat{y}=\mathbf{a}_{L-1}$ ``` Algorithm 1Forward Propagation Where $f_{i}$ is a non-linear activation shared by all the neurons in layer $n$. We emphasize that, under this model, multiplicative relationships between different elements of $a_{n}$ cannot be easily modeled. We will come back to this later. The parameters $W_{i}$ are chosen so as to minimize a given loss $\sum_{\mathbf{x}}\mathcal{L}(\hat{y}(\mathbf{x}),y(\mathbf{x}))$ on the predicted output. For regression analysis, it is desired that the predictions $\hat{y}(\mathbf{x})$ be noisy estimates of the true outputs $y(\mathbf{x})$, leading to the mean square error (MSE) loss function (when the noise is Gaussian). Since $\mathcal{L}$ is always chosen to be differentiable, this minimization can be carried out using e.g., Stochastic Gradient Descent. ### Implementation details Before explaining the details of our MLP implementation (hyperparameters, feature transformation, non-linearities), it can be useful to review the existing literature about GPU performance modeling. A comprehensive review of analytical performance models for compute-bound GPU kernels was offered by Volkov in his doctoral dissertation [(16)]. A common strategy for estimating the average arithmetic and memory throughput (in instructions/cycles) of the target kernel $K$ is: \[t_{\text{arith}}(n) =\max\Big{(}\frac{\text{alu\_latency}}{n},\text{alu\_throughput} \Big{)}\] (2) \[t_{\text{mem}}(n) =\max\Big{(}\frac{\text{mem\_latency}}{n},\text{mem\_throughput} \Big{)}\] Where $n$ is the mean occupancy (in warps per multi-processor), and alu_throughput, mem_throughput are underlying hardware characteristics. The total execution time $t(n)$ of $K$ is then: (3) \[t(n)=\max(t_{\text{arith}}(n)i_{\text{arith}},\quad t_{\text{mem}}(n)i_{\text{ mem}})\] Where $i_{\text{arith}}$ and $i_{\text{mem}}$ are respectively the number of arithmetic and memory instructions in K. The entire premise of our approach is that all the quantities involved in these computations depend – more or less strongly – on the relationship between _hidden_ hardware features (e.g., number of ALU, memory bandwidth, maximum throughput, banking structures) and _known_ input/tuning parameters (e.g., tensor shapes, tile sizes). A successful MLP should (implicitly) learn not only these relationships but also the corresponding hidden variables. As suggested by (2) and (3), it is expected that the relationships between all these variables include multiplications, divisions and maximums. Because, as mentioned previously, neural networks are not naturally designed to handle multiplications between different features, setting $\mathbf{a}_{-1}=\log(\mathbf{x})$ greatly improved the performance of our system. Furthermore, choosing the rectified linear unit (relu) activation function $f_{i}(\mathbf{z}_{i})=\max(0,\mathbf{z}_{i})$ seems appropriate to handle maximums. It may seem at first sight that deeper and wider MLPs would lead to higher runtime latency. However, research on neural networks inference tends to show that it is preferrable to train larger networks even if it means pruning or binarizing them afterwards [(6)]. ### Accuracy A common criticism of neural networks is that they are hard to engineer, hence this section attempts to provide insights on how good MLP architectures may be designed for our problem, as well as intuition regarding the amount of training data necessary to achieve good performance. We used matrix multiplication for our analysis, but the same qualitative behavior was observed in convolutions. Table 2 shows the cross-validation MSE of several MLP architectures, as measured on a fixed set of $10,000$ data-points separate from the $200,000$ samples used for training. Unsurprisingly, deeper networks seem to perform much better than shallower one (given a fixed amount of parameters). The accuracy of the network can be adjusted by adding (moderately wider) layers, at the cost of longer training and higher runtime latency. We emphasize the importance of the logarithmic feature transformation exposed in the previous subsection, without which our system would converge to much worse solutions – if at all. Hidden layer sizes \| #weights \| MSE (no log) ---\|---\|--- 64 \| 1k \| 0.17 (1.2) 512 \| 10k \| 0.13 (1.0) 32, 64, 32 \| 5k \| 0.088 (0.80) 64, 128, 64 \| 17k \| 0.08 (0.75) 32, 64, 128, 64, 32 \| 21k \| 0.073 (-) 64, 128, 256, 128, 64 \| 83k \| 0.067 (-) 64, 128, 192, 256, 192, 128, 64 \| 163k \| 0.062 (-) Table 2. Cross-validation MSE for various MLP architectures Figure 5 shows the evolution of our most accurate MLP’s accuracy as the amount of training data available grows. As expected, collecting more data does not seem to provide much benefits beyond a certain point (150,000 samples for GEMM, or $\sim 6$ hours of data collection). <figure><img src="content_image/1802.05371/x5.png"><figcaption>Figure 5. Cross-validation MSE for various data-set sizes</figcaption></figure> ## 6. Runtime Kernel Inference At this point, we possess a trained regression model that can predict the performance of any combination of input and tuning parameters. This model can be evaluated very quickly, in parallel, and with constant latency. This differs from actual kernel executions on a GPU, which may be slow, lock the device or even time-out when very inefficient kernels meet large problems. At runtime, the input parameters are provided by the user and fixed. Our model can be optimized over the remaining (i.e., tuning) parameters. Any discrete optimization method (e.g., simulated annealing, genetic algorithm, exhaustive search) may be used for this purpose. In this paper, we have opted for an exhaustive search, as it has several attractive properties when the number of tuning parameters is low enough: * It is guaranteed to find the global optimum within the specified search range. * The search is highly parallelizable. Up to a million different configurations per second can be evaluated – and potentially more, shall our system be bootstrapped in the process. * It is trivial to obtain the 100 (or more) fastest configurations for our model, and re-evaluate them on the target GPU to smooth out the inherent noise of our predictive model. The cost of exhaustive runtime inference, while high – up to a few seconds – is several orders of magnitude faster than running an exhaustive search on the target hardware (which can take up to 10 hours). The resulting predictions may be used directly in applications where this latency would be negligible (e.g., Deep Learning), cached on the filesystem, or even used as a kernel generation backend for low-latency libraries such as cuBLAS or cuDNN. ## 7. Numerical Experiments This section explores the performance of ISAAC for various input shapes, data-types and transposition layouts covering typical problem dimensions found in industrial benchmarks (LINPACK), scientific computing (LAPACK), deep learning (DeepBench) and signal processing (Independent Component Analysis). ### Hardware architectures While the focus of this paper is set on input-awareness, it is important that our framework be performance-portable across existing and future micro-architectures, hence our numerical experiments will be repeated on two distinct GPUs: \| Maxwell \| Pascal ---\|---\|--- GPU \| GTX 980 TI \| Tesla P100 (PCIE) Market Segment \| Consumer \| Server Micro-architecture \| GM200 \| GP100 CUDA cores \| 2816 \| 3584 Boost frequency \| 1075 MHz \| 1353 MHZ Processing Power \| 5.8 TFLOPS \| 9.7 TFLOPS Memory quantity \| 6 GB \| 16 GB Memory Type \| GDDR5 \| HBM2 Memory Bandwidth \| 336 GB/S \| 732 GB/s TDP \| 250W \| 250W Table 3. Test platforms hardware These two devices, though both designed by NVIDIA within a span of two years, differ in many ways. First, the Tesla P100 offers much more processing power than the GTX980 TI, as its higher power-efficiency allows it to carry more CUDA cores running at a higher frequency. Second, the P100 offers two times the bandwidth of the GTX980 TI – and again, these gains stem from major technological improvements. It is worth pointing out that HBM2 (large bus width, low frequency) and GDDR5 (small bus width, high frequency) handle memory transfers in a radically different way, to the point where IO-bound code designed for GDDR5 is not guaranteed to perform well with HBM2. ### Experimental protocol We compare our framework against cuBLAS 8.0 and cuDNN v6.0, which are the latest versions available at the time this paper is written. Despite a lot of research in automatic performance tuning, these two libraries have remained the gold standard for Linear Algebra and Deep Learning. Both libraries rely on handcrafted heuristics for choosing among a set of statically optimized assembly implementations. The cuBLAS API exposes functionalities to manually call individual kernels via the cublasGemmEx function, effectively allowing us to bypass any existing heuristics. We use this feature (under the label “Best Kernel“) to discriminate bad heuristical choices from missing tiling schemes. We use the flag IMPLICIT_PRECOMP_GEMM to force cuDNN to use of the algorithm presented in Section 3, with a scratch space of 64MB that remains on the device throughout the entire duration of our benchmark. ### GEMM Performance General Matrix Multiplication (GEMM) sits at the heart of High-Performance Computing, and is crucial to many applications, including supercomputer performance assessment, machine learning, signal processing and scientific computing. In this section, we evaluate our proposed framework on a set of input configurations (see Table 4) that we believe are representative of its practical usage. M \| N \| K \| A-T \| B-T \| Description ---\|---\|---\|---\|---\|--- LINPACK 512 \| 512 \| 512 \| No \| Yes \| Square case 1024 \| 1024 \| 1024 \| No \| Yes \| Square case 2048 \| 2048 \| 2048 \| No \| Yes \| Square case DeepBench 2560 \| 16 \| 2560 \| {No, Yes} \| No \| {Forw/Back} Propagation 2560 \| 32 \| 2560 \| {No, Yes} \| No \| {Forw/Back} Propagation 2560 \| 64 \| 2560 \| {No, Yes} \| No \| {Forw/Back} Propagation 2560 \| 128 \| 2560 \| {No, Yes} \| No \| {Forw/Back} Propagation Independent component analysis (ICA) 32 \| 32 \| 60000 \| No \| Yes \| 32-channels 64 \| 64 \| 60000 \| No \| Yes \| 64-channels 256 \| 256 \| 60000 \| No \| Yes \| 256-channels LAPACK (Blocked SVD – block-size 32 (lahabar09, )) 4096 \| 4096 \| 32 \| No \| Yes \| Iteration 0 3456 \| 3456 \| 32 \| No \| Yes \| Iteration 64 896 \| 896 \| 32 \| No \| Yes \| Iteration 100 Table 4. Tasks considered for the evaluation of ISAAC on GEMM. The column ’A-T’ (resp. ’B-T’) is marked as ’Yes’ if A (resp. B) is transposed, and ’No’ otherwise. #### 7.3.1. Gtx 980 Ti The results of our benchmarks are shown in Figure 6. LINPACK Our system rivals cuBLAS’s assembly kernels for large, square matrices (a case the library is specifically optimized for, due to its importance in performance assesment) and even outperforms it by almost 25% when M=N=K=512. <figure><img src="content_image/1802.05371/x6.png"><figcaption>Figure 6. SGEMM performance on the GTX 980 TI</figcaption></figure> DeepBench (Forward) The benefits of input-aware auto-tuning become more apparent for problems involving irregular input shapes. Our benchmark shows $80\%$ speed-ups on DeepBench for $N=16$ (here we show $M=N=2560$, but our results hold as long as $M,N$ are big enough to make GPU execution meaningful). These gains vanish as the batch size approaches tiling factors provided explicitly by cuBLAS ($N_{L}\in\{64,128\}$). It should be nonetheless noted that large batch sizes are rarely used in practice due to bad convergence properties [(7)]. We note poor heuristical kernel selection for cuBLAS when $N\in\{32,64\}$. Further investigation revealed that it was due to poor handling of reduction-splitting in the library’s heuristics. DeepBench (Backward) We found reduction splitting ($K_{L}>1$, $K_{G}>1$) to be even more necessary for achieving good performance on DeepBench’s back-propagation problems. This is due to unfavorable access patterns which requires both A and B to be internally transposed in shared memory prior to any computation. The latency of these transpositions can be hidden by using more warps, which is the exact purpose of reduction-splitting. All things considered, our framework outperforms cuBLAS’s best kernel by $65\%$ when $N=16$ and by $35\%$ when $N=128$. ICA It is known that cuBLAS implements some form of global reduction splitting ($K_{G}>1$) to handle cases where $K$ is large and $M.N$ is small. There seems to be several instances in which the library’s heuristics fail to properly leverage this feature, resulting in drastic slow-downs (over an order of magnitude) in our ICA benchmarks. Even after bypassing kernel selection, cuBLAS remains $10\%$ slower than ISAAC, which is attributed to cuBLAS not implementing reduction splitting within streaming multi-processors ($K_{L}>1$). LAPACK Minor performance gains (10%) are observed for packed outer-products commonly found in blocked linear algebra algorithms (e.g., householder bi-diagonalization in SVD). <figure><img src="content_image/1802.05371/x7.png"><figcaption>Figure 7. SGEMM performance on the Tesla P100</figcaption></figure> #### 7.3.2. Tesla P100 Single Precision Figure 6 showed that cuBLAS achieves more than 90% of Maxwell’s peak performance on large, square matrices. This efficiency does not seem to carry over to Pascal, as cuBLAS saturates at 85% of the P100’s peak performance. On the other hand, our system’s efficiency is constant ($85\%$), leading to performance parity with cuBLAS in our most pessimistic benchmarks. The automation inherent to our approach also allows for shorter development cycles – the tuning procedure only takes a few hours – which could facilitate the deployment of software updates following the release of a new architecture. The performance gains of ISAAC over cuBLAS’s best kernel remain otherwise consistent with those observed in the previous subsection, reaching 25% on LINPACK, 80% on DeepBench, 5% on ICA and 30% on LAPACK. The heuristics used by cuBLAS seems to retained the same deficienies as for Maxwell. <figure><img src="content_image/1802.05371/x8.png"><figcaption>Figure 8. H/DGEMM performance on the Tesla P100</figcaption></figure> Half/Double Precision Our numerical experiments would be incomplete without a proper account of ISAAC’s half and double precision performance, as the usage of single precision arithmetics is discouraged in both Deep Learning (where half precision is sufficient) and Scientific Computing (where double precision is necessary). Fortunately, our approach is not bound to any particular data-type, hence we re-evaluate GEMM in half and double precision on the Tesla P100, which provides respectively 0.5x and 2x single precision peak performance for these cases. Half precision is used for DeepBench and LINPACK; double precision is used for the rest. As shown in figure 8, our framework retains significant performance gains over cuBLAS in double precision, averaging 5% on LINPACK, 40% on ICA and 15% on LAPACK. A major advantage of our framework is its ability to generate many different kernels at a very low cost. This inherent flexibility translates into tremendous performance gains in cases where adding support for new tiling schemes and/or specialized instructions is cumbersome, and apparently not implemented in cuBLAS. As a result, ISAAC is able to leverage the “fp16x2“ instructions across the entire input-space, resulting in 2.5-3x speedups over cuBLAS on DeepBench. The near-optimal half-precision performance of NVIDIA’s library on LINPACK underlines the existence of a limited set of NVIDIA kernels implementing this feature. ### CONV Performance The rise of Deep Learning over the last 5 years [(8)] has made fast convolution routines not only desirable but also necessary to the rapid evolution of the field as a whole. CuDNN offers state-of-the-art performance for this algorithm, and is used in all major Deep Learning Frameworks (e.g., Tensorflow, Theano, Pytorch…). In this section, we show that input-aware auto-tuning can be used to produce compute kernels sometimes faster than cuDNN. The network architectures considered in this section were extracted from the DeepBench suite so as to span 6 different concrete applications. The corresponding data shapes are shown in Table 5. Recall that cuDNN treats (N, P, Q, K, C, R, S) convolutions as implicit (NPQ, K, CRS) matrix multiplications. N \| P \| Q \| K \| C \| R \| S \| NPQ \| CRS \| Name ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- DeepSpeech 16 \| 79 \| 341 \| 32 \| 1 \| 5 \| 20 \| 431024 \| 100 \| Conv1 16 \| 38 \| 166 \| 32 \| 32 \| 5 \| 10 \| 100928 \| 1600 \| Conv2 OCR 16 \| 24 \| 240 \| 32 \| 16 \| 3 \| 3 \| 92160 \| 144 \| Conv3 16 \| 12 \| 120 \| 64 \| 32 \| 3 \| 3 \| 23040 \| 288 \| Conv4 Face Recognition 8 \| 54 \| 54 \| 64 \| 64 \| 3 \| 3 \| 23328 \| 576 \| Conv5 8 \| 27 \| 27 \| 128 \| 128 \| 3 \| 3 \| 5832 \| 1152 \| Conv6 16 \| 14 \| 14 \| 48 \| 512 \| 5 \| 5 \| 3136 \| 12800 \| Conv7 16 \| 7 \| 7 \| 128 \| 832 \| 5 \| 5 \| 784 \| 20800 \| Conv8 Vision 8 \| 112 \| 112 \| 128 \| 64 \| 3 \| 3 \| 100352 \| 576 \| Conv9 8 \| 56 \| 56 \| 256 \| 128 \| 3 \| 3 \| 25088 \| 1152 \| Conv10 Speaker ID 16 \| 128 \| 39 \| 174 \| 64 \| 5 \| 5 \| 79872 \| 1600 \| Conv11 16 \| 256 \| 19 \| 87 \| 128 \| 5 \| 5 \| 77824 \| 3200 \| Conv12 ResNET 16 \| 7 \| 7 \| 512 \| 512 \| 3 \| 3 \| 784 \| 4608 \| Conv13 16 \| 7 \| 7 \| 2048 \| 1024 \| 1 \| 1 \| 784 \| 1024 \| Conv14 Table 5. Tasks considered for evaluating ISAAC on CONV <figure><img src="content_image/1802.05371/x9.png"><figcaption>Figure 9. SCONV performance on the GTX 980 TI</figcaption></figure> #### 7.4.1. Gtx 980 The performance benefits of ISAAC (see Figure 9) are noticeable but not as large as they were for GEMM. This is because cuDNN was optimized from the ground up with both Maxwell and DeepBench-like problems in mind (Large NPQ, small K and intermediate CRS). Nonetheless, we note substantial performance gains ($1.5\times$ to $2\times$) over cuDNN for the deep reductions found in Conv7 and Conv8. Note that cuDNN provides no public way of benchmarking individual kernels, hence it is difficult to say whether these gains come from poor heuristical choices or missing tiling configurations. We also note appreciable speed-ups ($\sim 10\%$) when NPQ is small and the operation does not degenerate to direct matrix multiplication ($RS>1$, Conv13). #### 7.4.2. Tesla P100 Figure 10 and 11 show the performance of ISAAC for single- and half-precision convolutions, respectively. We observe large performance gains (more than $5\times$ for Conv8 and $70\%$ for Conv13) that we attribute to cuDNN’s heuristics and kernels being tailored to Maxwell rather than Pascal. <figure><img src="content_image/1802.05371/x10.png"><figcaption>Figure 10. SCONV performance on the Tesla P100</figcaption></figure> <figure><img src="content_image/1802.05371/x11.png"><figcaption>Figure 11. HCONV performance on the Tesla P100</figcaption></figure> As for HCONV, ISAAC’s ability to easily support many tiling schemes result in almost consistently faster half-precision convolution routines than cuDNN. ## 8. Analysis The encouraging results shown in Section 7 beg for a thorough analysis of our system’s performance: How exactly are such speed-ups achieved? What constitutes good parameter choices for our kernel generator? Why is PTX necessary to obtain good performance? This section addresses these three questions, in order. We believe that the resulting insights could help library developers enhance existing software, like cuBLAS, which rely on a small set of statically generated kernels. ### DeepBench (Forward) Our previous benchmarks showed that, even in the presence of optimal kernel selection heuristics, cuBLAS could be up to 2x slower than ISAAC in single-precision, and 3x in half-precision. In order to explain why this is the case, this subsection provides a detailed comparison of ISAAC and cuBLAS’s best kernel when (M, N, K) = (2560, 32, 2560), for the Tesla P100. The first thing to note is that this input configuration is only IO-bound under strong latency hiding assumptions: should the arithmetic operations not properly overlap with data-transfers, cycles will be lost and the achieved effective bandwidth reduced. Hence, it is unfortunate that cuBLAS only provides 64- and 128- way tiling along the N dimension, as it precludes the launch of enough warps to sustain high enough GPU occupancy (optimality is not a hard requirement since the problem is ideally still IO-bound). \| ISAAC \| cuBLAS ---\|---\|--- TFLOPS \| 3.73 \| 2.56 ML \| 64 \| 128 NL \| 32 \| 64 KL \| 4 \| 5 P \| 16 \| 8 Shared Memory \| 12.25kB \| 12.25kB Registers Count \| 72 \| 120 Occupancy \| 17% \| 10% L2 hit rate \| 32% \| 24% The main problem of cuBLAS’s best kernel is that it assigns a large number of (de facto counter-productive) threads to an unexisting portion (64≤N<128) of the result matrix. This has two adverse effects, which conjunctly explain the relatively bad performance we observed: 1. By using smaller tiling factors, ISAAC can decrease register/shared memory pressure, resulting in higher occupancy and therefore better latency hiding. 2. When higher occupancy no longer translates to improved performance, ISAAC (automatically) learns to use resources still available to instead pre-fetch more data into shared memory (i.e., larger U), resulting in better cache-hit rate (i.e., higher effective bandwidth). Reduction-spltting is, as mentioned previously, an alternative way to increase occupancy. Both ISAAC and cuBLAS use this method, although cuBLAS uses KL=1. ### 8.2. Kernel Selection The previous section may have given the reader a sense of what differentiates good from bad parameter values: (1) tile sizes should be small enough to guarantee high occupancy, but large enough to retain opportunities for ILP; (2) reduction splitting can be leveraged to further improve latency hiding, at the cost of diminished write bandwidth (via atomics) and/or additional shared memory usage, and (3) the pre-fetching factor U can be increased to improve effective bandwidth when higher occupancy is no longer beneficial. In order to better comprehend these trade-offs, Table 6 shows the parameterization choices made by ISAAC for the aforementioned problem sizes. Problem \| Ms \| Ns \| ML \| NL \| U \| Ks \| KL \| KG ---\|---\|---\|---\|---\|---\|---\|---\|--- LINPACK (512) \| 2 \| 8 \| 32 \| 32 \| 8 \| 1 \| 1 \| 1 LINPACK (2048) \| 8 \| 8 \| 64 \| 64 \| 8 \| 1 \| 1 \| 1 DeepBench-F (16) \| 2 \| 4 \| 64 \| 16 \| 16 \| 1 \| 1 \| 4 DeepBench-F (128) \| 4 \| 4 \| 64 \| 32 \| 8 \| 1 \| 1 \| 2 DeepBench-B (16) \| 4 \| 2 \| 16 \| 16 \| 16 \| 1 \| 8 \| 1 DeepBench-B (128) \| 4 \| 4 \| 64 \| 64 \| 8 \| 1 \| 1 \| 4 ICA (32) \| 2 \| 4 \| 32 \| 32 \| 8 \| 1 \| 4 \| 32 ICA (256) \| 4 \| 4 \| 32 \| 64 \| 8 \| 1 \| 1 \| 8 LAPACK (896) \| 8 \| 4 \| 64 \| 64 \| 8 \| 1 \| 1 \| 1 LAPACK (4096) \| 8 \| 16 \| 64 \| 128 \| 4 \| 1 \| 1 \| 1 Table 6. Parameterization choices of ISAAC The main problem of cuBLAS’s best kernel is that it assigns a large number of (de facto counter-productive) threads to an unexisting portion ($64\leq N<128$) of the result matrix. This has two adverse effects, which conjunctly explain the relatively bad performance we observed: 1. By using smaller tiling factors, ISAAC can decrease register/shared memory pressure, resulting in higher occupancy and therefore better latency hiding. 2. When higher occupancy no longer translates to improved performance, ISAAC (automatically) learns to use resources still available to instead pre-fetch more data into shared memory (i.e., larger $U$), resulting in better cache-hit rate (i.e., higher effective bandwidth). Reduction-spltting is, as mentioned previously, an alternative way to increase occupancy. Both ISAAC and cuBLAS use this method, although cuBLAS uses $K_{L}=1$. ### Kernel Selection The previous section may have given the reader a sense of what differentiates good from bad parameter values: (1) tile sizes should be small enough to guarantee high occupancy, but large enough to retain opportunities for ILP; (2) reduction splitting can be leveraged to further improve latency hiding, at the cost of diminished write bandwidth (via atomics) and/or additional shared memory usage, and (3) the pre-fetching factor $U$ can be increased to improve effective bandwidth when higher occupancy is no longer beneficial. In order to better comprehend these trade-offs, Table 6 shows the parameterization choices made by ISAAC for the aforementioned problem sizes. Problem \| Ms \| Ns \| ML \| NL \| U \| Ks \| KL \| KG ---\|---\|---\|---\|---\|---\|---\|---\|--- LINPACK (512) \| 2 \| 8 \| 32 \| 32 \| 8 \| 1 \| 1 \| 1 LINPACK (2048) \| 8 \| 8 \| 64 \| 64 \| 8 \| 1 \| 1 \| 1 DeepBench-F (16) \| 2 \| 4 \| 64 \| 16 \| 16 \| 1 \| 1 \| 4 DeepBench-F (128) \| 4 \| 4 \| 64 \| 32 \| 8 \| 1 \| 1 \| 2 DeepBench-B (16) \| 4 \| 2 \| 16 \| 16 \| 16 \| 1 \| 8 \| 1 DeepBench-B (128) \| 4 \| 4 \| 64 \| 64 \| 8 \| 1 \| 1 \| 4 ICA (32) \| 2 \| 4 \| 32 \| 32 \| 8 \| 1 \| 4 \| 32 ICA (256) \| 4 \| 4 \| 32 \| 64 \| 8 \| 1 \| 1 \| 8 LAPACK (896) \| 8 \| 4 \| 64 \| 64 \| 8 \| 1 \| 1 \| 1 LAPACK (4096) \| 8 \| 16 \| 64 \| 128 \| 4 \| 1 \| 1 \| 1 Table 6. Parameterization choices of ISAAC ISAAC seems to properly learn to make sensible choices for all cases considered: (1) it chooses smaller tiles for smaller problems, (2) always split deep reductions problems (the proper trade-off is found between $K_{L}>1$ increases resources usage and $K_{G}>1$ which decreases write bandwidth) and (3) decreases $U$ appropriately to save hardware resources when good cache efficiency is not very important (see LAPACK). ### Advantages of PTX The first iteration of our software used CUDA-C and OpenCL for code-generation, but it was deprecated as adding bounds-checking resulted in a $15-20\%$ performance loss. Switching to PTX reduced this overhead to $2\%$. This is because modern NVIDIA hardware implement a mechanism called “predication“: each instruction is complemented with a binary mask that specifies which thread should or should not be active. This mechanism, which does not require any program counter modification and has virtually no latency, is exposed in PTX but not in CUDA C. ## 9. Conclusions In this paper, we have presented ISAAC, an open-source² framework for input-aware auto-tuning. Our tool relies on a versatile code generator able to adapt a wide range of problem sizes. We presented parameterization techniques for GEMM and CONV, used a multi-layer perceptron to model their behavior, and showed that features transformation was necessary to achieve proper convergence. We demonstrated how this model could be used to perform kernel selection at runtime, when input characteristics are fixed. Finally, we evaluated and analyzed the performance of our framework on a large variety of practical problems, and observed up to 3x performance gains over assembly-optimized vendor libraries. [FOOTNOTE:2][ENDFOOTNOTE] Still, we see several possible directions of future work. While the good performance of our system on square matrices suggests that there is little room for improvement in our kernel generation mechanisms, our performance model relies on a series of rather basic techniques. Data-generation could be improved using better generative modeling techniques (e.g., Markov random field), and more efforts could be spent tuning our regression network. Another valuable addition to our framework would be a more flexible front-end (possibly a Domain Specific Language) to allow its use on problems beyond GEMM and CONV. ## 10. Acknowledgments This work was supported by the National Science Foundation (IIS 1409097) and by IARPA (contract D16PC00002). ## References * (1)Agarwal, R. C., Balle, S. M., Gustavson, F. G., Joshi, M., and Palkar, P. A three-dimensional approach to parallel matrix multiplication. * (2)Bergstra, J., Pinto, N., and Cox, D. Machine learning for predictive auto-tuning with boosted regression trees. In _Innovative Parallel Computing (InPar), 2012_ (May 2012). * (3)Ding, Y., Ansel, J., Veeramachaneni, K., Shen, X., O’Reilly, U.-M., and Amarasinghe, S. Autotuning algorithmic choice for input sensitivity. In _Proceedings of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation_ (2015), PLDI 2015. * (4)Frigo, M., and Johnson, S. The design and implementation of fftw3. _Proceedings of the IEEE_ (2005). * (5)Fursin, G. G., O’Boyle, M. F. P., and Knijnenburg, P. M. W. Evaluating iterative compilation. In _Proceedings of the 15th International Conference on Languages and Compilers for Parallel Computing_, LCPC’02. * (6)Hubara, I., Courbariaux, M., Soudry, D., El-Yaniv, R., and Bengio, Y. Binarized neural networks. In _NIPS_ (2016), pp. 4107–4115. * (7)Keskar, N. S., Mudigere, D., Nocedal, J., Smelyanskiy, M., and Tang, P. T. P. On large-batch training for deep learning: Generalization gap and sharp minima. * (8)Krizhevsky, A., Sutskever, I., and Hinton, G. E. Imagenet classification with deep convolutional neural networks. In _Advances in Neural Information Processing Systems 25_. * (9)Lahabar, S., and Narayanan, P. Singular value decomposition on gpu using cuda. In _Parallel Distributed Processing, 2009. IPDPS 2009. IEEE International Symposium on_ (2009). * (10)Liu, Y., Zhang, E., and Shen, X. A cross-input adaptive framework for gpu program optimizations. In _Parallel Distributed Processing, 2009. IPDPS 2009. IEEE International Symposium on_ (2009). * (11)Magni, A., Grewe, D., and Johnson, N. Input-aware auto-tuning for directive-based gpu programming. In _Proceedings of the 6th Workshop on General Purpose Processor Using Graphics Processing Units_ (2013), GPGPU-6. * (12)Muralidharan, S., Shantharam, M., Hall, M., Garland, M., and Catanzaro, B. Nitro: A framework for adaptive code variant tuning. In _Proceedings of the 2014 IEEE 28th International Parallel and Distributed Processing Symposium_, IPDPS ’14. * (13)Püschel, M., Moura, J. M. F., Singer, B., Xiong, J., Johnson, J., Padua, D., Veloso, M., Johnson, R. W., Püschel, M., Moura, J. M. F., Singer, B., Xiong, J., Johnson, J., Padua, D., Veloso, M., and Johnson, R. W. Spiral: A generator for platform-adapted libraries of signal processing algorithms. _Journal of High Performance Computing and Applications 18_ (2004). * (14)Samadi, M., Hormati, A., Mehrara, M., Lee, J., and Mahlke, S. Adaptive input-aware compilation for graphics engines. In _Proceedings of the 33rd ACM SIGPLAN Conference on Programming Language Design and Implementation_ (2012), PLDI ’12. * (15)Tillet, P., Rupp, K., Selberherr, S., and Lin, C. T. Towards performance-portable, scalable, and convenient linear algebra. In _4th USENIX Workshop on Hot Topics in Parallelism, HotPar’12, Berkeley, CA, USA, June 7-8, 2012_. * (16)Volkov, V. Understanding latency hiding on gpus., 2016. * (17)Vuduc, R., Demmel, J. W., and Yelick, K. A. OSKI: A library of automatically tuned sparse matrix kernels. _Journal of Physics: Conference Series 16_, 1 (2005). * (18)Wang, Q., Zhang, X., Zhang, Y., and Yi, Q. Augem: Automatically generate high performance dense linear algebra kernels on x86 cpus. In _Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis_ (2013), SC ’13. * (19)Whaley, R. C., and Dongarra, J. J. Automatically tuned linear algebra software. In _Proceedings of the 1998 ACM/IEEE Conference on Supercomputing_ (1998), SC ’98. * (20)Xiong, J., Johnson, J., Johnson, R., and Padua, D. Spl: A language and compiler for dsp algorithms. In _Proceedings of the ACM SIGPLAN 2001 Conference on Programming Language Design and Implementation_ (2001), PLDI ’01. * (21)Yotov, K., Li, X., Ren, G., Cibulskis, M., DeJong, G., Garzaran, M., Padua, D., Pingali, K., Stodghill, P., and Wu, P. A comparison of empirical and model-driven optimization. _SIGPLAN Not._.
1806.09151	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 20298, "num_imgs": 5, "llama3_tokens_count": 7878 }	[ "content_image/1806.09151/mucase1n1.jpg", "content_image/1806.09151/betacase1n1.jpg", "content_image/1806.09151/alphacase1n1.jpg", "content_image/1806.09151/mucase2n1.jpg", "content_image/1806.09151/alphacase2n1.jpg" ]	# Spinor fields in spherically symmetric space-time Bijan Saha Laboratory of Information Technologies Joint Institute for Nuclear Research, Dubna 141980 Dubna, Moscow region, Russia and Institute of Physical Research and Technologies People’s Friendship University of Russia Moscow, Russia bijan@jinr.ru http://spinor.bijansaha.ru ###### Abstract Within the scope of a spherically symmetric space-time we study the role of a nonlinear spinor field in the formation of different configurations with spherical symmetries. The presence of the non-diagonal components of energy-momentum tensor of the spinor field leads to some severe restrictions on the spinor field itself. Since spinor field is the source of the gravitational one, the metric functions also changes in accordance with it. The system as a whole possesses solutions only in case of some additional conditions on metric functions. Spinor field, spherically symmetric model pacs: 98.80.Cq ## 1 Introduction In the recent past spinor description of matter and dark energy was used to draw the picture of the evolution of the Universe within the scope of Bianchi type anisotropic cosmological models [1; 2; 3; 4]. It was found that the approach in question gives rise to a variety of solutions depending on the choice of spinor field nonlinearity. Thanks to its sensitivity to gravitational field spinor field brings some unexpected nuances in the behavior of both the spinor and the gravitational fields. Taking this in mind in this paper we consider the nonlinear spinor field within the framework of spherically symmetric gravitational field. Since a variety of astrophysical systems such as stars, black holes are described by spherically symmetric configurations, the use of spinor field in this area might be very promising. ## 2 Basic Equation Let us consider a system of nonlinear spinor and spherically symmetric gravitational fields. The corresponding action we choose in the form \[{\cal S}(g;\psi,\bar{\psi})=\int\,\left(L_{\rm g}+L_{\rm sp} \right)\sqrt{-g}d\Omega\] (2.1) Here $L_{\rm g}$ corresponds to the gravitational field \[L_{\rm g}=\frac{R}{2\kappa},\] (2.2) where $R$ is the scalar curvature, $\kappa=8\pi G$ with $G$ being Newton’s gravitational constant and $L_{\rm sp}$ is the spinor field Lagrangian which we take in the form \[L_{\rm sp}=\frac{\imath}{2}\left[\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-\nabla _{\mu}\bar{\psi}\gamma^{\mu}\psi\right]-m\bar{\psi}\psi-F,\] (2.3) with the nonlinear term $F=F(K)$ and $K$ taking one of the following expressions: $\{I,\,J,\,I+J,\,I-J\}$. Here $I=\bar{\psi}\psi$ and $J=\imath\bar{\psi}\bar{\gamma}^{5}\psi$. Here $m$ is the spinor mass. The spinor field equations corresponding to the spinor field Lagrangian (2.3) are \[\imath\gamma^{\mu}\nabla_{\mu}\psi-m\psi-{\cal D}\psi-\imath{\cal G }\gamma^{5}\psi = 0,\] (2.4a) \[\imath\nabla_{\mu}\bar{\psi}\gamma^{\mu}+m\bar{\psi}+{\cal D}\bar {\psi}+\imath{\cal G}\bar{\psi}\gamma^{5} = 0,\] (2.4b) where we denote ${\cal D}=2SF_{K}K_{I}$ and ${\cal G}=2PF_{K}K_{J}$, with $F_{K}=dF/dK$, $K_{I}=dK/dI$ and $K_{J}=dK/dJ.$ In view of (2.4) it can be shown that \[L_{\rm sp}=2KF_{K}-F.\] (2.5) In the above expressions $\nabla_{\mu}\psi=\partial_{\mu}\psi-\Gamma_{\mu}\psi$ and $\nabla_{\mu}\bar{\psi}=\partial_{\mu}\bar{\psi}+\bar{\psi}\Gamma_{\mu}$ with $\Gamma_{\mu}$ being the spinor affine connection. The spherically-symmetric metric we choose in the form \[ds^{2}=e^{2\mu}dt^{2}-e^{2\alpha}dr^{2}-e^{2\beta}(d\vartheta^{2}+\sin^{2}{ \vartheta}d\varphi^{2}),\] (2.6) where the metric functions $\mu,\alpha,\beta$ depend on the spatial variable $r$ only. The spinor affine connection matrices are defined as \[\Gamma_{\mu}(x)=\frac{1}{4}g_{\rho\sigma}(x)\biggl{(}\partial_{\mu}e_{\delta}^ {b}e_{b}^{\rho}-\Gamma_{\mu\delta}^{\rho}\biggr{)}\gamma^{\sigma}\gamma^{ \delta},\] (2.7) where the tetrad $e_{b}^{\rho}$ correspond to the metric (2.6) we choose as follows: \[e_{0}^{(0)}=e^{\mu},\quad e_{1}^{(1)}=e^{\alpha},\quad e_{2}^{(2)}=e^{\beta}, \quad e_{3}^{(3)}=e^{\beta}\sin\theta.\] (2.8) The flat $\gamma$ matrices we choose in the from \[\bar{\gamma}^{0} = \left(\begin{array}[]{cc}I&0\\ 0&-I\end{array}\right),\quad\bar{\gamma}^{1}=\left(\begin{array}[]{cc}0&\sigma ^{1}\\ -\sigma^{1}&0\end{array}\right),\] \[\bar{\gamma}^{2} = \left(\begin{array}[]{cc}0&\sigma^{2}\\ -\sigma^{2}&0\end{array}\right),\quad\bar{\gamma}^{3}=\left(\begin{array}[]{cc }0&\sigma^{3}\\ -\sigma^{3}&0\end{array}\right).\] where \[I = \left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right),\quad\sigma^{1}=\left(\begin{array}[]{cc}\cos{\vartheta} &\sin{\vartheta}e^{-\imath\varphi}\\ \sin{\vartheta}e^{\imath\varphi}&-\cos{\vartheta}\end{array}\right),\] \[\sigma^{2} = \left(\begin{array}[]{cc}-\sin{\vartheta}&\cos{\vartheta}e^{- \imath\varphi}\\ \cos{\vartheta}e^{\imath\varphi}&\sin{\vartheta}\end{array}\right),\quad\quad \sigma^{3}=\left(\begin{array}[]{cc}0&\imath e^{-\imath\varphi}\\ -\imath e^{\imath\varphi}&0\end{array}\right).\] (2.9) Defining $\gamma^{5}$ as follows: \[\gamma^{5} = -\frac{i}{4}E_{\mu\nu\sigma\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{ \sigma}\gamma^{\rho},\quad E_{\mu\nu\sigma\rho}=\sqrt{-g}\varepsilon_{\mu\nu \sigma\rho},\quad\varepsilon_{0123}=1,\] \[\gamma^{5} = -i\sqrt{-g}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\,=\,-i\bar{ \gamma}^{0}\bar{\gamma}^{1}\bar{\gamma}^{2}\bar{\gamma}^{3}=\bar{\gamma}^{5},\] we obtain \[\bar{\gamma}^{5} = \left(\begin{array}[]{cc}0&-I\\ -I&0\end{array}\right).\] Taking into account that the functions $\alpha$, $\beta$ and $\mu$ depend on only $r$ ($x^{1}$) from (2.7) we find \[\Gamma_{0} = -\frac{1}{2}\mu^{\prime}\,e^{(\mu-\alpha)}\,\bar{\gamma}^{0}\bar{ \gamma}^{1},\] (2.10a) \[\Gamma_{0} = 0,\] (2.10b) \[\Gamma_{2} = \frac{1}{2}\beta^{\prime}\,e^{(\beta-\alpha)}\,\bar{\gamma}^{2} \bar{\gamma}^{1},\] (2.10c) \[\Gamma_{3} = \frac{1}{2}\beta^{\prime}\,\sin\theta e^{(\beta-\alpha)}\,\bar{ \gamma}^{3}\bar{\gamma}^{1}+\frac{1}{2}\cos\theta\bar{\gamma}^{3}\bar{\gamma}^ {2}.\] (2.10d) Let us now find the energy-momentum tensor of the spinor field which is given by \[T_{\mu}^{\,\,\,\rho} = \frac{\imath}{4}g^{\rho\nu}\left(\bar{\psi}\gamma_{\mu}\nabla_{ \nu}\psi+\bar{\psi}\gamma_{\nu}\nabla_{\mu}\psi-\nabla_{\mu}\bar{\psi}\gamma_{ \nu}\psi-\nabla_{\nu}\bar{\psi}\gamma_{\mu}\psi\right)\,-\delta_{\mu}^{\rho}L_ {\rm sp}\] (2.11) \[= \frac{\imath}{4}g^{\rho\nu}\left(\bar{\psi}\gamma_{\mu}\partial_{ \nu}\psi+\bar{\psi}\gamma_{\nu}\partial_{\mu}\psi-\partial_{\mu}\bar{\psi} \gamma_{\nu}\psi-\partial_{\nu}\bar{\psi}\gamma_{\mu}\psi\right)\] \[- \frac{\imath}{4}g^{\rho\nu}\bar{\psi}\left(\gamma_{\mu}\Gamma_{ \nu}+\Gamma_{\nu}\gamma_{\mu}+\gamma_{\nu}\Gamma_{\mu}+\Gamma_{\mu}\gamma_{\nu }\right)\psi\,-\delta_{\mu}^{\rho}L_{\rm sp}.\] On account of spinor field equations one finds the following non-trivial and linearly independent terms of the energy-momentum tensor \[T^{0}_{0} = F(K)-2KF_{K},\] (2.12a) \[T^{1}_{1} = mS+F(K),\] (2.12b) \[T^{2}_{2} = F(K)-2KF_{K},\] (2.12c) \[T^{3}_{3} = F(K)-2KF_{K},\] (2.12d) \[T^{0}_{1} = \frac{1}{4}\frac{\cos\theta}{\sin\theta}e^{(\alpha-\mu-\beta)}A^{ 3},\] (2.12e) \[T^{0}_{2} = -\frac{1}{4}\left(\mu^{\prime}-\beta^{\prime}\right)\,e^{(\beta- \alpha-\mu)}A^{3},\] (2.12f) \[T^{0}_{3} = \frac{1}{4}\left(\mu^{\prime}-\beta^{\prime}\right)\,e^{(\beta- \alpha-\mu)}\sin\theta A^{2}+\frac{1}{4}e^{-\mu}\cos\theta A^{1}.\] (2.12g) We consider the case when the spinor field depends on $r$ only. Then in view of (2.10) we have \[\imath e^{-\alpha}\bar{\gamma}^{1}\psi^{\prime}+\frac{\imath}{2} \left(\mu^{\prime}+2\beta^{\prime}\right)e^{-\alpha}\bar{\gamma}^{1}\psi+\frac {\imath}{2}\frac{\cos\theta}{\sin\theta}e^{-\beta}\bar{\gamma}^{2}\psi-m\psi-{ \cal D}\psi-\imath{\cal G}\gamma^{5}\psi = 0,\] (2.13a) \[\imath e^{-\alpha}\bar{\psi}^{\prime}\bar{\gamma}^{1}+\frac{ \imath}{2}\left(\mu^{\prime}+2\beta^{\prime}\right)e^{-\alpha}\bar{\psi}\bar{ \gamma}^{1}+\frac{\imath}{2}\frac{\cos\theta}{\sin\theta}e^{-\beta}\bar{\psi} \bar{\gamma}^{2}+m\bar{\psi}+{\cal D}\bar{\psi}+\imath{\cal G}\bar{\psi}\gamma ^{5} = 0.\] (2.13b) For the invariants of spinor field from (2.13) we find \[S^{\prime}+\left(\mu^{\prime}+2\beta^{\prime}\right)S-2e^{\alpha }{\cal G}A^{1} = 0,\] (2.14a) \[P^{\prime}+\left(\mu^{\prime}+2\beta^{\prime}\right)P+2e^{\alpha }\left(m+{\cal D}\right)A^{1} = 0,\] (2.14b) \[{A^{1}}^{\prime}+\left(\mu^{\prime}+2\beta^{\prime}\right)A^{1}+ \frac{\cos\theta}{\sin\theta}e^{\alpha-\beta}A^{2}+2e^{\alpha}\left(m+{\cal D} \right)P+2e^{\alpha}{\cal G}S = 0,\] (2.14c) \[{A^{2}}^{\prime}+\left(\mu^{\prime}+2\beta^{\prime}\right)A^{2}- \frac{\cos\theta}{\sin\theta}e^{\alpha-\beta}A^{1} = 0,\] (2.14d) where $A^{\mu}=\bar{\psi}\gamma^{5}\gamma^{\mu}\psi$ is the pseudovector. The foregoing system gives \[\left(SS^{\prime}-PP^{\prime}+A^{1}{A^{1}}^{\prime}+A^{2}{A^{2}}^ {\prime}\right)+\left(\mu^{\prime}+2\beta^{\prime}\right)\left(S^{2}-P^{2}+{A^ {1}}^{2}+{A^{2}}^{2}\right)=0,\] (2.15) with the relation \[\left(S^{2}-P^{2}+{A^{1}}^{2}+{A^{2}}^{2}\right)=C_{0}e^{-2\left( \mu+2\beta\right)},\] (2.16) On account of (2.12) we find the following system of Einstein equations \[\left(2\mu^{\prime}\beta^{\prime}+\beta^{\prime 2}\right)-e^{2( \alpha-\beta)} = -\kappa e^{2\alpha}\left(mS+F(K)\right),\] (2.17a) \[\left(\mu^{\prime 2}+\mu^{\prime}\beta^{\prime}-\mu^{\prime} \alpha^{\prime}+\beta^{\prime 2}-\beta^{\prime}\alpha^{\prime}+\mu^{\prime \prime}+\beta^{\prime\prime}\right) = -\kappa e^{2\alpha}\left(F(K)-2KF_{K}\right),\] (2.17b) \[\left(3\beta^{\prime 2}-2\beta^{\prime}\alpha^{\prime}+2\beta^{ \prime\prime}\right)-e^{2(\alpha-\beta)} = -\kappa e^{2\alpha}\left(F(K)-2KF_{K}\right),\] (2.17c) \[0 = \frac{\cos\theta}{\sin\theta}e^{(\alpha-\mu-\beta)}A^{3},\] (2.17d) \[0 = \left(\mu^{\prime}-\beta^{\prime}\right)\,e^{(\beta-\alpha-\mu)}A ^{3},\] (2.17e) \[0 = \left[\left(\mu^{\prime}-\beta^{\prime}\right)\,e^{(\beta-\alpha) }A^{2}+\frac{\cos\theta}{\sin\theta}A^{1}\right].\] (2.17f) From (2.17d) we obtain $A^{3}=0$ at least everywhere expect $\theta=\pi/2.$ Hence (2.17e) fulfills identically. Inserting $\frac{\cos\theta}{\sin\theta}e^{(\alpha-\beta)}A^{1}=-\left(\mu^{\prime}-\beta ^{\prime}\right)A^{2}$ from (2.17f) into (2.14d) for $A^{2}$ we find \[{A^{2}}^{\prime}+\left(2\mu^{\prime}+\beta^{\prime}\right)A^{2}=0,\] (2.18) with the solution \[A^{2}=z_{2}e^{-\left(2\mu+\beta\right)},\] (2.19) with $z_{2}$ being some integration constant. On account of (2.12) from Bianchi identity $\Gamma^{\nu}_{\mu;\nu}=0$ i.e., \[T^{\nu}_{\mu;\nu}=T^{\nu}_{\mu,\nu}+\Gamma^{\nu}_{\alpha\nu}T^{\alpha}_{\mu}- \Gamma^{\alpha}_{\mu\nu}T^{\nu}_{\alpha}=0\] (2.20) we find \[\left(mS+F\right)^{\prime}+\left(\mu^{\prime}+2\beta^{\prime} \right)\left(mS+2KF_{K}\right)=0.\] (2.21) Let us now consider two different cases. In case of $K=I=S^{2}$, on account of $F^{\prime}=2SF_{K}S^{\prime}$ from (2.21) we find \[\left(m+2SF_{K}\right)S^{\prime}+\left(\mu^{\prime}+2\beta^{ \prime}\right)\left(mS+2S^{2}F_{K}\right)=0,\] (2.22) which leads to \[S^{\prime}+\left(\mu^{\prime}+2\beta^{\prime}\right)S=0,\] (2.23) with \[S=c_{1}e^{-(\mu+2\beta)}.\] (2.24) In case of $K$ one of $\{J,\,I+J,\,I-J\}$ we consider the massless spinor field, as it was done in cosmology [1; 2]. Then on account of $F^{\prime}=F_{K}K^{\prime}$ we rewrite (2.21) as \[F_{K}K^{\prime}+2\left(\mu^{\prime}+2\beta^{\prime}\right)KF_{K} =0,\] (2.25) which leads to \[K^{\prime}+2\left(\mu^{\prime}+2\beta^{\prime}\right)K=0,\] (2.26) with \[K=c_{1}^{2}e^{-2(\mu+2\beta)}.\] (2.27) Thus we conclude that the relations (2.27) holds for a massless spinor field if $K$ takes one of $\{I,\,J,\,I+J,\,I-J\}$, whereas it is true for a non-trivial spinor mass, only if $K=I$. Now we can deal with the zero component of $A^{\mu}$. From Fierz identity we know \[S^{2}+P^{2}=-A_{\mu}A^{\mu}=-\left({A^{0}}^{2}+{A^{1}}^{2}+{A^{2 }}^{2}+{A^{3}}^{2}\right),\] (2.28) Subtraction of (2.28) from (2.16) in view of $A^{3}=0$ leads to \[{A^{0}}^{2}=-C_{0}e^{-2\left(\mu+2\beta\right)}-2P^{2},\] (2.29) whereas their addition yields \[{A^{0}}^{2}=C_{0}e^{-2\left(\mu+2\beta\right)}-2\left(S^{2}+{A^{1 }}^{2}+{A^{2}}^{2}\right).\] (2.30) Hence all the components of $A^{\mu}$ can be expressed in terms of metric functions. Thus the non-diagonal Einstein equations together with the equations for invariants of spinor field give us valuable information about the spinor field. On the other hand Bianchi identity relates the invariants with metric functions. Now we have only three diagonal Einstein equations left. Before dealing with diagonal Einstein equations let us go back to spinor field equations. As far as the spinor field equation (2.4a) is concerned, denoting $\phi=\psi e^{(\mu+2\beta)/2}$ it can be rewritten in the following matrix form \[\phi^{\prime}=B\phi,\] (2.31) where \[B=\left(\begin{array}[]{cc}Y_{3}\sigma^{1}+\imath Y_{1}\sigma^{3 }&Y_{2}\sigma^{1}\\ -Y_{2}\sigma^{1}&-Y_{3}\sigma^{1}+\imath Y_{1}\sigma^{3}\end{array}\right), \quad\phi={\rm col}\left(\phi_{1},\,\phi_{2},\,\phi_{3},\,\phi_{4}\right),\] \[,\] (2.32) and $Y_{1}=\left(\cos{\vartheta}/2\sin{\vartheta}\right)\exp{[\alpha-\beta]}$, $Y_{2}=\imath\left[m+{\cal D}\right]\exp{\alpha}$, and $Y_{3}={\cal G}\exp{\alpha}$. As one sees, $\det{B}=-Y_{1}^{2}+Y_{2}^{2}-Y_{3}^{2}\neq 0$. The solution to the equation (2.31) can be written in general form. Let us now go back to Einstein field equations. In order to solve these equations we need to consider some special cases. In what follows we consider a few of them. Case I Let us assume that \[\alpha=\mu+2\beta.\] (2.33) This type of assumption was consider in a number of papers [5; 6] and known as harmonic condition. Denoting $\beta^{\prime 2}+\beta^{\prime}\mu^{\prime}=U$ we rewrite the diagonal equations of Einstein system, i.e. (2.17a), (2.17b) and (2.17c) as follows: \[e^{-2\alpha}U-e^{-2\beta} = -\kappa\left(mS+F\right)\] (2.34a) \[e^{-2\alpha}\left(\mu^{\prime\prime}+\beta^{\prime\prime}-U\right) = \kappa\left(2KF_{K}-F\right)\] (2.34b) \[e^{-2\alpha}\left(2\beta^{\prime\prime}-U\right)-e^{-2\beta} = \kappa\left(2KF_{K}-F\right)\] (2.34c) Subtraction of (2.34c) from (2.34b) gives \[\mu^{\prime\prime}-\beta^{\prime\prime}+e^{2(\mu+\beta)}=0,\] (2.35) Subtraction of (2.34c) from (2.34a) gives \[\beta^{\prime\prime}-\beta^{\prime 2}-\beta^{\prime}\mu^{\prime}= \frac{\kappa}{2}e^{2(\mu+2\beta)}\left(mS+2KF_{K}\right)\] (2.36) As far as F is concerned, we may choose it in the form $F=\lambda K^{n}$, where $\lambda$ is the self-coupling constant. In case of $K=I=S^{2}$ we consider a massive spinor field, otherwise massless one. $K=I=S^{2}$ we find the following system of equations \[\mu^{\prime\prime}-\beta^{\prime\prime} = -e^{2(\mu+\beta)}\] (2.37a) \[\beta^{\prime\prime}-\beta^{\prime 2}-\beta^{\prime}\mu^{\prime} = \frac{\kappa}{2}\bigl{(}c_{1}me^{(\mu+2\beta)}+2\lambda nc_{1}^{2 n}e^{2(1-n)(\mu+2\beta)}\bigr{)}\] (2.37b) The foregoing system we solved numerically. In doing so we have give some concrete value of problem parameters as well as initial conditions. For simplicity we have chosen $\lambda=1,\,n=1,\,m=1,\,\kappa=1,\,c_{1}=1$. As initial conditions we have set $\mu(0)=1,\,\beta(0)=1,\,\mu^{\prime}(0)=0,\beta^{\prime}(0)=0$. Here our main aim was to find some solutions which we can use in our following detailed studies. In Figs. 1, 2 and 3 we have plotted the metric functions $\mu(r),\beta(r)$ and $\alpha(r)$, respectively. <figure><img src="content_image/1806.09151/mucase1n1.jpg"><figcaption>Figure 1: Behavior of μ(r) for λ=1,n=1,m=1,κ=1,c1=1 with the initialconditions μ(0)=1,β(0)=1,μ′(0)=0,β′(0)=0</figcaption></figure> <figure><img src="content_image/1806.09151/betacase1n1.jpg"><figcaption>Figure 2: Behavior of β(r) for λ=1,n=1,m=1,κ=1,c1=1 with the initialconditions μ(0)=1,β(0)=1,μ′(0)=0,β′(0)=0</figcaption></figure> <figure><img src="content_image/1806.09151/alphacase1n1.jpg"><figcaption>Figure 3: Behavior of α(r) for λ=1,n=1,m=1,κ=1,c1=1 with the initialconditions μ(0)=1,β(0)=1,μ′(0)=0,β′(0)=0</figcaption></figure> Case II As a second case we consider the widely used model setting $\beta=\ln r$. In this case the diagonal components of Einstein system takes the form \[e^{-2\alpha}\left(2\frac{\mu^{\prime}}{r}+\frac{1}{r^{2}}\right) -\frac{1}{r^{2}} = -\kappa\left(mS+F\right)\] (2.38a) \[e^{-2\alpha}\left(\mu^{\prime 2}+\frac{\mu^{\prime}}{r}-\mu^{ \prime}\alpha^{\prime}-\frac{\alpha^{\prime}}{r}+\mu^{\prime\prime}\right) = \kappa\left(2KF_{K}-F\right)\] (2.38b) \[e^{-2\alpha}\left(\frac{1}{r^{2}}-2\frac{\alpha^{\prime}}{r} \right)-\frac{1}{r^{2}} = \kappa\left(2KF_{K}-F\right)\] (2.38c) Subtracting (2.38c) from (2.38b) one finds \[e^{-2\alpha}\left(\mu^{\prime}+\alpha^{\prime}\right)=-\frac{ \kappa r}{2}\left(mS+2KF_{K}\right).\] (2.39) Inserting (2.39) into (2.38b) we find \[e^{-2\alpha}\left(2\mu^{\prime 2}+2\frac{\mu^{\prime}}{r}+\mu^{ \prime\prime}\right) = \kappa\left(2KF_{K}-F\right)-\frac{\kappa r}{2}\left(\mu^{\prime} +\frac{1}{r}\right)\left(mS+2KF_{K}\right).\] (2.40) It should be noted that in this case $S=(c_{1}/r^{2})e^{-\mu}$. As far as nonlinear term is concerned, as in previous case we consider the massive spinor field with $F=\lambda I^{n}=\lambda S^{2n}$. Inserting $F=\lambda I^{n}=\lambda S^{2n}$ into the equations finally we find the following system \[e^{-2\alpha}\left(\mu^{\prime}+\alpha^{\prime}\right) = -\frac{\kappa r}{2}\left(\frac{mc_{1}}{r^{2}}e^{-\mu}+\frac{2 \lambda nc_{1}^{2n}}{r^{4n}}e^{-2n\mu}\right)\] (2.41a) \[e^{-2\alpha}\left(2\mu^{\prime 2}+2\frac{\mu^{\prime}}{r}+\mu^{ \prime\prime}\right) = \kappa\left(\frac{(2n-1)\lambda c_{1}^{2n}}{r^{2n}}e^{-2\mu}\right)\] (2.41b) \[- \frac{\kappa r}{2}\left(\mu^{\prime}+\frac{1}{r}\right)\left( \frac{mc_{1}}{r^{2}}e^{-\mu}+\frac{2\lambda nc_{1}^{2n}}{r^{4n}}e^{-2n\mu}\right)\] This system can also be solved numerically to find $\mu$ and $\alpha$. Since in this case the point $r=0$ leads to singularity, we have to set the initial value at any point except this one. Like in the previous we consider the same problem parameters with the following initial conditions: $\alpha(0.1)=0.3,\,\mu(0.1)=0.5,$ and $\mu^{\prime}(0.1)=0.2$. The behavior of $\mu(r)$ and $\alpha(r)$ are given in the Figs. 4 and 5, respectively. It should be noted that this case was studied in [7] <figure><img src="content_image/1806.09151/mucase2n1.jpg"><figcaption>Figure 4: Behavior of μ(r) for λ=1,n=1,m=1,κ=1,c1=1 with the initialconditions μ(0.1)=0.5,α(0.1)=1,μ′(0.1)=0.2</figcaption></figure> <figure><img src="content_image/1806.09151/alphacase2n1.jpg"><figcaption>Figure 5: Behavior of α(r) for λ=1,n=1,m=1,κ=1,c1=1 with the initialconditions μ(0.1)=0.5,α(0.1)=1,μ′(0.1)=0.2</figcaption></figure> ## 3 Conclusion Within the scope of spherically symmetric gravitational the the role of nonlinear spinor field in the formation of different configuration is studied. Earlier it was found that the spinor field play a very important rope in the evolution of the Universe. This study can be taken as an attempt to exploit the spinor field in astrophysics. We hope to use these experience and results for the well studied astrophysical objects such as compact stars, black holes, wormholes etc. We plan to discus some aspects of modern astrophysics exploiting the spinor description of matter in our coming papers. ## Acknowledgments This work is supported in part by a joint Romanian-LIT, JINR, Dubna Research Project, theme no. 05-6-1119-2014/2018. ## References * (1) B. Saha and G.N. Shikin, Gen. Relat. Grav. 29, 1099 (1997) * (2) Bijan Saha, Phys. Rev. D 64, 123501 (2001) * (3) Bijan Saha, Phys. Rev. D 74, 124030 (2006) * (4) Bijan Saha, Phys. Part. Nucl. 49, 146 (2018) * (5) K.A. Bronnikov, In _Problems of Theory of Gravity and Elementary Particles_10, 37 Energoizdat Moscow (1979) * (6)Yu.P. Rybakov, G.N. Shikin and B. Saha, Int. J. Theor. Phys. 36, 1475 (1997) * (7) V.G. Krechet and I.V Sinilshikova, Izvestia VUZob, F izika, 57(7), 10 (2014)
1804.00814	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 30420, "num_imgs": 4, "llama3_tokens_count": 8431 }	[ "content_image/1804.00814/x1.png", "content_image/1804.00814/x2.png", "content_image/1804.00814/x3.png", "content_image/1804.00814/x4.png" ]	# Experimental signature of collective enhancement in nuclear level density Deepak Pandit deepak.pandit@vecc.gov.in Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India Srijit Bhattacharya Department of Physics, Barasat Govt. College, Barasat, N 24 Pgs, Kolkata - 700124, India Debasish Mondal Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India Pratap Roy Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India K Banerjee Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India S. Mukhopadhyay Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India Surajit Pal Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India A. De Department of Physics, Raniganj Girls’ College, Raniganj - 713358, India Balaram Dey Saha Institute of Nuclear Physics, 1/AF-Bidhannagar, Kolkata - 700064, India S. R. Banerjee (Ex)Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata - 700064, India February 26, 2024 ###### Abstract We present a probable experimental signature of collective enhancement in the nuclear level density (NLD) by measuring the neutron and the giant dipole resonance (GDR) $\gamma$ rays emitted from the rare earth ${}^{169}$Tm compound nucleus populated at 26.1 MeV excitation energy. An enhanced yield is observed in both neutron and $\gamma$ ray spectra corresponding to the same excitation energy in the daughter nuclei. The enhancement could only be reproduced by including a collective enhancement factor in the Fermi gas model of NLD to explain the neutron and GDR spectra simultaneously. The experimental results show that the relative enhancement factor is of the order of 10 and the fadeout occurs at $\sim$ 14 MeV excitation energy, much before the commonly accepted transition from deformed to spherical shape. We also explain how the collective enhancement contribution changes the inverse level density parameter ($k$) from 8 to 9.5 MeV observed recently in several deformed nuclei. pacs: 24.30.Cz,24.10.Pa,25.70.Gh, 21.10.Ma The atom, consisting of a tiny nucleus of protons and neutrons surrounded by a cloud of electrons, is responsible for nearly all the properties of matter that have shaped the world around us. Although, the atomic properties are governed by the electronic structure, its existence is decided by the nucleus. It is a complex quantal system which is held together by the strong nuclear force. The nucleus attains variety of configurations even if a small excitation energy is provided to it. The density of nuclear levels increases rapidly with increasing excitation energy [1; 2]. Thus, statistical models are not only appropriate but essential for the comprehension and prediction of different nuclear decays at moderate and high excitation energies. One of the important ingredients of the statistical model is the nuclear level density (NLD) which is defined as the number of excited levels per unit excitation energy. The NLD has important contribution in the calculations of explosive nuclear burning in astrophysical environments such as nuclear reaction rates in nucleosynthesis and reliable estimates of nuclear abundance [3; 4] as well as in nuclear fission [5], multifragmentation [6] and spallation reactions [7]. It also provides important information about the nuclear thermodynamic properties such as temperature (T), entropy and heat capacity [8]. The NLD is extracted experimentally from counting the levels, neutron resonance studies [9], Oslo technique [10], two-step cascade method [11], beta-Oslo method [12], $\gamma$-ray calorimetry [13] and particle evaporation spectra [14]. Theoretically, it has been characterized by phenomenological analytical expressions [1; 15; 16] as well as calculations based on different microscopic approaches [17; 18; 19; 20]. Apart from the intrinsic excitation, the nucleus also displays collective vibrational and rotational motion analogous to atomic and molecular physics. These collective degrees of freedom introduce new levels up to moderate excitation energies, and their contribution is described as collective enhancement factor in the NLD. The contribution of collectivity in the NLD $\rho(E^{},J)$ at excitation energy $E^{}$ and angular momentum $J$ is expressed phenomenologically [15] as, \[\rho(E^{},J)=\rho_{\textrm{\scriptsize{int}}}(E^{},J)K_{\textrm{\scriptsize {coll}}}\] (1) where $\rho_{\textrm{\scriptsize{int}}}(E^{},J)$ is the intrinsic single particle level density and $K_{\textrm{\scriptsize{coll}}}$ is the collective enhancement factor. Although, the NLD is indispensible in the study of nuclear decay, the collective enhancement in the NLD is still not a well understood topic due to the lack of experimental data. The magnitude and exact form of $K_{\textrm{\scriptsize{coll}}}$ still remains an open question. Several expressions for $K_{\textrm{\scriptsize{coll}}}$ exist in literature where the degree of enhancement varies from 10 to 100 [19; 21; 22; 23]. On the other hand, the earlier experimental studies have produced contradictory results on the collective enhancement and its fadeout [23; 24]. Quite recently, our extensive studies on neutron evaporation from several deformed nuclei have established the fact that the fadeout of collectivity is related to the nuclear shape phase transition and occurs at an excitation energy in the region of 14 - 21 MeV [25; 26]. While, a sharp change in the value of inverse level density parameter ($k$), within the initial compound nuclear excitation energy interval of 32-37 MeV, is observed for all the deformed nuclei (${}^{169}$Tm, ${}^{173}$Lu, ${}^{185}$Re), a weak effect is observed for the near spherical ${}^{201}$Tl nucleus [26]. Therefore, if there is an enhancement and its fadeout is in the region 14 - 21 MeV, then that should be directly evident in both neutron and giant dipole resonance (GDR) $\gamma$ decay spectra from the highly deformed rare-earth nuclei. The GDR is another collective mode of excitation of the nuclei which can be understood macroscopically as the out-of-phase oscillation between the protons and neutrons [27; 28]. Microscopically, it is conceived as the coherent superposition of particle-hole excitations. It is an indispensable tool in nuclear structure physics and has been utilized recently to determine the ratio of shear viscosity ($\eta$) to entropy density (s) of finite nuclear matter [29]. The GDR $\gamma$ emission occurs early in the decay of excited nuclei and also couples directly with the nuclear shape degrees of freedom. Thus, the investigation of its strength distribution should provide information about nuclear deformation and any enhanced yield will present an experimental signature of the collective enhancement in the NLD. The experiment was performed using the alpha beams (E${}_{Lab}$ = 28 MeV) from the K-130 cyclotron at the Variable Energy Cyclotron Centre, Kolkata by bombarding ${}^{165}$Ho target. The compound nucleus ${}^{169}$Tm (ground state deformation $\beta$$\sim$ 0.3 [30]) was populated at 26.1 MeV excitation energy. The critical angular momentum for the reaction was 11$\hbar$. The high-energy GDR $\gamma$ rays were detected at 90${}^{\circ}$ and 125${}^{\circ}$ with respect to the incident beam direction by employing the LAMBDA spectrometer [31], arranged in a 7x7 matrix, at a distance of 50 cm. The time-of-flight (TOF) technique was employed to discriminate the neutrons from the high-energy $\gamma$ rays. The pulse shape discrimination (PSD) technique was adopted to reject pile-up events in the individual detector elements by measuring the charge deposition over two time intervals (30 ns and 2 $\mu$s). However, the pile-up events are very few due to high granularity of the detector array LAMBDA [31]. The 50-element low-energy $\gamma$ multiplicity filter [32] was used (in coincidence with the high-energy $\gamma$ rays) to estimate the angular momentum populated in the compound nucleus in an event-by-event mode as well as to get the fast start trigger for the TOF measurements. The filter was split into two blocks of 25 detectors each of which were placed on the top and bottom of a specially designed scattering chamber at a distance of 4.5 cm from the target in staggered castle type geometry. The neutron evaporation spectra were measured using two 5${}^{\prime\prime}$ x 5${}^{\prime\prime}$ liquid-scintillator (BC501A) [33] detectors (in coincidence with the multiplicity filter) placed outside the scattering chamber at 120${}^{\circ}$ and 150${}^{\circ}$ with respect to the beam direction and at a distance of 150 cm from the target. The energy of the emitted neutrons was measured using the TOF technique whereas the neutron-$\gamma$ discrimination was achieved by both PSD and TOF. The time resolution of the neutron detectors was typically about 1.2 ns which give an energy resolution of about 0.9 MeV at 10 MeV for the present setup. To keep the background at a minimum level the beam dump was kept at 3 m away from the target and was well shielded with layers of lead and borated paraffin. The schematic view of the experimental setup is shown in Fig. 1. The details of the GDR [34; 35; 36; 37] and the neutron analyses [25; 26; 38] have already been discussed in our earlier papers. <figure><img src="content_image/1804.00814/x1.png"><figcaption>Figure 1: (color online) Schematic view of the experimental setup.</figcaption></figure> <figure><img src="content_image/1804.00814/x2.png"><figcaption>Figure 2: (color online) (a) The experimental neutron spectra measured at twoangles are compared with each other. (b) The experimental neutron spectrum iscompared with the CASCADE calculation. (c) The experimental γ-spectra measuredat two angles are compared with each other. (d) The experimental γ-spectrum iscompared with the CASCADE calculation plus bremsstrahlung component. Theenhancement in the spectra and the contribution from different nuclei areshown with arrows.</figcaption></figure> The neutron and the high-energy $\gamma$ ray spectra, each measured at two different angles, are shown in Fig. 2 (a) and (c), respectively. As can be seen, the two spectra almost overlap with each other, which indicates that they have originated from an equilibrated compound nucleus. Most noteworthy is the large yield in both neutron energy spectrum (beyond 6 MeV) and GDR $\gamma$ ray (around 16 MeV) spectrum. This high energy GDR $\gamma$ ray at 16 MeV can only arise from fully energy equilibrated compound nucleus since the nonfusion events are accompanied by $\gamma$ rays less than 10 MeV [39]. It is also interesting to note that the GDR and the neutron decay explore the same excitation energy region in the daughter nuclei ${}^{169}$Tm and ${}^{168}$Tm, respectively. In order to explain the experimental data, the neutron energy and the high-energy $\gamma$ ray spectra were calculated employing a modified version of statistical model code CASCADE [40; 41]. The shape of the particle spectra depends on the transmission coefficients of outgoing particles and the NLD of the residual nucleus. The transmission coefficients for statistical model calculation were obtained from the optical model where the potential parameters for neutron, proton and $\alpha$ were taken from Refs. [42], [43] and [44], respectively. The experimental fold distribution measured using the 50-element $\gamma$-multiplicity filter was converted to the spin distribution through comparison with a GEANT simulation and was used as input for the calculation [32]. The intrinsic level density used in the modified version of CASCADE code is based on the Fermi gas model [1] given as \[\rho_{\textrm{\scriptsize{int}}}(E^{},J)=\frac{2J+1}{12\theta^{3/2}}\sqrt{a} \frac{\exp{(2\sqrt{aU}})}{U^{2}}.\] (2) Here $U$ = $E^{}$ - $\frac{J(J+1)}{2I_{\textrm{\scriptsize{eff}}}}$ - $\Delta_{\textrm{\scriptsize p}}$ is the available thermal energy. $\frac{J(J+1)}{2I_{\textrm{\scriptsize{eff}}}}$ is the energy bound in rotation and $\theta$ = $\frac{2I_{\textrm{\scriptsize{eff}}}}{\hbar^{2}}$, where $I_{\textrm{\scriptsize{eff}}}$ is the effective moment of inertia. The excitation energy is shifted back by the pairing energy $\Delta_{\textrm{\scriptsize p}}$ which is calculated using the relation $\Delta_{\textrm{\scriptsize p}}$ = $\frac{12}{\sqrt{A}}$. The NLD parameter $a$ is related to the single-particle density of states at the Fermi energy. The prescription of Ignatyuk [45] was used for the level density parameter which is given as $a$=$\tilde{a}$[1+($\Delta$S/$U$)(1-exp(-$\gamma$$U$))] where $\tilde{a}$ = $A/k$, $\Delta$S is the shell correction and $\gamma$ is the shell damping factor. This parametrization takes into account the nuclear shell effects at low excitation energy and connects smoothly to the liquid drop value at high excitation energy and found to explain the GDR data well [41]. However, the shell correction factors for Tm isotopes are very small and less than 1.0 MeV [30]. It was observed that the variation in the transmission coefficients (using different prescriptions) and the deformation parameters were inconsequential. The shape of the neutron energy spectrum was determined by the inverse level density parameter only. Similarly, the $\gamma$ spectrum also depended only on the level density and the GDR parameters. However, it was not possible to explain the enhanced yield obtained in both neutron and $\gamma$ spectra by changing the $k$ value and the GDR parameters even after taking into account the shell and pairing effects in level density. Therefore, in order to explain the experimental data, the intrinsic NLD (Eq. 2) was multiplied by an energy dependent empirical enhancement factor parameterized as \[K_{\textrm{\scriptsize{coll}}}=1+Cexp{[-(U-E_{\textrm{\scriptsize{cr}}})^{2}/ 2\sigma^{2}]}.\] (3) where $C$, $E_{\textrm{\scriptsize{cr}}}$ and $\sigma$ are the magnitude, peak and width of the enhancement factor, respectively. At 26 MeV excitation energy, the neutron spectrum has contribution from 1n (${}^{168}$Tm) and 2n (${}^{167}$Tm) decay channels. But, the higher part of the spectrum ($\geq$ 5 MeV) is totally dominated from the first step decay. Hence, the neutron spectrum was analysed by including the enhancement factor in the NLD of ${}^{168}$Tm nucleus. The extracted parameters for ${}^{168}$Tm nucleus were $k$ = 8.0 $\pm$ 0.4 MeV, $C$ = 7 $\pm$ 2, $E_{\textrm{\scriptsize{cr}}}$ = 8.3 $\pm$ 0.5 MeV and $\sigma^{2}$ = 1.0 $\pm$ 0.3 MeV${}^{2}$. Next, the same parameters were used to explain the $\gamma$ spectra. As can be seen, the $\gamma$ spectrum could be explained below 14 MeV but it was not possible to explain the large yield at 16 MeV by varying the strength of the GDR component (red dotted line in Fig. 2d). Therefore, an enhancement was also included in the NLD of ${}^{169}$Tm nucleus as the high-energy GDR decay will be dominant from the first stage of the compound nuclear decay. The extracted parameters for ${}^{169}$Tm nucleus were $C$ = 11 $\pm$ 3, $E_{\textrm{\scriptsize{cr}}}$ = 9.0 $\pm$ 0.5 MeV and $\sigma^{2}$ = 1.0 $\pm$ 0.3 MeV${}^{2}$. The extracted GDR centroid energy ($E_{\textrm{\tiny{GDR}}}$), width ($\Gamma_{\textrm{\tiny{GDR}}}$) and strength ($S_{\textrm{\tiny{GDR}}}$) were $E_{\textrm{\tiny{GDR1}}}$ = 12.1 $\pm$ 0.4 MeV, $\Gamma_{\textrm{\tiny{GDR1}}}$ = 3.3 $\pm$ 0.6 MeV, $S_{\textrm{\tiny{GDR1}}}$ = 0.3 $\pm$ 0.04, $E_{\textrm{\tiny{GDR2}}}$ = 16.0 $\pm$ 0.5 MeV, $\Gamma_{\textrm{\tiny{GDR2}}}$ = 4.1 $\pm$ 0.7 MeV, $S_{\textrm{\tiny{GDR2}}}$ = 0.72 $\pm$ 0.05. We emphasize here that the extracted GDR centroid energies are very similar to the ground state values of ${}^{165}$Ho (12.2 and 15.8 MeV) measured by livermore group [46; 47] and also to those extracted for ${}^{166}$Er nuclei (having same deformation) at slightly higher temperature [48]. Our result supports the predictions of Brink-Axel hypothesis. The estimated deformation from the two GDR peaks is $\beta$ = 0.32 similar to the ground state deformation of Tm nuclei [30]. The bremsstrahlung component, as measured and observed in our earlier experiments at similar beam energy [36; 49], was parameterized by an exponential function (e${}^{-E_{\gamma}/E_{0}}$) where the slope parameter E${}_{0}$ was chosen according to the bremsstrahlung systematics [50]. Interestingly, the enhancement factor used for ${}^{168}$Tm to describe the neutron spectrum, simultaneously explains the $\gamma$ spectrum between E${}_{\gamma}$ = 7 and 11 MeV. This enhancement occurs due to the folding of the low energy tail of the 12.1 MeV GDR component with the enhanced level density region after the decay of one neutron populating ${}^{168}$Tm. Thus, almost similar enhancement was required in the level density of both ${}^{168}$Tm and ${}^{169}$Tm to simultaneously explain the neutron and the GDR spectra. It is also very interesting to note that no such enhancement in the $\gamma$ spectra was observed in our earlier experiments at similar excitation energies for near spherical nuclei ${}^{97}$Tc [36], ${}^{119}$Sb[49], and ${}^{201}$Tl[35]. It needs to be mentioned here that a similar enhancement in NLD was observed in the proton decay from ${}^{104}$Pd but at much lower effective excitation energy (below 6 MeV) [51]. The enhancement was explained considering pairing re-entrance at high angular momentum [52]. However, the pairing effect does not seem to be the plausible reason for the enhancement in our case, as it has been found to play an important role only below 6 MeV excitation energy and dominant in even-even nuclei [8; 19; 52]. Therefore, the enhancement in NLD for both ${}^{168}$Tm and ${}^{169}$Tm at similar excitation energy primarily appears to be due to the collective enhancement owing to large deformation of Tm nuclei (also observed experimentally via GDR). <figure><img src="content_image/1804.00814/x3.png"><figcaption>Figure 3: (color online) The symbols represent the neutron spectra (noenhancement in NLD) as calculated from CASCADE with k = 9.5 MeV for thereaction 4He(E\scriptsize{Lab}=40 MeV) + 165Ho studied earlier prat13 . Thesame calculation (continuous lines) but with k = 8.0 MeV and (a) no enhancmentin NLD of any nuclei and (b) enhancement in NLD of all the three nuclei.</figcaption></figure> <figure><img src="content_image/1804.00814/x4.png"><figcaption>Figure 4: (color online) The enhanced level densities of 169Tm (a) and 168Tm(b) at 11ℏ and 167Tm (c) at 16ℏ are shown as used in the CASCADE. The Fermigas level density is also displayed for comparison. The level densities arenot in absolute scale as they are not normalised to experimental data. Thelevel density of 170Yb is compared with 169Tm. (d) The relative enhancementfactors extracted as a function of excitation energy for three nuclei.</figcaption></figure> Recently, a sudden change in the value of $k$ from 8 to 9.5 MeV was obtained for several deformed nuclei indicating the appearance and fadeout of collectivity [25; 26]. We illustrate how the collective enhancement is manifested through the neutron evaporation spectra when populated in the excitation energy range 32 - 37 MeV. A statistical model calculation with $k$ = 9.5 MeV for the reaction ${}^{4}$He (E${}_{\textrm{\scriptsize{Lab}}}$ = 40 MeV) + ${}^{165}$Ho (performed earlier [25]) is shown in Fig. 3 (symbols). The same calculation with $k$ = 8.0 MeV (continuous line) is also displayed along with the contributions from different decay steps. As expected, the two calculations are completely different in the higher energy region. However, the two spectrum match very well when collective enhancement is included in the calculation. As can be seen from Fig. 3b, the 1n channel does not see the enhanced region of ${}^{168}$Tm and is unaffected. Interestingly, the cross section of the 2n channel in the higher energy region increases since it probes the enhanced level density region. Thus, the enhancement factor of ${}^{167}$Tm was extracted by fitting the spectra of $k$ = 9.5 MeV (enhancement not included) with $k$ = 8.0 MeV and the enhancement factor. The extracted parameters are $C$ = 9 $\pm$ 3, $E_{\textrm{\scriptsize{cr}}}$ = 11 $\pm$ 1 MeV and $\sigma^{2}$ = 1.0 $\pm$ 0.3 MeV${}^{2}$. It was not possible to extract the parameters of 3n channel decay populating ${}^{166}$Tm since its contribution was very small (Fig. 3) and same enhancement parameter as ${}^{167}$Tm was used for the calculation (E${}_{\textrm{\scriptsize{Lab}}}$ = 40 MeV). Thus, the sudden change in the value of $k$ observed in the experiments for deformed nuclei is due to the enhanced cross section of the 2n channel which changes the slope of the neutron spectra. This is compensated in the statistical calculations by changing the $k$ value when enhancement factor is not included. The slope of the neutron spectra is mostly decided by the 1n and 2n decay steps. Therefore, at further higher exciation energy, the first two steps do not see the enhanced level density region and thus, no signature of collective enhancement is observed in the neutron spectra at higher energies [26]. The enhanced level densities, for different Tm nuclei, used in the statistical model calculation are shown in Fig. 4. An indication of such enhancement in the level density beyond 7 MeV was also seen for ${}^{170}$Yb [53] obtained by the Oslo technique and is compared with ${}^{169}$Tm in Fig. 4a. The collective enhancement factors are also displayed independently in Fig. 4d (on a logarithmic scale) with excitation energy. Since, neutron evaporation and $\gamma$ ray emission in the statistical model is decided by the ratio of the level density of the daughter nucleus after particle/$\gamma$ ray emission to the compound nucleus, the observed $K_{\textrm{\scriptsize{coll}}}$ are relative collective enhancement factors. The magnitudes of $K_{\textrm{\scriptsize{coll}}}$ are similar to the microscopic shell model Monte Carlo calculations for ${}^{154}$Sm nucleus having similar deformation ($\beta\sim$ 0.27) [19]. It is also consistent with the prediction in terms of state density of nucleus and its redistribution [54]. The enhancement region for all the three Tm isotopes (having similar ground state deformation [30]) is almost same and the collectivity fades away beyond 14 MeV corresponding to the temperature T = 0.82 MeV (Fig. 4d). Interestingly, the deformation is observed directly via the splitting of the GDR strength but the enhanced yield is obtained only for the 16 MeV GDR component. This clearly points that the fadeout of the enhancement is indeed around 14 MeV excitation energy, else an enhancement in 12 MeV GDR component should also have been prominent. This is also corroborated by the neutron spectrum (Fig. 2c) where the enhancement is observed beyond 6 MeV which corresponds to 12 MeV excitation energy. Intriguingly, the result also suggests that the fadeout of the collective enhancement occurs much before the nuclear shape transition from deformed to spherical as predicted by theoretical calculations [19] and phenomenological estimations [22; 23]. One of the reasons for this behavior could be the thermal shape fluctuations ($\Delta\beta$) which increase with the increase in T, as explained earlier for the same fadeout zone for different deformations [26]. The calculations showed that the nuclear deformation persists at the ground state value up to T $\sim$ 0.8 MeV and then starts the gradual shape change and becomes spherical at T $\sim$ 1.7 MeV. Thus, at around T = 0.8 MeV, the ground state deformation starts to decrease and the thermal fluctuations become large ($\Delta\beta$/$\beta$ = 0.25). This convolutes the static ground state deformation which could lead to the loss of collectivity. Microscopically, the origin of this enhancement in NLD does not come from the levels or states created by deformation. It appears due to the rearrangement of the levels owing to deformation from higher energy to lower energy which are in the original basis [54]. Hence, when the ground state deformation changes slightly (T $\sim$ 0.8 MeV) and the role of thermal fluctuations becomes large, the energy levels may once again be redistributed leading to the decrease in levels at that particular E which will appear as loss of collectivity even in the presence of large deformation. However, further experimental and theoretical insights are required to understand the details of such a unique behavior. In deformed nucleus the enhancement also depends on J and K apart from U [18], and further work is required to see their influence on the enhancement factor. In summary, we present an experimental evidence of collective enhancement in NLD by measuring the GDR $\gamma$ rays and neutron decays from Tm nuclei. The relative enhancement factors estimated from the simultaneous analysis of GDR and neutron decays for all the three Tm isotopes are of the order of 10. Our technique only measures the apparent change in the enhancement factor and is not sensitive to the magnitude of the vibrational enhancement factor, unless it changes with energy. The experimental result also shows that the collective enhancement fades away beyond 14 MeV excitation energy which is much before the nuclear shape transition from deformed one to spherical predicted by theoretical models. The authors are thankful to VECC cyclotron operators for smooth running of the accelerator during the experiments. ## References * (1) H. A. Bethe, Phys. Rev. 50, 332 (1936). * (2) A. Bohr, B. R. Mottelson, Nuclear Structure (Benjamin, New York,1969), Vol.1,2. * (3) T. Rauscher and F. K. Thielemann, At. Data Nucl. Data Tables 75, 1 (2000). * (4) T. Rauscher, F. K. Thielemann, and K. L. Kratz, Phys. Rev. C 56, 1613 (1997). * (5) R. Capote _et al._, Nucl. Data Sheets 110, 3107 (2009). * (6) J.P. Bondorf _et al._, Nucl. Phys. A 443, 321 (1985). * (7) Davide Mancusi, Robert J. Charity and Joseph Cugnon, Phys. Rev. C 82, 044610 (2010). * (8) E. Melby _et al._, Phys. Rev. Lett 83, 3150 (1999). * (9) J. R. Huizenga and L. G. Moretto, Ann. Rev. Nucl. Sci. 22 427 (1972). * (10) A. Schiller _et al._, Nucl. Instrum. Methods Phys. Res. A 447, 498 (2000). * (11) F. Becvar _et al._, Phys. Rev. C 46, 1276 (1992). * (12) A. Spyrou _et al._, Phys. Rev. Lett 113, 232502 (2014). * (13) J. L. Ullman _et al._, Phys. Rev. C 87, 044607 (2013). * (14) A. V. Voinov _et al._, Phys. Rev. C 74, 014314 (2006). * (15) A. V. Ignatyuk, K.K. Istekov, G.N. Smirenkin, Sov. J. Nucl. Phys. 29, 450 (1979). * (16) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43, 1446 (1965). * (17) S. Goriely, S. Hilaire, and A. J. Koning, Phys. Rev. C 78, 064307 (2008). * (18) S. M. Grimes, Phys. Rev. C 88, 024613 (2013). * (19) C. Ozen, Y. Alhassid, and H. Nakada, Phys. Rev. Lett 110, 042502 (2013). * (20) N. Quang Hung, N. Dinh Dang and L. T. Quynh Huong, Phys. Rev. Lett 118, 022502 (2017). * (21) A. J. Koning, S. Hilaire, S. Goriely, Nucl. Phys. A 810, 13 (2008). * (22) G. Hansen and A.S. Jensen, Nucl. Phys. A 406, 236 (1983). * (23) A. R. Junghans _et al._, Nucl. Phys. A 629, 635 (1998). * (24) S. Komarov _et al._, Phys. Rev. C 75, 064611 (2007). * (25) Pratap Roy _et al._, Phys. Rev. C 88, 031601(R) (2013). * (26) K. Banerjee _et al._, Phys. Lett. B 772, 105 (2017). * (27) M. N. Harakeh and A. van der Woude, Giant Resonances, Fundamental High-frequency Modes of Nuclear Excitation, Clarendon Press, Oxford, (2001). * (28) J. J. Gaardhoje, Ann. Rev. Nucl. Part. Sci. 42 483 (1992). * (29) Debasish Mondal _et al._, Phys. Rev. Lett. 118, 192501 (2017). * (30) P. Moller _et al._, At. Data Nucl. Data Tables 59, 185 (1995) . * (31) S. Mukhopadhyay _et al._, Nucl. Instr. and Meth. A 582, 603 (2007). * (32) Deepak Pandit _et al._, Nucl. Instr. and Meth. A 624, 148 (2010). * (33) K. Banerjee _et al._, Nucl. Instr. and Meth. A 608, 440 (2009). * (34) Deepak Pandit _et al._, Phys. Lett. B 690, 473 (2010). * (35) Deepak Pandit _et al._, Phys. Lett. B 713, 434 (2012). * (36) Balaram Dey _et al._, Phys. Lett. B 731, 92 (2014). * (37) Debasish Mondal _et al._, Phys. Lett. B 763, 422 (2016). * (38) K. Banerjee _et al._, Phys. Rev. C 85, 064310 (2012). * (39) D. R. Chakrabarty, _et al._, Phys. Rev. C 85, 044619 (2012). * (40) Srijit Bhattacharya _et al._, Phys. Rev.C 78, 064601 (2008). * (41) Srijit Bhattacharya _et al._, Phys. Rev.C 90, 054319 (2014). * (42) C. M. Perey and F. G. Perey, At. Data Nucl. Data Tables 17, 1 (1976). * (43) F. G. Perey, Phys. Rev. 131, 745 (1963). * (44) L. Mcfadden and G. R. Satchler, Nucl. Phys. A 84, 177 (1966). * (45) A.V.Ignatyuk, G.N.Smirenkin, A.S.Tishin, Sov. J. Nucl. Phys. 21, 255 (1975), Yad.Fiz. 21, 485 (1975). * (46) B. L. Berman and S. C. Fultz, Reviews of Modern Physics 47, 713 (1975). * (47) A. V. Varlamov, V. V. Varlamov, D. S. Rudenko, and M. E. Stepanov, Atlas of Giant Dipole Resonances, INDC(NDS)-394, 1999 (unpublished); JANIS database. * (48) C. A. Gossett _et al._, Phys. Rev. Lett. 54, 1486 (1985). * (49) S. Mukhopadhyay et. al., Phys. Lett. B 709, 9 (2012). * (50) H. Nifennecker and J. A. Pinston, Annu. Rev. Nucl. Part.Sci. 40, 113 (1990). * (51) A. Mitra _et al._, Nucl. Phys. A 765, 277 (2006). * (52) N. Quang Hung _et al._, Acta Physica Polonica B Proceedings Supplement Vol8, 551 (2015). * (53) U. Agvaanluvsan_et al._, Phys. Rev. C 70, 054611 (2004). * (54) S. M. Grimes, Phys. Rev. C 78, 057601 (2008).
1409.3693	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 44590, "num_imgs": 6, "llama3_tokens_count": 14433 }	[ "content_image/1409.3693/x1.png", "content_image/1409.3693/x2.png", "content_image/1409.3693/x3.png", "content_image/1409.3693/x4.png", "content_image/1409.3693/x5.png", "content_image/1409.3693/x6.png" ]	# Estimation of the Extragalactic Background Light using TeV Observations of BL Lacs Atreyee Sinha¹ , S. Sahayanathan² , R. Misra³ , S. Godambe² , B. S. Acharya¹ [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] ###### Abstract The very high energy (VHE) gamma ray spectral index of high energy peaked blazars correlates strongly with its corresponding redshift whereas no such correlation is observed in the X-ray or the GeV bands. We attribute this correlation to a result of photon-photon absorption of TeV photons with the extragalactic background light (EBL) and utilizing this, we compute the allowed flux range for the EBL, which is independent of previous estimates. The observed VHE spectrum of the sources in our sample can be well approximated by a power-law, and if the de-absorbed spectrum is also assumed to be a power law, then we show that the spectral shape of EBL will be $\epsilon n(\epsilon)\sim klog(\frac{\epsilon}{\epsilon_{p}})$. We estimate the range of values for the parameters defining the EBL spectrum, $k$ and $\epsilon_{p}$, such that the correlation of the intrinsic VHE spectrum with redshift is nullified. The estimated EBL depends only on the observed correlation and the assumption of a power law source spectrum. Specifically, it does not depend on the spectral modeling or radiative mechanism of the sources, nor does it depend on any theoretical shape of the EBL spectrum obtained through cosmological calculations. The estimated EBL spectrum is consistent with the upper and lower limits imposed by different observations. Moreover, it also agrees closely with the theoretical estimates obtained through cosmological evolution models. galaxies: intergalactic medium, BL Lacertae objects: general, cosmology:cosmic background radiation, infrared: diffuse background ## 1 Introduction The extragalactic background light (EBL) is an isotropic diffuse radiation field extending from Ultraviolet (UV) to Infrared (IR) wavelength ($\lambda=0.1\,-\,1000\mu m$). It is the relic radiation containing information about the structure formation epoch of the universe and hence, is an important cosmological quantity (Dwek & Krennrich, 2013; De Angelis et al., 2013). The main contributors of the EBL spectrum are the stellar emission (peaking at optical-UV) and the dust emission (peaking at IR). Direct measurement of EBL is very difficult due to strong foreground contamination by the Galactic and zodiacal light, and depends on the choice of the zodiacal light models (Kelsall et al., 1998; Wright, 1998). However, different upper and lower limits on EBL, based on various observations and deep galaxy number counts, have been put forth (Hauser et al., 1998; Madau & Pozzetti, 2000; Domínguez et al., 2011; Helgason & Kashlinsky, 2012). Theoretical prediction of the spectral energy distribution (SED) of EBL can be obtained by evolving stellar populations and galaxies under various cosmological initial conditions (Primack et al., 2005; Stecker et al., 2006; Franceschini et al., 2008; Gilmore et al., 2009; Finke et al., 2010; Kneiske & Dole, 2010). However, such models involve a large number of parameters and the estimated EBL spectrum depends upon the underlying assumptions (Figure 1) (Hauser & Dwek, 2001; de Angelis et al., 2009; Dwek & Krennrich, 2013). Alternatively, indirect estimation of EBL intensity can be obtained by studying the very high energy (VHE) gamma ray (E $>100GeV$) spectrum of distant blazars, a class of Active Galactic Nuclei for which the relativistic jet is aligned close to the line of sight of the observer (Urry & Padovani, 1995). The VHE photons emitted from blazars are absorbed en-route by forming electron-positron pairs on interaction with the EBL photons, thereby causing the observed spectrum to differ significantly from the intrinsic one. The EBL intensity can thus be estimated from gamma ray observations of blazars under various assumptions of the intrinsic spectrum (Madau & Phinney, 1996; Coppi & Aharonian, 1999). Assuming that the EBL spectral shape is described by the theoretical estimates, Stanev & Franceschini (1998) used the VHE spectrum of Mkn501 during a flare to constrain the overall EBL intensity and corresponding spectral index. Aharonian et al. (7) set an upper limit on the EBL intensity by assuming that the intrinsic VHE spectral index of blazars cannot be harder than 1.5. Similar upper limits on EBL have been put forward by various authors, based on allowed hardness of the intrinsic VHE spectrum (Guy et al., 2000; Mazin & Raue, 2007; Orr et al., 2011). The EBL estimated from the VHE spectra of blazars often depends heavily on the underlying blazar emission models, though their broadband emission is still not well understood. The SED of blazars is dominated by a non-thermal spectrum extending from radio-to-gamma rays and are characterized by two peaks, with the low energy spectrum peaking at IR–X-ray and the high energy spectrum peaking at gamma rays (Ghisellini, 2011). They are further classified as BL Lac objects and flat spectrum radio quasars (FSRQ), where FSRQ show strong emission and/or absorption lines, while such features are absent/weak in the former. Depending upon the location of the low energy peak, BL Lacs are further subdivided into low energy peaked BL Lacs (LBL), intermediate energy peaked BL Lacs (IBL) and high energy peaked BL Lacs (HBL) (Fossati et al., 1998; Ghisellini, 2011). The low energy emission from BL Lacs is generally interpreted as synchrotron emission from a non-thermal distribution of electrons losing their energy in a magnetic field, while the high energy emission mechanism is still under debate. Under leptonic origin, this emission is modeled as inverse Compton scattering of soft target photons by the same electron distribution responsible for the low energy emission, whereas in hadronic models, the high energy emission is an outcome of hadronic processes from a region with energetic protons. The constraints available through present observations are not sufficient enough to differentiate between these models satisfactorily (Böttcher, 2007). Before 2000, the number of blazars detected at VHE energies were few($\sim 4$), primarily due to low sensitivity of first generation atmospheric Cherenkov telescopes (Costamante & Ghisellini, 2002). However, with the advent of new generation high sensitivity telescopes, namely VERITAS, MAGIC and HESS, the number of blazars detected at this energy are more than $50$¹. Hence the present period allows one to perform a statistical study of VHE blazars to estimate the EBL, independent of various emission models. A study of similar kind has been performed by Ackermann et al. (2012) using blazars detected by the _Fermi_-LAT, a satellite based gamma ray experiment. They used the GeV spectrum of $\sim 150$ blazars to estimate the EBL at UV–optical wavelengths. [FOOTNOTE:1][ENDFOOTNOTE] In this work, we utilize a novel method to estimate the EBL spectrum at IR energies from the observed VHE spectrum of HBL. First, we show that the observed VHE spectral index of HBL correlates well with the redshift. We attribute this correlation to a result of EBL absorption, since such correlations are not seen in other wavebands. The observed spectrum of all the sources in our sample can be well described by a power law. Considering the source spectrum also as a power law, we show that this is expected for a particular shape of EBL (Stecker & Scully, 2006). The parameters defining the EBL spectrum are then constrained by nullifying the correlations of the intrinsic VHE spectral index with redshift. In the next section, we present our correlation study between the observed spectral indices and redshift to show the presence of EBL induced absorption on VHE spectra of blazars. In §3, we describe the formalism used to estimate the EBL using the correlation study and in §4, we discuss the implications of the results. A cosmology with $\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$ and $H_{0}=71\,km\,s^{-1}\,Mpc^{-1}$ is used in this work. ## 2 EBL signature on VHE Spectra The effect of the absorption of VHE photons by the EBL is to steepen the VHE spectra, hence providing a signature of the EBL (Vassiliev, 2000; Mankuzhiyil et al., 2010). Since sources at higher redshifts are more affected by absorption; their average spectra are expected to be steeper than the lower redshift ones. To investigate this, we perform a correlation study between the VHE spectral index²$\Gamma$, and redshift for a homogeneous set of sources which are detected at VHE. We select all HBL detected by the HESS, MAGIC and VERITAS telescopes with known redshifts and measured spectral index. We restrict our sample to only HBL since an intrinsic systematic hardening with source type, from FSRQ to HBL, has been observed at the GeV energies (Ackermann et al., 2011); moreover, a non-homogeneous sample may lead to spurious correlations. This restricts the farthest source in the sample to be 1ES 0414+009 at a redshift of z = 0.287. In Table 1, we list all the HBL detected at VHE along with ones for which the redshift information is uncertain (lower group). Again from the list, we group 8 HBL (middle group), due to their unusual properties. The de-absorbed VHE spectral index of these sources, obtained considering various EBL models, suggests their spectrum is extremely hard with index $<2$(Tavecchio, 2014; Tanaka et al., 2014). Moreover, these sources are less luminous compared to other HBL with their synchrotron spectrum peaking at energies $>10keV$. Due to these peculiar properties, these sources have been classified as extreme HBL (EHBL) and occupy a distinct position in the so called blazar sequence (Costamante & Ghisellini, 2002). [FOOTNOTE:2][ENDFOOTNOTE] In Figure 2, we show the variation of $\Gamma$ with redshift for all the sources listed in Table 1. A Spearman rank correlation analysis on all these sources, with known redshift, shows that they are well correlated with a rank correlation coefficient, $rs=0.58$, corresponding to a null hypothesis probability of $P_{rs}=9.4\times 10^{-4}$. Repeating the study with EHBL removed from the list improves the correlation considerably, with $rs=0.75$, corresponding to $P_{rs}=8.02\times 10^{-5}$. Hence this study again suggests that, probably, EHBL can be treated as a separate class of HBL. However, poor statistics does not let one to assert this inference strongly. Although the redshift range of the sample is small, a positive correlation may also occur due to rapid redshift evolution of HBL, such that the intrinsic VHE spectral index increases with redshift. If so, then the redshift evolution should be expected to have an effect on the spectral shape at other wavelengths. To examine this possibility, we further studied the correlation between X-ray spectral indices with redshift for HBL ³ using a) 70 months of _Swift_-BAT catalog consisting of 27 HBL (Baumgartner et al., 2013), b) six year _Beppo_SAX catalog consisting of 39 HBL (Donato et al., 2005) and c) archival X-ray catalog from ASCA, EXOSAT, _Beppo_SAX, ROSAT and EINSTEIN consisting of 61 HBL (Donato et al., 2001). We found no evidence of any correlation of the X-ray spectral index with redshift and obtained the rank correlation coefficient and null hypothesis probability for the chosen set of catalogs as, a) $rs=0.05$ and $P_{rs}=0.79$ (_Swift_-BAT), b) $rs=-0.07$ and $P_{rs}=0.67$ (_Beppo_SAX) and c) $rs=0.03$ and $P_{rs}=0.7$ (archival). The plot of X-ray index vs. redshift for these three catalogs is given in Figure 3. Spearman rank correlation study was also performed between low energy gamma ray (GeV) spectral index and redshift for the 62 HBL listed in the second catalog of _Fermi_-LAT (Ackermann et al., 2011). For this, we obtained $rs=0.02$ with $P_{rs}=0.85$, suggesting that these quantities are uncorrelated. Hence, these studies violate the conjecture on redshift evolution of the spectral index of HBL, and instead support the steepening of VHE spectral index as a result of EBL absorption. [FOOTNOTE:3][ENDFOOTNOTE] The observed correlation between the VHE spectral index and redshift could be due to selection effects. The luminosity is expected to correlate with redshift due to Malmquist bias, and if the index correlates with VHE luminosity then a correlation with redshift may occur. However, for the HBL observed by MAGIC (de Angelis et al., 2009), while the VHE luminosity does correlate with redshift as expected, there is no significant correlation between the VHE index and luminosity. Here we restricted our sample only to MAGIC detected HBL as the threshold energy is different for each experiment. This correlation is shown in Figure 4 and a Spearman rank analysis gives $rs=0.26$ and $P_{rs}=0.34$. At X-ray energies, the correlation study between the spectral index and X-ray luminosity resulted in a) $rs=-0.07$ and $P_{rs}=0.15$ (_Swift_-BAT catalog) b) $rs=-0.20$ and $P_{rs}=0.16$ (_Beppo_SAX catalog) and c) $rs=-0.14$ and $P_{rs}=0.28$ (archival). Similarly, correlation study between GeV spectral index and luminosity for the HBL from _Fermi_-LAT catalog gives $rs=0.11$ and $P_{rs}=0.37$. Based on this study, we can exclude the possibility of selection effect in the observed correlation between the VHE spectral index and redshift. Significant correlation between the difference of the VHE index and the one measured by _Fermi_-LAT with redshift was reported by several authors (Stecker & Scully, 2010; Prandini et al., 2010; Sanchez et al., 2013). For the HBL listed in Table 1 (top group), we also observed significant correlation between these quantities with $rs=0.71$ and $P_{rs}=2.2\times 10^{-4}$; however, this correlation is weaker than the one between VHE spectral index and redshift. Based on these studies, it is quite evident that the correlation between VHE spectral index and redshift can be attributed solely due to the effect of EBL induced absorption and that the intrinsic spectral index is uncorrelated with redshift. ## 3 EBL Estimation The observed VHE spectra of the HBL are well reproduced by a power law, and hence the observed flux, $F_{o}(E_{i})$, from a source at redshift $z$ will be \[F_{o}(E_{i}) =F_{i}(E_{i})e^{-\tau(E_{i},z)}\] (1) \[\propto E_{i}^{-\Gamma}\] where $F_{i}(E_{i})$ is the de-absorbed flux at energy $E_{i}$ and $\tau$, the optical depth due to EBL absorption given by (Gould & Schréder, 46) \[\tau(E_{i},z)=\int\limits_{0}^{z}{d}z^{\prime}\frac{{d}l}{{d}z^{ \prime}}\int\limits_{-1}^{1}{d\mu}\frac{(1-\mu)}{2}\int\limits_{\epsilon_{th}} ^{\infty}{d}\epsilon_{z^{\prime}}n(\epsilon_{z^{\prime}},z^{\prime})\sigma_{ \gamma\gamma}(E_{i},\epsilon_{z^{\prime}},\mu)\] (2) Here, \[\frac{dl}{dz^{\prime}}=\frac{c}{H_{0}}\frac{1}{(1+z^{\prime})\sqrt{{\Omega}_{ \Lambda}+{\Omega}_{M}(1+z^{\prime})^{3}}},\] (3) is the distance traveled by a photon per unit redshift with $c$ as the velocity of light, $n$ is the number density of the EBL photon of energy $\epsilon_{z^{\prime}}$[$=\epsilon_{0}(1+z^{\prime})$] at redshift $z^{\prime}$, corresponding to a photon energy $\epsilon_{0}$ at $z=0$, $\epsilon_{th}$ [$=2m_{e}^{2}c^{4}\;(1+z^{\prime})/(E_{i}(1-\mu))$] is the threshold soft photon energy with $\mu$ the cosine of the interaction angle and the pair production cross section, $\sigma_{\gamma\gamma}$, is given by \[\sigma_{\gamma\gamma}(E,\epsilon,\mu)=\frac{2\pi\alpha^{2}}{3m_{e}^{2}}(1- \beta^{2})\times\left[2\beta(\beta^{2}-2)+(3-\beta^{4})\,ln\left(\frac{1+\beta }{1-\beta}\right)\right]\] (4) with $\beta(E,\epsilon)=\sqrt{1-(2\;m_{e}^{2}\;c^{4})/(\epsilon\,E\,(1-\mu))}$ being the speed of the electron/positron in the centre of mass frame, $\alpha$ is the fine structure constant and $m_{e}$ is the electron rest mass. Since VHE sources are detected only at low redshifts, one can neglect the evolution of EBL and hence (Madau & Phinney, 1996), \[n(\epsilon_{z},z)\approx(1+z)^{3}n(\epsilon_{z})\] (5) For an isotropic EBL distribution, the angle integrated cross section (Gould & Schréder, 47; Brown et al., 1973) \[\bar{\sigma}_{\gamma\gamma}(E\epsilon)=\frac{1}{2}\int\limits_{-1}^{1}{d\mu}(1 -\mu)\sigma_{\gamma\gamma}(E,\epsilon,\mu)\] (6) peaks at $E\epsilon=3.56\;m_{e}^{2}\;c^{4}$. Approximating $\bar{\sigma}_{\gamma\gamma}$ as a delta function along with $E_{z}\approx E_{i}$ and $\epsilon_{z}\approx\epsilon_{0}$, equation (2) can be simplified to \[\tau(E,z)\approx A_{\gamma\gamma}\,\epsilon\,n(\epsilon)f(z)\] (7) where $A_{\gamma\gamma}$($\approx 3.7\times 10^{-26}\,cm^{2}$) is a constant, $\epsilon\approx\frac{3.6\,m_{e}^{2}c^{4}}{E}$ and $f(z)$ is given by \[f(z) =\int\limits_{0}^{z}{d}z^{\prime}\frac{{d}l}{{d}z^{\prime}}(1+z^{ \prime})^{3}\] (8) \[\approx\frac{c}{H_{0}}z\] If the source spectrum is assumed to be a power-law, $F_{i}(E)\propto E^{-\zeta}$, then using equations (1) and (7) we obtain \[\epsilon\,n(\epsilon)=k\,ln\left(\frac{\,\epsilon_{p}}{\epsilon}\right)\] (9) where $k=\frac{\Gamma-\zeta}{A_{\gamma\gamma}f(z)}$ and $\epsilon_{p}\approx\frac{3.56\;m_{e}^{2}c^{4}}{E^{\star}}$ are source independent constants with $E^{\star}$ being the energy of the gamma ray photon at which the EBL induced absorption is negligible. While this form of the EBL spectrum is derived for the approximate optical depth equation (7), we have verified numerically that for the range of redshifts considered here, this EBL spectral shape will result in a nearly power-law observed spectrum. In particular, we have verified that if the EBL spectra is defined by equation (9) and the absorption optical depth is given by equation (2), the observed VHE spectrum for a source at $z=0.3$ can be well described as a power-law. The deviation from a power-law for sources with $z<0.3$ is less than 10 %. Hence for all calculations in this work we have used equation (2) for the optical depth. It may be noted that Stecker & Scully (2006) obtained a similar form of the optical depth while approximating a theoretical EBL spectrum by an analytic expression. However, here we arrive at this form of the EBL spectrum equation (9) only from the criterion that both the intrinsic and absorbed VHE spectra are well described by a power-law and hence our approach is independent of any cosmological calculations. The EBL spectrum given by equation (9) is characterized by two constants, namely $k$ and $\epsilon_{p}$, which in turn determines the source spectral index. Since the source spectral index should be uncorrelated with redshift (§2), the allowed range of $k$ and $\epsilon_{p}$ is restricted. For $k=2.4\times 10^{-3}cm^{-3}$ and $\epsilon_{p}=4.6$ eV, we found that the computed source spectral indices, for the sources listed in Table 1 (top group), turn out to be “most” uncorrelated with $rs=0.001$ and a maximum null hypothesis probability $P_{rs}=0.99$. Within $1-\sigma$ confidence limit, corresponding to $P_{rs}>0.33$, we obtained the range of $k$ and $\epsilon_{p}$ as $2.2^{+1.6}_{-0.7}\times 10^{-3}\,cm^{-3}$ and $4.6^{+4.4}_{-3.4}\,eV$, respectively. The resultant EBL spectrum, consistent with these values of $k$ and $\epsilon_{p}$, is plotted in Figure 1 along with the constraints derived from various observations (Dwek & Krennrich, 2013) and other theoretical estimates (Gilmore et al., 2009; Franceschini et al., 2008; Finke et al., 2010; Kneiske & Dole, 2010). If we consider all the HBL with known redshift in Table 1 (top and middle group), the $1-\sigma$ confidence limit of $k$ and $\epsilon_{p}$ are $2.8^{+2.6}_{-1.8}\times 10^{-3}\,cm^{-3}$ and $5.2^{+4.8}_{-4.0}\,eV$, respectively. Alternatively, a linear fit over source spectral index versus redshift can be used to constrain the constants $k$ and $\epsilon_{p}$. A linear fit between $\Gamma$ and redshift resulted in a straight line of slope $6.0\pm 1.1$ with reduced chi-square $\chi_{red}^{2}\approx 1.1$. Since the source spectral index is uncorrelated with the redshift (§2), a linear fit between these quantities should result in a constant line (a line with slope $0$). Within 1-$\sigma$ confidence limit, this corresponds to an EBL spectrum with $k=2.4^{+1.2}_{-0.8}\times 10^{-3}\,cm^{-3}$ and $\epsilon_{p}=5.2^{+3.8}_{-4.0}\,eV$, for which this condition can be achieved. We find that these constraints on $k$ and $\epsilon_{p}$ are consistent with the one obtained earlier through nullifying the correlation between source VHE spectral index and redshift. In Figure 5, we show the allowed range of $k$ and $\epsilon_{p}$ obtained by these two methods. ## 4 Discussions The EBL spectrum presented in this work is estimated directly from the observed VHE spectra of HBL with the condition that the source spectrum should be uncorrelated with redshift. The main uncertainty lies in the assumption that the source VHE spectrum is a power law, and that approximate EBL spectral shape is given by equation (9). However, the latter is verified numerically to reproduce the observed spectrum which can be well represented by a power law. Interestingly, the present estimation does not depend on the nature of the radiative process active in HBL or dust/stellar emission models from galaxies, yet still agrees well with other estimates as shown in Figure 1. Moreover, though the 1-$\sigma$ uncertainty range on the EBL spectrum is nearly a factor $\sim 4$, it is competitive compared to constraints put by observations and by other estimates. The predicted EBL spectrum is reasonably confined within the upper and lower limits (grey shaded area in Figure 1), obtained independently through observations (see §1). When compared with the other EBL estimates, obtained through cosmological evolution models, the present one predicts stronger emission at lower energies but closely agrees at higher energies, though the predicted spectrum is not well constrained in this regime. Deviation of the source spectrum from a power law may modify the EBL spectral shape described by equation (9) considerably. In such case, the present formalism needs to be modified by studying the correlation of other suitable observables instead of the power law spectral indices. However, the source spectrum of HBL (z<0.3) obtained using various EBL models are fitted resonably well by a power law and the one presented here agrees closely with these EBL models. To investigate further, we repeat the study considering the EBL spectral shape due to Franceschini et al. (2008), Gilmore et al. (2009), Finke et al. (2010) and Kneiske & Dole (2010). Following Abdo et al. (2010); Ackermann et al. (2012); H.E.S.S. Collaboration et al. (2013); Reesman & Walker (2013), we define the observed spectrum to be \[F_{o}(E)=F_{i}(E_{z})exp(-\tau_{theory}(E_{z},z)\times b)\] (10) where $\tau_{theory}$ is the optical depth predicted by the above mentioned theoretical models, and $b$ is a normalization factor required to assure that the de-absorbed spectral index is uncorrelated with the redshift. Then, $b=1.0$ would imply that the particular model is consistent with the non correlation of the source VHE index with redshift. For the models discussed above, we obtained $b_{Kneisnke}=1.5^{-0.6}_{+0.7}$, $b_{Franschini}=1.4^{-0.6}_{+0.7}$, $b_{Finke}=1.1^{-0.5}_{+0.6}$ and $b_{Gilmore}=1.6^{-0.5}_{+0.7}$. From these, one can argue that the EBL model due to Finke et al. (2010) very well supports the non correlation of the source VHE index with redshift; whereas, the deviation from this condition is observed to be maximum in case of Gilmore et al. (2009). The EBL spectrum, estimated in this work, can be used to find the intrinsic spectral index of HBL, which can then be compared with the one predicted by the radiative models of HBL. Under leptonic models, the spectral energy distributions of HBL are well reproduced by considering synchrotron and synchrotron self Compton emission from a broken power-law distribution of electrons. In such a case, the VHE spectrum corresponds to the high energy tail of the electron distribution. Similarly, the X-ray spectrum lying beyond the synchrotron peak is also governed by the high energy end of the electron distribution. Indeed, the X-ray-TeV correlation observed during flares further suggests that the same electron distribution is responsible for the emission at these energies (Takahashi et al., 1996). Hence, it can be argued that the spectral index at these energies is related to the high energy particle spectral index of the underlying electron distribution. If the Compton scattering responsible for the VHE emission occurs in the Thomson regime, then the corresponding spectral index $\zeta$ will be same as the X-ray spectral index, $\alpha_{2}$. On the other hand, if the scattering process happens at the extreme Klein-Nishina regime, then the VHE spectral index will be $2\alpha_{2}-\alpha_{1}$(Tavecchio et al., 1998), where $\alpha_{1}$ is the optical spectral index reflecting the low energy electron spectral index. In general, the VHE index is expected to lie in between these two limits. To examine this, we compare the intrinsic VHE spectral index, computed in this work, with the X-ray spectral indices of the sources for which simultaneous/contemporaneous observations are available from _Swift_-XRT/_Suzaku_/_XMM_-Newton/_Swift_-BAT observations (Table 1). In Figure 6, we plot the intrinsic VHE spectral indices against the X-ray spectral indices with the limiting lines corresponding to Thomson and extreme Klein-Nishina regimes. For the latter limit, we assume the optical spectral index as $1/3$ since this limits the hardest synchrotron spectrum attainable (Pacholczyk, 1973). Interestingly, all the sources are constrained well within these limits thereby supporting the afore mentioned interpretation. The analysis done in this work was possible because the intrinsic variation of spectral index for HBL is relatively small. The fractional root mean square deviation ($f_{rms}$) of the de-absorbed VHE indices is $0.16$, which is comparable to that of the X-ray ($f_{rms,X}=0.14$) and the low energy GeV gamma-rays ($f_{rms,GeV}=0.12$). The VHE index $f_{rms}$ is significantly less than the index change due to absorption $\Delta\Gamma\sim 2$ at a redshift of $z=0.266$. If the variation of index was comparable to the change due to absorption, the effect would not have been detectable. Since $f_{rms}$ is considerably smaller than $\Delta\Gamma$, this leaves the exciting possibility that the uncertainty in EBL, predicted by the present study, can be significantly reduced with increased number of blazars detected at VHE energies. The work presented here is similar to the EBL upper limits proposed by Schroedter (2005) and Finke & Razzaque (2009). They studied the steepening of VHE spectral index with increase in redshift and attributed it to the absorption by EBL. Schroedter (2005) suggested an upper limit on EBL by assuming that the source spectral index, $\zeta$, cannot be harder than $1.8$, whereas Finke & Razzaque (2009) considered limits for $\zeta$ as $1$ and $1.5$. Following a similar procedure, Yang & Wang (2010) proposed an EBL upper limit by considering _Fermi_-LAT spectral index as an allowed limit on $\zeta$. In this work, we systematically study the steepening of the VHE spectra of HBL with respect to redshift and exploit it to estimate the EBL spectrum. In addition, we do not impose any limits on $\zeta$; instead, the derived value of $\zeta$, using the current EBL, lies well within our present understanding of blazar emission models (Figure 6). Lately, various EBL estimates have been proposed, exploiting the properties of TeV blazars under unique techniques. Mazin & Raue (2007) and Meyer et al. (2012) estimated an upper limit on EBL by employing splines. From the observed VHE spectra of blazars, they converged to a particular shape of EBL, which leads to a de-absorbed spectra that are physically acceptable under the present understanding of blazars. Mankuzhiyil et al. (2010); Domínguez et al. (2013) and Singh et al. (2014) reproduced the broadband SED of VHE blazars under leptonic model and thereby predicting the intrinsic VHE spectra. Comparing this with the observed VHE spectra, they estimated the optical depth for the attenuation of VHE gamma rays. While Mankuzhiyil et al. (2010) used this to show the inconsistency among various EBL models interpreted theoretically, Singh et al. (2014) showed a systematic deviation of the optical depths towards high energy; between the estimated and the ones predicted by various EBL models. Domínguez et al. (2013) used the estimated optical depths to determine the cosmic gamma ray horizon. Reesman & Walker (2013) considered the EBL models by Franceschini et al. (2008); Gilmore et al. (2009); Domínguez et al. (2011) to estimate the optical depth for the VHE sources at $z\sim 0.1$. The optical depth is then scaled by a parameter to reproduce the observed flux for a range of de-absorbed spectral indices. Based on this scaling parameter, they concluded that these EBL models are consistent with the observed spectra, though the error on the parameter is large. Unlike these models, the work presented here does not have any bias on blazar emission models or a particular EBL shape. Instead, it relies upon the observed correlation between the VHE spectral index and redshift, along with the assumption that the de-absorbed VHE spectra is a power law. Finally, like other EBL models, the EBL spectrum presented in this work predicts very large opacities for VHE photons from distant sources, e.g. 3C 279 at $z=0.536$ and PKS 1424+240 at $z>0.6$(MAGIC Collaboration et al., 2008; Furniss et al., 2013). This is evident from Figure 2, where deviation of the observed VHE index from the best fit line is large for distant sources. This remains an open problem and may possibly be related to VHE emission through secondary processes resulting from the development of electromagnetic and hadronic cascades in the intergalactic medium (Essey & Kusenko, 2010) or more exotic scenarios associated with creation of axion like particles (de Angelis et al., 2009). With the help of present high sensitivity VHE telescopes and future telescopes, like CTA (Cherenkov Telescope Array), these uncertainties can be cleared, providing more insight into our cosmic evolution. ## 5 Acknowledgements The authors thank the anonymous referee for his valuable comments and suggestions. AS thanks Varsha Chitnis for helpful discussions and support. This research has made use of TeV catalog (http://tevcat.uchicago.edu/) created and maintained by Scott Wakely And Deirdre Horan and partially supported by NASA and the NSF. Source name \| z \| Γ \| EVHE \| α2 \| αf \| Ref ---\|---\|---\|---\|---\|---\|--- Mkn421 \| 0.031 \| 2.72 ± 0.12 \| 0.2 - 10 \| 2.58 ± 0.03 \| 1.771 ± 0.012 \| 1 Mkn501 \| 0.034 \| 2.79 ± 0.12 \| 0.1 - 2.0 \| 2.42 ± 0.01 \| 1.738 ± 0.027 \| 2 1ES 2344+514∗ \| 0.044 \| 2.95 ± 0.20 \| 0.2 - 2.0 \| 2.62 ± 0.50 \| 1.716 ± 0.08 \| 3 Mkn180 \| 0.045 \| 3.30 ± 0.70 \| 0.2 - 6.6 \| - \| 1.74 ± 0.083 \| 4 1ES 1959+650 \| 0.048 \| 2.58 ± 0.18 \| 0.2 - 2.5 \| 2.19 ± 0.02 \| 1.937 ± 0.031 \| 5 1ES 1727+502 \| 0.055 \| 2.70 ± 0.50 \| 0.15 - 2.0 \| - \| 2.0 ± 0.2 \| 6 PKS 0548−322∗ \| 0.069 \| 2.86 ± 0.34 \| 0.3 - 4.0 \| 2.28 ± 0.23 \| - \| 7 PKS 2005−489 \| 0.071 \| 3.00 ± 0.22 \| 0.4 - 4.0 \| 2.46 ± 0.01 \| 1.779 ± 0.047 \| 8 RGB J0152+017 \| 0.080 \| 2.95 ± 0.36 \| 0.1 - 4.0 \| - \| 1.788 ± 0.137 \| 9 BZB J0013-1854 \| 0.095 \| 3.4 ± 0.10 \| 0.2 - 2.0 \| - \| 1.96 ± 0.2 \| 10 1ES 1312-423 \| 0.105 \| 2.85 ± 0.7 \| 0.2 - 4.0 \| - \| 1.4 ± 0.4 \| 11 PKS 2155−304 \| 0.116 \| 3.34 ± 0.10 \| 1.0 - 10. \| 2.36 ± 0.01 \| 1.838 ± 0.015 \| 12 B3 2247+381 \| 0.119 \| 3.20 ± 0.60 \| 0.1 - 2.0 \| - \| 1.837 ± 0.113 \| 13 H 1426+428∗ \| 0.129 \| 3.50 ± 0.40 \| 0.3 - 2.00 \| 2.54 ± 0.24 \| 1.316 ± 0.123 \| 14 1ES 1215+304 \| 0.130 \| 2.96 ± 0.14 \| 0.1 - 1.51 \| 2.29 ± 0.16 \| 2.019 ± 0.036 \| 15 1ES 0806+524 \| 0.138 \| 3.60 ± 1.00 \| 0.3 - 1.02 \| 2.67 ± 0.08 \| 1.938 ± 0.057 \| 16 BZB J1010−3119 \| 0.143 \| 3.08 ± 0.42 \| 0.25 - 3.0 \| 2.15 ± 0.06 \| 2.239 ± 0.142 \| 17 RX J0648+1516 \| 0.179 \| 4.40 ± 0.80 \| 0.2 - 0.65 \| 2.51 ± 0.06 \| 1.737 ± 0.106 \| 18 RBS 0413 \| 0.190 \| 3.18 ± 0.68 \| 0.3 - 1.0 \| 2.22 ± 0.07 \| 1.551 ± 0.112 \| 19 1ES 1011+496 \| 0.212 \| 4.00 ± 0.50 \| 0.15 - 0.8 \| - \| - \| 20 PKS 0301−243 \| 0.266 \| 4.60 ± 0.70 \| 0.1 - 5 \| 2.51 ± 0.1 \| 1.938 ± 0.031 \| 21 IC310 \| 0.019 \| 1.96 ± 0.22 \| 0.12 - 8.1 \| - \| \| 22 RGB J0710+591∗ \| 0.125 \| 2.69 ± 0.22 \| 0.3 - 7.9 \| 2.29 ± 0.26 \| \| 23 1ES0229+200 \| 0.140 \| 2.50 ± 0.19 \| 0.5 - 15.0 \| 2.16 ± 0.28 \| \| 24 H 2356-309 \| 0.165 \| 3.09 ± 0.24 \| 0.2 - 1.04 \| 2.43 ± 0.11 \| \| 25 1ES 1218+304 \| 0.182 \| 3.08 ± 0.40 \| 0.1 - 2 \| - \| \| 26 1ES 1101−232 \| 0.186 \| 2.94 ± 0.20 \| 0.1 - 0.66 \| 2.32 ± 0.02 \| \| 27 1ES 0347−121∗ \| 0.188 \| 3.10 ± 0.23 \| 0.3 - 3 \| 2.27 ± 0.30 \| \| 28 1ES 0414+009 \| 0.287 \| 3.40 ± 0.50 \| 0.2 - 0.70 \| 2.40 ± 0.10 \| \| 29 HESS J1943+213 \| 0.14 \| 3.1 ± 0.30 \| \| \| \| 30 1ES 1440+122 \| 0.163 \| 3.4 ± 0.7 \| \| \| \| 31 PKS 0447-439 \| 0.175 \| 3.8 ± 0.4 \| \| \| \| 32 1ES 0502+675 \| 0.341 \| 3.9 ± 0.4 \| \| \| \| 33 PG 1553+113 \| 0.5 ± 0.08 \| 4.1 ± 0.3 \| \| \| \| 34 PKS 1424+240 \| 0.604 \| 4.2 ± 0.7 \| \| \| \| 35 Table 1: The list of HBL detected in VHE. The middle group lists the extreme HBLS and the bottom one with uncertain redshift. Column description 1: the Source name, 2: the redshift (z), 3: Observed VHE spectral index (Γ), 4: Observed VHE energy range (in TeV), 5: X-ray Spectral Index (quantities with * are obtained from Swift-BAT) 6. The Fermi-LAT average index, and 7: References: 1\. Aleksić et al. (2012c) 2\. Anderhub et al. (2009) 3\. Albert et al. (2007b) 4\. Albert et al. (2006) 5\. Tagliaferri et al. (2008) 6\. Aleksić et al. (2014a) 7\. Aharonian et al. (2010) 8\. H.E.S.S. Collaboration et al. (2011) 9\. Aharonian et al. (2008) 10\. H.E.S.S. Collaboration et al. (2013a) 11\. HESS Collaboration et al. (2013) 12\. Aharonian et al. (2009) 13\. Aleksić et al. (2012a) 14 .Petry et al. (2002) 15.Aleksić et al. (2012b) 16.Acciari et al. (2009a) 17\. H.E.S.S. Collaboration et al. (2012) 18\. Aliu et al. (2011) 19\. Aliu et al. (2012a) 20\. Albert et al. (2007a) 21\. H.E.S.S. Collaboration et al. (2013b) 22\. Aleksić et al. (2014b) 23\. Acciari et al. (2010) 24\. Aharonian et al. (2007b) 25\. Aharonian et al. (2006b) 26\. Acciari et al. (2009b) 27\. Reimer et al. (2008) 28\. Aharonian et al. (2007a) 29.Aliu et al. (2012b) 30 - 35. www.tevcat.edu <figure><img src="content_image/1409.3693/x1.png"><figcaption>Figure 1: The best fit EBL spectrum estimated in this work (thick black line)and the 1−σ (checkered orange region) and 2−σ constrains (striped pink region)compared with the different theoretical models of Franceschini (Franceschiniet al. (2008)), Gilmore (Gilmore et al. (2009)), Finke (Finke et al. (2010))and Kneiske (Kneiske & Dole (2010)) . The solid grey region shows the upperand lower limits estimated from various observations (Dwek & Krennrich, 2013).</figcaption></figure> <figure><img src="content_image/1409.3693/x2.png"><figcaption>Figure 2: Distribution of the observed VHE spectral index of the selected HBLwith redshift. The black stars correspond to extreme HBL and blue opendiamonds are the ones with uncertain redshifts. The lower limits on theredshifts have been shown with solid (blue) arrows. The solid line (green) isthe best fit straight line to the HBL denoted by filled circles (red).</figcaption></figure> <figure><img src="content_image/1409.3693/x3.png"><figcaption>Figure 3: Distribution of the observed X-ray spectral index of HBL withredshift. The blue diamonds are from the Swift-BAT catalog (Baumgartner etal., 2013), the green circles from the Beppo-SAX catalog (Donato et al.,2005), and the red stars from archival X-ray catalog (Donato et al., 2001)</figcaption></figure> <figure><img src="content_image/1409.3693/x4.png"><figcaption>Figure 4: Distribution of the observed VHE spectral index with luminosity ofthe HBL observed by the MAGIC telescope de Angelis et al. (2009).</figcaption></figure> <figure><img src="content_image/1409.3693/x5.png"><figcaption>Figure 5: The 1−σ confidence region for the parameters k and ϵp obtained fromthe correlation study (green forward stripes) and the straight line fit (redbackward stripes)</figcaption></figure> <figure><img src="content_image/1409.3693/x6.png"><figcaption>Figure 6: Distribution of the calculated intrinsic TeV indices with X-rayindices. Solid line (blue) denotes the Thomson regime, whereas the dashed line(green) denotes extreme Klein-Nishina regime.</figcaption></figure> ## References * Abdo et al. (2010) Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ, 723, 1082 * Acciari et al. (2009a) Acciari, V., Aliu, E., Arlen, T., et al. 2009a, ApJ, 690, L126 * Acciari et al. (2009b) Acciari, V. A., Aliu, E., Arlen, T., et al. 2009b, ApJ, 695, 1370 * Acciari et al. (2010) —. 2010, ApJ, 715, L49 * Ackermann et al. (2011) Ackermann, M., Ajello, M., Allafort, A., et al. 2011, ApJ, 743, 171 * Ackermann et al. (2012) —. 2012, Science, 338, 1190 * Aharonian et al. (2006a) Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006a, Nature, 440, 1018 * Aharonian et al. (2006b) —. 2006b, A&A, 455, 461 * Aharonian et al. (2007a) Aharonian, F., Akhperjanian, A. G., Barres de Almeida, U., et al. 2007a, A&A, 473, L25 * Aharonian et al. (2007b) —. 2007b, A&A, 475, L9 * Aharonian et al. (2008) —. 2008, A&A, 481, L103 * Aharonian et al. (2009) Aharonian, F., Akhperjanian, A. G., Anton, G., et al. 2009, ApJ, 696, L150 * Aharonian et al. (2010) —. 2010, A&A, 521, A69 * Albert et al. (2006) Albert, J., Aliu, E., Anderhub, H., et al. 2006, ApJ, 648, L105 * Albert et al. (2007a) —. 2007a, ApJ, 667, L21 * Albert et al. (2007b) —. 2007b, ApJ, 662, 892 * Aleksić et al. (2012a) Aleksić, J., Alvarez, E. A., Antonelli, L. A., et al. 2012a, A&A, 539, A118 * Aleksić et al. (2012b) —. 2012b, A&A, 544, A142 * Aleksić et al. (2012c) —. 2012c, A&A, 542, A100 * Aleksić et al. (2014a) Aleksić, J., Antonelli, L. A., Antoranz, P., et al. 2014a, A&A, 563, A90 * Aleksić et al. (2014b) —. 2014b, A&A, 563, A91 * Aliu et al. (2011) Aliu, E., Aune, T., Beilicke, M., et al. 2011, ApJ, 742, 127 * Aliu et al. (2012a) Aliu, E., Archambault, S., Arlen, T., et al. 2012a, ApJ, 750, 94 * Aliu et al. (2012b) —. 2012b, ApJ, 755, 118 * Anderhub et al. (2009) Anderhub, H., Antonelli, L. A., Antoranz, P., et al. 2009, ApJ, 705, 1624 * Baumgartner et al. (2013) Baumgartner, W. H., Tueller, J., Markwardt, C. B., et al. 2013, ApJS, 207, 19 * Böttcher (2007) Böttcher, M. 2007, Ap&SS, 307, 69 * Brown et al. (1973) Brown, R. W., Mikaelian, K. O., & Gould, R. J. 1973, Astrophys. Lett., 14, 203 * Coppi & Aharonian (1999) Coppi, P. S., & Aharonian, F. A. 1999, ApJ, 521, L33 * Costamante & Ghisellini (2002) Costamante, L., & Ghisellini, G. 2002, A&A, 384, 56 * De Angelis et al. (2013) De Angelis, A., Galanti, G., & Roncadelli, M. 2013, MNRAS, 432, 3245 * de Angelis et al. (2009) de Angelis, A., Mansutti, O., Persic, M., & Roncadelli, M. 2009, MNRAS, 394, L21 * Domínguez et al. (2013) Domínguez, A., Finke, J. D., Prada, F., et al. 2013, ApJ, 770, 77 * Domínguez et al. (2011) Domínguez, A., Primack, J. R., Rosario, D. J., et al. 2011, MNRAS, 410, 2556 * Donato et al. (2001) Donato, D., Ghisellini, G., Tagliaferri, G., & Fossati, G. 2001, A&A, 375, 739 * Donato et al. (2005) Donato, D., Sambruna, R. M., & Gliozzi, M. 2005, A&A, 433, 1163 * Dwek & Krennrich (2013) Dwek, E., & Krennrich, F. 2013, Astroparticle Physics, 43, 112 * Essey & Kusenko (2010) Essey, W., & Kusenko, A. 2010, Astroparticle Physics, 33, 81 * Finke & Razzaque (2009) Finke, J. D., & Razzaque, S. 2009, ApJ, 698, 1761 * Finke et al. (2010) Finke, J. D., Razzaque, S., & Dermer, C. D. 2010, ApJ, 712, 238 * Fossati et al. (1998) Fossati, G., Maraschi, L., Celotti, A., Comastri, A., & Ghisellini, G. 1998, MNRAS, 299, 433 * Franceschini et al. (2008) Franceschini, A., Rodighiero, G., & Vaccari, M. 2008, A&A, 487, 837 * Furniss et al. (2013) Furniss, A., Williams, D. A., Danforth, C., et al. 2013, ApJ, 768, L31 * Ghisellini (2011) Ghisellini, G. 2011, in American Institute of Physics Conference Series, Vol. 1381, American Institute of Physics Conference Series, ed. F. A. Aharonian, W. Hofmann, & F. M. Rieger, 180–198 * Gilmore et al. (2009) Gilmore, R. C., Madau, P., Primack, J. R., Somerville, R. S., & Haardt, F. 2009, MNRAS, 399, 1694 * Gould & Schréder (1967a) Gould, R. J., & Schréder, G. P. 1967a, Physical Review, 155, 1408 * Gould & Schréder (1967b) —. 1967b, Physical Review, 155, 1404 * Guy et al. (2000) Guy, J., Renault, C., Aharonian, F. A., Rivoal, M., & Tavernet, J.-P. 2000, A&A, 359, 419 * Hauser & Dwek (2001) Hauser, M. G., & Dwek, E. 2001, ARA&A, 39, 249 * Hauser et al. (1998) Hauser, M. G., Arendt, R. G., Kelsall, T., et al. 1998, ApJ, 508, 25 * Helgason & Kashlinsky (2012) Helgason, K., & Kashlinsky, A. 2012, ApJ, 758, L13 * H.E.S.S. Collaboration et al. (2011) H.E.S.S. Collaboration, Abramowski, A., Acero, F., et al. 2011, A&A, 533, A110 * H.E.S.S. Collaboration et al. (2012) —. 2012, A&A, 542, A94 * H.E.S.S. Collaboration et al. (2013a) —. 2013a, A&A, 554, A72 * H.E.S.S. Collaboration et al. (2013b) —. 2013b, A&A, 559, A136 * HESS Collaboration et al. (2013) HESS Collaboration, Abramowski, A., Acero, F., et al. 2013, MNRAS, 434, 1889 * H.E.S.S. Collaboration et al. (2013) H.E.S.S. Collaboration, Abramowski, A., Acero, F., et al. 2013, A&A, 550, A4 * Kelsall et al. (1998) Kelsall, T., Weiland, J. L., Franz, B. A., et al. 1998, ApJ, 508, 44 * Kneiske & Dole (2010) Kneiske, T. M., & Dole, H. 2010, A&A, 515, A19 * Madau & Phinney (1996) Madau, P., & Phinney, E. S. 1996, ApJ, 456, 124 * Madau & Pozzetti (2000) Madau, P., & Pozzetti, L. 2000, MNRAS, 312, L9 * MAGIC Collaboration et al. (2008) MAGIC Collaboration, Albert, J., Aliu, E., et al. 2008, Science, 320, 1752 * Mankuzhiyil et al. (2010) Mankuzhiyil, N., Persic, M., & Tavecchio, F. 2010, ApJ, 715, L16 * Mazin & Raue (2007) Mazin, D., & Raue, M. 2007, A&A, 471, 439 * Meyer et al. (2012) Meyer, M., Raue, M., Mazin, D., & Horns, D. 2012, A&A, 542, A59 * Orr et al. (2011) Orr, M. R., Krennrich, F., & Dwek, E. 2011, ApJ, 733, 77 * Pacholczyk (1973) Pacholczyk, A. G. 1973, Radio astrophysics. Non-thermal processes in galactic and extragalactic sources. * Petry et al. (2002) Petry, D., Bond, I. H., Bradbury, S. M., et al. 2002, ApJ, 580, 104 * Prandini et al. (2010) Prandini, E., Bonnoli, G., Maraschi, L., Mariotti, M., & Tavecchio, F. 2010, MNRAS, 405, L76 * Primack et al. (2005) Primack, J. R., Bullock, J. S., & Somerville, R. S. 2005, in American Institute of Physics Conference Series, Vol. 745, High Energy Gamma-Ray Astronomy, ed. F. A. Aharonian, H. J. Völk, & D. Horns, 23–33 * Reesman & Walker (2013) Reesman, R., & Walker, T. P. 2013, J. Cosmology Astropart. Phys, 12, 22 * Reimer et al. (2008) Reimer, A., Costamante, L., Madejski, G., Reimer, O., & Dorner, D. 2008, ApJ, 682, 775 * Sanchez et al. (2013) Sanchez, D. A., Fegan, S., & Giebels, B. 2013, A&A, 554, A75 * Schroedter (2005) Schroedter, M. 2005, ApJ, 628, 617 * Singh et al. (2014) Singh, K. K., Sahayanathan, S., Tickoo, A. K., & Bhatt, N. 2014, New A, 27, 34 * Stanev & Franceschini (1998) Stanev, T., & Franceschini, A. 1998, ApJ, 494, L159 * Stecker et al. (2006) Stecker, F. W., Malkan, M. A., & Scully, S. T. 2006, ApJ, 648, 774 * Stecker & Scully (2006) Stecker, F. W., & Scully, S. T. 2006, ApJ, 652, L9 * Stecker & Scully (2010) —. 2010, ApJ, 709, L124 * Tagliaferri et al. (2008) Tagliaferri, G., Foschini, L., Ghisellini, G., et al. 2008, ApJ, 679, 1029 * Takahashi et al. (1996) Takahashi, T., Tashiro, M., Madejski, G., et al. 1996, ApJ, 470, L89 * Tanaka et al. (2014) Tanaka, Y. T., Stawarz, L., Finke, J., et al. 2014, ArXiv e-prints, arXiv:1404.3727 * Tavecchio (2014) Tavecchio, F. 2014, MNRAS, 438, 3255 * Tavecchio et al. (1998) Tavecchio, F., Maraschi, L., & Ghisellini, G. 1998, ApJ, 509, 608 * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803 * Vassiliev (2000) Vassiliev, V. V. 2000, Astroparticle Physics, 12, 217 * Wright (1998) Wright, E. L. 1998, ApJ, 496, 1 * Yang & Wang (2010) Yang, J., & Wang, J. 2010, A&A, 522, A12
1211.1143	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 62877, "num_imgs": 7, "llama3_tokens_count": 16171 }	[ "content_image/1211.1143/Fig1.png", "content_image/1211.1143/Fig2.png", "content_image/1211.1143/x1.png", "content_image/1211.1143/x2.png", "content_image/1211.1143/Fig5.png", "content_image/1211.1143/Fig6.png", "content_image/1211.1143/x3.png" ]	# Frustration – induced inherent instability and growth oscillations in pollen tubes Mariusz Pietruszka¹ Laboratory of Plant Physiology, Faculty of Biology and Environmental Protection University of Silesia, ul. Jagiellońska 28, PL-40032 Katowice, Poland tel. +48322009453, E-mail: mariusz.pietruszka@us.edu.pl [FOOTNOTE:1][ENDFOOTNOTE] Abstract. In a seed plant a pollen tube is a vessel that transports male gamete cells to an ovule to achieve fertilization. It consists of one elongated cell, which exhibits growth oscillations, until it bursts completing its function. Up till now, the mechanism behind the periodic character of the growth has not been fully understood. It is shown that the mechanism of pressure – induced _symmetry frustration_ occurring in the wall at the perimeter of cylindrical and approximately hemispherical parts of a growing pollen cell, together with the addition of cell wall material, suffices to release and sustain mechanical self-oscillations and cell extension in pollen tubes. At the transition zone where symmetry frustration occurs and one cannot distinguish either of the involved symmetries, a kind of ’entangled state’ appears where either single or both symmetry(ies) can be realized by the system. We anticipate that testifiable predictions made by the model ($f\propto\sqrt{P}$) may deliver, after calibration, a new tool to estimate turgor pressure $P$ from oscillation frequency $f$ of the periodically growing cell. Since the mechanical principles apply to all turgor regulated walled cells including those of plant, fungal and bacterial origin, the relevance of this work is not limited to the case of the pollen tube. Keywords: cell wall; equilibrium equation; frustration potential; growth tensors; _Nicotiana tobaccum_; plant cytomechanics ## 1 Introduction ### General outline The pollen tube has become a widely used cellular model system. In addition to being the fastest growing plant cell, it features periodic oscillations of the growth rate that have attracted numerous attempts to model the process². While recent models have increasingly incorporated biological features such as ion transport and intracellular trafficking, a simple feature with potentially significant impact has been overlooked in past approaches: geometry. We modeled the strain rates in the cell wall caused by turgor pressure depending on the different symmetries present in the pollen tube and found that a crucial area on the cellular surface of the pollen tube is characterized by so termed _symmetry frustration_. This area represents the transition zone between hemisphere-shaped apex and cylindrical shank. From a biological point of view this zone is crucial since numerous molecular landmarks of polar growth are present on one side of this zone and absent from the other. [FOOTNOTE:2][ENDFOOTNOTE] The model predicts that the transition zone undergoes local peaks in strain rate opening intriguing avenues for research focusing on the polarity of the growth process. Furthermore, we propose that changes between different symmetry regimes might be the mechanical underpinning of periodic changes in growth rate and shape observed during oscillatory growth. We believe that our model make an important contribution to the field of plant cytomechanics in general and pollen tube growth in particular. ### Preliminaries Pollen tubes are rapidly growing plant cells whose morphogenesis is largely determined by spatial gradients in the biochemical composition of the cell wall. Pollen tube growth is a critical process in the life cycle of higher plants (Winship et al., 2010). It has garnered a lot of attention and is at the center of considerable controversy (Kroeger and Geitmann, 2011a). The pollen tube is the carrier of the male gametes in flowering plants. A controversy swirls around the modes of extension leading to periodicity in growth and growth rate (Winship et al., 2010). While some authors claim that hydrodynamics is the central integrator of pollen tube growth (Zonia and Munnik, 2007, 2009, 2011) leading to growth oscillations, the other couple the periodicity in growth dynamics to the changes in the wall material properties (Winship et al., 2010, 2011; Kroeger et al., 2011). Pollen tubes are tip growing cells, which means that cell wall unidirectional expansion is confined to the apex of the cell. They display extremely rapid growth that can be also reproduced under _in vitro_ conditions. All growth activity - delivery of new cell wall material and cell wall deformation - occurs at the tip of the cell (Geitmann and Cresti, 1998): the tube is capped by an approximately hemispherically shaped dome, the apex, to which all growth activity is confined (Fig. 1A). The deformation is driven by the turgor pressure, a hydrodynamic pressure inside the cell. Interestingly, pollen tube evolution displays characteristic oscillations in growth and growth rate (Plyushch et al., 1995; Hepler et al., 2001; Feijo et al., 2001). The pollen tube (e.g. _Lilium longiflorum_, _Nicotiana tobacum_) growth oscillation depends on many underlying phenomena, amongst others the ion and mass fluxes, wall mechanical properties, system symmetry and turgor pressure. In isotonic conditions (Zonia and Munnik, 2007) the average growth cycle period $T$ is about 50 s, while upon shifts to hypertonic or hypotonic conditions it is about 100 s and 25 s, respectively. The latter produce oscillations with typical frequencies ($f=1/T$): hypertonic – 0.01 Hz, isotonic – 0.02 Hz and hypotonic – 0.04 Hz. The longitudinal and transversal oscillation power spectrum of an individual _Nicotiana tabacum_ pollen tube (Haduch and Pietruszka, 2012) is visualized in Fig. 2, and can not solely be described by a double exponent model (Pietruszka, 2012; e.g. Fig. 8A), suitable for normal cell evolution, but incomplete for periodical growth. The cell wall is one of the structural key players regulating plant cell growth since plant cell expansion depends on an interplay between intracellular pressures and the controlled yielding of the wall (Geitmann, 2009). The cell wall may be treated as a polymer with the property of viscoelasticity (colloquially ”elasticity”), generally having notably low Young’s modulus $\epsilon$ and high yield strain, which is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length, compared with other materials. Cell wall polymers are amorphous polymers existing above their ’glass’ transition temperature, so that considerable segmental motion is possible. At ambient temperatures, the cell wall is thus relatively soft ($\epsilon\sim 1$ MPa) and easily deformed. Cells can grow to some extent simply by stretching their walls as they take up water. However, continued cell expansion involves synthesis of new wall material. Synthesis of cellulose at the plasma membrane and pectin and hemicelluloses components with Golgi apparatus deposits layers on the inside of the existing cell wall. A mechanical prerequisite for the unidirectional growth of pollen tubes for the (scalar) hydrostatic pressure is a softer cell wall at the tip of the cell, and more rigid at the distal part (Geitmann and Steer, 2006; Fayant et al., 2010). This gradient of mechanical properties is generated by the absence or scarcity of callose and cellulose at the tip (Aouar et al., 2009) as well as by the relatively high degree of esterification of the pectin polymers in this region. The gradient in cell wall composition from apical esterified to distal de-esterified is reported to be correlated with an increase in the degree of cell wall rigidity and a decrease of visco-elasticity (Parre and Geitmann, 2005). Also microindentation studies show (Fig. 2 in Winship et al., 2010) that the pollen tube tip is less rigid and that the distal stiffness may be opposed to apical softness. Needless to say, to sustain growth processes, a balance between the mechanical deformation of the viscoelastic cell wall and the addition of new cell wall material must be achieved (Kroeger and Geitmann, 2011b). Turgor pressure is the pressure of the cell sap against the wall in plant cells. This is a force exerted outward on a plant cell wall by the water and solutes contained in the cell. As a result it gives the cell rigidity. An excess turgor pressure or cell wall local weakening can result in the bursting of a cell. Both constituents, turgor pressure and the wall properties are decisive for the mechanical properties and dynamics of the developing plant cell. The osmotically maintained (hydrodynamic) turgor pressure in living plant cells and the mechanical properties of the cell wall itself are among the most fundamental physical factors that dictate both cell growth and cell morphogenesis in plants (Schopfer, 2006). In fact, the physical properties of the wall and the turgor pressure have pivotal functions since they represent the ’downstream parameters’ of all cellular signaling events (Chebli and Geitmann, 2007). For our future purposes we note that turgor pressure is high in pollen tubes: 0.1 – 0.4 MPa in lily (Benkert et al. 1997; Winship et al., 2010). The pollen tube geometry can be described by two different symmetries - a hemisphere shaped apex, and a cylindrical shank zone, that are connected by a transition zone between the two parts. In the present paper we propose a mechanism of _symmetry frustration_ occuring in this transition zone between the two involved symmetries as a possible mechanism responsible for growth rate oscillations. In simple terms by symmetry frustration we mean that a small ring of cell wall (hereafter referred to as interface $\Gamma$, Fig. 1B) is unable to ’decide’ if it should behave as an elastic cylinder or an elastic sphere. Following this hypothesis, oscillations may arise because this mesoscopic ring behaves as if it ’jumps’ periodically between the two mechanical states of different capacity of strain energy. The application of growth tensors to developing plant organs has been known for a long time (Kutschera, 1989). Such mathematical description has been utilized to apical meristems where the proliferating cells produce tissue stresses, which in turn influence the structure of the developing organ and, hence the principal directions of growth (Hejnowicz, 1984). A different situation is observed for elongating plant organs, such as coleoptyles or hypocotyles, or elongation zones in roots, where cell division rarely takes place. Also, the directional evolution of a single cell is primarily observed for the elongating pollen tube. As it may be expected, various stresses occur in the elongating cell because of different properties of cell walls exposed to the turgor pressure maintained by the gradient in the water potential (Kutschera, 2000). The distribution of the wall stresses as well as deformation of the particular wall layers can be calculated by solving equilibrium equations of elasticity theory. Such equations may help to locate deformation of a cell wall, exposed to an internal turgor pressure. The equilibrium equation may be derived both for materials deformed elastically (deformation vanishes when force equals zero) or non-elastically (plastic deformation survives, even when the acting force is removed). In this article we concentrate initially on elastic properties, because the highly controversial and uncertain mechanism of the oscillatory growth of pollen tubes is our main concern³ (Zonia and Munnik, 2011; Kroeger et al., 2011; Winship et al., 2011). Though, plastic properties are inherently present in the proposed model _via_ the derived (anharmonic) ’frustration potential’, and the assumed cell wall building processes located in the sub-apical, annular region, presumably at (about) the $\Gamma$ – interface (Zonia and Munnik, 2008; Geitmann and Ortega, 2009). [FOOTNOTE:3][ENDFOOTNOTE] Possible mechanisms have been proposed to account for the oscillation of pollen tube growth rate in quantitative terms (Bartnicki-Garcia et al., 2000; Feijo et al., 2001; Dumais et al., 2006). A model for calcium dependent oscillatory growth in pollen tubes has been put forward (Kroeger et al., 2008). More recently the finite elment technique was used (Fayant et al., 2010) to establish biomechanical model of polar growth in walled cells. Also, the chemically mediated mechanism of mechanical expansion of the pollen tube cell wall by which deposition causes turnover of cell wall cross-links thereby facilitating mechanical deformation was set forth (Rojas et al., 2011). The role of wall ageing in self-regulation in tip-growth was considered in (Eggen et al., 2011), while a model of plasma membrane flow and cytosis regulation in growing pollen tubes was discussed in (Chavarria-Krauser and Yejie, 2011). The irreversible expansion of the cell wall during growth as the extension of an inhomogeneous viscous fluid shell under the action of turgor pressure, fed by a material source in the neighborhood of the growing tip was examined in (Campas and Mahadevan, 2009). However, none of the models produced oscillations on mechanical basis⁴. [FOOTNOTE:4][ENDFOOTNOTE] In our approach, which does not contradicts previous achievements, we explore the relationship between turgor pressure and nontrivial cell geometry by changing loss of stability picture (Wei and Linthilac, 2007) to encompass cell wall mechanical properties in cylindrical and spherical geometries, both present in rapidly extending pollen tube. We base our physical model on the parametrized description of a tip growing cell that allows the manipulation of cell size, cell geometries, cell wall thickness, and local mechanical properties. However, the mechanical load (contrary to op. cit.) is applied in the form of hydrostatic (constant) pressure. An important feature of pollen tubes elongation is that growth rate oscillates and, additionally, many of the underlying processes also oscillate with the same period, but usually with different phase (e.g. Fig. 1c in (Zonia and Munnik, 2011)). However, the role of the oscillating ion gradients and fluxes in the control of pollen tube growth (Hepler and Winship, 2010) is beyond the scope of this paper and we will not discuss it here. Nonetheless, the outlined scenario leaves space for the periodical ion and mass fluxes in and out of the cell. We also share the fundamental view expressed in (Rojas et al., 2011; Proseus and Boyer, 2006) that the wall extension is primarily a biophysical (mechanical) process, although ultimately dependent on enzymatic activity, and that under conditions where the enzymatic background can be subtracted the biophysical process still proceedes normally. Any new model should deliver testifiable and quantitative predictions that can be validated by experimental data. In case of pollen tubes, it is necessary to present predictions that go beyond stress values which are inherently difficult to measure. The presented model satisfies this requirement offering, among others, an experimentally testifiable power law ($\omega\propto\sqrt{P}$) between the turgor pressure $P$ and growth oscillation angular frequency $\omega$. ## 2 Results and discussion In search for the cause of experimentally observed pollen tube growth oscillations we link analytic stress/strain relations with the mechanical properties of a tip growing cell, located at the perimeter of hemispherical dome and cylindrical shank. This is based on the observation that cell wall assembly by exocytosis occurs mainly at an annular region around the pole of the cell (Geitmann and Dumais, 2009; Zonia and Munnik, 2009) and that the concomittant turgor driven deformation of the cell wall causes characteristic strain in the hemisphere shaped apex of the cell (Fayant et al, 2010; Rojas et al., 2011). The dynamical properties of such a complex growing system should be found self-consistently (meaning that the turgor pressure and the wall mechanical properties are conjugate magnitudes that usually form coupled equations, which have to be solved by iteration methods). Nevertheless, in first approximation the following heuristic solution can be proposed. Assuming an intrinsic turgor pressure $P$, and a much smaller external pressure, of yet unspecified origin $p_{\mathrm{ext}}$ (it can be just atmospheric pressure) producing an effective pressure $\tilde{P}=P+p_{\mathrm{ext}}$) the equilibrium equation for the displacement vector $\vec{u}$, which is the shortest distance from the initial to the final position of a moving point, takes the form (Landau and Lifshitz, 1986): \[2\;(1-\nu)\;\mathrm{grad}(\mathrm{div}\;\vec{u})-(1-2\nu)\;\mathrm{curl}( \mathrm{curl}\;\vec{u})=\vec{0}\] (1) where $\nu$ is the Poisson coefficient, which is the ratio, when a sample object is stretched, of the contraction or transverse strain, to the extension or axial strain. From now on Young modulus $\epsilon(z)$ and Poisson coeficient $\nu$ are assumed as picewise constant functions, so they remain constant on the interface $\Gamma$. Note, that by acting divergence operator on both sides of Eq. (1) we receive $\bigtriangleup\;\mathrm{div}\;\vec{u}=0$, i.e. $\mathrm{div}\;\vec{u}$ denoting the volume change due to displacement field is a harmonic function satisfying Laplace’s equation. Eq. (1) may be solved analytically, providing that a problem exhibits a high degree of symmetry. In particular it can be solved exactly for spherical and cylindrical symmetries. Both symmetries are present in the description of pollen tube self-similar elongation, since the distal part resembles cylindrical tube while the apex is a hemispherical dome (Fig. 1B). Thus, the symmetries present in both subdomains should be utilized in the description of pollen tube shape and dynamical properties. By assuming cylindrical symmetry (for the displacement vector field $\vec{u}=(u_{r},u_{\phi},u_{z})=(u_{r},0,0)$, which is obtained under the assumption that the total length of the cylinder part remains constant (the axial elongation of the more rigid shank upon application of a constant internal pressure $P$ we assume as negligible), and hence we accept $u_{z}=0$ in the Ansatz $(u_{r},u_{\phi},u_{z})=(u_{r},0,0)$ instead of $(u_{r},u_{\phi},u_{z})=(u_{r},0,u_{z})$, which would lead to unavoidable numerical solution of Eq. (1)), and representing field operators (grad, div and curl) in cylindrical (polar) coordinates, Eq. (1) can be reduced to a much simpler form: \[\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}(ru_{r})\right]=0.\] (2) This differential equation can be solved for the displacement field $u_{r}$ to yield the displacement for the cylindrical symmetry \[u_{r}^{c}=ar+\frac{b}{r},\] (3) where $a$ and $b$ are constants to be determined from the boundary conditions (Landau and Lifshitz, 1986). Also, by introducing spherical coordinates with the origin in the center of a sphere the displacement field $\vec{u}$ is a function of the radius $r$: $\vec{u}=(u_{r},u_{\theta},u_{\phi})=(u_{r},0,0)$. Therefore $\mathrm{rot}\;\vec{u}=0$ and Eq. (1) reads: $\mathrm{grad}\;\mathrm{div}\;\vec{u}=0$. Hence, for the spherical symmetry we receive for displacement \[u_{r}^{s}=ar+\frac{b}{r^{2}},\] (4) where the upper index in Eqs (3) and (4) has been substituted to differentiate solutions for cylindrical (c) and spherical (s) geometries. The geometry of the elongating pollen tube can be described with a cylinder of radius $r_{T}$ capped by a half prolate spheroid with short radius $r_{T}$ and a long radius $r_{L}$ (see e.g. Fig. 1a in (Fayant et al., 2010)). Thus, as a model for normally growing tube we propose a thin-walled hollow cylinder ended by a hemispherical shell immersed in an external pool of pressure $p_{\mathrm{ext}}$ and filled with a cell sap with turgor pressure $P$ (we equate both radii $r_{T}=r_{L}$, for simplicity). The inner radius of the cylinder and a sphere is $r_{1}$, while the outer radius is $r_{2}$ (Fig. 1B). Another simplifying assumption of the model is that we deal with weak (elastic) interactions at the interface $\Gamma$, due to wall building processes occuring at this region, and hence the deflection field may slightly differ on both sides (from the mechanical point of view they may be treated as weakly coupled). It is consistent with the view that deposition chemically loosens the wall by breaking load–bearing cross–links while simultaneously creating new, load–free cross–links, thereby effecting a fail-safe scenario for mechanical expansion (Rojas et al., 2011). Based on the proposition that the mechanical cell wall properties at the growing tip must be different from those in the shank, it was suggested (Dumais et al., 2004; Kroeger et al., 2008) that an anisotropy in the cell wall elasticity is required to account for the transition between spherical and tubular shape at the tip of the cell. Also, it was found (Geitmann and Parre, 2004) that the rigidity of the tip of the pollen cell was an increasing function of the distance from the apex. Therefore, the elastic properties of the cylinder imitating the cell wall at the shank and the hemispherical tip are represented by two pairs of material constants: Young’s modulus $\epsilon$, also known as the tensile modulus, which is a measure of the stiffness of an elastic material and is a quantity used to characterize materials, and Poisson coefficient $\nu$. In further calculations, we assume different values for these coefficients for distal (thick and rigid) and apical (thin and elastic) walls of a pollen tube. Such assumptions, about different values of mechanical constants at the apical dome and cylindrical shank, are fully justified (see e.g. Fig. 4 in (Geitmann and Parre, 2004) , where a spatial distribution of the Young’s modulus is presented). Because we consider relatively small elastic deformations, the stress and strain tensors are related by the Hooke’s law of elasticity (which is an approximation that states that the extension of a spring is in direct proportion with the load applied to it) and makes the deformation reversible. For the radial part of the stress tensor $\sigma_{ij}$ we have: $\sigma_{rr}=-p_{\mathrm{ext}}$ at $r=r_{2}$ and $\sigma_{rr}=-P$ at $r=r_{1}$. Since the off-diagonal elements vanish, we are left with the strain $u_{rr}=\frac{du_{r}}{dr}=a-\frac{b}{r^{2}}$, $u_{\phi\phi}=\frac{u_{r}}{r}=a+\frac{b}{r^{2}}$ and $u_{zz}=0$, the interesting radial $\sigma_{rr}$ element of the stress tensor reads⁵: [FOOTNOTE:5][ENDFOOTNOTE] \[\sigma_{rr}=\frac{\epsilon}{(1+\nu)(1-2\nu)}\left[a-(1-2\nu)\frac{b}{r^{2}} \right].\] (5) By assuming boundary conditions as above, parameters $a$ and $b$ can be calculated (Lewicka and Pietruszka, 2009). They both depend on material constants: Young’s modulus $\epsilon$ and Poisson coefficient $\nu$, cylinder geometry ($r_{1}$, $r_{2}$ – radii) and pressure values $P$ and $p_{ext}$. Quantitative calculations steming from Eqs (3) – (5) are presented in Figs 3 – 4. In Fig. 3A we observe (a) lowering of deformation $u_{r}$ with radius $r$ of apical hemispherical shell (b) almost constant $u_{r}$ for a thin-walled cylinder. This fact produces a wall stress presented in Fig. 3B equal to about 60 MPa (for the model parameters), which seems big enough to cause cell wall instability at $\Gamma$ -- interface⁶. The calculated wall stress of the order of tenths of MPa is enough to cause local wall instability (radial strain), subsequent axial relaxation, wall building and unilateral cell expansion at the interface between the hemispherical apex and cylindrical shank. Unavoidable repetition of this process, owing to a constant effective pressure and a positive feedback mechanism necessary to drive oscillations (to overcome damping due to viscosity), may generate observable oscillations which continue until the wall building processes expire. Depending on the wall thickness, the calculated stress is equal to the distance between the curves (a) and (b) (inset). The tensile stress difference calculated at the apex and the distal part of the pollen tube cell wall ($\sigma_{rr}(\mathrm{apex})-\sigma_{rr}(\mathrm{distal})$) is shown in Fig. 3C. Here it is parametrized by the turgor pressure $P$ acting on the cell wall. The tensile stress in the wall clearly rises, as we increase the turgor pressure – this would, in turn, cause an increased growth oscillation frequency, as it is observed experimentally in the transition from hypertonic, through isotonic to hypotonic conditions (Zonia and Munnik, 2007). [FOOTNOTE:6][ENDFOOTNOTE] The influence of turgor on the oscillation period, as predicted by the model described in (Kroeger and Geitmann, 2011a), is also in agreement with our results presented in Fig. 4A and their consequences: the higher the pressure, the higher the oscillation frequency. Tensile stress difference at the transition zone, parametrized by the wall thickness, is presented in Fig. 4B. From the expression for tensile stress difference due to the symmetry change (note, curvature discontinuity and stress singularity occur at the transition zone) $\sigma_{rr}^{c}-\sigma_{rr}^{s}\equiv\sigma_{rr}(\mathrm{cylinder})-\sigma_{rr }(\mathrm{sphere})$ which equals (6) and from the opposite formula: $\sigma_{rr}^{s}-\sigma_{rr}^{c}$, it is shown that calculations performed for an infinitesimally narrow ring $\Gamma$, where both geometries (cylindrical and spherical) intervene, a _symmetry frustration_ – leading to oscillations of the radial part of the stress tensor – may take place. ’Frustration’ originates from the fact that none of the locally involved symmetries is distinguished. On the other hand, the calculated from Eq. (6) (by evaluating $E=\int\sigma_{rr}dr$) strain energy density reads: \[E_{\pm}\propto\pm P\;\frac{{r_{1}}^{2}{r_{2}}^{2}\left[2r^{3}+2r\left({r_{1}}^ {2}+{r_{1}}{r_{2}}+{r_{2}}^{2}\right)-{r_{1}}{r_{2}}({r_{1}}+{r_{2}})\right]}{ 2r^{2}\left({r_{1}}^{4}+{r_{1}}^{3}{r_{2}}-{r_{1}}{r_{2}}^{3}-{r_{2}}^{4} \right)}\] (7) The quasi–discrete energy levels $E_{-}$ and $E_{+}$ (possesing, however, a small dispersion $\delta E\cong 0.0003$ [energy u.]) presented in Fig. 5 are non–degenerate due to the existence of a constant turgor pressure $P$ which leads to the observed splitting (see also upper inset). Still, since both levels originate from the symmetry change at the critical limit considered in this work, they can be attributed to the oscillations in the pollen tube growth functions. Thus, the _resonating frequency_ of growth (growth rate) corresponds to the energy difference $E_{+}-E_{-}\cong 2E$ (since $E_{+}\cong-E_{-}$), which in turn is directly proportional to the turgor pressure $P$. Consequently, we may equate the transition energy $2E$ between the resonating levels (Fig. 5) with pollen tube oscillation frequency observed in experiments (Eq. (7) implies that if $P=0$ then the system exhibits no oscillations, which is exactly the case, see also the plot of the potential energy $U(r)$ at $r=r_{0}$ in Fig. 6). Notwithstanding, we note that the considered effect is exclusively connected with geometrically induced stress in the wall which may be linked with symmetry frustration⁷, and one can express it in measurable units [Pa m]. Indeed, calculating definite integral over the function expressed by Eq. (6): $\int_{r_{1}}^{r_{2}}[...]dr$ (with $r_{1}=5$ and $r_{2}=5.25$$\mu\mathrm{m}$) we receive the strain energy density: $E_{\pm}=\pm 0.256$ [MPa $\mu$m]. Therefore, the difference of strain energy density between the two levels is about 0.5 MPa for micrometer length scales typical for the width (which is about 250 nano-meters) of the pollen tube cell wall. Such energy density may lead to oscillations which are observable not only in growth rates. From our model it can also be deduced that the apical geometry oscillates (due to deformation $u_{r}$ located initially at $\Gamma$, compare Figs 3B and 4B in Pietruszka et al., 2012) to produce the so called pearled morphology (Rojas et al., 2011). Relicts (residues) of such deformations at $\Gamma$ are traced in Fig. 1c and 6a, c (ibid.) as crests smeared out on a distance $\lambda$, in agreement with our model⁸. However, what else draws our attention is that there is no sign of deflecions at a distance shorter than the tube radius $R$. The latter observation further supports the main idea presented in this paper of specific role of the $\Gamma$ – interface in initiating oscillations. By assuming, after Rojas et al. (2011), the value of the linear $v_{\mathrm{avg}}=0.2$ [$\mu$m/s] of the elongating cell and taking the average oscillation period $T=50$ s from Fig. 7 we receive the observed wavelength of about $\lambda=10$ [$\mu$m], which is a doubled value of the radius $R$, as it should be expected assuming the correctness of our approach. In this picture, the distance $\lambda$ comprise the local deflection/wall stress/stress relaxation/recovery through wall building processes for every period $T$. [FOOTNOTE:7][ENDFOOTNOTE] [FOOTNOTE:8][ENDFOOTNOTE] Transitions between the states of different symmetry are shown in Fig. 5. The system is pumped with energy to jump over the analytic discontinuity (energy gap) between the (hemi-)spherical and cylindrical geometry. As a positive feedback mechanism necessary to drive the oscillations (to prevent damping due to viscosity) the energy is absorbed (ATP–pumping (Rounds et al., 2011)) in the transition zone above the state $E_{-}$ producing the exited state $E_{+}$. Then the system returns (by spontaneous symmetry breaking, to reduce the energy of the overall system) to the lower symmetry (cylindrical) state $E_{-}$ stimulating axial expansion. Both transitions (up and down) close one growth cycle with the prediced transition’s rate $\omega\propto\frac{2\pi}{T}$, where $T$ is the period. The whole process is repeated at the expense of pressure $P$ and ATP–energy needed for wall synthesis (exhibiting growth oscillations), and eventually expires or reaches critical instability (the cell bursts). In a mechanical anharmonic oscillator, the relationship between force and displacement is not linear but depends upon the amplitude of the displacement. The nonlinearity arises from the fact that the spring (here: cell wall) is not capable of exerting a restoring force that is proportional to its displacement because of stretching in the material comprising the wall. As a result of the nonlinearity, the vibration frequency and amplitude can change, depending upon the system elements displacement upon pressure $P$. An approximate derivation performed in Appendix 1 delivers the analytic form of the (dual) ’frustration potential’, which is a sum of attractive and repulsive forces, possibly responsible for experimentally observed growth rate oscillations in pollen tubes. The latter, which is given by Eq. (12) and visualized in Fig. 6, we describe here shortly: Pollen tube oscillations are trapped at the potential well about the equilibrium point $r_{0}$ for the corresponding symmetric (harmonic) potential. Oscillation frequencies and amplitudes of the anharmonic potential depend upon the turgor pressure values, see Eqs (6) and (7), as it is observed (e.g. Fig. 4 in Kroeger et al., 2011; Kroeger and Geitmann, 2011a)). Wall expansion is allowed by molecular separation ($r$ – values) exceeding those of harmonic potential. Dissociation energy at zero potential level corresponds to system instability (burst at $\Gamma$). The constant turgor pressure, as taken from Fig. 5, induces the value of $U(r)$, hence the frequency of the oscillation and its amplitude. The low-lying (trapped) values deliver high frequencies and small amplitudes, while the higher-lying potential values – low frequencies and larger amplitudes of oscillations. Above the critical threshold (corresponding to ’zero energy’ at the vertical scale) a bond breaking occurs and the pollen tube burst at the transition zone, or deliver male gametes completing its function. The lower plot represents only one branch (of the prevailing cylindrical symmetry) of the full frustration potential; the second branch (above) is in ’dual’ subspace, and the oscillations take place between both branches. For the negative values of the cylindrical symmetry, as shown in the plot, the mechanism of symmetry breaking favorizes this ’lower order’ (cylindrical) symmetry for cell extension. The presented in Fig. 6 frustration potential is a more convenient model for vibrational structure of wall constituing molecules than harmonic oscillator potential, because it explicitly includes the effects of bond breaking and accounts for anharmonicity of real bonds in the extending cell wall. It is also responsible for the inherent instability at the $\Gamma$ – interface of a growing tube (and – in consequnce – polymer building process), which can be experimentally supported by the fact that the pollen tubes always rupture at the transition zone where the radial part of the strain tensor is considerable (Pietruszka et al., 2012, Movie S3). The form of the potential also contributes to the long debate among plant physiologists about the elastic/inelastic extension of plant cell wall in simple terms: any departure from parabola centered around $r_{0}$ will lead to plastic extension, corresponding to elongation growth (Fig. 6). In addition, the infinite potential barrier at low distance $r$ values prevents the growing cell wall from shrinking at a given pressure $P$. In order to calculate the value of the resonance angular frequency $\omega$ we momentarily accept the approximate (classical) relation: $E\propto\omega^{2}$. Assuming $P=0.3$ MPa, taking the approximate $A$ constant from the fit (see Fig. 7) we receive $f\cong 0.09$ Hz, a value which belongs to the observed frequency spectrum in pollen tube growth functions ($0.01$ Hz – $\sim 0.20$ Hz, (Haduch and Pietruszka, 2012) for tobacco, Fig. 2; and (McKenna et al., 2009) for lily). As an aside, we stress that the calculated from Eq. (7) resonance frequency satisfies a power law $\omega\propto\sqrt{P}$ (see Appendix 2 for detailed derivation). The application of this important relation to the experimental data is presented in Fig. 7. It is easy to notice, that this relation ($\omega=2\pi A\sqrt{P}$, or equivalently $f=A\sqrt{P}$, where $A$ is a constant – connected with the wall mechanical properties – to be determined from experiment, and [P] = MPa), if inverted, can serve (after calibration) to estimate difficult to measure turgor pressure $P$ values from easy to measure oscillation frequencies (or periods $T$). Furthermore, it is clear that the material properties of the cell wall in the apical region should not be homogeneous, and therefore a proper mechanical description of growth must involve a gradient in material properties from the apical to the distal region (Fayant et al., 2010; Eggen et al., 2011). It has been shown (Eggen et al., 2011) that the calculated ”expansion propensity” as a function of the distance from the apex measured in units of the tube radius $R$ (notation, as in (Eggen et al., 2011)) shrinks to an area near the apex. Closer examination reveals that the inflection point is located at about 1 pollen tube cylinder radius $R$, a place where we perform our calculations. The latter statement means that the slope is the greatest at $z\sim R$ (in axial direction). This, and the fact that the ’dilution’ sector is shown (Fig. 2 in Eggen et al., 2011)) exactly at the limit of the two considered axisymmetric zones, is consistent with the view of intense changes of the wall mechanical properties at the limit of the distal and apical part. The ’dilution’ effect is caused in our model by a rapid surface expansion due to displacement $u_{r}$ of the $\Gamma$ – interface (see also (Parre and Geitmann, 2005): Fig. 1 – a local dip in the wall stiffness appears at about 10 $\mu$m from the apex, a place where our calculations are performed; and Fig. 7 (2) showing the possible localization of the transition zone from the cylindrical to spherical symmetry). Corresponding radial strain may trigger exocytosis that results in delivery of new cell wall material which rejuvenates this area. However, we should note, what we observe is not only according to the mechanical properties gradient, but mainly due to the changing symmetry at this place and the analytical consequences (curvature discontinuity) of this fact (compare e.g. Eqs 3 and 4). Likewise, observations, together with the analysis of Flourescence Recovery After Photobleaching (FRAP) experiments (Geitmann and Dumais, 2009; Bove et al., 2008) indicate that exocytosis is likely to occur predominantly in the same annular region (cf. (Geitmann, 2010), Fig. 1) where wall expansion rates are greatest. It is concluded in the same work, that tip growth in plant cells does not seem to happen exactly at the tip. Further supporting data is provided in (Zonia and Munnik, 2008), where the vesicle fusion with the plasma membrane was shown to approach to within 2-5 $\mu$m distal to the apex. The observations that growth is mainly at an annular region around the pole are in accord with calculations presented in this article. Probing mechanical properties at the perimeter of cylindrical and spherical part may result in calculation of the local rate of exocytosis (the conventional explanation for this phenomenon, i.e. oscillating exocytosis rate (see e.g. Fig. 4A in (Cardenas et al., 2008), Fig. 2 in (McKenna et al., 2009), is widely accepted). This may be roughly estimated by taking the oscillation frequency from Fig. 7. The read off value: $f\sim 0.02\:-0.03$ Hz is in accord with the main observed periodical mode (Zonia and Munnik, 2007) in the longitudinal power spectrum of pollen tube oscillatory motion, and presumably may be equated with the rate of exocytosis and new cell wall assembly in _Nicotiana tobaccum_ pollen tubes and temporal variations in the secretion of cell wall precursors (Kroeger and Geitmann, 2011a). This periodical growth activity, in turn, could be related (among others) to the relaxation of the stress (invoked by the hydrostatic turgor pressure) in the $\Gamma$ – interface, i.e. in close proximity to the advancing apex of the cell, probably at the vesicle delivery zone in the subapex, where the vesicles are released into the cytoplasm (Fig. 3 in (Kroeger et al., 2009)). In addition, even small stress/strain fluctuations at this narrow cylindrical ring, belonging to both adjacent zones, localized on its circumference could lead to macrosopically observable change of orientation of this ring and consequently direction change of the elongating pollen tube. This is indeed the case – see e.g. Fig. 1F in (Zonia and Munnik, 2008), and even more pronounced in Fig. 1 in (Calder et al., 1997). It seems that polar growth in pollen tubes is associated with spatially confined dynamic changes in cell wall mechanical properties (Zerzour et al., 2009; and Appendix 2). Furthermore, the time course of an experiment, showing very little change in turgor pressure during cell growth, was measured by pressure probe monitoring of growing _Lilium longiflorum_ (Winship et al., 2010, Fig. 1a) and re-analysed in (Zonia and Munnik, 2011, Fig. 3b-c). Even though direct measurements fail to indicate large-scale turgor changes during growth, rapid small-scale pressure changes (jumps) are visible, which presumably⁹ may be caused by the change of orientation of the tilt angles of wall building cellulose microfibrils in subsequent growth cycles in the considered transition zone¹⁰. As stated (ibid.), the measured periodicity for pressure oscillations ranges from 12 s to 25 s, which is the same as the routinely reported for oscillatory dynamics in lily pollen tubes (McKenna et al., 2009; Zerzour et al., 2009) and approximately agrees with the calculated frequency ($f=1/12\cong 0.09$ Hz) from our model (see next paragraph). Closer examination of Figs 3a-c in (Zonia and Munnik, 2011) reveals, in addition, a slight but steady diminishing of turgor (negative slope) which can be a consequence of cell volume expansion in every cycle. The latter observation may be associated with the passive role of the turgor pressure in pollen tube growth, at least at unchanging osmotic potentials. In conlusion, we agree with the view, that in the growth process the (main) energy supply is derived from turgor, while the growth rate and direction from the wall local properties (Winship et al., 2010). [FOOTNOTE:9][ENDFOOTNOTE] [FOOTNOTE:10][ENDFOOTNOTE] Expansive growth in a plant cell relies on the interplay between the internal turgor and the forces in the cell wall opposing deformation. Which of the two parameters controls the dynamics of growth has been controversial in the case of pollen tube growth (Zerzour et al., 2009). The long-standing model of pollen tube growth considers that cyclic changes in cell wall properties initiate growth (Holdaway-Clarke and Hepler, 2003; Winship et al., 2010). A new model based on accumulating data from recent work indicates that oscillations in hydrodynamic flow and intracellular pressure initiate growth (Zonia 2010; Zonia and Munnik, 2011). Both models agree that once growth initiated, osmotic pressure drives cell elongation. We have shown that the rapid polar growth phases during oscillatory growth in pollen tubes may be preceded by a strain – induced softening of the cell wall at the brink of the apical and distal parts ($\Gamma$). We also showed that cellular turgor pressure does not need to undergo changes during these repeated growth phases to display periodicity in growth. However, turgor pressure still preserves an important role, together with the cell wall mechanical properties, in controlling the dynamics of pollen tube growth by changing wall stress and hence oscillation frequencies in the different osmotic environments – the frequency of oscillatory pollen tube growth in our model can be altered by changing the osmotic potential value of the surrounding medium (Kroeger and Geitmann, 2011a). There are many observations of oscillations that could affect growth rates (such as wall material deposition and extracellular ion fluxes). However, even from purely mechanical calculations, performed at the boundary between the wall cylinder shell and hemispherical shell at the apex, the following picture for the sequence of events for the elongating pollen tube emerges. A given (constant) turgor pressure produces _different_ strain at the apical and distal wall parts possessing various mechanical properties and different symmetries. This phenomenon is especially important at the narrow interface between these neighbouring parts. Consequently, a localized elevated stress of the order of tenths MPa is invoked in the wall causing serious instability at the brink of both sections generated by symmetry frustration, and the cell wall relaxation (loosening) process takes place in order to reduce tensile stress. This initiates a wall building process (which is an implicit assumtion of the model) in meridional direction. However, the effective turgor pressure produces strain at the similar location (in the co-moving – with the moving tip – reference frame) axially equidistant from the tip and the whole process/cycle repeats. This eventually results in time – periodicity in the growth dynamics recognized in the literature as pollen tube oscillations. ## 3 Final comment This article offers a nontrivial solution for a long – sought mechanism of pollen tubes growth oscillations, which is the subject of swirling controversy in the field. It is based on the phenomenon of _symmetry frustration_ of the cell wall in the apical region. This simple physical mechanism results in anlytically determined asymmetric ’frustration potential’, the appearance of the landscape of ’discrete’ energy levels at (different) constant pressures and the aforementioned oscillatory growth comes from the transitions between them. Moreover, a scaling relation between the turgor pressure $P$ and the angular frequency of the oscillations $\omega$ is derived, which is represented by a power 1/2 – law ($\omega\propto\sqrt{P}$). Later on, this prediction is successfully verified against a real plant physiological experimental data. ### Acknowledgements Author thank dr Paweł Gusin, string theory specialist, Institute of Theoretical Physics, University of Wrocław, Poland and Professor dr hab. Jan Sładkowski, Department of Astrophysics and Cosmology, Institute of Physics, University of Silesia, Katowice, Poland for critical reading of the manuscript. I feel specially indebted to David Logan, Coulson Professor of Theoretical Chemistry Oxford University, Physical and Theoretical Chemistry, Oxford, United Kingdom for expressing his kind opinion and encouragement about this work. ### Appendix 1 (i) The local force equation of motion in the mechanics of continuous bodies (due to Cauchy) reads (Lubliner, 2006): \[\partial_{j}\sigma_{ij}+F_{i}=0,\] (8) where i, j = 1, 2, 3, and Einstein’s summation convention is used. By ignoring any possible shear stresses and assuming for the infinitesimally small $\Gamma$ – interface $\sigma_{\phi\phi}=0$ and $\sigma_{zz}=0$ \[\sigma_{ij}=\left(\begin{array}[]{ccc}\sigma_{rr}&0&0\\ 0&\sigma_{\phi\phi}&0\\ 0&0&\sigma_{zz}\end{array}\right)\simeq\left(\begin{array}[]{ccc}\sigma_{rr}&0 &0\\ 0&0&0\\ 0&0&0\end{array}\right)\] (9) we receive $\sigma_{rr}=\sigma(r)$, which is the only matrix element that survives. From Eq. (8) we have \[\int\partial_{j}\sigma_{ij}dx^{i}+\int F_{i}dx^{i}=0\] (10) so since $\int F_{i}dx^{i}=-U(x)$ is the potential energy then $\int\partial_{j}\sigma_{ij}dx^{i}=U(x)$. Hence in our case $\int\partial_{r}\sigma_{rr}dr=\sigma_{rr}=U(r)$, and one gets: $U(r)=\sigma_{rr}=\sigma(r)$. (ii) By identyfying \[U(r)=\sigma(r)\] (11) we may follow Eq. (6) to receive \[\Delta\sigma^{\pm}=\pm\frac{\alpha}{r^{3}}\mp\frac{\beta}{r^{2}}+C\] (12) where $\alpha=P\frac{(r_{1}r_{2})^{3}}{r_{2}^{3}-r_{1}^{3}}$, $\beta=P\frac{(r_{1}r_{2})^{2}}{r_{2}^{2}-r_{1}^{2}}$ and $C$ is a constant. Next, in order to find $r_{0}$ we write $\Delta\sigma^{\prime}(r_{0})=0$ to get \[-\frac{3\alpha}{r^{4}}+\frac{2\beta}{r^{3}}=0\] (13) Hence \[r_{0}=\frac{3}{2}\frac{\alpha}{\beta}.\] (14) Since $U\equiv\Delta\sigma$ we can plot the ’symmetry frustrated’ potential $U(r)$, Fig. 6. (iii) Considering small oscillations around equilibrium $r_{0}$, we may introduce new coordinate $\rho$: $r=r_{0}-\rho$ and by substituting it to Eq. (12) receive \[U(x)=\frac{1}{r_{0}^{3}}\frac{\alpha}{(1-x)^{3}}-\frac{1}{r_{0}^{2}}\frac{ \beta}{(1-x)^{2}}\] (15) where $x=\rho/r_{0}$. By expanding both fractions for small $x$, we finally get the form for the harmonic potential: \[U(x)=\frac{4}{9}\frac{\beta^{3}}{\alpha^{2}}x^{2}-\frac{4}{27}\frac{\beta^{3}} {\alpha^{2}}\] (16) Comparing the above equation with the classical oscillator potential ($m=1$): \[U(x)=\frac{1}{2}\omega^{2}x^{2}+U_{0}\] (17) and using Eq. (12) we get $\omega^{2}\propto P$. Hence, the pollen tube oscillation frequency at the limit of small oscillations equals $\omega\propto\sqrt{P}$, where $P$ is the turgor pressure (compare also with Fig. 7, where the proportionality constant ($A$) is estimated from experiment). ### Appendix 2 Pollen tube oscillation: local deflection/wall stress/relaxation/recovery STAGES of one oscillation (mechanical view): 1. Recovery through wall building at $\Gamma$ – interface: visco-plastic process (elastic equations do not apply here, because of wall and mass production; the system is merely plastic and both subdomains are weakly coupled from the mechanical point of view). 2. Strain and deformation production on $\Gamma$ (the equations apply). 3. Wall stress production on both sides of $\Gamma$ and the resulting strain energy: visco-elastic process (equations apply). 4. Elastic strain energy relaxation in one cycle to produce elongation of one wave length λ (compare Rojas et al, 2011, Fig. 1C, Fig 6A): the phenomenology applies $f=A\sqrt{P}$, finding confirmation in comparison with authors’ (Haduch and Pietruszka, 2012) performed experiment, Fig. 7. The process repeats: next oscillation takes place (go to 1. to start another oscillation). ## References * [1] Aouar L, Chebli Y, Geitmann A (2009) Morphogenesis of complex plant cell shapes - the mechanical role of crystalline cellulose in growing pollen tubes. Sexual Plant Reprod 23: 15-27 * [2] Azuah RT, Kneller LR, Qiu Y, Tregenna-Piggott PLW, Brown CM, Copley JRD, Dimeo RM (2009) DAVE: A comprehensive software suite for the reduction, visualization, and analysis of low energy neutron spectroscopic data. J Res Natl Inst Stan Technol 114, 341 * [3] Bartnicki-Garcia S, Bracker C, Giertz G, Lopez-Franco R, Lu H (2000) Mapping the growth of fungal hyphae: orthogonal cell wall expansion during tip growth and the role of turgor. Biophys J 79: 2382-2390 * [4] Baskin T (2005) Anisotropic Expansion of the Plant Cell Wall. Annu Rev Cell Dev Biol 21: 203–22 * [5] Benkert R, Obermeyer G, Bentrup FW (1997) The turgor pressure of growing lily pollen tubes. Protoplasma 198: 1-8 * [6] Bove J, Vaillancourt B, Kroeger J, Hepler P, Wiseman PW, Geitmann A (2008) Magnitude and direction of vesicle dynamics in growing pollen tubes using spatio-temporal image correlation spectroscopy (STICS). Plant Phys 147: 1646-1658 * [7] Calder A, Franklin-Tong VE, Shaw PJ, Drobak BK (1997) Ca${}^{2+}$ oscillations in plant cells: initiation by rapid elevation in cytosolic free Ca${}^{2+}$ levels. Biochem Biophys Res Commun 29: 690-694. * [8] Campas O, Mahadevan L (2009) Shape and dynamics of tip-growing cells. Curr Biol 19: 2102-2107 * [9] Cardenas L, Lovy-Wheeler A, Kunkel JG, Hepler PK (2008) Pollen tube growth oscillations and intracellular calcium levels are reversibly modulated by actin polymerization. Plant Physiology 146: 1611-1621 * [10] Chavarria-Krauser A, Yejie D (2011) A model of plasma membrane flow and cytosis regulation in growing pollen tubes. J Theor Biol 285: 10-24 * [11] Chebli Y, Geitmann A (2007) Mechanical principles governing pollen tube growth. Funct Plant Sci Biotech 1: 232-245 * [12] Dumais J, SR Long, and SL Shaw (2004) The mechanics of surface expansion anisotropy in _Medicago truncatula_ root hairs. Plant Phys 136: 3266-3275 * [13] Dumais J, Shaw S, Steele C, Long S, Ray P (2006) Int J Dev Biol 50, 209-222 * [14] Eggen E., Keijzer MN, Mulder BM (2011) Self-regulation in tip-growth: the role of cell wall ageing. J Theor Biol 283: 113-121 * [15] Fayant P, Girlanda O, Chebli Y, Aubin CE, Villemure I, Geitmann A (2010) Finite element model of polar growth in pollen tubes. Plant Cell 22: 2579-93 * [16] Feijo JA, Sainhas J, Holdaway-Clarke TL, Cordeiro MS, Kunkel JG, Hepler PK (2001) Cellular oscillations and the regulation of growth: the pollen tube paradigm. BioEssays 23: 86-94 * [17] Geitmann A, Cresti M (1998) Ca${}^{2+}$ channels control the rapid expansions in pulsating growth of Petunia hybrida pollen tubes. J Plant Phys 152: 439-447. * [18] Geitmann A, Parre E (2004) The local cytomechanical properties of growing pollen tubes correspond to the axial distribution of structural cellular elements. Sex Plant Reprod 17:9-16 * [19] Geitmann A, Steer MW (2006) The architecture and properties of the pollen tube cell wall. In the pollen tube: A cellular and molecular perspective, Plant Cell Monographs, R Malho, ed (Berlin, Heidelberg: Springer Verlag), pp. 177-200 * [20] Geitmann A (2009) How to shape a cylinder - Pollen tube as a model system for the generation of complex cellular geometry. Sex Plant Reprod 23: 63–71 * [21] Geitmann A, Dumais J (2009) Not-so-tip-growth. Plant Signaling & Behavior 4: 136-138 * [22] Geitmann A, Ortega JKE (2009) Mechanics and modelling of plant cell growth. Trends Plant Sci 14: 467-478 * [23] Geitmann A (2010) Mechanical modeling and structural analysis of the primary plant cell wall. Current Opinion Plant Biol 13: 693-699 * [24] Haduch A, Pietruszka M (2012) Power spectrum, growth velocities and cross-correlations for longitudinal and transversal oscillations of _Nicotiana tabacum_ pollen tube – J Exp Bot, under review / asking for resubmission. Also: Power spectrum of _Nicotiana tabacum_ pollen tube. Int Conf Biophys Stud, 18.05-20.05.2012, Kraków, Poland * [25] Hejnowicz Z (1984) Trajectories of principal directions of growth, natural coordinate system in growing plant organ. Acta Soc Bot Pol 53: 29-42 * [26] Hepler PK, Vidali L, Cheung AY (2001) Polarized cell growth in higher plants. Annu Rev Cell Dev Biol 17: 159-187 * [27] Hepler PK, Winship LI (2010) Calcium and the cell wall - cytoplast interface. J Int Plant Biol 52: 147-160 * [28] Holdaway-Clarke, TL, Hepler PK (2003) Control of pollen tube growth: role of ion gradients and fluxes. New Phytol 159: 539-563 * [29] Kroeger JH, Geitmann A, Grant M (2008) Model for calcium dependent oscillatory growth in pollen tubes. J Theor Biol 253: 363-374 * [30] Kroeger JH, Daher FB, Grant M, Geitmann A (2009) Microfilament orientation constrains vesicle flow and spatial distribution in growing pollen tubes. Biophys J 97: 1822-1831 * [31] Kroeger JH, Zerzour R, Geitmann A (2011) Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth. PLoS One 6: e18549 * [32] Kroeger J, Geitmann A (2011a) Modeling pollen tube growth: feeling the pressure to deliver testifiable predictions. Plant Signaling & Behavior 6: 1828-1830 * [33] Kroeger J, Geitmann A (2011b) Pollen tube growth: getting a grip on cell biology through modeling. Mech Re Comm, doi:10.1016/j.mechrecomm.2011.11.005 * [34] Kutschera U (1989) Tissue stresses in growing plant organs. Physiol Plant 77: 157-163 * [35] Kutschera U (2000) Cell expansion in plant development. R Bras Fisiol 146: 126-132 * [36] Landau LD, Lifshitz EM. Theory of Elasticity. Vol. 7 (3rd ed.). Butterworth-Heinemann, 1986. * [37] Lewicka S, Pietruszka M (2009) Mathematical model of tissue stresses in growing plant cells and organs. Acta Soc Bot Pol 78: 19-23 * [38] Lubliner J. Plasticity theory. Pearson Education, Inc., Revised ed. 2006 * [39] McKenna ST, Kunkel JG, Bosch M, Rounds CM, Vidali L, Winship LJ, Hepler PK (2009) Exocytosis precedes and prdedicts the increase in growth in oscillating pollen tubes. Plant Cell 21, 3026-3040 * [40] Parre E, Geitmann A (2005) Pectin and the role of the physical properties of the cell wall in pollen tube growth of _Solanum chacoense_. Planta 220: 582-592 * [41] Pietruszka M (2012) A biosynthesis/inactivation model for enzymatic WLFs or non-enzymatically mediated cell evolution. J Theor Biol 315: 119-127 * [42] Pietruszka M, Lipowczan M, Geitmann A (2012) Persistent symmetry frustration in pollen tubes. PLoS ONE 7(11): e48087. doi:10.1371/journal.pone.0048087 * [43] Plyushch, T.A., Willemse, M.T.M., Franssen-Verheijen, M.A.W., and Reinders, M.C. (1995) Structural aspects of in vitro pollen tube growth and micropylar penetration in _Gasteria verrucosa_ (Mill.) H. Duval and _Lilium longiflorum_ thumb. Protoplasma 187: 13-21 * [44] Proseus TE, Boyer JS (2006) Identifying cytoplasmatic input to the cell wall of growing _Chara corallina_. J Exp Bot 57: 3231-3242 * [45] Rojas ER, Hotton S, Dumais J (2011) Chemically mediated mechanical expansion of the pollen tube cell wall. Biophysical J 101: 1844-1853 * [46] Rounds CM, Winship LJ, Hepler PK (2011) Pollen tube energetics: respiration, fermentation and the race to the ovule. AoB PLANTS 10.1093/aobpla/plr019 * [47] Schopfer P (2006) Biomechanics of plant growth. Am J Bot 93: 1415-1425 * [48] Wei C, Lintilhac PM (2007) Loss of stability: A new look at the physics of cell wall behaviour during plant cell growth. Plant Physiol 145:763-772 * [49] Winship LJ, Obermeyer G, Geitmann A, Hepler PK (2010) Under pressure, cell walls set the pace. Trends Plant Sci 15: 363-369 * [50] Winship LJ, Obermayer G, Geitmann A, Hepler PK (2011) Pollen tubes and the physical world. Trends Plant Sci 16: 353-355 * [51] Zerzour R, Kroeger JH, Geitmann A (2009) Polar growth in pollen tubes is associated with spatially confined dynamic changes in cell mechanical properties. Developmental Biol 334: 437-446 * [52] Zonia L, Munnik T (2007) Life under pressure: hydrostatic pressure in cell growth and function. Trends Plant Sci 12: 90-97 * [53] Zonia and Munnik (2008) Vesicle trafficing dynamics and visualization of zones of exocytosis and endocytosis in tobacco pollen tubes. J Exp Bot 59, 861-873 * [54] Zonia L, Munnik T (2009) Uncovering hidden treasures in pollen tube growth mechanics. Trends Plant Sci 14: 318-327 * [55] Zonia L (2010) Spatial and temporal integration of signaling networks regulating pollen tube growth. J Exp Bot 61: 1939-1967 * [56] Zonia L, Munnik T (2011) Understanding pollen tube growth: the hydrodynamic model versus the cell wall model. Trends Plant Sci 7: 1-6 <figure><img src="content_image/1211.1143/Fig1.png"><figcaption>Figure 1: Nicotiana tobaccum pollen tube apical region. (A) Microscopic view(B) Schematic view: radii of curvature r1 and r2, turgor pressure P and theinvestigated partition into two distinct regions (a narrow transition zone)are indicated in the chart.</figcaption></figure> <figure><img src="content_image/1211.1143/Fig2.png"><figcaption>Figure 2: Density plots of the longitudinal and transversal power spectrum ofNicotiana tabacum pollen tube obtained from the raw experimental data, biassubtracted (Haduch and Pietruszka, 2012) by Fourier analysis, calculated bythe power of the Nyquist criterion and Nyquist rate, for different osmoticenvironments (-1 corresponding to the hypotonic case, 0 – isotonic case, and2.5, 3 and 5 corresponding to 25, 30 and 50 mM NaCl in hypertonic conditions,respectively). A broad, narrowing valley at the centre of the lower plot isclearly visible – low frequencies seen for longitudinal modes are shiftedoutwards. Red colour indicate high intensity peaks. Interpolated by DAVE,developed at NIST (Azuah et al., 2009).</figcaption></figure> <figure><img src="content_image/1211.1143/x1.png"><figcaption>Figure 3: (A) Displacement ur due to the effective turgor pressure ~P (P=0.3MPa, pext=0.05 MPa) acting on the cell wall as a function of the radialdistance r from the pollen tube long axis. Different Young modulus ϵ andPoisson coefficient ν in both subsystems: (a) ur=usr for a hemispherical apex:r1=5μm, r2=5.25μm, ν=0.4, ϵ=0.2 [GPa] (b) ur=ucr for a cylindrical distalpart: r1=5μm, r2=5.25μm, ν=0.2, ϵ=1 [GPa]. (B) Tensile stress σrr due to theeffective turgor pressure ~P acting on the cell wall at the position where (a)the cylinder (shank) joins (b) the hemisphere (apex) as a function of theradial distance r from the pollen tube axis. Radial stress discontinuitybetween the distal wall and apical wall is proportional to the distancebetween the curves (a) and (b). The calculated maximum wall stress reachesabout 60 MPa for the simulation parameters. The strain energy leading tooscillations is proportional to the shaded area. (C) Tensile stress differenceσrr(apex)−σrr(distal) (in MPa) at the apex and the distal part parametrized bythe changing turgor pressure P acting on the cell wall: (a) P=0.5 MPa, (b)P=0.4 MPa and (c) P=0.3 MPa.</figcaption></figure> <figure><img src="content_image/1211.1143/x2.png"><figcaption>Figure 4: Tensile stress difference (A) σcrr−σsrr, upper curves, and theopposite case σsrr−σcrr, lower curves, calculated at the boundary zone betweenthe approximately hemispherical apical and the cylindrical distal part of thegrowing pollen tube. Parametrisation by the turgor pressure P acting on thecell wall (upper plots): P=0.1 (green), P=0.2 (blue) and P=0.3 MPa (violet).Remaining parameters for the respective wall geometries: r1=5 μm, r2=5.25 μm.(B) σcrr−σsrr, upper curves and the opposite σsrr−σcrr, lower curves,calculated at the boundary zone between the semispherical apical and thecylindrical distal part of the growing pollen tube, parametrized by the wallthickness (upper plots): r2=5.1 μm (green), r2=5.2 μm (blue) and r2=5.3 μm(violet). The inner wall radius: r1=5 μm; turgor pressure P=0.3 MPa.</figcaption></figure> <figure><img src="content_image/1211.1143/Fig5.png"><figcaption>Figure 5: Frustration energy EF splitting due to topological effects (Insets:bifurcation diagram and tensegrity force F=−∇U(r) balance diagram) in aquasi–2D biological system – pollen tube apical region. Corresponding symmetryexchange takes place between the resonating residual energy levels E−±δE andE+±δE of different major symmetry. Calculation performed at the transitionzone between the (hemi-spherical) apical and the (cylindrical) distal part ofa growing pollen tube (see Eq. (7)), at a constant turgor pressure P=0.3 MPa.The inner and the outer wall radius in both subsystems read: r1=5 μm, r2=5.25μm (wall thickness ∼250 nm), respectively. The dispersion of each energy levelδE≅0.0003.</figcaption></figure> <figure><img src="content_image/1211.1143/Fig6.png"><figcaption>Figure 6: Two branches of the anharmonic potential energy (frustrationpotential) U±(r)∝±αr3∓βr2, possibly leading to pollen tubes growth rateoscillations, as a function of wall constituing molecules separation r. (Here:α=β=1; in general the coefficients α and β are linear in P, see Appendix 1).Oscillations take place between U− and U+ potential energy level (by tunnelingthrough symmetry change) yielding ω, while the amplitude is determined by theactual pressure P level, in accord with experiments (Kroeger and Geitmann,2011a; Kroeger et al., 2011).</figcaption></figure> <figure><img src="content_image/1211.1143/x3.png"><figcaption>Figure 7: Least square fit of the experimental data (ω) as a function ofturgor pressure P for hypertonic (25 mM NaCl), isotonic and hypotonic (hypo-osmotic stress induced by the addition of water to the gel cultures) treatmentof Nicotiana tobaccum pollen tube (Haduch and Pietruszka, 2012) fitted to thesquare root function (Appendix 1) derived in this paper (ω=2πA√P; [A2]=m/kg).Stable turgor values correspond to those ranging between 0.1 and 0.4 MPa,which has been recorded using a turgor pressure probe, (Benkert et al., 1997).The initial point (0,0) added ’by hand’ in the chart is exact – the pollentube will not oscillate (ω=0) for the vanishing turgor pressure (P=0).</figcaption></figure>
1502.01611	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 56462, "num_imgs": 0, "llama3_tokens_count": 21587 }	[]	# Measure density for set decompositions and uniform distribution Maria Rita Iacò M.R. Iacò Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, 8010 Graz, Austria. iaco@math.tugraz.at Milan Paštéka M. Paštéka Pedagogická fakulta TU v Trnava, Priemyselná 4, P.O. BOX 9, Sk-918 43, Trnava, Slovakia. milan.pasteka@truni.sk Robert F. Tichy R. F. Tichy Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, 8010 Graz, Austria. tichy@tugraz.at ###### Abstract. The aim of this paper is to extend the concept of measure density introduced by Buck for finite unions of arithmetic progressions, to arbitrary subsets of ${\mathbb{N}}$ defined by a given system of decompositions. This leads to a variety of new examples and to applications to uniform distribution theory. Key words and phrases:Uniform distribution, measure theory 2010 Mathematics Subject Classification: 11K06, 11J71, 11A67, 28A05, 28C10 The first and third author are supported by the Austrian Science Fund (FWF): Project F5510, which is a part of the Special Research Program ”Quasi-Monte Carlo Methods: Theory and Applications” ## 1. Introduction and notation Let $S\subset\mathbb{N}$ be a subset of the set of positive integers. Then the limit \[d(S)=\lim_{N\to\infty}\frac{\#\{n\leq N;n\in S\}}{N}\] (if it exists) is called the _asymptotic density_ of $S$. Let us fix a positive integer $m\in{\mathbb{N}}$ and $a\in{\mathbb{N}}\cup\{0\}$. Clearly, \[a+(m)=\{x\in{\mathbb{N}};x\equiv a\ ({\rm mod}\ m)\}\ ,\] is an arithmetic progression, and $d(a+(m))=\frac{1}{m}$. Starting from the asymptotic density of finite unions of arithmetic progressions, Buck [2] defined the set function \[\mu^{\ast}(S)=\inf\left\{\sum_{k=1}^{n}\frac{1}{m}_{k};\ S\subset\bigcup_{k=1} ^{n}a_{k}+(m_{k})\right\}\ ,\] now called _Buck measure density_. In general, $d(S)\leq\mu^{\ast}(S)$ holds, but there are several examples of sets $S$ such that $d(S)\neq\mu^{\ast}(S)$. The system of sets defined by \[\text{$\mathcal{D}$}_{\mu}=\{S\subset{\mathbb{N}};\ \mu^{\ast}(S)+\mu^{\ast}({ \mathbb{N}}\setminus S)=1\}\] is an algebra of sets and its elements are called _Buck measurable sets_. Moreover, the restriction $\mu=\mu^{\ast}\|_{\text{$\mathcal{D}$}_{\mu}}$ is a finitely additive measure. Our aim is to extend the definition of $\mu(S)$ to a bigger class of sets $S$; in this context $\mu$ extends to a $\sigma$-additive measure. Let $\{s_{n}\}_{n\in\mathbb{N}}$ be a given sequence of numbers. Then the ”counting set function” $A(S;\{s_{n}\})$ is defined to be the set of positive integers given by (1.1) \[A(S;\{s_{n}\})=\{n\in{\mathbb{N}};s_{n}\in S\}\ .\] A sequence of positive integers $\{s_{n}\}$ is called _uniformly distributed in ${\mathbb{Z}}$_ (for short u.d. in ${\mathbb{Z}}$, details see in [16]) if and only if for every arithmetic progression $a+(m)$ we have \[d(A(a+(m));\{s_{n}\})=\frac{1}{m}\ .\] The following characterization of Buck measurability is proved in [21, Theorem 7, page 51]. Theorem 1.1.: _A set $S\subset{\mathbb{N}}$ belongs to $\text{$\mathcal{D}$}_{\mu}$ if and only if $d(\mathbb{A}(\{s_{n}\},S))=\mu^{\ast}(S)$ holds for every uniformly distributed sequence $\{s_{n}\}_{n\in\mathbb{N}}$ in ${\mathbb{Z}}$._ It is well-known that the uniform distribution property, introduced by H. Weyl [28] for sequences of real numbers in the unit interval, naturally extends to sequences on compact Hausdorff spaces and in topological groups (see e.g. [7, 15, 24]). In [25] the author provides a criterion for the uniform distribution of sequences in compact metric spaces. Let $({\mathcal{M}},\rho,P)$ be a compact metric space, with metric $\rho$ and a Borel probability measure $P$. A sequence $\{x_{n}\}_{n\in{\mathbb{N}}}$ in ${\mathcal{M}}$ is called _Buck uniformly distributed_ (for short B.u.d.) if and only if for every measurable set $H\subset{\mathcal{M}}$ with $P(\partial H)=0$ we have $A(H,\{x_{n}\})\in\text{$\mathcal{D}$}_{\mu}$ and $\mu(H,A(\{x_{n}\}))=P(H)$ (here $\partial H$ denotes as usual the boundary of $H$). The following theorem, proved in [19], is an analogue of Weyl’s criterion for B.u.d. sequences $\{x_{n}\}_{n\in\mathbb{N}}\in{\mathcal{M}}$. Theorem 1.2.: _A sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in ${\mathcal{M}}$ is Buck uniformly distributed if and only if for every continuous real valued function $f$ defined on ${\mathcal{M}}$ and for every sequence of positive integers $\{s_{n}\}_{n\in{\mathbb{N}}}$ uniformly distributed in ${\mathbb{Z}}$ we have_ \[\lim_{N\to\infty}\sum_{n=1}^{N}f(x_{s_{n}})=\int_{\mathcal{M}}fdP.\] In the same paper, the author proves the existence of B.u.d. sequences in ${\mathcal{M}}$. ## 2. General results Let $\mathbb{X}$ be an arbitrary set. For a fixed $n\in\mathbb{N}$, we denote by $\mathcal{E}_{n}=\{A_{1}^{(n)},\dots,A_{k_{n}}^{(n)}\}$ a system of disjoint decompositions of $\mathbb{X}$, i.e. $A_{i}^{(n)}\cap A_{j}^{(n)}=\emptyset$ and $\cup_{i=1}^{k_{n}}A_{i}^{(n)}=\mathbb{X}$. Each system of decompositions satisfies the following conditions extending the properties of arithmetic progressions: * For every family of sets $A_{h_{1}}^{(j_{1})},\dots,A_{h_{m}}^{(j_{m})}$ there exists an $s$ such that each of these sets is a union of sets belonging to $\mathcal{E}_{s}$. * If $\{h_{n}\}$ is an arbitrary sequence of indices, then the intersection $\bigcap_{n=1}^{\infty}A_{h_{n}}^{(n)}$ contains at most $1$ element. Let us denote by $\text{$\mathcal{D}$}_{0}$ the system of all sets of the form $A_{h_{1}}^{(j_{1})}\cup...\cup A_{h_{m}}^{(j_{m})}$. Condition (i) assures that $\text{$\mathcal{D}$}_{0}$ is an algebra of sets and let $\Delta:\text{$\mathcal{D}$}_{0}\to[0,1]$ be a finitely additive probability measure defined on $\text{$\mathcal{D}$}_{0}$. The set function \[\nu^{\ast}(S)=\inf\{\Delta(A);A\in\text{$\mathcal{D}$}_{0},S\subset A\}\] will be called the _measure density_ of the set $S$. Let us remark that this is the standard way of constructing the outer measure $\nu^{}$ starting from the finitely additive measure $\Delta$. In particular, the following result holds. Theorem 2.1.: _Let $\{c_{n}\}_{n\in\mathbb{N}}$ be a sequence in ${\mathbb{N}}$ such that for every $A\in\text{$\mathcal{D}$}_{0}$, there exists $n_{0}$ such that $A$ is a union of sets from $\mathcal{E}_{c_{n}}$, for $n\geq n_{0}$. Then for arbitrary $S\subset\mathbb{X}$ we have_ \[\nu^{\ast}(S)=\lim_{n\to\infty}\sum_{S\cap A_{j}^{(c_{n})}\neq\emptyset}\Delta (A_{j}^{(c_{n})})\ .\] _Moreover, if a set $S\subset\mathbb{X}$ has non-empty intersection with every set from $\mathcal{E}_{n},n=1,2,\dots$, then $\nu^{\ast}(S)=1$._ A set $S\subset\mathbb{X}$ is called $\nu^{\ast}$_-measurable_ if and only if $\nu^{\ast}(S)+\nu^{\ast}(\mathbb{X}\setminus S)=1$. We denote the Carathéodory extension of $\text{$\mathcal{D}$}_{0}$ by $\text{$\mathcal{D}$}_{\nu}$. By definition it is the system of all $\nu^{\ast}$-measurable subsets of $\mathbb{X}$. It follows from general measure theory that the system $\text{$\mathcal{D}$}_{\nu}$ is an algebra of sets and the restriction $\nu=\nu^{\ast}\|_{\text{$\mathcal{D}$}_{\nu}}$ is a finitely additive probability measure on $\text{$\mathcal{D}$}_{\nu}$ (see e.g. [10]). Now we provide some examples of systems of decompositions and related systems $\text{$\mathcal{D}$}_{\nu}$ of $\nu^{\ast}$-measurable subsets. Example 2.2.: _Take $\mathbb{X}={\mathbb{N}}$ and $\mathcal{E}_{n}=\{A_{1}^{(n)},\dots,A_{n!}^{(n)}\},n\in{\mathbb{N}}$, where $A_{j}^{(n)}=j-1+(n!),j=1,2,\dots,n!$. Then $\nu^{\ast}$ is the Buck measure density defined in [2]._ Example 2.3.: _Again let $\mathbb{X}={\mathbb{N}}$. Consider the system of decompositions $\mathcal{E}_{n}=\{A_{1}^{(n)},\dots,A_{n}^{(n)}\},n\in{\mathbb{N}}$. where $A_{1}^{(n)}={\mathbb{N}}\setminus\{1,\dots,n-1\},A_{j}^{(n)}=\{j-1\}$, $j=2,\dots,n$. In this case $\text{$\mathcal{D}$}_{0}$ consists of all subsets of ${\mathbb{N}}$ which are finite or have finite complement. Let $S\in\text{$\mathcal{D}$}_{0}$, put $\Delta(S)=1$ if $S$ is infinite and $\Delta(S)=0$ for $S$ finite. In this case $\nu^{\ast}(A)=1$ if and only if $A$ is an infinite set, and the system $\text{$\mathcal{D}$}_{\nu}$ coincides with $\text{$\mathcal{D}$}_{0}$._ Example 2.4.: _Let $\mathbb{X}=[0,1)\cap{\mathbb{Q}}$, and $\mathcal{E}_{n}=\{[\frac{k-1}{n},\frac{k}{n})\cap{\mathbb{Q}},\ k=1,\dots,n\}, n\in{\mathbb{N}}$ and $\Delta([\frac{k-1}{n},\frac{k}{n})\cap{\mathbb{Q}})=\frac{1}{n}$. Then $\nu^{\ast}$ is the Jordan upper measure defined on the system of subsets of $\mathbb{X}$. Here $\text{$\mathcal{D}$}_{\nu}$ is the system of Jordan measurable sets._ It is well-known that for every compact group there exists a probability measure defined on the system of its Borel subsets, invariant with respect the group operation (the normalized Haar measure, see for instance [10, 12]). In [8] the authors study certain finitely additive measures on topological groups and rings. Example 2.5.: _Let us first assume that $\mathbb{X}=\text{$\mathbb{G}$}$ is an infinite multiplicative locally compact abelian group and $\mathbb{H}$ a subgroup of finite index. Then the quotient $\text{$\mathbb{G}$}/\text{$\mathbb{H}$}$ is a finite group. Denote by $\Delta^{\ast}$ the normalized Haar measure measure on $\mathbb{G}$. Since it is invariant with respect the group operation we have_ \[\Delta^{\ast}(\text{$\mathbb{H}$})=\frac{1}{\|\text{$\mathbb{G}$}/\text{$ \mathbb{H}$}\|}\ ,\] _where $\|X\|$ denotes the cardinality of $X$. Let $S=\{\text{$\mathbb{H}$}_{n};n=1,2,\dots\}$ be a system of subgroups of $\mathbb{G}$ of finite index such that $\text{$\mathbb{H}$}_{i}\cap\text{$\mathbb{H}$}_{j}=\text{$\mathbb{H}$}_{k}$, for every $i,j,k$ and $\cap_{n=1}^{\infty}H_{n}=\{e\}$, where $e$ is the neutral element of $\mathbb{G}$. Thus for every $n$ we have a finite decomposition_ \[\mathcal{E}_{n}=\{a_{1}^{(n)}H_{n},\dots,a_{k_{n}}^{(n)}H_{n}\},\ k_{n}=\|\text {$\mathbb{G}$}/\text{$\mathbb{H}$}_{n}\|.\] _The system $\text{$\mathcal{D}$}_{0}$ consists of all sets of the form $g_{1}\text{$\mathbb{H}$}_{n}\cup\dots\cup g_{k}\text{$\mathbb{H}$}_{n}$, $g_{i}\in\text{$\mathbb{G}$}$, $n=1,2,\dots$. Let $\Delta$ be the restriction of $\Delta^{\ast}$ to $\text{$\mathcal{D}$}_{0}$ and $\nu^{\ast}$ the corresponding measure density._ _Let $\mathbb{G}$ be the free abelian group with countable set of generators $\{p_{1},p_{2},\dots\}.$ Let $\text{$\mathbb{H}$}_{n}$, $n=1,2,\dots$, be the subgroups generated by $\{p_{1}^{n},p_{2}^{n},\dots,p_{n}^{n},p_{n+1},p_{n+2},\dots\}$. Since every element of $\mathbb{G}$ can be written as product of a finite number of generators, we get the following disjoint decomposition_ \[\text{$\mathbb{G}$}=\bigcup_{0\leq j_{i}<n}p_{1}^{j_{1}}p_{2}^{j_{2}}\dots p_{ n}^{j_{n}}\text{$\mathbb{H}$}_{n}.\] _Thus $\text{$\mathbb{G}$}/\text{$\mathbb{H}$}_{n}$ contains $n^{n}$ elements and therefore $\Delta(a\text{$\mathbb{H}$}_{n})=\frac{1}{n^{n}}$, for $a\in\text{$\mathbb{G}$}$._ _In particular, if $\text{$\mathbb{G}$}={\mathbb{Q}}^{\ast}$ is the multiplicative group of positive rational numbers, then it can be considered as the free abelian group generated by all primes. In this case, the measurability is not compatible with the natural order relation on ${\mathbb{Q}}$. The inclusion_ \[(0,1]\cap{\mathbb{Q}}^{+}\subset\cup_{i=1}^{m}a_{i}\text{$\mathbb{H}$}_{n}\] _implies that the numbers $a_{i}$ take all the values $p_{1}^{j_{1}}p_{2}^{j_{2}}\dots p_{n}^{j_{n}},0\leq j_{i}<n$, thus $\nu^{\ast}((0,1]\cap{\mathbb{Q}}^{+})=1$. Analogously, we can show that $\nu^{\ast}((1,\infty)\cap{\mathbb{Q}}^{+})=1$. Thus these sets are not measurable._ Example 2.6.: _If $\text{$\mathbb{G}$}=\prod_{j=1}^{\infty}\text{$\mathbb{G}$}_{j}$ is the direct product of finite groups, then we can take $\text{$\mathbb{H}$}_{n}=\prod_{j=n+1}^{\infty}\text{$\mathbb{G}$}_{j}$ and in this case_ \[\Delta(a\text{$\mathbb{H}$}_{n})=\frac{1}{\|G_{1}\cdot...\cdot G_{n}\|},\ a\in \text{$\mathbb{G}$}\ .\] Let us return to the general setting. We will start by constructing a compact metric space containing $\mathbb{X}$ as dense subset. Then we define a Borel probability measure induced by $\Delta$. First, we define the metric on $\mathbb{X}$ based on the system of decompositions $\mathcal{E}_{n},n=1,2,\dots$. Let $x,y\in\mathbb{X}$ and put $\psi_{n}(x,y)=0$ if $x,y$ belong to the same set of $\mathcal{E}_{n}$, and $\psi_{n}(x,y)=1$ otherwise (for $n=1,2,\dots$). Define \[\rho(x,y)=\sum_{n=1}^{\infty}\frac{\psi_{n}(x,y)}{2^{n}}\ ,\] and $\rho$ is a metric on $\mathbb{X}$. In particular, (2.1) \[\rho(x,y)\leq\frac{1}{2^{N}}\] if and only if $x,y$ belong to the same set of every decomposition $\mathcal{E}_{n},n=1,\dots,N$. From condition (i) it follows that a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ of elements in $\mathbb{X}$ converges to an element $x\in\mathbb{X}$ if and only if for every $s=1,2,\dots$ there exists $n_{0}$ such that for every $n\geq n_{0}$ the elements $x_{n}$ and $x$ belong to the same set of $\mathcal{E}_{s}$. Similarly, one can define the concept of Cauchy sequence which leads to the completion of $\mathbb{X}$ in the usual way. Let $\bar{\mathbb{X}}$ be the completion of the metric space $(\mathbb{X},\rho)$ and for $S\subset\bar{\mathbb{X}}$ let $\bar{S}$ be its closure in $\bar{\mathbb{X}}$. Then, clearly \[\bar{\mathbb{X}}=\bar{A}_{1}^{(n)}\cup\dots\cup\bar{A}_{k_{n}}^{(n)}\] for $n=1,2,\dots$. Since a sequence of elements of $\mathbb{X}$ is defined to be fundamental if and only if for every $s=1,2,\dots$ there exists $n_{0}$ such that for $m,n\geq n_{0}$ the elements $x_{m}$ and $x_{n}$ belong to the same set of $\mathcal{E}_{s}$, then the sets $\bar{A_{1}^{(n)}},\dots,\bar{A}_{k_{n}}^{(n)}$,$n=1,2,\dots$ are disjoint. Thus they are open and closed and, by condition (i) and inequality (2.1), it follows that for every $N=1,2,\dots$ there exists a finite $\frac{1}{2^{N}}$-net. This shows that the metric space $\bar{\mathbb{X}}$ is compact. We construct a $\sigma-$additive Borel probability measure on $\bar{\mathbb{X}}$. The compactness of $\bar{X}$ implies that the extension of $\Delta$ to sets of the form $\{\bar{A};A\in D_{0}\}$ which are open and closed is a $\sigma$-additive probability measure, since $\Delta(\bar{A})=\Delta(A)$. Then \[P^{\ast}(B)=\inf\left\{\sum_{n=1}^{\infty}\Delta(\bar{A}_{n});B\subset\bigcup_ {n=1}^{\infty}\bar{A}_{n},\ \bar{A}_{n}\in{\mathcal{S}}\right\}\] is an outer measure on $\bar{\mathbb{X}}$ and ${\mathcal{S}}_{P^{\ast}}=\{B;P^{\ast}(B)+P^{\ast}(\bar{\mathbb{X}}\setminus B) =1\}$, the system of $P^{\ast}$-measurable sets, is a $\sigma$-algebra. Therefore, the restriction $P$ of $P^{\ast}$ on ${\mathcal{S}}_{P^{\ast}}$ is a $\sigma$-additive probability measure on ${\mathcal{S}}_{P^{\ast}}$. Moreover, since ${P^{\ast}}$ is, by definition, an outer measure, ${\mathcal{S}}_{P^{\ast}}$ contains all open sets. Thus $P$ is a Borel probability measure on the compact metric space $\bar{\mathbb{X}}$. Following the usual procedure, we have a compact metric space and a Borel probability measure defined on it. We can introduce a suitable definition of uniform distribution of a sequence $\{\alpha_{n}\}_{n\in\mathbb{N}}$ in $\bar{\mathbb{X}}$ with respect to $P$, namely Buck uniform distribution. Since the set $\mathbb{X}$ is dense in its completion, there exists a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $\mathbb{X}$ such that $\lim_{n\to\infty}\rho(x_{n},\alpha_{n})=0$. Since every continuous function on $\bar{\mathbb{X}}$ is uniformly continuous, $\{x_{n}\}_{n\in\mathbb{N}}$ is also a B.u.d. sequence. Considering a set $C$ with $\bar{C}\in\mathcal{S}$ and $\partial\bar{C}=\emptyset$, yealds $A(C,\{x_{n}\})\in\text{$\mathcal{D}$}_{\mu}$ and \[\mu(A(C,\{x_{n}\}))=\Delta(C).\] A sequence of elements of $\mathbb{X}$ fulfilling this condition will be called $\nu^{\ast}$_-B.u.d._. Moreover, it is easy to see that for every $S\in\text{$\mathcal{D}$}_{\nu}$ the set $A(S,\{x_{n}\})$ is measurable in sense of Buck and \[\mu(A(S,\{x_{n}\}))=\nu(S).\] Thus by Theorem 2.1 we have \[\nu(S)=\lim_{N\to\infty}\frac{1}{N}\left\|\left\{n\leq N;x_{s_{n}}\in S\right\}\right\|\] for $S\in\text{$\mathcal{D}$}_{\nu}$ and $\{s_{n}\}$ a sequence of positive integers u.d. in ${\mathbb{Z}}$. Therefore the measure density can be represented in certain sense as ”limit” density. Consider now a uniformly continuous function $f:\mathbb{X}\to[0,1]$ and a B.u.d. sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $\mathbb{X}$. Then for every real valued continuous function $g$ defined on $[0,1]$ we have (2.2) \[\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}g(f(x_{k_{n}}))=\int f\circ g\ dP,\] where $\{k_{n}\}$ is an arbitrary sequence of positive integers u.d. in ${\mathbb{Z}}$. More generally, let $\mathbb{Y}\neq\emptyset$ be a set and $\ell^{\ast}$ be a pre-measure defined on the ring of all subsets of $\mathbb{Y}$. Assume further that $\ell^{\ast}$ is a strong submeasure on $\text{$\mathcal{D}$}_{\ell}$ (a strong subadditive pre-measure, see[23]), i.e. \[\ell^{\ast}(A\cap B)+\ell^{\ast}(A\cup B)\leq\ell^{\ast}(A)+\ell^{\ast}(B)\ ,\] with $\ell^{\ast}(\mathbb{Y})=1,\ell^{\ast}(\emptyset)=0$. Consider $\text{$\mathcal{D}$}_{\ell}$, the system of all $C\subset\mathbb{Y}$ such that $\ell^{\ast}(C)+\ell^{\ast}(\mathbb{Y}\setminus C)=1$. Clearly, $\text{$\mathcal{D}$}_{\ell}$ is a set algebra and the restriction $\ell=\ell^{\ast}\|_{\text{$\mathcal{D}$}_{\ell}}$ is a finitely additive probability measure on $\text{$\mathcal{D}$}_{\ell}$. Following the proofs from [18, 21] we can derive the following result. Theorem 2.7.: _Let $g:\mathbb{X}\to\mathbb{Y}$ be a bijective mapping. The following statements are equivalent_ 1. $g(S)\in\text{$\mathcal{D}$}_{\ell}\land\ell(g(S))=\nu(S)$_, for all_ $S\in\text{$\mathcal{D}$}_{\nu}$_;_ 2. $\ell^{\ast}(g(A))\leq\Delta(A)$_, for all_ $A\in\text{$\mathcal{D}$}_{0}$_;_ 3. $ì\ell^{\ast}(g(B))\leq\nu^{\ast}(B)$_, for all_ $B\subset\mathbb{X}$_._ Analogously we can extend the concept of B.u.d. for this case. A sequence $\{x_{n}\}_{n\in\mathbb{N}}$ is said to be $\ell^{\ast}$-_B.u.d._ if and only it for every $B\in\text{$\mathcal{D}$}_{\ell}$ we have $A(B,\{x_{n}\})\in\text{$\mathcal{D}$}_{\mu}$ and $\mu(A(B,\{x_{n}\}))=\ell(B)$. In particular the following result holds. Theorem 2.8.: _If $g:\mathbb{X}\to\mathbb{Y}$ is a bijective mapping fulfilling condition (1) of Theorem 2.7 and $\{x_{n}\}$ is a $\ell^{\ast}$-B.u.d. sequence of elements in $\mathbb{Y}$ then $g^{-1}(x_{n})$ is a $\nu^{\ast}$-B.u.d. sequence of elements in $\mathbb{X}$._ For every bijection $g$, all $x\in\mathbb{X}$ and $S\subset\mathbb{X}$, we have $g^{-1}(x)\in S\Leftrightarrow x\in g(S)$. Thus for each sequence $\{x_{n}\}_{n\in\mathbb{N}}$ we have $A(S,\{g^{-1}(x_{n})\})=A(g(S),\{x_{n}\})$. This yields Corollary 2.9.: _If $g$ preserves measure density then for every $\nu^{\ast}$-B.u.d. sequence $\{x_{n}\}_{n\in\mathbb{N}}$ the sequence $\{g^{-1}(x_{n})\}_{n\in\mathbb{N}}$ is also $\nu^{\ast}$-B.u.d.._ _Remark 2.10_.: Functions $g^{-1}$ such that the sequence $\{g^{-1}(x_{n})\}_{n\in\mathbb{N}}$ is u.d. for every u.d. sequence $\{x_{n}\}_{n\in\mathbb{N}}$ are called uniform distribution preserving mappings (for short: u.d.p. mappings). They are of particular interest since they are maps generating u.d. sequences for every u.d. sequence $\{x_{n}\}_{n\in\mathbb{N}}$. In [26] the authors establish general criteria on u.d.p. transformations on compact metric spaces and a full characterization of u.d.p. maps on $[0,1]$. Therefore, in view of Theorem 2.7 we have the following Corollary 2.11.: _Let $g$ be a bijection. Then the following statements are equivalent._ 1. $g$ _preserves the measure density;_ 2. $\nu^{\ast}(g(A^{(n)}_{i}))\leq\Delta(A^{(n)}_{i})$_, for all_ $n\in{\mathbb{N}}$ _and all_ $i\leq k_{n}$_;_ 3. $\nu^{\ast}(g(S))\leq\nu^{\ast}(S)$_, for all_ $S\subset\mathbb{X}$_._ Example 2.12.: _Consider again $\mathbb{X}=\text{$\mathbb{G}$}$ a locally compact abelian group. The mapping $x\to x^{-1}$ defined on $G$ is a bijection fulfilling condition (2) of Corollary 2.11 by applying the decomposition $A^{(n)}_{j}=a_{j}H_{n}$. Thus Corollary 2.9 implies that each sequence $\{x_{n}\}_{n\in\mathbb{N}}$ of elements of $\mathbb{G}$ is $\nu^{\ast}$-B.u.d. sequence if and only if $\{x_{n}^{-1}\}$ is $\nu^{\ast}$-B.u.d.. Analogously we can consider the mapping $x\to ax$, for a fixed $a\in G$. Then each sequence $\{x_{n}\}_{n\in\mathbb{N}}$ of elements of $\mathbb{G}$ is $\nu^{\ast}$-B.u.d. sequence if and only if $\{ax_{n}\}$ is $\nu^{\ast}$-B.u.d.._ We now conclude this section with a theorem that can be considered as a generalization of the construction of Haar measure with the help of Kakutani’s fixed point Theorem (see e.g. [9]). Theorem 2.13.: _Let $g$ be a permutation defined on $\mathbb{X}$ such that $g(S)\in\text{$\mathcal{D}$}_{0}$ for every $S\in\text{$\mathcal{D}$}_{0}$, where $\text{$\mathcal{D}$}_{0}$ is a countable $\sigma$-algebra of $\mathbb{X}$. Then there exists a finite probability measure $\Delta$ such that for every $A\in\text{$\mathcal{D}$}_{0}$_ \[\Delta(g(A))=\Delta(A).\] Proof.: Denote by $\mathcal{B}$ the set of all bounded real valued and finitely additive set functions defined on $\text{$\mathcal{D}$}_{0}$. Then $\mathcal{B}$ is a linear space. Let $\mathcal{R}$ be the subset of $\mathcal{B}$ consisting of all finitely additive probability measures. It is easy to check that $\mathcal{R}$ is convex set. Define a topology on $\mathcal{B}$ by \[\Lambda_{n}\rightarrow\Lambda\quad\Leftrightarrow\quad\forall S\in\text{$ \mathcal{D}$}_{0}\quad\lim_{n\to\infty}\Lambda_{n}(S)=\Lambda(S)\ .\] Consider a sequence $\{\Delta_{n}\}$ of elements in $\mathcal{R}$. Since $\text{$\mathcal{D}$}_{0}$ is a countable set, we can iteratively select a sequence of indices $\{n_{k}\}$ such that $\{\Delta_{n_{k}}(S)\}$ converges for every $S\in\text{$\mathcal{D}$}_{0}$. Thus $\mathcal{R}$ is sequentially compact with respect to this topology. Let us define a linear mapping $\tilde{g}\colon\mathcal{B}\rightarrow\mathcal{B}$, with $\tilde{g}(\Lambda)(S)=\Lambda(g(S))$. Then $\tilde{g}(\mathcal{R})\subset\mathcal{R}$ and $\tilde{g}$ is continuous with respect to the topology under consideration. Put \[\tilde{g}_{n}(\Delta)=\frac{1}{n}\sum_{k=1}^{n}\tilde{g}^{n}(\Delta)\] for $\Delta\in\mathcal{R}$, $n=1,2,\dots$. Since $\mathcal{R}$ is sequentially compact, every countable centered system of closed sets has non empty intersection. Thus, by an application of Markov-Kakutani fixed point theorem, follows the assertion. ∎ The following example provides an explicit construction of a finite additive probability measure on the algebra $\text{$\mathcal{D}$}_{0}$. Example 2.14.: _Let $\mathcal{C}$ be the set of all real-valued uniformly continuous functions defined on $\mathbb{X}$. Since these functions are bounded we can define the norm_ \[\|\|f\|\|=\sup\{f(x);\ x\in\mathbb{X}\},\] _where $f\in\text{$\mathcal{C}$}$. It can be seen easily that $(\text{$\mathcal{C}$},\|\|\cdot\|\|)$ is a Banach space._ _Let $\text{$\mathcal{C}$}^{\ast}$ be the dual space to $\mathcal{C}$. Denote by ${\mathcal{P}}$ the set of all $\varphi\in\text{$\mathcal{C}$}^{\ast}$ that $\varphi(1)=1$ and $\varphi(f)\geq 0$ for $f\geq 0$. Then $\Delta_{\varphi}(A)=\varphi(\chi_{A}),A\in\text{$\mathcal{D}$}_{0}$, $\varphi\in{\mathcal{P}}$, is a finitely additive probability measure on $\text{$\mathcal{D}$}_{0}$ and $\varphi(f)=\int fd\Delta_{\varphi}$._ _Assume that $g:\mathbb{X}\to\mathbb{X}$ is a permutation such that $g(A)\in\text{$\mathcal{D}$}_{0}$ for $A\in\text{$\mathcal{D}$}_{0}$. We want to show that $g^{-1}\circ f\in\text{$\mathcal{C}$}$ for every $f\in\text{$\mathcal{C}$}$. $f\in\text{$\mathcal{C}$}$ if and only for every $\varepsilon>0$ there exists a step function $\sum_{i=1}^{k}c_{i}\chi_{A_{i}},A_{i}\in\text{$\mathcal{D}$}_{0}$ that $\|\|f-\sum_{i=1}^{k}c_{i}\chi_{A_{i}}\|\|<\varepsilon$. Hence $\|\|g^{-1}\circ f-\sum_{i=1}^{k}c_{i}\chi_{g(A_{i})}\|\|<\varepsilon$, since $g(A_{i})\in\text{$\mathcal{D}$}_{0}$. Thus $g^{-1}\circ f$ is uniformly continuous._ ## 3. Buck uniform distribution mod 1 In this section we study the connection between systems of measurable sets of positive integers and sets of real numbers in the unit interval. We will use the so-called radical-inverse function which is an important function in the theory of uniform distribution and in the study of low-discrepancy sequences (see for instance [7, 15]). Finally, the notion of upper Jordan measure mentioned in Example 2.4, here denoted with $\ell^{\ast}$, instead of $\nu^{\ast}$, will be relevant. Let $\mathbb{X}={\mathbb{N}}$ and $p$ a prime. Let us consider the arithmetic progression $r+(m)$. This leads to the system of decompositions \[\mathcal{E}_{n}=\{r+(p^{n});r=0,\dots,p^{n}-1\}\ ,\quad n=1,2,\dots.\] If $\Delta(r+(p^{n}))=\frac{1}{p^{n}}$, $n=1,2,\dots$, then the corresponding measure density $\nu^{\ast}$ will be the covering density with respect to the system $\{p^{n};n\in{\mathbb{N}}\}$, (see [17]). Now, let us recall that every $n\in{\mathbb{N}}$ has a unique $p$-adic expansion , i.e. $n$ can be written as \[n=a_{0}(n)+a_{1}(n)p+\dots+a_{s}(n)p^{s}\ ,\quad 0\leq a_{i}(n)<p\ ,\quad i=1, \dots,s.\] The radical-inverse function $g_{p}:{\mathbb{N}}\rightarrow[0,1)$ is defined by \[g_{p}(n)=\frac{a_{0}(n)}{p}+\frac{a_{1}(n)}{p^{2}}+\dots+\frac{a_{s}(n)}{p^{s+ 1}}\ .\] This function maps ${\mathbb{N}}$ to the set of $p$-adic rationals ${\mathbb{J}}_{p}=\{\frac{r}{p^{s}};r=0,\dots,p^{s}-1\}$ in $[0,1)$. Therefore the image of ${\mathbb{N}}$ under $g_{p}(n)$ is dense in $[0,1)$. Since every number from ${\mathbb{J}}_{p}$ has finite $p$-adic expansion we obtain that the mapping $g_{p}:{\mathbb{N}}\to{\mathbb{J}}_{p}$ is a bijection. The properties of $p$-adic expansions provide that \[g_{p}(r+(p^{n}))=\left[\frac{a}{p^{n}},\frac{a+1}{p^{n}}\right)\cap{\mathbb{J} }_{p}\] for $0\leq r<p^{n}$ and $\frac{a}{p^{n}}=g_{p}(r)$. Let $\text{$\mathcal{D}$}_{\ell}$ be the system of all $S\subset{\mathbb{J}}_{p}$ such that $\ell^{\ast}(S)+\ell^{\ast}({\mathbb{J}}_{p}\setminus S)=1$. Then $g_{p}$ and $\ell^{\ast}$ satisfy condition (2) of Theorem 2.7 which is equivalent to condition (1). Let us remark that the sequence $(g_{p}(n))_{n\in\mathbb{N}}$, with $p$ not necessarily prime, is called the van der Corput sequence in base $p$ and it is a well-known example of u.d. sequence in $[0,1]$ (see [7, 15]). Moreover, the above construction has been considered and extended to more general systems of numeration by several researchers (see e.g. [5, 13]). Recently, this method has been applied to obtain the so-called $LS$-sequences (see [5]). These sequences were first introduced in [4] as sequences of points associated to the so-called $LS$-sequences of partitions of $[0,1[$. The latter being obtained as a particular case of a splitting procedure introduced by Kakutani [14] and generalized in [27], for a particular choice of the parameters $L$ and $S$. Moreover, this construction has been generalized to the multidimensional case in [6]. Finally, let us note that when $L=p$ and $S=0$ the $LS$-sequence coincides with the van der Corput sequence in base $p$ (see [1]). Now, let us consider the Cantor expansion. By this expansion every $x\in{\mathbb{N}}$ is uniquely given in the form \[x=b_{1}(x)1!+b_{2}(x)2!+\dots+b_{s}(x)s!,\ s\in{\mathbb{N}},0\leq b_{i}(x)\leq i ,\ i=1,\dots,s.\] Then we define a generalization of the radical-inverse function by (3.1) \[g_{v}(x)=\frac{b_{1}(x)}{2!}+\frac{b_{2}(x)}{3!}+\dots+\frac{b_{s}(x)}{(s+1)!}.\] Consider \[\mathcal{E}_{n}=\{r+(n!);r=0,\dots,n!-1\}\ ,\quad n=1,2,\dots,\] as system of decompositions of ${\mathbb{N}}$, then $\nu^{\ast}=\mu^{\ast}$-Buck measure density. Since every rational number in $[0,1)$ has finite Cantor expansion we observe that $g_{v}:{\mathbb{N}}\to{\mathbb{J}}$ is a bijective mapping. Clearly, for $n=1,2,\dots,$ and $r=0,\dots,n!-1$ we have \[g_{v}(r+(n!))=\left[\frac{b}{n!},\frac{b+1}{n!}\right)\cap{\mathbb{J}},\qquad g _{v}(r)=\frac{b}{n!}.\] Again $\text{$\mathcal{D}$}_{\ell}$ is the set of all $S\subset{\mathbb{J}}$ such that $\ell^{\ast}(S)+\ell^{\ast}({\mathbb{J}}\setminus S)=1$, then $\ell^{\ast}$ and $g_{v}$ fulfill the condition $(2)$ of Theorem 2.7. Moreover, we observe that both $g_{v}$ and $g^{-1}_{v}$ satisfy $(1)$ of Theorem 2.7. Therefore Theorem 2.8 assures that a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ of elements of ${\mathbb{J}}$ is a B.u.d. sequence if and only if $\{g^{-1}_{v}(x_{n})\}$ is a B.u.d. sequence in ${\mathbb{N}}$. Let us remark that the above example of Cantor expansion can be extended to general Cantor series, as shown in the following example. Let $\left\{Q_{n}^{(k)}\right\}$ be increasing sequences of positive integers, $k=1,\dots,s$ such that $Q_{n}^{(k)}\|Q_{n+1}^{(k)},n=1,2,\dots$. Then every positive integer $a$ has a unique representation in Cantor series of the form (see [3]) \[a=\sum_{j=1}^{m_{k}}a^{k}_{j}Q_{j}^{(k)},\qquad 0\leq a^{k}_{j}<\frac{Q_{j+1}^ {(k)}}{Q_{j}^{(k)}}.\] Put \[\gamma_{k}(a)=\sum_{j=1}^{m_{k}}\frac{a^{k}_{j}}{Q_{j+1}^{(k)}}\ .\] We can associate to $a$ a point in the $s$-dimensional unit interval \[\gamma(a)=\left(\gamma_{1}(a),\dots,\gamma_{s}(a)\right).\] If $J\subset[0,1]^{s}$ is the set of the form $J=J_{1}\times\dots\times J_{s}$ then \[A(\{\gamma(n)\},J)=\bigcap_{k=1}^{s}A\left(\{\gamma_{k}(n)\},J_{k}\}\right).\] For $J_{k}=\left[\frac{i}{Q_{n}^{(k)}},\frac{i+1}{Q_{n}^{(k)}}\right[$ we have \[A(\{\gamma_{k}(n)\},J_{k})=b+\left(Q_{n}^{(k)}\right)\ ,\] with $b$ a suitable positive integer, and $b+\left(Q_{n}^{(k)}\right)$ the residue class of $b$ mod $Q_{n}^{(k)}$. Thus from the Chinese remainder theorem, if $Q_{n}^{(k_{1})},Q_{n}^{(k_{2})}$, $k_{1}\neq k_{2}$ are coprime, then $A(\{\gamma(n)\},J)$ is Buck measurable and (3.2) \[\mu(A(\{\gamma(n)\},J)=\|J_{1}\|\dots\|J_{s}\|.\] Since the set of the points $\left(\frac{i_{1}}{Q_{n}^{(1)}},...,\frac{i_{s}}{Q_{n}^{(s)}}\right)$ is dense in $[0,1]^{s}$ we can conclude that (3.2) holds for arbitrary intervals $J_{k}$, $k=1,\dots,s$. The above statements can be adapted to a more general setting. Let $f$ be a non-decreasing real-valued function on ${\mathbb{J}}$, with $f(0)=0$, $f(1)=1$. For every $S\subset{\mathbb{J}}$, we can associate, in the usual way, the Jordan-Stieltjes upper measure $\ell^{\ast}_{f}(S)$ defined as $\ell^{\ast}_{f}([a,b)\cap\mathbb{J})=f(b)-f(a)$, with $a,b\in{\mathbb{Q}}$. By the generalized radical-inverse function $g$ defined in (3.1), we can associate a finite additive measure on the system $\text{$\mathcal{D}$}_{0}$, where $\Delta_{f}(r+(n!))=\ell^{\ast}_{f}(g_{v}(r+(n!)))$. On the other hand, if a finitely additive probability measure $\Delta$ on $\text{$\mathcal{D}$}_{0}$ is given, we can define a non-decreasing function $f(x)=\Delta(g^{-1}_{v}([0,x)\cap{\mathbb{J}})),x\in{\mathbb{J}}$, since every rational number can be expressed in the form $x=\frac{a}{n!}$ and so the preimage of $[0,x)\cap{\mathbb{J}}$ is a union of a finite number of sets of the form $r+(n!)$. Thus $\Delta=\Delta_{f}$. If we denote by $\nu^{\ast}_{f}$ the corresponding measure density, condition (1) of Theorem 2.7 is fulfilled and Theorem 2.8 yields the following Corollary 3.1.: _A sequence of positive integers $\{y_{n}\}$ is $\nu_{f}$-B.u.d. if and only if $\{g_{v}(y_{n})\}$ is $\ell_{f}$-B.u.d.._ Moreover, we can prove the following result Lemma 3.2.: _Let $f$ and $\Delta_{f}$ be as above. Then $f$ is uniformly continuous on ${\mathbb{J}}$ if and only if_ (3.3) \[\lim_{n\to\infty}\Delta_{f}(r+(n!))=0\] _uniformly for $r=0,1,\dots$._ Proof.: Suppose that $[x_{1},x_{2})\in{\mathbb{J}}$. Clearly, (3.4) \[f(x_{2})-f(x_{1})=\Delta_{f}(g^{-1}_{v}([x_{1},x_{2})\cap{\mathbb{J}})\ .\] Let $x_{2}-x_{1}<\frac{1}{(n+1)!}$ for some $n\in{\mathbb{N}}$. Then $[x_{1},x_{2})\subset[\frac{c}{n!},\frac{c+1}{n!})$ for some $c\in{\mathbb{N}}$ with $0<c\leq n!$. Thus $g^{-1}_{v}([x_{1},x_{2})\cap{\mathbb{J}}\subset r+(n!)$ for some $r\in{\mathbb{N}}$. Using (3.4), this yields $f(x_{2})-f(x_{1})\leq\Delta_{f}(r+(n!))$. Hence $f$ is uniformly continuous on ${\mathbb{J}}$. The other implication immediately follows from \[\Delta_{f}(r+(n!))=\ell_{f}(g_{v}(r+(n!))=\ell_{f}\Big{(}\Big{[}\frac{c^{ \prime}}{n!},\frac{c^{\prime}+1}{n!}\Big{)}\cap{\mathbb{J}}\Big{)}=f\left( \frac{c^{\prime}+1}{n!}\right)-f\left(\frac{c^{\prime}}{n!}\right)\ ,\] with $r,n\in{\mathbb{N}}$, for a suitable non-negative integer $c^{\prime}$. ∎ Since a uniformly continuous function on ${\mathbb{J}}$ can be extended to a continuous function on $[0,1]$, Lemma 3.2 has the following immediate consequence (see [21, page 54]). Corollary 3.3.: _If $\{y_{n}\}$ is a $\nu$-B.u.d. sequence of positive integers fulfilling equation (3.3), then the sequence $\{g_{v}(y_{n})\}$ is Buck measurable and its Buck distribution function is the continuous extension of $f$ on $[0,1]$._ In the same way, one can prove the following result. Theorem 3.4.: _Let $g:\mathbb{X}\to[0,1]$ be an injective function such that $g(\mathbb{X})$ is dense in $[0,1]$ and assume that $g(A_{r}^{(n)})=I_{r}^{(n)}\cap g(\mathbb{X})$, with $I_{r}^{(n)}$ right half-open intervals, and_ \[\lim_{n\to\infty}\ell(I_{r}^{(n)})=0\] _uniformly for $r\in{\mathbb{N}}$. Denote $f(x)=\Delta(g^{-1}([0,x)\cap g(\mathbb{X}))$ for every right endpoint $x$ of $I_{r}^{(n)}$ and for all $r,n\in{\mathbb{N}}$. Then $f$ is uniformly continuous on $g(\mathbb{X})$ if and only_ \[\lim_{n\to\infty}\Delta(A_{r}^{(n)})=0\] _uniformly for $r\in{\mathbb{N}}$. In this case for every $\nu$-B.u.d. sequence $\{y_{n}\}$ the sequence $\{g(y_{n})\}$ is Buck measurable and its Buck distribution function is the continuous extension of $f$ to $[0,1]$._ It is well-known that a real-valued uniformly continuous function $f$ on a metric space $(\mathbb{X},\rho)$ can be extended to a continuous function on a compact space $\bar{\mathbb{X}}$ (see [22]). In particular, one can define the concept of Riemann integrability by defining the Riemann upper and lower sums associated to the decompositions $\mathcal{E}_{n},n=1,2,\dots$ and to the finitely additive measure $\Delta$. More precisely, we have the following definition. Definition 3.5.: _Let $\{c_{n}\}$ be a sequence of positive integers such that for every $A\in\text{$\mathcal{D}$}_{0}$ there exists $n_{0}$ such that $A$ is a union of sets from $\mathcal{E}_{c_{n}}$, for $n\geq n_{0}$. Then a function $f$ is said to be Riemann integrable if and only if there exists a real number $S$ such that for every system of finite sequences $\{a^{(n)}_{i};i=1,\dots,k_{c_{n}}\}$ with $a^{(n)}_{i}\in A^{(c_{n})}_{i}$ we have_ \[\lim_{n\to\infty}\sum_{i=1}^{k_{c_{n}}}\Delta(A_{i}^{(c_{n})})f(a^{(n)}_{i})=S.\] _In this case $S=\int f$._ _Remark 3.6_.: Let $B\subset\mathbb{X}$ then $B$ is a $\nu^{\ast}$-measurable set if and only if its indicator function $\chi_{B}$ is Riemann integrable and in this case \[\nu^{\ast}(B)=\int\chi_{B}.\] In this way, a sequence $\{x_{n}\}_{n\in{\mathbb{N}}}$ in $\mathbb{X}$ is $\nu^{\ast}$-B.u.d. if and only if for every Riemann integrable function $f$ we have \[\lim_{N\to\infty}\sum_{n=1}^{N}f(x_{s_{n}})=\int f\] for every sequence of positive integers $\{s_{n}\}$ uniformly distributed in ${\mathbb{Z}}$. Now, since $f$ is a real valued function uniformly continuous with respect to the metric $\rho$, we can extend it to a continuous function on $\bar{\mathbb{X}}$ and \[\int f=\int fdP.\] Thus Theorem 2.7 can be extended to uniformly continuous functions $f$ with respect to $\rho$. Under the assumption of continuity we can restate Theorem 2.7. Theorem 3.7.: _Let $g$ be a bijection such that $g^{-1}$ is uniformly continuous with respect to the metric $\rho$. Then $g$ preserves measure density if and only if there exists at least one $\nu^{\ast}$-B.u.d. sequence $\{x_{n}\}_{n\in\mathbb{N}}$ such that $\{g^{-1}(x_{n})\}$ is also $\nu^{\ast}$-B.u.d.._ ## 4. Buck uniform distribution on a free semigroup Let $\mathbb{X}=\text{$\mathbb{F}$}$ be a free semigroup generated by a countable set of generators $\{p_{1},p_{2},\dots,p_{n},\dots\}$. Let $a\in\text{$\mathbb{F}$}$ and denote by ${\mathcal{U}}(a)$ the set of all divisors of $a$. We say that a set $S\subset\text{$\mathbb{F}$}$ has a _divisor density_ if and only if there exists \[{\mathbb{v}}(S)=\lim_{n\to\infty}\frac{\|S\cap{\mathcal{U}}(p_{1}^{n}...p_{n}^{ n})\|}{(n+1)^{n}}\ .\] We denote by $\text{$\mathcal{D}$}_{{\mathbb{v}}}$ the family of all sets having a divisor density. It is easy to show that ${\mathbb{v}}$ is a finitely additive probability measure. Let us consider some examples. If $\mathcal{F}$ denotes the set of square free elements, then $\|\mathcal{F}\cap{\mathcal{U}}(p_{1}^{n}...p_{n}^{n})\|=2^{n}$, hence $\mathcal{F}\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(\mathcal{F})=0$. Let $\text{$\mathbb{F}$}^{s}=\{a^{s},a\in{\mathbb{N}}\}$, $s=2,3,\dots$. For $n$ sufficiently large we have $\|\text{$\mathbb{F}$}^{s}\cap{\mathcal{U}}(p_{1}^{n}...p_{n}^{n})\|=([n/s]+1)^{n}$. Thus $\text{$\mathbb{F}$}^{s}\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(\text{$\mathbb{F}$}^{s})=0$. For $r\in{\mathbb{N}}$, denote by ${\mathbb{O}}_{r}$ the set of all elements of $\mathbb{F}$ of the form $p_{i_{1}}^{\beta_{1}}\dots p_{i_{s}}^{\beta_{s}}$, where $\beta_{i}\geq r$. It can be easily seen that $\|{\mathbb{O}}_{r}\cap{\mathcal{U}}(p_{1}^{n}\dots p_{n}^{n})\|=(n-r+1)^{n}$. Thus ${\mathbb{O}}_{r}\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}({\mathbb{O}}_{r})=e^{-r}$, where $e$ is Euler’s number. Proposition 4.1.: _Let $S\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and $a\in\text{$\mathbb{F}$}$. Then $aS\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(aS)={\mathbb{v}}(S)$._ Proof.: It is sufficient to prove the assertion for $a=p$, one of the generators. Assume that $p$ occurs in $p_{1},\dots,p_{n}$. If $d\in S\cap{\mathcal{U}}(p_{1}^{n}\dots p_{n}^{n})$ and $d=p_{1}^{j_{1}}\dots p_{n}^{j_{n}}$ then $pd$ does not divide $p_{1}^{n}\dots p_{n}^{n}$ only in the case when the exponent of $p$ is $n$. Thus $\|pS\cap{\mathcal{U}}(p_{1}^{n}\dots p_{n}^{n})\|=S\cap{\mathcal{U}}(p_{1}^{n} \dots p_{n}^{n})-(n+1)^{(n-1)}$. ∎ Corollary 4.2.: _For $b\in\text{$\mathbb{F}$}$ we have $b\text{$\mathbb{F}$}\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(b\text{$\mathbb{F}$})=1$. For every $S\subset\text{$\mathbb{F}$}$, $b\in\text{$\mathbb{F}$}$ we have that $(b\text{$\mathbb{F}$})\cap S\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ implies $S\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(S)={\mathbb{v}}((b\text{$\mathbb{F}$})\cap S)$._ Proposition 4.3.: _Let $H\subset\text{$\mathbb{F}$}$ be the semigroup generated by the generators$\{p_{1}^{\alpha_{1}},\dots,p_{k}^{\alpha_{k}},p_{k+1},p_{k+2},\dots.\}$ (for given positive integers $\alpha_{i}$). Then $H\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(H)=\frac{1}{\alpha_{1}\dots\alpha_{k}}$ ._ Proof.: Let $n>\alpha_{i},i=1,\dots,k$, $n>k$. Then $H\cap{\mathcal{U}}(p_{1}^{n}...p_{n}^{n})$ contains the elements $p_{1}^{s_{1}\alpha_{1}},\dots,p_{k}^{s_{k}\alpha_{k}}p_{k+1}^{j_{k+1}}\dots p_ {n}^{j_{n}}$, where $s_{i}\alpha_{i}\leq n,\ i=1,\dots,k$ and $j_{i}\leq n,\ i=k+1,\dots,n$. Thus \[\|H\cap{\mathcal{U}}(p_{1}^{n}...p_{n}^{n})\|=([n/\alpha_{1}]+1)\dots([n/\alpha_ {k}]+1)(n+1)^{(n-k)}.\] Thus \[\lim_{n\to\infty}\frac{\|H\cap{\mathcal{U}}(p_{1}^{n}\dots p_{n}^{n})\|}{(n+1)^{ n}}=\frac{1}{\alpha_{1}\dots\alpha_{k}}.\] ∎ A mapping $x:\text{$\mathbb{F}$}\to[0,1]$ is called ${\mathbb{v}}$-_uniformly distributed_ if and only if for every subinterval $I\subset[0,1)$ the set $x^{-1}(I)$ belongs to $\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(x^{-1}(I))=\ell(I)$. Now we can formulate the following generalization of Weyl’s criterion. Proposition 4.4.: _Let $P_{n}=p_{1}^{n}\dots p_{n}^{n}$, $n=1,2,\dots$. A mapping $x:\text{$\mathbb{F}$}\to[0,1)$ is ${\mathbb{v}}$- uniformly distributed if and only if one of the following conditions holds:_ (4.1) \[\lim_{n\to\infty}(n+1)^{-n}n\sum_{d\|P_{n}}f(d)=\int^{1}_{0}f(x)dx\] _for every Riemann integrable function $f$ (or for continuous $f$) on $[0,1]$;_ (4.2) \[\lim_{n\to\infty}(n+1)^{-n}n\sum_{d\|P_{n}}e^{2\pi ihx(d)}=0\] _for integers $h\neq 0$._ Let $\text{$\mathbb{F}$}_{n}$ be the semigroup generated by $\{p_{1}^{n},p_{2}^{n},\dots p_{n}^{n},p_{n+1},p_{n+2},\dots\}$, $n=1,2,\dots$. Proposition 4.3 implies that $a\text{$\mathbb{F}$}_{k}\in\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(a\text{$\mathbb{F}$}_{k})=\frac{1}{k^{k}}$. Let $\text{$\mathcal{D}$}_{0}$ be the system of all subsets of $\mathbb{F}$ of the form $a_{1}\text{$\mathbb{F}$}_{n}\cup\dots\cup a_{k}\text{$\mathbb{F}$}_{n}$, where $a_{j}\in\{p_{1}^{j_{1}}p_{2}^{j_{2}}\dots p_{n}^{j_{n}},0\leq j_{i}<n\}$. Put $\Delta(A)={\mathbb{v}}(A)$ for $A\in\text{$\mathcal{D}$}_{0}$. For $S\subset\text{$\mathbb{F}$}$ the value \[\nu^{\ast}_{e}(S)=\inf\{\Delta(A);S\subset A,A\in\text{$\mathcal{D}$}_{0}\}\] is called _divisor measure density_ of the set $S$. We denote by $\text{$\mathcal{D}$}_{\nu_{e}}$ the system of measurable sets. It can be easily deduced that $\text{$\mathcal{D}$}_{\nu_{e}}\subset\text{$\mathcal{D}$}_{{\mathbb{v}}}$ and ${\mathbb{v}}(S)=\nu_{e}(S)$ for $S\in\text{$\mathcal{D}$}_{\nu_{e}}$. Let us consider again $\mathcal{F}$, the set of all square free elements of $\mathbb{F}$. This set has non-empty intersection only with the sets $aF_{n}$, $n=1,2,\dots$ and $a=p_{1}^{i_{1}}p_{2}^{i_{2}}\dots p_{n}^{i_{n}}$, where $i_{j}=0,1$. Hence, $\nu^{\ast}_{e}(\mathcal{F})\leq\frac{2^{n}}{n^{n}}$, and for $n\to\infty$ we get $\nu^{\ast}_{e}(\mathcal{F})=0$. This yields $\nu^{\ast}_{e}(\text{$\mathbb{F}$}\setminus\mathcal{F})=1$. So $\mathcal{F}$ is measurable and $\nu_{e}(\mathcal{F})=0$. Let us remark that the set of square free integers is not Buck measurable, its Buck measure density is $\frac{6}{\pi^{2}}$, and its complement has Buck measure density $1$. However, the set of square free numbers has asymptotic density $\frac{6}{\pi^{2}}$. Example 4.5.: _For the set $\mathcal{F}$ we have $\int\chi_{\mathcal{F}}d\nu_{e}=0$. However, this function is discontinuous at each point $a\in\mathcal{F}$._ Denote by $[S:\text{$\mathbb{F}$}_{n}]$ the number of sets $a_{j}\text{$\mathbb{F}$}_{n}$ such that $S\cap a_{j}\text{$\mathbb{F}$}_{n}\neq\emptyset$. Theorem 2.1 implies Proposition 4.6.: _Let $\{b_{k}\}$ be a sequence of positive integers such that for every $d\in{\mathbb{N}}$ there exists $k_{0}$ such that for $k>k_{0}$$d\|b_{k}$. Then for $S\subset\text{$\mathbb{F}$}$_ \[\nu^{\ast}_{e}(S)=\lim_{k\to\infty}\frac{[S:\text{$\mathbb{F}$}_{b_{k}}]}{b_{k }^{b_{k}}}.\] In the sequel $\{b_{k}\}$ will be a sequence as in Proposition 4.6. Denote $\text{$\mathbb{F}$}^{s}=\{a^{s};s\in\text{$\mathbb{F}$}\}$ for $s\in{\mathbb{N}}$. Suppose that $s\|b_{k}$. Then the intersection $\text{$\mathbb{F}$}^{s}$ with $p_{1}^{j_{1}}p_{2}^{j_{2}}\dots p_{b_{k}}^{j_{b_{k}}}\text{$\mathbb{F}$}_{b_{k}}$ is non-empty only if $s\|j_{i}$ for all $i=1,\dots,b_{k}$. Therefore \[\nu^{\ast}(\text{$\mathbb{F}$}^{s})=\lim_{k\to\infty}\frac{(b_{k}/s)^{b_{k}}}{ b_{k}^{b_{k}}}=\lim_{k\to\infty}\frac{1}{s^{b_{k}}}\] and so for $s>1$ we get $\nu^{\ast}_{e}(\text{$\mathbb{F}$}^{s})=0$. Let $\text{$\mathbb{F}$}^{\prime}$ be the semigroup generated by $p_{j_{1}},p_{j_{2}},\dots,p_{j_{n}},\dots$. Then $p_{1}^{j_{1}}p_{2}^{j_{2}}\dots p_{b_{k}}^{j_{b_{k}}}\linebreak\text{$\mathbb{ F}$}_{b_{k}}\cap\text{$\mathbb{F}$}^{\prime}\neq\emptyset$ if and only if the exponents $j_{i}\neq 0$ are exactly the generators occurring in the sequence of generators of $\text{$\mathbb{F}$}^{\prime}$. Denote by $R(k)$ the number of $r_{n}$ belonging to $1,2,\dots,b_{k}$. Then \[\nu^{\ast}_{e}(\text{$\mathbb{F}$}^{\prime})=\lim_{k\to\infty}\Big{(}\frac{{b_ {k}}^{R(k)}}{b_{k}^{b_{k}}}\Big{)}.\] And so if $R(k)<b_{k}$ then $\nu^{\ast}_{e}(\text{$\mathbb{F}$}^{\prime})=0$. Denote by ${\mathbb{O}}_{r}$ for given $r$ - positive integer the set of all elements of $\mathbb{F}$ in the form $p_{i_{1}}^{\beta_{1}}...p_{i_{s}}^{\beta_{s}}$ where $\beta_{i}\geq r$. Then ${\mathbb{O}}_{r}\cap p_{1}^{j_{1}}...p_{n}^{j_{n}}\text{$\mathbb{F}$}_{n}\neq\emptyset$ only when $j_{i}>r$ or $j_{i}=0$. Thus for $n>r$ we have $[{\mathbb{O}}_{r}:F_{n}]=(n-r)^{n}$. By application of Proposition I we get $\nu^{\ast}_{e}({\mathbb{O}}_{r})={\mathbb{e}}^{-r}$, ${\mathbb{e}}$ is Euler’s number. From the other side $(\text{$\mathbb{F}$}\setminus{\mathbb{O}}_{r})\cap p_{1}^{j_{1}}...p_{n}^{j_{n }}\text{$\mathbb{F}$}_{n}\neq 0$ for all cases. Thus $\nu^{\ast}_{e}(\text{$\mathbb{F}$}\setminus{\mathbb{O}}_{r})=1$, ${\mathbb{O}}_{r}$ is not measurable. Choose a sequence of positive integers $\{n_{k}\}$ with $n_{k}^{n_{k}}\|n_{k+1}^{n_{k+1}}$, in the same way as in [21, page 42]. It can be proved that $\nu_{e}$ has the Darboux property on $\text{$\mathcal{D}$}_{\nu_{e}}$, (see also [20]). Thus $\{\nu_{e}(A);A\in\text{$\mathcal{D}$}_{\nu_{e}}\}=[0,1]$. The following result follows immediately from Definition 3.5. Proposition 4.7.: _A real valued function $f$ defined on $\mathbb{F}$ is Riemann integrable if and only if there exists a real number $\alpha$ such that for every system of finite sequences $\{a^{(n)}_{i};i=1,\dots,b_{n}^{b_{n}}\}$ with $a^{(n)}_{i}\in a_{i}\text{$\mathbb{F}$}_{b_{n}}$_ \[\lim_{n\to\infty}b_{n}^{-b_{n}}\sum_{i=1}^{b_{n}^{b_{n}}}f(a^{(n)}_{i})=\alpha\] _holds. In this case $\alpha=\int fd\nu_{e}$._ A mapping $x:\text{$\mathbb{F}$}\to[0,1]$ is called _uniformly divisor measurable_ if and only if for every subinterval $I\subset[0,1)$ the set $x^{-1}(I)$ belongs to $\text{$\mathcal{D}$}_{\nu_{e}}$ and $\nu_{e}(x^{-1}(I))=\ell(I)$. The mentioned mapping is a real valued net defined on $\mathbb{F}$ (considered as a partially ordered set). From the definition it follows immediately: If $\{y_{n}\}$ is $\nu_{e}$-B.u.d. sequence in $\mathbb{F}$ and $x:\text{$\mathbb{F}$}\to[0,1]$ is a uniformly divisor measurable map, then $\{x(y_{n})\}$ is a B.u.d. sequence in $[0,1]$. Example 4.8.: _Let $\text{$\mathbb{F}$}={\mathbb{N}}$ and $\{b_{k}\}$ fulfilling additionally the ralations $b_{k}^{b_{k}}\|b_{k+1}^{b_{k+1}}$ for $k\in{\mathbb{N}}$. Consider the following partition of the unit interval_ \[[0,1)=\bigcup_{j=1}^{b_{k}^{b_{k}}}I_{j}^{(k)},\ I_{j}^{(k)}=\left[\frac{j-1}{ b_{k}^{b_{k}}},\frac{j}{b_{k}^{b_{k}}}\right)\] _for $k=1,2,\dots$._ _Suppose that the sequence $\{y_{n}\}$ of elements $[0,1)$ satisfies :_ \[y_{n}\in I_{j}^{(k)}\Longleftrightarrow n\in a_{j}F_{b_{k}}.\] _Then $y^{-1}(I_{j}^{(k)})=A\left(\{y_{n}\},I_{j}^{(k)}\right)=a_{j}F_{b_{k}}$. Thus $A(\{y_{n}\},I_{j}^{(k)})\in\text{$\mathcal{D}$}_{\nu_{e}}$ and $\nu_{e}(A(\{y_{n}\},I_{j}^{(k)}))=\ell(I_{j}^{(k)})$. The set $\left\{\frac{j}{b_{k}^{b_{k}}};j=1,\dots,b_{k}^{b_{k}},k=1,2,\dots\right\}$ is dense in $[0,1)$ and so $\{y_{n}\}$ is uniformly divisor measurable. For the details see [11]._ For every mapping $x:\text{$\mathbb{F}$}\to[0,1]$ we have $\chi_{x^{-1}(I)}(a)=\chi_{I}(x(a))$ for every $a$ and $I\subset[0,1)$. We can state the generalization of Weyl’s criterion in this case. Proposition 4.9.: _A mapping $x:\text{$\mathbb{F}$}\to[0,1)$ is uniformly divisor measurable if and only for every system of finite sequences $\{a^{(n)}_{i};i=1,\dots,b_{n}^{b_{n}}\}$ with $a^{(n)}_{i}\in a_{i}\text{$\mathbb{F}$}_{b_{n}}$_ \[\lim_{n\to\infty}b_{n}^{-b_{n}}\sum_{i=1}^{b_{n}^{b_{n}}}f(x(a^{(n)}_{i}))= \int^{1}_{0}f(x)dx\] _holds for every Riemann integrable function $f$ on $[0,1]$._ Proposition 4.10.: _Let $H\subset\text{$\mathbb{F}$}$ be the semigroup generated by the generators $\{p_{1}^{\alpha_{1}},\dots,p_{k}^{\alpha_{k}},p_{k+1},p_{k+2},\dots\}$, $\alpha_{i}>0$. Then $H\in\text{$\mathcal{D}$}_{\nu_{e}}$ and $\nu_{e}(H)=\frac{1}{\alpha_{1}...\alpha_{k}}$ ._ Proof.: Consider $n>k$ and $n$ divisible by all $\alpha_{i}$. The $\text{$\mathbb{F}$}_{n}\subset H$ and \[H=\bigcup p_{1}^{j_{1}\alpha_{1}}...p_{k}^{j_{k}\alpha_{k}}p_{k+1}^{j_{k+1}}.. .p_{n}^{j_{n}}\text{$\mathbb{F}$}_{n}\] where $j_{1}\alpha_{1}<n,\dots,j_{k}\alpha_{k}<n,j_{k+1}<n,\dots,j_{n}<n$. Thus $H\in\text{$\mathcal{D}$}_{\nu_{e}}$ and \[\nu_{e}(H)=\frac{n}{\alpha_{1}}\dots\frac{n}{\alpha_{k}}n^{(n-k)}\cdot n^{-n}= \frac{1}{\alpha_{1}...\alpha_{k}}.\] ∎ Proposition 4.11.: _Let $S\in\text{$\mathcal{D}$}_{\nu_{e}}$ and $a\in F$. Then $aS\in\text{$\mathcal{D}$}_{\nu_{e}}$ and $\nu_{e}(aS)=\nu_{e}(S)$._ Proof.: It is sufficient to prove this statement for $a=p_{i}$, one of the generators. Let $n>i$. If the exponent of $p_{i}$ in $b$ does not exceed $n-1$ then $p_{i}b\text{$\mathbb{F}$}_{n}\in\text{$\mathcal{D}$}_{\nu_{e}}$. If $b=p_{i}^{n-1}b^{\prime}$ and $p_{i}$ not occurring in $b^{\prime}$, then $p_{i}b\text{$\mathbb{F}$}_{n}\subset b^{\prime}F_{n}$. Thus $\nu^{\ast}_{e}(p_{i}b\text{$\mathbb{F}$}_{n})\leq n^{-n}$. Suppose that for $\varepsilon>0$ there exists $n>i$ such that \[\bigcup_{j=1}^{k}a_{j}F_{n}\subset S\subset\bigcup_{j=1}^{s}a_{j}F_{n},\ \frac {s-k}{n^{n}}<\varepsilon.\] Then \[\bigcup_{j=1}^{k}p_{i}a_{j}F_{n}\subset p_{i}S\subset\bigcup_{j=1}^{s}p_{i}b_{ j}F_{n}.\] The sequences $a_{j}$, $j=1,\dots,k$, $b_{j},j=1,\dots,s$ contain at most $n^{n-1}$ elements divisible by $p_{i}^{n-1}$. Thus \[\nu^{\ast}_{e}\Big{(}\bigcup_{j=1}^{s}p_{i}b_{j}F_{n}\setminus\bigcup_{j=1}^{k }p_{i}a_{j}F_{n}\Big{)}\leq\frac{s+n^{n-1}}{n^{n}}-\frac{k-n^{n-1}}{n^{n}}< \varepsilon+\frac{2}{n}.\] ∎ Let $g$ be a bijection on the set of generators. We can extend this mapping to an automorphism of $\mathbb{F}$. Proposition 4.6 and Proposition 4.7 provide that $g$ fulfills condition (2) of Corollary 2.11. We have proved that $g$ preserves divisors measure density. On the other hand each automorphism $\mathbb{F}$ is uniquely determined by its values on the set of generators. Thus each automorphism on $\mathbb{F}$ preserves divisor measure density or $\nu_{e}$ does not depend on the order of generators. Corollary 4.12.: _For $b\in\text{$\mathbb{F}$}$ it holds $b\text{$\mathbb{F}$}\in\text{$\mathcal{D}$}_{\nu_{e}}$ and $\nu_{e}(b\text{$\mathbb{F}$})=1$. If $S\subset\text{$\mathbb{F}$}$ and $b\in\text{$\mathbb{F}$}$ then $S\in\text{$\mathcal{D}$}_{\nu_{e}}$ if and only if $(b\text{$\mathbb{F}$})\cap S\in\text{$\mathcal{D}$}_{\nu_{e}}$. In this case $\nu_{e}(S)=\nu_{e}(b\text{$\mathbb{F}$}\cap S)$._ ## Acknowledgments The first and third author would like to acknowledge the support of the Austrian Science Fund (FWF): F5510. They are also grateful to Prof. O. Strauch from the Slovak Academy of Science and Prof. V. Baláž from Comenius University in Bratislava for the fruitful discussions they had during their visit in Bratislava in November 2014. ## References [1] C. Aistleitner, M. Hofer, and V. Ziegler. On the uniform distribution modulo 1 of multidimensional LS-sequences. _Ann. Mat. Pura Appl. (4)_, 193(5):1329–1344, 2014. * [2] R. C. Buck. The measure theoretic approach to density. _Amer. J. Math._, 68:560–580, 1946. * [3] G. Cantor. Über die einfachen zahlensysteme. _Zeitschrift für Math. und Physik_, 14:121–128, 1869. * [4] I. Carbone. Discrepancy of $LS$-sequences of partitions and points. _Ann. Mat. Pura Appl. (4)_, 191(4):819–844, 2012. * [5] I. Carbone. How to construct generalized van der Corput sequences. _arXiv preprint arXiv:1304.5083_, 2013. * [6] I. Carbone and A. Volčič. Kakutani’s splitting procedure in higher dimensions. _Rend. Istit. Mat. Univ. Trieste_, 39:119–126, 2007. * [7] M. Drmota and R. F. Tichy. _Sequences, discrepancies and applications_, volume 1651 of _Lecture Notes in Mathematics_. Springer-Verlag, Berlin, 1997. * [8] S. Frisch, M. Paštéka, R. F. Tichy, and R. Winkler. Finitely additive measures on groups and rings. _Rend. Circ. Mat. Palermo (2)_, 48(2):323–340, 1999. * [9] A. Granas and J. Dugundji. _Fixed point theory_. Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. * [10] P. R. Halmos. _Measure Theory_. D. Van Nostrand Company, Inc., New York, N. Y., 1950. * [11] Z. Hedrlín. On integration in compact metric spaces. _Comment. Math. Univ. Carolinae_, 2(4):17–19, 1961. * [12] E. Hewitt and K. A. Ross. _Abstract harmonic analysis. Vol. I_, volume 115 of _Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_. Springer-Verlag, Berlin-New York, second edition, 1979. Structure of topological groups, integration theory, group representations. * [13] M. Hofer, M. R. Iacò, and R. F. Tichy. Ergodic properties of $\beta$-adic Halton sequences. _Ergodic Theory and Dynamical Systems_, FirstView:1–15, 1 2015. * [14] S. Kakutani. A generalization of Brouwer’s fixed point theorem. _Duke Math. J._, 8:457–459, 1941. * [15] L. Kuipers and H. Niederreiter. _Uniform distribution of sequences_. Wiley-Interscience [John Wiley & Sons], New York, 1974. * [16] I. Niven. Uniform distribution of sequences of integers. _Trans. Amer. Math. Soc._, 98:52–61, 1961. * [17] M. Paštéka. Covering densities. _Math. Slovaca_, 42(5):593–614, 1992. * [18] M. Paštéka. Note on the permutations which preserve Buck’s measure density. _Mediterr. J. Math._, 6(1):125–134, 2009. * [19] M. Paštéka. On Buck’s uniform distribution in compact metric spaces. _Tatra Mt. Math. Publ._, 56:61–66, 2013. * [20] M. Paštéka and T. Šalát. Buck’s measure density and sets of positive integers containing arithmetic progression. _Math. Slovaca_, 41(3):283–293, 1991. * [21] M. Paštéka. _On four approaches to density_, volume 3. Peter Lang, Frankfurt am Main, 2013. * [22] W. Rudin. _Principles of mathematical analysis_. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, third edition, 1976. International Series in Pure and Applied Mathematics. * [23] J. Šipoš. Nonlinear integrals. _Math. Slovaca_, 29(3):257–270, 1979. * [24] O. Strauch and $\check{S}$. Porubský. _Distribution of Sequences: A Sampler_. Peter Lang, Frankfurt am Main, 2005. * [25] R. F. Tichy. A criterion for the uniform distribution of sequences in compact metric spaces. _Rend. Circ. Mat. Palermo (2)_, 36(2):332–342, 1987. * [26] R. F. Tichy and R. Winkler. Uniform distribution preserving mappings. _Acta Arith._, 60(2):177–189, 1991. * [27] A. Volčič. A generalization of Kakutani’s splitting procedure. _Ann. Mat. Pura Appl. (4)_, 190(1):45–54, 2011. * [28] H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins. _Mathematische Annalen_, 77:313–352, 1916.
1610.07246	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 100715, "num_imgs": 17, "llama3_tokens_count": 28754 }	[ "content_image/1610.07246/x1.png", "content_image/1610.07246/x2.png", "content_image/1610.07246/x4.png", "content_image/1610.07246/x10.png", "content_image/1610.07246/x11.png", "content_image/1610.07246/x12.png", "content_image/1610.07246/x13.png", "content_image/1610.07246/x15.png", "content_image/1610.07246/x16.png", "content_image/1610.07246/x17.png", "content_image/1610.07246/x18.png", "content_image/1610.07246/x19.png", "content_image/1610.07246/x20.png", "content_image/1610.07246/x21.png", "content_image/1610.07246/x24.png", "content_image/1610.07246/x25.png", "content_image/1610.07246/x26.png" ]	# Relaxation of photoexcitations in polaron-induced magnetic microstructures Thomas Köhler Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Sangeeta Rajpurohit Institut für Theoretische Physik, TU Clausthal, 38678 Clausthal-Zellerfeld, Germany Ole Schumann Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Daimler AG, Wilhelm-Runge-Str. 11, 89081 Ulm, Germany Sebastian Paeckel Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Fabian R. A. Biebl Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Math2Market, Richard-Wagner-Straße 1, 67655 Kaiserslautern, Germany Mohsen Sotoudeh Institut für Theoretische Physik, TU Clausthal, 38678 Clausthal-Zellerfeld, Germany Stephan C. Kramer Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, 67663 Kaiserslautern, Germany Peter E. Blöchl Institut für Theoretische Physik, TU Clausthal, 38678 Clausthal-Zellerfeld, Germany Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Stefan Kehrein Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Salvatore R. Manmana Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany February 26, 2024 ###### Abstract We investigate the evolution of a photoexcitation in correlated materials over a wide range of time scales. The system studied is a one-dimensional model of a manganite with correlated electron, spin, orbital, and lattice degrees of freedom, which we relate to the three-dimensional material Pr${}_{1-x}$Ca${}_{x}$MnO${}_{3}$. The ground-state phases for the entire composition range are determined and rationalized by a coarse-grained polaron model. At half-doping a pattern of antiferromagnetically coupled Zener polarons is realized. Using time-dependent density-matrix renormalization group (tDMRG), we treat the electronic quantum dynamics following the excitation. The emergence of quasiparticles is addressed, and the relaxation of the nonequilibrium quasiparticle distribution is investigated via a linearized quantum-Boltzmann equation. Our approach shows that the magnetic microstructure caused by the Zener polarons leads to an increase of the relaxation times of the excitation. Manganite, Photoexcitation, DMRG, MPS, quasiparticle, Hubbard, PCMO, quarterfilling, MD, LBE ## I Introduction The relaxation of optical excitations in materials is a process that is central for energy conversion in materials. In particular, short length and time scales reveal a whole range of interesting physical effects. In the presence of strong correlations, this topic is one of ongoing experimental[1; 2; 3; 4; 5; 6] and theoretical studies.[7; 8; 9; 10] A detailed understanding of these processes is expected to open doors for new technological applications. For example, the controlled application of pump-probe setups on the femtosecond time scale has lead to interesting discoveries, such as the formation of metastable states, some of which are described to be superconducting.[11] Light irradiation of interfaces of correlated materials has shown the possibility to realize unconventional photovoltaic effects[12; 13; 14; 15; 16; 17; 18] that are not based on the formation of excitons, but rather of polarons, i.e., quasiparticles consisting of electrons and phonons that are formed or excited by light absorption. Pump-probe experiments in manganites have produced evidence for long-lived hot-polaron states.[19] Such long-lived states are of interest, because they have the potential to overcome the Shockley-Queisser limit[20] for the efficiency of solar cells. The relaxation processes of excitations span a wide range of time scales: The absorption process of light in pump-probe experiments can last as short as femtoseconds, whereas the perturbed polaronic order may persist up to time scales in the range of nanoseconds. The focus on strong correlations is particularly promising in the context of identifying slow relaxation processes. Finding a unifying description of the evolution of these excitations, particularly in the presence of strong correlations, is a major challenge. In this paper, we combine a number of theoretical approaches to cover the large range of time scales of a relaxation process for a specific material. The material chosen has been inspired by manganites, a class of materials with strong correlations between electrons, spins and phonons. In order to make the problem tractable, however, we have chosen a one-dimensional model-manganite system, which nevertheless contains many of the relevant properties of the real materials. We investigate the ground-state properties of this model system and find that they are well rationalized in terms of polaronic order. Using the time-dependent density-matrix renormalization group (tDMRG) in a matrix product state (MPS) formulation,[21; 22; 23; 24; 25; 26] we then investigate the time evolution following a dipole excitation, by which we model the effect of a photoexcitation on one of the polaronic ground states. This allows us to study the role of the electron-electron interaction in the short-time dynamics after the excitation. The long-time behavior of the electron relaxation is then investigated using a linearized quantum Boltzmann equation (LBE).[27] The tDMRG and the LBE approaches both show that the relaxation time scales increase with the strength of the magnetic microstructure, which is induced by the polaronic order. The paper is organized as follows. A 1D tight-binding model for the model-manganite, its polaronic and magnetic order, and the resulting effective Hubbard-type model are presented in Sec. II. In Sec. III we present the tDMRG results for the short-time dynamics of the local density and of the momentum distribution function following a photoexcitation, which we model by polarizing a single dimer in the center of the system. In Sec. IV we discuss how to estimate the quasiparticle momentum distribution from the numerical tDMRG results and the computation of the quasiparticle relaxation rates from a linearized Boltzmann equation ansatz, followed by Sec. V, in which we summarize. The considerations on the LBE are complemented by AppendixA. In AppendixB, details on the effect of boundary conditions on the momentum distribution are discussed. Finally, in AppendixC details for the MPS computations at finite temperatures and for the estimate of the energy density of the excitation are presented. ## II Polaronic order and effective model In Ref. [28], a tight-binding model for the strongly correlated electronic, spin, and lattice degrees of freedom of the three-dimensional manganite Pr${}_{1-x}$Ca${}_{x}$MnO${}_{3}$ is developed. Models of this type have been described by Hotta.[29] The parameters of the model have been extracted from first-principles calculations.[28] Because the simulation of the full quantum dynamics following a photoexcitation is out of reach for the three-dimensional material, we replace it by a fictitious one-dimensional manganite, which still exhibits the main properties of the three-dimensional system. As Ref. [28] already details the derivation of the microscopic model in the three-dimensional case, we sketch the basic ideas here only briefly, and mention the differences leading to the 1D model treated here. ### Tight-binding model for manganites <figure><img src="content_image/1610.07246/x1.png"><figcaption>Figure 1: One-dimensional chain of corner connected MnO6 octahedra of themodel manganite in the coordinate system chosen.</figcaption></figure> <figure><img src="content_image/1610.07246/x2.png"><figcaption>Figure 2: Degrees of freedom of the tight-binding model. (top) Orbitaldegrees of freedom of the eg electrons, which are treated explicitly, andclassical spin degree of freedom related to the t2g electrons. (bottom)Breathing mode Q1 and Jahn-Teller-active phonon modes Q2, Q3.</figcaption></figure> The one-dimensional material consists of a chain of corner connected MnO${}_{6}$ octahedra. The coordinate system has been chosen, as shown in Fig.1, with the $z$ axis along the chain, and the $x$ and $y$ axis along the orthogonal octahedral axes. The dynamic electronic, spin and lattice degrees of freedom of the tight-binding model, sketched in Fig.2, are the following: * Electrons: The relevant electrons are those in the two $e_{g}$ orbitals of the Mn-ions. In the spirit of density-functional theory, the $e_{g}$ electrons are described by one-particle wave functions. The one-particle wave function with band index $n$ is expressed as \[\|\psi_{n}\rangle=\sum_{\sigma,\alpha,R}\|\chi_{\sigma,\alpha,R}\rangle\psi_{ \sigma,\alpha,R,n}\] (1) in terms of local spin-orbitals $\|\chi_{\sigma,\alpha,R}\rangle$. The spin orbitals at Mn-site $R$ have spin $\sigma\in\{\uparrow,\downarrow\}$ and spatial orbital character $\alpha$, denoting the $d_{x^{2}-y^{2}}$ orbital for $\alpha=a$ and the $d_{3z^{2}-r^{2}}$ orbital for $\alpha=b$. * Spins: The three low-lying, spin-aligned electrons in the Mn-$t_{2g}$ states at site $R$ are described by a classical spin $\vec{S}_{R}$ of size $\frac{3}{2}\hbar$. Recent calculations indicate this to be an excellent approximation.[30] * Lattice: The relevant phonons are the two Jahn-Teller active distortions of the MnO${}_{6}$ octahedra. Their phonon amplitudes are denoted by $Q_{2,R}$ and $Q_{3,R}$.[31] The mode $Q_{3,R}$ describes the oblate and prolate distortion of the octahedron at site R along the $z$ direction. The mode $Q_{2,R}$ describes the simultaneous elongation along the $x$ and contraction along the $y$ direction, and vice versa. We refer here to the local Cartesian coordinates aligned along the octahedral axes. All other degrees of freedom are either absorbed into the dynamical variables of the model, or they are considered as a bath and not treated explicitly. The total-energy functional of density-functional theory[32; 33] is replaced by the potential energy of the model, which takes the form \[E_{pot}=E_{e}+E_{S}+E_{ph}+E_{e-ph}+E_{e-S}\;,\] (2) where $E_{e}$ is the energy of the electronic subsystem in the Mn-$e_{g}$ orbitals, $E_{S}$ describes the antiferromagnetic interaction of the $t_{2g}$ electrons on neighboring Mn sites, and $E_{ph}$ is the energy of the Jahn-Teller active phonons. The coupling of electrons with the spin of the $t_{2g}$ electrons and the lattice vibrations is described by $E_{e-S}$ and $E_{e-ph}$, respectively. We treat the different terms in the following way: * Electronic energy $E_{e}$: The electronic energy contribution $E_{e}=E_{kin}+E_{U}$ consists of the kinetic energy $E_{kin}$ of the $e_{g}$ electrons and their interaction $E_{U}$. In the tight-binding model, the interaction $E_{U}$ is treated in a Hartree-Fock-like manner, analogous to hybrid density functionals[34], LDA+$U$[35] and GW[36] calculations. All on-site matrix elements of the Coulomb interaction within a $e_{g}$ shell are considered. They can be parameterized by two Kanamori parameters[37]$U$ and $J_{xc}$. The hopping amplitude of the kinetic energy term is denoted as $t_{hop}$. * Spin energy $E_{S}$: The spin energy describes the coupling between the $t_{2g}$ states of neighboring Mn sites. The coupling is antiferromagnetic and described by the parameter $J_{AF}$. * Phonon energy $E_{ph}$: The phonon energy describes a restoring force term which restores the perfect octahedron in the absence of other interactions. Note that the octahedral distortions of different sites are not independent but strongly coupled via oxygen atoms shared between two octahedra. * Electron-phonon coupling $E_{e-ph}$: The electron phonon coupling is responsible for the Jahn-Teller distortions of the octahedra in the presence of electrons. * Electron-spin coupling $E_{e-S}$: The electron spin coupling is responsible for the Hund’s coupling $J_{H}$ between electrons in the Mn-$e_{g}$ orbitals, which are described explicitly, and the Mn-$t_{2g}$ electrons represented by classical spins. This term is the origin of superexchange and double exchange, which are responsible for the complex magnetic properties of manganites. The parameters of the model are extracted from _ab initio_ calculations of the three-dimensional manganites by using the projector-augmented wave method[38] in combination with the local hybrid density functional PBE0${}^{\text{r}}$ (for details, see Ref. [28]). Compared to the treatment in Ref. [28], we include two changes: First, we ignore the breathing distortion $Q_{1}$ used in the original 3D model and, second, we increase the antiferromagnetic coupling for the one-dimensional model from 12 meV to 32.6 meV. The latter was necessary to avoid a ferromagnetic ground state, while the 3D material exhibits a complex antiferromagnetic order. The set of parameters obtained is reproduced in Table 1. JAF \| 32.6 \| meV \| g\definecolor[named]pgfstrokecolorrgb1,1,1\pgfsys@color@gray@stroke1\pgfsys@color@gray@fill1†JT \| 2.113 \| eV/Å ---\|---\|---\|---\|---\|--- JH \| 0.653 \| eV \| kJT \| 5.173 \| eV/Å2 U \| 2.514 \| eV \| thop \| 0.585 \| eV Jxc \| 0.692 \| eV \| \| \| Table 1: Model parameters for the one-dimensional model situation, based on the first-principle calculations on Pr1−xCaxMnO3 of Ref. sotoudeh17_prb95_235150, . JAF describes the antiferromagnetic coupling between the t2g states of neighboring Mn sites; JH is the Hund’s coupling; U and Jxc are the Kanamori parameters for electron-electron interaction between eg electrons; gJT and kJT parametrize the electron-phonon interaction; thop is the hopping amplitude of the eg electrons. With two exceptions described in the text, they are identical to the values extracted in Ref. sotoudeh17_prb95_235150, . ### Polaron and magnetic order in 1D manganites In this section, we describe the ground-state configurations of the one-dimensional manganite chain as obtained from the tight-binding model Eq. (2). Beyond the purpose of providing the ground states of the one-dimensional model, the motivation for the present study has been to explore how the complex phase diagram and the polaron arrangement of manganites in general can be described and analyzed. The polaron model derived in the following is a promising approach towards this goal. The ground states have been determined in a two step approach: First, stable and metastable configurations are obtained using Car-Parrinello-like dynamics [39] with friction. ¹ Second, the emerging patterns and their building blocks are identified. Moving towards a higher-level description, the total energy is then expressed in terms of the energies of these building blocks. The magnetic order and the resulting polaron composition are presented in Table 2. The electronic structure, i.e., the density of states, of the various polaron types is provided in Fig. 3. [FOOTNOTE:1][ENDFOOTNOTE] Ne \| magnetic order \| composition \| (ETB−EPM)[meV] \| ETB[eV] ---\|---\|---\|---\|--- 0 \| ↑↓↑↓↑↓↑↓↑↓↑↓ \| V12 \| 0 \| -0.3912 1 \| ↓↑↑↑↓↑↓↑↓↑↓↑ \| PeV9 \| 0 \| -1.98129 2 \| ↓↑↑↑↓↑↑↑↓↑↓↑ \| (PeV)2V4 \| 0.5 \| -3.57088 3 \| ↓↑↑↑↓↑↑↑↓↑↑↑ \| (PeV)3 \| 10.7 \| -5.15075 4 \| ↑↑↑↓↓↓↑↑↑↓↓↓ \| Pe4 \| 59.3 \| -6.69231 5 \| ∗1 noncollinear \| PZ3Pe2 \| -16.2 \| -8.08389 6 \| ↑↑↓↓↑↑↓↓↑↑↓↓ \| PZ6 \| 0 \| -9.38383 7 \| ↑↑↓↓↑↓↓↑↑↓↓↓ \| PZ4PhPJT \| -46.7 \| -10.46353 8 \| ∗2↑↑↑↓↓↑↓↑↑↓↑↑ \| PZ3PhPJT3 \| -12.0 \| -11.51915 9 \| ∗3↑↑↓↑↑↓↑↓↑↑↑↓ \| Pz2PhPJT5 \| -68.9 \| -12.51556 10 \| ↑↓↑↓↓↑↓↑↑↓↑↓ \| Pz2PJT8 \| -74.4 \| -13.51869 11 \| ↓↑↑↑↓↑↓↑↓↑↓↑ \| PhPJT9 \| 0 \| -14.47726 12 \| ↑↓↑↓↑↓↑↓↑↓↑↓ \| PJT12 \| 0 \| -15.47453 ∗1\- All the angles between classical spin vectors lie in the range of ∼(162−175)o. ∗2\- The average angle within the trimer is ∼<51o> and other angles are in the range ∼(157−166)o. ∗3 \- The average angle within the trimer is ∼<39.5o> and other angles are in the range ∼(162−175)o. Table 2: Magnetic orders, polaron composition, deviation EPM−ETB of the energy EPM from the polaron model, Eq. (3), and energy ETB from the tight- binding model Eq. (2) for different numbers of electrons in a 12-site unit cell of the 1D manganite. <figure><img src="content_image/1610.07246/x4.png"><figcaption>Figure 3: Projected density of states of the tight-binding model for thepolarons in the 1D manganite chain as a function of energy in eV. Top left:Two adjacent unoccupied sites V; top-right: two adjacent Jahn-Teller polaronsPJT; middle left: electron polaron Pe; middle right: hole polaron Ph; bottomZener polaron PZ. The horizontal line indicates the Fermi level. Empty andfilled y lines indicate d3z2−r2 orbitals pointing along the chain and dx2−y2orbitals orthogonal to the chain, respectively. The density of states isbroadened by 0.05 eV.</figcaption></figure> We obtained the following dominant patterns, which we describe as polarons: 1. There are sites $V$ without $e_{g}$ electrons, which we denote as vacant sites. The spins on these sites interact only weakly with each other via the antiferromagnetic Heisenberg exchange coupling $J_{AF}$. In this language, the electron-poor manganite – analogous to CaMnO${}_{3}$ – consists of tightly packed $V$ sites. 2. The electron polaron $P^{e}$ is a trimer of ferromagnetically aligned Mn sites occupied by a single $e_{g}$ electron. $P^{e}$ is analogous to an electron polaron in CaMnO${}_{3}$. $P^{e}$ has three electron states in the majority-spin direction: The lowest state is occupied and fully bonding. The second state is unoccupied and nonbonding. This state is distributed over the two outer sites of the trimer. The third state of the electron polaron is fully antibonding. 3. The Zener polaron[41; 42]$P^{Z}$ is a dimer of ferromagnetically aligned Mn sites, which share a single $e_{g}$ electron. The half-doped 1D material – analogous to Pr${}_{1/2}$Ca${}_{1/2}$MnO${}_{3}$ (PCMO) – can be described as a crystal of antiferromagnetically coupled Zener polarons. The Zener polaron has two states in the majority spin direction: a filled bonding and an empty antibonding state. 4. The hole polaron $P^{h}$ is a trimer of ferromagnetically aligned Mn sites occupied by two $e_{g}$ electrons. $P^{h}$ is analogous to a hole polaron in PrMnO${}_{3}$. It has the same three states as the electron polaron, however the second, nonbonding state is occupied as well. 5. The Jahn-Teller polaron $P^{JT}$ is an $e_{g}$ electron that occupies a single site. A crystal of Jahn-Teller polarons is analogous to PrMnO${}_{3}$. In order to extract the energies for these structural units, we start out by setting the reference $\mu_{0}$ for the electron chemical potential to the coexistence value of the electron-poor $(N_{e}=0)$ and the electron-rich $(N_{e}=N_{s})$ systems that are analogous to CaMnO${}_{3}$ and PrMnO${}_{3}$, respectively. That is, instead of the energy $E$, we consider the energy $E-\mu_{0}N_{e}$ of the manganite together with a conveniently chosen electron reservoir. With $N_{s}$, we denote the number of sites in the unit cell and with $N_{e}$ the number of electrons per unit cell. Then, we identify the polaron composition from the magnetic order. The formation energies of the polarons are determined in such a way that the energy \[\begin{split} E_{PM}[n_{V},n_{e},n_{Z},n_{h},n_{JT}]=\\ \sum_{j\in\{e,Z,h,JT\}}n_{j}(E_{f}^{(j)}+\mu_{0}N^{(j)}_{e}-J_{AF })\end{split}\] (3) of the polaron model matches the total energies obtained from the tight-binding calculation given in Table 2. With $n_{j}$, we denote the number of polarons $P^{(j)}$, where $j\in\{V,e,Z,h,JT\}$ denotes the polaron type. With $E_{f}^{(j)}$, we denote the polaron formation energy and with $N^{(j)}_{e}$ the number of $e_{g}$ electrons in the respective polaron. The values are presented in Table 3. Specifically, the polaron-formation energies have been extracted so that the energies of the model calculations are reproduced by Eq. (3) for $N_{e}=0,1,6,11,$ and $12$. Figure 4 compares the energetics of the configurations in Table 2 with that of the polaron model, Eq. (3), using the values from Table 3. It is evident that the simple polaron model captures the major features of the ground state energetics. <figure><img src="content_image/1610.07246/x10.png"><figcaption>Figure 4: Energy per Mn site of the model calculation (line) as a function ofthe electron occupation 1−x=Ne/Ns compared to the sum of polaron energiesgiven in Eq. (3) (symbols). The energy (1−x)μ0 of the particle reservoir hasbeen included.</figcaption></figure> polaron \| V \| Pe \| PZ \| Ph \| PJT ---\|---\|---\|---\|---\|--- E(j)f[meV] \| 0 \| -398.3 \| -274.4 \| -324.9 \| 0 N(j)s \| 1 \| 3 \| 2 \| 3 \| 1 N(j)e \| 0 \| 1 \| 1 \| 2 \| 1 Table 3: Formation energies of polarons, number N(j)s of sites occupied by the polaron Pj, and number N(j)e of eg electrons on it. Interestingly, we can attribute a definite size to the electron polaron: The mechanism limiting the size of the electron polaron $P^{e}$ is the competition of the kinetic energy with the antiferromagnetic coupling. Increasing the size of the electron polaron on one side lowers the kinetic energy of the electron, because it can spread over a larger region. On the other side, there is a penalty for aligning more sites ferromagnetically. The maximum size of the electron polaron is reached when the delocalization energy gained by extending the electron polaron by one site is exceeded by the antiferromagnetic coupling. In the limit of large Hund’s coupling $J_{H}$, the size of the polaron is thus determined by the ratio $t_{hop}/J_{AF}$ of hopping parameter and antiferromagnetic coupling. With our set of parameters, this maximum size is three sites. In the dilute limit, i.e., for $N_{e}\leq 3$, we find that the polarons are separated by at least one vacant site $V$, as if there were a nearest-neighbor repulsion between adjacent polarons. We attribute this repulsion to the Coulomb interaction between the wave function tails, which extend into a neighboring polaron, with the electrons already belonging to this neighboring polaron. Because of the kinetic energy cost, the smaller polarons, such as $P^{Z}$, are energetically less favorable than larger polarons, such as $P^{e}$. Thus, they become relevant only when the electron density is such that the larger polarons, namely the electron polaron $P^{e}$, are densely packed. For our system, this occurs at $N_{e}/N_{s}=1/3$. Beyond this value, electron polarons $P^{e}$ and Zener polarons $P^{Z}$ coexist until Zener polarons are densely packed. This is the case for half doping, i.e., $N_{e}/N_{s}=1/2$. This is the doping used for the study of the optical excitation, which will be discussed in the following section. In the electron-rich phase with $N_{e}/N_{s}=1$, the system forms a solid of antiferromagnetically coupled Jahn-Teller polarons $P^{JT}$. When doping the electron-rich phase with holes, the preferred way is via a Zener polaron $P^{Z}$. In analogy with the electron-poor manganite, one would have expected that the extended defect $P^{h}$ was favorable compared to the smaller Zener polaron $P^{Z}$. The reason for the preference of the Zener polaron is that the formation of hole polarons $P^{h}$ from Zener and Jahn-Teller polarons requires substantial energy \[P^{JT}+P^{Z}\to P^{h}-50.5\leavevmode\nobreak\ \text{meV }\;.\] (4) Nevertheless, we encounter hole polarons in our calculations. They are formed in response to spin frustration. The insertion of a hole into the electron rich material by forming a Zener polaron would, at the same time, introduce a domain wall into the antiferromagnetic order. An isolated domain wall can either annihilate with another domain wall or combine with a Zener polaron to form a hole polaron. With the hole polaron, we identified a structural unit that does not contribute to the ground state at zero Kelvin, but that plays an important role for the interconversion of polarons. The nature of an isolated domain wall can be rationalized from the point of view of a Zener polaron. An abrupt domain wall in the electron-rich material is equivalent to a Zener polaron with one additional electron. The additional electron enters into an antibonding state, which is energetically highly unfavorable. By forming a hole polaron, this electron is transferred into the nonbonding state of the hole polaron, which is energetically favorable. The occurrence of noncollinear spin arrangements indicates that the phase boundary can also delocalize and form a spin spiral. The energy scale of forming these polarons is of the order of 0.3 eV, while that of the interaction between polarons is much smaller, i.e., of the order of 10 meV. There is an analogy of the description of the order in manganites in terms of various types of weakly interacting polarons with molecules in chemistry. A polaron is the analogon of a molecule. Electrons that delocalize over several Mn-sites are analogous to a chemical bond. Similar to molecules, which can arrange into molecular crystals, the polarons arrange in various patterns, which give rise to the complex phase diagram of manganites. The conversion of polarons into each other is then analogous to a chemical reaction. An example for such a reaction between polarons is the formation of a hole polaron from a Jahn-Teller and a Zener polaron in Eq. (4) described above. <figure><img src="content_image/1610.07246/x11.png"><figcaption>Figure 5: Sketch of the effective Hubbard-type many-body model derived inthis section. The unit cell consists of four sites, with each site hosting aspin-up and a spin-down state. The two states are energetically separated bythe Hund’s splitting Δ=2JH. Electrons can hop from one site to another withthe hopping amplitude thop, provided the two states have the same spin. Forthe sake of clarity this is only shown for the spin-↓ direction. If twoelectrons are located on the same site, they feel the screened Coulombrepulsion U (depicted by the curly line). Otherwise, this repulsion isscreened by the positively charged atoms.</figcaption></figure> ### Hamiltonian for a frozen lattice of Zener polarons In the previous section, we explored the ordered phases of the 1D model manganite at low temperatures described by Eq. (2). In order to study the light-absorption process and the electronic relaxation, we focus on the electronic degrees of freedom. In the following, we therefore freeze the spin and lattice degrees of freedom in the ground state. The only dynamical entities in this model are the $e_{g}$ electrons. Furthermore, the Hilbert space for the $e_{g}$ electrons has been limited to two $d_{3z^{2}-r^{2}}$ spin orbitals per Mn site, i.e., $\|\chi_{\sigma,b,R}\rangle$, which makes the model similar to a single-band Hubbard model with spatially varying magnetic fields. Furthermore, we will focus on the half-doped system, because it allows us to study the role of the magnetic microstructures formed by antiferromagnetically coupled Zener polarons on the relaxation dynamics of a photoexcitation. This order corresponds to $N_{e}=6$ in Table 2. As shown in Fig.3, such a Zener polaron consists of two neighboring Mn sites, which share a single $e_{g}$ electron that is uniformly delocalized over both sites. The Mn ions inside a Zener polaron are ferromagnetically aligned, and, without loss of generality, we choose the spins to point along the $z$ axis, that is $S_{x}=S_{y}=0$. This leads to the spin configuration on the four Mn-sites of the unit cell, \[\Bigl{(}S_{z,1},S_{z,2},S_{z,3},S_{z,4}\Bigr{)}=\frac{3\hbar}{2} \Bigl{(}-1,-1,+1,+1\Bigr{)}\,.\] (5) The spin distribution is periodic, so that $S_{z,R+4}=S_{z,R}$. This means that the electrons experience the spin and lattice degrees of freedom as a staggered magnetic field. Because of the restriction to a collinear spin distribution of the classical spins $\vec{S}_{R}$, the two spin directions of the electrons decouple, except through the electron-electron interaction. Since the Jahn-Teller distortions do not modulate the potential for the remaining orbitals of the state under investigation, we also omit the electron-phonon coupling. As a result, the total energy can be expressed in the form of a one-band Hubbard model with a staggered magnetic field and total energy \[E=E_{kin}+E_{U}+E_{e-S}\;.\] (6) Formulated in second quantization, we thus obtain the simplified many-electron Hamiltonian for a half-doped 1D manganite, \[\hat{H}=\sum_{R} \biggl{\{} -t_{hop}\sum_{\sigma}\Bigl{(}\hat{c}^{\dagger}_{\sigma,R+1}\hat{c }_{\sigma,R}+\hat{c}^{\dagger}_{\sigma,R}\hat{c}_{\sigma,R+1}\Bigr{)}\] (7) \[+ U\hat{n}_{\uparrow,R}\hat{n}_{\downarrow,R}+\frac{\Delta}{3\hbar }S_{z,R}\left(\hat{n}_{\uparrow,R}-\hat{n}_{\downarrow,R}\right)\biggr{\}}\;,\] with $\hat{c}^{(\dagger)}_{\sigma,R}$ the annihilation (creation) operator for an electron of spin $\sigma$ at position $R$, and the local spin occupation $\hat{n}_{\sigma,R}:=\hat{c}^{\dagger}_{\sigma,R}\hat{c}_{\sigma,R}$. Using the values of Table 1, we obtain \[U\approx 4.3t_{hop}\] (8) for the Hubbard interaction and \[\Delta:=2{J_{H}}\approx 2.3t_{hop}\;\] (9) for the Hund’s splitting. The resulting Hubbard-type model thus has a unit cell of four sites and is sketched in Fig. 5. In relation to PCMO, it will be interesting to study the photoexcitation for the set of parameters (8) – (9). However, model Eq. (7) allows us to go beyond and tune the values of $U/t_{hop}$ and $\Delta/t_{hop}$ independently from each other. Consequently, this model realizes a minimal model for a manganite system to study the interplay between the Hund’s coupling and the electron-electron interaction after a photoexcitation in such systems. In the following, we will hence study the time evolution after a photoexcitation for the parameter values (8) – (9) using MPS and LBE techniques, and also the results when changing the values of $\Delta/t_{hop}$ and $U/t_{hop}$. #### ii.3.1 Band structure of non-interacting electrons in the lattice of frozen Zener polarons Before discussing the photoexcitations, let us first explore the basic features of model Eq. (7) without Coulomb interaction, i.e., the case $U=0$. The band structure of the noninteracting system will elucidate the role of the Hund’s splitting $\Delta$, which acts as a staggered magnetic field on the electronic structure. We obtain \[\epsilon_{\nu}(k) = s_{1,\nu}t_{hop}\sqrt{2+\tilde{\Delta}^{2}+s_{2,\nu}2\sqrt{\cos^ {2}(2ka)+\tilde{\Delta}^{2}}}\;,\] (10) where $\tilde{\Delta}=\frac{\Delta}{2t_{hop}}$, $k$ is the momentum in the reduced Brillouin zone, $\nu$ labels the bands in this reduced Brillouin zone, and $(s_{1,\nu},s_{2,\nu})=(-1,+1),(-1,-1),(+1,-1)$ and $(+1,+1)$ for $\nu=1,2,3,4$. The spacing between the Mn ions is denoted by $a$. For the details of the derivation, see AppendixA. In Fig.6, this band structure is shown for different values of $\Delta/t_{hop}$. Without Hund’s splitting, the system is equivalent to a single-band Hubbard chain, which has the band structure \[\epsilon(k)=-2t_{hop}\cos(ka)\,.\] (11) In the setting of the four-site unit cell, this band structure is folded back twice into the smaller reciprocal unit cell as shown in Fig.6. The lowest of the four bands is occupied. In the limit of infinite Hund’s splitting $\Delta$, the band structure develops into four nearly dispersionless bands. The nature of the states in this limit can be identified with those of a Hubbard dimer, respectively a hydrogen molecule. The isolated Hubbard dimer has a bonding and an antibonding state for each spin direction. The energetic separation of the two bands is given by the hopping parameter as $2t_{hop}$. The electrons of one spin direction experience a downward Zeeman-like shift by $J_{H}=\frac{1}{2}\Delta$ on one Zener polaron and a similar upward shift on the other Zener polaron. The resulting states are at $-\frac{1}{2}\Delta\pm t_{hop}$ and $\frac{1}{2}\Delta\pm t_{hop}$. The bandwidth of the four bands is given by the ability of electrons to tunnel between two second-nearest neighbor Zener polarons, which have the same spin orientation. The tunneling probability in turn acts as an effective hopping for the molecular states. As seen in Fig. 6, intermediate Hund’s splitting leads to a coexistence of gaps, flat bands, and bands with large dispersion. Thus, the behavior of the band structure will be non-trivial and probably most interesting for intermediate Hund’s splitting. The parameters in Table 1 show that PCMO lies in this regime. <figure><img src="content_image/1610.07246/x12.png"><figcaption>Figure 6: One-particle band structure of PCMO for different values of theHund’s splitting Δ, which is measured in units of thop. Γ denotes the originof the k points and X=π/4a the zone boundary with the Mn-Mn spacing a. One cansee that the distance between the center of the upper two bands and the centerof the lower two bands is close to Δ for large values of Δ. Furthermore, inthe same limit, the distance of the upper two bands (as well as the one of thelower two bands) is approximately 2thop.</figcaption></figure> ## III Photoexcitation dynamics ### Treatment of light-matter interaction A simple way to model the photoexcitation is to assume it to create particle-hole like excitations.[43; 44; 45; 7; 46; 7] Here, we start from a ground state described in terms of Zener polarons, in which the electron density is equally distributed. We then model the photoexcitation as inducing an electric dipole on a single polaron. The conceptually simplest operator then is \[\hat{Y}_{R}=\sum_{\sigma}\hat{c}^{\dagger}_{\sigma,R+1}\hat{c}_{\sigma,R}\;,\] (12) with $R$ and $R+1$ being lattice sites both located on the same polaron. In this paper, we will treat a single excitation on lattices with typically 40 sites. As discussed in AppendixC, assuming a light pulse of duration of $1$ fs, this corresponds to an intensity of $\sim 10^{8}$ W/mm. ### Details of the tDMRG calculations We use the two-site time evolution matrix-product operator (MPO) introduced in Ref. [47] with a time step of $\Delta t=0.05$. In this approach, the operator exponential of the propagator \[\hat{U}(t)=e^{-\frac{i}{\hbar}\hat{H}\Delta t}\] (13) is given by an MPO. In Ref. [47] two representations are introduced; we chose the one denoted as $\hat{W}^{II}$, which is considered to be more accurate. The Hamiltonian is given in terms of finite state machines and is subsequently transformed into the MPO form. [48; 49] Two main error sources need to be considered: First, the error due to the truncation of the MPS matrices to dimension $\chi_{\text{MPS}}$. In all simulations, the entanglement induced by the perturbation, as quantified by the von Neumann entropy,[50] is rather small. Therefore, a matrix dimension of $\chi_{\text{MPS}}=512$ for systems with up to $L=40$ lattice sites was sufficient to obtain a discarded weight $\varepsilon\sim 10^{-8}$ ($\sim 10^{-4}$) at the end of the time evolutions for $\Delta/t_{hop}=8$ ($\Delta=0$). The second source of error is due to the approximation of the operator exponential and is explained in detail in Ref. [47]. As it is much smaller than the error due to the truncation, this error is negligible. The MPS code used is implemented using the SciPAL[(51)] library, which is a framework based on C++ expression templates, and provides the possibility to use CPUs as well as GPUs by calling efficient implementations of BLAS and cuBLAS² functions. [FOOTNOTE:2][ENDFOOTNOTE] ### Short-time dynamics after the photoexcitation <figure><img src="content_image/1610.07246/x13.png"><figcaption>Figure 7: Time evolution of the local density ⟨^ni⟩ following an excitationby applying operator Eq. (12) at the center of the system. The panels showtDMRG results for different values of Δ/thop for chains with L=40 latticesites. Left side: U=0; right side: U/thop=4.3. The solid lines indicate themaximal group velocity of the excited electrons obtained from thenoninteracting band structure Eq. (10), assuming that one electron getsexcited from the first to the second band. The dotted and dashed linesindicate the phase velocity at the k-value with the maximal group velocity, asdiscussed in the text.</figcaption></figure> <figure><img src="content_image/1610.07246/x15.png"><figcaption>Figure 8: Time evolution of the local density ⟨^nR+1⟩ obtained by tDMRG for asystem with L=40 lattice sites at the right site of the Zener polaron at whichoperator Eq. (12) was applied to, R=L/2+1. Green: U=0; purple: U/thop=4.3. Thelines for Δ/thop=8 show a fit using a function of the formf(x)=14(cos(ax)+cos(bx))+12.</figcaption></figure> In Fig.7, we show the tDMRG results for the time evolution of the local densities $\langle\hat{n}_{R}\rangle:=\langle\hat{n}_{\uparrow,R}+\hat{n}_{\downarrow,R}\rangle$ following a photoexcitation. This is modeled by applying operator Eq. (12) at the center of the system to the ground state obtained from a DMRG calculation, which induces a local dipole on the Zener polaron at the center of the system. We display results for $\Delta/t_{hop}=0,\,2.3,\ 8$ and compare the cases $U=0$ (left panels) to the case $U/t_{hop}=4.3$ (right panels). The case with $\Delta/t_{hop}=2.3$ and $U/t_{hop}=4.3$ corresponds to the values of Table 1. Let us start the discussion with the behavior at $\Delta=0$. In the ground state we observe Friedel-like density oscillations caused by the open boundary conditions used.[53; 54] They are typical for the Luttinger liquid phase[55] realized in the Hubbard chain at this value of the filling.[56] These Friedel-like oscillations are stable and do not change with time. On top, we see that the local excitation created at the center of the system spreads through the lattice with constant maximum speed. This light-cone behavior is captured by a Lieb-Robinson bound,[57] which states that in nonrelativistic quantum lattice systems with a short-ranged Hamiltonian information spreads with a finite maximal velocity. In this case, for $U=0$ and $\Delta=0$, the maximal group velocity allowed by the band structure Eq. (10) is the Fermi velocity $v_{F}=2\frac{t_{hop}\,a}{\hbar}$. In the units used ($a=t_{hop}=\hbar=1$), this leads to a slope of $2$ in the lightcone, which is what is seen in Fig. 7 for $\Delta=U=0$. For $U>0$ and $\Delta=0$, the velocity gets modified by the interaction, but as expected from Luttinger liquid theory,[55] the system will always show ballistic motion of the excitation, i.e., it will propagate with a constant maximal velocity through the system. For finite values of $\Delta$ the Friedel-like oscillations disappear. This is expected, since for any finite value of $\Delta$ a band gap is formed so that the Fermi surface vanishes, and with it the Luttinger liquid phase and the Friedel-like oscillations. By increasing the value of $\Delta/t_{hop}$, the velocity of the spread of the excitation is seen to decrease. For $U=0$ this is expected from the single-particle band structure Eq. (10), in which the bands become flatter with increasing $\Delta/t_{hop}$, which also reduces the maximal group velocity. For the times shown $t/t_{hop}\leq 20$ (corresponding to $\sim 23$ fs using the values of Table 1), for $\Delta/t_{hop}=8$ the speed of the excitation is close to zero, since the group velocities obtained from the band structure are very small already (e.g., the maximal group velocity for an electron excited to the second band is $v\approx 0.08\frac{t_{hop}a}{\hbar}$). At the site of the excitation, the dipole-like density oscillations become clearly weaker with time for $\Delta/t_{hop}=2.3$ as the energy is transferred to the neighboring sites. For the largest Hund’s splitting shown, $\Delta/t_{hop}=8$, the dipole oscillations remain concentrated on the central site on the time scale shown. While for $\Delta=0$ the noninteracting electrons move with the expected Fermi velocity $v_{F}=2\frac{t_{hop}a}{\hbar}$, for the intermediate value $\Delta/t_{hop}=2.3$ an interesting structure emerges, which is apparently caused by the presence of both dipole-like oscillations of the electron on the excited Zener polaron and the relatively small tunneling barrier between the polarons: When the electron reaches the boundary between two Zener polarons, it gets partially reflected, but can also partially tunnel to the next polaron. This happens again for both the transmitted as well as the reflected part of the electron when they reach the border to the next polaron, and so on. The result is the intricate pattern seen in Fig. 7, in which the excited electron seems to spread through the system in a ping-pong or billiard-like manner for $U=0$ and $\Delta/t_{hop}=2.3$. However, now a further interesting effect comes into play, which leads to linear structures with a slope substantially larger than the maximal group velocity allowed by the band structure. This was discussed in Ref. [58] in the context of interacting Mott insulators: The spread of information through the lattice is governed by the Lieb-Robinson velocity, which here can be estimated as the maximal group velocity determined by $v_{g,\nu}=\partial\epsilon_{\nu}(k)/\partial k$, with the band $\nu$, to which the electron is excited to. However, as described in Ref. [58], within the light cone and in its vicinity it is possible to have linear structures with a slope corresponding to the maximal _phase velocity_ instead, which is determined via $v_{p,\nu}=\epsilon_{\nu}(k^{})/k^{}$, where $k^{}$ is the momentum, at which $v_{g,\nu}$ is maximal. The phase velocity can be substantially larger than the maximal group velocity. This corresponds to what is seen in Fig. 7 for $U=0$ and $\Delta/t_{hop}=2.3$: The excitation causes linear structures, whose slope is in excellent agreement with the maximal phase velocity obtained from the band $\nu=2$ in Eq. (10). However, the structure is seen to be strong only as long as it is within or close to the light cone, which is obtained from the maximal group velocity determined from $\epsilon_{2}(k)$ in Eq. (10). As soon as they reach the border of the light cone, their amplitude decays quickly, so that they do not contribute to the spread of information through the lattice. In the presence of repulsive $U$, it is an interesting question whether the ballistic transport will prevail, or if the interparticle scattering might change its speed, e.g., inhibiting transport by slowing down the spreading of the excitation, or enhancing transport by increasing its velocity. Also, it is possible that transport at finite $U$ could change its nature from ballistic to diffusive, or that even at a relatively small value of $\Delta/t_{hop}$ the excitation might get trapped. The right side of Fig. 7 shows results for $U/t_{hop}=4.3$. For $\Delta=0$, as discussed above, the ballistic motion prevails, as expected for a Luttinger liquid. At finite $\Delta/t_{hop}$, however, the behavior changes significantly when comparing to the corresponding $U=0$ cases: At $\Delta/t_{hop}=2.3$, the ping-pong-like structure disappears and is replaced by a more diffuse looking behavior. This is captured by the following scenario: Due to the rather strong interaction, the electron scatters as soon as it tunnels to the neighboring Zener polaron, since there the electron is of opposite spin, so that the Hubbard term comes into play. This scattering induces on one hand a dipole oscillation also on this Zener polaron, and on the other hand a partial tunneling of the electron of opposite spin to the neighboring lattice site. There, the mechanism repeats, and again a dipole-like oscillation also on this Zener polaron is excited, and partial tunneling of the electron with opposite spin direction to the further Zener polaron is induced, and so on. The resulting picture is a sequence of dipole oscillations formed on each Zener polaron, with an amplitude decreasing the further one moves away from the site of the excitation. This sequence of dipole oscillations seems to replace the ping-pong pattern observed at $U=0$. It is difficult to judge whether the motion of the original excitation through the system remains ballistic, or if it might change its nature. However, the strongest features are deep inside the light cone prescribed by the group velocity of the noninteracting system and seem to move with a smaller velocity, or in a diffusive manner. Also at large $\Delta/t_{hop}=8$, the effect of a finite value of $U$ is significant: While at $U=0$, on the time scales shown, there was essentially no spread of the excitation to the neighboring sites, now the dynamics is clearly composed of the dipole oscillation on the excited dimer, plus additional dipole oscillations on the close lying neighboring dimers. Again, it is difficult to conclude whether transport might be diffusive or ballistic. We leave this interesting aspect for future research. We complement this discussion by considering the time evolution of the local density $\langle\hat{n}_{R}\rangle$ on the excited dimer in more detail. In Fig. 8, we show our tDMRG results at $U=0$ and $4.3$ for the different values of $\Delta/t_{hop}$ indicated there. In contrast to the different behavior seen in Fig. 7 when comparing the results for $U=0$ to the ones for $U/t_{hop}=4.3$, in all cases shown and on the time scale displayed, the time evolution on the site of the excitation is qualitatively similar with and without interaction. On the time scale shown, three different types of behavior seem to exist: For $\Delta/t_{hop}=8$ the value of the local density shows a coherent oscillation for all times shown $t/t_{hop}\leq 20$ (corresponding to $\approx 23$ fs using the values of Table 1). The amplitude of this oscillation decays only slowly. As can be seen in Fig. 7, the reason for this are the dipole oscillations on the Zener polaron where the excitation was created, which are present for both values of $U/t_{hop}$. As the group velocity for the excitation moving away from this place is so small in this case, the dipole oscillations decay only slowly. For the local density, the effect of $U$ is to weakly dampen its oscillation. In the other extreme case displayed at $\Delta=0$, one sees that the coherent oscillation of the local density is completely suppressed, and the value of the local density drops very quickly to the equilibrium value $0.5$ and then shows only tiny oscillations around this value. This drop happens on a time scale $t/t_{hop}<10$, corresponding to $\sim 11$ fs using the parameters of Table 1. The reason for this is that the excitation moves freely through the system, as discussed above, so that at the site of the excitation the local density relaxes quickly to the equilibrium value, up to the small oscillations seen in Fig. 8. This is also true at finite $U$, where the system is in a Luttinger liquid phase.[55] For intermediate values of $\Delta/t_{hop}$, the time evolution of the local density on this time scale $\lesssim 30$fs reflects both aspects: At short times, coherent oscillations are seen, which are indicative for the dipole oscillation of the excited electron, whereas at later times the local density relaxes to its equilibrium value of $0.5$, since the excitation then is spreading through the system. Interestingly, the amplitudes of the oscillations around the equilibrium value are larger than for $\Delta=0$ and do not depend on the system size, as can be seen in Fig. 9, so that finite size effects seem to be excluded as cause for this behavior. <figure><img src="content_image/1610.07246/x16.png"><figcaption>Figure 9: Time evolution of the local density ⟨^nR+1⟩ after the operator Eq.(12) was applied to R=L/2+1 for Δ/thop=2.3 and U/thop=4.3, which is close tothe parameters of table 1. The plot compares tDMRG results for systems withL=16 (violet boxes), L=24 (green circles), L=32 (blue up-pointing triangles),L=40 (red down-pointing triangles), L=48 (yellow diamonds), and L=64 (darkblue pentagons). The results displayed are obtained with MPS matrix dimensionχMPS=5000. Additionally the time evolution for Δ=0 and U/thop=4.3 for a systemwith L=40 and χMPS=500 is plotted (black pluses).</figcaption></figure> ### Time evolution of the electronic momentum distribution function <figure><img src="content_image/1610.07246/x17.png"><figcaption>Figure 10: Momentum distribution for a system with L=40, Δ/thop=2.3, andU/thop=4.3 before (magenta) and just after (green) the photoexcitation byapplying operator Eq. (12) at the center of the system as obtained by theDMRG.</figcaption></figure> In this section, we present the time evolution of the momentum distribution at short times using the tDMRG, from which we obtain the time evolution of the electronic one-particle reduced density matrix \[\varrho_{\sigma,i,j}(t)=\langle\hat{c}^{\dagger}_{\sigma,i}\hat{c }_{\sigma,j}\rangle(t)\;.\] (14) The time evolution of the momentum distribution is obtained by Fourier-transforming the one-particle reduced density matrix by projecting onto the four bands of the noninteracting system. The momentum distribution of each band $\nu\in\mathds{B}=\{1,2,3,4\}$ is then obtained by the corresponding transformation of the creation and annihilation operators (see AppendixA), leading to (15) where the matrices $T_{\sigma\nu j}(k)$ are the unitary matrices holding the eigenvectors of the Hamiltonian of a single unit cell, as derived in detail in AppendixA. In Fig.10, we compare the momentum distribution of the ground state with the one obtained directly after the excitation. The system is excited by applying operator Eq. (12) to the central site of the system for $\Delta/t_{hop}=2.3$ and $U/t_{hop}=4.3$. Note, that the excitation affects predominantly one spin direction, which is due to the spin polarization of the polaron on which the excitation takes place. Let us first discuss the momentum distribution of the ground state. Because we are at quarter filling, as expected, the first band $\nu=1$ is highest populated, and the population of the higher bands is negligibly small but finite since $U>0$. Note that at $U/t_{hop}=4.3$ the populations are slightly inverted, so that the momentum distribution at $k=0$ is somewhat smaller than at finite $k$. This is absent at $U=0$, as further discussed in AppendixB. At finite $U$, we associate this effect with the projection onto the noninteracting band structure. It would be interesting to compare to the one-particle spectral function $A(k,\omega)$ at finite $U$, which can be measured in ARPES[59] experiments and which provides details of the band structure in the interacting case. As this exceeds the scope of this paper, we leave this aspect for future investigations. Here, we pursue a simpler path and consider the time evolution of the noninteracting bands and their populations as indicators for the strength of the scattering between the bands and for time scales emerging in the course of the time evolution. The photoexcitation moves particles from the lowest band to the higher ones. As we model it as strongly localized in real space, the excitation here transfers all possible momenta in contrast to light, for which the momentum transfer is negligible. For $\Delta/t_{hop}=2.3$, the second and third band get a higher population, whereas the one of the fourth band remains very small. For the largest value of the Hund’s splitting, $\Delta/t_{hop}=8$ treated in the previous section, the most affected band is the second one; the population of the two highest bands remains very small. Hence, the lowest band $\nu=1$ is highest populated in the ground state and remains highest populated also after the excitation in all cases treated. <figure><img src="content_image/1610.07246/x18.png"><figcaption>Figure 11: Time evolution of the population ∑knelσ,ν(k,t) of each bandfollowing the photoexcitation for Δ/thop=2.3 and U/thop=4.3 as obtained by thetDMRG. Additionally the population of the thermal state is given by thehorizontal lines.</figcaption></figure> <figure><img src="content_image/1610.07246/x19.png"><figcaption>Figure 12: Time evolution of the population ∑knelσ,ν(k,t) of each bandfollowing the photoexcitation as in Fig. 11, but for Δ/thop=8, as obtained bythe tDMRG. Additionally the population of the thermal state is given by thehorizontal lines.</figcaption></figure> Due to the finite value of $U/t_{hop}$, we expect the electrons to scatter so that the population of the four bands changes in time. In Figs.12 and 11, we show the time evolution of the populations of both spin directions for each of the four bands $\sum_{k}n_{\sigma,\nu}^{\text{el}}(k,t)$ for $\Delta/t_{hop}=2.3$ and $\Delta/t_{hop}=8$, respectively. Clearly, scattering between the bands takes place. In contrast to the time evolution of the local densities treated in the previous section, the band populations in Figs. 11 and 12 are indicative for bulk behavior and hence are better suitable for identifying time scales, on which the excitation evolves. As displayed in Fig. 11 for $\Delta/t_{hop}=2.3$, the populations of the first and second band seem to relax to a stationary value of $\sim 9.45$ and $\sim 0.35$ on a time scale of $\sim 5t_{hop}$ (corresponding to $\sim 6$fs using the parameters of Table 1). The populations of both spin directions relax to the same value and afterwards show rather small oscillations around these values. Similar behavior is also seen in the third and fourth band. For $\Delta/t_{hop}=8$, instead, relaxation happens only at a time $t/t_{hop}>30$. The first two bands seem to reach a population of $\sim 9.8$ and $\sim 0.2$, respectively. The third and fourth band have very small populations. The population of both spin directions seems to relax to the same value, even though at $t=0$ they significantly differ. As seen in Figs. 11 and 12 the spin moment inside the bands seems to relax on a short time scale $\lesssim 50$ fs. These results indicate that the relaxation time increases with the value of $\Delta/t_{hop}$. However, it is still possible that further aspects can become important for the lifetimes of the excitations. The question arises, if one can make a quantitative prediction for the lifetime of the excitation in the presence of $U$ and $\Delta$ also in cases, which are not amenable to the tDMRG. As much longer times are barely accessible to the tDMRG, we therefore now turn over to the LBE treatment, which is suitable to extract lifetimes of the excitations. ## IV Quasiparticle relaxation In this section, we use the numerically exact results for the time evolution obtained by MPS to estimate the quasiparticle content needed for a quantum Boltzmann equation (BE). The BE will then provide us with information about the long-time behavior after the excitation, which is inaccessible using the tDMRG because of the fast growth of entanglement[50] with time. ### Calculation of the quasiparticle momentum distribution from the tDMRG results The applicability of the BE for our one dimensional model is justified by the same reasoning as in Refs. [60; 61; 27], see also Refs. [62; 63]: The quasifree property of the system persists up to time scales $\propto U^{-2}$ and intervening scattering processes with a rate $\propto U^{2}$ allow one to use the fermionic BE on all time scales. One important point to realize is that within Fermi liquid theory, the BE requires the quasiparticle distribution function as input and not the distribution function for the electrons themselves [64] (notice that otherwise the zero temperature ground state of an interacting Fermi liquid would not be a fixed point of the BE). So as a first step, we need to find this quasiparticle momentum distribution from the tDMRG results. For the sake of simplicity we suppress the band and the spin index in the following. The equilibrium distribution function of the electrons $n^{\rm el}(k)$ is defined via Eq. (15). The quasiparticle distribution function $n^{\rm qp}(k)$ is the Fermi-Dirac distribution for the noninteracting band structure Eq. (10), here at $T=0$. For a Fermi liquid at zero temperature, the relation to $n^{\rm el}(k)$ in the vicinity of the Fermi surface is then: \[\lim_{k\to k_{F}}\left\{n^{\rm el}(k)-\frac{1}{2}-Z\,\left(n^{\rm qp}( k)-\frac{1}{2}\right)\right\}=0\,.\] (16) If we use this relation away from the Fermi surface, it defines a $k$-dependent quasiparticle residue $Z(k)$, which describes the spectral weight of the pole in the one-particle Green’s function for momentum $k$ \[n^{\rm el}(k)-\frac{1}{2}=Z(k)\,\left(n^{\rm qp}(k)-\frac{1}{2}\right),\] (17) which gives us the means to determine $Z(k)$ from the equilibrium distribution \[Z(k)=\frac{n^{\rm el}(k)-\frac{1}{2}}{n^{\rm qp}(k)-\frac{1}{2}}\ .\] (18) We now apply (17) to the nonequilibrium situation as well: \[n^{\rm el}(k,t)-\frac{1}{2} = Z(k)\,\left(n^{\rm qp}(k,t)-\frac{1}{2}\right)\] (19) \[\Rightarrow\quad n^{\rm qp}(k,t) = \frac{1}{2}+\frac{1}{Z(k)}\left(n^{\rm el}(k,t)-\frac{1}{2}\right)\] (20) \[= \frac{1}{2}+\frac{n^{\rm qp}(k)-\frac{1}{2}}{n^{\rm el}(k)-\frac{ 1}{2}}\left(n^{\rm el}(k,t)-\frac{1}{2}\right).\] (21) This yields the desired relation between the distribution function of the electrons $n^{\rm el}(k,t)$ measured by tDMRG and the quasiparticle distribution function of the quasiparticles $n^{\rm qp}(k,t)$ as input for the BE. The justification for the step from (17) to (19) comes from the continuous unitary transformation approach as used in Ref. [65]: $Z(k)$ describes the spectral weight of the electron with quasimomentum $k$ which propagates coherently between scattering processes described by the BE. This reasoning is a good approximation as verified by comparison with numerically exact results in Ref. [65]. It is important to note that Eq. (19) is only applicable after the short-time regime in which quasiparticles form. #### iv.1.1 DMRG results for the Momentum-distribution function of the quasiparticles In Fig.13, we show the quasiparticle distribution obtained from Eq. (19) at the beginning of the time evolution after applying operator Eq. (12) at the center of the system. Although we expect the quasiparticle picture to be better applicable at later times, it is nevertheless instructive to compare these electronic and quasiparticle distributions to each other at $t=0$. According to Eq. (19), we expect a renormalization by a $k$-dependent quasiparticle residue $Z(k)$, which is, however, constant in time. Thus, the time evolution of the quasiparticle momentum distribution will be similar to the one of the electrons, if the renormalization is not too strong. For both values of $\Delta$ shown, the renormalization of the electronic momentum distribution is very small. This is an important finding and justifies the following treatment with LBE. <figure><img src="content_image/1610.07246/x20.png"><figcaption>Figure 13: Comparison of the electronic and the quasiparticle momentumdistributions obtained with DMRG via Eq. (19) for a system with L=40,Δ/thop=2.3, and U/thop=4.3 just after the photoexcitation by applying operatorEq. (12) at the center of the system.</figcaption></figure> ### Linearized multi-band Boltzmann equation for long-time relaxation Based on the effective model from Sec.II.3, Eq. (7), we can investigate the relaxation of the electrons due to electron-electron interactions by means of a quantum BE. We use it in a similar manner as Biebl and Kehrein in Ref. [27], who investigated the thermalization rates of a Hubbard model with next-nearest-neighbor hopping. Furthermore, we perform a linearization of the BE to determine the relaxation rates. To investigate the relaxation of the quasi momentum distribution (QMD) $n^{\text{qp}}_{\sigma,\nu}(k,t)$, we use the multiband quantum Boltzmann equation (BE) \[\dot{n}^{\text{qp}}_{\sigma,\nu}\hskip-1.0pt(k,t)=\mathcal{I}_{ \text{coll}}[n^{\text{qp}}]_{\sigma,\nu}\hskip-1.0pt(k,t)\,,\] (22) with the collision term $\mathcal{I}_{\text{coll}}[n^{\text{qp}}]_{\sigma,\nu}(k,t)$. With the BE, we can estimate arbitrary time scales, which can also be longer than the spin relaxation time. In the following we assume that the $\uparrow$ particles have the same quasiparticle momentum distribution $n^{\text{qp}}_{\sigma,\nu}(k,t)$ as those with spin $\downarrow$. Hence, the spin index $\sigma$ is not written explicitly any more, i.e., $n^{\text{qp}}_{\nu}(k,t)=n^{\text{qp}}_{\uparrow,\nu}(k,t)=n^{\text{qp}}_{ \downarrow,\nu}(k,t)$. The collision term is \[\mathcal{I}^{(\text{coll})}_{\nu_{1}}(k_{1})=\left(\frac{4a}{2\pi }\right)^{2}\frac{\pi U^{2}}{\hbar t_{hop}}\int dk_{2}dk_{3}dk_{4}\sum_{\nu_{2 },\nu_{3},\nu_{4}\in\mathds{B}}\] \[\times\bigl{\|}\Phi_{{\vec{\nu}},{\vec{k}}}\bigr{\|}^{2}\sum_{G} \delta(P_{{\vec{k}}}\!+\!G)\delta\bigl{(}W_{\!{{\vec{\nu},\vec{k}}}}\bigr{)}\] \[\times\Bigl{\{}\underbrace{\bigl{[}1\!-\!n^{\text{qp}}_{\nu_{1}} \hskip-1.0pt(k_{1},t)\bigr{]}\bigl{[}1\!-\!n^{\text{qp}}_{\nu_{2}}\hskip-1.0pt (k_{2},t)\bigr{]}n^{\text{qp}}_{\nu_{3}}\hskip-1.0pt(k_{3},t)\,n^{\text{qp}}_{ \nu_{4}}\hskip-1.0pt(k_{4},t)}_{\text{gain term}}\] \[\hphantom{\times\Bigl{\{}}\underbrace{-\,n^{\text{qp}}_{\nu_{1}} \hskip-1.0pt(k_{1},t)\,n^{\text{qp}}_{\nu_{2}}\hskip-1.0pt(k_{2},t)\bigl{[}1\! -\!n^{\text{qp}}_{\nu_{3}}\hskip-1.0pt(k_{3},t)\bigr{]}\bigl{[}1\!-\!n^{\text{ qp}}_{\nu_{4}}\hskip-1.0pt(k_{4},t)\bigr{]}}_{\text{loss term}}\Bigr{\}}\,,\] with $\vec{\nu}=\left(\nu_{1},\ldots,\nu_{4}\right),\vec{k}=\left(k_{1},\ldots,k_{4}\right)$. Furthermore, we define $\Phi_{{\vec{\nu}},{\vec{k}}}$ as the matrix element of the interaction taken at four momenta in four bands, see Eq. (A). The momentum is conserved by $\delta(P_{{\vec{k}}})$ with $P_{{\vec{k}}}=k_{1}+k_{2}-k_{3}-k_{4}$ and the reciprocal lattice vector $G$. It is important to note that the summation over $G\in\mathds{Z}$ is not an artificial addition but emergent from the derivation of $\mathcal{I}_{\text{coll}}$. We express the energy conservation via $W_{\!{{\vec{\nu},\vec{k}}}}:=\frac{1}{t_{hop}}\left(\epsilon_{1}+\epsilon_{2}- \epsilon_{3}-\epsilon_{4}\right)$ obtained from the one-particle band structure Eq. (10). The collision integral of a model describing an infinite lattice naturally allows for Umklapp processes. We account for this by integrating $k_{4}$ over a region larger than only the first Brillouin zone. We want to emphasize that this is an exact reformulation of the derived collision integral. In order to investigate the long time relaxation, we linearize the BE around the thermal distribution $f_{\nu}\hskip-1.0pt(k)=1/\{1+\exp[\beta(\epsilon_{\nu}(k)-\mu)]\}$. Here $\mu$ is the chemical potential and $\beta$ is the inverse final temperature. It can be determined via DMRG by setting the total energy of the system after the photoexcitation equal to the corresponding thermal expectation value, $\langle\hat{H}\rangle(t)=\frac{\rm Tr\left(\exp\left[-\beta\hat{H}\right]\hat{ H}\right)}{\rm Tr\left(\exp\left[-\beta\hat{H}\right]\right)}$, for details see AppendixC. In order to perform the linearization, we define a perturbation $\phi_{\nu}(k,t)$ by[(66)] \[n^{\text{qp}}_{\nu}(k,t)=\frac{1}{\ 1+\exp\{\beta[\epsilon_{\nu} (k)-\mu]-\phi_{\nu}(k,t)\}\ }.\] (24) Note that at this point we could also use the numerical results for $n^{\text{qp}}_{\nu}(k,t)$. However, as discussed in Sec. IV.1.1, the quasiparticle distribution obtained from the tDMRG is very similar to the electronic one. The conceptually simplest approach is therefore to follow Ref. [27] and assume the quasiparticles to possess a distribution function as in Eq. (24). In future investigations, this can be refined by directly using the numerical results at large-enough times. Equation (24) leads to the linearized BE \[\dot{\phi}_{\nu}\hskip-1.0pt(k,t)=\hat{\mathcal{L}}[\phi]_{\nu}(k ,t).\] (25) The operator $\hat{\mathcal{L}}$ acts on the perturbation $\phi$ and returns the change of the perturbation: \[\hat{\mathcal{L}}[\phi]_{\nu_{1}} =\left(\frac{4a}{2\pi}\right)^{2}\frac{\pi U^{2}}{\hbar t_{hop}} \int\!dk_{2}dk_{3}dk_{4}\sum_{\nu_{2},\nu_{3},\nu_{4}\in\mathds{B}}\] (26) \[\times\bigl{\|}\Phi_{{\vec{\nu}},{\vec{k}}}\bigr{\|}^{2}\delta(P_{{ \vec{k}}})\,\delta(P_{{\vec{k}}}\!+\!G)\] \[\quad\times\frac{\bigl{[}1\!-\!f_{\nu_{2}}\hskip-1.0pt(k_{2},t) \bigr{]}f_{\nu_{3}}\hskip-1.0pt(k_{3},t)f_{\nu_{4}}\hskip-1.0pt(k_{4},t)}{f_{ \nu_{1}}\hskip-1.0pt(k_{1},t)}\] \[\quad\times\bigl{[}\phi_{\nu_{1}}\hskip-1.0pt(k_{1},t)\!+\!\phi_{ \nu_{2}}\hskip-1.0pt(k_{2},t)\!-\!\phi_{\nu_{3}}\hskip-1.0pt(k_{3},t)\!-\!\phi _{\nu_{4}}\hskip-1.0pt(k_{4},t)\bigr{]}.\] $\hat{\mathcal{L}}$ is hermitian in the scalar product \[\langle\phi,\psi\rangle:= 4a\int\frac{dk}{2\pi}\,\sum_{\mathclap{\nu\in\mathds{B}}}\phi_{ \nu}\hskip-1.0pt(k)\ f_{\nu}\hskip-1.0pt(k)\bigl{[}1-f_{\nu}\hskip-1.0pt(k) \bigr{]}\ \psi_{\nu}\hskip-1.0pt(k),\] (27) which induces the norm $\\|\phi\\|=\sqrt{\langle\phi,\phi\rangle}$. Therefore, we can represent the perturbation $\phi_{\nu}\hskip-1.0pt(k,t)$ by the eigenfunctions ${\text{\raisebox{1.8pt}{$\chi$}}^{(j)}_{\nu}\hskip-1.0pt(k)}$ and eigenvalues $\lambda_{j}$ of $\hat{\mathcal{L}}$: \[\phi_{\nu}\hskip-1.0pt(k,t)=\sum_{j}A_{j}(0)\ e^{-\lambda_{j}t}\ {\text{\raisebox{1.8pt}{$\chi$}}^{(j)}_{\nu}\hskip-1.0pt(k)}.\] (28) The amplitudes $A_{j}(0)=\langle\phi_{\nu}\hskip-1.0pt(k,0),\text{\raisebox{1.8pt}{$\chi$}}^{( j)}\rangle/\\|\text{\raisebox{1.8pt}{$\chi$}}^{(j)}\\|^{2}$ are the overlaps of the respective eigenfunction and the initial perturbation $\phi_{\nu}\hskip-1.0pt(k,0)$ at time $t=0$. The eigenvalues of $\hat{\mathcal{L}}$ are the relaxation rates of the corresponding contribution $A_{j}(0)$. One can proof that $\hat{\mathcal{L}}$ is positive definite, i.e., its eigenvalues are non-negative. As long as the eigenvalues are positive, the factor $e^{-\lambda_{j}t}$ leads to the decay of the corresponding contribution of the perturbation $\phi_{\nu}\hskip-1.0pt(k,t)$. For $\lambda_{j}=0$, the respective part of the perturbation $\phi_{\nu}\hskip-1.0pt(k,t)$ does not decay. There are two eigenvalues that are zero for any choice of the model parameters: ${\text{\raisebox{1.8pt}{$\chi$}}^{(1)}_{\nu}\hskip-1.0pt(k)}=\operatorname{const}$, and ${\text{\raisebox{1.8pt}{$\chi$}}^{(2)}_{\nu}\hskip-1.0pt(k)}=\epsilon_{\nu}(k)$. For both, the factor $[\phi_{1}+\phi_{2}-\phi_{3}-\phi_{4}]$ in Eq.26 vanishes. They correspond to conservation of particles and conservation of energy, respectively. ### Relaxation rates from Linearized Boltzmann Equations <figure><img src="content_image/1610.07246/x21.png"><figcaption>Figure 14: Relaxation rates as obtained from the linearized Boltzmannequation approach for Δ=1,2,4 from left to right. The eigenvalues λn of ^L aresorted by their magnitude, and we plot the lowest ones (n=1,2,3,...) as afunction of inverse temperature β. In all of the plots, the full lines arecalculations with the lowest two bands, the dashed lines indicate acalculation with the three lowest bands, and the dotted lines denote acomputation with all four bands. The eigenvalues n=1,2,3 are zero withinnumerical precision. Note that, for the 3\- and 4-band calculations, there isan additional zero eigenvalue, which we omit for comparability. As twoexamples for relaxation times, we consider u=U/thop=4.3 at room temperatureβ=thop/(kB⋅300K)≈23 and at the temperature after excitation, which we estimatein Appendix C, β=1.4. With Δ=2.3, thop≈0.585eV, and t0:=ℏ/πthop≈0.12fs thesmallest relaxation time is 1/λ4=109t0/u2≈6.5ns for room temperature and1/λ4=103t0/u2≈6.5fs for the temperature after the excitation treated in Sec.III.</figcaption></figure> The relaxation rates are found by diagonalizing the dimensionless linear operator $t_{0}(t_{hop}/U)^{2}\hat{\mathcal{L}}[\phi]$ with the time scale $t_{0}=\hbar/2\pi t_{hop}\approx 0.18\,\mathrm{fs}$. The result of our numerical evaluation is shown in Fig.14. There, we plot the results for a $2$-band, a $3$-band, and a $4$-band calculation, in which we always use the lowest bands possible. For final inverse temperature above $\beta\approx 1$, the calculations reveal the same relaxation rates. This means that the upper bands are not involved in the relaxation for low temperatures. Moreover, for low temperature, the relaxation rates decay exponentially in $\beta$. For inverse temperatures larger than $\beta=30$, the lowest eigenvalues are zero within numerical precision. Thus, for low temperatures, the corresponding contributions to the perturbation $\phi_{\nu}\hskip-1.0pt(k,t)$ become frozen. The eigenvalues $n=1,2,3$ depicted in Fig.14 are numerically zero for every $\beta$. For the $3$-band and $4$-band calculations, there is an additional zero eigenvalue $\lambda_{+}$, which we exclude from Fig.14 for a better comparison of the other eigenvalues. We can explain these zero relaxation rates analytically. They correspond to the eigenfunctions \[\bigl{(}{\text{\raisebox{1.8pt}{$\chi$}}^{(1)}_{\nu}\hskip-1.0pt( k)}\bigr{)}_{\nu=1,..,4} =\bigl{(}1,1,1,1\bigr{)}\,\forall k,\] (29) \[\bigl{(}{\text{\raisebox{1.8pt}{$\chi$}}^{(2)}_{\nu}\hskip-1.0pt( k)}\bigr{)}_{\nu=1,..,4} =\bigl{(}\epsilon_{1}(k),\epsilon_{2}(k),\epsilon_{3}(k),\epsilon _{4}(k)\bigr{)}\,\forall k,\] \[\bigl{(}{\text{\raisebox{1.8pt}{$\chi$}}^{(3)}_{\nu}\hskip-1.0pt( k)}\bigr{)}_{\nu=1,..,4} =\bigl{(}1,0,1,0\bigr{)}\,\forall k,\] \[\bigl{(}{\text{\raisebox{1.8pt}{$\chi$}}^{(+)}_{\nu}\hskip-1.0pt( k)}\bigr{)}_{\nu=1,..,4} =\bigl{(}1,1,0,0\bigr{)}\,\forall k.\] Because $\hat{\mathcal{L}}$ is linear, all combinations of these eigenfunctions are conserved. They correspond to quantum-mechanical state-space operators of the form $\hat{\Phi}[\psi]=\int dk\,\psi_{\nu}\hskip-1.0pt(k)\,n^{\text{qp}}_{\nu}(k)$: \[\hat{\Phi}[\text{\raisebox{1.8pt}{$\chi$}}^{(1)}] =\hat{N}\ \ \text{ (total number of particles)},\] (30) \[\hat{\Phi}[\text{\raisebox{1.8pt}{$\chi$}}^{(2)}] =\hat{H}+\mathcal{O}(U)\ \ \text{ (total energy)},\] \[\hat{\Phi}[\text{\raisebox{1.8pt}{$\chi$}}^{(3)}] =\hat{N}_{1}+\hat{N}_{3},\] \[\hat{\Phi}[\text{\raisebox{1.8pt}{$\chi$}}^{(+)}] =\hat{N}_{1}+\hat{N}_{2},\] with the band number operators $\hat{N}_{\nu}=\int dk\,n^{\text{qp}}_{\nu}(k)$. Hence, a contribution of type ${\text{\raisebox{1.8pt}{$\chi$}}^{(1)}_{\nu}\hskip-1.0pt(k)}$ leads to a change of the total number of particles $\langle\hat{N}\rangle$. Likewise, a contribution of type ${\text{\raisebox{1.8pt}{$\chi$}}^{(2)}_{\nu}\hskip-1.0pt(k)}$ changes the energy density. However, by construction of (24) the initial perturbation $\phi_{\nu}(k,0)$ has zero overlap with ${\text{\raisebox{1.8pt}{$\chi$}}^{(1)}_{\nu}\hskip-1.0pt(k)}$ and ${\text{\raisebox{1.8pt}{$\chi$}}^{(2)}_{\nu}\hskip-1.0pt(k)}$ and we can ignore these two eigenfunctions with vanishing rates. The eigenfunction ${\text{\raisebox{1.8pt}{$\chi$}}^{(3)}_{\nu}\hskip-1.0pt(k)}$ means that the number of particles in the first plus those in the third band cannot be changed during the relaxation process. Similarly, ${\text{\raisebox{1.8pt}{$\chi$}}^{(+)}_{\nu}\hskip-1.0pt(k)}$ is related to the conservation of particles in the first plus the second band. Obviously, this eigenfunction is the same as ${\text{\raisebox{1.8pt}{$\chi$}}^{(1)}_{\nu}\hskip-1.0pt(k)}$, if we do not include the bands $3$ and $4$. This is the reason for the additional zero eigenvalue $\lambda_{+}$ in the $3$- and $4$-band case. A contribution from either of the eigenfunctions ${\text{\raisebox{1.8pt}{$\chi$}}^{(3)}_{\nu}\hskip-1.0pt(k)}$ and ${\text{\raisebox{1.8pt}{$\chi$}}^{(+)}_{\nu}\hskip-1.0pt(k)}$ leads to at least two different chemical potentials in the long time limit. The cause of this is the relatively large value of $\Delta$. Therefore, the gaps between the bands are so large that some of the two-particle scattering processes are forbidden by energy conservation. More specifically, an interband relaxation requires the energy to be picked up by other interband or multiple intraband excitations. The limitation of scattering processes to two-electron processes, as in our study, suppresses the possibility of interband relaxation at the expense of increasing the intraband temperature. Higher-order scattering processes will eventually become important for low final temperature. The relaxation rates depend very sensitively on the final temperature and thereby the energy density of the photoexcitation. In the case treated in Sec. III, we treat systems with typically 40 lattice sites, in which one electron gets excited. As discussed in more detail in AppendixC, this corresponds to an energy density leading to a final inverse temperature $\beta\approx 1.4$. For such high temperatures, the relaxation rates obtained from the LBE lead to a relaxation time scale $\sim 5-100$ fs, depending in detail on the values of $\Delta/t_{hop}$ and $U/t_{hop}$. This prediction can now be compared to the numerical tDMRG results. As can be seen in Figs. 11 and 12, the tDMRG results indicate that band occupations of the first band seem to relax to expectation values, which agree with the thermal expectation values up to a few percent. Particle number conservation then leads to a difference of the band occupations in the other bands of similar absolute magnitude. This discrepancy can be due to the choice of boundary conditions and finite size effects, so that the results seem to be in good agreement with the corresponding thermal state. As the LBE predicts a relaxation to a thermal state on comparable time scales, we conclude that the LBE treatment has predictive power for estimating relaxation rates also in other cases. For example, when choosing an energy density such that the final temperature is of the order of room temperature ($\beta\approx 23$), for $\Delta/t_{hop}=2.3$ and $U/t_{hop}=4.3$, we obtain life times of several nanoseconds. We believe this estimate for time scales can be useful to guide future experiments on similar systems. ## V Conclusions and summary We present a combined theoretical approach to treat typical aspects of the relaxation behavior of photoexcitations in correlated materials over a wide range of time scales. Specifically, we combined tight-binding models, which describe the interplay of electrons, spins and phonons, with numerically exact tDMRG studies and kinetic calculations using the linearized quantum Boltzmann equation. In order to alleviate the difficulties related to higher dimensions, we performed our study on a hypothetical one-dimensional manganite. This limitation to a one-dimensional material simplifies the description in several points: It helps to visualize the complex polaron and spin orders, it permits the study of the initial relaxation processes using tDMRG, which works particularly well in one-dimensional systems, and finally, it simplifies the high-dimensional integrals required for the collision term in the linearized quantum Boltzmann equation. The tight-binding calculations showed that a polaronic microstructure is realized. This can be described in an effective way in terms of the aggregation of various types of polarons, such as electron, hole, Zener, and Jahn-Teller polarons. Hence, the low-energy scale of 1D manganites can be well described in terms of polarons as basic entities, their reactions and interactions. This description provides a blueprint on how to rationalize the complex orbital, polaron, and spin orders in real materials in higher dimensions. Furthermore, it is a promising route towards more coarse grained simulations of the relaxation dynamics on the very long time scale dominated by the polaronic order. The subsequent calculations have been performed with the spin and lattice degrees of freedom frozen in. Consistent with the electronic structure obtained from the tight-binding model, the electrons experience a sequence of Zener polarons, which are Mn dimers. Each dimer has two ferromagnetically coupled Mn sites, while two Zener polarons are antiferromagnetically coupled with each other. The resulting Hubbard-like model is controlled by three parameters, the hopping $t_{hop}$ between two Mn sites, the Hund’s splitting $\Delta$ between spin up and spin-down electrons, and the Coulomb interaction $U$ between the electrons. The excitation and the short-time initial relaxation has been studied using tDMRG simulations. In the absence of an interaction, the excitation of a specific Zener polaron produces internal dipole oscillations, which can be described as an electron-hole pair. Since the excitation is local in real space, it is spread over the entire reciprocal unit cell in momentum space. Electrons and holes propagate with a velocity determined by the slopes of the band structure. An intricate pattern of dipole oscillations inside the light cone of the excitation emerges at intermediate values of $\Delta/t_{hop}$, with structures propagating with the phase velocity at the $k$ value of the maximal group velocity, rather than with the group velocity itself. The group velocity decreases with increasing Hund’s splitting, as it effectively decouples Zener polarons. The electron-electron interaction $U$ induces a coupling of the dipole oscillations with different spins. It thus is responsible for a very rapid relaxation of the magnetic moment of the individual bands. While the role of $U$ is secondary in equilibrium, it has a pronounced effect on the dynamics, which differs strongly from that of noninteracting electrons. The tDMRG results for the momentum distribution clearly show that scattering due to the electron interaction leads to a redistribution of the electrons on the four bands on a femtosecond time scale. This timescale increases with $\Delta$ for fixed $U$. The estimate of the quasiparticle content from the electronic momentum distributions reveals that only a small renormalization is present, so that the time evolution of the quasiparticle momentum distribution is very similar to the one of free electrons. The relaxation rates have been determined with the linearized quantum Boltzmann equation. For all values of Hund’s splitting, the lifetimes scale as $\sim t_{hop}/U^{2}$. A large Coulomb interaction increases the rate of scattering and thus increases the relaxation rate. As shown in Fig. 14, a strong polaron microstructure expressed by the Hund’s splitting $\Delta$ leads to an enhanced lifetime of the excitations. This finding is in agreement with the tDMRG results. The thermalization rates in our PCMO model are always exponentially suppressed as a function of the inverse final temperature (this is different from Ref. [27]), which would be a clear experimental signature if initially the sample is at sufficiently low temperature. This exponential suppression is a specific consequence of the one-dimensional nature of our PCMO model. However, it should be noted that depending on the strength of the electron-phonon coupling this signature might be hidden by coupling to the phonon bath. Realistic modeling and inclusion of the phonon degrees of freedom is left to future work. ###### Acknowledgements. We acknowledge fruitful discussions with Z. Lenarčič, F. Heidrich-Meisner, E. Jeckelmann, B. Lenz, L. Cevolani, N. Abeling, M. Schmitt, and H.-G. Evertz. Financial support from the Deutsche Forschungsgemeinschaft (DFG) through SFB/CRC1073 (projects B03 and C03) and Research Unit FOR 1807 (project P7) is gratefully acknowledged. S.R.M. acknowledges hospitality of the Kavli Institute for Theoretical Physics (KITP), Santa Barbara, where part of this research was accomplished and supported in part by the NSF under Grant No. NSF PHY11-25915. We acknowledge numerous insightful discussions with Thomas Pruschke (deceased). S.R. and P.B. thank Robert Schade for his help with the tight-binding code. We thank Christian Jooss for numerous discussions on the manganites, and Michael Seibt and Tobias Meyer for discussions on ongoing experiments. ## Appendix A Boltzmann equation for electron relaxation Based on the effective model from Sec.II.3, Eq. (7), we investigate the relaxation of the electrons due to electron-electron interactions by means of a BE. We use it in a similar manner as Biebl and Kehrein,[27] who investigated the thermalization rates of a Hubbard model. Furthermore we perform a linearization of the BE to find the relaxation rates. ### Multi-band Boltzmann equation In this appendix, we describe the definitions used in setting up the BE. A first step towards calculating the BE is finding the one-particle bands $\epsilon_{\nu}(k)$ as eigenvalues of the noninteracting Hamiltonian $\hat{H}_{0}$. We consider a unit cell with four Mn-sites. The noninteracting Hamiltonian for a system with N unit cells has the form \[\hat{H}_{0}=\sum_{\sigma\in\{\uparrow,\downarrow\}}\sum_{j,j^{ \prime}=1}^{4}\sum_{\ell,\ell^{\prime}=1}^{N}h_{\sigma,j,j^{\prime},\ell,\ell^ {\prime}}\hat{c}^{\dagger}_{\sigma,j,\ell}\hat{c}_{\sigma,j^{\prime},\ell^{ \prime}}\;,\] (31) where $\sigma\in\{\uparrow,\downarrow\}$ is the spin index, $j\in\{1,2,3,4\}$ is the site index in the unit cell and $\ell$ is the index of the lattice translation $\tau_{\ell}=4a\ell$, where $a$ is the Mn-Mn distance and $\ell$ is an integer. We consider periodic boundary conditions with $N$ unit cells. The limit $N\rightarrow\infty$ is taken. The position of a Mn site is $R_{j,\ell}=aj+\tau_{\ell}$. With $\hat{c}^{\dagger}_{\sigma,j,\ell}$ and $\hat{c}_{\sigma,j,\ell}$, we denote the creation and annihilation operators of the Mn-$e_{g}$ orbital at $R_{j,\ell}$ pointing along the chain. In a Bloch representation, the creation and annihilation operators $\hat{b}^{\dagger}_{\sigma,j}(k)$ and $\hat{b}_{\sigma,j}(k)$ are defined via \[\hat{b}^{\dagger}_{\sigma,j}(k):=\sqrt{\frac{4a}{2\pi}}\sum_{\ell =1}^{N}e^{ik\tau_{\ell}}\hat{c}^{\dagger}_{\sigma,j,\ell}\,,\] (32) and the $k$-points spacing is $\Delta_{k}=\frac{2\pi}{4aN}$. The $k$ points are chosen from the interval $k\in[-\frac{\pi}{4a},\frac{\pi}{4a}]$. In this basis, the noninteracting Hamiltonian is \[\hat{H}_{0} = \int dk\;\sum_{\sigma\in\{\uparrow,\downarrow\}}\sum_{j,j^{\prime }=1}^{4}h_{\sigma,j,j^{\prime}}(k)\hat{b}^{\dagger}_{\sigma,j}(k)\hat{b}_{ \sigma,j^{\prime}}(k),\] (33) with \[h_{\sigma,j,j^{\prime}}(k)=\sum_{\ell=1}^{N}h_{\sigma,j,j^{ \prime},\ell,0}e^{ik\tau_{\ell}}\;.\] (34) The $k$-dependent Hamiltonian has the form \[h_{\sigma}(k)=\left(\begin{array}[]{cccc}\sigma\Delta/2&-t_{hop} &0&-t_{hop}\mathrm{e}^{i4ak}\\ -t_{hop}&\sigma\Delta/2&-t_{hop}&0\\ 0&-t_{hop}&-\sigma\Delta/2&-t_{hop}\\ -t_{hop}\mathrm{e}^{-i4ak}&0&-t_{hop}&-\sigma\Delta/2\\ \end{array}\right)\;,\] where we use $\sigma=1$ for the Hamiltonian of the spin-up electrons and $\sigma=-1$ for that of the spin down electrons. Diagonalization yields the eigenvalues $\epsilon_{\sigma,\nu}$ and the unitary matrix $T_{\sigma,j,\nu}$ holding the eigenvectors \[\sum_{j^{\prime}=1}^{4}h_{\sigma,j,j^{\prime}}(k)T_{\sigma,j^{ \prime},\nu}(k)=T_{\sigma,j,\nu}(k)\epsilon_{\sigma,\nu}(k)\,.\] (36) This leads to the band structure Eq. (10). The Hamilton operator is brought into the diagonal form using the creation and annihilation operators for specific bands \[\hat{a}^{\dagger}_{\sigma,\nu}(k)=\sum_{j=1}^{4}\hat{b}^{\dagger} _{\sigma,j}(k)T_{\sigma,j,\nu}(k)\;.\] (37) This yields \[\hat{H}_{0}=\sum_{\nu}\sum_{\sigma}\int dk\;\epsilon_{\nu}(k)\hat {n}_{\sigma,\nu}(k)\;,\] (38) where $\hat{n}_{\sigma,\nu}(k):=\hat{a}^{\dagger}_{\sigma,\nu}(k)\hat{a}_{\sigma,\nu} (k)$ is the number operator for a particle in band $\nu$ and with wave vector $k$. The interaction has the Hubbard form \[\hat{H}_{\text{int}} = \sum_{\ell=1}^{N}\sum_{j=1}^{4}U\hat{c}^{\dagger}_{\uparrow,j, \ell}\hat{c}^{\dagger}_{\downarrow,j,\ell}\hat{c}_{\downarrow,j,\ell}\hat{c}_{ \uparrow,j,\ell}\;,\] (39) which yields \[\hat{H}_{\text{int}} = \frac{4aU}{2\pi}\sum_{\nu_{1},\nu_{2},\nu_{3},\nu_{4}}\int dk_{1} \ldots\int dk_{4}\] (40) \[\times \Phi_{\vec{\nu},\vec{k}}\sum_{n\in\mathds{Z}}\delta(P_{\vec{k}}+G _{n})\] \[\times \hat{a}^{\dagger}_{\uparrow,\nu_{1}}(k_{1})\hat{a}^{\dagger}_{ \downarrow,\nu_{2}}(k_{2})\hat{a}_{\downarrow,\nu_{3}}(k_{3})\hat{a}_{\uparrow ,\nu_{4}}(k_{4})\;,\] with \[\Phi_{\vec{\nu},\vec{k}}:=\sum_{j=1}^{4}T^{}_{\uparrow,j,\nu_{1} }(k_{1})T^{}_{\downarrow,j,\nu_{2}}(k_{2})T_{\downarrow,j,\nu_{3}}(k_{3})T_{ \uparrow,j,\nu_{4}}(k_{4})\] and \[P_{\vec{k}}:=k_{1}+k_{2}-k_{3}-k_{4}\,.\] (42) With $G_{n}=\frac{2\pi}{4a}n$, we denote the reciprocal lattice vectors. The terms with nonzero reciprocal lattice vectors describe Umklapp processes, for which part of the momentum of the scattering particles is absorbed by the lattice. Band structure and interaction determine the collision term $\mathcal{I}_{\text{coll}}$ of the Boltzmann equation, leading to Eq. (IV.2). ## Appendix B Momentum distribution for $U=0$ <figure><img src="content_image/1610.07246/x24.png"><figcaption>Figure 15: Momentum distribution for a system with open boundary conditions,L=40, Δ/thop=2.3, and U/thop=0 before (magenta) and just after (green) thephotoexcitation by applying operator Eq. (12) at the center of the system asobtained by the DMRG.</figcaption></figure> <figure><img src="content_image/1610.07246/x25.png"><figcaption>Figure 16: Momentum distribution as in Fig. 15 but with periodic boundaryconditions.</figcaption></figure> In this appendix we show that the population inversion seen in Fig. 10 at $U/t_{hop}=4.3$ vanishes for $U=0$. In Fig.15 the momentum distribution for a system similar to Fig.10 with open boundary conditions, but with $U=0$, is shown. Without interactions, we expect at quarter filling that before the excitation the lowest band is completely filled and the other three bands completely empty. We obtain small differences to this expectation, which we associate with the choice of boundary conditions: In Fig.16 we present results for the same parameters, but with periodic boundary conditions. As can be seen, here the expectation is perfectly matched. Note that the effect of the excitation is independent of the boundary conditions used. We see that for $\Delta/t_{hop}=2.3$ at $U=0$ particles are excited from the lowest band to all higher bands. The resulting distributions show a peak at the $\Gamma$-point in the first, second, and fourth band, while in the third band a minimum is obtained. The differences to this behavior visible in Fig. 10 we associate to the effect of a finite value of $U/t_{hop}$. ## Appendix C Estimating temperature and energy density of the excitation using MPS In this appendix we discuss how we obtained the values for the inverse temperature $\beta$ and the energy density due to the excitation, Eq. (12). ### Final temperature of the excited state We use the purification approach discussed, e.g., in Refs. [26] to compute the properties of the equilibrium state at finite temperatures. In this approach the complete calculation takes place in an enlarged Hilbert space, which is created by adding an ancilla site to each physical site. To obtain the state at a given temperature we perform an imaginary time evolution starting from a suitable state at infinite temperature ($\beta=0$). Reference [67] presents a way to obtain such a state, which conserves the two $U(1)$ symmetries of the model (total spin and particle number conservation) independently within the physical and the ancilla system. The idea is to formulate a so-called entangler Hamiltonian or entangler, whose ground state is the desired state at $\beta=0$. Note that the entangler is constructed only by fixing the particle statistics, i.e., $S=1/2$ fermions in our case. Therefore the entangler can be used for any Hubbard like system to obtain an infinite temperature state. We follow Ref. [67] and use the entangler Hamiltonian \[\hat{H}_{\text{C2}}^{\text{Spin-}\frac{1}{2}\text{-fermions}}=-\sum_{i\neq j, \,\sigma=\uparrow,\downarrow}\hat{\Lambda}^{\dagger}_{\sigma,i}\hat{\Lambda}_{ \sigma,j}+h.c.\] (43) with $\hat{\Lambda}_{\sigma,i}=\hat{c}_{\sigma,i}\hat{c}_{\bar{\sigma},a(i)}\hat{P}^ {\sigma}_{i}$ and $\hat{P}^{\sigma}_{i}=\lvert 1-\hat{n}_{\bar{\sigma},i}-\hat{n}_{\sigma,a(i)}\lvert$. The index $i$ labels a site in the physical space, the index $a(i)$ the corresponding site on the ancilla space. In our MPS approach, we first formulate this long-range-interaction Hamiltonian as a finite state automaton (see Ref. [49] for details). In order to do so, we need to rewrite the projector, because it is not possible to evaluate the absolute value of a sum of operators within the finite state automaton framework. This leads to \[\hat{P}^{\sigma}_{i} = \lvert 1-\hat{n}_{\bar{\sigma},i}-\hat{n}_{\sigma,a(i)}\lvert=( \hat{P}^{\sigma}_{i})^{2}\] (44) \[= 1-2\hat{n}_{\bar{\sigma},i}-2\hat{n}_{\sigma,a(i)}+\hat{n}_{\bar {\sigma},i}\hat{n}_{\sigma,a(i)}\] \[+\hat{n}_{\sigma,a(i)}\hat{n}_{\bar{\sigma},i}+\hat{n}_{\bar{ \sigma},i}^{2}+\hat{n}_{\sigma,a(i)}^{2}\] \[= 1-\hat{n}_{\bar{\sigma},i}-\hat{n}_{\sigma,a(i)}+2\hat{n}_{\bar{ \sigma},i}\hat{n}_{\sigma,a(i)}\;.\] In Fig.17 the imaginary time evolution starting from an infinite temperature state obtained as ground state of (43) is shown for different values of $\Delta/t_{hop}$ and $U/t_{hop}$. <figure><img src="content_image/1610.07246/x26.png"><figcaption>Figure 17: Energy of systems with L=40, Δ/thop=2.3,8, and U/thop=0,8 as afunction of the inverse temperature β. The results are obtained using DMRG byan imaginary time evolution starting from the ground state of (43), which is asuitable state with β=0 in the physical space. The dashed horizontal linesindicate the energy after the excitation, which is obtained by computing theexpectation value Eexc=⟨ψ(t)\|^H\|ψ(t)⟩=const., with ^H the Hamiltonian (7) and\|ψ(t)⟩ the state after the excitation at time t. The intersection of thefinite-temperature results and Eexc indicates the value of β, which can beassociated to the energy of the excitation.</figcaption></figure> In thermal equilibrium, the value of $E(\beta)$ shown in Fig.17 and of the energy of the excited state $E_{\rm exc}=\langle\psi(t)\|\hat{H}\|\psi(t)\rangle=const.$ is the same for the Hamiltonian (7). Hence, the value of $\beta$ for which $E(\beta)=E_{\rm exc}$ corresponds to the temperature of the system after equilibration. The values for $E_{\rm exc}$ are displayed in table4, the corresponding values of $\beta$ are shown in table5. Eexc/thop \| U/thop=0 \| U/thop=4.3 ---\|---\|--- Δ/thop=2.3 \| −47.302 \| −45.354 Δ/thop=8 \| −100.90 \| −100.55 Table 4: Values of the energy of the excited states. β \| U/thop=0 \| U/thop=4.3 ---\|---\|--- Δ/thop=2.3 \| 1.56 \| 1.43 Δ/thop=8 \| 1.45 \| 1.44 Table 5: Values of the inverse temperature β at which E(β)=Eexc. ### Estimating the energy density of the excitation We estimate the energy density of the excitation by considering the difference of the energy of the excited state to the ground state, $E_{\rm exc}-E_{0}$, and dividing it by the length of the finite system used in our simulations. In this approach, we assume that the finite system considered represents a typical part of the lattice, which is excited by the incoming light, so that the energy density of the finite system would correspond to the one of an infinite lattice. Furthermore, we assume that the intensity $I$ of the incoming light amounts to the same energy density. We hence obtain \[I=\frac{E_{\text{groundstate}}-E_{\text{excited state}}}{a\,L\, \tau}\;.\] (45) We use as value for the Mn-Mn distance $a=3.818$ Å, see Ref. [68]. The duration of the light pulse is estimated to be $\tau=1$ fs. The values for the ground state energies are given in table6 and are obtained via DMRG for chains with $L=40$ sites. Egs/thop \| U/thop=0 \| U/thop=4.3 ---\|---\|--- Δ/thop=2.3 \| −48.753 \| −46.973 Δ/thop=8 \| −102.11 \| −101.76 Table 6: Energy of the ground states. This leads to an intensity $\sim 10^{8}$ W/mm. In a pump-probe setup, this would be the intensity of the pump laser in case of perfect absorption of the pump pulse. This value hence serves as a lower bound for the intensity needed to reproduce a scenario similar to the one discussed in this paper. As the intensity of lasers with ultrashort light pulses can reach $\sim 10$ TW/mm, the estimate shows that similar investigations are within reach of typical pump-probe setups. ## References (1) M. Rini, R. Tobey, N. Dean, J. Itatani, Y. Tomioka, Y. Tokura, R. W. Schoenlein, and A. Cavalleri, Nature 449, 72 (2007). * (2) I. Avigo, S. Thirupathaiah, M. Ligges, T. Wolf, J. Fink, and U. Bovensiepen, New Journal of Physics 18, 093028 (2016). * (3) F. Schmitt, P. S. Kirchmann, U. Bovensiepen, R. G. Moore, L. Rettig, M. Krenz, J.-H. Chu, N. Ru, L. Perfetti, D. H. Lu, M. Wolf, I. R. Fisher, and Z.-X. Shen, Science 321, 1649 (2008). * (4) Z. Tao, C. Chen, T. Szilvási, M. Keller, M. Mavrikakis, H. Kapteyn, and M. Murnane, Science 353, 62 (2016). * (5) M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Perucchi, S. Lupi, P. Di Pietro, D. Pontiroli, M. Riccò, S. R. Clark, D. Jaksch, and A. Cavalleri, Nature (London) 530, 461 (2016). * (6) W. Hu, S. Kaiser, D. Nicoletti, C. R. Hunt, I. Gierz, M. C. Hoffmann, M. Le Tacon, T. Loew, B. Keimer, and A. Cavalleri, Nat Mater 13, 705 (2014). * (7) Z. Lenarčič and P. Prelovšek, Phys. Rev. Lett. 111, 016401 (2013). * (8) Z. Lenarčič, M. Eckstein, and P. Prelovšek, Phys. Rev. B 92, 201104 (2015). * (9) M. Eckstein and P. Werner, Physical review letters 110, 126401 (2013). * (10) P. Werner and M. Eckstein, Structural Dynamics 3, 023603 (2016). * (11) D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Science 331, 189 (2011). * (12) K. Zhao, Y.-H. Huang, H.-B. Lü, M. He, K.-J. Jin, Z.-H. Chen, Y.-L. Zhou, B.-L. Cheng, S.-Y. Dai, and G.-Z. Yang, Chinese Physics 14, 420 (2005). * (13) K. Zhao, K.-j. Jin, H. Lu, Y. Huang, Q. Zhou, M. He, Z. Chen, Y. Zhou, and G. Yang, Applied Physics Letters 88, 141914 (2006). * (14) H. Ni, Z. Yue, K. Zhao, W. Xiang, S. Zhao, A. Wang, Y.-C. Kong, and H.-K. Wong, Opt. Express 20, A406 (2012). * (15) H. J. Snaith, The Journal of Physical Chemistry Letters 4, 3623 (2013). * (16) J. Gong, S. B. Darling, and F. You, Energy Environ. Sci. 8, 1953 (2015). * (17) G. Saucke, J. Norpoth, C. Jooss, D. Su, and Y. Zhu, Phys. Rev. B 85, 165315 (2012). * (18) B. Ifland, P. Peretzki, B. Kressdorf, P. Saring, A. Kelling, M. Seibt, and C. Jooss, Beilstein J. Nanotechnol. 6, 1467 (2015). * (19) D. Raiser, S. Mildner, B. Ifland, M. Sotoudeh, P. Blöchl, S. Techert, and C. Jooss, Adv. Energy Mater. 7, 1602174 (2017). * (20) W. Shockley and H. Queisser, J. Appl. Phys 32, 510 (1961). * (21) S. R. White, Phys. Rev. Lett. 69, 2863 (1992). * (22) S. R. White, Phys. Rev. B 48, 10345 (1993). * (23) U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005). * (24) A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, J. Stat. Mech.: Theor. Exp. 2004, P04005 . * (25) S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004). * (26) U. Schollwöck, Annals of Physics 326, 96 (2011). * (27) F. R. A. Biebl and S. Kehrein, Phys. Rev. B 95, 104304 (2017). * (28) M. Sotoudeh, S. Rajpurohit, P. Blöchl, D. Mierwaldt, J. Norpoth, V. Roddatis, S. Mildner, B. Kressdorf, B. Ifland, and C. Jooss, Phys. Rev. B 95, 235150 (2017). * (29) T. Hotta and E. Dagotto, in _Colossal Magnetoresistive Manganites_, edited by T. Chatterji (Springer Netherlands, Dordrecht, 2004), pp. 207–262. * (30) D. R. Neuber, M. Daghofer, H. G. Evertz, W. von der Linden, and R. M. Noack, Phys. Rev. B 73, 014401 (2006). * (31) E. Dagotto, _Nanoscale Phase Separation and Colossal Magnetoresistance_ (Springer, Berlin, Heidelberg, New York, 2003). * (32) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). * (33) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). * (34) A. D. Becke, J. Chem. Phys. 98, 1372 (1993). * (35) V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991). * (36) L. Hedin, Phys. Rev. 139, A796 (1965). * (37) J. Kanamori, Progress of Theoretical Physics 30, 275 (1963). * (38) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). * (39) R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985). * (40) We have done calculations for different unit cells. We choose a twelve-site model because it produces only a few frustrated structures ($N_{s}$ must be divisible by $2$, $3$, and $4$.) The size of the unit cell has been chosen small to avoid being trapped in metastable states. * (41) C. Zener, Phys. Rev. 82, 403 (1951). * (42) J.-S. Zhou and J. B. Goodenough, Phys. Rev. B 62, 3834 (2000). * (43) L. G. G. V. Dias da Silva, K. A. Al-Hassanieh, A. E. Feiguin, F. A. Reboredo, and E. Dagotto, Physical Review B 81, 125113 (2010). * (44) F. Hofmann and M. Potthoff, Phys. Rev. B 85, 205127 (2012). * (45) S. Wall, D. Brida, S. R. Clark, H. P. Ehrke, D. Jaksch, A. Ardavan, S. Bonora, H. Uemura, Y. Takahashi, T. Hasegawa, _et al._, Nature Physics 7, 114 (2011). * (46) K. A. Al-Hassanieh, F. A. Reboredo, A. E. Feiguin, I. González, and E. Dagotto, Phys. Rev. Lett. 100, 1 (2008). * (47) M. P. Zaletel, R. S. Mong, C. Karrasch, J. E. Moore, and F. Pollmann, Physical Review B 91, 165112 (2015). * (48) G. M. Crosswhite and D. Bacon, Phys. Rev. A 78, 012356 (2008). * (49) S. Paeckel, T. Köhler, and S. R. Manmana, SciPost Phys. 3, 035 (2017). * (50) L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008). * (51) S. C. Kramer, _CUDA-based Scientific Computing: Tools and Selected Applications_ (Niedersächsische Staats- und Universitätsbibliothek, Göttingen, 2013). * (52) cuBLAS is an implementation of the BLAS library especially for the usage on NVIDIA CUDA devices. * (53) G. Bedürftig, B. Brendel, H. Frahm, and R. M. Noack, Phys. Rev. B 58, 10225 (1998). * (54) S. R. White, I. Affleck, and D. J. Scalapino, Phys. Rev. B 65, 165122 (2002). * (55) T. Giamarchi, _Quantum Physics in One Dimension_, Vol. 121 of _International Series of Monographs on Physics_ (Oxford University Press, Oxford, 2004). * (56) F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, _The One-Dimensional Hubbard Model_ (Cambridge University Press, Cambridge, 2005). * (57) E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972). * (58) L. Cevolani, J. Despres, G. Carleo, L. Tagliacozzo, and L. Sanchez-Palencia, arXiv:1706.00838 (2017). * (59) A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003). * (60) M. L. R. Fürst, C. B. Mendl, and H. Spohn, Phys. Rev. E 86, 031122 (2012). * (61) M. L. R. Fürst, C. B. Mendl, and H. Spohn, Phys. Rev. E 88, 012108 (2013). * (62) L. Erdös, M. Salmhofer, and H.-T. Yau, Journal of Statistical Physics 116, 367 (2004). * (63) J. Lukkarinen and H. Spohn, Journal of Statistical Physics 134, 1133 (2009). * (64) A. Kamenev and A. Levchenko, Advances in Physics 58, 197 (2009). * (65) F. H. L. Essler, S. Kehrein, S. R. Manmana, and N. J. Robinson, Phys. Rev. B 89, 165104 (2014). * (66) H. Haug and A.-P. Jauho, _Quantum Kinetics in Transport and Optics of Semiconductors_ (Springer, Berlin, Heidelberg, 1996). * (67) A. Nocera and G. Alvarez, Phys. Rev. B 93, 045137 (2016). * (68) Z. Jirák, S. Krupička, Z. Šimša, M. Dlouhá, and S. Vratislav, Journal of Magnetism and Magnetic Materials 53, 153 (1985).
1802.09296	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 6067, "num_imgs": 0, "llama3_tokens_count": 1311 }	[]	Inferencesubsec:inference We rely on an approximate inference procedure, Markov Chain Monte Carlo in particular Andrieuetal2003. The method performs iterative inference for exploring the state space of possible question interpretations by proposing concrete changes to sets of variables that define a proposal distribution. The inference procedure performs an iterative local search and can be divided into (i) generating possible successor states for a given state by applying changes, (ii) scoring the states using the model score, and (iii) deciding which proposal to accept as successor state. A proposal is accepted with a probability that is proportional to the likelihood assigned by the distribution $\pi$. To compute the logical form of a question, we run two inference procedures using two different models. The first model L2KB is trained using a linking objective that learns to map open class words to KB identifiers. The MCMC sampling process is run for $m$ steps for the L2KB model; the top $k$ states are used as an input for the second inference model called QC that assigns meanings to closed class words to yield a full fledged semantic representation of the question. Both inference strategies generate successor states by exploration based on edges in the dependency parse tree. We explore only the following types of edges: Core arguments, Non-core dependents, Nominal dependents defined by Universal Dependencies\url{http://universaldependencies.org/u/dep/index.html}nd nodes that have the following POS tags: NOUN, VERB, ADJ, PRON, PROPN, DET. In both inference models, we alternate across iterations between using the probability of the state given the model and the objective score to decide which state to accept. Initially, all partial assignments $\alpha_{0},\beta_{0},\gamma_{0}$. are empty. [FOOTNOTE:\url{http://universaldependencies.org/u/dep/index.html}][ENDFOOTNOTE] We rely on an inverted index to find all KB IDs for a given query term. The inverted index maps terms to candidate KB IDs for all 3 languages. It has been created taking into account a number of resources: names of DBpedia resources, Wikipedia anchor texts and links, names of DBpedia classes, synonyms for DBpedia classes from WordNet miller1995wordnet,kilgarriff2000wordnet, as well as lexicalizations of properties and restriction classes from DBlexipedia walter2015dblexipedia. Entries in the index are grouped by DUDES type, so that it supports type-specific retrieval. The index stores the frequency of the mentions paired with KB ID. During retrieval, the index returns a normalized frequency score for each candidate KB ID. L2KB: Linking to Knowledge Basesubsubsec:l2kb Proposal Generation: The L2KB proposal generation proposes changes to a given state by considering single dependency edges and changing: i) the KB IDs of parent and child nodes, ii) the DUDES type of parent and child nodes, and iii) the argument index attached to the edge. The Semantic Type variables range over the 5 basic DUDES types defined, while the argument index variable ranges in the set {1,2}. The resulting partial semantic representations for the dependency edge are checked for satisfiability with respect to the knowledge base, pruning the proposal if it is not satisfiable. Figure edgeExplorer depicts the local exploration of the dobj-edge between Wikipedia and created. The left image shows an initial state with empty assignments for all hidden variables. The right image shows a proposal that is changed the KB IDs and DUDE types of the nodes connects by the dobj edge. The inference process has assigned the KB ID dbo:author and the Property DUDES type to the created node. The Wikipedia nodes gets assigned the type Resource DUDES as well as the KB ID dbr:Wikipedia. The dependency edge gets assigned the argument index 1, representing that dbr:Wikipedia should be inserted at the subject position of the dbo:author property. The partial semantic representation represented by this edge is the one depicted at the end of Section 2.2. As it is satisfiable, it is not pruned. In contrast, a state in which the edge is assigned the argument index 2 would yield the following non-satisfiable representation, corresponding to things that were authored by Wikipedia instead of things that authored Wikipedia: _v:- vs:{} l:1_$\text{\tt dbo:author}(y,dbr:Wikipedia)$ 1$(y,a_{2},2)$ [FIGURE:fig1][ENDFIGURE] Objective Function:As objective for the L2KB model we rely on a linking objective that calculates the overlap between inferred entity links and entity links in the gold standard SPARQL query. All generated states are ranked by the objective score. Top-k states are passed to the next sampling step. In the next iteration, the inference is performed on these $k$ states. Following this procedure for $m$ iterations yields a sequence of states $(s_{0},\dots,s_{m})$ that are sampled from the distribution defined by the underlying factor graphs. ## QC: Query Construction Proposal Generation: Proposals in this inference layer consist of assignments of the type _QueryVar DUDES_ to nodes for class words, in particular determiners, that could fill the argument position of a parent with unsatisfied arguments. Objective Function:As objective we use an objective function that measures the (graph) similarity between the inferred SPARQL query and the gold standard SPARQL query. Figure Objective Function: shows an input state and a sampled state for the QC inference layer of our example query: _Who created Wikipedia?_. The initial state (see Left) has Slot 1 assigned to the edge _dobj_. Property DUDES have 2 slots by definition. The right figure shows a proposed state in which the argument slot 2 has been assigned to the nsubj edge and the _QueryVar DUDES_ type has been assigned to node _Who_. This corresponds to the representation and SPARQL queries below: _v:- vs:{y} l:1_$\text{\tt dbo:author}(dbr:Wikipedia,y)$ 1 SELECT DISTINCT ?y WHERE { dbr:Wikipedia dbo:author ?y .} [FIGURE:theparagraph-at-IDa.fig1][ENDFIGURE]
1907.01230	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 19852, "num_imgs": 4, "llama3_tokens_count": 5859 }	[ "content_image/1907.01230/x1.png", "content_image/1907.01230/x2.png", "content_image/1907.01230/x3.png", "content_image/1907.01230/x4.png" ]	Evolution and pulsations of population I post–AGB stars © 2019 г. Yu. A. Fadeyev* [FOOTNOTE:][ENDFOOTNOTE] _Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya ul. 48, Moscow, 119017 Russia_ Received June 4, 2019; revised June 11, 2019; accepted June 25, 2019 Abstract* — Evolutionary calculations of population I stars with initial masses $M_{0}=1M_{\odot}$, $1.5M_{\odot}$ and $2M_{\odot}$ were carried out up to the stage of the proto–planetary nebula. Selected models of post–AGB evolutionary sequences with effective temperatures $3.6\times 10^{3}\,\mathrm{K}\lesssim T_{\mathrm{eff}}\lesssim 2\times 10^{4}\, \mathrm{K}$ were used as initial conditions in calculations of self–escited stellar oscillations. For the first time the sequences of hydrodynamic models of radially pulsating post–AGB stars were computed using the self–consistent solution of the equations of radiation hydrodynamics and time–dependent convection. Within this range of effective temperatures the post–AGB stars are the fundamental mode pulsators with period decreasing as the star evolves from $\Pi\approx 300$ day to several days. Period fluctuations are due to nonlinear effects and are most prominent at effective temperatures $T_{\mathrm{eff}}<5000$ K. The amplitude of bolometric light variations is $\Delta M_{\mathrm{bol}}\approx 1$ at $T_{\mathrm{eff}}\lesssim 6000$ K and rapidly decreases with increasing $T_{\mathrm{eff}}$. The theoretical dependence of the pulsation period as a function of effective temperature obtained in the study can be used as a criterion for the evolutionary status of pulsating variables suspected to be post–AGB stars. Keywords: _stars: variable and peculiar_ ## introduction The red giant evolutionary stage of stars with solar composition and the zero–age main sequence mass $M_{0}\lesssim 9M_{\odot}$ is completed due to the strong stellar wind and the loss of the major fraction of the hydrogen envelope. The star leaves the asymptotic giant branch (AGB) and crosses the Hertzsprung–Russel diagram (HRD) at nearly constant luminosity towards the region of planetary nebula cores with effective temperatures $T_{\mathrm{eff}}\sim 10^{5}$ K. An idea that the planetary nebulae originate from red giants was suggested by Shklovsky (1956) and was later confirmed by evolutionary computations (Paczyński, 1971; Schönberner, 1979; 1981; Wood, Faulkner, 1986; Vassiliadis, Wood, 1994; Blöcker, 1995; Weiss, Ferguson, 2009; Miller Bertolami, 2016). Formation of the opaque gas–dust envelope surrounding the star on the tip of the AGB substantially restricts abilities of optical observations, so that the evolutionary transition to the post–AGB stage still remains unclear. The strong stellar wind on the tip of the AGB seems to be due to nonlinear stellar pulsations that lead to dynamical instability of the outer stellar layers (Tuchman et al., 1978). Therefore, large infrared (IR) excesses detected in post–AGB stars (Volk, Kwok, 1989; Hrivnak et al., 1989; 1994; Ikonnikova et al., 2018) indicate existence of circumstellar dust grains condensed during the preceding evolutionary stage with high mass loss rates. The photometric variability due to stellar pulsations is a typical property of post–AGB stars. The period of light variations ranges from several dozen days (Arkhipova et al., 2010; Hrivnak et al.2015; 2018) to $\Pi\approx 200$ day (Arkhipova et al., 2009; 2016; Ikonnikova et al., 2018). Moreover, the pulsating variable AI CMi with period $\Pi\approx 310$ day seems to be the early post–AGB star (Arkhipova et al., 2017). A common feature of post–AGB stars is a lack of strict repetition in light variations, so that observational estimates of the period are often highly uncertain. The post–AGB stars are on the late stage of stellar evolution so that comparison of observations with results of evolutionary computations is often accompanied with difficulties arising from uncertainties of the theory (e.g., the mass loss rate on the AGB stage). This problem can be clarified with application of the stellar pulsation theory. Results of the linear analysis of pulsational instability were presented by Zalewski (1985; 1993) and Gautschy (1993) who showed that the region of pulsational instability of post–AGB stars on the HRD extends to effective temperatures as high as $T_{\mathrm{eff}}\approx 10^{4}$ K. However one should bear in mind that at luminosity to mass ratios typical for post–AGB stars ($L/M\sim 10^{4}L_{\odot}/M_{\odot}$) the low gas density and the small adiabatic exponent in the pulsating envelope imply high nonadiabaticity and strong nonlinearity of stellar oscillations. Results of hydrodynamic computations show that radial oscillations of low–mass supergiants of intermediate spectral types have the large amplitude and absence of strict periodicity is due to nonlinear effects (Fadeyev, Tutukov, 1981; Fadeyev, 1982; 1984; Aikawa, 1985a; 1985b; 1991; 1993; Fokin et al., 2001). Unfortunately, in all these works the authors ignored convection and assumed that energy transfer in the pulsating envelope is due to radiation. However the role of convection in stellar pulsation motions becomes important at effective temperatures $T_{\mathrm{eff}}<5000$ K, i.e. at pulsation periods $\Pi\gtrsim 50$ day. The goal of the present work is to investigate nonlinear oscillations of post–AGB stars with effective temperatures $3.6\times 10^{3}\,\mathrm{K}\leq T_{\mathrm{eff}}\leq 2\times 10^{4}\,\mathrm{K}$. The equations of radiation hydrodynamics and time–dependent convection are solved with initial conditions obtained from selected models of previously calculated evolutionary sequences. Computations of stellar evolution were done with the MESA code version 10398 (Paxton et al., 2018) from the main sequence to the stage of the proto–planetary nebula with effective temperature $T_{\mathrm{eff}}\approx 2\times 10^{4}$ K. Details of evolutionary computations (the nuclear network, convective mixing, mass loss rates) are described in our previous study (Fadeyev, 2018). The initial composition of stellar material was assumed to correspond to population I stars with fractional mass abundances of hydrogen and helium $X=0.70$ and $Y=0.28$, respectively. Abundances of elements heavier than helium were scaled according to Grevesse and Sauval (1998). Equations of hydrodynamics and parameters of the time–dependent convection theory (Kuhfuß, 1986) are discussed in our earlier papers (Fadeyev, 2013; 2015). ## evolutionary sequences and initial conditions In the present study we computed three evolutionary sequences of stars with initial masses on the main sequence $M_{0}=1M_{\odot}$, $1.5M_{\odot}$ and $2M_{\odot}$. Evolutionary tracks used in the present study for determination of initial conditions are shown in Fig. 1. Increase of luminosity in the right part of the figure represents the final phase of the AGB stage after the last thermal flash of the helium burning shell source. During this phase the mass of the hydrogen envelope $M_{\mathrm{env}}$ rapidly decreases, whereas the stellar luminosity is generated in the hydrogen burning shell. According to Miller Bertolami (2016) we assume that the the post–AGB stage begins when the hydrogen envelope mass to star mass ratio is $M_{\mathrm{env}}/M=0.01$. The onset of the post–AGB stage is indicated for each track in Fig. 1 by the vertical dash where for the sake on convenience the evolutionary time is set to zero ($t_{\mathrm{ev}}=0$). Further stellar evolution proceeds towards higher effective temperatures at insignificantly changing luminosity. Main properties of stars at the beginning of the post–AGB stage are listed in Table 1, where $M_{0}$ is the initial stellar mass, $t_{}$ is the star age measured from the zero age main sequence, $M_{}$ is the star mass, $M_{\mathrm{CO},}$ is the mass of the degenerate carbon–oxygen core, $L_{}$ is the stellar luminosity, $T_{\mathrm{eff},}$ is the effective temperature. The last column in Table 1 gives the evolutionary time $\Delta t_{\mathrm{ev}}$ when the effective temperature becomes as high as $T_{\mathrm{eff}}=2\times 10^{4}$ K. For the stellar initial mass increasing from $M_{0}=1M_{\odot}$ to $M_{0}=2M_{\odot}$ the evolutionary time $\Delta t_{\mathrm{ev}}$ reduces by a factor of $\approx 20$. Plots of the effective temperature $T_{\mathrm{eff}}$ as a function of evolutionary time $t_{\mathrm{ev}}$ are shown in Fig. 2. ## hydrodynamic models of post–AGB stars The stellar radius as well as the mass of hydrogen and helium ionization zones decrease as the post–AGB star evolves on the HRD towards the area of planetary cores. Therefore, evolutionary decrease of the pulsation period is accompanied by decreasing pulsation instability growth rate and diminishing role of nonlinear effects in stellar oscillations. At effective temperatures $T_{\mathrm{eff}}<4000$ K radial pulsations are driven in the hydrogen ionization zone which encompasses the substantial part of the stellar envelope. In stars with higher effective temperatures radial oscillations are driven in the helium ionization zones. Unfortunately, absence of strict repetition of pulsation motions does not allow us to calculate the mechanical work $\oint PdV$ done by each mass zone of the hydrodynamic model (here $P$ is the total pressure and $V$ is the specific volume) in order to evaluate its contribution into excitation or suppression of pulsational instability. As in our earlier studies devoted to nonlinear pulsations of AGB stars (Fadeyev, 2017; 2018) the pulsation period $\Pi$ was determined using the discrete Fourier transform of the kinetic energy of the pulsating stellar envelope. Thus, the period estimate of each hydrodynamic model was obtained by averaging over the time interval of the solution of the equations of hydrodynamics. For most hydrodynamic models the solution of the Cauchy problem comprised nearly a hundred pulsation cycles. A reduction of the role of nonlinear effects during evolution of the post–AGB star is illustrated in Fig. 3 where the plots of normalized power spectrum of the pulsation kinetic energy are shown for two hydrodynamic models of the evolutionary sequence $M_{0}=1.5M_{\odot}$. At the stellar effective temperature $T_{\mathrm{eff}}=4800$ K the growth rate of kinetic energy is $\eta=\Pi d\ln E_{\mathrm{K}}/dt=1.3$ and the mean fundamental mode period is $\Pi_{0}=85$ day. As can be seen in Fig. 3, cycle–to–cycle variations of the period due to nonlinear effects range within $\approx 20\%$ of the mean period. In the star with effective temperature $T_{\mathrm{eff}}=6500$ K the growth rate decreases up to $\eta=0.04$ whereas cycle–to–cycle variations of the period around its mean value $\Pi_{0}=31$ reduce to a few per cent. The plots of the mean radial pulsation period as a function of effective temperature for evolutionary sequences $M_{0}=1M_{\odot}$, $1.5M_{\odot}$ and $2M_{\odot}$ are shown in Fig. 4. At the beginning of the post–AGB stage the pulsation period is $\Pi\approx 300$ in all evolutionary sequences. While the effective temperature increases up to $T_{\mathrm{eff}}=2\times 10^{4}$ K the period decreases by more than two orders of magnitude. For more convenient comparison of theoretical dependences with observations the star age $t_{\mathrm{ev}}$ and the pulsation period $\Pi$ are listed for each evolutionary sequence in Table 2 as a function of the effective temperature. The presented tabular data were obtained by nonlinear interpolation of the results of evolutionary and hydrodynamic computations. In the last column of Table 2 we give typical amplitude of the bolometric light $\Delta M_{\mathrm{bol}}$. In stars with effective temperature $T_{\mathrm{eff}}<6000$ K modulation of radiative flux takes place in the hydrogen ionization zone, so that the amplitude of bolometric light is $\Delta M_{\mathrm{bol}}\approx 1$ mag. At effective temperatures $T_{\mathrm{eff}}\approx 8\times 10^{3}$ K the principal role in modulation of the radiative flux belongs to helium ionization zones and the light amplitude reduces to $\Delta M_{\mathrm{bol}}\approx 0.1$. Further evolutionary increase of effective temperature is accompanied by decrease of the mass of the pulsating envelope so that the bolometric light amplitude becomes less than 0.01 mag for $T_{\mathrm{eff}}>10^{4}$ K. ## conclusions In this paper we presented results of stellar evolution and nonlinear stellar pulsation calculations that describe the evolutionary change of the pulsation period of post–AGB stars. Self–consistent solution of the equations of radiation hydrodynamics and time–dependent convection allowed us to compute sequences of hydrodynamic models for effective temperatures ranging from 3600 K to $2\times 10^{4}$ K. Nearly twentyfold change of the evolution rate within the initial mass interval $1M_{\odot}\leq M_{0}\leq 2M_{\odot}$ allows us to conclude that most of observed post–AGB stars originate from stars with intial mass $M_{0}\approx 1M_{\odot}$. For more detailed comparison of the theory with observations the existing grids of evolutionary and hydrodynamic models of post–AGB stars should be extended due to different values of initial metal abundances $Z_{0}$ and mass loss rates. Dependences given in Fig. 4 and in Table 2 can be used for determination of the evolutionary status of the observed pulsating variable which is suspected to be the post–AGB star. For example, let us consider the pulsating variable AI CMi. According to the General Catalogue of Variable Stars (Samus’ et al., 2017) AI CMi is the semiregular pulsating variable of the spectral type G5Iab with period $\Pi\approx 230$ day. Assuming that the mean effective temperatures corresponding to the spectral type of AI CMi is $4500\,\mathrm{K}\lesssim T_{\mathrm{eff}}\lesssim 5000\,\mathrm{K}$ we find from Table 2 that the upper limit of the pulsation period of the post–AGB star is nearly two times less, that is $\Pi\approx 100$ day. It should be noted that the main obstacle in determination of the evolutionary status of the suspected post–AGB star is uncertainty in observational estimate of the period. Therefore, contradiction between observational estimate of the period and evolutionary status of AI CMi as the post–AGB star might be avoided due to smaller observational estimates of $\Pi$. ## references 1. T. Aikawa, Astrophys. Space Sci. 112, 125 (1985a). 2. T. Aikawa, Astrophys. Space Sci. 116, 401 (1985b). 3. Т. Aikawa, Astrophys. J. 374, 700 (1991). 4. T. Aikawa, MNRAS 262, 893 (1993). 5. V.P. Arkhipova, V.F. Esipov, N.P. Ikonnikova, G.V. Komissarova, A.M. Tatarnikov, and B.F. Yudin, Astron. Lett. 35, 764 (2009). 6. V.P. Arkhipova, N.P. Ikonnikova, and G.V. Komissarova, Astron. Lett. 36, 269 (2010). 7. V.P. Arkhipova, O.G. Taranova, N.P. Ikonnikova, V.F. Esipov, G.V. Komissarova, V.I. Shenavrin, and M.A. Burlak, Astron. Lett. 42, 756 (2016). 8. V.P. Arkhipova, N.P. Ikonnikova, V.F. Esipov, and G.V. Komissarova, Astron. Lett. 43, 416 (2017). 9. T. Blöcker, Astron. Astrophys. 299, 755 (1995). 10. Yu.A. Fadeyev, Astrophys Space Sci. 86, 143 (1982). 11. Yu.A. Fadeyev, Astrophys Space Sci. 100, 329 (1984). 12. Yu.A. Fadeyev and A.V. Tutukov, MNRAS 195, 811 (1981). 13. Yu.A. Fadeyev, Astron. Lett. 39, 306 (2013). 14. Yu.A. Fadeyev, MNRAS 449, 1011 (2015). 15. Yu.A. Fadeyev, Astron. Lett. 43, 602 (2017). 16. Yu.A. Fadeyev, Astron. Lett. 44, 546 (2018). 17. A.B. Fokin, A. Lébre, H. Le Coroller, and D. Gillet, 2001, Astron. Astrophys. 378, 546 (2001). 18. A. Gautschy, MNRAS 265, 340 (1993). 19. N. Grevesse and A.J. Sauval, Space Sci. Rev. 85, 161 (1998). 20. B.J. Hrivnak, S. Kwok, and K.M. Volk, Astrophys. J. 346, 265 (1989). 21. B.J. Hrivnak, S. Kwok, and T.R. Geballe, Astrophys. J. 420, 783 (1994). 22. B.J. Hrivnak, W. Lu, and K.A. Nault, Astron. J. 149, 184 (2015). 23. B.J. Hrivnak, G. Van de Steene, H. Van Winckel, W. Lu, and J. Sperauskas, Astron. J. 156, 300 (2018). 24. N.P. Ikonnikova, O.G. Taranova, V.P. Arkhipova, G.V. Komissarova, V.I. Shenavrin, V.F. Esipov, M.A. Burlak, and V.G. Metlov, Astron. Lett. 44, 457 (2018). 25. R. Kuhfuß, Astron. Astrophys. 160, 116 (1986). 26. M.M. Miller Bertolami, Astron. Astrophys. 588, A25 (2016). 27. B. Paczyński, Acta Astron. 21, 417 (1971). 28. B. Paxton, J. Schwab, E.B. Bauer, L. Bildsten, S. Blinnikov, P. Duffell, R. Farmer, J.A. Goldberg, et al., Astropys. J. Suppl. Ser. 234, 34 (2018). 29. N.N. Samus’, E.V. Kazarovets, O.V. Durlevich, N.N. Kireeva, and E.N. Pastukhova, Astron. Rep. 61, 80 (2017)]. 30. D. Schönberner, Astron. Astrophys. 79, 108 (1979). 31. D. Schönberner, Astron. Astrophys. 103, 119 (1981). 32. I.S. Shklovsky, Astronomicheskii Zhurnal 33, 315 (1956). 33. Y. Tuchman, N. Sack, and Z. Barkat, Astrophys. J. 219, 183 (1978). 34. E. Vassiliadis and P.R. Wood, Astrophys. J. Suppl. Ser. 92, 125 (1994). 35. K.M. Volk and S. Kwok, Astrophys. J. 342, 345 (1989). 36. A. Weiss and J.W. Ferguson, Astron. Astrophys. 508, 1343 (2009). 37. P.R. Wood and D.J. Faulkner, Astrophys. J. 307, 659 (1986). 38. J. Zalewski, Acta Astron. 35, 51 (1985). 39. J. Zalewski, Acta Astron. 43*, 431 (1993). M0/M⊙ \| t∗,109 yr \| M∗/M⊙ \| MCO,∗/M⊙ \| L∗/L⊙ \| Teff,∗, K \| Δtev, yr ---\|---\|---\|---\|---\|---\|--- 1.0 \| 12.607 \| 0.542 \| 0.493 \| 3322 \| 3707 \| 7485 1.5 \| 3.040 \| 0.590 \| 0.550 \| 5971 \| 3678 \| 1077 2.0 \| 1.352 \| 0.615 \| 0.581 \| 8169 \| 3620 \| 342 Table 1: Properties of evolutionary models at the onset of the post–AGB stage Teff, K \| 1M⊙ \| 1.5M⊙ \| 2M⊙ \| ΔMbol ---\|---\|---\|---\|--- \| tev, yr \| Π, day \| tev, yr \| Π, day \| tev, yr \| Π, day \| 3600 \| -1642 \| 320.9 \| -146 \| 346.4 \| -11 \| 298.1 \| 1.0 3800 \| 953 \| 180.8 \| 171 \| 207.1 \| 77 \| 207.8 \| 4000 \| 2191 \| 123.1 \| 357 \| 149.5 \| 128 \| 185.6 \| 4200 \| 2866 \| 100.1 \| 466 \| 121.5 \| 161 \| 145.7 \| 4500 \| 3456 \| 82.7 \| 567 \| 100.7 \| 192 \| 104.7 \| 5000 \| 3957 \| 57.5 \| 656 \| 73.4 \| 220 \| 81.7 \| 6000 \| 4307 \| 29.8 \| 707 \| 36.7 \| 237 \| 39.9 \| 0.5 8000 \| 4461 \| 15.2 \| 722 \| 20.9 \| 240 \| 26.6 \| 0.1 10000 \| 4630 \| 9.4 \| 729 \| 15.7 \| 241 \| 19.4 \| 0.01 12000 \| 5033 \| 5.8 \| 759 \| 10.7 \| 252 \| 13.3 \| 15000 \| 5927 \| 2.7 \| 868 \| 5.7 \| 282 \| 7.3 \| Table 2: The evolutionary time tev and the period of radial oscillations Π of post–AGB stars <figure><img src="content_image/1907.01230/x1.png"><figcaption>Figure 1: Evolutionary tracks of stars with initial masses M0=1M⊙, 1.5M⊙ and2M⊙ during transition from AGB to post–AGB. M∗ is the stellar mass at theonset of the post–AGB stage indicated on the plots by vertical dashes.</figcaption></figure> <figure><img src="content_image/1907.01230/x2.png"><figcaption>Figure 2: Effective temperature Teff of post–AGB stars as a function ofevolutionary time tev for evolutionary sequences M0=1M⊙, 1.5M⊙ and 2M⊙.</figcaption></figure> <figure><img src="content_image/1907.01230/x3.png"><figcaption>Figure 3: Normalized power spectra of the pulsation kinetic energy EK forhydrodynamic models with initial mass M0=1.5M⊙ at effective temperaturesTeff=4800 K (solid line) and Teff=6500 K (dotted line). The mean period of thefundamental mode Π0 is given near the plot.</figcaption></figure> <figure><img src="content_image/1907.01230/x4.png"><figcaption>Figure 4: The period of radial pulsations Π as a function of effectivetemperature Teff for evolutionary sequnces with initial masses M0=1M⊙ (solidline), 1.5M⊙ (dashed line) and 2M⊙ (dorred line). Filled circles representmean pulsation periods of hydrodynamic models.</figcaption></figure>
1205.0884	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 22538, "num_imgs": 2, "llama3_tokens_count": 6529 }	[ "content_image/1205.0884/vkin.png", "content_image/1205.0884/velschw.png" ]	# Kinematic relative velocity with respect to stationary observers in Schwarzschild spacetime Vicente J. Bolós Dpto. Matemáticas para la Economía y la Empresa, Facultad de Economía, Universidad de Valencia. Avda. Tarongers s/n. 46022, Valencia, Spain. e-mail: vicente.bolos@uv.es ###### Abstract We study the kinematic relative velocity of general test particles with respect to stationary observers (using spherical coordinates) in Schwarzschild spacetime, obtaining that its modulus does not depend on the observer, unlike Fermi, spectroscopic and astrometric relative velocities. We study some fundamental particular cases, generalizing some results given in other work about stationary and radial free-falling test particles. Moreover, we give a new result about test particles with circular geodesic orbits: the modulus of their kinematic relative velocity with respect to any stationary observer depends only on the radius of the circular orbit, and so, it remains constant. ## 1 Introduction The concept of “relative velocity” of a test particle with respect to an observer in general relativity is only well defined when the observer and the test particle are in the same event. Nevertheless, the notion of relative velocity of a distant test particle is fundamental in physics and so, it was revised by the IAU using reference systems adapted to the solar system (see [1, 2]). Thereby, some authors have introduced new geometric concepts motivated by the coordinate-dependence of some definitions; for example, scaled Fermi-Walker derivatives let us define geometrically local notions of velocities of a test particle with respect to a congruence of observers (see [3]). Moreover, four different intrinsic geometric definitions of relative velocity of a distant test particle with respect to a single observer were introduced in [4]. These definitions are strongly associated with the concept of simultaneity: _kinematic_ and _Fermi_ in the framework of “spacelike simultaneity”, _spectroscopic_ and _astrometric_ in the framework of “lightlike simultaneity”. These four concepts each have full physical sense, and have proved to be useful in the study of properties of particular spacetimes (see [4, 5, 6, 7]). Following this line, we are going to study the kinematic relative velocity of test particles with respect to stationary observers in Schwarzschild spacetime. This velocity shows a kind of “Newtonian behavior” in this spacetime unlike the other three velocities, and some interesting properties about stationary observers hold, as we are going to develop in the present work. This paper is organized as follows. In Section 2, we establish notation and define the concept of kinematic relative velocity. In Section 3 we introduce the Schwarzschild metric in spherical coordinates and their corresponding stationary observers, giving some lemmas that are applied in Section 4 to obtain the main result: the expression of the modulus of the kinematic relative velocity of a general test particle with respect to any stationary observer. We also study in this section some fundamental examples that were previously introduced in [4], generalizing the results obtained in that paper, and we present a new example about circular geodesic orbits. Finally, Section 5 gives concluding remarks that lead to a property that extends to general relativity the main result. ## 2 Definitions and notation We work in a Lorentzian spacetime manifold $\left(\mathcal{M},g\right)$, with $c=1$ and using the ‘mostly plus’ signature convention $(-,+,+,+)$. We suppose that $\mathcal{M}$ is a convex normal neighborhood; thus, given two events $p$ and $q$ in $\mathcal{M}$, there exists a unique geodesic joining them (results on the existence of convex normal neighborhoods in semi-Riemannian manifolds are given in [8], pp. 129–131; see Remark 2.1 for a discussion about working in a non convex normal neighborhood). The parallel transport from $q$ to $p$ along this geodesic is denoted by $\tau_{qp}$. If $\beta:I\rightarrow\mathcal{M}$ is a curve with $I\subseteq\mathbb{R}$ a real interval, we identify $\beta$ with the image $\beta I$ (that is a subset in $\mathcal{M}$), in order to simplify the notation. Vector fields are denoted by uppercase letters and vectors (defined at a single point) are denoted by lowercase letters. Moreover, if $x$ is a spacelike vector, then $\\|x\\|:=g\left(x,x\right)^{1/2}$ is the modulus of $x$. If $X$ is a vector field, $X_{p}$ denotes the unique vector of $X$ in $T_{p}\mathcal{M}$. In general, we say that a timelike world line $\beta$ is an _observer_ (or a _test particle_). Nevertheless, we say that a future-pointing timelike unit vector $u$ in $T_{p}\mathcal{M}$ is an _observer at $p$_, identifying the observer with its 4-velocity. The _Landau submanifold_$L_{p,u}$ (also called _Fermi surface_) is given by all the geodesics starting from $p$ and orthogonal to $u$ (see [9, 10, 6]). ### Kinematic relative velocity Throughout the paper, we consider an observer $\beta$ and a test particle $\beta^{\prime}$ (parameterized by their proper times) with 4-velocities $U$ and $U^{\prime}$ respectively. Let $u:=U_{p}$ be the 4-velocity of $\beta$ at an event $p$ and let $q$ be the event of $\beta^{\prime}$ such that there exists a spacelike geodesic $\psi$ orthogonal to $u$ joining $p$ and $q$ (see Figure 1). Note that since we work in a convex normal neighborhood, this event is unique and it is given by $q:=L_{p,u}\cap\beta^{\prime}$. We denote $u^{\prime}:=U^{\prime}_{q}$ in order to simplify the notation. <figure><img src="content_image/1205.0884/vkin.png"><figcaption>Figure 1: Scheme of the elements involved in the definition of the kinematicrelative velocity of β′ (test particle) with respect to β (observer). Thecurve ψ is a spacelike geodesic orthogonal to the 4-velocity of β at p,denoted by u. The vector u′ is the 4-velocity of β′ at q.</figcaption></figure> The _kinematic relative velocity of $u^{\prime}$ with respect to $u$_ is the vector \[v_{\mathrm{kin}}:=\frac{1}{-g\left(\tau_{qp}u^{\prime},u\right)}\tau_{qp}u^{ \prime}-u.\] (1) In the case $p=q$, this definition coincides with the usual concept of relative velocity \[v=\frac{1}{-g\left(u^{\prime},u\right)}u^{\prime}-u,\] (2) which is only defined when $u$ and $u^{\prime}$ are in the same tangent space. Note that $v_{\mathrm{kin}}$ is spacelike and orthogonal to $u$, and the square of its modulus is given by \[\\|v_{\mathrm{kin}}\\|^{2}=g\left(v_{\mathrm{kin}},v_{\mathrm{kin}}\right)=1- \frac{1}{g\left(\tau_{qp}u^{\prime},u\right)^{2}}=1-\frac{1}{g\left(u^{\prime} ,\tau_{pq}u\right)^{2}},\] (3) since parallel transport conserves the metric. Varying $p$ along $\beta$, we construct the vector field $V_{\mathrm{kin}}$ defined on $\beta$, representing the _kinematic relative velocity of $\beta^{\prime}$ with respect to $\beta$_ (see [4, 11]). Throughout the paper we are going to denote $v_{\mathrm{kin}}:=V_{\mathrm{kin}\,p}$ as we have already done in this section. Remark 2.1: The concept of kinematic relative velocity can be extended to non convex normal neighborhoods. If $p$ is an event of the observer $\beta$ with 4-velocity $u$, $q$ is an event of the test particle $\beta^{\prime}$ with 4-velocity $u^{\prime}$, and $\psi$ is a spacelike geodesic segment orthogonal to $u$ joining $p$ and $q$, expression (1) defines a kinematic relative velocity of $u^{\prime}$ with respect to $u$, where $\tau_{qp}$ represents the parallel transport from $q$ to $p$ along $\psi$; in this case, there exists a unique $v_{\mathrm{kin}}$ associated to the set $\left\{p,q,\psi\right\}$. Working in a convex normal neighborhood implies that given $p$, there exists a unique set $\left\{p,q,\psi\right\}$ satisfying the above conditions, and so there is a unique $v_{\mathrm{kin}}$ associated to this $p$. But if we work in a non convex normal neighborhood, we could find different sets $\left\{p,q,\psi^{\prime}\right\}$, $\left\{p,q^{\prime},\psi^{\prime\prime}\right\}$, $\ldots$ satisfying the above conditions, and hence, there would be a different $v_{\mathrm{kin}}$ for each set. This extension can be also done for the other concepts of relative velocity and, for example, in the framework of lightlike simultaneity, if there is gravitational lensing then each image of the observed object has a different spectroscopic and astrometric relative velocity. ## 3 Stationary observers in Schwarzschild spacetime The Schwarzschild metric in spherical coordinates $\left\{t,r,\theta,\varphi\right\}$ is given by the line element \[ds^{2}=-\left(1-\frac{2m}{r}\right)\mathrm{d}t^{2}+\left(1-\frac{2m}{r}\right) ^{-1}\mathrm{d}r^{2}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d} \varphi^{2}\right),\] (4) where the parameter $m$ is interpreted as the mass of the gravitating object, $r>2m$ is the radial coordinate, and $0<\theta<\pi$. From now on we are going to suppose that the coordinates hold these restrictions. In the framework of this coordinate system, a _stationary observer_ is an observer with constant spatial coordinates and its 4-velocity is given by \[U=\left(1-\frac{2m}{r}\right)^{-1/2}\frac{\partial}{\partial t}=\left(\left(1- \frac{2m}{r}\right)^{-1/2},0,0,0\right).\] (5) Note that stationary observers are not geodesic, but they are useful in the description and interpretation of the Schwarzschild spacetime because the vector field $\frac{\partial}{\partial t}$ is Killing. Moreover, we can consider _asymptotic exterior observers_ making $r\rightarrow+\infty$ from stationary observers. Next, we are going to give some lemmas that will be applied in the next section. Lemma 3.1: _Let $p$ be an event of a stationary observer, and let $u=U_{p}$ be its 4-velocity at $p$, see (5). Then, the Landau submanifold $L_{p,u}$ is contained in the hypersurface $t=t_{0}$, where $t_{0}$ is the coordinate time of $p$._ Proof.: The Landau submanifold is formed by all the spacelike geodesics starting from $p$ and orthogonal to $u$ at $p$; let $\psi$ be one of these geodesics (affinely parameterized) and let $S$ be its tangent vector field. Since the vector field $\frac{\partial}{\partial t}$ is Killing and taking into account (5), we have \[g\left(S,\frac{\partial}{\partial t}\right)=g\left(S_{p},\left.\frac{\partial} {\partial t}\right\|_{p}\right)=\left(1-\frac{2m}{r_{0}}\right)^{1/2}g\left(S_{ p},u\right)=0,\] (6) where $r_{0}$ is the radial coordinate of $p$, and $S_{p}$ is orthogonal to $u$. Hence, from (6) and taking into account that the metric (4) is diagonal, we have that $S^{t}=0$, and so $\psi^{t}$ is constant, concluding that $\psi$ is in the hypersurface $t=t_{0}$. ∎ Lemma 3.2: _The vector field $U$ given by (5) is parallel transported along curves in hypersurfaces of the form $t=\mathrm{constant}$, i.e. curves with constant time component._ Proof.: Let $\varphi$ be a curve in a hypersurface of the form $t=\mathrm{constant}$ and let $S$ be its tangent vector field. Hence $S^{t}=0$, and taking into account the Christoffel symbols of the metric, there is only one nontrivial equation for the parallel transport of $U$ along $\varphi$: \[\frac{\mathrm{d}U^{t}}{\mathrm{d}r}S^{r}+\Gamma^{t}_{rt}S^{r}U^{t}=0\quad \Longrightarrow\quad\left(\frac{\mathrm{d}U^{t}}{\mathrm{d}r}+\frac{m}{r\left( r-2m\right)}U^{t}\right)S^{r}=0.\] (7) Since $U^{t}=\left(1-\frac{2m}{r}\right)^{-1/2}$, equation (7) holds for any $S^{r}$. ∎ ## 4 Modulus of the kinematic relative velocity with respect to stationary observers In this section, we are going to work in the Schwarzschild metric using spherical coordinates. Moreover, we are going to consider a stationary observer containing $p=\left(t_{0},r_{0},\theta_{0},\varphi_{0}\right)$, whose 4-velocity at $p$ is $u=U_{p}$, where $U$ is the vector field given by (5). Theorem 4.1: _Given a test particle with 4-velocity $u^{\prime}$ at $q=\left(t_{0},r_{1},\theta_{1},\varphi_{1}\right)$, we have_ \[\\|v_{\mathrm{kin}}\\|^{2}=\\|v\\|^{2}=1-\frac{r_{1}}{\left(r_{1}-2m\right)\left(u ^{\prime t}\right)^{2}},\] (8) _where $v_{\textrm{kin}}$ is the kinematic relative velocity of $u^{\prime}$ with respect to the stationary observer $u$, and $v$ is the usual relative velocity of $u^{\prime}$ with respect to $U_{q}$._ Proof.: Using Lemmas 3.1 and 3.2, we have that $\tau_{pq}u=U_{q}$. Then, by (2), (3) and (5) the result holds. ∎ Since (8) does not depend on $u$, the kinematic relative velocity shows a kind of “Newtonian behavior” when it is measured by stationary observers, unlike the other three relative velocities (Fermi, spectroscopic and astrometric), that do not have this behavior in general, (see Figure 2). Moreover, making $r_{0}\rightarrow+\infty$ we obtain that Theorem 4.1 also holds for asymptotic exterior observers. Remark 4.1: Schwarzschild spacetime is not a convex normal neighborhood and then we have to take into account Remark 2.1. Following the notation of that remark, Lemma 3.1 assures that given $p\in\beta$ there exists a unique $q\in\beta^{\prime}$ such that $q\in L_{p,u}$ (because $p$ and $q$ must have the same coordinate time), and viceversa; but there could exist different spacelike geodesics orthogonal to $u$ joining $p$ and $q$, and consequently, different kinematic relative velocities of $u^{\prime}$ with respect to $u$. Nevertheless, we can conclude from Theorem 4.1 that all of these velocities have the same modulus. Next, we are going to study some fundamental particular cases. ### Stationary test particles Taking into account Theorem 4.1 and (5), the kinematic relative velocity of a stationary observer (in the role of a test particle) with respect to any other stationary observer is zero (in [4] it is proved in the particular case of stationary observers aligned with the origin, i.e. with the same $\theta$ and $\varphi$ coordinates). In fact, it can be proved that the Fermi and astrometric relative velocities are also zero, because they are based on changes of relative position. On the other hand, the spectroscopic relative velocity is not zero and produces the gravitational shift (see [12, 13, 4]). ### Radially inward free-falling test particles In [4] it is also studied the case of a radially inward free-falling test particle at $r_{1}$ with respect to a stationary observer at $r_{0}\geq r_{1}$ and aligned with the test particle (i.e. with the same $\theta$ and $\varphi$ coordinates). Without loss of generality we can consider that the test particle has $\theta=\pi/2$ (i.e. it is equatorial) and $\varphi=0$ (because it is radial), and so its 4-velocity at $q=\left(t_{0},r_{1},\pi/2,0\right)$ is given by \[u^{\prime}=\left(\frac{E}{1-\frac{2m}{r_{1}}},-\sqrt{E^{2}-\left(1-\frac{2m}{r _{1}}\right)},0,0\right),\] (9) where $E:=\left(\frac{1-2m/r_{\textrm{ini}}}{1-v_{\textrm{ini}}^{2}}\right)^{1/2}$ is a constant of motion, $r_{\textrm{ini}}$ is the radial coordinate at which the fall begins, and $v_{\textrm{ini}}$ is the initial velocity (see [14]). It was obtained that the square of the modulus of the kinematic relative velocity with respect to an aligned stationary observer is \[\left\\|v_{\mathrm{kin}}\right\\|^{2}=1-\frac{1-\frac{2m}{r_{1}}}{E^{2}}.\] (10) From Theorem 4.1 and the expression of $u^{\prime t}$ given in (9), we can generalize this result obtaining that (10) is also valid for any stationary observer not necessarily aligned with the test particle. ### Test particles with circular geodesic orbits Another important and interesting case is that of a test particle with circular geodesic orbit at radius $r_{1}>3m$, that we can suppose equatorial (without loss of generality) and whose 4-velocity is then given by \[U^{\prime}=\left(\sqrt{\frac{r_{1}}{r_{1}-3m}},0,0,\frac{1}{r_{1}}\sqrt{\frac{ m}{r_{1}-3m}}\right).\] (11) This case is very hard to study analytically and only numerical results have been obtained. Nevertheless, taking into account Theorem 4.1 and the expression of $U^{\prime t}$ given in (11), the square of the modulus of the kinematic relative velocity of the test particle with respect to any stationary observer is constant and given by \[\\|V_{\mathrm{kin}}\\|^{2}=\frac{m}{r_{1}-2m}.\] (12) Since (12) only depends on $r_{1}$, the behavior of the kinematic relative velocity is even more “Newtonian” compared with the other three relative velocities whose moduli are not constant (see Figure 2). <figure><img src="content_image/1205.0884/velschw.png"><figcaption>Figure 2: Retarded comparison (see [7]) of the moduli of the kinematic, Fermi,spectroscopic and astrometric relative velocities of a test particle withequatorial circular geodesic orbit with radius r1=4, θ1=π/2 and φ1=0 at t=0,with respect to a stationary observer at r0=3 (left) and r0=8 (right), θ0=π/2and φ0=0, in the Schwarzschild metric with m=1. The modulus of the kinematicrelative velocity (dashed line) remains constant and equal to √1/2. They havebeen numerically computed with a relative error less than 10−6.</figcaption></figure> ## 5 Final remarks Theorem 4.1 assures that the modulus of the kinematic relative velocity of a test particle with respect to a stationary observer can be computed choosing any stationary observer. Moreover, stationary observers are _kinematically comoving_ (see [4]) between them as it is proved in Section 4.1. In fact, from Theorem 5.1 we obtain the next result that holds in any metric, generalizing Theorem 4.1: given a congruence of kinematically comoving observers, the modulus of the kinematic relative velocity of a test particle with respect to any observer of the congruence remains constant. Theorem 5.1: _Let $U$ be a congruence of kinematically comoving observers. Given a test particle with 4-velocity $u^{\prime}$ at $q$ and given $p$ such that $q\in L_{p,u}$ (with $u:=U_{p}$), we have_ \[\left\\|v_{\mathrm{kin}}\right\\|=\left\\|v\right\\|,\] (13) _where $v_{\textrm{kin}}$ is the kinematic relative velocity of $u^{\prime}$ with respect to $u$, and $v$ is the usual relative velocity of $u^{\prime}$ with respect to $U_{q}$._ Proof.: Since $U$ is a congruence of kinematically comoving observers, the kinematic relative velocity of $U_{q}$ with respect to $u$ is zero, and so, by (3) we have that $g\left(U_{q},\tau_{pq}u\right)^{2}=1$. Hence, $g\left(U_{q},\tau_{pq}u\right)=-1$ because they are timelike and future-pointing, and then $\tau_{pq}u=U_{q}$, because they are also unit. Finally, by (2) and (3) we have $\left\\|v_{\mathrm{kin}}\right\\|^{2}=1-\frac{1}{g\left(u^{\prime},U_{q}\right)^ {2}}=\left\\|v\right\\|^{2}$ and so, (13) holds. ∎ Theorem 5.1 can be also expanded to spectroscopic relative velocity taking into account a congruence of _spectroscopically comoving_ observers and the past-pointing horismos submanifold $E^{-}_{p}$ (see [15, 4]) instead of $L_{p,u}$; but stationary observers are not spectroscopically comoving between them in Schwarzschild spacetime, and so, we can not apply it in our case. On the other hand, this result does not hold in general for Fermi or astrometric relative velocities: for example, the congruence of stationary observers is also _Fermi comoving_ and _astrometrically comoving_ (see Section 4.1), but the modulus of the Fermi or the astrometric relative velocity of a test particle depends on the chosen stationary observer (see Figure 2). All these facts and the results obtained in Section 4 (specially in Section 4.3, where it is shown that the modulus of the kinematic relative velocity of a test particle with circular geodesic orbit with respect to a stationary observer remains constant) leads to the conclusion that the kinematic relative velocity can be interpreted as the most “Newtonian-like” velocity of the four geometric velocities introduced in [4]. ## References * [1] M. Soffel, _et al_. The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: explanatory supplement. _Astron. J._126 (2003), 2687–2706 (arXiv:astro-ph/0303376). * [2] L. Lindegren, D. Dravins. The fundamental definition of ‘radial velocity’. _Astron. Astrophys._401 (2003), 1185–1202 (arXiv:astro-ph/0302522). * [3] M. Carrera, D. Giulini. On Doppler tracking in cosmological spacetimes. _Class. Quantum Grav._23 (2006), 7483–7492 (arXiv:gr-qc/0605078). * [4] V. J. Bolós. Intrinsic definitions of “relative velocity” in general relativity. _Commun. Math. Phys._273 (2007), 217–236 (arXiv:gr-qc/0506032). * [5] D. Klein, P. Collas. Recessional velocities and Hubble’s law in Schwarzschild-de Sitter space. _Phys. Rev. D_81, 063518 (2010) (arXiv:1001.1875). * [6] D. Klein, E. Randles. Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies. _Ann. Henri Poincaré_12 (2011), 303–328 (arXiv:1010.0588). * [7] V. J. Bolós, D. Klein. Relative velocities for radial motion in expanding Robertson-Walker spacetimes. _Gen. Relativ. Gravit._44 (2012), 1361–1391 (arXiv:1106.3859). * [8] B. O’Neill. Semi-Riemannian Geometry With Applications to Relativity. Pure and Applied Mathematics, Volume 103. Elsevier Science, San Diego (1983). * [9] E. Fermi. Sopra i fenomeni che avvengono in vicinanza di una linea oraria. _Atti R. Accad. Naz. Lincei, Rendiconti, Cl. sci. fis. mat & nat._31 (1922), 21–23, 51–52, 101–103. * [10] V. J. Bolós, V. Liern, J. Olivert. Relativistic simultaneity and causality. _Internat. J. Theoret. Phys._41 (2002) 1007–1018 (arXiv:gr-qc/0503034). * [11] V. J. Bolós. A note on the computation of geometrically defined relative velocities. _Gen. Relativ. Gravit._44 (2012), 391–400 (arXiv:1109.0131). * [12] J. V. Narlikar. Spectral shifts in general relativity. _Am. J. Phys._62 (1994), no. 10, 903–907. * [13] V. J. Bolós. Lightlike simultaneity, comoving observers and distances in general relativity. _J. Geom. Phys._56 (2006), 813–829 (arXiv:gr-qc/0501085). * [14] P. Crawford, I. Tereno. Generalized observers and velocity measurements in general relativity. _Gen. Relativ. Gravit._34 (2002), no. 12, 2075–2088 (arXiv:gr-qc/0111073). * [15] J. K. Beem, P. E. Ehrlich. Global Lorentzian Geometry. Marcel Dekker, New York (1981).
0711.1075	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 48532, "num_imgs": 5, "llama3_tokens_count": 14403 }	[ "content_image/0711.1075/x1.png", "content_image/0711.1075/x2.png", "content_image/0711.1075/x4.png", "content_image/0711.1075/x6.png", "content_image/0711.1075/x8.png" ]	# Combining CPT-conjugate Neutrino channels at Fermilab Andreas Jansson${}^{1}$, Olga Mena${}^{2}$, Stephen Parke${}^{1}$ and Niki Saoulidou${}^{1}$ ${}^{1}$Fermi National Accelerator Laboratory P.O.Box 500, Batavia, IL 60510, USA ${}^{2}$INFN Sez. di Roma, Dipartimento di Fisica, Università di Roma“La Sapienza”, P.le A. Moro, 5, I-00185 Roma, Italy February 23, 2024 ###### Abstract We explore an alternative strategy to determine the neutrino mass hierarchy by making use of possible future neutrino facilities at Fermilab. Here, we use CPT-conjugate neutrino channels, exploiting a $\nu_{\mu}$ beam from the NuMI beamline and a $\bar{\nu}_{e}$ beam from a betabeam experimental setup. Both experiments are performed at approximately the same $\langle E\rangle/L$. We present different possible accelerator scenarios for the betabeam neutrino setup and fluxes. This CPT-conjugate neutrino channel scenario can extract the neutrino mass hierarchy down to $\sin^{2}2\theta_{13}\approx 0.02$. pacs: 14.60Pq FERMILAB-PUB-07-456-APC-E-T [-0.1in] ROMA-TH-1458 [-0.1in] ## I Introduction During the last several years the physics of neutrinos has achieved remarkable progress. The experiments with solar [1; 2; 3; 4; 5; 6], atmospheric [7], reactor [8], and also long-baseline accelerator [9; 10; 11] neutrinos, have provided compelling evidence for the existence of neutrino oscillations, implying non zero neutrino masses. The present data require two large ($\theta_{12}$ and $\theta_{23}$) and one small ($\theta_{13}$) angles in the neutrino mixing matrix [12], and at least two mass squared differences, $\Delta m_{ji}^{2}\equiv m_{j}^{2}-m_{i}^{2}$ (where $m_{j}$’s are the neutrino masses), one driving the atmospheric ($\Delta m_{31}^{2}$) and the other one the solar ($\Delta m_{21}^{2}$) neutrino oscillations. The mixing angles $\theta_{12}$ and $\theta_{23}$ control the solar and the dominant atmospheric neutrino oscillations, while $\theta_{13}$ is the angle limited by the data from the CHOOZ and Palo Verde reactor experiments [13; 14]. The Super-Kamiokande (SK) [7] and K2K [9] data are well described in terms of dominant $\nu_{\mu}\rightarrow\nu_{\tau}$ ($\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\tau}$) vacuum oscillations. A recent global fit [15] provides the following $3\sigma$ allowed ranges for the atmospheric mixing parameters \[\|\mbox{$\Delta m_{31}^{2}$}\|=(2-3.2)\times 10^{-3}{\rm eV^{2}},~{ }~{}~{}~{}0.32<\sin^{2}\theta_{23}<0.64~{}.\] (1) The sign of $\Delta m_{31}^{2}$, sign$(\mbox{$\Delta m_{31}^{2}$})$, cannot be determined with the existing data. The two possibilities, $\mbox{$\Delta m_{31}^{2}$}>0$ or $\mbox{$\Delta m_{31}^{2}$}<0$, correspond to two different types of neutrino mass ordering: normal hierarchy and inverted hierarchy. In addition, information on the octant in which $\theta_{23}$ lies, if $\sin^{2}2\theta_{23}\neq 1$, is beyond the reach of present experiments. The 2-neutrino oscillation analysis of the solar neutrino data, including the results from the complete salt phase of the Sudbury Neutrino Observatory (SNO) experiment [6], in combination with the KamLAND spectrum data [16], shows that the solar neutrino oscillation parameters lie in the low-LMA (Large Mixing Angle) region, with best fit values [15]$\mbox{$\Delta m_{21}^{2}$}=7.9\times 10^{-5}~{}{\rm eV^{2}}$ and $\sin^{2}\theta_{12}=0.30$. A combined 3-neutrino oscillation analysis of the solar, atmospheric, reactor and long-baseline neutrino data [15] constrains the third mixing angle to be $\sin^{2}\theta_{13}<0.04$ at the $3\sigma$ C.L. However, the bound on $\sin^{2}\theta_{13}$ is dependent on the precise value of $\Delta m^{2}_{31}$. The future goals for the study of neutrino properties is to precisely determine the already measured oscillation parameters and to obtain information on the unknown ones: namely $\theta_{13}$, the CP–violating phase $\delta$ and the type of neutrino mass hierarchy (or equivalently sign$(\mbox{$\Delta m_{31}^{2}$})$). In the presence of matter effects, the neutrino (antineutrino) oscillation probability gets enhanced [17; 18] for the normal (inverted) hierarchy. Making use of the different matter effects for neutrinos and antineutrinos seems, in principle, the most promising way to distinguish among the two possibilities: normal versus inverted hierarchy. However, the sensitivity to the mass hierarchy determination from the neutrino-antineutrino comparison is highly dependent on the value of the CP violating phase. Thus, possible alternative methods were first proposed in Ref. [19]. In this paper we concentrate on the extraction of the neutrino mass hierarchy by combining a $\nu_{\mu}\to\nu_{e}$ experiment with its CPT conjugated channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$, see Ref. [19]. More recently, it is primarily the CPT-conjugate channel pairs that give the CERN-MEMPHYS proposal sensitivity to the hierarchy, see Ref. [20]. If nature respects CPT symmetry, then, at the same $E/L$ the only difference between the two flavor transitions can come from matter effects and that near the first oscillation maximum \[P(\nu_{\mu}\to\nu_{e}) > P(\bar{\nu}_{e}\to\bar{\nu}_{\mu})\quad{\rm for~{}Normal~{}Hierarchy}\] \[{\rm and}\quad P(\nu_{\mu}\to\nu_{e}) < P(\bar{\nu}_{e}\to\bar{\nu}_{\mu})\quad{\rm for~{}Inverted~{} Hierarchy,}\] i.e. for the normal hierarchy the neutrino channel is enhanced and the antineutrino CPT conjugate channel suppressed and vice versa for the inverted hierarchy. This is the effect that will be exploited in this paper to determine the neutrino mass hierarchy. We will show that the combination of the Phase I (neutrino-data only) of the long-baseline $\nu_{e}$ appearance experiment NO$\nu$A [21], exploiting the off-axis technique¹ with a possible future betabeam facility [23; 24; 25; 26; 27] at Fermilab exploiting a $\bar{\nu}_{e}$ neutrino beam from radiative ion decays could help enormously in measuring the neutrino mass hierarchy. For our analysis, unless otherwise stated, we will use a representative value of $\|\mbox{$\Delta m_{31}^{2}$}\|=2.5\times 10^{-3}\ \rm{eV}^{2}$ and $\sin^{2}2\theta_{23}=1$. For the solar oscillation parameters $\Delta m_{21}^{2}$ and $\theta_{12}$, we will use the best fit values quoted earlier in this section. The structure of the paper is as follows. In Section II we present the general physics strategy used to determine the neutrino mass hierarchy including the CPT conjugate channels used in this paper. Section III contains a realistic description of possible future betabeam facilities at Fermilab. The different scenarios deal with different ions, baselines and luminosities, and the performance of the strategy followed here in each of these scenarios is illustrated in Section IV. The sensitivity curves for the several scenarios will be presented in Section V and the final remarks are summarized in Section VI. In the Appendix A, we discuss the details associated with comparing CPT conjugate neutrino oscillation probabilities. [FOOTNOTE:1][ENDFOOTNOTE] ## II Combining Neutrino channels The strategy we have introduced in the previous section and we explain in detail here is different from the usual one, which exploits the combination of the neutrino and antineutrino oscillation channels. Typically, the proposed long baseline neutrino oscillation experiments have a single far detector and plan to run with the beam in two different modes, muon neutrinos and muon antineutrinos. In principle, by measuring the probability of neutrino and antineutrino flavor conversion, the values of the CP–violating phase $\delta$ and the sign$(\mbox{$\Delta m_{31}^{2}$})$ could be extracted, since, in the presence of matter effects there will be two allowed regions for each type of hierarchy, normal or inverted, in the $P(\nu_{\mu}\to\nu_{e})$ versus $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ plane. In practice, the neutrino–antineutrino comparison does not provide the ideal tool to extract the neutrino mas hierarchy, as we explain below. Suppose we compute the oscillation probabilities $P(\nu_{\mu}\to\nu_{e})$ and $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ for a given set of oscillation parameters and the CP-phase $\delta$ is varied between $0$ and $2\pi$: we obtain a closed CP trajectory (an ellipse) in the bi–probability space of neutrino and antineutrino conversion [28]. Matter effects are responsible for the departure of the center of the ellipses from the diagonal line in the bi–probability plane for normal and inverted hierarchy. In Figure 1, we have illustrated the case for $E=2.0$ GeV and $L=810$ km, which roughly correspond to those of the NO$\nu$A experiment [21]. The distance between the center of the ellipse for the normal hierarchy (lower blue) and that for the inverted hierarchy (upper red) is governed by the size of the matter effects. Notice that the ellipses overlap for a significant fraction of values of the CP–phase $\delta$ for every allowed value of $\sin^{2}2\theta_{13}$. This makes the determination of sign$(\mbox{$\Delta m_{31}^{2}$})$ extremely difficult, i. e., the sign$(\mbox{$\Delta m_{31}^{2}$})$-extraction is not free of degeneracies. <figure><img src="content_image/0711.1075/x1.png"><figcaption>Figure 1: The bi–probability plot for P(νμ→νe) versus P(¯νμ→¯νe) at a baselineof 810 km and an energy of 2.0 GeV for the normal (blue) and the inverted(red) hierarchies. The smaller, lower (larger, upper) ellipses are forsin22θ13=0.02 ( 0.10).</figcaption></figure> <figure><img src="content_image/0711.1075/x2.png"><figcaption>Figure 2: (a) The left panel is the bi–probability plot for P(νμ→νe) versusP(νμ→νe) with baselines 295 km and 810 km for the normal (blue) and theinverted (red) hierarchies. The smaller, lower (larger, upper) ellipses arefor sin22θ13=0.02 ( 0.10). The mean neutrino energies are chosen such that the⟨E⟩/L for the two experiments are approximately identical. (b) The right panel is the bi–probability plot for P(νμ→νe) versus P(¯νe→¯νμ)for the normal (blue) and the inverted (red) hierarchies. The baseline andmean neutrino energy for both experiments are 810 km and ∼ 2 GeV,respectively. The smaller, lower (larger, upper) squashed ellipses are forsin22θ13=0.02 ( 0.10).</figcaption></figure> Following the line of thought developed by Minakata, Nunokawa and Parke [19], we exploited in a previous work [29; 30] the neutrino data only from two experiments at different distances and at different off-axis locations, such that the $\langle E\rangle/L$ is the same for the two experiments (see also Refs. [31; 32; 33; 34; 35]). In the case of bi–probability plots for neutrino–neutrino modes at different distances (which will be referred as near (N) and far (F)), the CP–trajectory is also elliptical. In Figure 2 (a) we present the bi–probability plot for the mean energies and baselines of the $\nu_{e}$ appearance experiments T2K [36] and NO$\nu$A [21]. The overlap of the two ellipses, which implies the presence of a degeneracy of the type of hierarchy with other parameters, is determined by their width and the difference in the slopes. Using the fact that matter effects are small ($aL\ll\Delta_{31}$, being $a=G_{F}N_{e}/\sqrt{2}\approx(4000~{}km)^{-1}$ the matter parameter), we can perform a perturbative expansion and assuming that the $\langle E\rangle/L$ of the near and far experiments is the same², at first order, the ratio of the slopes reads [19] [FOOTNOTE:2][ENDFOOTNOTE] \[\frac{\alpha_{+}}{\alpha_{-}}\simeq 1+4\left(a_{\rm N}L_{\rm N}-a _{\rm F}L_{\rm F}\right)\left(\frac{1}{\Delta_{31}}-\frac{1}{\tan(\Delta_{31}) }\right)~{},\] (2) where $\alpha_{+}$ and $\alpha_{-}$ are the slopes of the center of the ellipses as one varies $\theta_{13}$ for normal and inverted hierarchies, $a_{\rm F}$ and $a_{\rm N}$ are the matter parameters, and $L_{\rm F}$ and $L_{\rm N}$ are the baselines for the two experiments. The separation between the center of the ellipses for the two hierarchies increases as the difference in the matter parameter times the path length, ($aL$), for the two experiments increases. Also, since $(\Delta^{-1}-\cot{\Delta})$ is a monotonically increasing function of $\Delta$, we conclude that the smaller the energy, the larger the ratio of slopes, assuming the same $\langle E\rangle/L$. However the width of the ellipses is crucial: even when the separation between the central axes of the two regions is substantial, if the ellipses for the normal and inverted hierarchy overlap, the hierarchy cannot be resolved for values of the CP phase, $\delta$, for which there is overlap. The width of the ellipses is determined by the difference in the $\langle E\rangle/L$ of the two experiments. In the case of bi–probability plots for the $\nu_{\mu}\to\nu_{e}$ and its CPT conjugated channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ at the same energy divided by baseline,$\langle E\rangle/L$, the CPT–trajectory collapses to a line (see Figure 2 (b)). As for the neutrino-neutrino case, we can perform a perturbative expansion, and, assuming that the $\langle E\rangle/L$ of the CPT conjugated channels is the same (to minimize the ellipses width), at first order, the ratio of the slopes reads (see Appendix and also Ref. [19]) \[\frac{\alpha_{+}}{\alpha_{-}}\simeq 1+4\left(aL+a_{\rm CPT}L_{\rm CPT }\right)\left(\frac{1}{\Delta_{31}}-\frac{1}{\tan(\Delta_{31})}\right)~{},\] (3) where $\alpha_{+}$ and $\alpha_{-}$ are the slopes of the center of the ellipses as one varies $\theta_{13}$ for normal and inverted hierarchies, $a$ and $a_{\rm CPT}$ are the matter parameters and $L$ and $L_{\rm CPT}$ are the baselines for the two experiments which exploit the $\nu_{\mu}\to\nu_{e}$ and its CPT conjugated channel ($\bar{\nu}_{e}\to\bar{\nu}_{\mu}$). Notice that, compared to the neutrino–neutrino case given by Eq. (2), the separation between the center of the ellipses for the two hierarchies increases as the sum of the matter parameter times the baseline, $aL$, for both experiments does. Here the ratio of the slopes is enhanced by matter effects for both $\nu_{\mu}\to\nu_{e}$ and its CPT conjugated channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$. Figure 2 (b) shows the bi–probability curves for the combination of these two flavor transitions, assuming that the two experiments are performed at the same mean energy and baseline. If the $\langle E\rangle/L$ of both experiments is the same, the ellipses will become lines with a negligible width. The separation of the lines for the normal and inverted hierarchy grows as the matter effects for both experiments increase. Consequently, the comparison of CPT conjugated channels is more sensitive to the neutrino mass hierarchy than the neutrino–neutrino one. ## III Beta Beams at Fermilab A betabeam facility exploits a beam of electron neutrinos (antineutrinos) from boosted-ion $\beta^{+}$ ($\beta^{-}$) decays in the straight section of a storage ring [23; 24]. The idea of considering higher $\gamma$ factors (and, consequently, longer detector baselines) was first proposed in Ref. [25]. An extensive phenomenological work has been devoted in order to optimize the betabeam physics reach, analyzing several scenarios with different $\gamma$ factors, boosted-ions and/or detector baselines [26]. Early on, ${}^{6}$He and ${}^{18}$Ne were identified as optimal ions, because of the low Q factor of their decay. The lower the neutrino energy is in the rest frame, the more boost is needed to get to a given energy, and since the angular spread of the beam goes as $1/\gamma$ this yields a more focused neutrino beam, which in turn produces more events in the far detector. Recently, it was proposed to use ${}^{8}$Li and ${}^{8}$B, which could potentially be produced in large amounts using a small storage ring with an internal gas target[38]. Since these ions have larger Q factor, they produce fewer neutrinos per ion in the far detector for a fixed neutrino energy and baseline. However, because less boost is needed, a smaller accelerator would be needed to achieve the same neutrino energy, as compared to the case of ${}^{6}$He and ${}^{18}$Ne. In this section we present possible betabeam scenarios at Fermilab, exploiting its current accelerator facilities. Since the analysis considered in this paper only requires anti-neutrinos from a beta-beam, we concentrate on ${}^{6}$He and ${}^{8}$Li. A rather thorough study of achievable ion intensities has been done at CERN [37]. The CERN scenario uses the existing PS and SPS accelerators, and would in addition require a proton source (e.g. the proposed Superconducting Proton Linac), target station, ion source, ion linac and Rapid Cycling Synchrotron (RCS), as well as a decay ring operating at SPS top energy. Based on a ${}^{6}$He ion production rate of $2\times 10^{13}$/s, ions decaying in the straight section directed towards the experiment would produce approximately $3\times 10^{18}$ antineutrinos per year. We consider two possible scenarios at Fermilab, namely: 1) accelerating ${}^{6}$He to Tevatron top energy and 2) accelerating ${}^{8}$Li to Main Injector (MI) top energy. These two scenarios produce neutrinos of comparable energies. The ions would be generated using a proton source (e.g. the Project X linac) and accelerated using e.g. a linac and a small RCS before being injected into the existing Booster. Possibly, the Recycler could also be used to accumulate bunches while the MI is ramping. In both cases, a new decay ring would be needed to store the ions ³. [FOOTNOTE:3][ENDFOOTNOTE] Extrapolating from the work done at CERN, it appears reasonable to expect a useful ion decay rate (decays in the direction of the experiment) of about $1\times 10^{18}$${}^{6}$He per year in the Tevatron case. At this intensity, the average power deposition in the Tevatron would be about $1$ W/m, which is a generally accepted limit for hands-on-maintenance. Preliminary simulations indicate that the Tevatron magnets would be able to handle the distributed energy deposition from decay products in the arcs, but special care would have to be taken to cope with the build-up of decay products in the straight sections. In the case of ${}^{8}$Li in MI, the injectors could operate with a significantly higher duty factor, since there is no need to wait for the slow Tevatron ramp. However, at repetition rates and intensities corresponding to a useful ion decay rate of about $1\times 10^{19}$ per year, activation of the Booster from decay products would become a serious issue. A very important property of the neutrino beam is the duty factor, defined as the relative fraction of time occupied by the neutrino pulse. This is used to suppress background by gating the data acquisition in the experiment. In the CERN study, a duty factor of a few per mil was obtained with considerable difficulty. A small duty factor is challenging because it requires the ions to be concentrated in very few bunches, which among other things can cause space-charge problems, in particular at low energies. Using the CERN number of $1\times 10^{12}$ ions per RCS cycle, and a Booster injection energy of around $500$ MeV/u, the beam must be distributed over about 10 bunches in the Booster to keep the space charge tune shift at an acceptable level. Approximately eight transfers from the booster per MI cycle would be required to obtain $1\times 10^{19}$ useful ${}^{8}$Li decays per year at MI top energy. Without RF manipulations, this would yield a neutrino beam duty factor of about $10\%$, but a duty factor of order $1\%$ could likely be obtained by coalescing bunches at MI top energy⁴. In the case of ${}^{6}$He, about twenty booster injections per cycle would be required per cycle, in order to compensate for the long Tevatron cycle and obtain a useful ${}^{6}$He decay rate of $1\times 10^{18}$ per year. Assuming bunches are coalesced in the MI, this should also yield a neutrino beam duty factor of about $1\%$. Space charge is not expected to be an issue in the MI or Tevatron at these bunch intensities. [FOOTNOTE:4][ENDFOOTNOTE] For the ${}^{6}$He case, therefore, it appears possible to generate $10^{18}$ useful ion decays per year with a maximum gamma of $\gamma_{\rm He}=350$. For the ${}^{8}$Li case, at a maximal gamma of $\gamma_{\rm Li}=55$, the rate could be higher (as explained above). We will explore an optimistic scenario of $5\times 10^{19}$ useful ion decays per year, as well a more pessimistic scenario of $10^{19}$ useful ion decays per year from ${}^{8}$Li decays. We will assume a duty factor of $1\%$ for both ion species. Table 1 shows the maximum Lorentz gamma factors in the Fermilab machines for the ${}^{6}$He and ${}^{8}$Li, as well as other ions considered in the literature. Machine \| 6He2+ \| 8Li3+ \| 18Ne10+ \| 8B5+ ---\|---\|---\|---\|--- Linac \| 1.6 \| 1.07 \| 1.15 \| 1.19 Booster \| 3.3 \| 3.7 \| 5.4 \| 6.1 Main Injector \| 54 \| 60 \| 90 \| 101 Tevatron \| 351 \| 395 \| 586 \| 659 Table 1: Maximum Lorentz gamma factors obtainable in the Fermilab machines for different ions of interest. ## IV Possible experimental setups at Fermilab As we discussed in Section II, the most sensitive, degeneracy free method, to extract the neutrino mass hierarchy exploiting a future high intensity conventional neutrino beam ($\nu_{\mu}\to\nu_{e}$) is the combination of this channel with its CPT conjugate $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$. Future facilities like betabeams (neutrino factories) can produce neutrino beams which are entirely (partially) composed of $\nu_{e}$ or $\bar{\nu}_{e}$. The experimental strategy that we follow here is to combine the NO$\nu$A experiment, which will measure the flavor transitions $\nu_{\mu}\to\nu_{e}$, with its CPT conjugated channel. The NO$\nu$A experiment is expected to run at least five years with neutrinos. A $30$ kton low density tracking calorimeter with an efficiency of $24\%$ would be located at a baseline of $810$ km and at $12$ km off-axis distance from the beam center, resulting in a mean muon neutrino energy of $2.0$ GeV. For the CPT conjugate channel, $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$, we exploit possible, future betabeam facilities at Fermilab described in the previous section for two antineutrino emitters: ${}^{6}$He and ${}^{8}$Li. A future neutrino factory exploiting neutrino fluxes from muon decays could also provide the $\bar{\nu}_{e}$ CPT conjugate channel, if $\mu^{-}$’s are stored in the decay ring. In the present study we do not explore this possibility. For the ${}^{6}$He ion case, (maximum $\gamma_{\textrm{He}}=350$), the mean electron antineutrino energy is $\sim 1.2$ GeV ($\langle E_{\nu}\rangle\simeq\gamma_{\textrm{He}}E_{0,\textrm{He}}$, $E_{0,\textrm{He}}=3.5+m_{e}$ MeV, being the electron end-point energy for ${}^{6}$He). We present the results for two possible experimental ${}^{6}$He setups. In the first scenario, we consider a single baseline $L=810$ km, at which the NO$\nu$A detector would be located, with a total detector mass of $40$ kton. This first scenario could be easily achieved adding to the $30$ kton NO$\nu$A far detector a second $10$ kton detector at the NO$\nu$A far site. Possible detector technologies are Liquid Argon or iron calorimeter detectors. Liquid Argon (LAr) detectors have excellent efficiency, background rejection and energy resolution, but they could suffer from a large atmospheric neutrino background, which could be overcome only if the beam duty cycle is $<10^{-4}$ [24]. In the second scenario, we consider a detector similar to the one of the MINOS [10] experiment ($5$ kton) at $735$ km but twice in size. If the ion luminosity could be improved by a factor of two, MINOS far detector would be sufficient. This scenario benefits from the lower atmospheric neutrino backgrounds at the MINOS site. The beam duty cycle would not, therefore, be a major issue, and a duty factor of $\sim 1\%$ would be sufficient to overcome the atmospheric neutrino background in this case. For the ${}^{8}$Li ion case, (maximum $\gamma_{\textrm{Li}}=50$), the mean electron antineutrino energy is $\sim 1.8$ GeV ($\langle E_{\nu}\rangle\simeq\gamma_{\textrm{Li}}E_{0,\textrm{Li}}$, $E_{0,\textrm{Li}}=16$ MeV, being the electron end-point energy for ${}^{8}$Li). In order to ensure an almost degeneracy free hierarchy measurement, the $\langle E\rangle/L$ of the $\nu_{\mu}\to\nu_{e}$ channel from NO$\nu$A and its CPT conjugate channel should be similar, therefore the detector should be located at $L=300$ km. We will consider $5\times 10^{19}$ ($1\times 10^{19}$) useful ion decays per year with a $10$ ($50$) kton detector respectively. As previously discussed, considering $5\times 10^{19}$ useful ion decays is a quite optimistic assumption. However, notice that the same statistics could be achieved with the more conservative assumption of $1\times 10^{19}$ useful ion decays per year, with a larger $50$ kton detector (these two configurations provide the same statistics). Due to the lower neutrino energies, the detector could be a Liquid Argon TPC, a NO$\nu$A-like Totally Active Scintillator Detector (TASD), or a water Cherenkov. For the NUMI off-axis neutrino beam, we have not considered a binning in the signal, since neutrino events will lie in a very narrow energy window [1.5,2.5] GeV. The ${}^{6}$He and ${}^{8}$Li antineutrino betabeams have been divided in three energy bins, assuming an energy detection threshold of $0.5$ GeV. For the ${}^{6}$He case, we divide the signal in three energy bins, in the energy ranges [0.5,1.0], [1.0,1.5] and [1.5,2.4] GeV, respectively. In the ${}^{8}$Li case, the energy ranges are [0.5,0.75], [0.75,1.0] and [1.0,1.5] GeV, respectively. <figure><img src="content_image/0711.1075/x4.png"><figcaption>Figure 3: (a) The allowed regions in the bi–event plot for N(νμ→νe) for NOνAversus N(¯νe→¯νμ) for a betabeam experiment which exploits antineutrinos from6He decays with γHe=350, and a detector of 40 kton located at a distance ofL=810 km. The blue (red) ellipses denote normal (inverted) hierarchies. Frombottom up, the ellipses correspond to sin22θ13 varying from 0.01 to 0.1. Thesolid (dashed) ellipses illustrate the third (second) energy bins of thebetabeam spectrum. (b) Same as (a) but with antineutrino fluxes resulting from8Li decays, and a detector of 10 kton at 300 km.</figcaption></figure> Figure 3 depict the bi–event plots for the combination of the NO$\nu$A neutrino events ($\nu_{\mu}\to\nu_{e}$) with its CPT conjugated channel ($\bar{\nu}_{e}\to\bar{\nu}_{\mu}$) from the betabeam experiment, resulting from the decays of ${}^{6}$He (left panel) and ${}^{8}$Li (right panel), for both normal and inverted hierarchies. The statistics considered for NO$\nu$A correspond to Phase I of the experiment (neutrino running only). For the ${}^{6}$He betabeam experiment, we assume a number of useful ion decays of $10^{18}$ per year, five years of data taking, and a $40$ kton detector located at $810$ km. For the ${}^{8}$Li betabeam experiment, we assume $5\times 10^{19}$ ($1\times 10^{19}$) useful ion decays per year, ten years of data taking, and a $10$ ($50$) kton detector located at $300$ km. Figure 3 (a) shows that, for the combination of NO$\nu$A off-axis neutrino events with the antineutrino events from ${}^{6}$He decays, the separation between the bi–event contours for the normal and inverted hierarchies is larger than in the case of ${}^{8}$Li generated antineutrino events, as seen in Figure 3 (b). As previously explained, the difference in the slopes of the two hierarchies is proportional to the sum of the size of matter effects times the baseline, $a_{NO\nu A}L_{NO\nu A}+a_{CPT}L_{CPT}$. The product $a_{CPT}L_{CPT}$ is larger for the ${}^{6}$He betabeam $\bar{\nu}_{e}$ events (with a baseline of $810$ km), than for the ${}^{8}$Li betabeam $\bar{\nu}_{e}$ events (with a baseline of $300$ km). The solid (dashed) contours in Figure 3 show the number of betabeam antineutrino events in the second (third) energy bin. When the $\langle E\rangle/L$ of the $\nu_{\mu}\to\nu_{e}$ and its CPT conjugated channel are similar, the ellipses width is minimal (they collapse to a line) and therefore the elliptical contours for the normal and inverted hierarchies will not overlap, regardless the value of the CP violating phase $\delta$. For the combination of NO$\nu$A off-axis neutrino events with the ${}^{6}$He betabeam antineutrino events, there exists a clear difference between the second and third energy bins in the bi–event contours: while they are ellipses in the former, they are almost lines in the latter. Only in the third energy bin ([1.5,2.5] GeV) is the $\langle E\rangle/L$ the same for the ${}^{6}$He betabeam $\bar{\nu}_{e}$ and for the NO$\nu$A $\nu_{\mu}$ events. For the ${}^{8}$Li case the ellipses width is minimal for both the second and third energy bins, since both bins are close to $E\sim 0.8$ GeV, the energy at which the $(\langle E\rangle/L)_{Li}$ equals the $(\langle E\rangle/L)_{NO\nu A}$. ## V Sign $\Delta m^{2}_{31}$ sensitivities In this section we present the physics results from the combination of antineutrino data, resulting from several possible betabeam setups, with the NO$\nu$A neutrino data. Figure 4 (a) shows the $90\%$ C.L mass hierarchy sensitivity, assuming two degrees of freedom statistics (2 d.o.f, that is, $\Delta\chi^{2}>4.21$) for the combination of NO$\nu$A neutrino data with the ${}^{6}$He betabeam antineutrino data, neglecting the background in the $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ channel. In order to safely neglect the atmospheric neutrino background, one would need a very small, experimentally challenging duty factor for the betabeam neutrino fluxes. The baseline for the two experiments is fixed at $810$ km and the $\gamma_{\textrm{He}}=350$, in order to have a similar $\langle E\rangle/L$ in both the muon neutrino and the electron antineutrino channels. The binning of the signal has been chosen as quoted in the previous section. We have considered a flux of $10^{18}$ antineutrinos per year, and five years of data taking. The blue (red) lines assume normal (inverted) hierarchy, and the solid (dotted) lines depict the results for a betabeam antineutrino experiment with a 40 (10) kton detector. For the $40$ kton detector, the sensitivity is better for the inverted mass hierarchy, since in this case statistics is dominated by the antineutrino channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$. For the $10$ kton detector, the two channels (i.e., the neutrino channel from NO$\nu$A and the antineutrino channel from the betabeam experiment) will have similar statistics and therefore the sensitivity for the normal and the inverted mass hierarchies are similar. As a comparison, we also show the results for the NO$\nu$A experiment (upgraded by a factor of five in statistics), assuming five years of neutrino and five years of antineutrino data taking, see the dashed curves in Figure 4. The setup proposed here improves the sensitivity of the NO$\nu$A upgraded experiment by an order of magnitude, and more importantly, eliminates the dependence of the mass hierarchy determination on the value of the CP violating phase $\delta$. Figure 4 (b) shows the $90\%$ C.L mass hierarchy sensitivity, assuming two degrees of freedom statistics (2 d.o.f, that is, $\Delta\chi^{2}>4.21$) for the combination of NO$\nu$A neutrino data with the ${}^{6}$Li betabeam antineutrino data, neglecting the background in the $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ channel. The baseline for the betabeam experiment is $L=300$ km and the binning of the signal has been chosen as described in the previous section. We assume $5\times 10^{19}$${}^{8}$Li generated antineutrinos per year, and ten years of data taking. The blue (red) lines assume normal (inverted) hierarchy, and the the solid (dotted) lines depict the results for a betabeam antineutrino experiment with a 10 (2) kton detector. For the case of ${}^{8}$Li betabeam experiment, the sensitivity is similar for the normal and inverted mass hierarchy, and smaller than in the case of ${}^{6}$He betabeam experiment. This is expected, since at a shorter baseline ($300$ km) the product of the matter potential times the distance is reduced. Again, the combination of the NO$\nu$A neutrino data only with the ${}^{8}$Li betabeam antineutrino data provides a much better sensitivity to the mass hierarchy than the NO$\nu$A upgraded experiment alone. However, as previously stated, the beam duty cycle needed in order to neglect the atmospheric neutrino background is highly challenging. For a MINOS-like detector, there are $30$ atmospheric neutrino interactions per kton-year which could mimic a muon coming from the oscillated $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ [39]. In order to avoid such a large background, we have assumed a betabeam duty cycle $\sim 10^{-2}$ , which seems experimentally achievable. Figure 5 shows the equivalent to Figure 4 when adding the atmospheric neutrino background quoted above, rescaled accordingly to the detector sizes and the exposure times. Notice that the presence of a non negligible atmospheric neutrino background in the antineutrino channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ reduces the sensitivity reach especially for the case of the inverted mass hierarchy. However, even in the presence of a non negligible background, and with the very conservative assumption of a $10^{-2}$ beam duty cycle, the combination of the betabeam antineutrino data with the NO$\nu$A Phase I neutrino data, provides better sensitivity than the NO$\nu$A experiment alone (upgraded by a factor of five and running in both neutrino and antineutrino mode). If a smaller duty factor $<10^{-2}$ could be achievable (as commonly assumed, following Ref. [24]) the sensitivity to the mass hierarchy would lie within the limits illustrated in Figure 4 (the most optimistic case with no atmospheric neutrino induced background) and the limits depicted in Figure 5 (the most pessimistic case with atmpospheric neutrino backgrounds and a beam duty cycle $\sim 10^{-2}$). <figure><img src="content_image/0711.1075/x6.png"><figcaption>Figure 4: (a) The 90% CL (2 d.o.f) hierarchy resolution curves for differentexposures for the 6He betabeam ¯νe fluxes at 810 km, combined with five yearsof neutrino data only from the NOνA far detector, located 12 km off-axis at810 km. Only backgrounds in the NOνA experiment have been included. The blue(red) curves assume normal (inverted) hierarchy. The solid (dotted) linedepicts the results for 2×1020 (5×1019) useful ion decays times kton. The blue(red) dashed curve shows the sensitivity reach at 90% CL (2 d.o.f.) from thecombination of neutrino and antineutrino data from the NOνA experiment,assuming five years running in both neutrinos and antineutrinos with a factorof five increase in statistics and normal (inverted) hierarchy. (b) The 90% CL(2 d.o.f) hierarchy resolution curves for different exposures for the 8Libetabeam electron antineutrino fluxes at 300 km, combined with muon neutrinodata only from the NOνA far detector at 12 km off-axis at 810 km. Again, onlybackgrounds in the NOνA experiment have been included.The blue (red) curvesassume normal (inverted) hierarchy. The solid (dotted) line depicts theresults for 5×1021 (1021) useful ion decays times kton.</figcaption></figure> <figure><img src="content_image/0711.1075/x8.png"><figcaption>Figure 5: Same as Figs. 4 but including backgrounds in the betabeam electronantineutrino data, see text for details.</figcaption></figure> ## VI Conclusions We have explored an alternative strategy for measuring the neutrino mass hierarchy. Unlike the approach followed by future long baseline neutrino oscillation experiments that combine the neutrino–antineutrino data, the combination of the CPT conjugated channels that we study here provides an almost degeneracy free determination of the neutrino mass hierarchy, provided the two channels have similar $\langle E\rangle/L$. Future neutrino facilities at Fermilab could provide these CPT neutrino–conjugated channels. The NO$\nu$A $\nu_{e}$ off-axis appearance experiment could provide the $\nu_{\mu}\to\nu_{e}$ channel. A future betabeam facility based at Fermilab could provide the CPT-conjugated $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ channel. A realistic estimate of the expected electron antineutrino fluxes from boosted ion decays is presented. We propose two possible accelerator scenarios for generating the betabeam electron antineutrino fluxes: the Tevatron, which could accelerate ${}^{6}$He ions, and the Main Injector, which could accelerate ${}^{8}$Li ions. In the case of the Tevatron, the decay ring would be very large and possibly prohibitively expensive. The first scenario could benefit from the NO$\nu$A far detector at $L=810$ km, (but the decay ring needed would be very large) ; for the second scenario, an additional, although smaller $2-10$ kton MINOS like detector at a shorter baseline, $L=300$ km, would be necessary (the decay ring needed in this case would be smaller, though). In the more pessimistic case, with a modest beam duty cycle of $10^{-2}$ and including realistic atmospheric neutrino backgrounds, the neutrino mass hierarchy could be determined for $\sin^{2}2\theta_{13}>0.01$, independently of the value of the CP violating phase $\delta$, for both accelerator possibilities. These two alternative choices could improved by an order of magnitude the sensitivity to the neutrino mass hierarchy obtained by a future NO$\nu$A upgraded experiment exploiting both neutrinos and antineutrinos. ## Acknowledgments We wish to thank A. Donini for useful comments on the manuscript. OM is supported by the European Programme “The Quest for Unification” contract MRTN-CT-2004-503369. Fermilab is operated by FRA under DOE contract DE-AC02-07CH11359. OM would like to thank the Theoretical Physics Department at Fermilab for hospitality and support. ## Appendix A CPT Conjugate Probabilities: The amplitudes for $\nu_{\mu}\to\nu_{e}$ and $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ consists of two terms, one associated with the atmospheric $\delta m^{2}$ scale and the other associated with the solar $\delta m^{2}$ scale. Thus, the probability for these the CPT-conjugate processes contain three terms; the square of each of the amplitudes plus the interference term between the two amplitudes which depends on the CP phase $\delta$. For the normal (upper sign) and inverted (lower sign) hierarchy, the $\nu_{\mu}\to\nu_{e}$ and $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ appearance probabilities are given by \[P(\nu_{\mu}\to\nu_{e}) = X_{\pm}\theta^{2}\pm 2\sqrt{X_{\pm}}\sqrt{P_{\odot}}~{}\theta \cos(\pm\Delta_{31}+\delta)+P_{\odot}\] \[\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu}) = X_{\mp}\theta^{2}\pm 2\sqrt{X_{\mp}}\sqrt{P_{\odot}}~{}\theta \cos(\pm\Delta_{31}+\delta)+P_{\odot}.\] (4) The coefficients $P_{\odot}$ and $X_{\pm}$ are simply \[\sqrt{P_{\odot}} = \cos\theta_{23}\sin 2\theta_{12}\frac{\sin(aL)}{(aL)}~{}\Delta_{2 1},\] \[\sqrt{X_{\pm}} = 2\sin\theta_{23}\frac{\sin(\pm\Delta_{31}-aL)}{(\pm\Delta_{31}- aL)}~{}\Delta_{31},\] where $\Delta_{ij}=\|\delta m^{2}_{ij}\|L/4E$ and $a=G_{F}N_{e}/\sqrt{2}\approx(4000~{}km)^{-1}$. The atmospheric amplitude for $\nu_{\mu}\to\nu_{e}$ is $\pm\sqrt{X_{\pm}}\theta$ whereas the solar amplitude is $\sqrt{P_{\odot}}$ and the relative phase between these two amplitudes⁵ is $(\pm\Delta_{31}+\delta)$. In vacuum, $X_{+}=X_{-}\equiv X_{0}$ and the two probabilities are identical, as they must since they are CPT conjugates. [FOOTNOTE:5][ENDFOOTNOTE] The other related CPT conjugate pair of appearance probabilities, $P(\nu_{e}\to\nu_{\mu})$ and $\overline{P}(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$, can be obtained from the above by changing the sign of $\delta$, as follows \[P(\nu_{e}\to\nu_{\mu}) = X_{\pm}\theta^{2}\pm 2\sqrt{X_{\pm}}\sqrt{P_{\odot}}~{}\theta \cos(\pm\Delta_{31}-\delta)+P_{\odot}\] \[\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu}) = X_{\mp}\theta^{2}\pm 2\sqrt{X_{\mp}}\sqrt{P_{\odot}}~{}\theta \cos(\pm\Delta_{31}-\delta)+P_{\odot}.\] (5) The difference between the first two CPT conjugate appearance probabilities, is given by \[P(\nu_{\mu}\to\nu_{e})-\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{ \mu}) = \pm\theta~{}(\sqrt{X_{+}}-\sqrt{X_{-}})~{}\left[(\sqrt{X_{+}}+ \sqrt{X_{-}})\theta\pm 2\sqrt{P_{\odot}}\cos(\pm\Delta_{13}+\delta)\right].\] This quantity is positive for the normal hierarchy (NH) and negative for the inverted hierarchy (IH), if \[\sqrt{X_{+}} > \sqrt{X_{-}}\quad{\rm and}\quad\theta>2\sqrt{P_{\odot}}/(\sqrt{X_ {+}}+\sqrt{X_{-}})\approx\sqrt{P_{\odot}}/\sqrt{X_{0}},\] (6) for all values of the CP phase $\delta$. The constraint on $\theta$ requires⁶ [FOOTNOTE:6][ENDFOOTNOTE] \[\sin^{2}2\theta_{13}>\frac{\sin^{2}2\theta_{12}\Delta^{2}_{21}}{ \tan^{2}\theta_{23}\sin^{2}\Delta_{31}}\sim 0.001-0.002,\] (7) whereas the constraint, $\sqrt{X_{+}}>\sqrt{X_{-}}$, is satisfied near the first oscillation maximum provided $(aL)\ll 1$, i.e. $L\ll 4000~{}km$. With these rather weak constraints then \[P(\nu_{\mu}\to\nu_{e}) > \overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu})\quad{\rm for~{}NH}\] (8) \[{\rm and}\quad P(\nu_{\mu}\to\nu_{e}) < \overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu})\quad{\rm for~{}IH}\] (9) for all values of the CP phase $\delta$. For the normal (inverted) hierarchy, the matter effect enhances (suppresses) the $P(\nu_{\mu}\to\nu_{e})$ channel and suppresses (enhances) the $\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu})$ channel, thus the matter effect in a sense is used twice. Of course, the difference between these two appearance probabilities is larger at larger values of $\theta$ and at larger values of the matter effect. This is the effect that is exploited here to determine the neutrino mass hierarchy. In the $P(\nu_{\mu}\to\nu_{e})$ versus $\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu})$ plane the trajectory for fixed value of $\theta$ as the CP phase $\delta$ is varied from 0 to $2\pi$ is in general an ellipse which collapses to a line if the $E/L$ of both channels is the same. The centre of this ellipse is given by \[(\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu}),P(\nu_{\mu}\to\nu_ {e}))=(X_{\mp}\theta^{2}+P_{\odot},~{}X_{\pm}\theta^{2}+P_{\odot}).\] (10) Thus, as $\theta$ is varied, the centre of the ellipses form lines with slope given by \[\alpha_{+} \equiv \frac{X_{-}}{X_{+}}\quad{\rm for~{}NH}\] \[{\rm and}\quad\alpha_{-} \equiv \frac{X_{+}}{X_{-}}\quad{\rm for~{}IH}.\] (11) If the matter effect is small, $(aL)\ll\Delta_{31}$, one can perform a Taylor series about the vacuum such that \[\alpha_{\pm}=1\mp 4(aL)[\Delta_{31}^{-1}-\cot\Delta_{31}]+{\cal O }(aL)^{2}.\] (12) It is the difference in the slopes of the two lines (for the normal hierarchy, $\alpha_{+}$ and for the inverted hierarchy, $\alpha_{-}$) which provides the separation between the allowed regions for two hierarchies in the $P(\nu_{\mu}\to\nu_{e})$ versus $\overline{P}(\bar{\nu}_{e}\to\bar{\nu}_{\mu})$ plane. ## References * (1) B. T. Cleveland _et al._, Astrophys. J. 496, 505 (1998); Y. Fukuda _et al._ [Kamiokande Collaboration], Phys. Rev. Lett. 77, 1683 (1996); J. N. Abdurashitov _et al._ [SAGE Collaboration], J. Exp. Theor. Phys. 95, 181 (2002); W. Hampel _et al._ [GALLEX Collaboration], Phys. Lett. B 447, 127 (1999); T. A. Kirsten [GNO Collaboration], Nucl. Phys. Proc. Suppl. 118, 33 (2003). * (2) S. Fukuda _et al._ [Super-Kamiokande Collaboration], Phys. Lett. B 539, 179 (2002). * (3) Q. R. Ahmad _et al._ [SNO Collaboration], Phys. Rev. Lett. 87, 071301 (2001). * (4) Q. R. Ahmad _et al._ [SNO Collaboration], Phys. Rev. Lett. 89, 011301 (2002) and _ibid._89, 011302 (2002). * (5) S. N. Ahmed _et al._ [SNO Collaboration], Phys. Rev. Lett. 92, 181301 (2004). * (6) B. Aharmim _et al._ [SNO Collaboration], Phys. Rev. C 72, 055502 (2005). * (7) Y. Ashie _et al._ [Super-Kamiokande Collaboration], Phys. Rev. D 71, 112005 (2005). * (8) K. Eguchi _et al._ [KamLAND Collaboration], Phys. Rev. Lett. 90, 021802 (2003). * (9) M. H. Ahn _et al._ [K2K Collaboration], Phys. Rev. D 74, 072003 (2006). * (10) E. Ables _et al._ [MINOS Collaboration], FERMILAB-PROPOSAL-0875. * (11) D. G. Michael _et al._ [MINOS Collaboration], Phys. Rev. Lett. 97, 191801 (2006). * (12) B. Pontecorvo, Sov. Phys. JETP 6, 429 (1957) [Zh. Eksp. Teor. Fiz. 33, 549 (1957)] and _ibid._7, 172 (1958) [_ibid._34 (1958) 247]; Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962). * (13) M. Apollonio _et al._ [CHOOZ Collaboration], Phys. Lett. B 466, 415 (1999). * (14) F. Boehm _et al._, Phys. Rev. Lett. 84, 3764 (2000) and Phys. Rev. D 62, 072002 (2000). * (15) M. C. Gonzalez-Garcia and M. Maltoni, arXiv:0704.1800 [hep-ph]. * (16) T. Araki _et al._ [KamLAND Collaboration], Phys. Rev. Lett. 94, 081801 (2005). * (17) L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); V. D. Barger, K. Whisnant, S. Pakvasa and R. J. N. Phillips, Phys. Rev. D 22, 2718 (1980); S. P. Mikheev and A. Y. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985); S. J. Parke, Phys. Rev. Lett. 57, 1275 (1986); H. W. Zaglauer and K. H. Schwarzer, Z. Phys. C 40, 273 (1988); M. C. Banuls, G. Barenboim and J. Bernabeu, Phys. Lett. B 513, 391 (2001); M. Freund _et al._, Nucl. Phys. B 578, 27 (2000). * (18) J. Arafune, M. Koike and J. Sato, Phys. Rev. D 56, 3093 (1997) [Erratum-ibid. D 60, 119905 (1999)]; H. Minakata and H. Nunokawa, Phys. Lett. B 413, 369 (1997); A. Donini _et al._, Nucl. Phys. B 574, 23 (2000). * (19) H. Minakata, H. Nunokawa and S. J. Parke, Phys. Rev. D 68, 013010 (2003). * (20) A. de Bellefon _et al._, arXiv:hep-ex/0607026; J. E. Campagne, M. Maltoni, M. Mezzetto and T. Schwetz, JHEP 0704, 003 (2007); T. Schwetz, JHEP 0705, 093 (2007). * (21) D. S. Ayres _et al._ [NOvA Collaboration], hep-ex/0503053. FERMILAB-PROPOSAL-0929, March 21, 2005. Revised NO$\nu$A Proposal available at http://www-nova.fnal.gov/NOvA_Proposal/Revised_NOvA_Proposal.html * (22) A. Para and M. Szleper, hep-ex/0110032. * (23) P. Zucchelli, Phys. Lett. B 532, 166 (2002). * (24) M. Mezzetto, J. Phys. G 29, 1771 (2003). * (25) J. Burguet-Castell _et al._, Nucl. Phys. B 695, 217 (2004); * (26) C. H. Albright _et al._, arXiv:physics/0411123; J. Burguet-Castell _et al._, Nucl. Phys. B 725, 306 (2005). P. Huber _et al._, Phys. Rev. D 73, 053002 (2006); A. Donini and E. Fernandez-Martinez, Phys. Lett. B 641, 432 (2006); A. Donini _et al._, Eur. Phys. J. C 48, 787 (2006); A. Blondel _et al._, Acta Phys. Polon. B 37, 2077 (2006); S. K. Agarwalla, S. Choubey and A. Raychaudhuri, Nucl. Phys. B 771, 1 (2007); S. K. Agarwalla _et al._, Phys. Rev. D 75, 097302 (2007); A. Donini _et al._, arXiv:hep-ph/0703209. * (27) ISS Working Group, arXiv:0710.4947v1 [hep-ph]. * (28) H. Minakata and H. Nunokawa, JHEP 0110, 001 (2001). * (29) O. Mena, H. Nunokawa and S. J. Parke, Phys. Rev. D 75, 033002 (2007) * (30) O. Mena, arXiv:hep-ph/0609031. * (31) P. Huber, M. Lindner and W. Winter, Nucl. Phys. B 654, 3 (2003). * (32) V. Barger, D. Marfatia and K. Whisnant, Phys. Lett. B 560, 75 (2003). * (33) O. Mena Requejo, S. Palomares-Ruiz and S. Pascoli, Phys. Rev. D 72, 053002 (2005). * (34) O. Mena, S. Palomares-Ruiz and S. Pascoli, Phys. Rev. D 73, 073007 (2006). * (35) M. Ishitsuka _et al._, Phys. Rev. D 72, 033003 (2005); K. Hagiwara, N. Okamura and K. i. Senda, Phys. Lett. B 637, 266 (2006). * (36) Y. Hayato _et al._, Letter of Intent, available at http://neutrino.kek.jp/jhfnu/ * (37)http://cern.ch/beta-beam * (38) C. Rubbia _et al._, Nucl. Instrum. Meth. A 568, 475 (2006). [arXiv:hep-ph/0602032]. * (39) A. Blake, private communication.
1509.03092	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 26289, "num_imgs": 4, "llama3_tokens_count": 8832 }	[ "content_image/1509.03092/Y.jpg", "content_image/1509.03092/example-1.jpg", "content_image/1509.03092/pic.jpg", "content_image/1509.03092/pic2.jpg" ]	# On edge-decomposition of cubic graphs into copies of the double-star with four edges † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] S. Akbari${}^{\mathsf{a},\mathsf{c}}$, H. R. Maimani${}^{\mathsf{b},\mathsf{c}}$ and A. Seify${}^{\mathsf{b},\mathsf{c}}$, ${}^{\mathsf{a}}$_Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran_ ${}^{\mathsf{b}}$_Department of Science, Shahid Rajaee Teacher Training University, Tehran, Iran_ ${}^{\mathsf{c}}$_School of Mathematics, Institute for Research in Fundamental Sciences (IPM), ${}^{\mathsf{}}$P.O. Box 19395-5746, Tehran, Iran._ _E-mail addresses_: $\mathsf{s\_akbari@sharif.edu}$, $\mathsf{maimani@ipm.ir}$ and $\mathsf{abbas.seify@gmail.com}$. ###### Abstract A tree containing exactly two non-pendant vertices is called a double-star. Let $k_{1}$ and $k_{2}$ be two positive integers. The double-star with degree sequence $(k_{1}+1,k_{2}+1,1,\ldots,1)$ is denoted by $S_{k_{1},k_{2}}$. If $G$ is a cubic graph and has an $S$-decomposition, for a double-star $S$, then $S$ is isomorphic to $S_{1,1}$, $S_{1,2}$ or $S_{2,2}$. It is known that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching. In this paper, we study the $S_{1,2}$-decomposition of cubic graphs. First, we present some necessary conditions for the existence of an $S_{1,2}$-decomposition in cubic graphs. Then we prove that every $\{C_{3},C_{5},C_{7}\}$-free cubic graph of order $n$ with $\alpha(G)=\frac{3n}{8}$ has an $S_{1,2}$-decomposition, where $\alpha(G)$ denotes the independence number of $G$. Finally, we obtain some results on the $S_{1,r-1}$-decomposition of $r$-regular graphs. ## 1 Introduction Let $G=(V(G),E(G))$ be a graph and $v\in V(G)$. We denote the set of all neighbors of $v$ by $N(v)$ and for $X\subseteq V(G)$ we define $N(X)=\cup_{x\in X}N(x)$. Also, we denote the neighbors of $X$ in $S$ by $N_{S}(X)=N(X)\cap S$. An _independent set_ is a set of vertices in a graph such that no two of which are adjacent. The _independence number_$\alpha(G)$ is the size of a largest independent set in $G$. A _dominating set_ of $G$ is a subset $D$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The _domination number_$\gamma(G)$ is the size of a smallest dominating set in $G$. A subset $S\subseteq V(G)$ in which all components of $G\setminus S$ are cycles is called a _cycling set_. Moreover, if $S$ is an independent set, then we say that $S$ is an _independent cycling set_. We denote the number of path components of $G$ by $n(P,G)$. A subset $M\subseteq E(G)$ is called a _matching_, if no two edges of $M$ are incident. A matching $M$ is called a _perfect matching_, if every vertex of $G$ is incident with some edge in $M$. Hall proved that a bipartite graph $G=(A,B)$ has a matching saturates $A$ if and only if for every $S\subseteq A$ we have $\|N_{B}(S)\|\geq\|S\|$, see [1] and [3]. A graph $G$ has an $H$-_decomposition_, if all edges of $G$ can be decomposed into subgraphs isomorphic to $H$. If $G$ has an $H$-decomposition, then we say that $G$ is $H$-_decomposable_. A tree with exactly two non-pendant vertices is called a double-star. Let $k_{1}$ and $k_{2}$ be two positive integers. The double-star with degree sequence $(k_{1}+1,k_{2}+1,1,\ldots,1)$ is denoted by $S_{k_{1},k_{2}}$. A vertex of degree $i$ is called an $i$-_vertex_. If $G$ is an $r$-regular graph with an $S_{1,r-1}$-decomposition and $S\subseteq V(G)$ is the set of all $r$-vertices of this decomposition, then we say that $G$ is $(S_{1,r-1},S)$-decomposable. <figure><img src="content_image/1509.03092/Y.jpg"><figcaption> Figure1. S1,2.</figcaption></figure> Tree decomposition of highly connected graphs is studied in [2], [6] and [7]. In [2] it has been shown that every 191-edge-connected graph, whose size is divisible by 4 has an $S_{1,2}$-decomposition. Let $G$ be a cubic graph. If $G$ is $S$-decomposable and $S$ is a double-star, then $S$ is isomorphic to $S_{1,1}$ or $S_{1,2}$ or $S_{2,2}$, because otherwise $S$ has a vertex of degree at least four. It was proved that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching, see [5]. In this paper, we study edge-decomposition of cubic graphs into copies of $S_{1,2}$. This paper is organized as follows. In Section 2, we study $S_{1,2}$-decomposition of cubic graphs and provide some necessary and some sufficient conditions for the existence of an $S_{1,2}$-decomposition in cubic graphs. In Section 3, we obtain some results on $S_{1,r-1}$-decomposition of $r$-regular graphs. ## 2 $S_{1,2}$-Decomposition of Cubic Graphs In this section, we present some necessary and some sufficient conditions for the existence of $S_{1,2}$-decompositions in cubic graphs. Finally, we study $R$-decomposition in cubic graphs. Let $G$ be a cubic graph and $S\subseteq V(G)$. The question is that whether $G$ is $(S_{1,2},S)$-decomposable or not? For giving a response to this question, we need a new bipartite graph $H=(S,L)$, in which $S$ is the set of all 3-vertices of $S_{1,2}$-trees and for each edge $e\in E(G\setminus S)$, we put a vertex $u_{e}$ in $L$. Two vertices $s_{i}$ and $u_{e_{j}}$ are adjacent in $H$ if and only if there exists an edge $e\in E(G)$ such that one end of $e$ is $s_{i}$ and moreover $e$ and $e_{j}$ have a common end vertex. This means that $u_{e_{j}}$ and $s_{i}$ are adjacent in $H$ if and only if we can obtain an $S_{1,2}$ by adding $e_{j}$ to a claw containing $s_{i}$ as a central vertex. We have the following. Lemma 1.: _Let $G$ be a cubic graph of order $n$. Then $G$ is $(S_{1,2},S)$-decomposable if and only if $\|S\|=\frac{3n}{8}$ and $H=(S,L)$ has a perfect matching._ Proof.: Clearly, if $G$ is $(S_{1,2},S)$-decomposable, then $\|S\|=\frac{3n}{8}$. Also, note that if $e_{s}=uv$ is an edge in some $S_{1,2}$ with $s$ as a 3-vertex and moreover $u$ and $v$ are 2-vertex and 1-vertex of this double-star, respectively. Then $e_{s}\in L$ and $\{(s,e_{s}):s\in S\}$ is a perfect matching in $H$. Conversely, if there exists a perfect matching $M=\{(s,e_{s}):s\in S\}$, then one can obtain an $S_{1,2}$-decomposition. ∎ In the following lemma we provide some necessary conditions for $S_{1,2}$-decomposition of cubic graphs. Lemma 2.: _Let $G$ be a cubic graph of order $n$ which has a $S_{1,2}$-decomposition. Then the following hold: (i) $8\;\|\;n$. (ii) There exists an independent set $S\subset V(G)$ such that:_ _1- $\|S\|\geq\frac{3n}{8}$,_ _2- Each component of $G\setminus S$ is either a cycle or a tree,_ _3- No component of $G\setminus S$ has two $3$-vertices. (iii) There exists an independent set $T\subseteq V(G\setminus S)$ with $\|T\|=\frac{n}{4}$._ Proof.: Let $G$ be $(S_{1,2},S)$-decomposable. Then from the preceding lemma, (i) and the first part of (ii) are clear. Suppose that $F$ is a given component of $G\setminus S$. If $F$ is neither a tree nor a cycle, then it has a cycle like $C:v_{1},e_{1},v_{2},e_{2},\ldots,v_{t},e_{t},v_{1}$ and an edge $e=v_{i}w$, where $1\leq i\leq t$ and $w\in V(F)$. Consider the set $A=E(C)$. Since $G$ is cubic, each vertex in cycle $C$ has at most one neighbor in $S$ and $v_{i}$ has no neighbor in $S$. Hence $\|N_{H}(A)\|\leq\|A\|-1$, which contradicts Hall’s condition and so by Lemma 1$G$ has no $(S_{1,2},S)$-decomposition, a contradiction. If there exist two 3-vertices $u$ and $v$ in some component $F$, then there exists a $(u,v)$-path $P:u=v_{1},e_{1},\ldots,e_{t},v_{t}=v$ in $F$. Now, let $A=E(P)$. Similar to the proof of the previous part, one can show that $\|N_{H}(A)\|\leq\|A\|-1$, which contradicts Hall’s condition and so by Lemma 1$G$ has no $(S_{1,2},S)$-decomposition, a contradiction. For (iii), let $T$ be the set of vertices which are only used as a pendant vertex in $S_{1,2}$-trees in the given decomposition. It is easy to see that $T$ is an independent set and $\|T\|=\frac{n}{4}$. Now, the proof is complete. ∎ These necessary conditions are not sufficient. Some examples are given as follows. <figure><img src="content_image/1509.03092/example-1.jpg"><figcaption> Figure 2.</figcaption></figure> Now, we provide some sufficient conditions for the existence of $S_{1,2}$-decomposition in cubic graphs and in the next section we will generalize them for the $S_{1,r-1}$-decomposition of $r$-regular graphs. By Lemma 2, if $G$ is an $S_{1,2}$-decomposable cubic graph, then $\alpha(G)\geq\frac{3n}{8}$. We consider the case $\alpha(G)=\frac{3n}{8}$ and find two sufficient conditions for the existence of an $S_{1,2}$-decomposition in this case. Theorem 1.: _Let $G$ be a cubic graph of order $n$ with $\alpha(G)=\frac{3n}{8}$. Suppose that there exists an independent cycling set $S\subseteq V(G)$ such that $\|S\|=\frac{3n}{8}$ and moreover, no vertex of $S$ is contained in a triangle. Then $G$ is $(S_{1,2},S)$-decomposable._ Proof.: Suppose that $C_{i}$ ($1\leq i\leq t$) are cycle components of $G\setminus S$. We claim that there are no two non-isolated vertices in $G\setminus S$ which have the same neighbor in $S$. Consider two non-isolated vertices $u$ and $v$ in $G\setminus S$. If $u$ and $v$ are adjacent, then since no vertex of $S$ is contained in a triangle, we are done. Now, suppose that $u$ and $v$ are not adjacent. If $N_{S}(u)=N_{S}(v)=\{s\}$, then $S^{\prime}=(S\setminus\{s\})\cup\{u,v\}$ is an independent set and $\|S^{\prime}\|>\frac{3n}{8}$, a contradiction. Now, for each component $C:v_{1},e_{1},\ldots,v_{t},e_{t},v_{1}$ of $G\setminus S$ there exist distinct vertices $s_{i_{1}},\ldots,s_{i_{t}}$ in which $v_{k}$ is adjacent to $s_{i_{k}}$ in $S$. By adding $e_{j}$ to a claw containing $s_{i_{j}}$ as a central vertex we obtain an $S_{1,2}$-decomposition. ∎ As a special case, we have the following result. Corollary 1.: _Let $G$ be a triangle-free cubic graph with $\alpha(G)=\frac{3n}{8}$ and there exists an independent cycling set $S\subseteq V(G)$ such that $\|S\|=\frac{3n}{8}$. Then $G$ is $(S_{1,2},S)$-decomposable._ Another interesting result in the case of $\alpha(G)=\frac{3n}{8}$ is as follows. Theorem 2.: _Let $G$ be a cubic graph of order $n$ with $\alpha(G)=\frac{3n}{8}$ and moreover there exists an independent set $S\subseteq V(G)$ such that $\|S\|=\frac{3n}{8}$ and no vertex of $S$ is contained in a triangle, $C_{5}$ or $C_{7}$. Then $G$ is $(S_{1,2},S)$-decomposable._ Proof.: We divide the proof into four claims. Claim 1. Each component of $G\setminus S$ is a path or a cycle. If there exists a vertex of degree 3 in $G\setminus S$, then by adding this vertex to $S$ we obtain an independent set $S^{\prime}$ such that $\|S^{\prime}\|=\frac{3n}{8}+1$, a contradiction. Claim 2. Let $u,v\in V(G\setminus S)$ be two vertices of degree two in $G\setminus S$. Then $N_{S}(v)\neq N_{S}(u)$. Let $u$ and $v$ be two vertices in $G\setminus S$ such that $d_{G\setminus S}(u)=d_{G\setminus S}(v)=2$ and $N_{S}(u)=N_{S}(v)=\{s\}$. If $u$ and $v$ are adjacent, then $s$ is contained in a triangle, a contradiction. Also, if $u$ and $v$ are not adjacent, then $S^{\prime}=(S\setminus\{s\})\cup\{u,v\}$ is an independent set and $\|S^{\prime}\|=\frac{3n}{8}+1$, a contradiction. So, the claim is proved. Now, we check the Hall’s condition for the edges of $G\setminus S$. Suppose that $L=\{e_{1},\ldots,e_{l}\}\subseteq E(G\setminus S)$. Let $P_{1},\ldots,P_{k}$ be all path components of $G\setminus S$. Now, we consider two cases: Case 1. No $P_{i}$ is contained in $\langle L\rangle$. Note that for each edge $e\in L$, one of its endpoints has degree 2 in $G\setminus S$. Because if both endpoints are of degree 1 in $G\setminus S$, then the induced subgraph on this edge is a path component of $G\setminus S$. Now, we show that for each edge $e_{i}\in L$, one can find $v_{e_{i}}\in V(G\setminus S)$ such that $d_{G\setminus S}(v_{i})=2$, $v_{i}$ is an endpoint of $e_{i}$ and if $i\neq j$, then $v_{e_{i}}\neq v_{e_{j}}$. For $e_{1}$ define $v_{e_{1}}$ one of the its endpoints whose degree is 2. If $v_{e_{1}}$ is not one of the endpoints of $e_{i}$, $2\leq i\leq l$, then define $v_{e_{2}}$ as one of its endpoints which has degree 2 in $G\setminus S$. Otherwise, suppose that $v_{e_{1}}$ is one of the endpoints of $e_{j}=\{v_{e_{1}},u\}$. If $d_{G\setminus S}(u)=1$, then $e_{i}$ and $e_{j}$ induce a $P_{3}$-component in $G\setminus S$, a contradiction. Hence, $d_{G\setminus S}(u)=2$ and define $v_{e_{j}}=u$. By repeating this procedure for each edge $e\in L$ one can find $\{v_{e_{1}},\ldots,v_{e_{l}}\}$. Now, Claim 2 implies that for each $e\in L$ there exists a distinct vertex in $S$ which is adjacent to $v_{e}$ and so in this case Hall’s condition holds. Case 2. There exist $i_{1},\ldots,i_{t}$ such that $1\leq i_{j}\leq k$ and $P_{i_{1}},\ldots,P_{i_{t}}$ are all path components of $\langle L\rangle$. We have the following. Claim 3. Let $v\in V(G\setminus S)$ such that $d_{G\setminus S}(v)=1$ and $N_{S}(v)=\{x,y\}$. Then both $x$ and $y$ are not adjacent to the vertices of degree two in $G\setminus S$. Let $x$ and $y$ be adjacent to $v_{x}$ and $v_{y}$ in $G\setminus S$, respectively, and $d_{G\setminus S}(v_{x})=d_{G\setminus S}(v_{y})=2$. Note that $v$ is not adjacent to $v_{x}$ and $v_{y}$, since otherwise there exists a triangle containing $v$, a contradiction. Now, if $v_{x}$ and $v_{y}$ are adjacent, then $C:v,x,v_{x},v_{y},y,v$ is a cycle of length 5, a contradiction. If $v_{x}$ and $v_{y}$ are not adjacent, then define $S^{\prime}=(S\setminus\{x,y\})\cup\{v,v_{x},v_{y}\}$. It can be easily seen that $S^{\prime}$ is an independent set and $\|S^{\prime}\|=\frac{3n}{8}+1$, a contradiction. Now, we can prove that in the second case, $L$ satisfies the Hall condition. It suffices to show that the edges of $P_{i_{1}},\ldots,P_{i_{t}}$ satisfy Hall’s condition. Because, similar to the proof of the first case, one can see that other edges have distinct neighbors in $S$ and we are done. Now, Claim 2 implies that we can find $\sum_{j=1}^{t}(\|E(P_{i_{j}})\|-1)$ vertices in $S$ which are adjacent to the vertices of degree 2 in the path components. Let $T\subseteq S$ be the set of vertices in $S$ which are adjacent to the end vertices of $P_{i_{1}}\ldots,P_{i_{t}}$ and they are adjacent to no vertex of degree 2 in $G\setminus S$. It suffices to show that $\|T\|\geq t$. By contrary, suppose that $\|T\|\leq t-1$. Then Claim 3 implies that each end vertex of paths has a neighbor in $T$. Let $A$ be the set of end vertices of paths that have one neighbor in $T$ and let $B$ be the set of end vertices which have two neighbors in $T$. We have the following. $\|A\|+\|B\|=2t\;\;,\;\;\|A\|+2\|B\|\leq 3t-3.$ Hence, we conclude that $\|A\|\geq t+3$. Now, we prove the following claim. Claim 4. If $u,v\in A$, then $N_{T}(u)\cap N_{T}(v)=\emptyset$. First, note that if $u$ and $v$ are adjacent, then we are done. So, we may assume that $u$ and $v$ are not adjacent. Let $N_{T}(u)=N_{T}(v)=\{w\}$. Suppose that $N_{S}(u)=\{w,x\}$ and $N_{S}(v)=\{w,y\}$. By the definition of $T$, we conclude that $x$ and $y$ are adjacent to some vertices of degree 2 in $G\setminus S$ say $v_{x}$ and $v_{y}$, respectively. Notice that if $x=y$ and $\{u,v,v_{x}\}$ is not independent set, then one can find a triangle contains a vertex of $S$, a contradiction. Thus, $\{u,v,v_{x}\}$ is an independent set. Now, $S^{\prime}=(S\setminus\{u,x\})\cup\{u,v,v_{x}\}$ is an independent set of size $\frac{3n}{8}+1$, a contradiction. Hence, $x\neq y$. We show that $\{u,v,v_{x},v_{y}\}$ is an independent set. Since no vertex of $S$ is contained in a triangle, $u$ and $v_{x}$ are not adjacent (similarly, $v$ and $v_{y}$ are not adjacent). So, suppose that $v$ and $v_{x}$ are adjacent. Then $C:u,x,v_{x},v,w,u$ is a cycle of length 5 which contains vertices of $S$, a contradiction. Also, note that $v_{x}$ and $v_{y}$ are not adjacent. Since, otherwise $C:u,x,v_{x},v_{y},y,v,w,u$ is a cycle of length 7, a contradiction. This implies that $\{u,v,v_{x},v_{y}\}$ is an independent set. Now, let $S^{\prime}=(S\setminus\{x,y,w\})\cup\{u,v,v_{x},v_{y}\}$. Then $S^{\prime}$ is an independent set and $\|S^{\prime}\|>\frac{3n}{8}$, contradiction and this completes the proof of the claim. Now, Claim 4 implies that for every $v\in A$ we have a distinct neighbor $t_{v}\in T$ and this implies that $\|T\|\geq t+3$, a contradiction. This completes the proof. ∎ Now, we have an immedaite corollary. Corollary 2.: _Let $G$ be a $\{C_{3},C_{5},C_{7}\}$-free cubic graph of order $n$ with $\alpha(G)=\frac{3n}{8}$. Then $G$ has an $S_{1,2}$-decomposition._ Now, we obtain another sufficient condition for the existence of an $S_{1,2}$-decomposition in a cubic graph. If $G$ is a cubic graph, then $\gamma(G)\geq\frac{n}{4}$. In the following theorem, we provide a necessary and sufficient condition on the existence of a $S_{1,2}$-decomposition for cubic bipartite graph $G$ under which $\gamma(G)=\frac{n}{4}$. Theorem 3.: _Let $G=(A,B)$ be a cubic bipartite graph of order $n$ such that $8\|n$. Then $\gamma(G)=\frac{n}{4}$ if and only if there exists $S\subseteq A$ of size $\frac{3n}{8}$ such that $G$ is both $(S_{1,2},S)$-decomposable and $(S_{1,2},N(A\setminus S))$-decomposable._ Proof.: Let $D$ be a dominating set of $G$ of size $\frac{n}{4}$. Then vertices of $D$ has no common neighbors in $V(G)\setminus D$. Now, let $D_{1}=D\cap A$ and $D_{2}=D\cap B$ and $\|D_{1}\|=a,\|D_{2}\|=b$. Since $D$ is a dominating set of size $\frac{n}{4}$ we have: $a+b=\frac{n}{4}$ , $3a+b=\frac{n}{2}$. Then $a=b=\frac{n}{8}$. Now, let $S=N(D_{1})$. We show that $G$ has an $(S_{1,2},S)$-decomposition. Clearly, $\|S\|=\frac{3n}{8}$ and $E(G\setminus S)$ is exactly the edges between $D_{2}$ and $N(D_{2})$. Note that if $v\in N(D_{2})$, then $d_{S}(v)=2$. Now, it is not hard to see that the graph $H=(S,L)$, defined in Lemma 1, is a 2-regular bipartite graph and hence it has a perfect matching. So, by Lemma 1, $G$ is $(S_{1,2},S)$-decomposable. Notice that if we consider $T=N(D_{2})$, then similarly $G$ is $(S_{1,2},T)$-decomposable. Since $T=N(A\setminus S)$, this completes the proof of the one side of the theorem. Conversely, suppose that there exists $S\subseteq A$ such that satisfies the conditions. Note that each vertex in $A\setminus S$ is a 3-vertex in $G\setminus S$. Now, Lemma 2 implies that each of them is in a different component of $G\setminus S$ and so they have no common neighbors. By a similar method, one can show that the vertices of $B\setminus N(A\setminus S)$ have no common neighbors. Now, $D=(A\setminus S)\cup(B\setminus N(A\setminus S))$ is a dominating set of size $\frac{n}{4}$ and this completes the proof. ∎ Now, we state the following corollary. Corollary 3.: _Let $G=(A,B)$ be a bipartite cubic graph of order $n$. If $\gamma(G)=\frac{n}{4}$, then $G$ is $S_{1,2}$-decomposable._ As an example of this result we can obtain that $Q_{3}$ is $S_{1,2}$-decomposable, because it is bipartite and $\gamma(Q_{3})=2=\frac{n}{4}$. Now, we provide another sufficient condition for the existence of an $S_{1,2}$-decomposition in bipartite cubic graphs. Theorem 4.: _Let $G=(A,B)$ be a bipartite cubic graph of order $n$ and $S\subseteq A$ be of size $\frac{3n}{8}$. Then $G$ is $(S_{1,2},S)$-decomposable if and only if there exists a perfect matching between $S$ and $N(A\setminus S)$._ Proof.: Necessity. First suppose that $G$ is $(S_{1,2},S)$-decomposable. Then the second part of Lemma 2 indicates that no component of $G\setminus S$ has two 3-vertices. This implies that no two vertices of $A\setminus S$ have a common neighbor in $B$. So, $\|N(A\setminus S)\|=\frac{3n}{8}$. Now, note that if Hall’s condition does not hold for $S$ and $N(A\setminus S)$, then Hall’s condition does not hold in $H=(S,L)$, too. This is a contradiction and this completes the proof of the one side of theorem. Sufficiency. Suppose that there exists a perfect matching between $S$ and $N(A\setminus S)$. Then $\|N(A\setminus S)\|=\frac{3n}{8}$ which implies that no two vertices of $A\setminus S$ have a common neighbor in $B$. For each vertex $v\in N(A\setminus S)$, there exists a unique edge $e_{v}\in E(G\setminus S)$ in which $v$ is one of its end points. Let $M=\{(u_{i},v_{i})\|\;i=1,2,\ldots,\frac{3n}{8}\}$ be a matching between $S$ and $N(A\setminus S)$. Then by adding edge $e_{v_{i}}$ to a claw containing $u_{i}$ as a 3-vertex, one can obtain an $S_{1,2}$-decomposition. ∎ ## 3 $S_{1,r-1}$-Decomposition of $r$-Regular Graphs In this section, we generalize the results of the previous section to $S_{1,r-1}$-decomposition of $r$-regular graphs. Similar to the cubic graphs we find some necessary and some sufficient conditions for the existence of $S_{1,r-1}$-decomposition in $r$-regular graphs. The following holds. Remark 1.: _Let $G$ be an $r$-regular graph of order $n$ which is $S_{1,r-1}$-decomposable. Then $2(r+1)\|rn$ and $\alpha(G)\geq\frac{rn}{2(r+1)}$._ The following theorem is a generalization of Theorem 1. But in this case we do not need the condition $\alpha(G)=\frac{rn}{2(r+1)}$. Theorem 5.: _Let $G$ be an $r$-regular graph ($r\geq 4$) of order $n$ and there exists an independent cycling set $S\subseteq V(G)$ such that $\|S\|=\frac{rn}{2(r+1)}$ and moreover no vertex of $S$ is contained in a triangle. Then $G$ is $(S_{1,r-1},S)$-decomposable._ Proof.: We check Hall’s condition for $H=(S,L)$ defined in Lemma 1. Let $e=uv\in E(G\setminus S)$. Since $G$ is $r$-regular and $S$ is a cycling set, each end of $e$ has exactly $r-2$ neighbors in $S$. No vertex of $S$ is contained in a triangle and this yields that $u$ and $v$ have no common neighbor in $S$. So, we conclude that $\|N_{S}(\{u,v\})\|=2r-4$. Now, let $M=\{e_{1},\ldots,e_{t}\}\subseteq E(G\setminus S)$ and $V_{1}$ be the set of all end points of the edges of $M$. We have: $\|N_{S}(V_{1})\|\geq\frac{(2r-4)t}{r}.$ Now, since $r\geq 4$, $2r-4\geq r$ and this implies that in $H=(S,L)$ we have $\|N_{S}(M)\|\geq\|M\|$, for every $M\subseteq L$ . So, $H=(S,L)$ satisfies the Hall condition and this yields that $G$ is $S_{1,r-1}$-decomposable. ∎ We have the following result in regular bipartite graphs which makes a connection between the domination number and the existence of $S_{1,r-1}$-decomposition. The proof is the same as the case of cubic bipartite graphs and we omit it. Theorem 6.: _Let $G$ be a bipartite $r$-regular graph of order $2n$ such that $r+1\|n$ and $\gamma(G)=\frac{2n}{r+1}$. Then $G$ is $S_{1,r-1}$-decomposable._ ## 4 Questions In Theorem 2, we show that every $\{C_{3},C_{5},C_{7}\}$-free graph of order $n$ with $\alpha=\frac{3n}{8}$ is $S_{1,2}$-decomposable. In [4], it was shown that if $G$ is a planar triangle-free graph with maximum degree at most $3$, then $\alpha(G)\geq\frac{3n}{8}$. There are several examples with $\alpha=\frac{3n}{8}$, containing $C_{5}$ or $C_{7}$ and have an $S_{1,2}$-decomposition in [4]. <figure><img src="content_image/1509.03092/pic.jpg"><figcaption> Figure 3.</figcaption></figure> Figure 3 shows that the conditions of being $\{C_{5},C_{7}\}$-free in Theorem 2 are not necessary. Also, there exists a $\{C_{5},C_{7}\}$-free graph with $\alpha=\frac{3n}{8}$, containing a triangle and has no $S_{1,2}$-decomposition, see Figure 4. <figure><img src="content_image/1509.03092/pic2.jpg"><figcaption> Figure 4.</figcaption></figure> This shows that the condition of being triangle-free in Theorem 2 is necessary. Now, the following question is natural. Question 1. Let $G$ be a triangle-free cubic graph of order $n$ with $\alpha(G)=\frac{3n}{8}$. Is it true that $G$ is $S_{1,2}$-decomposable? Figure 2 and Figure 4 show that being connected and 2-connected are not sufficient. Also, Figure 2 shows that there exists a triangle-free connected cubic graph with no $S_{1,2}$-decomposition. Question 2. Does there exist a triangle-free 2-connected cubic graph of order divisible by 8 which has no $S_{1,2}$-decomposition? Question 3. Does there exist a (triangle-free) 3-connected cubic graph of order divisible by 8 which has no $S_{1,2}$-decomposition? Another open question is as follows. Question 4. Is it true that every bipartite cubic graph of order divisible by 8 is $S_{1,2}$-decomposable? ## References * [1] J. Akiyama and M. Kano, Factors and Factorizations of Graphs, _Springer_, 2011. * [2] J. Barat, D. Gerbner, Edge-decomposition of graphs into copies of a tree with four edges, _The Electronic Jounal of Combinatorics_, 21(1), 2014. * [3] A. Bondy, U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics, _Springer_, 2008. * [4] C. C. Heckman, R. Thomas, Independent sets in triangle-free cubic planar graphs, _J. Combin. Theory, Ser. B_, 96 (2006) 253-275. * [5] A. Kötzig, Aus der Theorie der endlichen regulären Graphen dritten und vierten Grades, _Časopis. Pěst. Mat._ 82 (1957) 76-92. * [6] C. Thomassen, Edge-decompositions of highly connected graphs, _Abh. Math. Semin. Univ. Hamburg_, 18 (2008), 17-26. * [7] C. Thomassen, Decompositions of highly connected graphs into paths of length 3, _J. Graph Theory_, 58 (2008), 286-292.
1708.00224	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 29373, "num_imgs": 8, "llama3_tokens_count": 7484 }	[ "content_image/1708.00224/normal.png", "content_image/1708.00224/test.png", "content_image/1708.00224/pca_mrf.jpg", "content_image/1708.00224/test.png", "content_image/1708.00224/test.png", "content_image/1708.00224/test.png", "content_image/1708.00224/test.png", "content_image/1708.00224/pose.png" ]	# Fast Preprocessing for Robust Face Sketch Synthesis Yibing Song${}^{1}$, Jiawei Zhang${}^{1}$, Linchao Bao${}^{2}$, and Qingxiong Yang${}^{3}$ ${}^{1}$City University of Hong Kong ${}^{2}$Tencent AI Lab ${}^{3}$University of Science and Technology of China ###### Abstract Exemplar-based face sketch synthesis methods usually meet the challenging problem that input photos are captured in different lighting conditions from training photos. The critical step causing the failure is the search of similar patch candidates for an input photo patch. Conventional illumination invariant patch distances are adopted rather than directly relying on pixel intensity difference, but they will fail when local contrast within a patch changes. In this paper, we propose a fast preprocessing method named Bidirectional Luminance Remapping (BLR), which interactively adjust the lighting of training and input photos. Our method can be directly integrated into state-of-the-art exemplar-based methods to improve their robustness with ignorable computational cost¹. [FOOTNOTE:1][ENDFOOTNOTE] ## 1 Introduction Exemplar-based face sketch synthesis has received much attention in recent years ranging from digital entertainment to law enforcement [6, 13, 15, 21, 8, 7, 16]. Typically these methods usually consist of two steps. In the first step, all photos (including a given input photo and all training photos) are divided into local patches, and a K-NN patch search is performed among all training photos for each input photo patch. The second step is to merge the corresponding sketch patches (according to the photo patch search results) into an output sketch image via global optimization [12, 18, 22, 14, 20] or local fusion [10]. However, these methods usually fail when input photos are captured differently from training photos which only contain faces in normal lighting. The critical step causing the failure is the search of similar patch candidates for a given input photo patch. <figure><img src="content_image/1708.00224/normal.png"><figcaption>Figure 1: An example of varying lighting conditions. An input photo in (a) iscaptured in the same condition with training photos. The sketches generated bystate-of-the-art MRF, MWF and SSD methods are in (b)-(d). (e) is a synthesizedinput photo in a different lighting and background condition. (f)-(h) are theresults generated by these methods. Our method can be integrated into existingmethods to improve the output quality as shown in (j)-(l).</figcaption></figure> Most state-of-the-art methods (e.g., MRF [12], MWF [22], and SSD [10]) adopt either $L_{1}$ or $L_{2}$ norm based on pixel luminance differences during photo patch search. They perform well on ideal cases where both input and training photos are captured in the same lighting condition. However, for input photos which are captured in different lighting conditions from training photos, these distance metrics often cause incorrect matchings of photo patches and thus lead to erroneous sketch synthesis. Fig. 1 shows an example. A direct amending to these methods is to replace the metrics of the pixel luminance difference with illumination invariant ones based on gradient (like DoG [18]) or correlation (like NCC [11]). However, illumination invariant patch distances will fail when local contrast within a patch changes. For example, if the background is brighter than facial skin in input photos while the background is darker than facial skin in training photos, the photo patches near face boundaries are difficult to locate training correspondences (e.g., the left ear region in Fig. 4(e)). Meanwhile, illumination invariant methods [2, 17] for face recognition are not suitable for face sketch synthesis. They only focus on face region where hair and background are not included. To enable similar statistics of the face and non-face regions between input and training photos, we propose a novel method, namely _Bidirectional Luminance Remapping_ (BLR), to interactively adjust the lighting of both input and training photos. First, the BLR method adapt the lighting of the input photo according to training photos, then it utilizes the offline pre-computed alpha matte information of training photos to recompose them according to the adapted input photo. The advantage of BLR is that it formulate online foreground/background segmentation into offline alpha matting, which enables efficient and accurate patch search. It can be integrated into existing face sketch synthesis methods with ignorable computational cost. ## 2 Proposed Algorithm In this section, we present the details of how BLR handles lighting variations. Meanwhile, we describe the details for how to integrate BLR into existing methods. ### Bidirectional Luminance Remapping (BLR) When an input photo with a different lighting from the training photos is given, a straightforward solution is to perform a global linear luminance remapping (LR) on the input photo to make it contain the same luminance statistics (_e.g._, mean and variance) with those in training photos [3, 18]. However, the global mapping scheme is not applicable for many cases (e.g., when the background has different intensities), and thus, the result is erroneous shown in Fig. 2(b). We now present BLR to make the luminance statistics of the face and non-face regions individually consistent between input and training photos. Each photo consists of the face and non-face regions and the remapping algorithm is performed in two steps. First, we perform a global linear luminance remapping on the input photo according to training photos. Note that this global remapping is only based on the luminance in the face region. It is computed regardless of the non-face region in the training photos, and the non-face region of the input photo can be remapped to arbitrary luminance. In the second step, we adjust the luminance of non-face region in each training photo (using offline pre-computed alpha matte) to make the overall statistics of training photos consistent with those of the input photo obtained in the first step. In this way, the luminance statistics of the face and non-face regions are adjusted similar between input and training photos. <figure><img src="content_image/1708.00224/test.png"><figcaption>Figure 2: Improvement with BLR intergration. MRF sketch synthesis method isused in this example. (a) is a challenging input photo captured in darklighting condition and textured background. (b) is the result with luminanceremapping. (c) is the result of the first step of BLR (Sec. 2.1.1). (d) is theresult with BLR.</figcaption></figure> #### 2.1.1 Luminance Remapping on Input Photo We perform luminance remapping on the input photo to enable it contains similar luminance statistics of the face region with those of training photos. The face region is approximately obtained using facial landmarks. We denote x as the input photo, $\textbf{X}=a_{i}\cdot\textbf{x}+b_{i}$ as the adapted photo (where $a_{i}$ and $b_{i}$ are two scalars), and y as all the training photos. We denote $\mu_{\textbf{x}}$, $\mu_{\textbf{X}}$ and $\mu_{\textbf{y}}$ as the mean of the face photo(s) x, X and y, respectively. We denote $\sigma_{\textbf{x}}$, $\sigma_{\textbf{X}}$ and $\sigma_{\textbf{y}}$ as the corresponding standard deviations of x, X and y, respectively. Our remapping transforms the luminance statistics of the input photo as: \[\mu_{\textbf{X}} = a_{i}\cdot\mu_{\textbf{x}}+b_{i};\] (1) \[\sigma_{\textbf{X}}^{2} = a_{i}^{2}\sigma_{\textbf{x}}^{2};\] (2) The remapping parameters $a_{i}$ and $b_{i}$ are computed based on the face region between input and training photos. We denote $\textbf{X}_{f}$ and $\textbf{y}_{f}$ as the face region in the adapted photo X and the training photos y, respectively. We set $\mu_{\textbf{X}_{f}}=\mu_{\textbf{y}_{f}}$ and $\sigma_{\textbf{X}_{f}}=\sigma_{\textbf{y}_{f}}$ to enable similar luminance statistics on the face region between input and training photos. As a result, parameters $a_{i}$ and $b_{i}$ are computed as follows: \[a_{i} = \frac{\sigma_{\textbf{y}_{f}}}{\sigma_{\textbf{x}_{f}}},\] (3) \[b_{i} = \mu_{\textbf{y}_{f}}-a_{i}\cdot\mu_{\textbf{x}_{f}}.\] (4) We use parameters $a_{i}$ and $b_{i}$ to adjust the input photo while not altering training photos at present. #### 2.1.2 Luminance Remapping on Training Photos After conducting luminance remapping in Sec. 2.1.1, we are confident that the luminance statistics of the face region in adapted input photo X are similar with those of the training photos. The remaining problem resides in the boundary between the face and non-face regions, which may lead to incorrect patch search and thus the erroneous boundary occurs in the results shown in Fig. 2(f). We decompose each training photo into portrait image, non-portrait image and alpha map using matting algorithm [5] with manually labeled trimap. The portrait image contains the whole human portrait region while the non-portrait image contains the background region. The non-portrait image is used to approximate the non-face region and the matting operation is done offline. We keep the portrait image fixed and a luminance remapping on the non-portrait image is performed to enable the overall statistics of the training photos similar to those of the adapted input photo obtained in Sec. 2.1.1. We denote $\textbf{y}_{p}$, $\textbf{y}_{n}$ and $\alpha$ as the portrait images, non-portrait images and alpha maps in the training images. So training images y can be written as \[\textbf{y}=\alpha\cdot\textbf{y}_{p}+(1-\alpha)\cdot\textbf{y}_{n}.\] (5) We denote Y as the adapted training images with luminance remapped non-portrait region, then Y \[= \alpha\cdot\textbf{y}_{p}+(1-\alpha)(a_{t}\cdot\textbf{y}_{n}+b_{ t})\] (6) \[= \alpha\cdot\textbf{y}_{p}+a_{t}\cdot(1-\alpha)\cdot\textbf{y}_{n} +b_{t}\cdot(1-\alpha),\] where $a_{t}$ and $b_{t}$ are parameters to adjust the non-portrait regions. We denote $\textbf{P}=\alpha\cdot\textbf{y}_{p}$, $\textbf{N}=(1-\alpha)\cdot\textbf{y}_{n}$, and $\textbf{A}=(1-\alpha)$. The adapted training images Y can be written as: \[\textbf{Y}=\textbf{P}+a\cdot\textbf{N}+b\cdot\textbf{A}\] (7) and its mean can be computed as: \[\mu_{\textbf{Y}}=\mu_{\textbf{P}}+a\cdot\mu_{\textbf{N}}+b\cdot\mu_{\textbf{A}}.\] (8) We denote $\overline{\textbf{Y}}$ as the mean operator on photos Y. So we compute the variance of Y as: \[\sigma_{\textbf{Y}}^{2} = \overline{(\textbf{Y}-\mu_{\textbf{Y}})^{2}}\] (9) \[= \overline{\left((\textbf{P}-\mu_{\textbf{P}})+a_{t}\cdot(\textbf{ N}-\mu_{\textbf{N}})+b_{t}\cdot(\textbf{A}-\mu_{\textbf{A}})\right)^{2}}\] \[= \sigma_{\textbf{P}}^{2}+a_{t}^{2}\sigma_{\textbf{N}}^{2}+b_{t}^{2 }\sigma_{\textbf{A}}^{2}+2a_{t}\cdot\sigma_{\textbf{P},\textbf{N}}+2b_{t}\cdot \sigma_{\textbf{P},\textbf{A}}\] \[+2a_{t}b_{t}\cdot\sigma_{\textbf{N},\textbf{A}},\] where $\sigma_{\textbf{x},\textbf{y}}$ corresponds to the covariance between x and y. We set $\mu_{\textbf{X}}=\mu_{\textbf{Y}}$ and $\sigma_{\textbf{X}}=\sigma_{\textbf{Y}}$ to enable the luminance statistics of adapted input photo similar with the adapted training photos. The parameters $a_{t}$ and $b_{t}$ can be computed by solving the above two quadratic equations. In practice, we notice that parameter $b$ is normally small, and thus we can approximate Eq. (6) by Y \[= \alpha\cdot\textbf{y}_{p}+a_{t}\cdot\left((1-\alpha)\cdot\textbf{ y}_{n}\right)+b_{t}\] (10) \[= \textbf{P}+a_{t}\cdot\textbf{N}+b_{t}.\] Then we have \[\mu_{\textbf{X}}=\mu_{\textbf{Y}} = \mu_{\textbf{P}}+a_{t}\cdot\mu_{\textbf{N}}+b_{t}\] (11) \[\sigma_{\textbf{X}}^{2}=\sigma_{\textbf{Y}}^{2} = \overline{(\textbf{Y}-\mu_{\textbf{Y}})^{2}}\] (12) \[= \overline{\left((\textbf{P}-\mu_{\textbf{P}})+a_{t}\cdot(\textbf{ N}-\mu_{\textbf{N}})\right)^{2}}\] \[= \sigma_{\textbf{N}}^{2}\cdot a_{t}^{2}+2\sigma_{\textbf{P}, \textbf{N}}\cdot a_{t}+\sigma_{\textbf{P}}^{2}.\] Parameter $a_{t}$ can then be computed by solving the quadratic equation in Eq. (12) and then used to solve for parameter $b_{t}$ in Eq. (11). There will be two possible solutions for the linear transform. To attenuate noise, we choose the positive $a_{t}$ value that minimizes parameter $b_{t}$: \[a_{t} = \frac{-\sigma_{\textbf{P},\textbf{N}}+\sqrt{\sigma_{\textbf{P}, \textbf{N}}^{2}-\sigma_{\textbf{N}}^{2}\sigma_{\textbf{P}}^{2}+\sigma_{\textbf {N}}^{2}\sigma_{\textbf{X}}^{2}}}{\sigma_{\textbf{N}}^{2}},\] (13) \[b_{t} = \mu_{\textbf{X}}-\mu_{\textbf{P}}-a_{t}\cdot\mu_{\textbf{N}}.\] (14) After obtaining parameters $a_{t}$ and $b_{t}$ we perform remapping on the non-portrait images. Then we recompose training photos using adapted non-portrait image, portrait image, and alpha mat. As a result, we enable similar luminance statistics of face and non-face regions between input and training photos. The photo patch search has been accurate for existing face sketch synthesis methods to synthesize sketches. ### Practical Issues #### 2.2.1 Side Lighting In practice, side lighting may occur in input photos. We use Contrast Limited Adaptive Histogram Equalization (CLAHE) [9] to reduce the effect but find that shadows may still exist around facial components. Then we remap shadow region under the guidance of its symmetric normal lighting region on the face. Specifically, we use landmarks to divide the whole face region into two symmetric parts, i.e, shadow region, and normal lighting region. For each patch in the shadow region, we perform a $1$-NN search in the normal lighting region around the corresponding symmetric position using normalized cross correlation (NCC). Then we remap the luminance of pixels in the shadow region using gamma correction. We denote $T_{p}$ as a patch centered at a pixel $p$ in the shadow region and $T_{p^{\prime}}$ as the most similar patch centered at $p^{\prime}$ in the normal lighting region. The gamma correction can be written as: \[I_{p}=I_{p}^{\mu_{T_{p}}/\mu_{T_{p^{\prime}}}}\] (15) where $I_{p}$ is the luminance of $p$. $\mu_{T_{p}}$ and $\mu_{T_{p}^{\prime}}$ are the mean luminance of patch $T_{p}$ and $T_{p}^{\prime}$, respectively. #### 2.2.2 Pose Variance In addition to lighting problem, the patch appearance distortion due to pose variations also degrades the quality of selected patch. We precompute the average position of each facial landmark from all training photos to generate a triangulated face template. Given an input photo, we detect its landmarks and compute the local affine transform. Through this transform, input photo is warped to a pose corrected photo, and $K$-NN patch search is then performed between the pose corrected photo and training photos. After sketch synthesis, we warp it back to the original pose. #### 2.2.3 Implementation Details In our implementation, we precompute the facial landmarks, portrait, and non-portrait images, alpha mattes for all training photos in advance. Given an input photo, we first detect facial landmarks using the algorithm in [4] and perform local affine transform illustrated in Sec 2.2.2 to warp the input photo into a pose corrected one for further processing. The landmark detection and local affine transform can be conducted in real-time. Second, side lighting is handled as described in Sec 2.2.1. Then BLR is applied to adapt both input and training photos. After BLR, preprocessed input and training photos can be adopted by existing face sketch synthesis algorithms to synthesize sketch images. Finally, the sketch image is mapped back using local affine transform to yield the final sketch result. <figure><img src="content_image/1708.00224/pca_mrf.jpg"><figcaption>Figure 3: Quantitative evaluation on synthetic CUHK dataset. We use σF and σBto adjust the foreground and background lightings of input photos,correspondingly. Our integration improves the robustness of MRF, MWF and SSDregarding to different lightings. It performs favorably against luminanceremapping integration and original RMRF method.</figcaption></figure> ## 3 Experiments We conduct experiments using state-of-the-art face sketch synthesis methods including MRF [12], RMRF [18], MWF [22] and SSD [10]. The focus is to demonstrate the improvement after integrating BLR into existing methods. The experiments are conducted on the benchmarks including CUHK [12], AR [1], and FERET datasets [19]. The number of photo-sketch pairs for CUHK, AR and FERET are 188, 123 and 1165, respectively. The photos in these three datasets are captured in frontal view and neutral expression. In CUHK dataset the lighting condition is similar for all the photos. In AR dataset the lighting condition is also similar among the photos. However, the lighting condition of CUHK dataset is different from that of AR dataset. For FERET the lighting varies in different photos within this dataset. In addition, we also conduct experiments on the CUHK side lighting and pose variation datasets, which belong to CUHK dataset. ### Varying Lighting Conditions #### 3.1.1 Synthetic Experiments We first carry out quantitative and qualitative evaluations for synthetic lighting conditions. The synthetic evaluations are conducted on modified CUHK dataset. We split the CUHK dataset into 88 training photo-sketch pairs and 100 input pairs and then generate synthetic input photos in varying lightings as follows. We use matting algorithm [5] to divide each input photo into foreground, background and alpha matte images. Then we separately adjust the luminance of foreground and background using two scalars (i.e., $\sigma_{F}$ and $\sigma_{B}$). The luminance values of all foreground pixels are multiplied by $\sigma_{F}$ and those of background are multiplied by $\sigma_{B}$. Then we combine adjusted foreground and background images with alpha matte to generate synthetic input photos. We compare our method with baseline luminance remapping (LR) [3] preprocessing. Note that RMRF is specially designed for improving the robustness of MRF, we compare with RMRF when evaluating the improvement on MRF. In addition, RMRF can be treated as an independent method which can be integrated with our algorithm. So we first evaluate the original performance of MRF, MWF, SSD and RMRF. Then we compare the improvements of the LR and our integration of these methods. Quantitative evaluation of face sketch synthesis methods can be conducted through face sketch recognition as suggested in [12]. For each input photo, the synthesized sketch should be matched to the corresponding sketch drawn by the artist. If an algorithm achieves higher sketch recognition rates, it suggests that this method is more robust to synthesize sketches. Fig. 3 shows the performance of quantitative evaluation. The foreground of input photos is adjusted by three values of $\sigma_{F}$, i.e., 0.5, 1.0, and 1.5. These values simulate the dark, normal, and bright foreground of input photos, respectively. For each $\sigma_{F}$ we adjust $\sigma_{B}$ from 0.5 to 1.5 incremented by 0.1, which simulates the varying background lightings. The results show that MRF, MWF, and SSD are not robust to synthesize sketches from photos captured in different lightings. Due to its global normalization scheme, LR preprocessing cannot robustly handle all lighting conditions. Our algorithm can consistently improve the performance of existing methods. Compared with RMRF, our algorithm is more robust against extreme cases (the first row of Fig. 3). Moreover, our algorithm can be integrated with RMRF to improve its robustness (the last row of Fig. 3). <figure><img src="content_image/1708.00224/test.png"><figcaption>Figure 4: An example of synthetic lighting experiments. (a) is the input photoconsists of dark foreground and bright background. (b)-(e) are the results ofexisting methods. (f)-(h) are the results of improved existing methods withluminance remapping integration. (i)-(l) are the results of improved existingmethods with our integration.</figcaption></figure> Fig. 4 shows one example of the visual comparison for the synthetic evaluation. The input photo consists of dark foreground and bright background. As the foreground differs from training photos patch candidates can not be correctly matched, which results in blurry and artifacts as shown in (b)-(d). LR based on global luminance statistics fails to correct the lightings and thus produces erroneous results as shown in (f)-(h). In comparison, BLR adapts both input and training photos to enable more accurate patch search in the face and non-face regions. As a result, the accuracy of $K$-NN patch searching is improved and the obtained sketch results achieve ideal performance as shown in (i)-(l). Meanwhile, the local contrast within photo patch is reduced through our integration and thus the result in (i) is improved around face boundary. #### 3.1.2 Cross-Dataset Experiments We notice that CUHK and AR datasets are captured in different lightings. Thus we evaluate the robustness of BLR using CUHK as training and AR as input and vice versa. Fig. 5 shows the visual comparison where BLR can consistently improve existing methods. Although ethnic facial difference exists between two datasets, BLR can still robustify sketch synthesis of existing methods. #### 3.1.3 Real Lighting Experiments We conduct an evaluation of BLR on FERET dataset. Different from the previous two datasets FERET contains photos captured in real world varying lighting conditions. We randomly select 100 photo-sketch pairs as training and use the remaining 1065 pairs as input. Fig. 6 shows one example of the visual evaluation. The lighting is different in both foreground and background regions, which leads to artifacts on the synthesized sketches of existing methods. Through our integration, the statistics of the face and non face regions are adjusted similarly among input and training photos. It enables existing methods to robustify sketch synthesis. <figure><img src="content_image/1708.00224/test.png"><figcaption>Figure 5: An example of cross-dataset experiments (CUHK as training while ARas input). (a) is an input photo. (b)-(l) are with the same meaning as Fig. 4.</figcaption></figure> <figure><img src="content_image/1708.00224/test.png"><figcaption>Figure 6: An example of experiments on FERET dataset. (a) is an input photo.(b)-(l) are with the same meaning as Fig. 4.</figcaption></figure> #### 3.1.4 Side Lighting Experiments We conduct experiments on CUHK side lighting dataset [18] which contains two different types of side lighting (dark left / dark right) photos for each subject. As the input photo contains shadows in the facial region shown in Fig. 7, existing methods cannot find correctly matched photo patches around these shadow regions. It leads to blur and artifacts shown in (b)-(d). In comparison, Our method can locally adjust input photo to make an improvement. ### Varying Poses We perform experiments on CUHK pose variation dataset [18] where subjects are in varying poses. Note that some methods [10, 22] tend to increase search range for handling varying poses. Thus we also compare BLR with existing methods using extended search range. Fig. 8 shows an example of the visual evaluation result. Our algorithm favorably improves the robustness of existing methods. <figure><img src="content_image/1708.00224/test.png"><figcaption>Figure 7: An example of side lighting experiments. (a) is an input photo.(b)-(l) are with the same meaning as Fig. 4.</figcaption></figure> <figure><img src="content_image/1708.00224/pose.png"><figcaption>Figure 8: An example of varying pose experiments. (a) is an input photo.(b)-(d) are the synthesized sketches. (e)-(h) are the results synthesized withextended search range. (i)-(l) are the sketches synthesized with ourintegration.</figcaption></figure> \| MRF \| MWF \| RMRF \| SSD ---\|---\|---\|---\|--- Original \| 38.4 \| 35.6 \| 88.2 \| 4.5 Original (ext.)† \| 94.5 \| 93.8 \| 252.3 \| 13.4 Original + Ours \| 38.6 \| 35.9 \| 93.5 \| 4.7 †With extended search range (see Sec. 3.2). Table 1: Runtime (seconds) for a CUHK input image. ### Computational Cost Table 1 shows the runtime of existing methods to process a CUHK input image (obtained from a 3.4GHz Intel i7 CPU). It shows that the additional computation cost brought by BLR is ignorable compared with the original time cost of existing methods. Note that the reason why RMRF needs more additional computational cost is that we need to extract features online of the recomposed training photos. ## 4 Concluding Remarks We propose BLR, which interactively adjusts the lighting of training and input photos. It moves online face image segmentation to offline using human supervised alpha matting. The experiments demonstrate that BLR improves the robustness of existing methods with ignorable computational cost. ## References * [Aleix and Robert1998] Martínez Aleix and Benavente Robert. The ar face database. Technical Report CVC Tech Report 24, Purdue University, 1998. * [Han _et al._2010] Hu Han, Shiguang Shan, Laiyun Qing, Xilin Chen, and Wen Gao. Lighting aware preprocessing for face recognition across varying illumination. In _European Conference on Computer Vision_, 2010. * [Hertzmann _et al._2001] Aaron Hertzmann, Charles E Jacobs, Nuria Oliver, Brian Curless, and David H Salesin. Image analogies. _ACM Transactions on Graphics (SIGGRAPH)_, 2001. * [Kazemi and Sullivan2014] Vahid Kazemi and Josephine Sullivan. One millisecond face alignment with an ensemble of regression trees. In _IEEE Conference on Computer Vision and Pattern Recognition_, 2014. * [Levin _et al._2008] Anat Levin, Dani Lischinski, and Yair Weiss. A closed-form solution to natural image matting. _IEEE Transactions on Pattern Analysis and Machine Intelligence_, 2008. * [Liu _et al._2007] Wei Liu, Xiaoou Tang, and Jianzhuang Liu. Bayesian tensor inference for sketch-based facial photo hallucination. In _International Joint Conference on Artificial Intelligence_, 2007. * [Peng _et al._2016a] Chunlei Peng, Xinbo Gao, Nannan Wang, and Jie Li. Graphical representation for heterogeneous face recognition. _IEEE Transactions on Pattern Analysis and Machine Intelligence_, 2016. * [Peng _et al._2016b] Chunlei Peng, Nannan Wang, Xinbo Gao, and Jie Li. Face recognition from multiple stylistic sketches: Scenarios, datasets, and evaluation. In _European Conference on Computer Vision Workshop_, 2016. * [Pizer _et al._1987] Stephen M Pizer, E Philip Amburn, John D Austin, Robert Cromartie, Ari Geselowitz, Trey Greer, Bart ter Haar Romeny, John B Zimmerman, and Karel Zuiderveld. Adaptive histogram equalization and its variations. _Computer vision, graphics, and image processing_, 1987. * [Song _et al._2014] Yibing Song, Linchao Bao, Qingxiong Yang, and Ming-Hsuan Yang. Real-time exemplar-based face sketch synthesis. In _European Conference on Computer Vision_, 2014. * [Szeliski2010] Richard Szeliski. _Computer vision: algorithms and applications_. Springer, 2010. * [Wang and Tang2009] Xiaogang Wang and Xiaoou Tang. Face photo-sketch synthesis and recognition. _IEEE Transactions on Pattern Analysis and Machine Intelligence_, 2009. * [Wang _et al._2012] Shenlong Wang, Lei Zhang, Yan Liang, and Quan Pan. Semi-coupled dictionary learning with applications in image super-resolution and photo-sketch synthesis. In _IEEE Conference on Computer Vision and Pattern Recognition_, 2012. * [Wang _et al._2013] Nannan Wang, Dacheng Tao, Xinbo Gao, Xuelong Li, and Jie Li. Transductive face sketch-photo synthesis. _IEEE Transactions on Neural Networks and Learning Systems_, 2013. * [Wang _et al._2014] Nannan Wang, Dacheng Tao, Xinbo Gao, Xuelong Li, and Jie Li. A comprehensive survey to face hallucination. _International Journal of Computer Vision_, 2014. * [Wang _et al._2017] Nannan Wang, Xinbo Gao, Leiyu Sun, and Jie Li. Bayesian face sketch synthesis. _IEEE Transactions on Image Processing_, 2017. * [Xie _et al._2008] Xiaohua Xie, Wei-Shi Zheng, Jianhuang Lai, and Pong C. Yuen. Face illumination normalization on large and small scale features. In _IEEE Conference on Computer Vision and Pattern Recognition_, 2008. * [Zhang _et al._2010] Wei Zhang, Xiaogang Wang, and Xiaoou Tang. Lighting and pose robust face sketch synthesis. In _European Conference on Computer Vision_, 2010. * [Zhang _et al._2011] Wei Zhang, Xiaogang Wang, and Xiaoou Tang. Coupled information-theoretic encoding for face photo-sketch recognition. In _IEEE Conference on Computer Vision and Pattern Recognition_, 2011. * [Zhang _et al._2015a] Shengchuan Zhang, Xinbo Gao, Nannan Wang, and Jie Li. Face sketch synthesis from a single photo-sketch pair. _IEEE Transactions on Circuits and Systems for Video Technology_, 2015. * [Zhang _et al._2015b] Shengchuan Zhang, Xinbo Gao, Nannan Wang, Jie Li, and Mingjin Zhang. Face sketch synthesis via sparse representation-based greedy search. _IEEE Transactions on Image Processing_, 2015. * [Zhou _et al._2012] Hao Zhou, Zhanghui Kuang, and Kenneth Wong. Markov weight fields for face sketch synthesis. In _IEEE Conference on Computer Vision and Pattern Recognition_, 2012.
1706.04931	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 99698, "num_imgs": 21, "llama3_tokens_count": 33932 }	[ "content_image/1706.04931/SSRDpar.png", "content_image/1706.04931/x1.png", "content_image/1706.04931/DD1.png", "content_image/1706.04931/lux_lambdaL.png", "content_image/1706.04931/x2.png", "content_image/1706.04931/luxvxA.png", "content_image/1706.04931/omegaA2.png", "content_image/1706.04931/luxsinA.png", "content_image/1706.04931/ESDMLOWLux.png", "content_image/1706.04931/x3.png", "content_image/1706.04931/x4.png", "content_image/1706.04931/x6.png", "content_image/1706.04931/x7.png", "content_image/1706.04931/x11.png", "content_image/1706.04931/x13.png", "content_image/1706.04931/x15.png", "content_image/1706.04931/x17.png", "content_image/1706.04931/x18.png", "content_image/1706.04931/x20.png", "content_image/1706.04931/x21.png", "content_image/1706.04931/LFV_A.png" ]	###### Abstract We investigate the electroweak vacuum stability in an extended version of the Standard Model which incorporates two additional singlet scalar fields and three right handed neutrinos. One of these extra scalars plays the role of dark matter while the other scalar not only helps in making the electroweak vacuum stable but also opens up the low mass window of the scalar singlet dark matter ($<$ 500 GeV). We consider the effect of large neutrino Yukawa coupling on the running of Higgs quartic coupling. We have analyzed the constraints on the model and identify the range of parameter space which is consistent with neutrino mass, appropriate relic density and direct search limits from the latest XENON 1T preliminary result as well as realizing the stability of the electroweak vacuum upto the Planck scale. Study of Electroweak Vacuum Stability from Extended Higgs Portal of Dark Matter and Neutrinos Purusottam Ghosh${}^{a,}$¹, Abhijit Kumar Saha${}^{a,}$², Arunansu Sil${}^{a,}$³ [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [3mm] _${}^{a}$ Department of Physics, Indian Institute of Technology Guwahati, 781039 Assam, India_ ## 1 Introduction The discovery of the Higgs boson[1, 2, 3] has been considered as the greatest triumph in present day particle physics. Although the experimental search is still on in order to investigate the Higgs boson’s properties, several theoretical and phenomenological reasons are there to push us toward hunting for an enlarged Higgs sector compared to the one present in Standard model (SM). For example the Higgs quartic coupling $\lambda_{H}$ in SM becomes negative at a high energy scale ($\Lambda_{I}^{\textrm{SM}}\sim 10^{10}$ GeV) leading to a possible instability of Higgs vacuum. The present measured values of Higgs mass$\sim 125.09$ GeV[4] and top mass $\sim 173.2$ GeV[4], suggest that the electroweak (EW) vacuum can be metastable [5, 6, 7, 8, 9, 10, 11]. However the conclusion exclusively depends on the precise measurement of the top mass[12, 13]. Also the metastability of the Universe is not a very robust situation in the context of cosmological inflation[14]. One of the possible solutions to this is to introduce new physics between EW scale and $\Lambda_{I}^{\textrm{SM}}$. In view of SM’s incompetence in resolving some of the other issues like dark matter, neutrino mass, matter antimatter symmetry, inflation etc, the introduction of new physics is of course a welcome feature. In particular, SM fails to accommodate a significant share in terms of its content called dark matter(DM). The most economical and popular scenario is the singlet scalar extension of the SM[15, 16, 17, 18, 19, 20, 21, 22, 23] having Higgs portal interaction. The stability of the dark matter is ensured by imposing a $Z_{2}$ symmetry on it. The relic abundance and corresponding direct detection cross section are solely determined by the DM (the scalar singlet) mass and its coupling with Higgs ( portal coupling). However present experiments, LUX[24] and XENON1T[25], strongly disfavor the model below $m_{\textrm{DM}}<500$ GeV except the resonance region. The bound on Higgs invisible decay width further constrain the model for $m_{\textrm{DM}}<62.5$ GeV[26]. Hence a large range of DM mass seems to be excluded within this simplest framework which otherwise would be an interesting region of search for several ongoing and future direct[24, 25, 27] and indirect experiments[28]. It is interesting to note that the presence of extra scalar in the form of DM can shift the instability scale ($\Lambda_{I}$) toward larger value compared to the SM one ($\Lambda_{I}>\Lambda_{I}^{\textrm{SM}}$)[29, 30, 31, 32, 33, 34, 35]. On the other hand to accommodate non-zero neutrino mass via type-I seesaw mechanism, one can extend SM with three right handed(RH) neutrinos. The RH neutrinos, being SM singlet, will have standard Yukawa like coupling involving Higgs and lepton doublets. The presence of the neutrino Yukawa coupling affects the running of the Higgs quartic coupling similar to the top Yukawa coupling. In fact with neutrino Yukawa coupling, $Y_{\nu}$, of $\mathcal{O}(1)$, $\Lambda_{I}$ could be lower than $\Lambda_{I}^{\textrm{SM}}$[36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], that might lead to an unstable Universe. The situation does not alter much even if one includes scalar singlet DM ($m_{\textrm{DM}}\leq 500$ GeV) in this framework [34, 36, 37, 48, 49, 50]. So the combined framework of RH neutrinos and scalar singlet DM excludes a significant range of DM mass ($<500$ GeV) while keeping the EW vacuum on the verge of being unstable. With an endeavour to make the EW vacuum absolutely stable upto the planck scale $M_{P}$, in a scenario that can accommodate both the DM and massive neutrinos with large $Y_{\nu}$ (in type-I seesaw) and simultaneously to reopen the window for lighter scalar singlet DM mass ($<$ 500 GeV), we incorporate two SM real singlet scalars and three SM singlet RH neutrinos in this work. Similar models to address DM phenomenology involving additional scalars (without involving RH neutrinos) have been studied [51, 35, 52, 53, 54, 55], however with different motivations. Our set up also differs from them in terms of inclusion of light neutrino mass through type-I seesaw. The proposed model has several important ingredients which are mentioned below along with their importance. * One of the additional SM singlet scalars is our DM candidate whose stability is achieved with an unbroken $Z_{2}$ symmetry. * The other scalar would acquire a nonzero vacuum expectation value (vev). This field has two fold contributions in our analysis: (i) it affects the running of the SM Higgs quartic coupling and (ii) the dark matter phenomenology becomes more involved due to its mixing with the SM Higgs and the DM. * The set up also contains three RH neutrinos in order to generate non-zero light neutrino mass through type-I seesaw mechanism. Therefore, along with the contributions from the additional scalar fields, neutrino Yukawa coupling, $Y_{\nu}$, is also involved in studying the running of the Higgs quartic coupling. We observe that the presence of the scalar⁴ with non-zero vev affects the DM phenomenology in such a way that $m_{\textrm{DM}}$ less than 500 GeV becomes perfectly allowed mass range considering the recent XENON-1T result[25], which otherwise was excluded from the DM direct search results[56]. We also include XENON-nT[25] prediction to further constrain our model. On the other hand, we find that the SM Higgs quartic coupling may remain positive till $M_{P}$ (or upto some other scale higher than $\Lambda_{I}^{\textrm{SM}}$) even in presence of large $Y_{\nu}$, thanks to the involvement of the scalar with non-zero vev. We therefore identify the relevant parameter space (in terms of stability, metastability and instability regions) of the model which can allow large $Y_{\nu}$ (with different mass scales of RH neutrinos) and scalar DM below 500 GeV. Bounds from other related aspects, $e.g.$ lepton flavor violating decays, neutrinoless double beta decay etc., are also considered. The set-up therefore demands rich phenomenology what we present in the following sections. [FOOTNOTE:4][ENDFOOTNOTE] The paper is organized as follows. In section 2, we discuss the set-up of our model and in section 3, we include the constraints on our model parameters. Then in the subsequent sections 4 and 5, we discuss the DM phenomenology and vacuum stability respectively in the context of our model. In section 6, we discuss connection of the model with other observables. Finally we conclude in section 7. ## 2 The Model As mentioned in the introduction, we aim to study how the EW vacuum can be made stable in a model that would successfully accommodate a scalar DM and neutrino mass. For this purpose, we extend the SM by introducing two SM singlet scalar fields, $\phi$ and $\chi$, and three right-handed neutrinos, $N_{i=1,2,3}$. We have also imposed a discrete symmetry, $Z_{2}\times Z^{\prime}_{2}$. The field $\phi$ is odd (even) under $Z_{2}$ ($Z^{\prime}_{2}$) and $\chi$ is even (odd) under $Z_{2}$ ($Z^{\prime}_{2}$) while all other fields are even under both $Z_{2}$ and $Z^{\prime}_{2}$. There exists a non-zero vev associated with the $\chi$ field. The unbroken $Z_{2}$ ensures the stability of our dark matter candidate $\phi$. On the other hand, the inclusion of $Z^{\prime}_{2}$ simplifies the scalar potential in the set-up⁵. The RH neutrinos are included in order to incorporate the light neutrino mass through type-I seesaw mechanism. [FOOTNOTE:5][ENDFOOTNOTE] The scalar potential involving $\phi,\chi$ and the SM Higgs doublet ($H$) is given by \[V=V_{\textrm{I}}+V_{\textrm{II}}+V_{\textrm{III}}+V_{\textrm{H}},\] (1) where \[V_{\rm{I}} = \frac{1}{2}\mu_{\phi}^{2}\phi^{2}+\frac{1}{4!}\lambda_{\phi}\phi^ {4}+\frac{1}{2}\lambda_{\phi H}\phi^{2}H^{\dagger}H;\] \[V_{\rm{II}} = -\frac{1}{2}\mu_{\chi}^{2}\chi^{2}+\frac{\lambda_{\chi}}{4!}\chi^ {4}+\frac{\lambda_{\chi H}}{2}\chi^{2}\|H\|^{2};\] \[V_{\rm{III}} = \frac{1}{4}\lambda_{\chi\phi}\phi^{2}\chi^{2},\leavevmode\nobreak \ \leavevmode\nobreak\ {\rm{and}}\leavevmode\nobreak\ \leavevmode\nobreak\ V_{ \rm{H}}=-\mu^{2}_{H}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}.\] The relevant part of the Lagrangian responsible for neutrino mass is given by \[-\mathcal{L}_{\nu}=Y_{\nu_{ij}}\bar{l_{L}}_{i}\tilde{H}{N}_{j}+\frac{1}{2}{M_{ N}}_{ij}N_{i}N_{j},\] where ${l_{L}}_{i}$ are the left-handed lepton doublets, $M_{N}$ is the Majorona mass matrix of the RH neutrinos. This leads to the light neutrino mass, $m_{\nu}=Y_{\nu}^{T}{M_{N}}^{-1}Y_{\nu}\frac{v^{2}}{2}$ with $v=246$ GeV as the vacuum expectation value of the SM Higgs. Minimization of the potential $V$ leads to the following vevs of $\chi$ and $H^{0}$ (the neutral component of $H$), as given by⁶ [FOOTNOTE:6][ENDFOOTNOTE] \[v_{\chi}^{2}=6\frac{2\mu_{\chi}^{2}\lambda_{H}-\mu_{H}^{2} \lambda_{\chi H}}{2\lambda_{H}\lambda_{\chi}-3\lambda_{\chi H}^{2}},\] (2) \[v^{2}=2\frac{\mu_{H}^{2}\lambda_{\chi}-3\mu_{\chi}^{2}\lambda_{ \chi H}}{2\lambda_{H}\lambda_{\chi}-3\lambda_{\chi H}^{2}}.\] (3) So after $\chi$ gets the vev and electroweak symmetry is broken, the mixing between $H^{0}$ and $\chi$ will take place and new mass or physical eigenstates, $H_{1}$ and $H_{2}$, will be formed. The two physical eigenstates are related with $H^{0}$ and $\chi$ by \[H_{1}=H^{0}\cos\theta-\chi\sin\theta,\] \[H_{2}=H^{0}\sin\theta+\chi\cos\theta,\] (4) where the mixing angle $\theta$ is defined by \[\tan 2\theta=\frac{\lambda_{\chi H}vv_{\chi}}{-\lambda_{H}v^{2}+ \frac{\lambda_{\chi}v_{\chi}^{2}}{6}}.\] (5) Similarly the mass eigenvalues of these physical Higgses are found to be \[m_{H_{1}}^{2}=\frac{\lambda_{\chi}}{6}v_{\chi}^{2}(1-\sec 2 \theta)+\lambda_{H}v^{2}(1+\sec 2\theta),\] (6) \[m_{H_{2}}^{2}=\frac{\lambda_{\chi}}{6}v_{\chi}^{2}(1+\sec 2 \theta)+\lambda_{H}v^{2}(1-\sec 2\theta).\] (7) Using Eqs.(5,6,7), the couplings $\lambda_{H}$, $\lambda_{\chi}$ and $\lambda_{\chi H}$ can be expressed in terms of the masses of the physical eigenstates $H_{1}$ and $H_{2}$, the vevs ($v$, $v_{\chi}$) and the mixing angle $\theta$ as \[\lambda_{H}= \frac{m_{H_{1}}^{2}}{4v^{2}}(1+\cos 2\theta)+\frac{m_{H_{2}}^{2}} {4v^{2}}(1-\cos 2\theta),\] (8) \[\lambda_{\chi}= \frac{3m_{H_{1}}^{2}}{2v_{\chi}^{2}}(1-\cos 2\theta)+\frac{3m_{H_ {2}}^{2}}{2v_{\chi}^{2}}(1+\cos 2\theta),\] (9) \[\lambda_{\chi H}= \sin 2\theta\Big{(}\frac{m_{H_{2}}^{2}-m_{H_{1}}^{2}}{2vv_{\chi}} \Big{)}.\] (10) Among $H_{1}$ and $H_{2}$, one of them would be the Higgs discovered at LHC. The other Higgs can be heavier or lighter than the SM Higgs. Below we proceed to discuss the constraints to be imposed on the couplings and mass parameters of the model before studying the DM phenomenology and vacuum stability in the subsequent sections. ## 3 Constraints Here we put together the constraints (both theoretical and experimental) that we will take into account to find the parameter space of our model. * In order to keep the entire potential stable, one needs to maintain the following conditions involving the couplings present in $V$ (considering all couplings as real) ST \[{}_{1,2,3}\] : \[\lambda_{H}>0,\textrm{ }\lambda_{\chi}>0,\textrm{ }\lambda_{\phi} >0,\] ST \[{}_{4,5,6}\] : \[\lambda_{\chi H}+\sqrt{\frac{2}{3}\lambda_{H}\lambda_{\chi}}>0, \textrm{ }\lambda_{\phi H}+\sqrt{\frac{2}{3}\lambda_{H}\lambda_{\phi}}>0, \textrm{ }3\lambda_{\chi\phi}+\sqrt{\lambda_{\chi}\lambda_{\phi}}>0,\] ST \[{}_{7}\] : \[\sqrt{\lambda_{H}\lambda_{\chi}\lambda_{\phi}}+\lambda_{\chi H} \sqrt{\frac{3}{2}\lambda_{\chi}}+3\lambda_{\phi H}\sqrt{\lambda_{H}}+3\lambda_ {\chi\phi}\sqrt{\lambda_{H}},\] \[+3\Big{[}\Big{(}\lambda_{\chi H}+\sqrt{\frac{2}{3}\lambda_{H} \lambda_{\chi}}\Big{)}\Big{(}\lambda_{\phi H}+\sqrt{\frac{2}{3}\lambda_{H} \lambda_{\phi}}\Big{)}\Big{(}\lambda_{\chi\phi}+\frac{1}{3}\sqrt{\lambda_{\phi }\lambda_{\chi}}\Big{)}\Big{]}^{1/2}>0,\] (11) which followed from the co-positivity of the mass-squared matrix involving $H,\leavevmode\nobreak\ \chi$ and $\phi$[66, 67]. * In addition, the perturbative unitarity associated with the $S$ matrix corresponding to 2 $\rightarrow$ 2 scattering processes involving all two particles initial and final states [68, 69] are considered. In the specific model under study, there are eleven neutral and four singly charged combinations of two-particle initial/final states. The details are provided in Appendix A. It turns out that the some of the scalar couplings of Eq.(1) are bounded by \[\lambda_{H}\leavevmode\nobreak\ <4\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{\phi H }\leavevmode\nobreak\ <8\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{\chi H}\leavevmode\nobreak \ <8\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{\chi\phi}\leavevmode\nobreak\ <8\pi\rm.\] (12) The other scalar couplings are restricted (in form of combinations among them) from the condition that the roots of a polynomial equation should be less than 16$\pi$ (see Eq.(A.9) of Appendix A). * To maintain the perturbativity of all the couplings, we impose the condition that the scalar couplings should remain below 4$\pi$ while Yukawa couplings are less than $\sqrt{4\pi}$ till $M_{P}$. An upper bound on $\tan\beta(=v/v_{\chi})$ follows from the perturbativity of $\lambda_{\chi}$[70] with a specific choice of $m_{H_{2}}$. * Turning into the constraints obtained from experiments, we note that the observed signal strength of the 125 GeV Higgs boson at LHC [73, 74, 75, 76, 77, 71] provides a limit on $\sin\theta$ as, $\|\sin\theta\|\lesssim 0.36$ with $m_{H_{2}}\gtrsim 150$ GeV. The analysis in [72] shows that $\sin\theta$ is restricted significantly ($\|\sin\theta\|\lesssim 0.3$) by the direct Higgs searches at colliders [73, 74, 75, 76, 77] and combined Higgs signal strength [78] for 150 GeV $<m_{H_{2}}<300$ GeV while for 300 GeV $<m_{H_{2}}<$ 800 GeV, it is the NLO contribution to the $W$ boson mass [70] which restricts $\sin\theta$ in a more stipulated range. Corrections to the electroweak precision observables through the $S,T,U$ parameters turn out to be less dominant compared to the limits obtained from $W$ boson mass correction [70]. For our purpose, we consider $\sin\theta\lesssim 0.3$ for the analysis. Apart from these, we impose the constraints on $Y_{\nu}$ from lepton flavor violating decays. Also phenomenological limits obtained on the scalar couplings involved in order to satisfy the relic density $(0.1175\leq\Omega h^{2}\leq 0.1219)$[79] and direct search limits[25] by our dark matter candidate $\phi$ are considered when stability of the EW minimum is investigated. ## 4 Dark matter phenomenology The scalar field $\phi$ playing the role of dark matter has a mass given by $m^{2}_{\textrm{DM}}=(\mu_{\phi}^{2}+\frac{1}{2}\lambda_{\phi H}v^{2})$ as followed from Eq.(1). Before moving toward the relic density calculation in our model, we would like to comment on the simplest $Z_{2}$ odd scalar dark matter scenario in view of recent XENON 1T[25] result. Note that for the purpose of DM phenomenology in this case, the only relevant parameters are given by $m_{\textrm{DM}}$ and the Higgs portal coupling $\lambda_{\phi H}$ (or $\mu_{\phi}$ and $\lambda_{\phi H}$). <figure><img src="content_image/1706.04931/SSRDpar.png"><figcaption>Figure 1: (Left panel:) relic density contour plot in mDM−λϕH plane; the redportion corresponds to the disfavored range of parameters by recent directdetection results, while the blue portion stands for the allowed region ofparameters consistent with direct detection results. (Right panel:) Spinindependent cross section is plotted (blue line) for relic density allowedpoints as a function of mDM, where the LUX 2016 and XENON 1T limits areindicated by blue and red dashed lines respectively.</figcaption></figure> In Fig.1 (left panel), we provide a contour plot for relic density consistent with the Planck result[79] in the $\lambda_{\phi H}-m_{\textrm{DM}}$ plane indicated by the blue solid line. In the right panel of Fig.1, we provide the DM-nucleon cross section evaluated with the value of $\lambda_{\phi H}$ corresponding to the $m_{\textrm{DM}}$ value as obtained from the left panel plot. We then incorporate the direct search limits on the DM-nucleon cross section as obtained from LUX 2016 [24], and the most recent XENON 1T[25] result [25] in the same plot denoted by blue and red dashed lines respectively. We conclude that the dark matter mass below 500 GeV is excluded from the present XENON 1T[25] result. This result is indicated by the red portion of the contour line in the left panel, while the remaining blue portion of the contour plot ( of the left panel) represents the allowed range of $m_{\textrm{DM}}$ satisfying both the relic density and direct search constraints. Let us now move to the relic density estimate in our set-up with the extra scalar $\chi$ and compare the phenomenology with the simplest scalar DM scenario in the light of the mixing between the SM Higgs and $\chi$. Using Eq.(2) and inserting them into the SM Lagrangian along with the ones mentioned in Eq.(1), we obtain the following list of interaction vertices involving two Higgses ($H_{1}$ and $H_{2}$), dark matter field ($\phi$) and several other SM fields. \[H_{1}f\bar{f},H_{2}f\bar{f} : \frac{m_{f}}{v}\cos\theta,\frac{m_{f}}{v}\sin\theta\] \[H_{1}ZZ,H_{2}ZZ : \frac{2m_{Z}^{2}}{v}\cos\theta g^{\mu\nu},\frac{2m_{Z}^{2}}{v} \sin\theta g^{\mu\nu}\] \[H_{1}W^{+}W^{-},H_{2}W^{+}W^{-} : \frac{2m_{W}^{2}}{v}\cos\theta g^{\mu\nu},\frac{2m_{W}^{2}}{v} \sin\theta g^{\mu\nu}\] \[\phi\phi H_{1} : -v_{\chi}\lambda_{\chi\phi}\sin\theta+v\lambda_{\phi H}\cos\theta \equiv\lambda_{1}\] \[\phi\phi H_{2} : v_{\chi}\lambda_{\chi\phi}\cos\theta+v\lambda_{\phi H}\sin\theta \equiv\lambda_{2}\] \[\phi\phi H_{1}H_{1} : \lambda_{\phi H}\cos^{2}\theta+\lambda_{\chi\phi}\sin^{2}\theta\] \[\phi\phi H_{2}H_{2} : \lambda_{\phi H}\sin^{2}\theta+\lambda_{\chi\phi}\cos^{2}\theta\] \[\phi\phi H_{1}H_{2} : (\lambda_{\phi H}-\lambda_{\chi\phi})\sin\theta\cos\theta\] \[H_{1}H_{1}H_{1} : [6v\lambda_{H}\cos^{3}\theta-3v_{\chi}\lambda_{\chi H}\cos^{2} \theta\sin\theta+3v\lambda_{\chi H}\cos\theta\sin^{2}\theta-v_{\chi}\lambda_{ \chi}\sin^{3}\theta]\] \[H_{2}H_{2}H_{2} : [6v\lambda_{H}\sin^{3}\theta+3v_{\chi}\lambda_{\chi H}\cos\theta \sin^{2}\theta+3v\lambda_{\chi H}\cos^{2}\theta\sin\theta+v_{\chi}\lambda_{ \chi}\cos^{3}\theta]\] \[H_{1}H_{1}H_{2} : [2v(3\lambda_{H}-\lambda_{\chi H})\cos^{2}\theta\sin\theta+v \lambda_{\chi H}\sin^{3}\theta+v_{\chi}(\lambda_{\chi}-2\lambda_{\chi H})\cos \theta\sin^{2}\theta\] \[+v_{\chi}\lambda_{\chi H}\cos^{3}\theta]\] \[H_{1}H_{2}H_{2} : [2v(3\lambda_{H}-\lambda_{\chi H})\cos\theta\sin^{2}\theta+v \lambda_{\chi H}\cos^{3}\theta-v_{\chi}(\lambda_{\chi}-2\lambda_{\chi H})\cos^ {2}\theta\sin\theta\] \[-v_{\chi}\lambda_{\chi H}\sin^{3}\theta].\] Following Eq.(4) we draw the Feynman diagrams for DM annihilation channels into SM particles and to the second Higgs in Fig.2. It is expected that the DM candidate is in thermal equilibrium with the SM degrees of freedom in the early universe. We therefore proceed to evaluate their abundance through the standard freeze-out mechanism. The Boltzmann equation, \[\dot{n}_{\phi}+3\textrm{H}n_{\phi}=-\langle\sigma v_{\phi\phi} \rangle\left(n_{\phi}^{2}-{n_{\phi}^{eq}}^{2}\right),\] (14) is employed for this purpose, where $n_{\phi}$ is the number density of the dark matter $\phi$, H is the Hubble parameter, $\langle\sigma v_{\phi\phi}\rangle$ represents the total annihilation cross-section as given by $\langle\sigma v_{\phi\phi}\rangle=\langle\sigma v_{\phi\phi\to SM,SM}\rangle+ \langle\sigma v_{\phi\phi\to H_{1}H_{2}}\rangle+\langle\sigma v_{\phi\phi\to H _{2}H_{2}}\rangle$. We consider here the RH neutrinos to be massive enough compared to the DM mass. So RH neutrinos do not participate in DM phenomenology. We have then used the MicrOmega package[80] to evaluate the final relic abundance of DM. <figure><img src="content_image/1706.04931/x1.png"><figcaption>Figure 2: Diagrams contributing to ϕϕ annihilation to SM particles and theother Higgs.</figcaption></figure> We have the following parameters in our set-up, \[\{m_{H_{1}},m_{H_{2}},m_{\textrm{DM}},\sin\theta,\lambda_{\chi \phi},\lambda_{\phi H},v,\tan\beta,\lambda_{\phi}\}.\] (15) The parameters $v_{\chi}$ is involved in the definition of $\tan\beta=v/v_{\chi}$. Parameters $(\lambda_{H},\lambda_{\chi},\lambda_{\chi H})$ can be written in terms of other parameters as shown in Eqs.(8,9,10). Among all the parameters in Eq.(15), $\lambda_{\phi}$ does not play any significant role in DM analysis. We first assume $H_{1}$ as the Higgs discovered at LHC _i.e._$m_{H_{1}}\sim$125.09 GeV[4] and the other Higgs is the heavier one ($m_{H_{2}}>m_{H_{1}}$). It would be appealing in view of LHC accessibility to keep $m_{H_{2}}$ below 1 TeV. In this case limits on $\sin\theta,\tan\beta$ are applicable as discussed in Section 3 depending on specific value of $m_{H_{2}}$[72]. Now in this regime (where $m_{H_{2}}$ is not too heavy, in particular $m_{H_{2}}<1$ TeV), $\sin\theta$ is bounded by $\sin\theta\lesssim 0.3$[72] and we have taken here a conservative choice by fixing $\sin\theta=0.2$. Note that in the small $\sin\theta$ approximation, $H_{1}$ is mostly dominated by the SM Higgs doublet $H$. In this limit the second term in Eq.(8) effectively provides the threshold correction to $\lambda_{H}$[82, 57, 81] which helps in achieving vacuum stability as we will see later. Furthermore considering this threshold effect to be equal or less than the first term in Eq.(8) (_i.e._ approximately the SM value of $\lambda_{H}$), we obtain an upper bound on $m_{H_{2}}$ as $m_{H_{2}}<\frac{m_{H_{1}}}{\tan\theta}$. Therefore in case with $m_{H_{2}}>m_{H_{1}}$, our working regime of $m_{H_{2}}$ can be considered within $\frac{m_{H_{1}}}{\tan\theta}>m_{H_{2}}>m_{H_{1}}$. We take $m_{H_{2}}$ to be 300 GeV for our analysis. Note that with small $\theta$, $\lambda_{\chi}$ almost coincides with the second term in Eq.(9). It is quite natural to keep the magnitude of a coupling below unity to maintain the perturbativity limit for all energy scales including its running. Hence with the demand $\lambda_{\chi}<1$, one finds $v_{\chi}>\sqrt{3}m_{H_{2}}$. To show it numerically, let us choose $\sin\theta=0.2$, then we obtain $125\textrm{ GeV}<m_{H_{2}}<620$ GeV. Therefore with $m_{H_{2}}=300$ GeV, a lower limit on $v_{\chi}\geq 520$ GeV can be set. We consider $v_{\chi}$ to be 800 GeV so that $\tan\beta$ turns out to be 0.307. On the other hand, if we consider the other Higgs to be lighter than the one discovered at LHC, we identify $m_{H_{2}}$ to be the one found at LHC and hence $m_{H_{1}}\leq 125$ GeV. Then Eq.(2) suggests $\sin\theta\to 1$ as the complete decoupling limit of the second Higgs. Following the analysis in [83, 72, 84, 85, 86, 87], we infer that most of the parameter space except for a very narrow region both in terms of mixing angle ($\sin\theta\sim 0.9$) and mass of the lighter Higgs ($m_{H_{1}}\sim 85-100$) GeV, is excluded from LEP and LHC searches. <figure><img src="content_image/1706.04931/DD1.png"><figcaption>Figure 3: Feynman diagram for DM Direct Detection.</figcaption></figure> Such a range is not suitable for our purpose as can bee seen from Eq.(8). In this large $\sin\theta$ limit, $\lambda_{H}$ gets the dominant contribution from the second term in Eq.(8) where the first term serves the purpose of threshold effect on $\lambda_{H}$. However $m_{H_{1}}$ being smaller than $m_{H_{2}}$ (the SM like Higgs), this effect would not be sufficient to enhance $\lambda_{H}$ such that its positivity till $M_{P}$ can be ensured. Therefore we discard the scenario $m_{H_{1}}<m_{H_{2}}$ (SM like Higgs) from our discussion. Hence the DM phenomenology basically depends on $m_{\textrm{DM}},\sin\theta,\lambda_{\chi\phi}$ and $\lambda_{\phi H}$. In a direct detection experiment, the DM scatters with the nucleon through the exchange of $H_{1}$ and $H_{2}$ as shown schematically in Fig.3 . The resulting spin-independent cross-section of DM-nucleon elastic scattering is given by [35] : \[\sigma_{n}^{SI}=\frac{f_{n}^{2}\mu_{n}^{2}m_{n}^{2}}{4\pi v^{2}m_ {\textrm{DM}}^{2}}\Big{[}\frac{\lambda_{1}\cos\theta}{m_{H_{1}}^{2}}+\frac{ \lambda_{2}\sin\theta}{m_{H_{2}}^{2}}\Big{]}^{2},\] (16) where $\mu_{n}=\frac{m_{n}m_{\textrm{DM}}}{m_{n}+m_{\textrm{DM}}},f_{n}=0.284$[88, 89]. The couplings appeared as $\lambda_{1},\lambda_{2}$ are specified in the list of vertices in Eq.(4). Below we discuss how we can estimate the relevant parameters ($\lambda_{\phi_{H}}$,$\lambda_{\chi\phi}$ and $m_{\textrm{DM}}$) from relic density and direct search limits. For this purpose, we consider $m_{H_{2}}=300$ GeV and $v_{\chi}=800$ GeV as reference values, unless otherwise mentioned. ### DM mass in region R1: [$150\textrm{ GeV}<m_{\textrm{DM}}\leq 500$ GeV] In this region any decay mode of $H_{1}$ and $H_{2}$ into DM is kinematically forbidden following our consideration for $m_{H_{2}}=300$ GeV. As stated before, we consider $m_{H_{1}}$ to be the SM like Higgs discovered at LHC, with $v_{\chi}=800$ GeV and $\tan\beta$ is fixed at 0.307. Therefore in order to satisfy the relic density $\Omega h^{2}=0.1161\pm 0.0028$[79], we first scan over $\lambda_{\phi H}$ and $\lambda_{\chi\phi}$ for different ranges of dark matter mass where $\sin\theta$ is kept fixed at 0.2. The allowed range of parameter space contributing to the relic abundance satisfying the correct relic density is indicated on $\lambda_{\phi H}-\lambda_{\chi\phi}$ plane in Fig.4 (in the top left panel), where different coloured patches indicate different ranges of $m_{\textrm{DM}}$. In the upper-right plot of Fig.4, the corresponding direct search cross sections for the relic density satisfied points obtained from the upper left plot (including the variation of $\lambda_{\phi H},\lambda_{\chi\phi}$) are provided. It can be clearly seen that many of these points lie below the LUX 2016[24] experimental limit for a wide range of dark matter mass (indicated by the colors depicted in the inset of Fig.4, upper left panel). <figure><img src="content_image/1706.04931/lux_lambdaL.png"><figcaption>Figure 4: Top left: Allowed points on λϕH-λχϕ plane for DM having mass150<mDM<500 GeV to satisfy correct order of relic density. Top right: Spinindependent nucleon cross section of DM has been plotted against the DM mass.Bottom panel: The top left plot has been constrained using recent LUX2016[24], Xenon 1T[25] limits to produce bottom-left figure and Xenon nT[27]predictions to get bottom-right figure.</figcaption></figure> From the top left panel of Fig.4, the relic density contour plot (with a particular $m_{\textrm{DM}}$) in $\lambda_{\chi\phi}$-$\lambda_{\phi H}$ plane shows that there exists a range of $\lambda_{\phi H}$ for which the plot is (almost) insensitive to the change in $\lambda_{\chi\phi}$. This becomes more prominent for plots associated with higher dark matter mass. In particular, the contour line satisfying the correct relic density with $m_{\textrm{DM}}=500$ GeV depicts a sharp variation in $\lambda_{\chi\phi}$ (below 0.4) with almost no variation of $\lambda_{\phi H}$ around 0.13. We now discuss the reason behind such a behaviour. We note that for $\lambda_{\phi H}>0.13$, the total annihilation cross section satisfying the relic density is mostly dominated by the $\phi\phi\rightarrow$ SM, SM process, specifically $\phi\phi\to WW,ZZ$ dominate. In our scenario, $\phi\phi\to WW,ZZ$ processes are mediated by both the Higgses, $H_{1}$ and $H_{2}$. Although $\lambda_{\chi\phi}$ is involved in the vertices characterizing these processes, it turns out that once both the $H_{1},H_{2}$ contributions are taken into account, the $\lambda_{\chi\phi}$ dependence is effectively canceled leaving the $\phi\phi\to WW,ZZ$ annihilation almost independent of $\lambda_{\chi\phi}$. Hence $\phi\phi\rightarrow$ SM, SM depends mostly on $\lambda_{\phi H}$. The other processes like $\phi\phi\to H_{1}H_{2}(H_{2}H_{2})$ are subdominant (these are allowed provided $m_{\textrm{DM}}>212.5(300)$ GeV) in this region with large $\lambda_{\phi H}$. Then the total cross section $\langle\sigma v_{\phi\phi}\rangle$ and hence the relic density contour line becomes insensitive to the change in $\lambda_{\chi\phi}$ as long as it remains below 0.4 while $\lambda_{\phi H}>0.13$. This is evident in the top left panel of Fig.4. Similar effects are seen in case of lower $m_{\textrm{DM}}$ ($<500$ GeV) as well. Once we keep on decreasing $\lambda_{\phi H}$ below 0.13, it turns out that $\phi\phi\rightarrow$ SM, SM becomes less important compared to the $\phi\phi\to H_{2}H_{2}$ (in particular the $t$ channel) with $\lambda_{\chi\phi}$ beyond 0.4 (in case of $m_{\textrm{DM}}=500$ GeV). Note that the plot shows the insensitiveness related to $\lambda_{\phi H}$ in this low $\lambda_{\phi H}$ region for obvious reason. Similar results follow with $m_{\textrm{DM}}<300$ GeV also, where $\phi\phi\to H_{1}H_{2}$ provides the dominant contribution in $\langle\sigma v_{\phi\phi}\rangle$. Based on our discussion so far we note that for $\lambda_{\chi\phi}\gg\lambda_{\phi H}$ the channels with Higgses in the final states contribute more to total $\langle\sigma v_{\phi\phi}\rangle$. On the other hand for low values of $\lambda_{\chi\phi}$ (although comparable to $\lambda_{\phi H}$), the model resembles the usual Higgs portal dark matter scenario where W bosons in the final state dominate. To summarize, * $150\textrm{ GeV}<m_{\textrm{DM}}<212.5$ GeV: For low $\lambda_{\chi\phi}$, $\phi\phi\to W^{+}W^{-}$ dominates. However for large $\lambda_{\chi\phi}$, $\phi\phi\to H_{1}H_{1}$ becomes the main annihilation channel. * $212.5\textrm{ GeV}<m_{\textrm{DM}}<300$ GeV: New annihilation process $\phi\phi\to H_{1}H_{2}$ opens up. This with $\phi\phi\to H_{1}H_{1}$ contribute dominantly for large $\lambda_{\chi\phi}$. Otherwise the channels with SM particles in final states dominate. * $300\textrm{ GeV}<m_{\textrm{DM}}<500$ GeV: The annihilation channel $\phi\phi\to H_{2}H_{2}$ opens up in addition to $H_{1}H_{1}$ and $H_{1}H_{2}$ in the final states. Their relative contributions to total $\langle\sigma v_{\phi\phi}\rangle$ again depend on the value of $\lambda_{\chi\phi}$. <figure><img src="content_image/1706.04931/x2.png"><figcaption>Figure 5: DM relic density contour lines in λϕH-λχϕ plane with mDM=299 (red),305 GeV (green).</figcaption></figure> In the top left panel of Fig.4, we also note the existence of a small overlapped region when $\lambda_{\phi H}\ll\lambda_{\chi\phi}$ for the dark matter mass regions between 280-300 GeV and 300-310 GeV. This has been further clarified in Fig.5, where we note that relic density contour lines with $m_{\textrm{DM}}=299$ GeV and $m_{\textrm{DM}}=305$ GeV intersect each other around $\lambda_{\phi H}\sim 0.05$ and $\lambda_{\chi\phi}\sim 0.21$. Note that when DM mass $m_{\textrm{DM}}\geq m_{H_{2}}=300$ GeV, in addition to the $\phi\phi\to SM,SM$ and $\phi\phi\to H_{1}H_{2}$ annihilation processes, $\phi\phi\to H_{2}H_{2}$ opens up and contribute to the total annihilation cross section ( this new channel can be realized through both $H_{1}$ and $H_{2}$ mediation). Then total annihilation cross section will be enhanced for $m_{\textrm{DM}}>300$ GeV case, _i.e_$\langle\sigma v_{\phi\phi}\rangle=\langle\sigma v\rangle_{\phi\phi\to SM ,SM}+\langle\sigma v\rangle_{\phi\phi\to H_{1},H_{2}}+\langle\sigma v \rangle_{\phi\phi\to H_{2},H_{2}}$ becomes large compared to the $280\textrm{ GeV}<m_{\textrm{DM}}<300$ GeV mass range where $\langle\sigma v\rangle_{\phi\phi\to H_{2}H_{2}}$ is not present. This enhancement has to be nullified in order to realize the correct relic density and this is achieved by reducing $\lambda_{\chi\phi}$ compared to its required value for a fixed $\lambda_{\phi H}$ and $m_{\textrm{DM}}$ in $280\textrm{ GeV}\leq m_{\textrm{DM}}<300$ region. Note that in view of our previous discussion, we already understand that $\phi\phi\to H_{2}H_{2}$ becomes important compared to $\phi\phi\rightarrow$ SM, SM process in the region with $\lambda_{\chi\phi}\gg\lambda_{\phi H}$. Hence the two mass regions (below and above 300 GeV) overlap in $\lambda_{\phi H}-\lambda_{\chi\phi}$ plane as seen in the top left panel of Fig.4 as well in Fig.5. The total annihilation cross section of DM depends on its mass also. However the small mass differences between the two overlapped regions have very mild effect on $\langle\sigma v\rangle_{\textrm{Tot}}$. Similar effect should be observed below and above $m_{\textrm{DM}}\sim(m_{H_{1}}+m_{H_{2}})/2=$ 212.5 GeV as $\phi\phi\to H_{1}H_{2}$ opens up there. However we find that around the $m_{\textrm{DM}}=212.5$ GeV, even with $\lambda_{\chi\phi}\gg\lambda_{\phi H}$, the contribution from this particular channel to $\langle\sigma v\rangle_{\textrm{Tot}}$ is negligible as compared to $\phi\phi\rightarrow$ SM SM contribution and hence we do not observe any such overlapped region there. In the top right panel of Fig.4 we provide the spin-independent (SI) direct detection (DD) cross sections corresponding to the points in the left panel satisfying relic density data having different range of dark matter masses as indicated by the colored patches. We further put the LUX 2016[24], XENON 1T[25] and nT (expected) lines on it. As known, for a lowerer cross section, it reaches the neutrino floor where signals from DM can not be distinguished from that of neutrino. We find that the scenario works with reasonable values of the parameters, _i.e._ not with any un-naturally small or large values of couplings. Note that once we use the XENON 1T[25] and projected XENON nT[27] limits on the scattering cross section, we would obtain more restricted region of parameter space for $\lambda_{\phi H}-\lambda_{\chi\phi}$ as shown in left (with XENON 1T[25]) and right (with XENON nT[27]) figures of the bottom panel. From the plot with XENON-nT prediction, we find that the scenario works even with reasonably large values of $\lambda_{\phi H},\leavevmode\nobreak\ \lambda_{\chi\phi}$ required for satisfying the relic density, although they are comparable to each other. This is because of the fact that to keep the direct detection cross section relatively small (even smaller than the XENON nT), it requires a cancellation between $\lambda_{\phi H}$ and $\lambda_{\chi\phi}$ as can be seen from Eq.(16) in conjugation with definition of $\lambda_{1}$ and $\lambda_{2}$ for a specific $\sin\theta=0.2$ value. Such a cancellation is not that important for plots with LUX 2016[24] or XENON 1T[25] results and hence showing a wider region of parameter space for $\lambda_{\chi\phi}$ and $\lambda_{\phi H}$. <figure><img src="content_image/1706.04931/luxvxA.png"><figcaption>Figure 6: Allowed parameter space to satisfy correct relic abundance inλϕH−λχϕ plane with different vχ for mDM=300 GeV . Other parameters mH2=300 GeVand and sinθ=0.2 have been kept fixed. The LUX 2016[24] allowed region arealso accommodated (solid black region) in the figures.</figcaption></figure> It can be concluded from upper panel of Fig.4 that the presence of additional singlet scalar field $\chi$ helps in reducing the magnitude of $\lambda_{\phi H}$ that was required (say $\lambda^{0}_{\phi H}$) to produce correct relic density in minimal form of singlet scalar DM or in other words it dilutes the pressure on $\lambda_{\phi H}$ to produce correct relic density and to satisfy DD cross section simultaneously. For illustrative purpose, let us choose a dark matter mass with 500 GeV. From Fig.1, we found that in order to satisfy the relic density, we need to have a $\lambda^{0}_{\phi H}\sim$ 0.15 which can even be 0.02 in case with large $\lambda_{\chi\phi}\sim 0.6$. Similarly we notice that for $m_{\textrm{DM}}=300$ GeV, $\lambda_{\phi H}^{0}$ was 0.086 in order to produce correct relic density which however was excluded from direct search point of view. This conclusion changes in presence of $\lambda_{\chi\phi}$ as we can see from Fig.4, (left panel) that $m_{\textrm{DM}}=300$ GeV can produce correct relic density and evade the direct search limit with smaller $\lambda_{\phi H}:0.065-0.086$. This is possible in presence of nonzero $\lambda_{\chi H}$ and small $\sin\theta$($\sim 0.2$ here) which redistribute the previously obtained value of $\lambda^{0}_{\phi H}$ into $\lambda_{\phi H}$ and $\lambda_{\chi\phi}$ while simultaneously brings the direct search cross section less than the experimental limit due to its association with $\sin\theta$ (see the definition of $\lambda_{1}$ and $\lambda_{1}$). <figure><img src="content_image/1706.04931/omegaA2.png"><figcaption>Figure 7: Relic density vs mDM plot in the combined set up of SM+DM+RHneutrinos and χ field for two different specified range of of λϕH and λχϕ asmentioned within the inset of figures. Two resonances are clearly visible atmDM=mH1/2 and mH2/2 respectively Blue patch represents the favoured region byLUX 2016 direct detection cross section limit whereas red patch is excluded byLUX 2016.</figcaption></figure> <figure><img src="content_image/1706.04931/luxsinA.png"><figcaption>Figure 8: Allowed parameter space to satisfy correct relic abundance inλϕH−λχϕ plane with different values of sinθ for mDM=300 GeV. Other parametersmH2=300 GeV and vχ=800 GeV have been kept fixed. The LUX 2016[24] allowedregion are also accommodated (solid black region) in the figures. The blue dotpoint denoted by X in right panel will be used as a reference point for studyon Higgs vacuum stability.</figcaption></figure> In Fig.7 (left panel), we show the relic density versus $m_{\textrm{DM}}$ plot with our chosen set of parameters, $\{m_{H_{2}}=300\textrm{ GeV}$ ,$m_{H_{1}}=125.09$ GeV, $\tan\beta=0.307$, $\sin\theta=0.2\}$ while varying $\lambda_{\chi H}$ and $\lambda_{\phi H}$ within $0.16\leq\lambda_{\chi\phi}\leq 0.17$ and $0.05\leq\lambda_{\phi H}\leq 0.06$. Similarly in right panel, we provide the relic density vs $m_{\textrm{DM}}$ plot for a different range of $\lambda_{\chi H}$ and $\lambda_{\phi H}$. We note that there are two resonance regions, one at $m_{H_{1}}/2$ for SM like Higgs and other at $m_{H_{2}}/2$ with heavy Higgs⁷ mass at 300 GeV. In left panel for DM heavier than 150 GeV, we find $m_{\textrm{DM}}\sim 300$ GeV can correctly produce the relic density in the observed range and simultaneously evade the DD limit set by LUX 2016[24]. This result is consistent with the plot in Fig.4. Similarly $m_{\textrm{DM}}\sim 500$ GeV is in the acceptable range, which is in line with observation in Fig.4. In the left panel of Fig.7 we also have another region of DM mass$\sim 75$ GeV having correct relic abundance however discarded by LUX 2016. The region was not incorporated in top left panel of Fig.4 as we have started with $m_{\textrm{DM}}$ bigger than 150 GeV only. The possibility of having dark matter lighter than 150 GeV in the present scenario will be discussed in the next subsection. Since in obtaining the Fig.4, we have fixed $\sin\theta,\tan\beta$ and $m_{H_{2}}$, below in Fig.6 and 8, we provide the expected range of two couplings $\lambda_{\chi H}$ and $\lambda_{\phi H}$ when $\sin\theta,\tan\beta$ are varied for dark matter mass $m_{\textrm{DM}}=300$ GeV . We find the variation is little sensitive with the change of both $v_{\chi}$ and $\sin\theta$. As $v_{\chi}$ or $\sin\theta$ increases for $m_{\textrm{DM}}=300$ GeV, it requires less $\lambda_{\chi\phi}$ for a particular $\lambda_{\phi H}$ to satisfy the relic density. We have also applied the LUX 2016[24] DD cross section limit in those plots and are indicated by solid black patches. In Fig.8, one dark blue dot has been put on the $\sin\theta=0.2$ contour which will be used in study of Higgs vacuum stability as a reference point. [FOOTNOTE:7][ENDFOOTNOTE] <figure><img src="content_image/1706.04931/ESDMLOWLux.png"><figcaption>Figure 9: Relic density satisfied points in λϕH−λχϕ plane for mDM<150 GeV withDD cross section consistent with [left panel] LUX 2016[24] and [right panel]XENON 1T limit[25]. Benchmark points: mH2=300 GeV, vχ=800 GeV, sinθ=0.2.</figcaption></figure> <figure><img src="content_image/1706.04931/x3.png"><figcaption>Figure 10: mDM vs sinθ plot for a fixed λχϕ and λϕH as mentioned in the figureto satisfy the correct relic abundance and direct detection cross sectionconsistent with LUX 2016 limit. Values of other parameters: mH2=300 GeV,λχϕ=0.2 and vχ=800 GeV.</figcaption></figure> ### DM mass in region R2: ($m_{\textrm{DM}}<150$ GeV) Here we briefly discuss the DM phenomenology in the low mass region $m_{\textrm{DM}}<\frac{m_{H_{2}}}{2}=150$ GeV. In this region, the decay process of heavy higgs to DM ($H_{2}\rightarrow\phi\phi$) will be active. For further low $m_{\textrm{DM}}<m_{H_{1}}/2\simeq 62.5$, both $H_{2}\rightarrow\phi\phi$ and $H_{1}\rightarrow\phi\phi$ decay modes will be present. We perform a scan over the $\lambda_{\phi H}-\lambda_{\chi\phi}$ region to find the correct relic density satisfied parameter space with allowed direct detection cross section from LUX 2016[24] and XENON 1T experiments[25]. The results are shown in Fig.9, left and right panels where DD limits from LUX 2016 [left panel] and XENON 1T (preliminary) [right panel] are considered separately. In doing these plots, we have considered different mass ranges as indicated by different colors. The color codes are depicted within the inset of each figures. We note that the required $\lambda_{\chi\phi}$, $\lambda_{\phi H}$ values are almost in the similar range as obtained in Fig.4. We also note that there exists a resonance region through $H_{1}$ near $m_{\textrm{DM}}\sim 63$ GeV, indicated by the blue patch. In this resonance region, the relic density becomes insensitive to the coupling and hence the blue patch is extended over the entire region of $\lambda_{\chi\phi}$, $\lambda_{\phi H}$ in the Fig.9. Finally we attempt to estimate the $\sin\theta$ required to provide the correct amount of modification over the minimal version of a real singlet DM having interaction with SM Higgs only in order to revive the ‘below 500 GeV’ DM into picture. In other words, the amount of $\sin\theta$ should be enough to satisfy correct relic abundance and DD cross section limits of LUX 2016[24] and XENON 1T[25] for this particular mass range. To do the analysis, we fix $\lambda_{\chi\phi}\sim 0.2$ while three different values of $\lambda_{\phi H}$ at 0.04, 0.08 and 0.10 are considered for the study. We then provide the $\sin\theta$ versus $m_{\textrm{DM}}$ plot in Fig.10 which is consistent with relic density and LUX 2016 limits. We infer that a sizable value of $\sin\theta$ is required for this. With $\lambda_{\phi H}=0.1$, we have noted earlier from Fig. 1 that it alone reproduces the desired relic density with a 330 GeV dark matter, although excluded by LUX 2016 limits. Now we observe from Fig. 10 that in order to make this as a viable DM mass, we need to have a $\sin\theta=\mathcal{O}$(0.1) with $\lambda_{\phi H}=0.1$. Such a moderate value of $\sin\theta$ is compatible with LEP and LHC results. A larger value of $\sin\theta$$\sim\mathcal{O}$(0.3) with $\lambda_{\phi H}=0.1$ can accommodate DM mass around 440 GeV as seen from the Fig. 10. Similarly, we indicate that with $\lambda_{\phi H}=0.08[0.04]$ (for which DM mass $\sim$$270$ GeV and $110$ GeV satisfy the relic density as seen from Fig. 1), $\sin\theta$ variation covers a range of DM mass $\sim$ 330-370 GeV [240-290 GeV] provided we restrict ourselves upto $\sin\theta=0.3$. ## 5 Vacuum stability In this section, we will discuss how the EW vacuum stability can be achieved in our model. For clarification purpose and a comparative study of it, we first discuss how the presence of different ingredients (three RH neutrinos, DM and extra scalar $\chi$) can affect the running of the Higgs quartic coupling when added one after other. We first comment on the inclusion of the RH neutrinos and investigate the running of $\lambda_{H}$. Then we study how the involvement of the scalar singlet DM field $\phi$ can alter the conclusion. Finally we discuss the result corresponding to our set-up, $i.e.$ including the $\chi$ field as well. In doing this analysis, the absolute stability of the Higgs vacuum is ensured by $\lambda_{H}(\mu)>0$ for any energy scale $\mu$ where the EW minimum of the scalar potential is the global minimum. However there may exist another minimum which is deeper than the EW one. In that case we need to calculate the tunneling probability of the EW vacuum to the second minimum. The Universe will be in metastable state provided the decay time of EW vacuum is longer than the age of the universe. The tunneling probability is given by[5, 6], \[\mathcal{P}=T_{U}^{4}\mu_{B}^{4}e^{-\frac{8\pi^{2}}{3\|\lambda_{H} (\mu_{B})\|}},\] (17) where $T_{U}$ is the age of the universe. $\mu_{B}$ is the scale at which probability is maximized, determined from $\beta_{\lambda_{H}}(\mu_{B})=0$. Hence for metastable Universe requires[5] \[\lambda_{H}(\mu_{B})>\frac{-0.065}{1-0.01\ln\Big{(}\frac{v}{\mu_{ B}}\Big{)}},\] (18) where $T_{U}\simeq 10^{14}$ yr is used. As noted in [6], for $\mu_{B}>M_{P}$, one can safely consider $\lambda_{H}(\mu_{B})=\lambda_{H}(M_{P}$). Before proceeding further, some discussion on the involvement of light neutrino mass in the context of vacuum stbaility is pertinent here. As stated before, the light neutrino mass is generated through type-I seesaw for which three RH neutrinos are included in the set up. We now describe the strategy that we adopt here in order to study their impact on RG evolution. For simplicity, the RH neutrino mass matrix $M_{N}$ is considered to be diagonal with degenerate entries, _i.e._$M_{i=1,2,3}=M_{R}$. As we will see, it is $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ which enters in the $\beta$ function of the relevant couplings. In order to extract the information on $Y_{\nu}$, we employ the type-I mass formula $m_{\nu}=Y_{\nu}^{T}Y_{\nu}\frac{v^{2}}{2M_{R}}$. Naively one would expect that large Yukawas are possible only with very large RH neutrino masses. For example with $M_{R}\sim 10^{14}$ GeV, $Y_{\nu}$ comes out to be 0.3 in order to obtain $m_{\nu}\simeq 0.05$ eV. Contrary to our naive expectation, it can be shown that even with smaller $M_{R}$ one can achieve large values of $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ once a special flavor structure of $Y_{\nu}$ is considered[38]. Note that we aim to study the EW vacuum stability in presence of large value of $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$. For this purpose, we use the parametrization by [90] and write $Y_{\nu}$ as \[Y_{\nu}=\sqrt{2}\frac{\sqrt{M_{R}}}{v}\mathcal{R}\sqrt{m_{\nu}^{ d}}\leavevmode\nobreak\ U^{\dagger}_{\textrm{PMNS}},\] (19) where $m_{\nu}^{d}$ is the diagonal light neutrino mass matrix and $U_{\textrm{PMNS}}$ is the unitary matrix diagonalizing the neutrino mass matrix $m_{\nu}$ such that $m_{\nu}=U^{}_{\textrm{PMNS}}m_{\nu}^{d}U^{\dagger}_{\textrm{PMNS}}$. Here $\mathcal{R}$ represents a complex orthogonal matrix which can be written as $\mathcal{R}=O\textrm{exp}(i\mathcal{A})$ with $O$ as real orthogonal and $\mathcal{A}$ as real antisymmetric matrices respectively. Hence one gets \[\textrm{ Tr}[Y_{\nu}^{\dagger}Y_{\nu}]=\frac{2M_{R}}{v^{2}} \textrm{Tr}\Big{[}\sqrt{m_{\nu}^{d}}e^{2i\mathcal{A}}\sqrt{m_{\nu}^{d}}\Big{]}.\] (20) Note that the real antisymmetric matrix $\mathcal{A}$ does not appear in the seesaw expression for $m_{\nu}=\frac{Y_{\nu}^{T}Y_{\nu}v^{2}}{2M_{R}}$. Therefore with any suitable choice of $\mathcal{A}$, it would actually be possible to have sizeable Yukawas even with light $M_{R}$ and hence this can affect the RG evolution of $\lambda_{H}$ significantly. As an example, let us consider magnitude of all the entries of $\mathcal{A}$ to be equal, say $a$ with all diagonal entries as zero. Then with $M_{R}$ = 1 TeV, $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ can be as large as 1 with $a=8.1$[90, 91]. Below we specify the details of Higgs vacuum stability in presence of RH neutrinos only. ### Higgs vacuum stability with right-handed neutrinos In presence of the RH neutrino Yukawa coupling $Y_{\nu}$, the renormalization group (RG) equation of SM couplings will be modified[92]. Below we present the one loop beta functions of Higgs quartic coupling $\lambda_{H}$, top quark Yukawa coupling $y_{t}$ and neutrino Yukawa coupling $Y_{\nu}$, \[\frac{d\lambda_{H}}{d\textrm{ln}\mu}=\frac{1}{16\pi^{2}}\{\beta_{ \lambda_{H}}^{SM}+\beta_{\lambda_{H}}^{\textrm{I}}\}\textrm{ with }\beta_{ \lambda_{H}}^{\textrm{I}}=4\lambda_{H}\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]-2 \textrm{Tr}[(Y_{\nu}^{\dagger}Y_{\nu})^{2}]\,,\] (21) \[\frac{dy_{t}}{d\textrm{ln}\mu}=\frac{1}{16\pi^{2}}\{\beta_{y_{t}} ^{SM}+\beta_{y_{t}}^{\textrm{I}}\}\textrm{ with }\beta_{y_{t}}^{\textrm{I}}= \textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]y_{t},\] (22) \[\frac{d\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]}{d\textrm{ln}\mu}= \frac{1}{16\pi^{2}}\beta_{\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]}^{\textrm{I}}= \frac{1}{16\pi^{2}}\Big{\{}(6y_{t}^{2}+2\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]- \frac{3}{2}g_{1}^{2}-\frac{9}{2}g_{2}^{2})\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu} ]+3\textrm{Tr}[(Y_{\nu}^{\dagger}Y_{\nu})^{2}]\Big{\}},\] (23) where $\beta_{\lambda_{H}}^{SM}$ and $\beta_{y_{t}}^{SM}$ represent the $\beta$ functions of $\lambda_{H}$ and $y_{t}$ respectively in SM. The $Y_{\nu}$ dependence is to be evaluated in accordance with the type-I seesaw expression, $m_{\nu}=Y_{\nu}^{T}Y_{\nu}\frac{v^{2}}{M_{R}}$. <figure><img src="content_image/1706.04931/x4.png"><figcaption>Figure 11: RG running of λH with energy scale μ in SM + RH neutrinos;[Leftpanel]: different RH neutrino mass scales MR are considered with fixedmt=173.2 GeV, [Right panel]: different top masses are considered with MR=108GeV.</figcaption></figure> Also with large $a$ (elements of $\mathcal{A}$), it is found [38] that Tr$[(Y_{\nu}^{\dagger}Y_{\nu})^{2}]\simeq\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]^{2}$ and we will be using this approximated relation in obtaining the running of the couplings through Eqs.(21,22,23). Here we have used the best fit values of neutrino oscillation parameters for normal hierarchy[93, 94]. We have also considered the mass of lightest neutrino to be zero. <figure><img src="content_image/1706.04931/x6.png"><figcaption>Figure 12: Region plot for mt-Tr[Y†νYν] in the SM, extended with RH neutrinoshaving degenerate mass MR=108 GeV. The plane is divided into three categories(i) absolute stability, (ii) metastability and (iii) instability.</figcaption></figure> Note that just like the top quark Yukawa coupling, the neutrino Dirac Yukawa is having a similar impact on the Higgs quartic coupling, in particular with large $Y_{\nu}$. Also the top quark Yukawa would have a contribution dependent on $Y_{\nu}$. This has been studied in several works[36, 38, 39, 40, 41, 42, 43, 44, 45, 46]. We summarize here the results with some benchmark values of RH neutrino masses. These will be useful for a comparative study with the results specific to our model. In Fig.11 (left panel), we have plotted running of the Higgs quartic coupling $\lambda_{H}$ against energy scale $\mu$ till $M_{P}$ for different choices of $M_{R}=10^{3},10^{8}$ and $10^{14}$ GeV with $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]=0.5$ denoted by red, black and green solid lines respectively. The pink shaded portion represents the instability region given by the inequality [5]$\lambda_{H}\leq{-0.065}/[1-0.01\textrm{ln}\Big{(}\frac{v}{\mu}\Big{)}]$. As expected, we find that the Higgs quartic coupling enters into the instability region well before the Planck scale. Scale \| yt \| g1 \| g2 \| g3 \| λH ---\|---\|---\|---\|---\|--- μ=mt \| 0.93610 \| 0.357606 \| 0.648216 \| 1.16655 \| 0.125932 Table 1: Values of the relevant SM couplings (top-quark Yukawa yt, gauge couplings gi and λH) at energy scale μ=mt=173.2 GeV with mh=125.09 GeV and αS(mZ)=0.1184. In the right panel of Fig.11, the effect of choosing different $m_{t}$ within the present $2\sigma$ uncertainty is shown for a fixed $M_{R}=10^{8}$ GeV. The black solid, dashed and dotted lines represent the $\lambda_{H}$ running with $m_{t}$ as 173.2 GeV, 177 GeV and 171 GeV respectively. In doing this analysis, we fix the initial values of all SM couplings [6] as given in Table 1 at an energy scale $\mu=m_{t}$. Here we consider $m_{h}=125.09$ GeV, $m_{t}=173.2$ GeV and $\alpha_{s}=0.1184$. In Fig.12, we have shown a region plot for $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ and $m_{t}$ with fixed $M_{R}$ at $10^{8}$ GeV in terms of stability ($\lambda_{H}$ remains positive all the way upto $M_{P}$), metastability and instability of the EW vacuum of the SM. The top quark mass is varied between 168 GeV to 178 GeV. The region in which EW vacuum is stable is indicated by green and the metastable region is indicated by white patches. The instability region is denoted with pink shaded part. It can be noted that the result coincides with the one obtained in [41]. We aim to discuss the change obtained over this diagram in the context of our model. ### Higgs vacuum stability from Higgs Portal DM and RH neutrinos Here we discuss the vacuum stability scenario in presence of both the scalar DM ($\phi$) and three RH neutrinos ($N$). In that case, effective scalar potential becomes $V_{\textrm{I}}+V_{H}$ only. Note that the DM phenomenology is essentially unaffected from the inclusion of the heavy RH neutrinos with the assumption $M_{R}\gg m_{\textrm{DM}}$. On the other hand combining Eq.(21,22,23), we obtain the corresponding beta functions for the couplings as provided below; \[\frac{d\lambda_{H}}{dt} =\frac{1}{16\pi^{2}}\{\beta_{\lambda_{H}}^{SM}+\beta_{\lambda_{H} }^{\textrm{I}}+\beta_{\lambda_{H}}^{\textrm{II}}\}\textrm{ where }\beta_{ \lambda_{H}}^{\textrm{II}}=\frac{\lambda_{\phi H}^{2}}{2},\] (24) \[\frac{d\lambda_{\phi H}}{dt} =\frac{1}{16\pi^{2}}\beta_{\lambda_{\phi H}}^{\textrm{I}}=\frac{1 }{16\pi^{2}}\Big{\{}12\lambda_{H}\lambda_{\phi H}+\lambda_{\phi}\lambda_{\phi H }+4\lambda_{\phi H}^{2}+6y_{t}^{2}\lambda_{\phi H}-\frac{3}{2}g_{1}^{2}\lambda _{\phi H}-\frac{9}{2}g_{2}^{2}\lambda_{\chi H}+2\textrm{Tr}[Y_{\nu}^{\dagger}Y _{\nu}]\lambda_{\phi H}\Big{\}},\] (25) \[\frac{d\lambda_{\phi}}{dt} =\frac{1}{16\pi^{2}}\beta_{\lambda_{\phi}}^{\textrm{I}}=\frac{1}{ 16\pi^{2}}\Big{\{}3\lambda_{\phi}^{2}+12\lambda_{\phi H}^{2}\Big{\}}.\] From the additional term $\beta_{\lambda_{H}}^{\textrm{II}}$, we expect that the involvement of DM would affect the EW vacuum stability in a positive way ($i.e.$ pushing the vacuum more toward the stability) as shown in [29, 30, 31, 32, 33, 34] whereas we noted in the previous subsection that the Yukawa coupling (if sizable) has a negative impact on it. The interplay between the neutrino Yukawa coupling and Higgs portal coupling with DM is shown in Fig. 13, left and right panels (top and bottom). For the purpose of comparison, we have kept the same set of choices of parameters as in Fig.11, (left and right panels ). For the top panels, we consider mass of the dark matter to be $m_{\textrm{DM}}=300$ GeV and for the bottom set, $m_{\textrm{DM}}=920$ GeV is taken. The choice of $m_{\textrm{DM}}$ could in turn fix the $\lambda_{\phi H}$ coupling from the relic density plot of Fig.1. For example with $m_{\textrm{DM}}=300$ GeV $\lambda_{\phi H}$ is 0.075 and for $m_{\textrm{DM}}=920$ GeV, $\lambda_{\phi H}$ is given by 0.286 value. It is evident that the presence of Higgs portal coupling only has a mild effect as compared to the impact created by the neutrino Yukawa coupling. Finally in Fig.14 we provide the region plot in Tr[$Y^{\dagger}_{\nu}Y_{\nu}$] - $m_{t}$ plane where the stable and instable regions are indicated by green and pink patches. This plot while compared with Fig.12, indicates that there is no such noticeable improvement except the mild enhancement of the metastable region due to the involvement of singlet scalar (DM) with Higgs portal coupling. With an aim to accommodate both the massive neutrinos and a relatively light dark matter ($<$ 500 GeV), we move on to the next section where the $\chi$ field is included. <figure><img src="content_image/1706.04931/x7.png"><figcaption>Figure 13: RG evolution of λH with energy scale μ with SM+DM+RH Neutrinos withλϕ=0.7, mH=125.09 GeV and αS(mZ)=0.1184.: (a) [top panels]: mDM=300 GeV, and(b) [bottom panels]: mDM=920 GeV. In left panels mt is fixed at 173.2 GeV andplots are there with different MR while in right panel MR is fixed at 108 GeVand different mt values are considered.</figcaption></figure> <figure><img src="content_image/1706.04931/x11.png"><figcaption>Figure 14: Regions of Stability, metastability and instability in SM+DM+RHneutrinos case in the Tr[Y†νYν] -mt plane for mDM=300 GeV (left panel) and 920GeV (right panel). We consider λϕ=0.7, mH=125.09 GeV and αS(mZ)=0.1184 forboth the figures.</figcaption></figure> ### EW vacuum stability in extended Higgs portal DM and RH neutrinos Turning into the discussion on vacuum stability in our framework of extended Higgs portal having three RH neutrinos, DM and the $\chi$ fields, we first put together the relevant RG equations (for $\mu>m_{\textrm{DM}},m_{H_{2}}$) as given by, \[\frac{d\lambda_{H}}{dt} =\frac{1}{16\pi^{2}}\{\beta_{\lambda_{H}}^{SM}+\beta_{\lambda_{H} }^{\textrm{I}}+\beta_{\lambda_{H}}^{\textrm{II}}+\frac{\lambda_{\chi H}^{2}}{2 }\},\] (27) \[\frac{d\lambda_{\phi H}}{dt} =\frac{1}{16\pi^{2}}\{\beta_{\lambda_{\phi H}}^{\textrm{I}}+ \lambda_{\chi\phi}\lambda_{\chi H}\},\] (28) \[\frac{d\lambda_{\phi}}{dt} =\frac{1}{16\pi^{2}}\{\beta_{\lambda_{\phi}}^{\textrm{I}}+3 \lambda_{\chi\phi}^{2}\},\] (29) \[\frac{d\lambda_{\chi H}}{dt} =\frac{1}{16\pi^{2}}\Big{\{}12\lambda_{H}\lambda_{\chi H}+\lambda _{\chi}\lambda_{\chi H}+4\lambda_{\chi H}^{2}+6y_{t}^{2}\lambda_{\chi H}-\frac {3}{2}g_{1}^{2}\lambda_{\chi H}-\frac{9}{2}g_{2}^{2}\lambda_{\chi H}+\lambda_{ \chi\phi}\lambda_{\phi H}+2\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]\lambda_{\chi H }\Big{\}},\] \[\frac{d\lambda_{\chi}}{dt} =\frac{1}{16\pi^{2}}\Big{\{}3\lambda_{\chi}^{2}+12\lambda_{\chi H }^{2}+3\lambda_{\chi\phi}^{2}\Big{\}},\] \[\frac{d\lambda_{\chi\phi}}{dt} =\frac{1}{16\pi^{2}}\Big{\{}4\lambda_{\chi\phi}^{2}+\lambda_{\chi \phi}(\lambda_{\phi}+\lambda_{\chi})+4\lambda_{\phi H}\lambda_{\chi H}\Big{\}}.\] (30) We note that the couplings $\lambda_{\chi\phi},\lambda_{\phi H}$ and $\lambda_{\chi H}$ which played important role in DM phenomenology, are involved in the running of couplings as well. From the discussion of the DM section, we have estimated these parameters in a range so as to satisfy the appropriate relic density and be within the direct search limits for a specific choice of other parameters at their reference values: $m_{H_{2}}=300$ GeV and $v_{\chi}=800$ GeV, $\sin\theta=0.2$ (henceforth we describe this set as $A$). In particular an estimate for $\lambda_{\chi\phi},\lambda_{\phi H}$ are obtained from Fig.4 (for 150 GeV$<m_{\textrm{DM}}<$ 500 GeV) and from Fig. 9 (for $m_{\textrm{DM}}<$ 150 GeV) having different choices of $m_{\textrm{DM}}$ and $\sin\theta$. The parameter $\lambda_{\chi H}$ dependence is mostly realized through $\sin\theta$ following Eq.(10), where $m_{H_{2}},\tan\beta$ are fixed from set $A$. This $\sin\theta$ is the most crucial parameter which control both the DM phenomenology and the vacuum stability. We have already seen that it allows the scalar singlet DM to be viable for the low mass window by relaxing $\lambda_{\phi H}$ from its sole role in case of single scalar singlet DM. On the other hand, a non-zero $\sin\theta$ provides a positive shift (it is effectively the threshold effect in the small $\theta$ limit as seen from Eq. (8)) to the Higgs quartic coupling and hence guides the $\lambda_{H}$ toward more stability. Hence $\sin\theta$ would be a crucial parameter in this study. Note that the RH neutrinos being relatively heavy as compared to the DM, neutrino Yukawa coupling does not play much role in DM phenomenology. <figure><img src="content_image/1706.04931/x13.png"><figcaption>Figure 15: RG running of λH vs μ in the combined scenario of SM+RHNeutrinos+DM+χ field with mDM=300 GeV, sinθ=0.2 and mH2=300 GeV. In [leftpannel] mt (∼173.2 GeV) is kept fixed, MR is varied, and in [right panel] MR(∼108 GeV) is fixed, mt has been varied. Point X (λϕH=0.06, λχϕ=0.135) fromFig.8 and λϕ=0.7 have been used as benchmark points.</figcaption></figure> Assuming the validity of this extended SM (with three RH neutrinos and two singlets, $\phi,\chi$) upto the Planck scale, <figure><img src="content_image/1706.04931/x15.png"><figcaption>Figure 16: RG running of λH with energy scale μ for different values of sinθin the combined set up of SM+DM+RH neutrinos+χ field where in [left panel]MR=108 GeV and in [right panel] MR=103 GeV. Other reference values: mDM=300GeV, mH2=300 GeV, Tr[Y†νYν]=0.5 and λϕH=0.06 and λχϕ=0.135.</figcaption></figure> we study the running of the Higgs quartic coupling $\lambda_{H}$ from EW scale to $M_{P}$ as shown in Fig.15. In obtaining the running, we have considered $m_{H_{2}}=300$ GeV, $\sin\theta=0.2$ and $m_{\textrm{DM}}$ is considered to be 300 GeV. The values of $\lambda_{\chi\phi}$ and $\lambda_{\phi H}$ are fixed at 0.135 and 0.06 respectively (this particular point is denoted by a blue dot, named $\small{X}$, on Fig.8 ). It turns out that any other set of $\lambda_{\chi\phi}$ and $\lambda_{\phi H}$ other than this blue dot from Fig. (while $m_{\textrm{DM}}=300$ GeV is fixed) would not change our conclusion significantly as long as $\sin\theta$ is considered at 0.2. In order to compare the effect of the extra scalar $\chi$ in the theory, we keep the neutrino parameters Tr[$Y^{\dagger}_{\nu}Y_{\nu}$] and $M_{R}$ at their respective values considered in Figs.11, 13. <figure><img src="content_image/1706.04931/x17.png"><figcaption>Figure 17: Stability, metastability and instability region on Tr[Y†νYν] -mtplane for MR=108 GeV in the extended scenario of SM with 3 RH neutrinos, DMand χ. We have used point X (λϕH=0.06, λχϕ=0.135) from Fig.8, sinθ=0.2,mH2=300 GeV, vχ=800 GeV, mDM=300 GeV and λϕ=0.7 as benchmark points.</figcaption></figure> <figure><img src="content_image/1706.04931/x18.png"><figcaption>Figure 18: Stability, metastability and instability region on Tr[Y†νYν] -mtplane in the extended scenario of SM with 3 RH neutrinos, DM and χ for (rightpanel) MR=103 GeV and (left panel) MR=1014 GeV. We have used Point X(λϕH=0.06, λχϕ=0.135) from Fig.8, sinθ=0.2, mH2=300 GeV, vχ=800 GeV, mDM=300GeV and λϕ=0.7 as benchmark points.</figcaption></figure> In the left panel of Fig.15, the running is performed for three different choices of $M_{R}$, specifically at 1 TeV, $10^{8}$ GeV and $10^{14}$ GeV while top mass is fixed at 173.2 GeV. A similar plot is exercised in right panel of Fig.15 where three different choices of $m_{t}=(171,173.2,177)$ GeV are considered while $M_{R}$ is fixed at $10^{8}$ GeV. Contrary to our previous finding in section (see Fig.11, 13 ), we clearly see here that with $M_{R}=10^{14}$ GeV and $m_{t}=171$ GeV, $\lambda_{H}$ remains positive upto $M_{P}$ even in presence of large Tr[$Y^{\dagger}_{\nu}Y_{\nu}$] $\sim\mathcal{O}(1)$. Hence EW vacuum turns out to be absolutely stable. Although there exists other values of $M_{R}$ and/or $m_{t}$, for which EW vacuum still remains unstable, the scale at which $\lambda_{H}$ enters into the instable region is getting delayed with a noticeable change from earlier cases (Figs.11, 13). This becomes possible due to the introduction of the $\chi$ field having contribution mostly from the $\sin\theta$ parameter. In order to show its impact on stability, in Fig.16 (left panel), we plot $\lambda_{H}$ running with different choices of $\sin\theta=0.1,0.2,0.3$ for $M_{R}=10^{8}$ GeV, $m_{t}=173.2$ GeV and $m_{\textrm{DM}}=300$ GeV while keeping $\textrm{Tr}[Y_{\nu}Y_{\nu}^{\dagger}]=0.5$ (same as in Fig.15, left panel, black solid line). It shows that while $\sin\theta=0.2$ (black solid line) can not make the EW vacuum absolutely stable till $M_{P}$, an increase of $\sin\theta$ value $\sim$ 0.3 can do it (dotted line). Similarly in Fig. 16 (right panel), we consider a lowerer $M_{R}$ as 1 TeV. We have already noticed that such a low $M_{R}$ with large $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]=0.5$ pushes EW vacuum toward instability at a much lower scale $\sim 10^{6}$ GeV. In order to make the EW vacuum stable with such an $M_{R}$ and $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$, one requires $\sin\theta\sim 0.4$ as seen from the right panel of Fig.16 (dotted line). However such a large $\sin\theta$ is ruled out from the experimental constraints [72]. For representative purpose, we also include study with other $\sin\theta=0.2,0.3$ denoted by dashed and solid lines. We provide Fig.17 where the regions with stability, meta-stability and instability are marked green, white and pink patches in the plane containing Tr[$Y^{\dagger}_{\nu}Y_{\nu}$] and $m_{t}$. With $M_{R}=10^{3}$ GeV and $M_{R}=10^{14}$ GeV, similar plots are shown in Fig.18, left and right panels. Finally in Fig.19, we have shown the RG evolution of all the stability conditions in Eq.(3) from $m_{t}$ to $M_{P}$ to check their validity all the way upto $M_{P}$. For this purpose, we have considered the initial values of the parameters involved in the following way. For values of $\lambda_{\phi H}$ and $\lambda_{\chi\phi}$ corresponding to $\sin\theta=0.2,v_{\chi}=800$ GeV and $m_{\textrm{DM}}=300$ GeV, we have considered the benchmark point values as indicated by a blue dot named $X$ in Fig.8 . The value of $\lambda_{\chi}$ is then followed from Eq.(9) and $\lambda_{\phi}$ is chosen to be at 0.7. Values of Tr[$Y^{\dagger}_{\nu}Y_{\nu}]=0.24$ and $m_{t}=173.2$ GeV are chosen for this purpose from Fig.17 (here the benchmark values are denoted by a black dot $Y$). We conclude that all the stability criteria are fulfilled within the framework. Lastly we comment that instead of picking up the point X from relic density contour with $\sin\theta\sim 0.2$ in Fig.8 to study vacuum stability in our model, we could have chosen any other point from that curve. As the stability of Higgs vacuum primarily depends on the value of $\theta$, our conclusion would not change much. However choice of any point having large $\lambda_{\chi\phi}$ could make it reaching Landau pole well before $M_{P}$ in its RG running through Eq.(30). To avoid that one can reduce the value of $\lambda_{\phi}$$\sim\mathcal{O}(10^{-2})$ or less (earlier it was 0.7) which has no direct connection or impact on DM phenomenology and vacuum stability analysis in the proposed set up. In Fig.20, we have shown the running of all parameters from $M_{R}$ to $M_{P}$ involved in perturbative unitarity bound for the benchmark point: $m_{H_{2}}=300$ GeV, $\tan\beta=0.30$, $\sin\theta=0.2$, $m_{\textrm{DM}}=300$ GeV, $\lambda_{\phi H}$=0.06, $\lambda_{\chi\phi}=0.135$, $M_{R}=10^{8}$ GeV and Tr$[Y_{\nu}^{\dagger}Y_{\nu}]=0.24$ with $m_{t}=173.2$ GeV. The parameters never exceed the upper limits coming from the unitarity bound. We have also confirmed that any other benchmark points wherever mentioned in our analysis satisfy the perturbativity unitarity limit. <figure><img src="content_image/1706.04931/x20.png"><figcaption>Figure 19: Evolution of stability parameters (Eq.3) for the point Y (mt=173.2GeV, Tr[Y†νYν]=0.24) from Fig.18 (top right panel). Benchmark points: Point X(λϕH=0.06, λχϕ=0.135) from Fig.8, MR=108 GeV, sinθ=0.2, mH2=300 GeV, vχ=800GeV, mDM=300 GeV and λϕ=0.7 have been used.</figcaption></figure> <figure><img src="content_image/1706.04931/x21.png"><figcaption>Figure 20: Evolution of parameters required to satisfy the perturbativityunitarity limit (Eq.(12)) for the point Y (mt=173.2 GeV, Tr[Y†νYν]=0.24) fromFig.18 (top right panel). Benchmark points: Point X (λϕH=0.06, λχϕ=0.135) fromFig.8, MR=108 GeV, sinθ=0.2, mH2=300 GeV, vχ=800 GeV, mDM=300 GeV and λϕ=0.7have been considered.</figcaption></figure> We end this section by comparing the results of vacuum stability in presence of (i) only RH neutrinos, (ii) RH neutrinos + DM and (iii) RH neutrinos + DM + extra scalar with non-zero vev, where in each cases neutrino Yukawa coupling $Y_{\nu}$ has sizeable contributions. For this purpose, we consider $m_{t}=173.2$ GeV and $M_{R}=10^{8}$ GeV. From Fig.12, for SM + RH neutrinos, we see that stability can not be achieved. The metastability scenario is still valid in this case upto Tr$[Y_{\nu}^{\dagger}Y_{\nu}]<0.26$. Next we add a singlet scalar DM candidate with nonzero Higgs portal coupling to SM with RH neutrinos. Fig.14 (left panel) shows, for $m_{\textrm{DM}}=300$ GeV, stability of EW vacuum still remains elusive. On the other hand the metastability bound on Tr$[Y_{\nu}^{\dagger}Y_{\nu}]$ increases slightly from previous limit to 0.28. So DM with mass 300 GeV has mild impact on study of vacuum stability. Finally we add the extra scalar singlet with non zero vev to the SM with RH neutrinos and scalar DM. We have fixed the heavier Higgs mass $m_{H_{2}}=300$ GeV and $\sin\theta=0.2$. Now in the combined set up of SM, scalar DM, scalar with non zero vev and RH neutrinos, the situation changes drastically from previous case as seen in Fig.17. For the same top and RH neutrino masses, we can now achieve absolute stability upto Tr$[Y_{\nu}^{\dagger}Y_{\nu}]<0.3$ and the metastability bound on Tr$[Y_{\nu}^{\dagger}Y_{\nu}]$ further improved to 0.41. Overall notable enhancement in the stability and metastability region has been observed in Tr$[Y_{\nu}^{\dagger}Y_{\nu}]-m_{t}$ plane compared to the earlier cases. Hence, the numerical comparison clearly shows that the extra scalar having non zero mixing with SM Higgs effectively plays the leading role to get absolute vacuum stability in our model. ## 6 Connection with other observables In this section, we first discuss in brief the constraints on the parameters of the model that may arise from lepton flavor violating (LFV) decays. The most stringent limit follows from $\mu\to e\gamma$ decay process. The branching ratio of such decay process in our set-up is given by[95, 96, 97] \[\textrm{Br}(\mu\to e\gamma)=\frac{3\alpha_{e}v^{4}}{16\pi M _{R}^{4}}\|Y_{\nu_{ei}}^{\dagger}Y_{\nu_{i\mu}}\|^{2}\|f(x)\|^{2},\] (31) where $\alpha_{e}=\frac{e^{2}}{4\pi}$ is the fine sructure constant, $i$ runs from 1 to 3, $x=\frac{M_{R}^{2}}{m_{W}^{2}}$ and \[f(x)=\frac{x\left(2x^{3}+3x^{2}-6x-6x^{2}\ln x+1\right)}{2(1-x)^ {4}}.\] (32) The current experimental limit on LFV branching ratio is[4] \[\textrm{Br}(\mu\to e\gamma)<5.7\times 10^{-13}.\] (33) <figure><img src="content_image/1706.04931/LFV_A.png"><figcaption>Figure 21: LFV and absolute vacuum stability constraint on Tr[Y†νYν]−MR in thecombined set up of SM+DM+RH neutrinos+χ field where mDM=300 GeV, mH2=300 GeV,sinθ=0.2, λϕH=0.06 and λχϕ=0.135.</figcaption></figure> Using this limit, we therefore obtain bounds on $\|(Y_{\nu}^{\dagger}Y_{\nu})_{e\mu}\|$ corresponding to a fixed $M_{R}$ value which can be converted to constrain $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ in our set up. In obtaining limits on $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ (for fixed $M_{R}$), first note that $Y_{\nu}^{\dagger}Y_{\nu}$ remains function of $M_{R}$ and parameter $a$ only (see Eq.(19) with $O=\mathbb{I}$), once the best fit values of neutrino mixing angles[93, 94] are used to evaluate $U_{\textrm{PMNS}}$. Hence LFV limit basically constrains the parameter $a$ which in turn is used to obtain $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$. This limit is shown on Fig.21 by the brown solid line, the left side of which is the disallowed region by LFV. In the same plane of Fig.21 we also include the region of the parameter space allowed by both stability and metastability criteria. The green shaded region denotes the absolute stability of Higgs vacuum while the white region satisfies the metastability condition. We also indicate the instability region by pink patch in the same figure under discussion. For this purpose we have used $m_{t}=173.2$ GeV and $m_{\textrm{DM}}=300$ GeV, $\sin\theta=0.2$, $\lambda_{\phi H}=0.06$ and $\lambda_{\chi\phi}=0.135$ (corresponding to the benchmark point indicated by X in Fig.8). The brown shaded region is disfavored by the LFV constraint. Hence from Fig.21 we infer that for low $M_{R}$, LFV constraints turn out to be stronger one and for high $M_{R}$ values, $\textrm{Tr}[Y_{\nu}^{\dagger}Y_{\nu}]$ is mostly restricted by the stability issue. It turns out that the proposed scenario does not provide any significant contribution to neutrinoless double beta decay[98, 99, 100, 101, 102, 103] even for relatively low RH neutrino mass ($\sim 10^{3}$ GeV). This is in line with the observation made in [40]. Before concluding the section, it is perhaps important to comment on the possibility of explaining the baryon asymmetry of the Universe (BAU). The involvement of RH neutrinos would make the leptogenesis natural candidate to explain BAU from the completion point of view. However with the exactly degenerate RH neutrinos (we consider this for simplicity though), it is not possible. Once a small mass-splitting $\Delta M_{R}$ between two heavy RH neutrinos can be introduced (for example by radiative effect [104, 105, 106]), resonant leptogenesis mechanism [107, 108, 109] can be succesfully implemented[110]. Apart from this, provided one can extend our vacuum stability analysis in presence of non-degenerate RH neutrinos[44] with DM and $\chi$ field, usual thermal leptogenesis can also be employed to explain the BAU of the universe. ## 7 Conclusions We have considered an extension of the SM by three RH neutrinos and two scalar singlets with an aim to study the EW vacuum stability in a framework that can incorporate a stable light DM within the reach of collider experiments and to explain the light neutrino mass. A $Z_{2}\times Z^{\prime}_{2}$ symmetry is imposed of which $Z^{\prime}_{2}$ is broken from the vev of one of the scalars. It is known that with a real scalar singlet DM model, present experimental limits by LUX 2016 and XENON 1T rule out DM mass below $m_{\textrm{DM}}=500$ GeV. Also its presence does not modify the fate of EW vacuum much and hence keep it metastable only. Although metastability is acceptable, it however leaves some unwanted questions if we include primordial inflation in the picture. So an absolute stability of the EW vacuum is more favourable. On the other hand, introduction of RH neutrinos would have large impact on the running of the Higgs quartic coupling due to the neutrino Yukawa interaction. Provided the neutrino Yukawa coupling is as large as $\mathcal{O}$(1) or more, it can actually destabilize the EW vacuum. Hence we have tried here achieving the stability of the EW vacuum in presence of RH neutrinos and DM. We also plan to find the possibility of a light scalar DM below 500 GeV. For this purpose, we have introduced additional scalar field which gets a vev. The other scalar among the two introduced does not get a vev and thereby is a good candidate for being a dark matter. The presence of the singlet with non-zero vev helps achieving the vacuum stability through a threshold like correction to $\lambda_{H}$. So in this particular scenario _i.e._ SM extended by DM, three RH neutrinos plus one extra scalar, we have studied the Higgs vacuum stability issue considering large Yukawa coupling and variation of $m_{t}$ within $2\sigma$ range of uncertainty. We have found the stability region in the Tr$[Y_{\nu}^{\dagger}Y_{\nu}]-m_{t}$ plane has been significantly increased in presence of $\chi$. Simultaneously mixing of this extra scalar with SM Higgs doublet ensures its involvement in the DM annihilations. This mixing is effectively controlled by the Higgs portal coupling of the scalar which also enters into the running of the Higgs quartic coupling. Hence an interplay between the two conditions: one is to achieve the EW vacuum stability and the other is to find a viable DM below 500 GeV, can actually constrain the parameters involved to some extent. Since the set-up involves several new particles, finding their existence in future and ongoing experiments would be an interesting possibility to search for. Here we have assumed the physical Higgs other than the SM one is heavier. The other situation where the second physical Higgs is lighter than the Higgs discovered at 125 GeV. However this case is not of very interest in the present study as following from Eq.(8), it can be seen that the effective Higgs quartic coupling becomes less than the SM one in this case and this would not help making EW vacuum stable. Also the $\sin\theta$ allowed region for $m_{H_{2}}<m_{H_{1}}/2$ is almost excluded from the decay of $H_{2}\to H_{1}H_{1}$. Hence we discard this possibility. One interesting extension of our work could be the study of a SM gauge extension where the involvement of gauges bosons can modify our result. We keep it for a future study. ## Appendix A Unitarity Constraints In this section we draw the perturbative unitarity limits on quartic couplings present in our model. Scattering amplitude for any $2\to 2$ process can be expressed in terms of Legendre polynomial as[68, 69] \[\mathcal{M}^{2\to 2}=16\pi\sum^{\infty}_{l=0}{a_{l}}(2l+1)\mathit{P}_{ l}(\cos\theta),\] where, $\theta$ is the scattering angle and $P_{l}(\cos\theta)$ is the Legendre polynomial of order $l$. In high energy limit only s wave ($l=0$) partial amplitude $a_{0}$ will determine the leading energy dependence of the scattering processes[68, 69]. The unitarity constraint says \[\lvert\rm Re\leavevmode\nobreak\ a_{0}\rvert<1/2.\] (A.1) This constraint Eq.(A.1) can be further translated to a bound on the scattering amplitude $\mathcal{M}$[68, 69]. \[\lvert\mathcal{M}\rvert<8\pi.\] (A.2) In our proposed model we have multiple possible $2\to 2$ scattering processes. Therefore we need to construct a matrix ($M^{2\to 2}_{i,j}=\mathcal{M}_{i\to j}$) considering all possible two particle states. Finally we need to calculate the eigenvalues of $M$ and employ the bound as in Eq.(A.2). In the high energy limit we express the SM Higgs doublet as $H^{T}=(w^{+},\frac{H^{0}+iz}{2})$. Then the scalar potential ($V$) in Eq.(1) gives rise to eleven neutral combination of two particle states \[w^{+}w^{-},\leavevmode\nobreak\ \frac{zz}{\sqrt{2}},\leavevmode \nobreak\ \frac{H^{0}H^{0}}{\sqrt{2}},\leavevmode\nobreak\ \frac{\chi\chi}{ \sqrt{2}},\leavevmode\nobreak\ \frac{\phi\leavevmode\nobreak\ \phi}{\sqrt{2}}, \leavevmode\nobreak\ H^{0}\chi,\leavevmode\nobreak\ H^{0}\leavevmode\nobreak\ \phi,\leavevmode\nobreak\ \chi\leavevmode\nobreak\ \phi,\leavevmode\nobreak\ z \leavevmode\nobreak\ H^{0},\leavevmode\nobreak\ z\leavevmode\nobreak\ \chi, \leavevmode\nobreak\ z\leavevmode\nobreak\ \phi),\] (A.3) and four singly charged two particle states \[w^{+}H^{0},\leavevmode\nobreak\ w^{+}\chi,\leavevmode\nobreak\ w ^{+}z,\leavevmode\nobreak\ w^{+}\phi.\] (A.4) Hence we can write the scattering amplitude matrix ($M$) in block diagonal form by decomposing it into neutral and singly charged sector as \[M_{15\times 15}=\left(\begin{array}[]{cc}M^{\textrm{n}}_{11 \times 11}&0\\ 0&M^{sc}_{4\times 4}\end{array}\right).\] (A.5) The submatrices are provided below : \[M^{n}_{11\times 11}=\left(\begin{array}[]{ccccccccccc}4\lambda_{ H}&\sqrt{2}\lambda_{H}&\sqrt{2}\lambda_{H}&\frac{\lambda_{\chi H}}{\sqrt{2}}& \frac{\lambda_{\phi H}}{\sqrt{2}}&0&0&0&0&0&0\\ \sqrt{2}\lambda_{H}&3\lambda_{H}&\lambda_{H}&\frac{\lambda_{\chi H}}{2}&\frac{ \lambda_{\phi H}}{2}&0&0&0&0&0&0\\ \sqrt{2}\lambda_{H}&\lambda_{H}&3\lambda_{H}&\frac{\lambda_{\chi H}}{2}&\frac{ \lambda_{\phi H}}{2}&0&0&0&0&0&0\\ \frac{\lambda_{\chi H}}{\sqrt{2}}&\frac{\lambda_{\chi H}}{2}&\frac{\lambda_{ \chi H}}{2}&\frac{\lambda_{\chi}}{2}&\frac{\lambda_{\chi\phi}}{2}&0&0&0&0&0&0 \\ \frac{\lambda_{\phi H}}{\sqrt{2}}&\frac{\lambda_{\phi H}}{2}&\frac{\lambda_{ \phi H}}{2}&\frac{\lambda_{\chi\phi}}{2}&\frac{\lambda_{\phi}}{2}&0&0&0&0&0&0 \\ 0&0&0&0&0&\lambda_{\chi H}&0&0&0&0&0\\ 0&0&0&0&0&0&\lambda_{\phi H}&0&0&0&0\\ 0&0&0&0&0&0&0&\lambda_{\chi\phi}&0&0&0\\ 0&0&0&0&0&0&0&0&2\lambda_{H}&0&0\\ 0&0&0&0&0&0&0&0&0&\lambda_{\chi H}&0\\ 0&0&0&0&0&0&0&0&0&0&\lambda_{\phi H}\\ \end{array}\right).\] (A.6) \[M^{sc}_{4\times 4}=\left(\begin{array}[]{cccc}2\lambda_{H}&0&0&0\\ 0&\lambda_{\chi H}&0&0\\ 0&0&2\lambda_{H}&0\\ 0&0&0&\lambda_{\phi H}\\ \end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .\] (A.7) The distinct eigen values of matrix Eq.(A.6) and Eq.(A.7) are following : \[2\lambda_{H},\leavevmode\nobreak\ \lambda_{\chi H},\leavevmode\nobreak\ \lambda_{\phi H},\leavevmode\nobreak\ \lambda_{\chi\phi}\textrm{ and }x_{1,2,3},\] where $x_{1,2,3}$ are the roots of the following polynomial equation, \[x^{3}+x^{2}(-12\lambda_{H}-\lambda\chi-\lambda\phi)+x\left(12 \lambda_{H}\lambda_{\chi}+12\lambda_{H}\lambda_{\phi}-4\lambda_{\chi H}^{2}- \lambda_{\chi_{\phi}}^{2}+\lambda_{\chi}\lambda_{\phi}-4\lambda_{\phi H}^{2}\right)\] \[+12\lambda_{H}\lambda_{\chi\phi}^{2}-12\lambda_{H}\lambda_{\chi} \lambda_{\phi}+4\lambda_{\chi H}^{2}\lambda_{\phi}+4\lambda_{\chi}\lambda_{ \phi H}^{2}-8\lambda_{\chi H}\lambda_{\chi\phi}\lambda_{\phi H}=0.\] (A.8) Therefore the unitarity constraints in the proposed set up are following: \[\lambda_{H}\leavevmode\nobreak\ <4\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{\phi H }\leavevmode\nobreak\ <8\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{\chi H}\leavevmode\nobreak \ <8\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{\chi\phi}\leavevmode\nobreak\ <8\pi\leavevmode \nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode \nobreak\ \leavevmode\nobreak\ x_{1,2,3}<16\pi\leavevmode\nobreak\ \leavevmode \nobreak\ .\] (A.9) ## References [1] G. Aad _et al._ [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]]. * [2] S. Chatrchyan _et al._ [CMS Collaboration], Phys. Lett. B 716, 30 (2012) doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]]. * [3] P. P. Giardino, K. Kannike, I. Masina, M. Raidal and A. Strumia, JHEP 1405, 046 (2014) doi:10.1007/JHEP05(2014)046 [arXiv:1303.3570 [hep-ph]]. * [4] K. A. Olive _et al._ [Particle Data Group], Chin. Phys. C 38, 090001 (2014). doi:10.1088/1674-1137/38/9/090001 * [5] G. Isidori, G. Ridolfi and A. Strumia, Nucl. Phys. B 609, 387 (2001) doi:10.1016/S0550-3213(01)00302-9 [hep-ph/0104016]. * [6] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio and A. Strumia, JHEP 1312, 089 (2013) doi:10.1007/JHEP12(2013)089 [arXiv:1307.3536 [hep-ph]]. * [7] L. A. Anchordoqui, I. Antoniadis, H. Goldberg, X. Huang, D. Lust, T. R. Taylor and B. Vlcek, JHEP 1302, 074 (2013) doi:10.1007/JHEP02(2013)074 [arXiv:1208.2821 [hep-ph]]. * [8] Y. Tang, Mod. Phys. Lett. A 28, 1330002 (2013) doi:10.1142/S0217732313300024 [arXiv:1301.5812 [hep-ph]]. * [9] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori and A. Strumia, JHEP 1208, 098 (2012) doi:10.1007/JHEP08(2012)098 [arXiv:1205.6497 [hep-ph]]. * [10] J. Ellis, J. R. Espinosa, G. F. Giudice, A. Hoecker and A. Riotto, Phys. Lett. B 679, 369 (2009) doi:10.1016/j.physletb.2009.07.054 [arXiv:0906.0954 [hep-ph]]. * [11] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, A. Riotto and A. Strumia, Phys. Lett. B 709, 222 (2012) doi:10.1016/j.physletb.2012.02.013 [arXiv:1112.3022 [hep-ph]]. * [12] V. Branchina, E. Messina and A. Platania, JHEP 1409, 182 (2014) doi:10.1007/JHEP09(2014)182 [arXiv:1407.4112 [hep-ph]]. * [13] G. Degrassi, Nuovo Cim. C 037, no. 02, 47 (2014) doi:10.1393/ncc/i2014-11735-1 [arXiv:1405.6852 [hep-ph]]. * [14] A. Kobakhidze and A. Spencer-Smith, Phys. Lett. B 722, 130 (2013) doi:10.1016/j.physletb.2013.04.013 [arXiv:1301.2846 [hep-ph]]. * [15] V. Silveira and A. Zee, Phys. Lett. 161B, 136 (1985). doi:10.1016/0370-2693(85)90624-0 * [16] J. McDonald, Phys. Rev. D 50, 3637 (1994) doi:10.1103/PhysRevD.50.3637 [hep-ph/0702143 [HEP-PH]]. * [17] J. M. Cline, K. Kainulainen, P. Scott and C. Weniger, Phys. Rev. D 88, 055025 (2013) Erratum: [Phys. Rev. D 92, no. 3, 039906 (2015)] doi:10.1103/PhysRevD.92.039906, 10.1103/PhysRevD.88.055025 [arXiv:1306.4710 [hep-ph]]. * [18] W. L. Guo and Y. L. Wu, JHEP 1010, 083 (2010) doi:10.1007/JHEP10(2010)083 [arXiv:1006.2518 [hep-ph]]. * [19] F. S. Queiroz and K. Sinha, Phys. Lett. B 735, 69 (2014) doi:10.1016/j.physletb.2014.06.016 [arXiv:1404.1400 [hep-ph]]. * [20] K. Ghorbani and H. Ghorbani, Phys. Rev. D 93, no. 5, 055012 (2016) doi:10.1103/PhysRevD.93.055012 [arXiv:1501.00206 [hep-ph]]. * [21] S. Bhattacharya, S. Jana and S. Nandi, Phys. Rev. D 95, no. 5, 055003 (2017) doi:10.1103/PhysRevD.95.055003 [arXiv:1609.03274 [hep-ph]]. * [22] S. Bhattacharya, P. Poulose and P. Ghosh, arXiv:1607.08461 [hep-ph]. * [23] J. A. Casas, D. G. Cerdeño, J. M. Moreno and J. Quilis, arXiv:1701.08134 [hep-ph]. * [24] D. S. Akerib _et al._ [LUX Collaboration], Phys. Rev. Lett. 118, no. 2, 021303 (2017) doi:10.1103/PhysRevLett.118.021303 [arXiv:1608.07648 [astro-ph.CO]]. * [25] E. Aprile _et al._ [XENON Collaboration], arXiv:1705.06655 [astro-ph.CO]. * [26] P. Athron _et al._ [GAMBIT Collaboration], arXiv:1705.07931 [hep-ph]. * [27] E. Aprile _et al._ [XENON Collaboration], JCAP 1604, no. 04, 027 (2016) doi:10.1088/1475-7516/2016/04/027 [arXiv:1512.07501 [physics.ins-det]]. * [28] J. Conrad, arXiv:1411.1925 [hep-ph]. * [29] N. Haba, K. Kaneta and R. Takahashi, JHEP 1404, 029 (2014) doi:10.1007/JHEP04(2014)029 [arXiv:1312.2089 [hep-ph]]. * [30] N. Khan and S. Rakshit, Phys. Rev. D 90, no. 11, 113008 (2014) doi:10.1103/PhysRevD.90.113008 [arXiv:1407.6015 [hep-ph]]. * [31] V. V. Khoze, C. McCabe and G. Ro, JHEP 1408, 026 (2014) doi:10.1007/JHEP08(2014)026 [arXiv:1403.4953 [hep-ph]]. * [32] M. Gonderinger, Y. Li, H. Patel and M. J. Ramsey-Musolf, JHEP 1001, 053 (2010) doi:10.1007/JHEP01(2010)053 [arXiv:0910.3167 [hep-ph]]. * [33] M. Gonderinger, H. Lim and M. J. Ramsey-Musolf, Phys. Rev. D 86, 043511 (2012) doi:10.1103/PhysRevD.86.043511 [arXiv:1202.1316 [hep-ph]]. * [34] W. Chao, M. Gonderinger and M. J. Ramsey-Musolf, Phys. Rev. D 86, 113017 (2012) doi:10.1103/PhysRevD.86.113017 [arXiv:1210.0491 [hep-ph]]. * [35] E. Gabrielli, M. Heikinheimo, K. Kannike, A. Racioppi, M. Raidal and C. Spethmann, Phys. Rev. D 89, no. 1, 015017 (2014) doi:10.1103/PhysRevD.89.015017 [arXiv:1309.6632 [hep-ph]]. * [36] C. S. Chen and Y. Tang, JHEP 1204, 019 (2012) doi:10.1007/JHEP04(2012)019 [arXiv:1202.5717 [hep-ph]]. * [37] I. Garg, S. Goswami, Vishnudath K.N. and N. Khan, Phys. Rev. D 96, no. 5, 055020 (2017) doi:10.1103/PhysRevD.96.055020 [arXiv:1706.08851 [hep-ph]]. * [38] W. Rodejohann and H. Zhang, JHEP 1206, 022 (2012) doi:10.1007/JHEP06(2012)022 [arXiv:1203.3825 [hep-ph]]. * [39] L. Delle Rose, C. Marzo and A. Urbano, JHEP 1512, 050 (2015) doi:10.1007/JHEP12(2015)050 [arXiv:1506.03360 [hep-ph]]. * [40] G. Bambhaniya, P. S. B. Dev, S. Goswami, S. Khan and W. Rodejohann, arXiv:1611.03827 [hep-ph]. * [41] M. Lindner, H. H. Patel and B. Radovčić, Phys. Rev. D 93, no. 7, 073005 (2016) doi:10.1103/PhysRevD.93.073005 [arXiv:1511.06215 [hep-ph]]. * [42] J. Chakrabortty, M. Das and S. Mohanty, Mod. Phys. Lett. A 28, 1350032 (2013) doi:10.1142/S0217732313500326 [arXiv:1207.2027 [hep-ph]]. * [43] A. Datta, A. Elsayed, S. Khalil and A. Moursy, Phys. Rev. D 88, no. 5, 053011 (2013) doi:10.1103/PhysRevD.88.053011 [arXiv:1308.0816 [hep-ph]]. * [44] C. Coriano, L. Delle Rose and C. Marzo, Phys. Lett. B 738, 13 (2014) doi:10.1016/j.physletb.2014.09.001 [arXiv:1407.8539 [hep-ph]]. * [45] J. N. Ng and A. de la Puente, Eur. Phys. J. C 76, no. 3, 122 (2016) doi:10.1140/epjc/s10052-016-3981-4 [arXiv:1510.00742 [hep-ph]]. * [46] C. Bonilla, R. M. Fonseca and J. W. F. Valle, Phys. Lett. B 756, 345 (2016)Bambhaniya:2016rbb doi:10.1016/j.physletb.2016.03.037 [arXiv:1506.04031 [hep-ph]]. * [47] S. Khan, S. Goswami and S. Roy, Phys. Rev. D 89, no. 7, 073021 (2014) doi:10.1103/PhysRevD.89.073021 [arXiv:1212.3694 [hep-ph]]. * [48] E. Ma, Phys. Rev. D 73, 077301 (2006) doi:10.1103/PhysRevD.73.077301 [hep-ph/0601225]. * [49] H. Davoudiasl and I. M. Lewis, Phys. Rev. D 90, no. 3, 033003 (2014) doi:10.1103/PhysRevD.90.033003 [arXiv:1404.6260 [hep-ph]]. * [50] N. Chakrabarty, D. K. Ghosh, B. Mukhopadhyaya and I. Saha, Phys. Rev. D 92, no. 1, 015002 (2015) doi:10.1103/PhysRevD.92.015002 [arXiv:1501.03700 [hep-ph]]. * [51] A. Abada and S. Nasri, Phys. Rev. D 88, no. 1, 016006 (2013) doi:10.1103/PhysRevD.88.016006 [arXiv:1304.3917 [hep-ph]]. * [52] A. Ahriche, A. Arhrib and S. Nasri, JHEP 1402, 042 (2014) doi:10.1007/JHEP02(2014)042 [arXiv:1309.5615 [hep-ph]]. * [53] N. Chakrabarty, U. K. Dey and B. Mukhopadhyaya, JHEP 1412, 166 (2014) doi:10.1007/JHEP12(2014)166 [arXiv:1407.2145 [hep-ph]]. * [54] D. Das and I. Saha, Phys. Rev. D 91, no. 9, 095024 (2015) doi:10.1103/PhysRevD.91.095024 [arXiv:1503.02135 [hep-ph]]. * [55] N. Chakrabarty and B. Mukhopadhyaya, Eur. Phys. J. C 77, no. 3, 153 (2017) doi:10.1140/epjc/s10052-017-4705-0 [arXiv:1603.05883 [hep-ph]]. * [56] G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo and F. S. Queiroz, arXiv:1703.07364 [hep-ph]. * [57] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, H. M. Lee and A. Strumia, JHEP 1206, 031 (2012) doi:10.1007/JHEP06(2012)031 [arXiv:1203.0237 [hep-ph]]. * [58] N. Okada and Q. Shafi, Phys. Lett. B 747, 223 (2015) doi:10.1016/j.physletb.2015.06.001 [arXiv:1501.05375 [hep-ph]]. * [59] A. K. Saha and A. Sil, Phys. Lett. B 765, 244 (2017) doi:10.1016/j.physletb.2016.12.031 [arXiv:1608.04919 [hep-ph]]. * [60] K. Bhattacharya, J. Chakrabortty, S. Das and T. Mondal, JCAP 1412, no. 12, 001 (2014) doi:10.1088/1475-7516/2014/12/001 [arXiv:1408.3966 [hep-ph]]. * [61] Y. Ema, K. Mukaida and K. Nakayama, Phys. Lett. B 761, 419 (2016) doi:10.1016/j.physletb.2016.08.046 [arXiv:1605.07342 [hep-ph]]. * [62] S. Baek, P. Ko, W. I. Park and E. Senaha, JHEP 1211, 116 (2012) doi:10.1007/JHEP11(2012)116 [arXiv:1209.4163 [hep-ph]]. * [63] G. R. Dvali, Z. Tavartkiladze and J. Nanobashvili, Phys. Lett. B 352, 214 (1995) doi:10.1016/0370-2693(95)00511-I [hep-ph/9411387]. * [64] A. Barroso, P. M. Ferreira, I. P. Ivanov and R. Santos, JHEP 1306, 045 (2013) doi:10.1007/JHEP06(2013)045 [arXiv:1303.5098 [hep-ph]]. * [65] A. Barroso, P. M. Ferreira, I. Ivanov and R. Santos, arXiv:1305.1235 [hep-ph]. * [66] K. Kannike, Eur. Phys. J. C 72, 2093 (2012) doi:10.1140/epjc/s10052-012-2093-z [arXiv:1205.3781 [hep-ph]]. * [67] J. Chakrabortty, P. Konar and T. Mondal, Phys. Rev. D 89, no. 9, 095008 (2014) doi:10.1103/PhysRevD.89.095008 [arXiv:1311.5666 [hep-ph]]. * [68] J. Horejsi and M. Kladiva, Eur. Phys. J. C 46, 81 (2006) doi:10.1140/epjc/s2006-02472-3 [hep-ph/0510154]. * [69] G. Bhattacharyya and D. Das, Pramana 87, no. 3, 40 (2016) doi:10.1007/s12043-016-1252-4 [arXiv:1507.06424 [hep-ph]]. * [70] D. López-Val and T. Robens, Phys. Rev. D 90, 114018 (2014) doi:10.1103/PhysRevD.90.114018 [arXiv:1406.1043 [hep-ph]]. * [71] M. J. Strassler and K. M. Zurek, Phys. Lett. B 661, 263 (2008) doi:10.1016/j.physletb.2008.02.008 [hep-ph/0605193]. * [72] T. Robens and T. Stefaniak, Eur. Phys. J. C 76, no. 5, 268 (2016) doi:10.1140/epjc/s10052-016-4115-8 [arXiv:1601.07880 [hep-ph]]. * [73] V. Khachatryan _et al._ [CMS Collaboration], JHEP 1510, 144 (2015) doi:10.1007/JHEP10(2015)144 [arXiv:1504.00936 [hep-ex]]. * [74] G. Aad _et al._ [ATLAS Collaboration], Eur. Phys. J. C 76, no. 1, 45 (2016) doi:10.1140/epjc/s10052-015-3820-z [arXiv:1507.05930 [hep-ex]]. * [75] S. Chatrchyan _et al._ [CMS Collaboration], Phys. Rev. D 89, no. 9, 092007 (2014) doi:10.1103/PhysRevD.89.092007 [arXiv:1312.5353 [hep-ex]]. * [76] CMS Collaboration (2012), CMS-PAS-HIG-12-045. * [77] CMS Collaboration (2013), CMS-PAS-HIG-13-003. * [78] ATLAS and CMS Collaborations (2015), ATLAS-CONF-2015-044 * [79] P. A. R. Ade _et al._ [Planck Collaboration], Astron. Astrophys. 571, A16 (2014) doi:10.1051/0004-6361/201321591 [arXiv:1303.5076 [astro-ph.CO]]. * [80] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 185, 960 (2014) doi:10.1016/j.cpc.2013.10.016 [arXiv:1305.0237 [hep-ph]]. * [81] O. Lebedev, Eur. Phys. J. C 72, 2058 (2012) doi:10.1140/epjc/s10052-012-2058-2 [arXiv:1203.0156 [hep-ph]]. * [82] L. Basso, O. Fischer and J. J. van Der Bij, Phys. Lett. B 730, 326 (2014) doi:10.1016/j.physletb.2014.01.064 [arXiv:1309.6086 [hep-ph]]. * [83] T. Robens and T. Stefaniak, Eur. Phys. J. C 75, 104 (2015) doi:10.1140/epjc/s10052-015-3323-y [arXiv:1501.02234 [hep-ph]]. * [84] G. Chalons, D. Lopez-Val, T. Robens and T. Stefaniak, PoS ICHEP 2016, 1180 (2016) [arXiv:1611.03007 [hep-ph]]. * [85] S. Dawson and I. M. Lewis, Phys. Rev. D 92, no. 9, 094023 (2015) doi:10.1103/PhysRevD.92.094023 [arXiv:1508.05397 [hep-ph]]. * [86] O. Fischer, Mod. Phys. Lett. A 32, no. 06, 1750035 (2017) doi:10.1142/S0217732317500353 [arXiv:1607.00282 [hep-ph]]. * [87] I. M. Lewis and M. Sullivan, arXiv:1701.08774 [hep-ph]. * [88] J. M. Alarcon, J. Martin Camalich and J. A. Oller, Phys. Rev. D 85, 051503 (2012) doi:10.1103/PhysRevD.85.051503 [arXiv:1110.3797 [hep-ph]]. * [89] J. M. Alarcon, L. S. Geng, J. Martin Camalich and J. A. Oller, Phys. Lett. B 730, 342 (2014) doi:10.1016/j.physletb.2014.01.065 [arXiv:1209.2870 [hep-ph]]. * [90] J. A. Casas and A. Ibarra, Nucl. Phys. B 618, 171 (2001) doi:10.1016/S0550-3213(01)00475-8 [hep-ph/0103065]. * [91] W. l. Guo, Z. z. Xing and S. Zhou, Int. J. Mod. Phys. E 16, 1 (2007) doi:10.1142/S0218301307004898 [hep-ph/0612033]. * [92] Y. F. Pirogov and O. V. Zenin, Eur. Phys. J. C 10, 629 (1999) doi:10.1007/s100520050602, 10.1007/s100529900035 [hep-ph/9808396]. * [93] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler and T. Schwetz, JHEP 1701, 087 (2017) doi:10.1007/JHEP01(2017)087 [arXiv:1611.01514 [hep-ph]]. * [94] P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola and J. W. F. Valle, arXiv:1708.01186 [hep-ph]. * [95] A. Ilakovac and A. Pilaftsis, Nucl. Phys. B 437, 491 (1995) doi:10.1016/0550-3213(94)00567-X [hep-ph/9403398]. * [96] D. Tommasini, G. Barenboim, J. Bernabeu and C. Jarlskog, Nucl. Phys. B 444, 451 (1995) doi:10.1016/0550-3213(95)00201-3 [hep-ph/9503228]. * [97] D. N. Dinh, A. Ibarra, E. Molinaro and S. T. Petcov, JHEP 1208, 125 (2012) Erratum: [JHEP 1309, 023 (2013)] doi:10.1007/JHEP09(2013)023, 10.1007/JHEP08(2012)125 [arXiv:1205.4671 [hep-ph]]. * [98] A. Ibarra, E. Molinaro and S. T. Petcov, JHEP 1009, 108 (2010) doi:10.1007/JHEP09(2010)108 [arXiv:1007.2378 [hep-ph]]. * [99] V. Tello, M. Nemevsek, F. Nesti, G. Senjanovic and F. Vissani, Phys. Rev. Lett. 106, 151801 (2011) doi:10.1103/PhysRevLett.106.151801 [arXiv:1011.3522 [hep-ph]]. * [100] M. Mitra, G. Senjanovic and F. Vissani, Nucl. Phys. B 856, 26 (2012) doi:10.1016/j.nuclphysb.2011.10.035 [arXiv:1108.0004 [hep-ph]]. * [101] J. Chakrabortty, H. Z. Devi, S. Goswami and S. Patra, JHEP 1208, 008 (2012) doi:10.1007/JHEP08(2012)008 [arXiv:1204.2527 [hep-ph]]. * [102] P. S. Bhupal Dev, S. Goswami and M. Mitra, Phys. Rev. D 91, no. 11, 113004 (2015) doi:10.1103/PhysRevD.91.113004 [arXiv:1405.1399 [hep-ph]]. * [103] P. S. Bhupal Dev, S. Goswami, M. Mitra and W. Rodejohann, Phys. Rev. D 88, 091301 (2013) doi:10.1103/PhysRevD.88.091301 [arXiv:1305.0056 [hep-ph]]. * [104] R. Gonzalez Felipe, F. R. Joaquim and B. M. Nobre, Phys. Rev. D 70, 085009 (2004) doi:10.1103/PhysRevD.70.085009 [hep-ph/0311029]. * [105] K. Turzynski, Phys. Lett. B 589, 135 (2004) doi:10.1016/j.physletb.2004.03.071 [hep-ph/0401219]. * [106] G. C. Branco, R. Gonzalez Felipe, F. R. Joaquim and B. M. Nobre, Phys. Lett. B 633, 336 (2006) doi:10.1016/j.physletb.2005.11.070 [hep-ph/0507092]. * [107] M. Flanz, E. A. Paschos, U. Sarkar and J. Weiss, Phys. Lett. B 389, 693 (1996) doi:10.1016/S0370-2693(96)01337-8, 10.1016/S0370-2693(96)80011-6 [hep-ph/9607310]. * [108] A. Pilaftsis, Phys. Rev. D 56, 5431 (1997) doi:10.1103/PhysRevD.56.5431 [hep-ph/9707235]. * [109] A. Pilaftsis and T. E. J. Underwood, Nucl. Phys. B 692, 303 (2004) doi:10.1016/j.nuclphysb.2004.05.029 [hep-ph/0309342]. * [110] G. C. Branco, R. Gonzalez Felipe, M. N. Rebelo and H. Serodio, Phys. Rev. D 79, 093008 (2009) doi:10.1103/PhysRevD.79.093008 [arXiv:0904.3076 [hep-ph]].
0801.3672	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 38598, "num_imgs": 7, "llama3_tokens_count": 9263 }	[ "content_image/0801.3672/x1.png", "content_image/0801.3672/x2.png", "content_image/0801.3672/x3.png", "content_image/0801.3672/x4.png", "content_image/0801.3672/x5.png", "content_image/0801.3672/x6.png", "content_image/0801.3672/x7.png" ]	# Cellular automata for the spreading of technologies in socio-economic systems Ferenc Kun feri@dtp.atomki.hu Department of Theoretical Physics University of Debrecen H-4010 Debrecen, P.O.Box: 5, Hungary ###### Abstract We introduce an agent-based model for the spreading of technological developments in socio-economic systems where the technology is mainly used for the collaboration/interaction of agents. Agents use products of different technologies to collaborate with each other which induce costs proportional to the difference of technological levels. Additional costs arise when technologies of different providers are used. Agents can adopt technologies and providers of their interacting partners in order to reduce their costs leading to microscopic rearrangements of the system. Analytical calculations and computer simulations revealed that starting from a random configuration of different technological levels a complex time evolution emerges where the spreading of advanced technologies and the overall technological progress of the system are determined by the amount of advantages more advanced technologies provide, and by the structure of the social environment of agents. We show that agents tend to form clusters of identical technological level with a power law size distribution. When technological progress arises, the spreading of technologies in the system can be described by extreme order statistics. keywords: Spreading, technology, cellular automata, extreme order statistics , Gergely Kocsis, and János Farkas ## 1 Introduction Recently, the application of statistical physics and of the theory of critical phenomena provided novel insight into the dynamics of socio-economic systems [1, 2, 3, 4, 5, 6, 7, 11]. Various types of models have been developed which capture important aspects of the emergence of communities [1], opinion spreading [2, 3, 4, 5, 6, 7] or the evolution of financial data [12]. The dynamics of innovation and the spreading of new technological achievements show also interesting analogies to complex physical systems [8, 9, 10, 11]. The process of innovation has recently been studied by introducing a technology space based on percolation theory [9]. In this model new inventions arise as a result of a random search in the technology space starting from the current best-practice frontier. The model could reproduce the interesting observation that innovations occur in clusters whose sizes are described by the Pareto distribution [9]. Another important aspect of technological development is the spreading of new technological achievements. In a socio-economic system different level technologies may coexist and compete as a result of which certain technologies proliferate while others disappear from the system. One of the key components of the spreading of successful technologies is the copying, _i.e._ members of the system adopt technologies used by other individuals according to certain decision mechanisms. Decision making is usually based on a cost-benefit balance so that a technology gets adopted by a large number of individuals if the upgrading provides enough benefits. The gradual adaptation of high level technologies leads to spreading of technologies and an overall technological progress of the socio-economic system. In the present paper we consider a simple agent-based model of the spreading of technological achievements in socio-economic systems. Agents of the model may represent individuals or firms which use certain technologies to collaborate with each other. For simplicity, we assume that costs of the cooperation arise solely due to the incompatibility of technologies used by the agents which then have two origins: on the one hand, difference of technological levels incurs cost, the larger the difference is, the higher the cost gets. On the other hand, technologies used by agents may belong to different providers which induce additional costs. Agents interacting with their social neighborhood can decrease their cost by adopting technologies of their interacting partners. The local rejection-adaptation strategy of agents can lead to interesting changes of the system on the meso- and macro-level, namely, agents can form clusters with identical technological levels, which can also be accompanied by an overall technological progress of the system. We analyze the time evolution of this model socio-economic system starting from a random configuration of technological levels and providers without considering the possibility of innovation. Based on analytic calculations and computer simulations we study how the adaptation of technologies of interacting partners leads to spreading of technological achievements. We characterize the microstructure of communities of agents, and the technological progress of the system on the macro level. ## 2 Model Our model captures some relevant features of the spreading of technological developments when they are mostly used for the cooperation of individuals. In the model we represent the socio-economic system by a set of agents which posses products of different technological levels and use it to cooperate with each other. Thinking in terms of telecommunication technologies, agents are characterized by two variables: the technological level of the product an agent has (the technological level of the device the agent uses for communication) is described by a real variable $\tau$ such that a larger value of $\tau$ stands for more advanced technologies. New technologies developed by the producers reach the agents through providers. For simplicity, we assume that there are at most two providers active in the system and each agent belongs to one of them. The provider of agents is characterized by an integer variable $S$ which can take two different values $S=\{-1,1\}$. The agents are assumed to cooperate with their social partners which is the easiest if the partners have products of the same technological level. Using technologies of different level can induce difficulties which may be realized by additional costs. It is reasonable to assume that the cost $C$ induced by the collaboration of agents $i$ and $j$ is a monotonous function of the difference of the technological levels $\|\tau_{i}-\tau_{j}\|$. For the purpose of the explicit mathematical analysis we consider the simplest functional form and cast the cost of cooperation into the following form \[C(i\to j)=a\|\tau_{i}-\tau_{j}\|+\frac{1}{2}\Delta(1-S_{i}S_{j}).\] (1) The equation expresses that being at different technological levels (having different $\tau$ values) incurs cost, the higher the difference is in $\tau$ the higher the costs are, while being at the same technological level is cost-free. This crude assumption models a socio-economic system which favors the local communities to be at the same technological level. The value of the multiplication factor $a$ has to be chosen to capture the effect that in case of different technological levels it is favorable for agents to be on a higher technological level than their interacting partners. It follows that the value of $a$ should depend on the relative technological level of the agents with the condition \[a=\left\{\begin{array}[]{lll}a_{1},&if&\tau_{i}>\tau_{j}\\ a_{2},&if&\tau_{i}<\tau_{j}\end{array}\ \ \ \ \ \ \mbox{where}\ \ \ \ \ a_{1}< a_{2}.\right.\] (2) The condition $a_{1}<a_{2}$ implies that being on a higher technological level, _i.e._ being more advanced than the surroundings $\tau_{i}>\tau_{j}$, can lower the costs compared to the opposite case. The second term of Eq. (1) takes into account that the cooperation of agents belonging to different providers implies additional expenses. We assume that this cost does not depend on the technological levels by setting $\Delta>0$ to a constant value. Note that $(1/2)(1-S_{i}S_{j})$ takes value 1 or 0 for agents of different and of the same providers, respectively, resulting in an additional cost $\Delta$ when agents of different providers collaborate. The arrow $\to$ in the argument of $C$ in Eq. (1) expresses that due to the condition Eq. (2) the cost function is not symmetric with respect to agents $i$ and $j$. Hence, $C(i\to j)$ defines the cost of agent $i$ arising due to the cooperation with agent $j$ and this is not equal to the cost of agent $j$, _i.e._$C(i\to j)\neq C(j\to i)$. If agent $i$ has $n$ collaborating partners with technological levels $\tau_{1},\tau_{2},\ldots,\tau_{n}$, the total cost of its collaboration can be obtained by summing up the cost function Eq. (1) over all connections \[C(i)=\sum_{j=1}^{n}C(i\to j).\] (3) ### Dynamics of the model system The system agents can change their technological levels with the aim to minimize their total costs $C(i)$ under the conditions set by their social environment. For simplicity, in the present form of the model no investment is considered, which means that new technologies cannot appear in the system. Agents reject their low level technologies and copy/adopt the more advanced ones of other agents in order to minimize their costs. We call this microscopic mechanism _rejection-adaptation_ which may also improve the global technological level of the system. Upgrading the technological level of agents does not induce cost, _i.e._ there is no resistance against the change; if the adaptation is advantageous it will be performed. The time evolution of the system proceeds as follows: at time $t$ the communication of agent $i$ with technological level $\tau_{i}^{t}$ and provider $S_{i}^{t}$ incurs the total cost $C^{t}(i)$. At time $t+1$ the agent can either keep its technological level or can take over the $\tau$ and $S$ values of one of its social partners, $\tau_{i}^{t+1}\in\{\tau_{i}^{t},\tau_{1}^{t},\tau_{2}^{t},\ldots,\tau_{n}^{t}\}$ and $S_{i}^{t+1}\in\{S_{i}^{t},S_{1}^{t},S_{2}^{t},\ldots,S_{n}^{t}\}$. The possibility which is actually realized is the choice that minimizes the total cost \[C^{t+1}(i)=\min\{C(\tau\in\{\tau_{i}^{t},\tau_{1}^{t},\tau_{2}^{ t},\ldots,\tau_{n}^{t}\},S\in\{S_{i}^{t},S_{1}^{t},S_{2}^{t},\ldots,S_{n}^{t} \})\}.\] (4) Based on this dynamics the time evolution can be followed by computer simulation treating the system as a cellular automaton, _i.e._, the dynamics Eq. (4) is evaluated for each agent at the same time (parallel dynamics) under the neighborhood conditions set by the structure of the socio-economic environment. The above dynamics results in a rather complex time evolution of the system during which certain technologies disappear while others survive and spread over the system. In the following we analyze the time evolution of the system on the micro and macro level by varying the parameters $a_{1}$, $a_{2}$, $\Delta$ and the topology of the social environment of agents. ## 3 Mean field versus local interaction In order to understand the decision mechanism how agents select the technology to adopt, it is useful to study simplified configurations by analytic calculations. For clarity, first we consider only one provider in the system, or analogously to set $\Delta=0$. Let us assume that the system is composed of a large number of agents which have randomly distributed technological levels in an interval $\tau_{min}\leq\tau\leq\tau_{max}$ with a probability density $p(\tau)$ and distribution function $P(\tau)=\int\limits_{\tau_{min}}^{\tau}p(\tau^{\prime})d\tau^{\prime}$. In the limiting case of an infinite range of interaction, all agents interact with all others so that the cost of interaction of an agent of technological level $\tau$ can be cast into the form \[C(\tau)=a_{1}\int_{\tau_{min}}^{\tau}(\tau-\tau^{\prime})p(\tau^ {\prime})d\tau^{\prime}+a_{2}\int^{\tau_{max}}_{\tau}(\tau^{\prime}-\tau)p( \tau^{\prime})d\tau^{\prime}\] (5) as a function of $\tau$. In the next time step the agent will change its technological level from $\tau$ to that $\tau^{}$, which minimizes the cost function Eq. (5), _i.e._$\frac{dc}{d\tau}\left\|{}_{\tau^{}}\right.=0$. The technology that optimizes the cost can finally be obtained as the solution of the equation \[P(\tau^{})=\frac{1}{1+\frac{1}{r}},\] (6) where $r=a_{2}/a_{1}$ is the ratio of the two cost factors $a_{1}$ and $a_{2}$. Due to the infinite range of interaction all agents make the same decision, thus after a single time step all agents adopt the same technology $\tau^{}$ and the evolution of the system stops. It can be seen from Eq. (6) that the optimal technology adopted by the entire system is just determined by the ratio $r=a_{2}/a_{1}$ which characterizes how much advantages the more advanced technology provides compared to the less advanced ones. In the special case of $r=1$ (_i.e._ being on a higher technological level does not provide any advantages), the system adopts the median $\tau^{}=m$ of the initial distribution of technologies $p(\tau)$[13]. It is interesting to note that the optimal choice $\tau^{}(r)$ is a monotonically increasing function of $r$; however (and surprisingly), the most advanced technology $\tau_{max}$ is solely chosen in the limiting case $\lim\limits_{r\to\infty}\tau^{}(r)=\tau_{max}$. At any finite value of $r>1$ the large number of agents of low level technologies can force the system to stay at a lower technological level. In the following let us consider a finite community of $n$ agents with technological levels $\tau_{1}<\tau_{2}<\ldots<\tau_{n}$ communicating with each other. The collaboration of agent $i$ of technological level $\tau_{i}$ with the other $n-1$ agents induce the cost \[C(\tau_{i})=a_{1}\sum\limits_{j=1}^{i-1}(\tau_{i}-\tau_{j})+a_{2 }\sum\limits_{j=i+1}^{n}(\tau_{j}-\tau_{i}).\] (7) In the next time step the agent decides to adopt that technology among the $n-1$ possibilities which minimizes the cost function Eq. (7). It can be obtained analytically that the decision is only determined by $r$, namely, the $i$th largest technological level is adopted $\tau^{}=\tau_{i}$ when $r$ falls in the interval \[\frac{i-1}{n-i+1} < r<\frac{i}{n-i}\ \ \ \ \mbox{for}\ \ \ 1\leq i<n,\] (8) \[n-1 < r\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{for}\ \ \ i=n.\] (9) It can be seen from the above equations that the limits of the subintervals of $r$ to choose the $i$th and $n-(i-1)$th largest $\tau$ are symmetric with respect to $r=1$. The most advanced technology $\tau^{}=\tau_{n}$ of the available ones is adopted only if $r$ exceeds the number of interacting partners $r>n-1$. Of course, the actual value of $\tau^{}$ is not determined by the above equations, so that in a system composed of a large number of local communities of agents with randomly distributed $\tau$ values a complex time evolution emerges, which is locally governed by Eqs. (8,9). ## 4 Agents on a square lattice In order to reveal the time evolution of an ensemble of a large number of interacting agents based on the rejection-adaptation mechanism of technologies, we consider a set of agents organized on a square lattice of size $L\times L$ with nearest-neighbor interactions. Initially agents have randomly distributed technological levels between 0 and 1 with a uniform distribution \[p_{0}(\tau)=1,\ \ \ \ \mbox{and}\ \ \ \ P_{0}(\tau)=\tau,\ \ \ \mbox{for}\ \ \ \ 0\leq\tau\leq 1.\] (10) Assuming periodic boundary conditions, all agents have four interacting partners. The rejection-adaptation dynamics based on the cost minimization Eq. (4) results in a non-trivial time evolution of the system which is followed by computer simulations treating the system locally as a cellular automaton. It has to be emphasized that in the simulations parallel update is used, _i.e._ the dynamic rule Eq. (4) is simultaneously applied to all agents keeping their interacting partners fixed. This parallel dynamics is one of the sources of the complex time evolution of the system. <figure><img src="content_image/0801.3672/x1.png"><figcaption>Figure 1: Evolution of cellular automata at different values of r=a2/a1:r=1.0, 2.0, 4.0 for the upper, middle and lower rows, respectively. Snapshotswere taken at the same times in the rows: t=1,2,4,15 from left to right. Thegray-scale illustrates the technological level of single agents.</figcaption></figure> Applying the analytic results Eqs. (8,9) for the specific case of $n=4$, the agents will always copy the 1th, 2nd, 3rd or 4th largest $\tau$ of their local interacting partners when the value of the parameter $r$ falls in the intervals $0<r<1/3$, $1/3<r<1$, $1<r<3$, $3<r$, respectively. Representative examples of the time evolution of a system of size $L=100$ are shown in Fig. 1 for different parameter values in the range $r\geq 1$. (Note that the behavior of the system is symmetric with respect to $r=1$.) Since the system dynamics favors local communities to use the same technology (to be at the same technological level), the agents tend to form clusters with equal $\tau$ at any value of $r$. For the case of $r=1.0$, when being more advanced than the surroundings does not provide any advantages, it can be seen that the system evolves into a frozen cluster structure. The technological level $\tau$ of these clusters covers practically the entire available range with a non-trivial distribution, _i.e._ communities of low level technologies can survive in the presence of highly advanced ones (see Fig. 1$a,b,c,d$). At $r=2.0$, where more advanced technologies are favored by the agents (locally the second largest $\tau$), the system converges into an almost completely homogeneous state of a relatively high technological level (see Fig. 1$e,f,g,h$). In the simulations, initially clusters of agents with identical $\tau$ grow and finally the entire system evolves into a homogeneous state with all agents adopting the same technology. <figure><img src="content_image/0801.3672/x2.png"><figcaption>Figure 2: Cluster size distributions nt(S) at different time values t for r=1.For t≥30 the distribution practically does not change, it converges to arapidly decreasing exponential form.</figcaption></figure> Locally the agents decide for the second largest $\tau$, and therefore both very low and very high level technologies disappear during the evolution. The gray-scale also illustrates that the limit value of $\tau$ adopted by almost all agents in Fig. 1$h$ falls between 0.8 and 1, _i.e._ it is smaller than the highest available value $\tau_{max}=1$. To reach the most advanced technologies, $r$ has to surpass the threshold value $r=3$. This regime is illustrated in Fig. 1$i,j,k,l$ for the specific case of $r=4$, where the white color in Fig. 1$l$ implies that the most advanced technology $\tau_{max}=1$ spraw onto the entire lattice. <figure><img src="content_image/0801.3672/x3.png"><figcaption>Figure 3: Cluster size distributions nt(S) at different time values t for r=2(a), and r=4 (b). At the parameter value r=2.0 (a) the distribution nt(S)evolves from an initially rapidly decreasing exponential form to a power lawdistribution. The power law form proved to be the limit distribution of thesystem over a broad range of cluster sizes. The value of the exponent α of thefitted power law (straight line in the figure) α=1.75±0.05. In both cases (a)and (b) only one cluster remains in the final state, i.e. all the agents havethe same technological level. This state is reached through a complexevolution of cluster sizes.</figcaption></figure> This microscopic restructuring and clusterization process of agents can be characterized by studying the distribution $n_{t}$ of cluster sizes $S$ at different times $t$. A cluster is identified on the lattice as a connected set of agents with the same technological level, where the number of agents defines the cluster size $S$. The numerical results are presented in Figs. 2 and 3 for a system of size $L=1500$. It can be seen in Fig. 2 that for $r=1$, after a few time steps the cluster size distribution converges to a rapidly decreasing exponential form, where even the largest cluster contains a relatively small number of agents. More interesting is the regime $1<r<3$ where agents locally always prefer to adopt the second highest technological level. In this case the initially exponential distribution tends to a power law as time elapses \[n_{t}(S)\sim S^{-\alpha},\] (11) and keeps this functional form over a broad range of time scales (see Fig. 3$a$). The final homogeneous state is reached when small clusters gradually disappear and only one large cluster remains, but the power law distribution remains the same for a long time. The value of the exponent $\alpha$ was determined numerically as $\alpha=1.75\pm 0.05$. For $r>3$ the convergence to the homogeneous final state is faster, but even in this case a power law emerges for intermediate times with the same exponent as before and gradually disappears as the system gets dominated by a single cluster (see Fig. 3$b$). ## 5 Extreme order statistics and technological progress In the previous section we have shown that our rejection-adaptation mechanism results in a complex time evolution with local rearrangements which then leads to an overall system change. A very interesting aspect of the model is that the disappearance of certain technologies and proliferation of others may give rise to a global technological progress. In order to give a quantitative characterization of this evolution process, we determined the distribution $p_{t}(\tau)$ of technological levels $\tau$ at different times, and the mean $\left<\tau^{t}\right>$ and standard deviation $\sigma^{t}$ of this distribution (12) Fig. 4 shows that for $r=1$, when higher level technologies do not provide advantages for agents, the distribution $p_{t}(\tau)$ rapidly attains a Gaussian shape. <figure><img src="content_image/0801.3672/x4.png"><figcaption>Figure 4: Rescaled distributions of technological levels for r=1. Thestandard Gaussian has a perfect agreement with the numerical results.</figcaption></figure> In order to demonstrate the validity of the Gaussian form, we plot in Fig. 4 the rescaled distributions $p_{t}(\tau)\sigma^{t}$ as a function of $\left(\tau-\left<\tau^{t}\right>\right)/\sigma^{t}$, which have an excellent agreement with the standard Gaussian $g(x)=1/\sqrt{2\pi}\exp{(-x^{2}/2)}$. The convergence to the Gaussian is very fast, practically after 30-40 iteration steps the system completely forgets its initial uniform state and $p_{t}$ attains the limit distribution. The Gaussian implies that the fraction of agents having very high and very low level technologies both decrease and agents tend to copy technologies in the vicinity of the distribution mean. Consequently, the system does not have any technological progress, the average technological level remains constant during the time evolution, and $\left<\tau^{t}\right>\to 0.5$ (see Fig. 6$a$). In addition, the standard deviation of $p_{t}$ attains a non-zero constant value in the large $t$ limit, $\sigma^{t}\to 0.12$ (Fig. 6$b$). In contrast, when agents locally prefer to adopt higher level technologies, namely, the largest or the second largest $\tau$ value of the neighborhood on the square lattice, the system undergoes a more complex time evolution involving also extreme order statistics. For $1<r<3$ all the agents adopt the second highest available technology; hence, in a large enough system the distribution of technological levels right after the first iteration step $p_{1}(\tau)$ is the $k=3$ rank extreme distribution $\phi^{k}_{M}$ of $M=4$ variables of uniform distribution. <figure><img src="content_image/0801.3672/x5.png"><figcaption>Figure 5: Distribution pt of the technological level τti of agents atdifferent time values t during the evolution of the system. For r=2 (a) andr=4 (b), the distribution takes an asymmetric form with an increasing averageand decreasing standard deviation. In (a) the continuous line of t=1 indicatesϕ34, furthermore, in (b) the lines of t=1,2,3,4,5 stand forϕ44,ϕ99,ϕ1616,ϕ2525,ϕ3636, which are in an excellent agreement with thenumerical results.</figcaption></figure> In general, the probability density function $\phi^{k}_{M}(x)$ of the $k$th largest value of $M$ realizations of the random variable $x$ which has a probability density $p(x)$ and a distribution function $P(x)$, can be cast into the form \[\phi^{k}_{M}(x)=\frac{M!}{(k-1)!(M-k)!}P(x)^{k-1}(1-P(x))^{M-k}p(x).\] (13) It can be seen in Fig. 5$a$ that by substituting the initial uniform distribution Eq. (10) into Eq. (13), a perfect agreement is obtained between $\phi^{3}_{4}$ and $p_{1}(\tau)$. Due to the overlap of the local neighborhoods of the lattice sites, however, at higher iteration steps the distributions $p_{t}$ do not follow Eq. (13) when we substitute $\phi_{M}^{k}$ and the corresponding distribution function recursively on the right hand side. By increasing the number of iterations, $p_{t}$ gets narrower and converges to a sharply peaked function as the final homogeneous state is approached (see Fig. 5$a$ and Figs. 1$e,f,g,h$). Consequently, the average value of the technological level increases and converges to a limit value which is smaller than the available maximum $\tau_{max}=1$. The standard deviation goes to zero indicating the homogeneity of the final state (see Fig. 6). <figure><img src="content_image/0801.3672/x6.png"><figcaption>Figure 6: Average technological level ⟨τt⟩ as a function of time t (left), andthe standard deviation σt (right). Note the symmetry of the values of ⟨τt⟩ forr and 1/r while the corresponding standard deviations fall on the top of eachother. For r=4 and r=0.25 the continuous lines in both figures indicate theanalytic results which have an excellent agreement with the numericalcalculations.</figcaption></figure> When the control parameter $r$ becomes larger than $3$, more advanced technologies provide so much benefit that it is always better for agents to adopt the available highest level technology in the local neighborhood. Consequently, $p_{t}(\tau)$ rapidly converges to a sharply peaked form at $\tau_{max}=1$ through extreme order distributions. It is interesting to note that contrary to the case of $1<r<3$, in this regime the distribution $p_{t}$ can be described by the extreme order density function $\phi^{k}_{M}$ Eq. (13) with $k=M$ at any time $t$ by taking into account that $M$ increases as a function of time. We found a recursive formula for the time dependence of the parameter $M$ \[M_{t+1}=M_{t}+5+2(t-1),\qquad\qquad\mbox{with}\qquad\qquad M_{1}=4,\] (14) which describes the spreading of successful technologies as a function of time. The continuous lines in Fig. 5$b$ demonstrate the excellent agreement of the above analytic prediction with the numerically obtained distribution functions. Note that due to the symmetry of the system with respect to the parameter value $r=1$, the same holds also for $r<1/3$ with $\phi^{1}_{M}$, where the smallest value ($k=1$) of $M_{t}$ variables given by Eq. (14) is selected. These results imply that the average technological level in these regimes can easily be obtained analytically, _i.e._ the average of the largest and of the smallest value of $M_{t}$ variables with uniform distribution can be cast into the form \[\left<\tau_{max}\right>=M_{t}/(M_{t}+1),\ \ \ \ \qquad\qquad\left<\tau_{min} \right>=1/(M_{t}+1).\] (15) Substituting the recursive formula of $M_{t}$ into Eq. (15) a perfect agreement is obtained with the numerical results of $\left<\tau^{t}\right>$ presented in Fig. 6$a$. ## 6 Two providers By now we have studied the behavior of the system without considering the effect of providers, _i.e._ all the agents belonged to the same provider. In the following we show that the presence of more than one provider results in a substantial change of the behavior of the system on the meso- and macro-levels. <figure><img src="content_image/0801.3672/x7.png"><figcaption>Figure 7: (left) The asymptotic value of the average technological level⟨τ⟩=limt→∞⟨τt⟩ as a function of Δ. For large Δ the value of ⟨τ⟩ converges to alimit value τ∗, which depends on r. (right) τ∗ as a function of r togetherwith the mean field result Eq. (6) and the analytic solution Eqs. (8,9) forn=4 obtained at Δ=0.</figcaption></figure> Let us assume that the initial technological levels are randomly distributed, as before, but in addition a number $n_{1}$ of agents have a provider $S=+1$ and a number $n_{2}$ has another one $S=-1$. The value of $S$ is independent of the technological levels $\tau$. The fraction of agents belonging to the two providers are $p=n_{1}/N$ and $q=1-p=n_{2}/N$, where $N=n_{1}+n_{2}$ is the total number of agents. In the case of an infinite range of interaction among agents (mean field approach) it is straightforward to show that the cost function $C(\tau,\Delta)$ at a finite value of $\Delta$ can be cast into the form \[C(\tau,\Delta)=C(\tau)+2pq\Delta,\] (16) where $C(\tau)$ denotes the mean field cost function Eq. (5) where no providers were considered. Based on the arguments presented in Sec. 3, it follows that after one iteration step the system minimizes the cost function by attaining a uniform state where the median $m$ of the initial distribution of technological levels is copied by all agents and they choose the same provider, namely, the one with the higher initial fraction. A more interesting (and more realistic) situation occurs when agents of the two providers separate according to the technological levels, _i.e._ we assume that the $n_{1}$ agents of the $S=-1$ provider have the same technological level $\tau_{1}$, while the $n_{2}$ agents of the $S=+1$ provider have a different one $\tau_{2}$, where $\tau_{1}>\tau_{2}$. We can determine analytically that even if higher level technologies provide advantages ($r>1$), the agents choose the lower level technology $\tau_{2}$ to minimize $C(\tau,\Delta)$ if the excess cost $\Delta$ induced by the interaction of agents of different providers surpass a threshold value \[\frac{n_{2}a_{2}-n_{1}a_{1}}{n_{1}-n_{2}}(\tau_{1}-\tau_{2})<\Delta.\] (17) In the specific case of $n_{1}=n_{2}$ the decision of agents does not depend on the value of $\Delta$, it is only determined by the ratio $r$, as it was the case when there was only one provider present in the system ($n_{1}=0$ or $n_{2}=0$). In local communities (smaller groups of agents) in an extended system such inhomogeneous configurations can frequently occur when two different providers have different number of agents with a clear separation of their technological levels. The above results imply that in such cases $\Delta$ can have a substantial effect on the behavior of the system by modifying the optimal decision of agents with respect to the $\Delta=0$ case. In order to give a quantitative characterization of the effect of $\Delta$ on the micro- and macro-level of the socio-economic system, we carried out computer simulations on a square lattice varying the value of $\Delta$ in a broad range for fixed $r$ parameters. The calculations started from a uniform distribution of technological levels where the two providers had the same fraction $p=0.5$ and $q=0.5$. Computer simulations revealed that as time elapses the average technological level $\left<\tau^{t}\right>$ converges to a limit value which depends both on $r$ and $\Delta$. Fig. 7$a$ presents representative examples of the large time limit of $\left<\tau^{t}\right>$ as a function of $\Delta$ for several different values of $r=a_{2}/a_{1}$ in the range $1<r<3$. It can be seen that at $\Delta=0$ all the curves obtained at different $r$ values start from the same point; however, as $\Delta$ increases the curves split up and the system follow $r$-dependent histories. For very large $\Delta$ the system converges to a limit value $\tau^{}(r)$ which solely depends on the parameter $r$. Fig. 7$b$ presents a comparison of the $\tau^{}$ obtained analytically, results of computer simulations performed on a square lattice with $\Delta=0$ (one provider), and the $\Delta\to\infty$ limit values $\tau^{}(r)$ of $\left<\tau\right>$ as a function of $r$. It is interesting to note that in the regime $r>1$ for most of the cases the presence of two providers makes possible a more intense technological progress, _i.e._ agents have a higher tendency to adopt technologies closer to the possible maximum compared to the case of a single provider. As $\Delta$ increases the system converges to a steady state where the average technological level is higher than it was with a single provider. ## 7 Discussion We presented an agent-based cellular automata model to study the spread of technological achievements in socio-economic systems. Agents of the model can represent individuals or firms which use different level technologies to collaborate with each other. Costs arise due to the incompatibility of technological levels and to different technological providers. Agents can reduce their costs by adopting the technologies and providers of their interacting partners. We showed by analytic calculations and computer simulations that the local adaptation-rejection mechanism of technologies results in a complex time evolution accompanied by microscopic rearrangements of technologies with the possibility of technological progress on the macrolevel. We showed that agents tend to form clusters of equal technological levels. If higher level technologies provide advantages for agents, the system evolves to a homogeneous state but clusters show a power law size distribution for intermediate times. The redistribution of technological levels involves extreme order statistics leading to an overall technological progress of the system. The presence of providers proved to play a substantial role in the time evolution. The competition of providers seems to make the system more sensitive to advantages provided by the higher level technologies and can lead to additional technological progress by forcing the agents to select locally the more advanced technology. Our model emphasizes the importance of copying in the spreading of technological achievements and considers one of the simplest possible dynamical rules for the decision mechanism. In the model calculations no innovation was considered, _i.e._, agents could not improve their technological level by locally developing a new technology instead of only taking over of the technology of others. Innovation in the model can be taken into account by randomly selecting agents to increase their technological level by a random amount according to some probability distribution. The generalization of the model in this direction is in progress. Our calculations show also the importance of the structure of local communities in the time evolution of the system which addresses interesting questions for future studies of the model varying the coordination number of the lattice, and on small-world and scale-free network topologies [4, 6, 7, 14, 15]. The emergence of power law size distribution of clusters of agents with equal technological level and the behavior of the exponents on different topologies can be relevant also for applications. Compared to opinion spreading models like the Sznajd-model [2, 3] and its variants [4, 6, 7], the main difference is that in our case the technological level of agents is a continuous random variable; furthermore, the decision making is not a simple majority rule but involves a minimization procedure. A closer analogy can be found when two providers are considered in the system so that the spreading of a provider could be interpreted as a success of one of two competing “opinions”. Opinion of individuals can also be represented by a continuous real variable which makes possible to study under which conditions consensus, polarization or fragmentation of the system can occur [8]. Such models show more similarities to our spreading model of technologies. It is interesting to note that our model captures some of the key aspects of the spreading of telecommunication technologies, where for instance mobile phones of different technological levels are used by agents to communicate/interact with each other. In this case, for example, the incompatibility of MMS-capable mobile phones with the older SMS ones may motivate the owner to reject or adopt the dominating technology in his social neighborhood by taking into account the offers of providers of the interacting partners. ## Acknowledgment: This work was carried out with the generous support of Toyota Central R&D Labs., Aichi, Japan. F. Kun was also supported by OTKA T049209. We would like to thank our referees for the valuable comments and suggestions. ## References [1]W. Weidlich, Sociodynamics: A systematic approach to mathematical modelling in the social sciences, (Dover Publications, Mineola, USA, 2000). * [2]K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157 (2000). * [3]K. Sznajd-Weron and R. Weron, Int. J. Mod. Phys. C 13, 115 (2002). * [4]D. Stauffer, Journal of Artificial Societies and Social Simulation 5, No. 1, paper 4. * [5]D. Stauffer, Int. J. Mod. Phys. C 13, 315 (2002). * [6]A. T. Bernardes, D. Stauffer, and J. Kertész, Eur. Phys. J. B 25, 123 (2002). * [7]K. Sznajd-Weron, Acta Physica Polonica B 36, 2537 (2005). * [8]R. Hegselmann and U. Krause, Journal of Artificial Societies and Social Simulations 5, No. 3, paper 2. * [9]G. Silverberg and B. Verspagen, Journal of Economic Dynamics and Control 29, 225 (2005). * [10]R. M. Ruiz, E. Albuquerque, L. C. Ribeiro, and A. T. Bernardes, AIP Conf. Proc. 779, 162 (2005). * [11] A. Arenas, A. Díaz-Guilera, C. J. Pérez, and F. Vega-Redondo, Phys. Rev. E 61, 3466 (2000). * [12]J.-P. Bouchaud and M. Potters, _Theory of financial risk and derivative pricing_, (Cambridge University Press, 2000). * [13] D. Sornette, _Critical Phenomena in Natural sciences_, (Springer Verlag, Berlin, 2000). * [14]A. L. Barabasi and R. Albert, Science 286, 509 (1999). * [15] M. E. J. Newman, SIAM Review 45, 167 (2003).
1411.3779	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 21303, "num_imgs": 5, "llama3_tokens_count": 5476 }	[ "content_image/1411.3779/x1.png", "content_image/1411.3779/x2.png", "content_image/1411.3779/x2.png", "content_image/1411.3779/x2.png", "content_image/1411.3779/x2.png" ]	# Two-step pulse observation for Raman-Ramsey coherent population trapping atomic clocks Yuichiro Yano${}^{1}$ Shigeyoshi Goka${}^{1}$ and Masatoshi Kajita${}^{2}$ ${}^{1}$Graduate School of Science and EngineeringGraduate School of Science and Engineering Tokyo Metropolitan University Tokyo Metropolitan University Minami-Osawa Minami-Osawa Hachioji Hachioji Tokyo 192-0397 Tokyo 192-0397 Japan ${}^{2}$National Institute of Information and Communications Technology Japan National Institute of Information and Communications Technology Koganei Koganei Tokyo 184-8795 Tokyo 184-8795 Japan E-mail: yano-yuichiro@ed.tmu.ac.jp Japan E-mail: yano-yuichiro@ed.tmu.ac.jp ###### Abstract We propose a two-step pulse observation method to enhance frequency stability for coherent population trapping (CPT) atomic clocks. The proposed method is a Raman-Ramsey scheme with low light intensity at resonance observation, and provides a Ramsey-CPT resonance with both reduced frequency sensitivity to the light intensity and a high signal-to-noise ratio by reducing the repumping into a steady dark state. The resonance characteristics were calculated based on density matrix analysis of a $\Lambda$-type three level system that was modeled on the ${}^{133}$Cs-D${}_{1}$ line, and the characteristics were also measured using a vertical-cavity surface-emitting laser and a Cs vapor cell. Atomic clocks based on coherent population trapping (CPT) resonance have attracted attention as a means of fabricating very small frequency references, such as chip-scale atomic clocks[1]. CPT atomic clocks are in great demand for many applications, such as telecommunications, navigation systems, and synchronization of networks[2], and such clocks are required for their high frequency stability. Frequency stability in atomic clocks is generally represented by the Allan deviation. The frequency stability can be classified into two types based on the averaging time of the Allan deviation, which are short-term and long-term frequency stability; these two frequency stability types are degraded by different mechanisms. To enhance the short-term frequency stability, a high signal-to-noise ratio (SNR) and a narrow CPT resonance linewidth are required because the short-term stability is determined by the product of the SNR and the Q value. For long-term frequency stability, the light shift (or the ac Stark shift), which is a frequency shift caused by a light field, is a major limiting factor[3]. Because the long-term stability degradation induced by the light shift is caused by laser light power fluctuations or aging of the optical elements, reduced frequency sensitivity to the light intensity is desired. The Raman-Ramsey (RR) scheme for enhancement of the frequency stability of CPT atomic clocks has been investigated by a number of researchers[4, 5, 6, 7, 8]. This scheme significantly reduces the variation in the light shift to be one or two orders of magnitude lower than that under continuous wave (cw) illumination[9, 8, 7]. Also, because this scheme suppresses power broadening, a narrow linewidth can be obtained[4]. Recently, a short-term stability of 3.2$\times 10^{-13}\tau^{-1/2}$ was obtained, and stability as low as 3$\times 10^{-14}$ was achieved at an averaging time of 200 s using a Cs vapor cell with the RR scheme[10]. Also, in the case where a cold Rb atom was used, short-term stability of 4$\times 10^{-11}\tau^{-1/2}$ was obtained, and a long-term frequency stability of 3$\times 10^{-13}$ was achieved for an averaging time of 5 h [11]. In our previous paper, we investigated the light shift in the RR scheme both theoretically and experimentally with the aim of enhancing the long-term frequency stability of CPT atomic clocks[8]. The results showed that the light shift with the RR scheme is significantly reduced when compared with that under cw illumination. We also found that the frequency variation of the light intensity is reduced by setting the observation time to be as short as possible. This main reason for this is that the atoms evolve towards a steady state during observation. In addition, we found that the light shift in the RR scheme is a nonlinear function of the light intensity, and reduced frequency sensitivity is obtained under a high light intensity beyond a set threshold value. Thus, the measurement conditions with both short observation times and high light intensity provide reduced frequency sensitivity to the light shift. However, the short observation time causes the short-term stability to degrade because of increases in both the Dick effect and the laser intensity noise[10]. In addition, while the high light intensity provides a high transition population, it also leads to a reduction of the resonance signal because of increased repumping into the steady dark state.[12, 13] Therefore, it is difficult to maintain the SNR of the resonance while simultaneously reducing the variation in the light shift. In this paper, we propose a two-step pulse observation method for enhanced frequency stability in CPT atomic clocks. The two-step pulse method is an RR scheme with low light intensity at observation. This method provides low variation in the light shift without reducing the SNR. We describe a light shift of the 6${}^{2}$S${}_{1/2}$ ($F=3\leftrightarrow 4$) state of ${}^{133}$Cs as observed through CPT using the two-step pulse observation method. The light shift with the RR scheme is calculated based on the density matrix. We also propose an estimation equation for the light shift with the RR scheme. This equation is useful for estimation of the light shift with the two-step pulse observation method. <figure><img src="content_image/1411.3779/x1.png"><figcaption>Figure 1: (Color online) Ramsey pulse sequence: τ is the excitation time, T isthe free evolution time, and τm is the resonance signal observation time.</figcaption></figure> The observation scheme for the two-step pulse observation method is shown in Fig. 1. The Ramsey-CPT resonance is observed by measuring the transmitted light intensity at the observation time $\tau_{m}$ after the pulse rise. The two-step pulse observation maintains low light intensity for $rI_{p}$ until the observation point. $r$ is defined as the observation intensity ratio, with a range of values from 0 to 1. When $r=1$ is set, the two-step pulse observation method is treated with the same scheme as that used for the conventional RR scheme. After the measurement is taken, the light intensity is changed into $I_{p}$ and the atoms are prepared for the next measurement. A laser pulse with duration of ($\tau-\tau_{m}$) irradiates the atoms for pumping of the steady dark state. After the free evolution time $T$, a laser pulse irradiates the atoms again with light intensity $rI_{p}$. The light intensity $I(t)$ for this scheme is written as \[I(t)=\left\{\begin{array}[]{l l}rI_{p}&(0<t\leq\tau_{m})\\ I_{p}&(\tau_{m}<t\leq\tau)\\ 0&(\tau<t\leq\tau+T)\\ \end{array}\right..\] (1) <figure><img src="content_image/1411.3779/x2.png"><figcaption>Figure 5: (Color online) Experimental results for light shift as a function ofthe light intensity for various values of r.</figcaption></figure> The calculated light shift in the RR scheme as a function of the light intensity is shown in Fig. 2. The D${}_{1}$-line of Cs was used as the excitation transition, and the 6${}^{2}$P${}_{1/2}$, $F=3$ state was selected as the excited state. The calculation method is shown in more detail in Ref. yano2014theoretical. The black solid line is the calculated light shift under cw illumination. The light shift under cw illumination is proportional to the light intensity ($\mathcal{S}_{\rm cw}=\alpha_{\rm cw}I_{p}$). $\alpha_{\rm cw}$ is the slope of the light shift. The red line is the calculated light shift with the RR scheme in the absence of the influence of observation $\mathcal{S}\|_{\tau_{m}=0}$. The free evolution time was set at 800 $\mu$s, and the $\tau\rightarrow\infty$ steady state solution is used as the initial condition to determine the evolution at later times[13]. The light shift in the RR scheme is a nonlinear function of the light intensity, and this light shift has two characteristics. The first characteristic is that the variation under the RR scheme is equal to that under cw illumination with low light intensity. The reason for this is that the effective light field applied under the RR scheme is the same as that applied under cw illumination with low light intensity, because the low light intensity leads to a long coherence time. The second characteristic is that the light shift under the RR scheme approaches a saturation value $\mathcal{S}_{\rm lim}$ as the light intensity $I_{p}$ approaches infinity. This saturation value $\mathcal{S}_{\rm lim}$ is inversely proportional to the free evolution time $T$ ($\mathcal{S}_{\rm lim}=C_{\rm lim}/T$). When taking both of these characteristics into account, the light shift estimation equation in the absence of an observation time, $\mathcal{S}\|_{\tau_{m}=0}$, is found to be as follows: \[\mathcal{S}\|_{\tau_{m}=0}=\frac{\alpha_{\rm cw}I_{p}}{\alpha_{\rm cw}C_{\rm lim }^{-1}TI_{p}+1}.\] (2) Under low light intensity conditions ($I_{p}\ll C_{\rm lim}/(\alpha_{\rm cw}T)$), because the first term of the denominator is negligible, the value of the light shift $\mathcal{S}$ approaches that produced under cw illumination. Also, under high light intensity conditions ($I_{p}\gg C_{\rm lim}/(\alpha_{\rm cw}T)$), because the second term of the denominator is negligible, the light shift $\mathcal{S}$ converges on $C_{\rm lim}/T$$(=\mathcal{S}_{\rm lim})$. Therefore, Eq. (2) is satisfied using the light shift characteristics above, under the RR scheme. Under the influence of observation ($\tau_{m}>0$), the light shift under the RR scheme then depends on $\tau_{m}$. For large $\tau_{m}$, the light shift under the RR scheme approaches that produced under cw illumination. Because the light intensity at observation is given by $rI_{p}$, the limit of $\mathcal{S}$ as the observation time $\tau_{m}$ approaches infinity is as follows: \[\mathcal{S}\|_{\tau_{m}\to\infty}=r\mathcal{S}_{\rm cw}=r\alpha_{\rm cw}I_{p}\] (3) From (2) and (3), the light shift is shown to be a function that varies from the initial state to reach a steady value with increasing $\tau_{m}$. We found that the time-varying function of the light shift is followed by a logistic function of $\tau_{m}$. The light shift represented by a general logistic function is as follows: \[\mathcal{S}=\frac{a_{1}}{a_{2}e^{-a_{3}\tau_{m}}+1},a_{3}>0\] (4) where, $a_{1},a_{2},a_{3}$ are values that do not depend on $\tau_{m}$. From the initial solution of Eq. (2) and the steady-state solution of Eq. (3), $a_{1}$ and $a_{2}$ equal $r\alpha_{\rm cw}I_{p}$ and $r(\alpha_{\rm cw}C_{\rm lim}^{-1}TI_{p}+1)-1$, respectively. When taking the time-dependent behavior of the CPT resonance into account, $a_{3}$ is equal to the inverse of the pumping time $\tau_{p}$, which is inversely proportional to the light intensity[12]. Thus, from Eqs. (2), (3), and (4), the light shift behavior in the two-step pulse observation $\mathcal{S}$ can be obtained as follows: \[\mathcal{S}=\frac{r\alpha_{\rm cw}I_{p}}{\bigl{\{}r(\alpha_{\rm cw}C_{\rm lim} ^{-1}TI_{p}+1)-1\bigr{\}}e^{-\tau_{m}/\tau_{p}}+1}\] (5) We found that $r$ has an optimal value for suppression of the frequency dependence of the observation time $\tau_{m}$. The optimal $r$ is derived as follows: \[r=\frac{\mathcal{S}\|_{\tau_{m}=0}}{\mathcal{S}_{\rm cw}}=\frac{1}{\alpha_{\rm cw }C_{\rm lim}^{-1}TI_{p}+1}\] (6) This means that $r$ has a value such that the light shift of the steady state $\mathcal{S}\|_{\tau_{m}\to\infty}$ is equal to the light shift in the absence of observation. Under this condition for $r$, the light shift is no longer dependent on $\tau_{m}$, and the light shift $\mathcal{S}$ equals $\mathcal{S}\|_{\tau_{m}=0}$. <figure><img src="content_image/1411.3779/x2.png"><figcaption>Figure 5: (Color online) Experimental results for light shift as a function ofthe light intensity for various values of r.</figcaption></figure> Light shifts as a function of the light intensity under different values of $r$ are shown in Fig. 3(a). The free evolution time $T$ was 800 $\mu$s, and the observation time $\tau_{m}$ was set to be 10 $\mu$s. The results were then calculated numerically from the instantaneous values by using density matrix analysis. The light shift was determined based on the condition that the dispersion spectrum given by the density matrix analysis is zero. The zero dispersion was found using Newton’s method. The light shift decreases with decreasing $r$. Because the pumping rate increases with increasing light intensity, two-step pulse observation with a small $r$ is effective for reduction of the light shift under high light intensity. The light shift variation is also reduced by the use of a small $r$. However, the offset shift remains at $r\approx 0.00$. Because the offset shift is mainly attributed to the free evolution time $T$, a long free evolution time is required to reduce the systematic frequency shift. Figure 3(b) shows the light shifts as a function of the observation time under different values of $r$. The light intensity $I_{p}$ was set to be 2.0 mW/cm${}^{2}$. The light shift is proportional to $\tau_{m}$ because the observation time $\tau_{m}$ is smaller than the pumping time $\tau_{p}$ in this range. The frequency variation of the observation time is strongly dependent on $r$ and decreases significantly with decreasing $r$ in the range from 0.25 to 1.00. Also, the sign of the observation variation changes in the $0.00<r<0.25$ range. These results indicate that there is an optimal value of $r$ for suppression of the variation in the range of $r$ from 0.00 to 0.25. The dashed line was plotted by setting $r=\mathcal{S}\|_{\tau_{m}=0}/\mathcal{S}_{\rm cw}$. $r$ was 0.07 under this condition. Because the light shift is independent of the observation time setting $r=\mathcal{S}\|_{\tau_{m}=0}/\mathcal{S}_{\rm cw}$, it is not necessary to shorten the observation time to reduce the light shift. Also, because a small $r$ leads to a low pumping rate, the resonance signal retains its high value, even if we set the observation time to be longer. Thus, the Ramsey-CPT resonance can be measured without reducing the resonance signal by two-step pulse observation. Comparison of the results of the numerical calculations performed by density matrix analysis and the estimated values from Eq. (5) shows that the relative error of the results is no more than 0.05%. The residual of the relative frequency corresponds to the 10${}^{-15}$ order, which is very small in the CPT atomic clocks field. Therefore, the proposed estimation equation is useful for practical purposes. <figure><img src="content_image/1411.3779/x2.png"><figcaption>Figure 5: (Color online) Experimental results for light shift as a function ofthe light intensity for various values of r.</figcaption></figure> A contour map of the frequency variation of the light intensity is shown in Fig. 4(a). Because this variation decreases with increasing light intensity, the smallest variation using the conventional scheme ($r=1.00$) is 0.562 Hz/(mW/cm${}^{2}$) at a light intensity of 2.4 mW/cm${}^{2}$. The variation with two-step pulse observation decreases with decreasing $r$, and the smallest variation is obtained at $r=0.00$. The variation at $r=0.00$ was two to seven times smaller than that obtained using the conventional scheme. The variation reduction effect increases with increasing light intensity. The smallest variation with the two step pulse observation method is 0.073 Hz/(mW/cm${}^{2}$), which is 7.6 times less than that obtained using the conventional scheme. The red dashed line indicates a locus of the condition $r=\mathcal{S}\|_{\tau_{m}=0}/\mathcal{S}_{\rm cw}$. The variation under this locus is higher than that under $r=0.00$; however, the difference is no more than 10%. Therefore, a large reduction in the variance can be expected under this locus. Figure 4(b) shows the frequency variation of the observation time. The variation when using the conventional scheme increases significantly with increasing light intensity. This is because the high light intensity leads not only to a high pumping rate but also to a large frequency difference between $\mathcal{S}_{\rm cw}$ and $\mathcal{S}\|_{\tau_{m}=0}$. The variation in the observation time is dramatically suppressed under the condition where $r=\mathcal{S}\|_{\tau_{m}=0}/\mathcal{S}_{\rm cw}$. Because the contour interval is shorter for higher light intensities, the two-step pulse observation method provides a particular advantage under high light intensity excitation. <figure><img src="content_image/1411.3779/x2.png"><figcaption>Figure 5: (Color online) Experimental results for light shift as a function ofthe light intensity for various values of r.</figcaption></figure> Figure 5 shows the experimental results for the light shifts as a function of light intensity for various $r$ values. These results were obtained using a Cs-D${}_{1}$ line vertical-cavity surface-emitting laser (VCSEL) and a Cs vapor cell. The light intensity was modulated by an acousto-optical modulator (AOM), and the two-step pulse shape is formed using the control voltage of the AOM. Because the absolute resonance signal decreases with decreasing $r$, the density of the neutral density filter that was located in front of the photodetector was adjusted so that the resonance signal with the two-step pulse observation method was equal to that obtained using the conventional RR scheme. In this experiment, $r$ was set at 0.25 so that it did not saturate the photodetector voltage during pumping. The free evolution time was 800 $\mu$s and the observation time was 10 $\mu$s. The experimental setup is described in detail in Ref. yano2014theoretical. The measured $\alpha_{\rm cw}$ was 28.4 Hz/(mW/cm${}^{2}$). The solid line is a fitting curve for the experimental data. The variation obtained using the conventional scheme was 1.62 Hz/(mW/cm${}^{2}$). The variation obtained with two-step pulse observation ($r=0.25$) is 0.42 Hz/(mW/cm${}^{2}$), which is almost one quarter of that obtained using the conventional scheme. This reduction is consistent with the calculated results. In summary, we propose a two-step pulse observation method to enhance the frequency stability of CPT atomic clocks. The two-step pulse observation method provides a low variation in the light shift without reducing the resonance signal by reducing the steady state repumping. The calculation results show that there is an optimal value of $r$ for suppression of the frequency variation of the observation time. We also derive an equation for estimation of the light shift in the RR scheme. The relative frequency differences between the results obtained using the estimation equation and those from the numerical calculations are of the order of $10^{-15}$, which means that the estimation equation is useful for practical purposes. The measured light shift is reduced by the two-step pulse observation method, and the reduction in the light shift is reproduced well by the calculations. The authors would like to thank the Ricoh Company, Ltd., for providing us with the Cs-D${}_{1}$ VCSEL. This work was supported by Grant-in-Aid for JSPS Fellows (no. 26$\cdot$6442). ## References * [1] S. Knappe, V. Shah, P. D. Schwindt, L. Hollberg, J. Kitching, L.-A. Liew and J. Moreland: Appl. Phys. Lett. 85 (2004) 1460. * [2] J. R. Vig: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40 (1992) 522. * [3] O. Kozlova, J.-M. Danet, S. Guérandel and E. de Clercq: IEEE Trans. Instrum. and Meas. 63 (2014) 1863. * [4] T. Zanon, S. Guerandel, E. De Clercq, D. Holleville, N. Dimarcq and A. Clairon: Phys. Rev. Lett. 94 (2005) 193002. * [5] F.-X. Esnault, E. Blanshan, E. Ivanov, R. Scholten, J. Kitching and E. A. Donley: Phys. Rev. A 88 (2013) 042120. * [6] C. Xi, Y. Guo-Qing, W. Jin and Z. Ming-Sheng: Chin. Phys. Lett. 27 (2010) 113201. * [7] I. Yoshida, N. Hayashi, K. Fujita, S. Taniguchi, Y. Hoshina and M. Mitsunaga: Phys. Rev. A 87 (2013) 023836. * [8] Y. Yano, W. Gao, S. Goka and M. Kajita: Phys. Rev. A 90 (2014) 013826. * [9] N. Castagna, R. Boudot, S. Guerandel, E. Clercq, N. Dimarcq and C. Clairon: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56 (2009) 246. * [10] J.-M. Danet, M. Lours, E. De Clercq, et al.: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 61 (2014) 567. * [11] E. A. Donley, E. Blanshan, F.-X. Esnault and J. Kitching: Proceedings of the 2014 IEEE International Frequency Control Symposium (2014) * [12] S. Guérandel, T. Zanon, N. Castagna, F. Dahes, E. de Clercq, N. Dimarcq and C. Clairon: IEEE Trans. Instrum. and Meas. 56 (2007) 383. * [13] T. Zanon-Willette, E. de Clercq and E. Arimondo: Phys. Rev. A 84 (2011) 062502.
1506.04541	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 42800, "num_imgs": 7, "llama3_tokens_count": 11286 }	[ "content_image/1506.04541/x1.png", "content_image/1506.04541/x2.png", "content_image/1506.04541/x3.png", "content_image/1506.04541/x4.png", "content_image/1506.04541/x5.png", "content_image/1506.04541/x6.png", "content_image/1506.04541/x7.png" ]	# Optimal Data Attacks on Power Grids: Leveraging Detection & Measurement Jamming Deepjyoti Deka, Ross Baldick and Sriram Vishwanath Department of Electrical & Computer Engineering, The University of Texas at Austin Email: deepjyotideka@utexas.edu, baldick@ece.utexas.edu, sriram@ece.utexas.edu ###### Abstract Meter measurements in the power grid are susceptible to manipulation by adversaries, that can lead to errors in state estimation. This paper presents a general framework to study attacks on state estimation by adversaries capable of injecting bad-data into measurements and further, of jamming their reception. Through these two techniques, a novel ‘detectable jamming’ attack is designed that changes the state estimation despite failing bad-data detection checks. Compared to commonly studied ‘hidden’ data attacks, these attacks have lower costs and a wider feasible operating region. It is shown that the entire domain of jamming costs can be divided into two regions, with distinct graph-cut based formulations for the design of the optimal attack. The most significant insight arising from this result is that the adversarial capability to jam measurements changes the optimal ’detectable jamming’ attack design only if the jamming cost is less than half the cost of bad-data injection. A polynomial time approximate algorithm for attack vector construction is developed and its efficacy in attack design is demonstrated through simulations on IEEE test systems. ## I Introduction As power grids around the world move towards smarter devices and distributed control, it has led to large scale placement of cyber meters like PMUs [1] for real-time data collection. This can have a variety of positive implications for the grid, notably monitoring of the grid state for improved reliability and optimal electricity prices. However, ‘smart’ meters and associated communication infrastructure are vulnerable to adversarial attacks by rogue agents and online viruses. Examples of these attacks include GPS spoofing attack on PMUs [4], ‘Dragonfly’ virus [2], Arora test attack [3] among others. Such data attacks can lead to incorrect estimation of the grid state and result to large scale blackouts. The extreme consequences of adversarial attacks and counter strategies has attracted significant interest from the research community. [5] first introduced the problem of undetectable data attacks that bypass standard bad-data tests present in the state estimator. The optimal attack vector comprising of the compromised measurements is constructed in [5] using projection matrices. Subsequent work has looked at the problem of constructing the optimal attack under different grid conditions and adversarial objectives. Attack construction that require minimum number of measurement corruptions are presented in [6] using $l_{0}-l_{1}$ relaxation. Reference [7] analyzed a system with phasor measurements and used mixed integer linear programming to create the optimal attack. For systems with phasor and line flow measurements and PMUs, [8, 9] discusses graph cut based attack designs on specific buses on the grid and associated protection strategies. Similarly, other protection schemes have been discussed in literature, including heuristic protection schemes [10], greedy schemes [6, 9] among others. It is worth noting that most research on power grid cyber-security has focussed on designing ‘hidden’ attack vectors that completely evade the bad-data detection tests at the grid’s state estimator. However, the authors of [11] showed that data ‘framing’ attacks can be constructed that changes the values in half of the measurements in the attack vector while damaging the other half. The attack is initially detected by the estimator but becomes feasible after the bad-data identifier removes the damaged measurements. In [12], a generalized ‘detectable’ attack model was presented for systems where a subset of the measurements are incorruptible. The authors in [12] showed that by focussing on the bad-data identifier, the cardinality of the optimal ‘detectable’ data attack in most cases can be reduced by greater than $50\%$ ($50\%$ in worst case) of that of ‘hidden’ attacks. More importantly, the ‘detectable’ attack framework in [12] is shown to produce feasible attacks in operating regimes that are secure against ‘hidden’ attacks. In this work, we consider the ‘detectable’ attack framework in [12] but with one major modification to the adversary’s capability. In addition to modifying insecure measurements (bad-data injection) as described in previous work, the adversary considered here is capable of jamming or blocking measurement communication to the state estimator. Note that measurement jamming can be conducted using commercial jammers (for wireless communication), Denial of Service attack [13] or by physically damaging the communication channel. Compared to bad-data injection that requires measurements to be changed by precise real values, measurement jamming is in fact less resource-intensive. One can make the realistic assumption that the non-negative cost of jamming lies in the range between $0$ and the cost of injecting bad-data into a measurement. The overarching goal of this work is thus to _study the impact of adding measurement jamming to the adversary’s arsenal on the design of the optimal ‘detectable’ data attacks._ Here, we formulate the optimal attack vector design as a graph cut problem based on the necessary and sufficient conditions for feasibility. We show that the entire range of values for measurement jamming cost can be divided into two intervals with different optimal attack formulations that lead to two distinct design strategies. Specifically, we prove that measurement jamming significantly alters the optimal ‘detectable’ attack design only if the jamming cost is less than half the cost of data-injection. In contrast, we show that for ‘hidden’ data attacks, measurement jamming leads to a single simple attack strategy independent of the jamming cost. We provide recursive min-cut based algorithms to design the optimal attack over the entire range of jamming cost values and show the cost improvement derived from measurement jamming through simulations on IEEE test cases [14]. By discussing the scope of measurement jamming as an adversarial strategy, our work thus provides a potent and realistic generalization of current data attack frameworks. Finally, we show that number of incorruptible measurements needed to prevent ‘detectable’ attacks scales at least with the total number of measurements. This is much higher than ‘hidden’ attacks where the security needs scale with the number of buses in the system [9]. Thus, in addition to significantly reducing the cost of data attacks, our attack framework also undermines measures of grid resilience based on ‘hidden’ attacks. The rest of this paper is organized as follows. The next section presents a description of the system models used in state estimation, bad-data detection and identification. The novel adversarial attack model with jamming is introduced in Section III along with conditions necessary for attack feasibility. Section IV analyzes how the cost of jamming affects the attack strategy and grid resilience and presents a graph theoretic formulation for the optimal attack design. Our algorithm to design an optimal attack vector is presented in Section V. Simulations of the proposed algorithm for the range of jamming and bad-data injection costs on IEEE bus systems and comparisons with existing work are shown in Section VI. Finally, concluding remarks and future directions of work are presented in Section VII. ## II State Estimation and Bad-Data Detection in Power Grids We denote the power grid by a set $V$ of buses (nodes) connected by a set $E$ of transmission lines (directed edges). Figure 1 shows the graph representation of the IEEE $14$ bus test system [14]. <figure><img src="content_image/1506.04541/x1.png"><figcaption>Fig. 1: IEEE 14-bus test system [14]</figcaption></figure> Measurement Model: We use DC power flow model [16] for the grid here where nodal line voltage magnitudes and line resistances are ignored. It is given by: \[z=Hx+e\] (1) Here $z\in\mathbb{R}^{m}$ is the $m$ length vector of measurements. We consider two kinds of measurements in the grid: a) flow measurements on lines and b) voltage phasor measurements on buses, measured by conventional meters and phasor measurement units. $x\in\mathbb{R}^{n}$ denotes the state vector of length $n=\|V\|$ that comprises of the phase angles at the buses in the grid. $H$ is the measurement matrix and $e$ is a zero mean Gaussian measurement noise vector with known covariance $\Sigma$. Let the $k_{1}^{th}$ and $k_{2}^{th}$ entries in $z$ represent the power flow on line $(i,j)$ from nodes $i$ to $j$ and the voltage phasor at node $i$ respectively. Then, $z(k_{1})=B_{ij}(x(i)-x(j)),~{}z(k_{2})=x(i)$. Here $B_{ij}$ is the susceptance of line $(i,j)$. The corresponding rows in $H$ thus have the following structure: \[H(k_{1})=[0..0~{}~{}B_{ij}~{}~{}0..0~{}~{}-B_{ij}~{}~{}0..0]\] (2) \[H(k_{2})=[0..0~{}~{}1~{}~{}0..0]\] (3) We assume $m>n$ and full column rank of $H$, without a loss of generality. Further, without a loss of generality, we introduce a $(n+1)^{th}$ reference bus with phase angle $0$ in our system and represent it by augmenting $0$ to the state vector $x$. Let $z$ include the phase angle measurement for some bus $i$. Note that the angle measured can be considered equivalent to a flow on a hypothetical line of unit conductance between bus $i$ and the reference bus (with phase $0$). Thus, we can add an extra binary valued column $h^{g}$ corresponding to the reference bus in matrix $H$ to get $z=Hx=[H\|h^{g}]\begin{bmatrix}x\\ 0\end{bmatrix}$. Here $h^{g}(k)=-1$ if $z(k)$ measures a phase angle and $0$ otherwise. Observe that after addition of the reference bus in the system, all measurements now correspond to flow measurements. Abusing notation, we use $x$ and $H$ to denoted the augmented state vector and measurement matrices respectively from this point. State Estimator: We consider a least-square state estimator in the grid as shown in Figure 2[15, 16]. <figure><img src="content_image/1506.04541/x2.png"><figcaption>Fig. 2: State Estimator for a power system [15, 16]</figcaption></figure> The state vector estimate $x^{}$ for a given measurement vector $z$ is generated by minimizing the weighted measurement residual $J(x,z)=\\|\Sigma^{-.5}(z-Hx)\\|_{2}$ over variable $x$. Following estimation, a threshold ($\lambda$) based bad-data detector determines the presence of erroneous measurements by the following test: \[\\|\Sigma^{-.5}(z-Hx^{})\\|_{2} \leq\lambda~{}~{}\text{accept~{}~{}~{}}x^{}\] \[>\lambda~{}~{}\text{detect bad-data}\] (4) If the test detects bad-data, the measurements are sent for eliminating the bad-data as described below, following which the state estimate is recomputed. Bad-data Removal:* Note that the measurement residue vector $r$ for measurement $z$ and estimated $x^{}$ is given by [15, 16]: \[r=z-Hx^{}=[I-H(H^{T}\Sigma^{-1}H)^{-1}H^{T}\Sigma^{-1}]z\] (5) with variance $R_{r}$. Assuming that each measurement is independently affected by natural bad data, the state estimator removes the least number of erroneous measurements such that the resulting residual satisfies the threshold condition in Eq. (4) while preserving full column rank in $H$. For a single removal, the optimal strategy is to remove the measurement with largest normalized residual [15]. However, for multiple bad-data entries, the optimal removal strategy is a non-convex problem [15, 12]. We assume in the remainder of this paper that the measurement data $z$, in the absence of any adversarial manipulation, is reasonably clean and capable of producing the correct state estimate $x^{}$ by passing the bad-data detection test. ### _Attack Models_ Let $a$ denote the injected adversarial attack vector that is added to correct measurements in $z$ to generate the compromised measurement vector $z+a$. Traditional attack models have focussed on bypassing the bad-data detector by ensuring that the measurement residual in Eq. (4) remains unchanged following the injection of bad-data. Mathematically, this requires $a=Hc\neq 0$ for some $c\in\mathbb{R}^{n}$ as $\\|\Sigma^{-.5}(z-Hx^{})\\|_{2}=\\|\Sigma^{-.5}(z+a-H(x^{}+c))\\|_{2}$. Thus, a ‘Hidden’ Attack* results that produces an erroneous state vector $x^{}+c$[5]. Next we describe ‘detectable’ data attacks [12] that are the focus of this paper. ‘Detectable’ Data Attack:* From the bad-data removal scheme described earlier, it is clear that an attack vector $a\neq 0$ will change the state estimate if removal of some other $k<\\|a\\|_{0}$ measurements (distinct from the attack vector) satisfies the bad-data detection test. For a nonzero $Hc$, consider the adversarial strategy that excludes (or does not corrupt) less than $50\%$ of the non-zero entries in $Hc$ from the attack vector $a$. Note that $a$ still gives a feasible ‘detectable’ attack as the non-zero terms in $(Hc-a)$ are identified as bad-data instead of vector $a$. This happens as $\\|a\\|_{0}>\\|Hc-a\\|_{0}$. In the next section, we formulate in detail the design of the optimal ‘detectable’ data attack and the use it to analyze changes that arise due to the adversarial capability to jam measurements. ## III ‘Detectable’ Attack with Measurement Jamming In a general setting, few of the measurements in the grid may be incorruptible due to geographical isolation or encryption. We denote this set of measurements secure from adversarial corruption by $S$. Note that measurements in $S$ suffer from normal bad-data arising from measurement noise. The remaining insecure measurements belong to set $S^{c}$. The measurements included in the minimum cost ‘detectable’ attack are given by non-zero terms in the optimal vector $d$ in the following optimization problem [12]: \[\smashoperator[l]{\min_{d\in\{0,1\}^{m},c\in\mathbb{R}^{n+1}}^{}} \\|d\\|_{0}\] (P-1) s.t. \[a=Hc,c\neq\textbf{0},c(n+1)=0\] \[d(i)=0~{}\forall i\in S_{m}~{}~{}(\text{secure measurements})\] \[\\|d\\|_{0}>\\|a\\|_{0}/2~{}~{}(\text{for feasibility})\] (6) \[rank(DH)=n,~{}diag(D)=\textbf{1}-(\textbf{1}-d)a_{spty}\] (7) Here, $ab$ refers to the element-wise multiplication between vector $a$ and $b$, while $a_{spty}$ denotes the sparsity pattern in vector $a$. Condition (6) ensures that the estimator removes measurement entries corresponding to non-zero terms in $(\textbf{1}-d)a$ as bad-data, instead of the data injected in $da$. $D$ is a diagonal matrix whose diagonal entries are $0$ for removed data and $1$ otherwise. $DH$ is the measurements matrix after bad-data removal. Condition (7) keeps it at full rank. The attack passes the bad-data detection test as it lies in the column space of $DH$. It is worth restating that as each row in augmented $H$ corresponds to a flow measurement, $H$ is equivalent to a susceptance weighted incidence matrix of a graph $G_{H}$ with $n+1$ nodes and edges given by rows in $H$. Due to this structure of $H$, it can be shown that [8, 9, 12] the optimal attack $a^{}=Hc$ corresponds to a $0-1$ binary valued nodal vector $c$. Further, the optimal attack strategy for Problem P-1 doesn’t change if $H$ is replaced by the un-weighted incidence matrix $A_{H}$ of graph $G_{H}$ ($A_{H}(i,j)=1(\hat{H}(i,j)>0)-1(\hat{H}(i,j)<0)$) as for a binary valued $c$, $A_{H}c$ and $Hc$ have the same set of non-zero terms (identical sparsity pattern). Note that non-zero values in $A_{H}c$ actually represents cut edges in graph $G_{H}$ between nodes marked $1$ and $0$. This leads to the following result (Theorem 2 in [12]) for optimal attack for Problem P-1. Theorem 1 ([12, Theorem 2]).: _Let $C^{}$ denote the minimum cardinality cut in $G_{H}$ with a minority of secure cut-edges ($\|C^{}\cap S\|<\|C^{}\|/2$ ). An optimal ‘detectable’ attack for Problem P-1 is given by any $\lfloor 1+\|C^{}\|/2\rfloor$ cut-edges in $C^{}\cap S^{c}$ (insecure cut edges)._ We ignore the proof here for space constraints. Observe that if $d$ is restricted to an all-$1$ vector, Problem P-1 reduces to the problem of determining the optimal ‘hidden’ attack. The optimal attack in that case is given by the minimum cardinality cut in $G_{H}$ that does not include any secure edge in $S^{m}$[8, 9]. ‘Detectable Jamming’ Attack:* We now analyze an adversary with the capacity to jam insecure measurements in addition to manipulating their values by bad-data injection. Secure measurements are assumed to be Let $p_{J}$ and $p_{I}$ be the cost associated with jamming and bad-data injection into an insecure measurement in the grid respectively. We assume that $0\leq p_{I}\leq p_{I}$ as the range of $p_{J}$ as jamming is less resource intensive than bad-data injection. This is a reasonable assumption as jamming can even be conducted by introducing garbage values through bad-data injection techniques. For ease of elucidation, we assume that the jamming and manipulation costs are uniform over all measurements in $S^{c}$, though all analysis follows immediately for variable costs as well. Consider a cut $C$ in graph $G_{H}$. Let $n^{C}_{S}$ and $n^{C}_{S^{c}}$ denote the number of secure and insecure edges in cut $C$ with $n^{C}_{S^{c}}>n^{C}_{S}$ as shown in Fig. 3. By Theorem 1, attack feasibility requires injection into $k^{C}$ ($k^{C}>\|C\|/2$) insecure edges at a cost of $p_{I}k^{C}$. Instead, consider a different strategy where the adversary jams $k^{C}_{J}$ insecure measurements. As jammed measurements are not received and ignored by the control center, the cut-size effectively reduces to $\|C\|-k^{C}_{J}$. If the remaining $n^{C}_{S^{c}}-k^{C}_{J}$ insecure edges in the cut are greater in number than the $n^{C}_{S}$ secure edges, the adversary can still attack $k^{C}_{I}\geq 1+\lfloor\frac{\|C\|-k^{C}_{J}}{2}\rfloor$ measurements and generate a feasible attack. As depicted in Fig. 3, the cost of this new attack is $p_{I}k^{C}_{I}+p_{J}k^{C}_{J}$. We term it a ‘detectable jamming’ attack to distinguish it from the original ‘detectable’ attack that doesn’t incorporate jamming. We formulate the design of the optimal ‘detectable jamming’ attack as follows: \[\smashoperator[l]{\min_{d_{J},d_{I}\in\{0,1\}^{m}}^{}}p_{J}\\|d_{J }\\|_{0}+p_{I}\\|d_{I}\\|_{0}\] s.t. \[a=A_{H}c,c\in\{0,1\}^{n+1}-{\textbf{0}},c(n+1)=0\] \[d_{J}+d_{I}\in\{0,1\}^{m}\] (8) \[d_{J}(i)=d_{I}(i)=0~{}\forall i\in S_{m}\] (9) \[\\|d_{I}\\|_{0}>(\\|a\\|_{0}-\\|d_{J}\\|_{0})/2~{}~{}(\text{for feasibility})\] (10) \[rank(DA_{H})=n\text{~{}~{}where~{}}diag(D)=\textbf{1}-(\textbf{1 }-d_{J}-d_{I})\|a\|\] (11) The non-zero values in optimal $d_{J}$ and $d_{I}$ give the measurements to jam and injection bad-data respectively in the optimal attack. Note that in Problem III, we replaced $H$ with incidence matrix $A_{H}$ and made $c$ a $0-1$ vector as discussed earlier. Here, condition 8 ensures data injection and jamming cannot occur at the same measurement. The remaining conditions arise from incorruptibility of secure measurements (9), feasibility of ‘detectable’ attack (10) and full system observability after bad-data removal (11). From the discussion preceding Problem III, it is clear that the optimal ‘detectable jamming’ attack has a graph-cut based construction as stated below. Lemma 1.: _Let $C$ denote a cut in $G_{H}$ with $(n^{C}_{S^{c}}>\|C\|/2)$ insecure cut-edges. A feasible attack is given by jamming $(k^{C}_{J}\geq 0)$ and injecting data into $(\lfloor 1+(\|C^{}\|-k^{C}_{J})/2\rfloor>0)$ of the $n^{C}_{S^{c}}$ insecure cut-edges at a cost of $p_{J}k^{C}_{J}+p_{I}\lfloor 1+(\|C^{}\|-k^{C}_{J})/2\rfloor$. The optimal ‘detectable jamming’ attack is given by minimizing the attack cost over variable $k^{C}_{J}$ (jammed edges) for all feasible cuts $C$._ It is noteworthy that if $k^{C}_{J}=0$ in Lemma 1, we obtain the optimal ‘detectable’ attack (no jamming) as a feasible ‘detectable jamming’ attack. This leads to following important properties. Corollary 1.: _The space of system configurations with feasible ‘detectable jamming’ attacks is identical to that of ‘detectable’ attacks and is a superset of that of hidden attacks._ * _The cost of the optimal ‘detectable jamming’ attack is never greater than the cost of optimal ‘detectable’ attack and never greater than_ $.5+1/\|C_{h}^{}\|$ _times the cost of optimal ‘hidden’ attack on a system,_ $\|C_{h}\|$ _being the cardinality of optimal ‘hidden’ attack._ The first property arises as the set of cuts with majority of edges in $S^{c}$ (feasibility requirement of ‘detectable’ and ‘detectable jamming’ attacks) is a superset of the set of cuts will all edges in $S^{c}$ (feasibility requirement of ‘hidden’ attacks). The second property has two parts: the first part follows from the fact that the optimal ‘detectable’ attack is a feasible ‘detectable jamming’ attack and hence not of lower cost that the optimal; the second part follows from the fact that injecting bad-data into $1+\lfloor\|C_{h}^{}\|\rfloor/2$ measurements of the optimal ‘hidden’ attack constitutes a feasible ‘detectable’ attack. It needs to be mentioned that these bounds reflect comparisons in the worst-case. The simulation results in Section VI demonstrate that the average impact of ‘detectable jamming’ attack is much more substantial. In the next section, we discuss the effect of jamming cost $p_{J}$ on the design of the optimal attack vector and its key properties. ## IV Effect of Jamming cost on Attack Construction <figure><img src="content_image/1506.04541/x3.png"><figcaption>Fig. 3: Effect of jamming cost pJ and bad-data injection cost pI on theminimum cost attack C∗ derived from a feasible cut C with nCS secure and nCScinsecure measurements. Secure, insecure but untouched, jammed, bad-datainjected measurements in the cut are represented by red, white, blue and greencolors respectively. When pJ<pI/2, attack cost is reduced by replacing onebad-data injection with jamming two measurements as shown in the cuts on theleft of C. For pJ≥pI/2, attack cost is reduced by replacing two jammedmeasurements by one measurement with bad-data injection while leaving theother untouched as shown on the right side of cut C. Optimal cuts C∗ got fromthis replacement are given by Theorem 2.</figcaption></figure> As mentioned earlier, we consider the jamming cost $p_{J}$ to lie in the interval $[0,p_{I}]$ where $p_{I}$ is the bad-data injection cost. Consider a feasible cut $C$ with $n^{C}_{S^{c}}$ insecure edges and $n^{C}_{S}$ secure edges in the measurement graph $G_{H}$. Here $n^{C}_{S^{c}}>n^{C}_{S}$ as shown in Fig. 3. By Theorem 1, a feasible ‘detectable jamming’ attack comprises of selecting $(k^{C}_{J}\geq 0)$ and $(k^{C}_{I}=\lfloor 1+(\|C\|-k^{C}_{J})/2\rfloor>0)$ insecure edges for jamming and bad-data injection respectively, at a overall cost of $p^{C}$ \[p^{C} =p_{J}k^{C}_{J}+p_{I}\lfloor 1+(\|C\|-k^{C}_{J})/2\rfloor\] \[=(p_{J}-p_{I}/2)k^{C}_{J}+p_{I}\frac{\|C\|+2-(\|C\|-k^{C}_{J})\mod 2} {2}\] (12) We divide the range of $p_{J}$ into two intervals: A ($p_{J}<p_{I}/2$) and B ($p_{I}/2\leq p_{J}\leq p_{I}$). Note that in interval A, the cost $p^{C}$ is a decreasing function of $k^{C}_{J}$. Therefore, the minimum cost attack for feasible cut $C$ is obtained by jamming $n^{C}_{S^{c}}-n^{C}_{S}-1$ (the maximum permissible number of) insecure edges . The remaining $n^{C}_{S}+1$ insecure edges, greater than the number of secure edges by one, are injected with bad-data. The attack cost is given by \[p^{C} =p_{J}(n^{C}_{S^{c}}-n^{C}_{S}-1)+p_{I}(n^{C}_{S}+1)\] \[=(p_{I}-p_{J})n^{C}_{S}+p_{J}n^{C}_{S^{c}}+(p_{I}-p_{J})\] (13) . Ignoring constant $(p_{I}-p_{J})$, this equals $C$’s cut-weight if secure and insecure edges are given weights of $(p_{I}-p_{J})$ and $p_{J}$ respectively. Thus, if $p_{J}<p_{I}/2$, the optimal ‘detectable jamming’ cut corresponds to the feasible cut $C^{}$ with lowest cut-weight in $G_{H}$, where secure and insecure edges have weights of $(p_{I}-p_{J})$ and $p_{J}$ respectively. Next consider interval B ($p_{I}/2<p_{J}\leq p_{I}$). In Eq. (12), if $k^{C}_{J}$ is reduced by $2$, the $(\|C^{}\|-k^{C}_{J})\mod 2$ term remains unchanged and the overall cost $p^{C}$ decreases. Hence the optical attack for cut $C$ corresponds to either $k^{C}_{J}=0$ or $k^{C}_{J}=1$, otherwise the attack cost can be reduced further. Checking the contribution of $(\|C^{}\|-k^{C}_{J})\mod 2$ term manually, we note that the optimal attack for cut $C$ is given by $(k^{C}_{J}=0,k^{C}_{I}=(1+\|C\|)/2)$ for odd $\|C\|$, and $(k^{C}_{J}=1,k^{C}_{I}=\|C\|/2)$ for even $\|C\|$. In either case, the optimal attack cost is an increasing function of the cut-size $\|C\|$ expressed below. \[p^{C} =p_{J}(1-\|C\|\mod 2)+p_{I}\lfloor(1+\|C\|)/2\rfloor\] (14) Thus, in interval B, the optimal ‘detectable jamming’ attack corresponds to the feasible cut $C^{}$ with lowest cut-size in $G_{H}$. We summarize this discussion by presenting our main theorem for optimal ‘detectable jamming’ attack construction. Theorem 2.: _The minimum cost ‘detectable jamming’ attack for measurement graph $G_{H}$ with jamming cost $p_{J}$ and bad-data injection cost $p_{I}$ is constructed as follows._ $p_{J}<p_{I}/2$_:_ _Give weights of_ $p_{I}-p_{J}$ _and_ $p_{J}$ _to secure and insecure edges respectively in_ $G_{H}$ _and find the minimum weight feasible cut_ $C^{}$ _with_ $n^{C^{}}_{S}$ _secure edges. Use_ $(n^{C^{}}_{S}+1)$ _insecure measurements for bad-data injection and jam the rest._ $p_{J}\geq p_{I}/2$_:_ _Find the minimum cardinality feasible cut_ $C^{}$ _in_ $G_{H}$_. Use_ $\lfloor(1+\|C^{}\|)/2\rfloor$ _insecure measurements for bad-data injection and jam_ $(1-\|C^{}\|\mod 2)$ _measurement._ The comparison of attack costs in ‘detectable jamming’ attacks with that of standard ‘detectable’ attacks is given by the following. Theorem 3.: _Let cut_ $C^{}$ _with_ $n^{C^{}}_{S}$ _secure and_ $n^{C^{}}_{S^{c}}$ _insecure edges correspond to the optimal ‘detectable’ attack (no jamming). The cost of optimal ‘detectable jamming’ attack satisfies the following bounds._ _For_ $p_{J}<p_{I}/2$_, the cost of the optimal ‘detectable jamming’ attack is less than that of the optimal ‘detectable’ attack cost by at least_ $(p_{I}-2p_{J})\lfloor\frac{n^{C^{}}_{S^{c}}-n^{C^{}}_{S}}{2}\rfloor+p_{J}(1- \|C^{}\|\mod 2)$_._ _For_ $p_{J}\geq p_{I}/2$_, the cost of the optimal ‘detectable jamming’ attack is less than that of ‘detectable’ attack by_ $p_{I}-p_{J}$ _(if_ $\|C^{}\|$ _is even), and equal otherwise._ Proof.: For $p_{J}\geq p_{I}/2$, using Theorem 1 and Theorem 2, it follows that the optimal cuts for ‘detectable’ and ‘detectable jamming’ attacks are identical. The difference is costs follows immediately from the attack construction using the optimal cut $C^{}$ in either case. For $p_{J}<p_{I}/2$, note that the minimum-cost ‘detectable jamming’ attack for feasible cut $C^{}$ is given by injecting bad-data into $n^{C^{}}_{S}+1$ edges and jamming the other insecure edges. The difference in cost between ‘detectable jamming’ attack and ‘detectable’ attack for cut $C^{}$ is thus: \[\delta =p^{I}\lfloor 1+\frac{n^{C^{}}_{S^{c}}+n^{C^{}}_{S}}{2}\rfloor- p_{I}(n^{C^{}}_{S}+1)-p_{J}(n^{C^{}}_{S^{c}}-n^{C^{}}_{S}-1)\] \[=(p_{I}-2p_{J})\lfloor\frac{n^{C^{}}_{S^{c}}-n^{C^{}}_{S}}{2} \rfloor+p_{J}(1-\|C^{}\|\mod 2)\] (15) As $C^{}$ is a feasible ‘detectable jamming’ attack (not necessarily optimal) in this case, Eq. 15 gives a lower bound on the difference in optimal costs. ∎ Further, the following statements holds: Corollary 2.: * _For_ $p_{J}=0$ _(minimum jamming cost), the optimal ‘detectable jamming’ attack corresponds to the cut_ $C^{}$_, which has the minimum number of secure edges among all feasible cuts._ _For_ $p_{J}=0$_, if a ‘hidden’ attack exists, an optimal ‘detectable jamming’ attack corresponds to the same cut_ $C^{}$_._ Finally, the following theorem presents the potency of ‘detectable jamming’ attacks by a lower bound on the number of secure measurements required for complete security. Theorem 4.: _A system is always vulnerable to ‘detectable jamming’ attacks if less than half the total number of measurements are secure._ Proof.: Consider the graph $G_{H}$ generated from the measurement system. A feasible ‘detectable jamming’ attack requires a cut in $G_{H}$ with a majority of insecure edges. As less than half of the measurements in $G_{H}$ are secure, there is at least one bus connected with a majority of insecure edges. Thus, a feasible ‘detectable jamming’ attack can be constructed using that bus’s edges as the cut. Hence proved. ∎ Note that Theorem 4 provides a $O(\|E\|)$ lower bound on the minimum number of secure measurements required for complete security, that scales with the total number of measurements. In contrast, complete protection from ‘hidden’ attacks require a maximum of $\|V\|-1$ secure measurements [5, 9], that is much lesser that in general graphs. In Section VI, we show simulations that confirm that ‘detectable jamming’ attacks are more resilience to presence of secure measurements than ‘hidden’ attacks. In the next Section, we present our algorithm to construct the optimal attack described in Theorem 2 and Corollary 2. ## V Algorithm For Attack Construction To confirm the existence of a feasible attack, we need to identify a feasible cut with a majority of insecure edges in the graph. Theorem $3$ in [12] proves that this is equivalent to the ‘ration-cut’ problem, a known NP-hard problem. Thus, the design of the optimal ‘detectable jamming’ attack, in the worst case, is hard as well. We now provide an approximate algorithm (Algorithm $1$) for attack vector construction. For $p_{J}<p_{I}/2$ (interval A), we create weighted graph $G_{H}$ with secure (insecure) edges having weight $p_{I}-p_{J}$ ($p_{J}$). For $p_{J}\geq p_{I}/2$ (interval B), we consider unweighted $G_{H}$. Using Theorem 2, the optimal attack, in either case, is given by the minimum weighted feasible cut in $G_{H}$. Working:* Algorithm $1$ proceeds by computing the minimum weight cut $C$ in $G_{H}$ (Step 1) and checks if it is a feasible cut (Step 3). If $C$ is infeasible, one secure edge is selected randomly in $C$ and its edge-weight is increased by $\beta$ (Step 4). We consider two cases, one where $\beta$ is taken as the secure edge-weight and the other where it is taken as $\infty$. Following the increase, the algorithm recomputes the minimum weight cut and checks for feasibility. This process is iterated until a feasible cut is obtained (construct the attack vector) or the cut-weight reaches a threshold $\gamma<\infty$(declare no solution). Input: Graph $G_{H}$ with secure and insecure edges weighted based on $p_{J},p_{I}$, $S,S^{c},\beta,\gamma$ ``` 1: Compute min-weight cut $C$ in $G_{H}$ 2: $w_{C}\leftarrow$ weight of $C$ 3: while ($w_{C}<\gamma,2\|C\bigcap S\|\geq\|C\|$) do 4: Randomly pick edge $i\in C\bigcap S$ and increase its weight by $\beta$ 5: Compute min-weight cut $C$ in $G_{H}$ 6: $w_{C}\leftarrow$ weight of $C$ 7: end while 8: if $2\|C\bigcap S\|<\|C\|$ then 9: Construct attack vector using Theorem 2 10: else 11: Declare no solution 12: end if ``` Algorithm 1 ‘Detectable Jamming’ Attack Construction Note that for $\beta=\infty$, in the worst case, there are $\|S\|$ min-cut computations (one for each secure edge) of complexity $O(\|V\|\|E\|+\|V\|^{2}\log\|V\|)$ giving the algorithm a computational complexity of $O(\|S\|\|V\|\|E\|+\|S\|\|V\|^{2}\log\|V\|)$. However, as the algorithm is approximate, it might not return a solution in every case. In the next section, we show simulation results on designing optimal attacks by Algorithm $1$ in IEEE test systems. We also demonstrate the capacity of ‘detectable jamming’ attacks in overcoming high placement of secure measurements in the systems considered. ## VI Results on IEEE test systems We discuss the performance of Algorithm $1$ in designing ‘detectable jamming’ attacks by simulations on IEEE $14$-bus and $57$-bus test systems [14]. In each simulation run, we put flow measurements on all lines in the test system considered and phase angle measurements on $60\%$ (randomly selected) of the system buses. Over multiple simulations, we vary the fraction of secure measurements and record the trends in average cost of constructing ‘detectable jamming’ attack. We consider either interval of jamming cost ($p_{J}=0$, $p_{J}<p_{I}/2$ and $p_{J}>p_{I}/2$), and different values of parameter $\beta$ (finite and $\infty$) in Algorithm $1$. The trends in average optimal cost of ‘detectable jamming’ attacks for the $14$ bus system are presented in Fig. 4 (for configurations that allow feasible ‘hidden’ attacks), and Fig. 5 (for configurations that are resilient to ‘hidden’ attacks). To demonstrate the efficacy of our approach, we compare the trends with average costs of constructing ‘hidden’ and ‘detectable’ (no jamming) attacks. Note that while the average attack cost is way below the upper bound (Corollary 1) in Fig. 4, it is observed to eventually decrease with increasing secure measurements in the system. This trend results from the fact that system configurations resilient to attacks that increase with increasing secure measurements are not accounted for in the plotted average attack costs. Further, it is apparent that changing the value of $\beta$ does not affect the performance of Algorithm $1$ much. Similarly, Fig. 6 includes the average cost trends for the $57$ bus system, with $\beta$ in Algorithm $1$ being taken equal to the weight of secure measurements. From the figures it is clear that jamming enabled attacks have significantly reduced costs over both ‘hidden’ and ‘detectable’ attacks. Finally, Fig. 7 plots the increase in number of completely resilient operating regimes (no feasible attack possible) with an increase in the number of secure measurements in the system. It is easily evidenced in Fig. 7 that compared to ‘hidden’ attacks, ‘detectable’ and ‘detectable jamming’ attacks pose a much greater threat to the grid vulnerability as the number of secure operating regimes in the latter hardly increases with an increase in the number of secure measurements. This is in line with the security needs highlighted in Theorem 4. The simulations prove the dual adversarial benefits created by ‘detectable jamming’ attacks: lowering of attack cost and increased insensitivity to deployment of incorruptible measurements. <figure><img src="content_image/1506.04541/x4.png"><figcaption>Fig. 4: Average cost of optimal attacks (‘hidden’, ‘detectable’ and‘detectable jamming’) produced for different values of β (size of secure edgeand ∞) by Algorithm 1 on the IEEE 14 bus test system with flow measurements onall lines, phasor measurements on 60% of the buses and protection on afraction of measurements selected randomly. The bad-data injection cost (pI)is taken as 1. For the ‘detectable jamming’ attack, the jamming costs (pJ)considered are 0,1/4(<pI/2),3/4(>pI/2). Only configurations where ‘hidden’attacks are possibly are considered to determine the average costs.</figcaption></figure> <figure><img src="content_image/1506.04541/x5.png"><figcaption>Fig. 5: Average cost of optimal attacks (‘detectable’ and ‘detectablejamming’) produced for different values of β (size of secure edge and ∞) byAlgorithm 1 on the IEEE 14 bus test system with flow measurements on alllines, phasor measurements on 60% of the buses and protection on a fraction ofmeasurements selected randomly. The bad-data injection cost (pI) is taken as1. For the ‘detectable jamming’ attack, the jamming costs (pJ) considered are0,1/4(<pI/2),3/4(>pI/2). Only configurations which are resilient against‘hidden’ attacks are considered to determine the average costs.</figcaption></figure> <figure><img src="content_image/1506.04541/x6.png"><figcaption>Fig. 6: Average cost of optimal attacks (‘detectable’ and ‘detectablejamming’) produced by Algorithm 1 (with finite β) on the IEEE 57 bus testsystem with flow measurements on all lines, phasor measurements on 60% of thebuses and protection on a fraction of measurements selected randomly. The bad-data injection cost (pI) is taken as 1. For the ‘detectable jamming’ attack,the jamming costs (pJ) considered are 0,1/4(<pI/2),3/4(>pI/2).</figcaption></figure> <figure><img src="content_image/1506.04541/x7.png"><figcaption>Fig. 7: Average fraction of simulated configurations with no feasible ‘hidden’and ‘detectable jamming’ attacks given by Algorithm 1 for different values ofβ in IEEE 14 and 57 bus test systems. Each test system has flow measurementson all lines, phasor measurements on 60% of the buses and protection on afraction of measurements selected randomly.</figcaption></figure> ## VII Conclusion We introduce a new data attack framework on power grids termed ‘detectable jamming’ attacks, where an adversary uses measurement jamming as a tool in addition to changing meter readings (bad-data injection). Through the use of these dual techniques on an optimal set of measurements, the adversary creates a violation of the bad-data detection test but still creates a change in the estimated state vector. This is ensured by leading the state estimator to incorrectly label uncorrupted correct data as bad-data. We show that the design of the minimum cost attack of this regime is shown to be equivalent to a constrained graph cut problem that takes two different forms, dependent on the relative values of jamming and data injection costs. We prove that even the worst-case attack cost of ‘detectable jamming’ attacks is approximately half of the optimal ‘hidden’ attack cost, while the capability to overcome incorruptible measurements is much more pronounced than in ‘hidden’ attacks. This is highlighted by the fact that the number of secure measurements required for complete resilience against ‘hidden’ attack is of the order of number of buses in the system, while complete resilience against ‘hidden’ attacks requires greater than half the measurements to be incorruptible and scales with the number of edges in the measurement graph. We further show that in comparison to ‘detectable’ (no jamming) attacks, our jamming reliant framework significantly alters the optimal attack (given by the optimal graph cut) only if the jamming cost is less than half the cost of bad-data injection. For values of jamming cost greater than half the injection cost, ‘detectable jamming’ attacks have a lower attack cost but correspond to the same optimal graph cut as ‘detectable’ attacks. As the design of the optimal attack is NP hard in general, we present an iterative min-cut based approximate algorithm with polynomial complexity to determine the optimal cut. We demonstrate the adversarial benefits of our proposed attack framework and performance of our approximate algorithms through simulations on IEEE test cases for different values of jamming costs and different system conditions. This paper exposes the adverse effects to grid security posed by measurement jamming when used as an adversarial tool to supplement ‘bad-data’ injection. Designing optimal security measures against this attack regime is the object of our current research in this domain. ## References * [1] A. G. Phadke, “Synchronized phasor measurements in power systems”, _IEEE Comput. Appl. Power_, vol. 6, 1993. * [2]http://www.nytimes.com/2014/07/01/technology/energy-sector-faces-attacks-from-hackers-in-russia.html * [3] J. Meserve, “Staged cyber attack reveals vulnerability in power grid”, CNN, 2007. Available: http://www.cnn.com/2007/US/ 09/26/power.at.risk/index.html. * [4] Shepard, D. P., Humphreys, T. E., and Fansler, A. A., “Evaulation of the Vulnerability of Phasor Measurement Units to GPS Spoofing”, _International Journal of Critical Infrastructure Protection_, 2012. * [5] Y. Liu, P. Ning, and M. K. Reiter, “False data injection attacks against state estimation in electric power grids”, _Proc. ACM Conf. Comput. Commun. Security_, 2009. * [6] T. Kim and V. Poor, “Strategic Protection Against Data Injection Attacks on Power Grids”, _IEEE Trans. Smart Grid_, vol. 2, no. 2, 2011. * [7] O. Vukovic, K. C. Sou, G. Dan, and H. Sandberg, “Network-aware mitigation of data integrity attack on power system state estimation”, _IEEE Journal on Selected Areas in Communications_, vol. 30, no. 6, 2012. * [8] D. Deka, R. Baldick, and S. Vishwanath, “Optimal Hidden SCADA Attacks on Power Grid: A Graph Theoretic Approach”, _ICNC_, 2014. * [9] D. Deka, R. Baldick, and S. Vishwanath, “Data Attack on Strategic Buses in the Power Grid: Design and Protection , _IEEE PES General Meeting_, 2014. * [10] O. Kosut, L. Jia, R. J. Thomas, and L. Tong, “Limiting false data attacks on power system state estimation”, _Proc. Conf. Inf. Sci. Syst._, 2010. * [11] J. Kim, L. Tong, and R. J. Thomas, “Data Framing Attack on State Estimation with Unknown Network Parameters”, _Asilomar Conference on Signals, Syst., and Computers_, 2013. * [12] D. Deka, R. Baldick, and S. Vishwanath, “Data Attacks on the Power Grid DESPITE Detection”, _IEEE PES Innovative Smart Grid Technologies_, 2015. * [13] L. Shichao, L. P. Xiaoping, and S. E. Abdulmotaleb, “Denial-ofservice (dos) attacks on load frequency control in smart grids”, _IEEE PES Innovative Smart Grid Technologies_, 2013. * [14] R. Christie, “Power system test archive”, Available: http://www.ee.washington.edu/research/pstca. * [15] A. Monticelli, “State estimation in electric power systems: a generalized approach”, _Kluwer Academic Publishers_, 1999. * [16] A. Abur and A. G. Exposito, “Power System State Estimation: Theory and Implementation”, _CRC_, 2000. * [17] M. R. Garey and D. S. Johnson, “Computers and Intractability: A Guide to the Theory of NP-Completeness”, _W. H. Freeman_, 1979. * [18] S. Boyd and L. Vandenberghe, “Convex Optimization”, _Cambridge University Press_, 2004. * [19] M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems”, _Journal of the ACM_, vol. 42, 1995.
0811.4057	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 54318, "num_imgs": 0, "llama3_tokens_count": 14337 }	[]	# Gamma-rays from the vicinity of accreting neutron stars inside compact high-mass X-ray binaries W. Bednarek Department of Astrophysics, University of Łódź, 90-236 Łódź, ul. Pomorska 149/153, Poland ¹ [FOOTNOTE:1][ENDFOOTNOTE] Received ; accepted Key Words.:gamma-rays: theory – radiation mechanisms: non-thermal – binary systems: close – neutron starsDense wind of a massive star can be partially captured by a neutron star (NS) inside a compact binary system. Depending on the parameters of NS and the wind, the matter can penetrate the inner NS magnetosphere. At some distance from the NS a very turbulent and magnetized transition region is formed due to the balance between the magnetic pressure and the pressure inserted by accreting matter. This region provides good conditions for acceleration of particles to relativistic energies. The matter at the transition region can farther accrete onto the NS surface (the accretor phase) or is expelled from the NS vicinity (the propeller phase). We consider the consequences of acceleration of electrons at the transition region concentrating on the situation in which at least part of the matter falls onto the NS surface. This matter creates a hot spot on the NS surface which emits thermal radiation. Relativistic electrons lose energy on the synchrotron process and the inverse Compton (IC) scattering of this thermal radiation. We calculate the synchrotron spectra (from X-rays to soft $\gamma$-rays) and IC spectra (above a few tens MeV) expected in such a scenario. It is argued that a population of recently discovered massive binaries by the INTEGRAL observatory, which contain neutron stars hidden inside dense stellar winds of massive stars, can be detectable by the recently launched _Fermi_ LAT telescope at GeV energy range. As an example, we predict the expected $\gamma$-ray flux from recently discovered source IGR J19140+0951. ## 1Introduction Massive binary systems, containing a compact object (neutron star or black hole), have been suspected as sites of particle acceleration up to at least TeV energies since early 80’ties. But only recently, three such type of objects have been confirmed as VHE $\gamma$-ray sources by the Cherenkov telescopes. It is clear that at least in one case the $\gamma$-ray emission is due to acceleration of particles as a result of interaction of the energetic pulsar wind with the wind of a massive star. Other TeV $\gamma$-ray binaries either operate under this same general scenario or the acceleration of particles occurs in the jet launched by the compact object. In the case of these TeV $\gamma$-ray binaries, the neutron star (pulsar) should rotate fast enough in order to produce energetic pulsar wind. In this paper we consider the case when the neutron star is not so energetic and the matter from the stellar wind can penetrate into the inner pulsar magnetosphere. At certain distance from the NS surface the magnetic pressure balances the pressure of accreting matter creating very turbulent transition region. Then, the NS is enshrouded in a dense cocoon and resembles objects called as the _hidden pulsars_ (see e.g. Tavani 1991, Tavani & Brookshow 1993) or cauldrons (Begelman & Rees 1984, Treves et al. 1993). Note that $\gamma$-ray emission has been predicted from such _hidden pulsars_ (Tavani 1993). We consider the situation in which electrons are accelerated in turbulent, strongly magnetized, transition region in the inner pulsar magnetosphere. In principle, electrons can even reach $\sim TeV$ energies but, as we show below, due to large synchrotron energy losses only $\gamma$-rays in the _Fermi_ LAT telescope energy range may be effectively produced. It looks that the INTEGRAL observatory has recently discovered a class of objects within the sample of compact high mass X-ray binaries (see e.g. Chaty et al 2007, Rodriquez & Bodaghee 2008) which might operate under the mechanism discussed by us. These newly discovered massive binaries are compact, with the orbital periods a few to several days. Some of them contain relatively slowly rotating neutron stars which may allow the matter to penetrate the inner NS magnetosphere. According to the classification scheme of X-ray binaries (e.g. Lipunov 1992) such binaries with slowly rotating neutron stars (NS) belong to the propeller and accretor class. In these objects, the matter of the dense stellar wind accrete onto the strongly magnetized NS, interacting with the rigidly rotating inner magnetosphere. In the case of accretor most of the matter reach the NS surface. In the case of propeller most of the matter is expelled from the NS vicinity due to the centrifugal force. It is proposed that such general scenario can be applied to a type of obscured compact objects inside massive binaries recently observed by the INTEGRAL observatory. We perform detailed calculations of the $\gamma$-rays spectra including in some cases also the inverse Compton (IC) e${}^{\pm}$ pair cascade initiated by relativistic electrons in the radiation field produced on the neutron star surface. We also take into account the synchrotron energy losses of primary electrons and cascade $e^{\pm}$ pairs, in order to obtain simultaneous synchrotron X-ray spectra. The acceleration of hadrons in such a scenario and possible production of high energy $\gamma$-rays, neutrinos and neutrons will be discussed in another paper. ## 2Description of the model We consider a compact binary system containing rotating neutron star (NS) and a massive companion of the O,B type star. It is assumed that a mass from the stellar wind is effectively captured by a strong gravitational potential of the NS. Such slowly rotating neutron stars appear at a certain stage of the evolution of the binary system due to the loss of angular momentum either by the pulsar mechanism or as a result of the torque exhibited by the matter accreted from the stellar wind. Depending on the rotational period and surface magnetic field of NS, the accretion process onto NS can occur in different phases. According to the classification scheme of Lipunov (1992), the accretion process can occur in the phase of accretor (for relatively slower rotators) or in the phase of propeller. For very energetic, strongly magnetized and short period neutron stars, the accretion process does not occur at all since the matter can not penetrate below the light cylinder radius. In the case of NS in the accretor and propeller phases, the matter from the stellar wind penetrate below the light cylinder radius of the rotating NS magnetosphere. This matter extracts rotational energy from the NS as a result of the interaction of a free falling matter with the rigidly rotating inner NS magnetosphere. In this paper, we consider such direct collision of the matter from the stellar wind with the NS magnetosphere (see general scenario shown in Fig. 1). As a result of such interaction, a very turbulent and magnetized transition region is formed. In the case of accretor, most of the matter falls onto the NS surface creating small hot region on the NS surface. In the case of propeller, most of the matter is expelled from the vicinity of the NS. It is expected that the matter might be able to accrete also in this stage but in a rather non-stationary form since due to gradual accumulation of the matter close to the transition region the pressure of the matter can overcome the pressure from the rotating magnetosphere. It is supposed that some amount of the matter can accrete but most of it is expelled from the binary system (forming a larger scale jets ?) for some specific transition parameters describing the accretor/propeller scenario. Note that in some massive binaries (e.g. Her X-1), the accretion of matter can occur also through the Roche lobe overflow. Then, the accretion disk around the NS is at first formed and the matter from the disk interacts directly with the NS magnetosphere. We do not discuss this possibility in this paper but rather concentrate on a simpler quasi-spherical accretion onto magnetized NS of the matter from the dense wind of the massive star. The accretion through the disk can be easily taken into our considerations in our scenario by introducing a geometrical factor which describes the part of the sphere in which the accretion occurs. Therefore, the accretion through the disk will correspond to one of our discussed cases with effectively larger accretion rate occurring only inside a limited region determined by the thickness of the accretion disk. [FIGURE:S2.F1][ENDFIGURE] Let us at first consider the case of the NS in the phase of pure accretor. The accretion rate of the matter onto the neutron star (${\dot{M}}_{\rm acc}=10^{16}M_{\rm 16}$ g s${}^{-1}$) can be estimated from the observed (in some cases) thermal X-ray emission ($L_{\rm X}=10^{36}L_{36}$ erg s${}^{-1}$). These two values can by related for the known radius and the mass of NS (we assume $R_{\rm NS}\approx 10^{6}$ cm and $M_{\rm NS}=1.4M_{\odot}$), \[{\dot{M}}_{\rm acc}\approx 5\times 10^{15}L_{\rm 36}~{}~{}~{}~{}{ \rm g~{}s^{-1}}.\] (1) This matter arrives onto the neutron star surface along the magnetic field lines. The distance at which the magnetic field starts to dominate the dynamics of the matter (the Alfven radius) can be estimated by comparing the magnetic field energy density with the kinetic energy density of the matter, \[B_{\rm A}^{2}/8\pi=\rho v_{\rm f}^{2}/2,\] (2) where $B_{\rm A}$ is the magnetic field in the inner neutron star magnetosphere, $\rho={\dot{M}}_{\rm acc}/(\pi R_{\rm A}^{2}v_{\rm f})$ is the density of accreting matter, $v_{\rm f}=(2GM_{\rm NS}/R_{\rm A})^{1/2}$ is the free fall velocity of accreting matter, $R_{\rm A}$ is the Alfven radius, and G is the gravitational constant. The medium in the transition region is very turbulent and strongly magnetized providing good conditions for the acceleration of particles to high energies. Let us estimate the location of this region from the surface of the neutron star by applying Eq. 2 and assuming that the magnetic field in the neutron star magnetosphere is of the dipole type, i.e. $B_{\rm A}=B_{\rm NS}(R_{\rm NS}/R_{\rm A})^{3}$, \[R_{\rm A}=4\times 10^{8}B_{\rm 12}^{4/7}M_{\rm 16}^{-2/7}~{}~{}~ {}~{}{\rm cm},\] (3) where the magnetic field at the neutron star surface is $B_{\rm NS}=10^{12}B_{12}$ (see also Baan & Treves 1973). Based on the known vale of $R_{\rm A}$, we can estimate the magnetic field strength at the transition region, \[B_{\rm A}=1.6\times 10^{4}M_{\rm 16}^{6/7}B_{\rm 12}^{-5/7}~{}~{ }~{}~{}{\rm G}.\] (4) Considered here accretor scenario can occur provided that the neutron star fulfills some conditions. At first, the radius of the transition region has to lay inside the light cylinder radius of the neutron star, i.e $R_{\rm A}<R_{\rm LC}=cP/2\pi$, where $P=1P_{1}$ s is the rotational period of the neutron star, and $c$ is the velocity of light. The above condition is fulfilled when, \[P_{1}>0.084B_{\rm 12}^{4/7}M_{\rm 16}^{-2/7}.\] (5) Therefore, only relatively slowly rotating neutron stars can be considered. At second, the rotational velocity of the magnetosphere at $R_{\rm A}$ has to be lower than the keplerian velocity of the accreting matter. The rotational velocity, \[v_{\rm rot}=2\pi R_{\rm A}/P\approx 2.5\times 10^{9}B_{12}^{4/7} M_{16}^{-2/7}/P_{1}~{}~{}~{}{\rm cm~{}s^{-1}}.\] (6) is lower than the keplerian velocity, \[v_{\rm k}=(GM_{\rm NS}/R_{\rm A})^{1/2}\approx 6.8\times 10^{8}B _{12}^{-2/7}M_{16}^{1/7}~{}~{}~{}{\rm cm~{}s^{-1}}.\] (7) for the NS with the periods, \[P_{1}>3.7B_{12}^{6/7}M_{16}^{-3/7}.\] (8) This last condition separates the population of NS in the propeller phase (lower periods) from these ones in the accretor phase (larger periods). Note, that discussed below scenario for acceleration of electrons can be applied also to the propeller stage of accretion onto the NS. However, in the propeller phase the radiation field created by the matter accreting onto the NS is not uniquely defined. A part of the matter accreting onto NS can not be at present stage of knowledge linked to the basic parameters describing the model. We have to introduce additional factor which describes the amount of the matter which falls onto the NS surface to the total amount of the matter penetrating the inner NS magnetosphere (i.e accreting and expelled from the vicinity of NS). The third condition relates the radius of the transition region, $R_{\rm A}$, to the capturing radius of the matter from the stellar wind. It is determined by the balance between the kinetic energy of the wind with its gravitational energy around the NS. The capturing radius is described by, \[R_{\rm c}=2GM_{\rm NS}/v_{\rm w}^{2}\approx 3.7\times 10^{10}v_{ 8}^{-2}~{}~{}{\rm cm},\] (9) where $v_{\rm w}=10^{8}v_{8}$ cm s${}^{-1}$ is the velocity of surrounding matter measured in respect to the NS. The accretion from the stellar wind occurs when $R_{\rm c}>R_{\rm A}$, which happens for, \[B_{12}<2.8\times 10^{3}v_{8}^{-7/2}M_{16}^{1/2}.\] (10) For likely parameters of the neutron stars (classical and millisecond pulsars), this limit is not restrictive. Note that $v_{\rm w}$ corresponds to the velocity of the stellar wind and/or the velocity of the neutron star on its orbit around the massive star. This last velocity can be estimated from, $v_{\rm NS}=2\pi D/T=5\times 10^{7}D_{2}/T_{10}$ cm s${}^{-1}$, where $D=100D_{2}R_{\odot}$ cm is the radius of the orbit in units of 100 solar radii $R_{\odot}$, and $T=10T_{10}$ days is the period of the binary system in units of 10 days. It is typically lower than the free fall velocity at the $R_{\rm A}$. Therefore, we neglected it when deriving Eq. 3. Neutron stars with the periods within the range defined by Eq. 5 and Eq. 8 are in the propeller phase, those ones with the periods above that given by Eq. 8 are in the accretor phase, and those ones with the periods shorter than given by Eq. 5 are in the ejector phase. As already noted, only in the accretor phase the accretion rate can be directly linked to the thermal X-ray emission from the NS surface. In the propeller phase, the amount of the matter which accrete onto the NS surface can not be uniquely determined from the basic parameters of the model. ### Acceleration of electrons In the conditions expected for the transition region (strongly magnetized and very turbulent medium), particles should be efficiently accelerated. In this paper we consider only acceleration of electrons. The energy gain rate of electrons with energy $E$ (and the Lorentz factor $\gamma$) is often parametrized by the Larmor radius of electrons and so called acceleration parameter, \[{\dot{P}}_{\rm acc}=\xi cE/r_{\rm L}\approx 2.6\times 10^{4}\xi_{ -1}M_{16}^{6/7}B_{12}^{-5/7}~{}~{}~{}{\rm erg~{}s^{-1}},\] (11) where $\xi=10^{-1}\xi_{-1}$ is the acceleration parameter, $c$ is the velocity of light, $r_{\rm L}=E/eB_{\rm A}$ is the Larmor radius, and $e$ is the electron charge. The acceleration parameter contains all the unknown details of the acceleration process. Since the plasma moves in the transition region with the velocity which takes a significant part of the velocity of light, it seems proper to consider the values of $\xi$ in the range $\sim 0.1-0.01$. During the acceleration process, electrons also suffer energy losses on the synchrotron process and the inverse Compton scattering of radiation from the massive star and the surface of the neutron star. These energy losses determine the maximum energies of accelerated electrons since their Larmor radius is typically much smaller than the characteristic dimensions of the considered scenario, i.e. $R_{\rm L}<R_{\rm A}$. This last condition allows in principle acceleration of electrons up to $E_{\rm e}\approx 2M_{16}^{4/7}B_{12}^{-8/7}$ PeV. However, as we show below, energy losses of electrons limit their maximum energies significantly below this value. Electrons lose energy on the IC process in the Thomson (T) and the Klein-Nishina (KN) regimes. In general, this process can occur on the stellar radiation and on the thermal X-ray radiation from the surface of the polar cap on the NS. Let us estimate the photon energy densities from the star ($\rho_{\star}$) and the polar cap ($\rho_{\rm cap}$) at the acceleration region, \[\rho_{\star}={{4\sigma T_{\star}^{4}}\over{c}}\left({{R_{\star}} \over{D}}\right)^{2}\approx 6.1\times 10^{3}T_{4}^{4}\left({{R_{\star}}\over{D }}\right)^{2}~{}~{}{\rm erg~{}cm^{-3}},\] (12) (where $T_{\star}=3\times 10^{4}T_{4}$ K and $\sigma$ is the Stefan-Boltzmann constant) and, \[\rho_{\rm cap}={{4\sigma T_{\rm cap}^{4}}\over{c}}{{R_{\rm cap}^{ 2}}\over{R_{\rm A}^{2}}}\approx 1.4\times 10^{8}M_{16}^{11/7}B_{12}^{-8/7}~{}~ {}{\rm{erg~{}cm^{-3}}},\] (13) where $T_{\rm cap}=10^{7}T_{7}$ K is the temperature of the polar cap on the NS surface, $R_{\rm cap}$ is the polar cap radius on the NS surface on which the matter accrete. In Eq. 13, we have assumed that the observed X-ray thermal emission is re-radiated from the region of the polar cap according to $L_{\rm X}=\pi R_{\rm cap}^{2}\sigma T_{\rm cap}^{4}$. In fact, the emission from the polar cap region is well described by the Black Body spectrum (see e.g. Zane et al. 2000). The radius of the polar cap region on the NS surface, on which the matter falls and from which the thermal X-ray emission is emitted, can be estimated from (assuming dipole structure of the magnetic field), \[R_{\rm cap}=(R_{\rm NS}^{3}/R_{\rm A})^{1/2}\approx 5\times 10^{ 4}B_{12}^{-2/7}M_{16}^{1/7}~{}~{}~{}{\rm cm}.\] (14) Then, the surface temperature of the polar cap is, \[T_{\rm cap}=(L_{\rm X}/\pi R_{\rm cap}^{2}\sigma)^{1/4}\approx 4 .7\times 10^{7}B_{12}^{1/7}M_{16}^{5/28}~{}~{}{\rm K}.\] (15) Note that, the energy density of radiation from the polar cap at the distance of the turbulent region ($R_{\rm A}$) can be explicitly expressed by the basic parameters of the model. [FIGURE:S2.F2][ENDFIGURE] Let us also estimate the energy density of the magnetic field at the transition region ($R_{\rm A}$ given by Eq. 3), \[\rho_{\rm B}=B^{2}_{\rm A}/8\pi\approx 10^{7}M_{16}^{12/7}B_{12}^ {-10/7}~{}~{}{\rm{erg~{}cm^{-3}}},\] (16) The energy losses of electrons for every considered process (synchrotron and IC in the T regime) can be calculated from, \[{\dot{P}}_{\rm loss}=(4/3)c\sigma_{\rm T}\rho\gamma^{2}\approx 2. 7\times 10^{-14}\rho_{\rm(cap,B,\star)}\gamma^{2}~{}~{}{\rm erg~{}s^{-1}},\] (17) where $\sigma_{\rm T}$ is the Thomson cross section. The energy losses of electrons on the radiation from the NS surface can dominate over energy losses on other targets only at low energies, i.e. when the scattering occurs in the T regime which happens for the Lorentz factors of electrons, $\gamma_{\rm T/KN}^{\rm X}<m_{\rm e}c^{2}/(3kT_{\star})\approx 200T_{7}^{-1}$. In the KN regime, the energy losses can be approximately estimated by introducing into Eq. 17 the value of the Lorentz factor corresponding to the transition between T and KN regimes, $\gamma_{\rm T/KN}^{\rm X}$. For the range of parameters defining the model, the IC energy losses of electrons in the KN regime becomes lower than the synchrotron energy losses for the Lorentz factors about ten times larger than $\gamma_{\rm T/KN}^{\rm X}$ (typically at $\sim$GeV energies). Therefore the maximum energies of accelerated electrons are determined by the balance between energy gains from the acceleration process (Eq. 11) and energy losses on synchrotron process (Eq. 17), \[\gamma_{\rm max}\approx 3\times 10^{5}\xi_{-1}^{1/2}B_{12}^{5/14} M_{16}^{-3/7}.\] (18) It is clear that electrons can reach even TeV energies (see Fig. 2a) for realistic parameters of the model. However, TeV $\gamma$-ray production will be strongly suppressed due to dominant synchrotron energy losses of electrons with TeV energies. We can also estimate the characteristic energies of synchrotron photons which can be expected in such a model by applying the approximate formula, \[\epsilon_{\rm x}\approx m_{\rm e}c^{2}(B_{\rm A}/B_{\rm cr}) \gamma_{\rm max}^{2}\approx 1.9\xi_{-1}~{}~{}{\rm MeV},\] (19) where $m_{\rm e}$ is the electron rest mass, $B_{\rm cr}=4.4\times 10^{13}$ G is the critical magnetic field, and the Lorentz factor of electrons $\gamma_{\rm max}$ is given by Eq. 18. We conclude that synchrotron emission can extend through the soft $\gamma$-ray energy range. it should be detectable by satellites sensitive at these energies, e.g. the INTEGRAL observatory. Note that the bremsstrahlung energy losses of relativistic electrons in the matter inside the transition region can be neglected since their energy loss rate is relatively low, \[{\dot{P}}_{\rm br}=m_{\rm e}c^{3}\rho_{\rm H}\gamma/X_{\rm o} \approx 6.5\times 10^{-5}\rho_{17}\gamma~{}~{}{\rm erg~{}s^{-1}},\] (20) where $X_{\rm o}=62$ g cm${}^{-2}$ is the radiation length for hydrogen, and $\rho_{\rm H}=10^{17}\rho_{17}$ cm${}^{-3}$ is the density of matter. We can neglect the bremsstrahlung energy losses in respect to other energy loss rates since typical density of matter in the transition region is $\rho\approx 1.5\times 10^{13}B_{12}^{-6/7}M_{16}^{10/7}$ cm${}^{-3}$ (estimated for the known accretion rate and the location of $R_{\rm A}$). ### Energetics The maximum power available for acceleration of electrons is limited by the energy extracted from the rotating neutron star by the in-falling matter. This matter from the stellar wind has to be accelerated to the velocity of the magnetic field lines at $R_{\rm A}$ in order to farther accrete onto the neutron star surface. The power which has to be transfered from the rotating NS to the matter can be estimated from \[L_{\rm acc}={\dot{M}}_{\rm acc}v_{\rm rot}^{2}/2\approx 3\times 1 0^{34}B_{12}^{8/7}M_{16}^{3/7}P_{1}^{-2}~{}~{}{\rm erg~{}s^{-1}}.\] (21) By using the limiting period given by Eq. 8, we get the upper limit on the available power in the accretor stage, \[L_{\rm acc}<2.2\times 10^{33}B_{12}^{-4/7}M_{16}^{9/7}~{}~{}{\rm erg ~{}s^{-1}}.\] (22) This power is shown in Fig. 2b for two example surface magnetic field strengths of the neutron star (classical NS: $B=3\times 10^{12}$ G and millisecond NS: $B=10^{9}$ G) as a function of the accretion rate. We assume that only a part, $\eta$, of this power (applying typically $\eta=0.1$) can be converted to relativistic electrons in the turbulent and magnetized plasma at $R_{\rm A}$. Note, that for reasonable accretion rates this power is greater than the minimum required power of the GeV $\gamma$-ray source which can be detected by the ${\it Fermi}$ LAT telescope and possibly also by the planned next generation Cherenkov telescopes system CTA whose sensitivity should be about one order of magnitude better than presently available. The model discussed above for the accretor stage of the binary system can also operate for the propeller stage in which only a part of the matter can eventually accrete onto the surface of the NS. However, in this case it is difficult to estimate the power which can be transfered to relativistic particles due to the unknown observational signatures of the amount of matter which accumulates close to the shock region. In summary, electrons can be accelerated even to TeV energies in the case of neutron stars inside X-ray binaries with relatively weak thermal X-ray emission from its surface in contrary to the powerful X-ray binaries in which only a few ten GeV electrons are expected. However, the situation is the opposite concerning the maximum power transfered to these relativistic electrons. Sources in which electrons are accelerated only to GeV energies should be powerful enough to be observable by the $\gamma$-ray telescopes. ## 3Production of radiation We assume that electrons are accelerated in the turbulent transition region with the power law spectrum to maximum energies estimated from Eq. 18. Eventually, the spectrum of electrons can be strongly peaked at the highest possible energies due to the synchrotron energy losses during acceleration process (the so called pile-up mechanism, see e.g. Protheroe 2004). As we have shown above, electrons lose energy on different radiation mechanisms but the dominant ones are: the synchrotron process (dominates at the highest energies); and the ICS of thermal radiation from the NS polar cap (contributes to the high energies in the KN regime and dominates at lower energies in the T regime). We neglect the production of $\gamma$-rays by electrons in the acceleration region in the scattering of stellar radiation since its energy density can be safely neglected in respect to the energy density of the magnetic field (at high energies) and the energy density of polar cap radiation (at low energies) (see Sect. 2.1). In order to check whether electrons can lose efficiently energy already in the transition region, we estimate their convection time scale with the matter falling onto the NS surface along the open magnetic field lines on, \[\tau_{\rm conv}=R_{\rm A}/v_{\rm rot}=P_{1}/2\pi~{}~{}~{}{\rm s}.\] (23) Let us compare this time scale with the radiation time scale, which (for electrons with low enough energies) is determined by the IC losses (in T regime) in the radiation field from the polar cap of the NS, \[\tau_{\rm rad}=m_{\rm e}\gamma/{\dot{P}}_{\rm rad}\approx 0.2M_{1 6}^{-11/7}B_{12}^{8/7}\gamma^{-1}~{}~{}~{}{\rm s}\] (24) From this comparison, we estimate lower limit on the Lorentz factor of electrons which lose efficiently energy before being convected onto the NS surface on, \[\gamma=1.3M_{16}^{-11/7}B_{12}^{8/7}P_{1}^{-1}.\] (25) It is concluded that relativistic electrons lose their energy close to the acceleration place. In order to calculate the $\gamma$-ray spectra produced by electrons inside the transition region, we simulate the energy loss process of electrons in the radiation field of the polar cap and at the magnetic field of the transition region by applying the Monte Carlo method. It is assumed that the magnetic field is very turbulent in the transition region (the distribution of electrons is isotropic) and that the radiation field comes from the region of the polar cap. For some range of model parameters, IC $\gamma$-rays can be farther absorbed in the radiation field of the polar cap. In this way the IC $e^{\pm}$ pair cascade is initiating. In order to determine the conditions for the cascade process, in the next Section we calculate the optical depths for $\gamma$-ray photons in the radiation of the polar cap. We calculate also the optical depths for $\gamma$-rays in the radiation field of the massive companion in order to check whether this $\gamma$-ray absorption process should be also taken into account when evaluating the $\gamma$-ray spectra escaping from the binary system. [FIGURE:S3.F3][ENDFIGURE] ## 4Optical depths for $\gamma$-rays As we have argued above, the cooling of electrons in such strong magnetic and radiation fields occurs very efficiently. High energy $\gamma$-ray produced in the ICS of thermal photons from the polar cap can be also absorbed in this same radiation field and, in principle, also in the radiation field of the close massive companion star. Below, we calculate the optical depths for the $\gamma$-ray photons in these two radiation fields. We show that for specific conditions (determined mainly by the accretion rate onto the NS and the parameters of the NS), the $\gamma$-ray spectrum emerging from the binary system is in fact formed in the cascade process occurring in the radiation field of the polar cap. ### Polar cap radiation We calculate the optical depths for $\gamma$-rays assuming that they are produced by electrons in the transition region at the distance $R_{\rm A}$ from the stellar surface. In these calculations we assume the diluted black body spectrum for the polar cap emission with the dilution factor at the production site of $\gamma$-rays estimated from $(R_{\rm cap}/R_{\rm A})^{2}$. The average optical depths (averaged over the isotropic injection of $\gamma$-rays) for different accretion rates onto the neutron star (i.e equivalent to different X-ray luminosities, see Eq. 1) are shown in Fig. 3. It is clear that for low accretion rates ($<10^{16}$ g s${}^{-1}$), when electrons can be accelerated to $\sim$TeV energies, the optical depths for $\gamma$-ray photons are relatively low, i.e. they are below unity for energies above $\sim 10$ GeV. In this case GeV-TeV $\gamma$-rays can escape from the radiation field of the polar cap without significant absorption. On the other hand, for large accretion rates ($>10^{16}$ g s${}^{-1}$), electrons are accelerated at most to a few tens of GeV. $\gamma$-rays produced by these electrons should be efficiently absorbed in the radiation of the polar cap. Therefore, in this case the escaping $\gamma$-ray spectra can be only obtained by calculating complicated IC $e^{\pm}$ pair cascade in the radiation of the polar cap. We conclude that for large accretion rates $\gamma$-rays, produced in this model, could be observable by the satellite telescopes (${\it Fermi}$ LAT and possibly AGILE). ### Massive star radiation [FIGURE:S4.F4][ENDFIGURE] The optical depths for $\gamma$-ray photons injected at an arbitrary distance from the surface of the massive star have been calculated for the first time in the general case (including also dimensions of the star) by Bednarek (1997, 2000). In fact, the optical depths for $\gamma$-rays close to the surface of other stars can be easily re-scaled from those early calculations. For example, $\gamma$-ray photons with energies, $E_{\gamma}^{\rm o}$, propagating at specific distances $D$ and at directions (defined by the angle $\alpha$) close to the star with specific parameters ($T_{\rm o}$ and $R_{\rm o}$) are related to the optical depths around arbitrary star with $T_{\star}$ and $R_{\star}$ in the following way, \[\tau(E_{\gamma}^{\star}={{E_{\gamma}^{\rm o}}\over{S_{\rm T}}},T_ {\star},R_{\star},D,\alpha)=S_{\rm T}^{3}S_{\rm R}\tau(E_{\gamma}^{\rm o},T_{ \rm o},R_{\rm o},D,\alpha)\] (26) where $S_{\rm T}=T_{\star}/T_{\rm o}$, $S_{\rm R}=R_{\star}/R_{\rm o}$, distance $D$ from the stellar surface of specific star is measured in the stellar radii. The example calculations of the optical depths for the case of one specific massive star (present inside the binary system IGR J19140+098) are shown in Fig 4. These optical depths become larger than unity for the $\gamma$-rays with energies above $\sim$20 GeV. As we show latter, $\gamma$-rays produced in the considered model have typically energy below $\sim$10 GeV. Their absorption in the radiation field of the massive star can be safely neglected. Therefore, the absorption effects of $\gamma$-rays should not introduce any modulation of the $\gamma$-ray signal with the period of the binary in the case of accreting neutron stars. This is in contrast to the TeV $\gamma$-ray binaries supposed to contain ejecting neutron stars (strong pulsar wind prevents the accretion of stellar wind), see e.g. the model considered by Sierpowska & Bednarek (2005). The eventual modulation of the $\gamma$-ray signal with the period of the binary can be only produced by the change of the accretion rate with the distance of the neutron star on its elliptic orbit around the massive star. ## 5Gamma-rays from the vicinity of accreting neutron star In the considered model, radiation is produced by electrons accelerated to maximum energies estimated by Eq. 18 in the synchrotron and IC process. The synchrotron energy losses dominate at the largest electron energies (close to $E_{\rm max}$) and the IC process can only dominate at lower energies due to the KN cross section. However, as we have shown above, for large accretion rates, $\gamma$-rays produced by electrons can be farther absorbed in the thermal radiation from the NS surface. We have shown in Fig. 3 that the average optical depths for $\gamma$-ray photons in the thermal radiation from the NS polar cap region in order to envisage for what parameters the cascading effects can become important. Therefore, in these cases we have to consider the production of the $\gamma$-rays in the IC $e^{\pm}$ pair cascade with additional synchrotron energy losses of primary electrons and secondary $e^{\pm}$ pairs. On the other hand, for small accretion rates, the cascade does not develop. $\gamma$-rays are produced in this case only as a result of cooling of primary electrons. In order to follow the process of $\gamma$-ray production, we developed the numerical code which simulate the cooling process of electrons taking into account not only the $\gamma$-ray production in the IC process but also the synchrotron energy losses of primary electrons and secondary cascade $e^{\pm}$ pairs. Note that the acceleration of electrons occurs inside the inner pulsar magnetosphere where the magnetic field is relatively strong and depends on the distance of the turbulent transition region from the NS surface. Since the synchrotron process dominates over the IC process at the high energy part of the injected electron spectrum, the $\gamma$-rays are rarely produced with energies comparable to energies of primary electrons. Our IC $e^{\pm}$ pair cascade code with synchrotron energy losses gives both, the synchrotron and the IC spectra expected in such a model. Two models for injection of relativistic electrons are considered: 1. Electrons injected with the power law spectrum up to the maximum Lorentz factor $\gamma_{\rm max}$ (see Eq. 18), as expected in the stochastic acceleration mechanism. The power law spectrum of primary electrons is normalized to a part, $\eta$, of the kinetic power which is transfered from the rotating neutron star to the matter in the transition region. 2. Electrons injected with the Lorentz factors $\gamma_{\rm max}$. Such injection spectrum can give good approximation in the case of acceleration process occurring with large radiative energy losses (the pile up mechanism at the end of accelerated power law spectrum of electrons, see e.g. Protheroe 2004). These two injection models are likely to be the limiting cases of the real acceleration process of electrons in the turbulent region of the matter accreting onto the magnetized neutron star. We investigate two general scenarios describing the cases of _pure_ accretor phase (all the matter arriving to the transition region falls onto the NS surface) and the intermediate accretor-propeller phase (only part of the matter arriving to the transition region accrete onto the NS surface and the rest of it is expelled from the vicinity of the NS in the propeller mechanism). ### The accretor stage [FIGURE:S5.F5][ENDFIGURE] The scenario for the accretion process onto neutron star in the accretor stage is much better defined since the thermal radiation field from the NS surface (depending on the dimension of the hot spot and its temperature) is uniquely determined by the parameters describing the model. Therefore, the synchrotron and IC spectra produced by electrons depends on a relatively small number of free parameters, i.e. the accretion rate onto NS, the surface magnetic field of NS, and the acceleration rate $\xi$. The observed X-ray to $\gamma$-ray power depends additionally on the parameter, $\eta D^{-2}$, which combines the energy conversion efficiency from the transition region to relativistic electrons, $\eta$, and the distance to the source $D=1D_{1}$ kpc. At first, we performed calculations for the power law spectrum of injected electrons $N(e)=AE^{-s}$ between $E_{\rm max}$ and $E_{\rm min}$, where $A$ is the normalization coefficient equal to $A=\eta L_{\rm acc}(2-s)/(E_{\rm max}^{(2-s)}-E_{\rm min}^{(2-s)})$. Note, that $A$ only weakly depends on $E_{\rm min}$ for the spectral indices not far from 2. In our example calculations, we apply the spectral index equal to 2.2 and choose $E_{\rm min}=30$ MeV since we cool electrons only to such minimum energies. Electrons with such energies are not able to produce $\gamma$-rays detectable by the Fermi LAT. The calculated spectra are also shown for different values of ${\dot{M}}$, $\xi$ and $B$ (see Fig. 5). In Figs. 5a-d, we consider the case of classical relatively young NS ($B\gg 10^{9}$ G), rotating with the limiting period defined by Eq. 8. When confronting with the sensitivity of the _Fermi_ LAT telescope, it becomes clear that $\gamma$-ray emission from such objects can be detected at energies below $\sim 1$ GeV in the case of large accretion rates (significantly above $10^{16}$ g s${}^{-1}$). Moreover, the power law synchrotron spectra, corresponding to these accretion rates, should be also easily detected by the soft and hard X-ray detectors presently on the orbit. Note that due to strong magnetic field and large energies of primary electrons, the synchrotron spectra can extend in some cases also through the soft $\gamma$-ray energy range. In Fig. 5e-f, we show the synchrotron and IC spectra for the case of the NS with low surface magnetic field, i.e. characteristic for so called millisecond pulsars. Also in the case of these slowly magnetized neutron stars, $\gamma$-ray emission can be detected by the _Fermi_ LAT telescope, provided that the accretion rate onto the NS is above a few $10^{14}$ g s${}^{-1}$. We conclude that $\gamma$-ray telescopes presently on the orbit should be able to detect $\gamma$-ray emission at a few hundred MeV from the class of accreting NS. These NS are characterized by the thermal emission from its surface at a level above $\sim 2\times 10^{36}$ erg s${}^{-1}$ in the case of strongly magnetized neutron stars, and above a few $10^{34}$ erg s${}^{-1}$ in the case of NS with the magnetic field characteristic for millisecond pulsars provided that the source is at the distance of 1 kpc. These limiting X-ray luminosities have been derived by using the relation between the accretion rate and the thermal emission from the NS surface (see Eq. 1). We also investigate how the detectability of such accreting NSs depends on the spectral index of the injected electrons. In Fig. 6 the synchrotron and IC $\gamma$-ray spectra are shown for the range of spectral indexes $s=1-4$ and for fixed other parameters describing the model: ${\dot{M}}=10^{17}$ g s${}^{-1}$, $\xi=0.1$, and $B=10^{12}$ G. Electron spectrum has to have spectral index lower than $\sim 3$ in order to produce IC $\gamma$-rays above the sensitivity of ${\it Fermi}$ LAT telescope. Such spectral indexes are expected in the stochastic models for particle acceleration in turbulent media. [FIGURE:S5.F6][ENDFIGURE] As already noted above, the acceleration of particles under strong radiative energy losses may result in their accumulation at the highest energies determined by the balance between energy gains from the acceleration mechanism and energy losses (see e.g. Protheroe 2004). In such cases, most of the energy is accumulated in particles with the highest possible energies. At first approximation, we can consider that the spectrum of particles is mono-energetic. Therefore, in Fig. 7 we show also the photon spectra expected in the model for the case of mono-energetic injection of electrons. In such a case, IC $\gamma$-ray spectra are flat. Their detectability is more difficult than in the case of injection of electrons with the power law spectra. This is due to the fact that electrons with maximum possible energies lose most of their energy on synchrotron process. Therefore the IC spectra are reduced and synchrotron spectra are enhanced (compare corresponding cases shown in Figs. 5 and 7). We conclude that accreting NS, which have the highest chance to be detected by the _Fermi_ LAT telescope, should have the intermediate values of $\xi$ and $B$ and large accretion rates (see Fig. 7c-d). Such parameters allow acceleration of electrons to energies of the order of a few tens GeV, i.e close to the energy region where ICS process starts to become comparably efficient to the synchrotron process. The $\gamma$-ray photons produced by electrons in the transition region of the accretion flow onto the NS originate close to the massive companion star which creates strong radiation field. In Sect. 4.2, we show the optical depths for $\gamma$-rays in the radiation field of the massive star with the parameters typical for massive binary systems (the example case of the massive binary IGR J19140+0951). As we have shown above, primary electrons accelerated to the maximum energies cool at first mainly on the synchrotron process. Therefore $\gamma$-rays produced by them in the IC process have energies typically below $\sim 10$ GeV. On the other hand, the optical depths in the massive star radiation (see Fig. 4) are quite large but at energies which are clearly above energies of $\gamma$-rays produced in our model. Therefore, we conclude that in the case of considered here model the absorption of $\gamma$-rays in the stellar radiation can be neglected. [FIGURE:S5.F7][ENDFIGURE] In Fig. 7a,c,e, we also show the spectra of thermal radiation from the NS polar cap region for the example accretion rate $M_{\rm acc}=10^{16}$ g s${}^{-1}$ and different values of the magnetic field on the NS surface ($B=10^{12}$ G (a), $10^{11}$ G (c), and $10^{9}$ G (e). These spectra clearly dominate over the nonthermal synchrotron spectra produced by relativistic electrons below a few keV. In fact, as we already noted above, such soft X-ray thermal excesses have been reported from some INTEGRAL hard X-ray massive binaries. In reality this soft thermal X-ray emission may be significantly modified due to the interaction with the matter accreting onto the NS surface, the matter of the stellar wind, and the matter onto the surface of the close massive companion. In our calculations we are not able to take these possible modifications into account. ### The intermediate accretor-propeller stage As noted above, in the _pure_ propeller stage the matter arriving to the transition region at the Alfven radius, $R_{\rm A}$, is expelled by the centrifugal source from the vicinity of the neutron star preferentially along the rotational axis. Here we consider the intermediate case in which a small part of the accreting matter is able to penetrate onto the NS surface but most of it is expelled outside the neutron star. It is not clear whether such process can last stationary in time or it occurs only non-stationary when the accretion rate onto the neutron star changes in time e.g. due to the clumpy wind or the elliptic orbit of the NS around the massive star. In order to take into account the effects of only partial accretion of matter onto the NS surface, we introduce the parameter, $\kappa$, which is the ratio of the matter accreting onto the NS surface to the whole amount of matter arriving to the transition region (i.e. the matter accreted onto the surface and expelled from the vicinity of NS). In such an intermediate case, the temperature of the polar cap region on the NS surface (given by Eq. 23) should be a factor $\kappa^{1/4}$ lower than estimated in the case of complete accretion. However, in principle the power available for acceleration of electrons can be larger than expected from period limiting the accretor and propeller stage (given by Eq. 8). Instead, it is rather limited by Eq. 5. In conclusion, as a result of only partial accretion of matter from the transition region onto the NS surface, the surface temperature of the polar cap is lower and the power transfered to electrons can be larger than expected for the _pure_ accretor stage. In Fig. 8 we show the example $\gamma$-ray spectra obtained in the case of the partial accretion of matter onto the NS surface assuming different accretion rates defined by the factor $\kappa$. We also scale the power transfered to relativistic electrons by the factor $\kappa^{-1}$ since more energy can be extracted from the transition region in this case. Based on these calculations, we expect that the $\gamma$-ray spectra produced in the intermediate stage are steeper, although they have larger luminosities. Therefore, they should be easier detected by the $\gamma$-ray telescopes (see Fig. 8 for the comparison with the sensitivity of ${\it Fermi}$ LAT telescope for the source at the distance of 1 kpc). [FIGURE:S5.F8][ENDFIGURE] ## 6Discussion and Conclusions We propose that accreting neutron stars inside compact massive binary systems produce $\gamma$-ray fluxes which can be observable by the satellite telescope ${\it Fermi}$ LAT. To show this we consider acceleration of electrons at the turbulent, strongly magnetized, transition region in the inner neutron star magnetosphere which appears as a result of the interaction of accreting matter with rotating NS magnetosphere. Relativistic electrons produce X-rays and $\gamma$-rays as a result of synchrotron and IC processes occurring on the thermal radiation from the NS surface. The cooling process of electrons is followed by applying the Monte Carlo method since some of produced $\gamma$-rays can be farther absorbed in the thermal radiation. We showed that synchrotron emission from primary electrons and secondary cascade $e^{\pm}$ pairs can extend up to the MeV energy range and the IC emission can appear up to a few GeV. A physical realization of such a scenario can be characteristic for the newly discovered by the INTEGRAL observatory a class of obscured massive binary systems which show: (1) a hard, power law X-ray emission and, (2) evidences of soft X-ray black body component. As an example, we consider here in a more detail the high mass X-ray binary system, IGR J19140+0951, discovered by the INTEGRAL (Hannikainen et al. 2003). A compact object in this system is likely to be a neutron star (Cabanac et al. 2005). IGR J19140+0951 is at the distance of $\sim 2-3$ kpc (Rahoui et al. 2008). The hard X-ray emission up to 100 keV with the spectral index $2.39\pm 0.11$ is observed from this source (Hannikainen et al. 2004). Its X-ray luminosity in the high state was $\sim 3.7\times 10^{37}\times(D/10kpc)^{2}$ erg s${}^{-1}$ (Rodriquez et al. 2005). Unfortunately, the basic parameters of the neutron star in IGR J19140+0951 are unknown. Therefore, we investigate the range of model parameters which are consistent with the hard power law spectrum in the X-ray energy range by the INTEGRAL. Comparison of the observations with the example calculations are shown in Fig. 9. It is clear that for some parameters the $\gamma$-ray flux predicted at a few hundred MeV can be detectable by the extensive _Fermi_ LAT observations. We performed calculations for the neutron stars with the parameters characteristic for the ”classical” radio pulsars (surface magnetic field strength $B\sim 10^{12}$ G and periods of the order of seconds) and for the millisecond pulsars ($B\sim 10^{9}$ G, periods of a few to several milliseconds). It is clear that in order to produce $\gamma$-ray fluxes observable by the ${\it Fermi}$ LAT telescope, the neutron stars in binary systems at the distance of a few kpc should collect the matter from the wind at a relatively large accretion rates ($>10^{16}$ g s${}^{-1}$). However, in the case of millisecond pulsars these accretion rates can be significantly lower (above a few $10^{14}$ g s${}^{-1}$). Therefore, in principle, the millisecond pulsars seems to be more favorite $\gamma$-ray sources. On the other hand, millisecond pulsars have only low mass companions, so the large accretion rates are not expected in such binary systems. Note, that synchrotron X-ray and IC $\gamma$-ray emission is produced in such a model almost isotropically. As we have shown above, we do not expect any modulation of the $\gamma$-ray signal with the period of the binary system due to the selective absorption in the companion star soft radiation, as it is expected in the case of TeV $\gamma$-ray binary systems (e.g. LS 5039 and LSI 61 303). However, the modulation might be related to the change of the accretion rate in the case of elliptic orbit of the neutron star. It is expected that the accretion rate should increase when the neutron star is closer to the companion star. Therefore, we predict that the largest fluxes of X-rays and $\gamma$-rays should be observed closer to the periastron passage of the neutron star. Moreover, the X-ray and $\gamma$-ray emission should be correlated (see spectra in Figs. 5-8). In the model considered here we do not take into account the scattering of nonthermal synchrotron radiation by the accelerated electrons. As we have shown in Fig. 7, the thermal radiation clearly dominates at energies below a few keV over the nonthermal synchrotron radiation produced by accelerated electrons and secondary cascade $e^{\pm}$ pairs. Since their energy densities at the acceleration site scales in a similar way with the distance from the NS, the energy losses on thermal radiation has to dominate over the energy losses on nonthermal radiation. Therefore, we can safely neglect the cooling of electrons on this nonthermal radiation. Note moreover, that the scattering of nonthermal X-rays with energies above a few keV becomes inefficient since it occurs in the Klein-Nishina regime. Similar processes to these considered in this paper are also expected in the case of non-spherical accretion of matter onto the rotating neutron star. For example, in the case of disk accretion, the corresponding accretion rate should be scaled by a part of sphere which is intercepted by the accretion disk. Then, the turbulent region on the inner disk age, in which acceleration of electrons occurs, is also limited to corresponding part of the sphere. Only the IC process may become more complicated in this case since thermal radiation from the hot spot on the neutron star surface, visible by relativistic electrons, can be partially obscured by the neutron star surface. In conclusion, we predict that present satellite telescopes can discover a new class of $\gamma$-ray sources, i.e. accreting neutron stars inside binary system. The emission from these sources should be characterized by a strong hard, synchrotron spectra and IC $\gamma$-rays limited to energies below $\sim 1$ GeV, due to the very strong synchrotron energy losses of relativistic electrons accelerated in the vicinity of neutron stars. [FIGURE:S6.F9][ENDFIGURE] ###### Acknowledgements. This work is supported by the Polish MNiSzW grant N N203 390834. ## References * (1) Baan, W.A., Treves, A. 1973 A&A 22, 421 * (2) Bednarek, W. 1997 A&A 322, 523 * (3) Bednarek, W. 2000 A&A 363, 646 * (4) Bednarek, W., Giovannelli, F. 2007, A&A 464, 437 * (5) Begelman, M.C., Rees, M. J. 1984 MNRAS 206, 209 * (6) Chaty, S. 2007 Chin.J.Astron.Astrophys., in press * (7) Cabanac, C., Rodriquez, J., Petrucci, P.-O., Henri, G., Hannikainen, D.C., Schultz, J., Lund, N., Durouchoux, P. 2005, Chin. J. Astron. Astrophys. 5, 93 * (8) Corbet, R.H.D., Hannikainen, D.C., Remillard, R. 2005 ATel 269 * (9) Hannikainen, D.C., Rawlings, M.G., Muhli, P., Vilhu, O., Schultz, J., Rodriquez, J. 2007, MNRAS 380, 665 * (10) Hannikainen, D.C., Rodriquez, J., Cabanac, C., Schultz, J., Lund, N., Vilhu, O., Petrucci, P.O., Henri, G. 2004, A&A 423, L17 * (11) Hannikainen, D.C., Rodriquez, J., Pottschmidt, K. 2003, IAUC, 8088 * (12) Lipunov, V.M. 1992, Astrophysics of neutron stars, Springer-Verlag, Heidelberg * (13) Protheroe, R.J. 2004 APh 21, 415 * (14) Rodriquez, J., Bodaghee, A. in The Nature and Evolution of X-ray Binaries in Diverse Environments, eds. Bandyopadhyay, S. et al. eds., arXiv:0801.0961 * (15) Rodriquez, J., Cabanac, C., Hannikainen, D.C., Beckmann, V., Shaw, S.E., Schultz, J. 2005, A&A 432, 235 * (16) Rahoui, F., Chaty, S., Lagage, P.-O., Pantin, E. 2008, A&A, submitted (arXiv:0802.1770) * (17) Sierpowska, A., Bednarek, W. 2005, MNRAS 356, 711 * (18) Tavani, M. 1991, ApJ 379, L69 * (19) Tavani, M. 1993, A&AS 97, 313 * (20) Tavani, M., Brookshaw, L. 1993 ApJ 413, L39 * (21) Treves, A., Colpi, M., Lipunov, V.M. 1993 A&A 269, 319 * (22) Zane, S., Turolla, R., Treves, A. 2000, ApJ 537, 387
1308.3219	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 186501, "num_imgs": 1, "llama3_tokens_count": 63068 }	[ "content_image/1308.3219/x1.png" ]	# A unifying perspective: the relaxed linear micromorphic continuum Patrizio Neff and Ionel-Dumitrel Ghiba¹ and Angela Madeo² and Luca Placidi³ and Giuseppe Rosi⁴ Patrizio Neff, Head of Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neff@uni-due.de [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] ###### Abstract We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict non-polar size-effects. It is intended for the homogenized description of highly heterogeneous, but non polar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models. Key words: micromorphic elasticity, symmetric Cauchy stresses, dynamic problem, dislocation dynamics, gradient plasticity, symmetric micromorphic model, dislocation energy, earthquake processes, generalized continua, non-polar material, microstructure, micro-elasticity, size effects, fracture, non-smooth solutions, gradient elasticity, strain gradient elasticity, couple stresses, Cosserat couple modulus, wave propagation, band gaps. “Das also war des Pudels Kern.”, _Faust I_, J.W. v. Goethe ###### Contents Contents * 1 Introduction * 1.1 Motivation * 1.2 Historical perspective * 1.3 Approach in this work * 1.4 Notation * 2 Formulation of the problem. Preliminaries * 2.1 Eringen’s linear asymmetric micromorphic elastodynamics revisited * 2.2 The relaxed micromorphic continuum model * 2.3 Mathematical analysis * 3 Another further relaxed problem * 3.1 Formulation of the problem * 3.2 Mathematical analysis * 4 New and/or existing relaxed models * 4.1 Kröner’s view * 4.2 Forest’s approach * 4.3 The asymmetric isotropic Eringen-Claus model for dislocation dynamics * 4.4 The linear isotropic Cosserat model in terms of the dislocation density tensor * 4.5 Lazar’s translational gauge theory of dislocations * 4.6 The symmetric earthquake structure model of Teisseyre * 4.7 The asymmetric microstretch model in dislocation format * 4.8 The microvoids model in dislocation format * 4.9 A glimpse on the isotropic strain gradient model * 5 Conclusion * 6 Outlook * A Some useful identities ## 1 Introduction ### Motivation Microstructural motions are observed to produce new effects that cannot be accounted for by classical translatory degrees of freedom (dof) used to formulate conventional theories. For instance, plane waves in an unbounded elastic medium propagate without dispersion, i.e. the wave speed is independent of the frequency. However, experiments with real solids disclose dispersive wave propagation. In order to incorporate the microstructure of the matter into the classical theory, generalized continuum models may be used. Among the various extended continuum theories we mention the higher gradient elasticity theories [108, 110, 131, 82, 3] and micromorphic models [47, 119, 101, 100, 99, 159]. General continuum models involving independent rotations have been introduced by the Cosserat brothers [34] at the beginning of the last century. A material point carrying three deformable directors introduces nine extra degrees of freedom besides the translational degrees of freedom from the classical theory. Many developments have been reported since the seminal work of the Cosserat brothers. The derived generalized theories are called polar, micropolar, micro-elastic, micromorphic, Cosserat, multipolar, oriented, complex, etc., according to the specifically considered kinematical variables and to the choice of the set of constitutive variables. All materials, whether natural or synthetic, possess microstructures if one considers sufficiently small scales. Viewed first as a formal theoretical investigation, the micromorphic models (12 dof) derived by Eringen and Suhubi [49], Mindlin [111, 106, 107, 109] and Toupin [164, 165] are justified recently as more realistic continuum models based on molecular dynamics and ensemble averaging [26, 27, 96, 168, 171]. A large class of engineering materials, porous solids with deformable grains and pores, composites, polymers with deformable molecules, crystals, solids with microcracks, dislocations and disclinations [105, 8], and biological tissues like “bones and muscles” may be modeled more realistically by means of the theory of micromorphic materials. This is the reason why micromorphic mechanics is a dynamic field of research both from a theoretical and practical point of view. Considerations of the format of balance laws in geometrically nonlinear micromorphic elasticity have been undertaken in [24, 100, 25, 36, 170]. The only known existence results for the static geometrically nonlinear formulation are due to Neff [121] and to Mariano and Modica [100]. In fact, Mariano and Modica [94] treat general microstructures described by manifold-valued variables, even if they discuss essentially what is called by Neff in [121] macro-stability (two other cases are treated in [121], one leads to fractures - a situation excluded in [100] - the other is left open). When the energy analyzed by Mariano and Modica is reduced to micromorphic materials in the splitted version considered by Neff [100], their coercivity assumptions result are more stringent than Neff’s ones (the blow up of the determinant of $\textrm{det}\,F$ a part), so they restrict the material response. However, the direct comparison of the two existence results is not completely straightforward. As for the numerical implementation, see [101] and the development in [83]. In [83] the original problem is decoupled into two separate problems. Corresponding domain-decomposition techniques for the subproblem related to balance of forces are investigated in [83]. The size effects involved in a natural way in the micromorphic models (see e.g. [156]) have recently received much attention in conjuction with nano-devices and foam-like structures. A geometrically nonlinear generalized continuum of micromorphic type in the sense of Eringen for the phenomenological description of metallic foams is given by Neff and Forest [127]. Moreover, in [127] the authors proved the existence of minimizers and they identified the relevant effective material parameters. A comparison of the geometrically nonlinear elastic micromorphic theories with affine microstructure with the intrinsically linear models of Mindlin and Eringen is given in [119, 127]. In the present paper, a useful decomposition (_mixed variant_) of the constitutive choice for the strain energy density is presented for the _classical_ linear-elastic micromorphic media of Mindlin-Eringen type. This decomposition allows to individuate, in the isotropic case, a unique parameter $\mu_{c}$ (called _Cosserat couple modulus_) which governs the asymmetry of the force stresses and which is strongly related to penalty formulations without intrinsic physical significance. This parameter is not included in the _relaxed model_ we introduce in the second part of the present paper. In the following, we refer to the classical micromorphic model, relaxed model etc. according to the following understanding * classical: Dirichlet boundary condition for $P$, the energy density is similar to \[\hskip-14.226378pt\mu_{e}\,\\|\operatorname{sym}(\nabla u-P)\\|^{2}+\framebox[93 .894094pt][l]{$\mu_{c}\,\\|\mathop{\rm skew}(\nabla u-P)\\|^{2}$}+\mu_{h}\,\\| \operatorname{sym}P\\|^{2}+\framebox[34.143307pt][l]{$\\|\nabla P\\|^{2}$}\,\,;\] * relaxed: tangential boundary condition for $P$, the energy density is similar to \[\hskip-102.429921pt\mu_{e}\\|\operatorname{sym}(\nabla u-P)\\|^{2}+\mu_{h}\\| \operatorname{sym}P\\|^{2}+\framebox[48.369685pt][l]{$\\|\operatorname{Curl}P\\|^ {2}$}\,\,;\] * mixed variant: tangential boundary condition for $P$, the energy density is similar to \[\mu_{e}\\|\operatorname{sym}(\nabla u-P)\\|^{2}+\framebox[93.894094pt][l]{$\mu_{ c}\,\\|\mathop{\rm skew}(\nabla u-P)\\|^{2}$}+\mu_{h}\\|\operatorname{sym}P\\|^{2} +\framebox[48.369685pt][l]{$\\|\operatorname{Curl}P\\|^{2}$}\,\,,\] where $u$ is the displacement and $P$ is the micro-distortion. The precise definitions will be given in the Subsections 2.1 and 2.2. In contrast to the Mindlin and Eringen models, we avoid the presence of the only one parameter in the force stress response which can not be directly related to simple experiments. However, the presence of the parameter $\mu_{c}$ may be necessary to completely describe the mechanical behavior of artificial metamaterials in which strong contrasts of the elastic properties are present at the microscopic level. This necessity is evident when studying e.g. phononic crystals which are especially designed to exhibit frequency band-gaps. This means that such metamaterials are conceived to block wave propagation in precise frequency ranges. As far as standard heterogeneous materials (natural and artificial) are concerned, our reduced model with symmetric stress is sufficient to fully describe their mechanical behavior. The new well-posedness results for the relaxed model include the well-posedness results for the classical model. ### Historical perspective The capability of continuum theories to describe the time evolution and the deformation of the micro-structure of complex mechanical systems was recognized in the very first formulations of continuum mechanics (see the pioneering work by Piola [148]). Piola was led by stringent physical considerations to consider gradients of displacement field higher than the first as needed independent variables in the constitutive equation for the deformation energy of continuous media (for a modern presentation of this subject see e.g. [42, 157, 41, 131, 159]). However, more or less in the same period in which Piola was producing his papers, Cauchy and Poisson managed to determine a very elegant and effective format for continuum mechanics in which: i) the only kinematical descriptor is the displacement from a reference configuration, ii) the crucial conceptual tool is the _symmetric Cauchy force stress tensor_$\sigma=\mathbb{C}.\varepsilon$ which is constitutively related only to the symmetrized gradient of displacement $\varepsilon=\operatorname{sym}\nabla u$, iii) the crucial postulates are those concerning balance of mass, linear and angular momentum and (eventually) energy. The Cauchy and Poisson format is very effective to describe the mechanical behavior of a very wide class of natural and also artificial materials. Nevertheless, when considering materials with well-organized microstructures subjected to particular loads and/or boundary conditions, a Cauchy continuum theory may fail to give accurate results. This is the case for some engineering materials showing high contrast of material properties (see e.g. [144, 21, 51]) or for some natural materials which show highly heterogeneous hierarchical microstructures (see e.g. [22]). In all these cases, the introduction of more sophisticated models becomes mandatory if one wants to catch all features of the mechanical behavior of such complex materials. About fifty years later, Piola’s ideas were developed by the Cosserat brothers, who were among the first authors who complemented the standard kinematics constituted by a placement field with additional independent kinematical fields. In their case, these suitable fields are given by rigid rotations of the microstructure with respect to the macroscopic continuum displacement. This introduces three additional dof into the theory. Cosserat contributions [34] were underestimated for another fifty years and only starting from 1960 a group constituted by relevant scientific personalities as Mindlin⁵[110, 107, 108], Green and Rivlin [70, 72, 69, 71], Toupin [164, 165], Eringen [47, 49, 50] and Germain [61, 62] managed to establish (with still some resistances) the formal validity of Cosserat’s point of view. [FOOTNOTE:5][ENDFOOTNOTE] Actually, the Cosserat’s approach must be further generalized as not only micro-rotations should be included in a macroscopic modelling picture, but also micro-stretches, micro-strains, micro-shear or concentrated micro-distortions. This can be done by introducing the so-called micro-structured or micromorphic continuum models, which are suitably formulated by means of a postulation process based on the principle of least action (see e.g. [4]) or on the principle of virtual works (see the beautiful works [84, 102, 105, 104, 170, 25]) even if later on the alternative postulation procedure based on “generalized balance laws” has been also attempted (see [47, 49, 50]). Indeed, as already remarked in [148], when starting from a discrete system characterized by a micro-structure spanning several length scales and strong contrast of the elastic properties it is rather unlikely to get as a suitable macroscopic model the simple standard Cauchy continuum. Whether the force stresses remain symmetric in such homogenization procedure is open [11, 144]. In addition, the evolution of non purely mechanical phenomena can be described within the framework of micromorphic continua theory: in this context we refer for instance to those observed in nematic liquid crystals (see e.g. [166]) where also electromagnetic descriptors need to be introduced to characterize completely the kinematics of the system. In such a case we speak of polar materials in which the force stress may clearly become non-symmetric. It has to be explicitly remarked that Piola’s and Cosserat’s models can be reconciled by means of the introduction of suitable “internal constraints” and “Lagrange multipliers” as clearly stated e.g. in [19]. Actually one can get Piola’s deformation energies depending on higher gradients of displacement as a limit of many different (and physically non-equivalent) more detailed micromorphic models. If in the classical micromorphic model, in the formal limits, we assume e.g. that the coefficients $\widehat{\mathbb{C}}\rightarrow\infty$ then this leads to the free energy (2.10) for the gradient elasticity model [129, 131]. On the other hand, we can always relax Piola’s gradient material into suitable micromorphic models. ### Approach in this work In the present paper we re-investigate the general micromorphic model with a focus on heterogeneous, but non-polar materials. More particularly, we start by recalling the classical Mindlin-Eringen model for micromorphic media with intrinsically non-symmetric force stresses. As our contribution, we propose a relaxed linear micromorphic model with symmetric Cauchy force stresses and curvature response only due to dislocation energy, and we formulate the initial–boundary value problems. We prove that this new model is still well-posed [66], i.e. we study the continuous dependence of solution with respect to the initial data and supply terms and existence and uniqueness of the solution. The main point in establishing the existence, uniqueness and continuous dependence results [66] is represented by the new coercive inequalities recently proved by Neff, Pauly and Witsch [135, 136, 137] and by Bauer, Neff, Pauly and Starke [7, 5, 6] (see also [66, 88]). This relaxed formulation of micromorphic elasticity has some similarities to recently studied models of gradient plasticity [45, 126, 139, 140, 33]. Indeed, in the static case, the micromorphic relaxed minimization problem has the same stored elastic energy. In gradient plasticity, however, the plastic distortion/micromorphic distortion is determined not by energy minimization, but instead by a flow rule. Our new approach, in sharp contrast to classical micromorphic models, features a symmetric force stress tensor, which we call Cauchy stress. Teisseyre [161, 162] in his model for the description of seismic wave propagation phenomena [44, 117, 118] also used a symmetric force stress tensor. In fact, Teisseyre’s model is a fully symmetric model and it is a particular case of the dislocation dynamics theory proposed by Eringen and Claus [31, 48, 32], where the relative force-stress is considered to be non-symmetric. The model of Eringen and Claus [31, 48, 32] contains the linear Cosserat model [120, 128, 81, 130] with asymmetric force stresses upon suitable restriction. This is a situation we avoid in the proposed relaxed micromorphic approach (see in Subsection 4.1 the motivation given by Kröner for symmetric force stresses in dislocation dynamics). In fact, it turns out (to our surprise) that our relaxed model is the Eringen-Claus model [31, 48, 32], albeit with symmetric Cauchy stresses and absent mixed coupling terms. In Section 4 we disclose the relation of our new relaxed model to the existing models in more detail. Our critical remarks concerning the linear Cosserat model leave open the possible usefulness of a geometrically nonlinear Cosserat model [120, 115] with symmetric Cauchy stresses. In contrast with the models considered until now, our free energy of the relaxed model is _not uniformly pointwise positive definite_ in the control of the constitutive variables. The proposed relaxed micromorphic model with symmetric force-stress may be thought to fully describe the mechanical behaviour of a great variety of natural and artificial microscopically heterogeneous materials. Granular assemblies are also a field of application of micromorphic models. The possible non-symmetry of the force stress tensor in such models has been discussed e.g. in [68] and it is proved that in the absence of intergranular contact moments the grain rotation makes no direct contribution to quasi-static contact work, and that the widely accepted formula based on volume averaging yields a symmetric Cauchy stress. On the other hand, we are aware of the possible usefulness of Cosserat model for what concerns the modeling of artificial engineering metamaterials with strong contrast at the microscopic level. The results established in our paper can be extended to theories which include electromagnetic and thermal interactions [60, 59, 73, 103]. For isotropic materials, the models presented in this paper involve only a reduced number of constitutive parameters. This fact will allow us to find exact solutions for wave propagation problems using analog methods as in [80, 30, 28, 29] and we may also compare the analytical solutions with experiments in order to identify the fewer relaxed constitutive coefficients. It is known since the pionering works of Mindlin [107] that two types of waves can propagate in a micromorphic continuum: _acoustic waves_, i.e. waves for which the frequency vanishes for vanishing wavenumbers (wavelength which tends to infinity), and _optic waves_, i.e. waves which have non-vanishing, finite frequency corresponding to vanishing wavenumbers (space independent oscillations). It can be shown that, for particular frequency ranges, also a third type of waves may exist in our relaxed micromorphic media, namely so-called _standing waves_, i.e. waves which do not propagate inside the medium but keep oscillating in a given region of space. These waves are impossible in the classical micromorphic model. Wave propagation in the considered relaxed micromorphic model and the precise effect of the considered elastic parameters will be carefully studied in a forthcoming paper to show the interest of using our model to proceed towards innovative technological applications. The paper [154] gives a rich reference list on the wave propagation in second-gradient materials and on generalized media, in general. Moreover, we will deal with the static model and consider the elliptic regularity question. The numerical treatment of our new model needs FEM-discretisations in ${\rm H}({\rm curl};\Omega)$, see [67]. This will be left for future work. ### Notation For $a,b\in\mathbb{R}^{3}$ we let ${\langle a,b\rangle}_{\mathbb{R}^{3}}$ denote the scalar product on $\mathbb{R}^{3}$ with associated vector norm $\\|a\\|_{\mathbb{R}^{3}}^{2}={\langle a,a\rangle}_{\mathbb{R}^{3}}$. We denote by $\mathbb{R}^{3\times 3}$ the set of real $3\times 3$ second order tensors, written with capital letters. The standard Euclidean scalar product on $\mathbb{R}^{3\times 3}$ is given by ${\langle X,Y\rangle}_{\mathbb{R}^{3\times 3}}=\textrm{tr}({XY^{T}})$, and thus the Frobenius tensor norm is $\\|X\\|^{2}={\langle X,X\rangle}_{\mathbb{R}^{3\times 3}}$. In the following we omit the index $\mathbb{R}^{3},\mathbb{R}^{3\times 3}$. The identity tensor on $\mathbb{R}^{3\times 3}$ will be denoted by $1\!\!1$, so that $\textrm{tr}({X})={\langle X,1\!\!1\rangle}$. We let ${\rm{Sym}}$ denote the set of symmetric tensors. We adopt the usual abbreviations of Lie-algebra theory, i.e., $\operatorname{\mathfrak{so}}(3):=\{X\in\mathbb{R}^{3\times 3}\;\|X^{T}=-X\}$ is the Lie-algebra of skew symmetric tensors and $\operatorname{\mathfrak{sl}}(3):=\{X\in\mathbb{R}^{3\times 3}\;\|\textrm{tr}({X })=0\}$ is the Lie-algebra of traceless tensors. For all vectors $\xi,\eta\in\mathbb{R}^{3}$ we have the tensor product $(\xi\otimes\eta)_{ij}=\xi_{i}\,\eta_{j}$ and $\epsilon_{ijk}$ is the Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, given by \[\epsilon_{ijk}=\left\{\begin{array}[]{ll}1&\text{if}\quad(i,j,k) \quad\text{ is an even permutation of}\quad(1,2,3)\\ -1&\text{if}\quad(i,j,k)\quad\text{ is an odd permutation of}\quad(1,2,3)\\ 0&\text{otherwise}.\end{array}\right.\] (1.1) For all $X\in\mathbb{R}^{3\times 3}$ we set $\operatorname{sym}X=\frac{1}{2}(X^{T}+X)\in{\rm{Sym}}$, $\mathop{\rm skew}X=\frac{1}{2}(X-X^{T})\in\operatorname{\mathfrak{so}}(3)$ and the deviatoric part $\operatorname{dev}X=X-\frac{1}{3}\;\textrm{tr}{X}\,1\!\!1\in\operatorname{ \mathfrak{sl}}(3)$ and we have the _orthogonal Cartan-decomposition of the Lie-algebra_$\operatorname{\mathfrak{gl}}(3)$ \[\operatorname{\mathfrak{gl}}(3) =\{\operatorname{\mathfrak{sl}}(3)\cap{\rm{Sym}}(3)\}\oplus \operatorname{\mathfrak{so}}(3)\oplus\mathbb{R}\!\cdot\!1\!\!1,\] \[X =\operatorname{dev}\operatorname{sym}X+\mathop{\rm skew}X+\frac{1 }{3}\textrm{tr}(X)\!\cdot\!1\!\!1\,.\] (1.2) By $C_{0}^{\infty}(\Omega)$ we denote infinitely differentiable functions with compact support in $\Omega$. We employ the standard notation of Sobolev spaces, i.e. $L^{2}(\Omega),H^{1,2}(\Omega),H_{0}^{1,2}(\Omega)$, which we use indifferently for scalar-valued functions as well as for vector-valued and tensor-valued functions. Throughout this paper (when we do not specify else) Latin subscripts take the values $1,2,3$. Typical conventions for differential operations are implied such as comma followed by a subscript to denote the partial derivative with respect to the corresponding cartesian coordinate, while $t$ after a comma denotes the partial derivative with respect to the time. The usual Lebesgue spaces of square integrable functions, vector or tensor fields on $\Omega$ with values in $\mathbb{R}$, $\mathbb{R}^{3}$ or $\mathbb{R}^{3\times 3}$, respectively will be denoted by $L^{2}(\Omega)$. Moreover, we introduce the standard Sobolev spaces [1, 67, 97] \[{\rm H}^{1}(\Omega)=\{u\in L^{2}(\Omega)\,\|\,{\rm grad}\,u\in L^{ 2}(\Omega)\},\ \ \ {\rm grad}=\nabla\,,\] \[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\|u\\|^{2}_{{\rm H}^{1}(\Omega)}:=\\| u\\|^{2}_{L^{2}(\Omega)}+\\|{\rm grad}\,u\\|^{2}_{L^{2}(\Omega)}\,,\] \[{\rm H}({\rm curl};\Omega)=\{v\in L^{2}(\Omega)\,\|\,{\rm curl}\,v \in L^{2}(\Omega)\},\ \ \ {\rm curl}=\nabla\times\,,\] (1.3) \[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\|v\\|^{2}_{{\rm H}({\rm curl} ;\Omega)}:=\\|v\\|^{2}_{L^{2}(\Omega)}+\\|{\rm curl}\,v\\|^{2}_{L^{2}(\Omega)}\,,\] \[{\rm H}({\rm div};\Omega)=\{v\in L^{2}(\Omega)\,\|\,{\rm div}\,v \in L^{2}(\Omega)\},\ \ \ {\rm div}=\nabla\cdot\,,\] \[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\|v\\|^{2}_{{\rm H}({\rm div}; \Omega)}:=\\|v\\|^{2}_{L^{2}(\Omega)}+\\|{\rm div}\,v\\|^{2}_{L^{2}(\Omega)}\,,\] of functions $u$ or vector fields $v$, respectively. Furthermore, we introduce their closed subspaces $H_{0}^{1}(\Omega)$, and ${\rm H}_{0}({\rm curl};\Omega)$ as completion under the respective graph norms of the scalar valued space $C_{0}^{\infty}(\Omega)$, the set of smooth functions with compact support in $\Omega$. Roughly speaking, $H_{0}^{1}(\Omega)$ is the subspace of functions $u\in H^{1}(\Omega)$ which are zero on $\partial\Omega$, while ${\rm H}_{0}({\rm curl};\Omega)$ is the subspace of vectors $v\in{\rm H}({\rm curl};\Omega)$ which are normal at $\partial\Omega$ (see [135, 136, 137]). For vector fields $v$ with components in ${\rm H}^{1}(\Omega)$ and tensor fields $P$ with rows in ${\rm H}({\rm curl}\,;\Omega)$, resp. ${\rm H}({\rm div}\,;\Omega)$, i.e., \[v=\left(\begin{array}[]{c}v_{1}\\ v_{2}\\ v_{3}\\ \end{array}\right)\,,v_{i}\in{\rm H}^{1}(\Omega),\ \quad P=\left(\begin{array} []{c}P_{1}^{T}\\ P_{2}^{T}\\ P_{3}^{T}\\ \end{array}\right)\,\quad P_{i}\in{\rm H}({\rm curl}\,;\Omega)\,\quad\ \text{ resp.}\quad P_{i}\in{\rm H}({\rm div}\,;\Omega)\] (1.4) we define \[{\rm Grad}\,v=\left(\begin{array}[]{c}{\rm grad}^{T}\,v_{1}\\ {\rm grad}^{T}\,v_{2}\\ {\rm grad}^{T}\,v_{3}\\ \end{array}\right)\,,\ \ \ \ {\rm Curl}\,P=\left(\begin{array}[]{c}{\rm curl}^ {T}\,P_{1}\\ {\rm curl}^{T}\,P_{2}\\ {\rm curl}^{T}\,P_{3}\\ \end{array}\right),\,\ \ \ \ {\rm Div}\,P=\left(\begin{array}[]{c}{\rm div}\,P _{1}\\ {\rm div}\,P_{2}\\ {\rm div}\,P_{3}\\ \end{array}\right).\] (1.5) We note that $v$ is a vector field, whereas $P$, ${\rm Curl}\,P$ and ${\rm Grad}\,v$ are second order tensor fields. The corresponding Sobolev spaces will be denoted by \[{\rm H}({\rm Grad}\,;\Omega)\ \ \ \text{and}\ \ \ \ {\rm H}({\rm Curl }\,;\Omega)\,.\] (1.6) Furthermore, if ${T}$ is a third order tensor then we define \[\operatorname{Div}T:=(\operatorname{Div}T_{1},\operatorname{Div}T _{2},\operatorname{Div}T_{3})^{T}\,,\] (1.7) where $T_{k}=(T_{ijk})\in\mathbb{R}^{3\times 3}$ are second order tensors. We recall that if $\mathbb{C}$ is a fourth order tensor and $X\in\mathbb{R}^{3\times 3}$, then $\mathbb{C}.X\in\mathbb{R}^{3\times 3}$ with the components \[(\mathbb{C}.X)_{ij}=\sum\limits_{k=1}^{3}\sum\limits_{l=1}^{3} \mathbb{C}_{ijkl}X_{kl}\,,\] (1.8) and $\mathbb{C}^{T}.X\in\mathbb{R}^{3\times 3}$ with the components \[(\mathbb{C}^{T}.X)_{kl}=\sum\limits_{i=1}^{3}\sum\limits_{j=1}^{3 }\mathbb{C}_{ijkl}X_{ij}\,.\] (1.9) If $\mathbb{G}$ is a fifth order tensor and $\mathbb{L}$ a sixth order tensor , then \[\mathbb{G}.\,Y\in\mathbb{R}^{3\times 3\times 3}\,\ \ \ \text{for all}\ \ Y\in\mathbb{R}^{3\times 3},\ \ \ (\mathbb{G}.\,Y)_{ijk}=\sum\limits_{m =1}^{3}\sum\limits_{n=1}^{3}\mathbb{G}_{mnijk}X_{mn},\] (1.10) and \[\mathbb{L}.Z\in\mathbb{R}^{3\times 3\times 3}\,\ \ \ \text{for all}\ \ Z\in\mathbb{R}^{3\times 3\times 3},\ \ \ (\mathbb{L}.Z)_{ijk}=\sum \limits_{m=1}^{3}\sum\limits_{n=1}^{3}\sum\limits_{p=1}^{3}\mathbb{L}_{ijkmnp} Z_{mnp}\,.\] (1.11) ## 2 Formulation of the problem. Preliminaries We consider a micromorphic continuum which occupies a bounded domain $\Omega$ and is bounded by the piecewise smooth surface $\partial\Omega$. Let $T>0$ be a given time. The motion of the body is referred to a fixed system of rectangular Cartesian axes $Ox_{i}$, $(i=1,2,3)$. ### Eringen’s linear asymmetric micromorphic elastodynamics revisited In this subsection, we present the initial-boundary value problem of the linear asymmetric micromorphic theory introduced by Eringen [47], which is basically identical to Mindlin’s theory of elasticity with microstructure [107]. The micro-distortion (plastic distortion) $P=(P_{ij}):\Omega\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ describes the substructure of the material which can rotate, stretch, shear and shrink, while $u=(u_{i}):\Omega\times[0,T]\rightarrow\mathbb{R}^{3}$ is the displacement of the macroscopic material points. In this dynamic micromorphic theory, the basic equations in strong form consist of the equations of motion \[\varrho\,{u}_{,tt} =\operatorname{Div}\widehat{\sigma}+f,\] (2.1) \[\varrho\,I.\,{P}_{,tt} =\operatorname{Div}\widehat{m}+\widehat{\sigma}-s+M,\text{\ \ in \ \ }\Omega\times[0,T],\] the constitutive equations \[\widehat{\sigma}=\widehat{\mathbb{C}}.\,e+\widehat{\mathbb{E}}.\, \varepsilon_{p}+\widehat{\mathbb{F}}.\,\gamma,\] \[s=\widehat{\mathbb{E}}^{T}.\,e+{\mathbb{H}}.\,\varepsilon_{p}+ \widehat{\mathbb{G}}.\,\gamma,\] (2.2) \[\widehat{m}=\widehat{\mathbb{F}}.\,e+\widehat{\mathbb{G}}.\, \varepsilon_{p}+\widehat{\mathbb{L}}.\,\gamma,\text{\ \ in \ \ }\overline{ \Omega}\times[0,T],\] exclusively depending on the set of _independent constitutive variables_ \[e:=\nabla u-P,\quad\quad\quad\varepsilon_{p}:=\operatorname{sym} P,\quad\quad\quad\gamma:=\nabla P,\ \ \ \ \text{\ \ in \ \ }\overline{\Omega} \times[0,T].\] (2.3) The symmetric part of $e$ corresponds to the difference of material strain $\varepsilon$ and microstrain $\varepsilon_{p}$, whereas its skew-symmetric part accounts for the relative rotation of the material with respect to the substructure. Various strain measures for the Cosserat continuum have been extensively discussed in [145, 146]. Using the assumption of small strains and assuming skew-symmetry of $P$ the strain measures (2.3) coincide with the natural Cosserat strain measures which are non-symmetric, in general. The quantities involved in the above system of equations have the following physical signification: * $(u,P)$ are the _kinematical variables_, * $\varrho$ is the reference mass density, * $I$ is the microinertia tensor (second order), * $\widehat{\sigma}$ is the force-stress tensor (second order, in general non-symmetric), * $s$ is the microstress tensor (second order, symmetric), * $\widehat{m}$ is the moment stress tensor (micro-hyperstress tensor, third order, in general non-symmetric), * $u$ is the displacement vector (translational degrees of freedom), * $P$ is the micro-distortion tensor (“plastic distortion”, second order, non-symmetric), * $f$ is the body force, * $M$ is the body moment tensor (second order, non-symmetric), * $e:=\nabla u-P$ is the elastic distortion (relative distortion, second order, non-symmetric), * $\varepsilon_{e}:=\operatorname{sym}e=\operatorname{sym}(\nabla u-P)$ is the elastic strain tensor (second order, symmetric), * $\varepsilon:=\operatorname{sym}\nabla u$ is the total strain tensor (material strain tensor, second order, symmetric), * $\varepsilon_{p}:=\operatorname{sym}P$ is the micro-strain tensor (“plastic strain”, second order, symmetric), * $\gamma:=\nabla P\in\mathbb{R}^{27}$ is the micro-curvature tensor (third order), * $\widehat{\mathbb{C}}=(\widehat{\mathbb{C}}_{ijmn})$, ${\mathbb{H}}=({\mathbb{H}}_{ijmn})$, $\widehat{\mathbb{E}}=(\widehat{\mathbb{E}}_{ijmn})$, $\widehat{\mathbb{F}}=(\widehat{\mathbb{F}}_{ijmnp})$ and $\widehat{\mathbb{G}}=(\widehat{\mathbb{G}}_{ijmnp})$ are tensors determining the constitutive coefficients which satisfy the symmetry relations \[\widehat{\mathbb{C}}_{ijmn} =\widehat{\mathbb{C}}_{mnij},\] \[{\mathbb{H}}_{ijmn} ={\mathbb{H}}_{mnij}={\mathbb{H}}_{jimn},\quad\quad\quad\widehat{ \mathbb{E}}_{mnij}=\widehat{\mathbb{E}}_{mnji},\quad\quad\quad\ \widehat{ \mathbb{G}}_{ijmnp}=\widehat{\mathbb{G}}_{jimnp},\] (2.4) * The tensor $\widehat{\mathbb{L}}$ determines various _characteristic length scales_ in the model, its unit is [$\mathrm{MPa}\,\!\cdot\!\,\mathrm{m}^{2}$] and it satisfies the symmetry relations \[\widehat{\mathbb{L}}_{ijkmnp}=\widehat{\mathbb{L}}_{mnpijk}\,.\] (2.5) The symmetries of $\widehat{\mathbb{E}}$ and $\widehat{\mathbb{G}}$ imply that $\widehat{\mathbb{E}},\widehat{\mathbb{G}}:\mathbb{R}^{3\times 3}\rightarrow{ \rm{Sym}}(3)$. Thus, the microstress tensor $s$ is always symmetric. In contrast, the symmetries of $\widehat{\mathbb{C}}$ do not imply that $\widehat{\mathbb{C}}$ maps symmetric matrices into symmetric matrices, while $\mathbb{H}:{\rm{Sym}}(3)\rightarrow{\rm{Sym}}(3)$ has this property, as the classical elasticity tensor. For micro-isotropic materials the microinertia tensor is given by $I=\frac{1}{3}J\cdot 1\!\!1$, where $J$ is a known scalar function on $\Omega$. Using the micro-strain tensor $\varepsilon_{p}=\operatorname{sym}P$ instead of $P$ itself in the list of independent constitutive variables (2.3) is mandatory for frame-indifference. The above equations lead to a system of 12 linear partial differential equations of Lamé type for the functions $u$ and $P$. In order to study the existence of solution of the resulting system, Hlaváček [74], Ieşan and Nappa [76] and Ieşan [77] considered null boundary conditions, i.e. \[u(x,t)=0\quad\quad\text{and the {\it strong anchoring condition} }\quad\quad\ P(x,t)=0,\ \ \text{on}\ \ \partial\Omega\times(0,T).\] (2.6) We adjoin the initial conditions \[u(x,0)=u^{0}(x),\quad\ \ \ P(x,0)=P^{0}(x),\quad\ \ \ \dot{u}(x, 0)=\dot{u}^{0}(x),\quad\ \ \ \dot{P}(x,0)=\dot{P}^{0}(x),\quad\ \ \text{on}\ \ \overline{\Omega},\] (2.7) where the quantities on the right-hand sides are prescribed, satisfying $u^{0}(x)=0$ and $P^{0}(x)=0$ on $\partial\Omega$. The system of governing equations (2.1) is derived from the following elastic free energy \[2 \mathcal{E}(e,\varepsilon_{p},\gamma)=\langle\widehat{\mathbb{C}} .\,(\nabla u-P),(\nabla u-P)\rangle+\langle{\mathbb{H}}.\,\operatorname{sym}P, \operatorname{sym}P\rangle+\langle\widehat{\mathbb{L}}.\,\nabla P,\nabla P\rangle\] (2.8) \[\quad\quad\quad\quad\quad\quad+2\langle\widehat{\mathbb{E}}.\, \operatorname{sym}P,(\nabla u-P)\rangle+2\langle\widehat{\mathbb{F}}.\,\nabla P ,(\nabla u-P)\rangle+2\langle\widehat{\mathbb{G}}.\,\nabla P,\operatorname{sym }P\rangle\,,\] \[\widehat{\sigma} =D_{e}\,\mathcal{E}(e,\varepsilon_{p},\gamma)\in\mathbb{R}^{3 \times 3},\quad\quad s=D_{\varepsilon_{p}}\,\mathcal{E}(e,\varepsilon_{p}, \gamma)\in{\rm{Sym}}(3),\quad\quad\widehat{m}=D_{\gamma}\,\mathcal{E}(e, \varepsilon_{p},\gamma)\in\mathbb{R}^{3\times 3\times 3}.\] Since the _elastic distortion_$e:=\nabla u-P$ is in general _non-symmetric_, in this model the relative force-stress tensor $\widehat{\sigma}$ is also non-symmetric. In case that $P$ is assumed to be purely skew-symmetric, this model turns into the linear Cosserat model after orthogonal projection of the equation for the micro-distortion to the skew-symmetric subspace (see the Subsection 4.4). The status of the linear Cosserat model as a useful description of real material behaviour is still doubtful⁶[120, 128] as far its application to classical heterogeneous materials is concerned even if the asymmetry of the stress tensor may be of use for some suitably conceived engineering metamaterials as e.g. phonon crystals. The existence results from [74, 76, 77] are established assuming that the energy $\mathcal{E}$ is a pointwise positive definite quadratic form in terms of the _independent constitutive variables_$e$, $\varepsilon_{p}$ and $\gamma$, i.e. there is a positive constant $c^{+}$ such that [FOOTNOTE:6][ENDFOOTNOTE] \[\widehat{\mathcal{E}}(\nabla u-P,\operatorname{sym}P,\nabla P) \geq c^{+}\left(\\|\nabla u-P\\|^{2}+\\|\operatorname{sym}P\\|^{2}+\\|\nabla P\\|^{2 }\right)\,.\] (2.9) A general feature of the asymmetric micromorphic model is its regularizing influence on the solution when coupled with other effects, e.g. incompressible plasticity is regularized by adding Cosserat effects, see [123, 124, 125, 134, 132, 116, 33]. When the body possesses a center of symmetry, the tensors $\widehat{\mathbb{F}}$ and $\widehat{\mathbb{G}}$ have to vanish. Thus, for centro-symmetric elastic materials the two mixed terms $\langle\widehat{\mathbb{F}}.\,\nabla P,\nabla u-P\rangle$ and $\langle\widehat{\mathbb{G}}.\,\nabla P,\operatorname{sym}P\rangle$ are absent. The centro-symmetry of the material does not imply that $\widehat{\mathbb{E}}$ vanishes [47]. In the following we omit for simplicity the mixed term $\langle\widehat{\mathbb{E}}.\,\operatorname{sym}P,\nabla u-P\rangle$ in the energy since on the one hand its physical significance is unclear and it would induce nonzero relative stress $\widehat{\sigma}$ for zero elastic distortion $e=\nabla u-P=0$. Moreover, we show in the Subsections 4.4, 4.7 and 4.8 how our energy without any mixed terms leads, in principle, to complete equations for the Cosserat model, the microstretch model and the microvoids model in dislocation format. This is a consequence of our choice of the _independent constitutive variables⁷_. However, mixed terms may appear if homogenization techniques are used, see [52, 53, 57]. Our mathematical analysis can be extended in a straightforward manner to the case when the mixed terms are also present in the total energy. [FOOTNOTE:7][ENDFOOTNOTE] If in the classical asymmetric micromorphic model, in the formal limits, we assume that the coefficients $\widehat{\mathbb{C}}\rightarrow\infty$ then this leads to the free energy from the gradient elasticity model [129, 131] in which we do not have mixed terms either. Indeed, in this case $P=\nabla u$ and in consequence, for centro-symmetric materials, the free energy will reduce to \[2\mathcal{E}(\operatorname{sym}\nabla u,\nabla(\nabla u)) =\langle{\mathbb{H}}.\,\operatorname{sym}\nabla u,\operatorname{ sym}\nabla u\rangle+\langle\widehat{\mathbb{L}}.\,\nabla(\nabla u),\nabla( \nabla u)\rangle\] (2.10) \[=\langle{\mathbb{H}}.\,\operatorname{sym}\nabla u,\operatorname{ sym}\nabla u\rangle+\langle\widehat{\mathbb{L}}.\,D^{2}u,D^{2}u\rangle\,.\] Hitherto, the asymmetric micromorphic model has best been seen and motivated as a higher gradient elasticity model, in which the second derivatives have been replaced by a gradient of a new field [157, 37, 39, 42, 38]. A distinctive feature of such a second gradient elasticity model is that the _local force stresses_ always remain _symmetric_[41, 160] and are characterized by the two classical Lamé constants $\mu,\lambda$ in the isotropic case. Therefore, the close connection between the classical micromorphic model and the gradient elasticity model is apparent. In many cases, the classical micromorphic model is thus used as a “cheap” 2nd order numerical replacement for the “expensive” 4th order model [129, 131, 138, 33]. Let us remark that if in the free energy from isotropic strain gradient elasticity we replace the terms \[\mathcal{E}(\nabla u,\nabla(\operatorname{sym}\nabla u))=\mu\,\\|\operatorname{ sym}\nabla u\\|^{2}+\frac{\lambda}{2}\,[\textrm{tr}(\nabla u)]^{2}+\mu L_{c}^{2 }\,\\|\nabla(\operatorname{sym}\nabla u)\\|^{2},\] (2.11) with \[\mathcal{E}(\nabla u,P,\nabla P)=\mu\,\\|\operatorname{sym}\nabla u\\|^{2}+\frac {\lambda}{2}\,[\textrm{tr}(\nabla u)]^{2}+\varkappa^{+}\mu\,\\|\operatorname{ sym}\nabla u-\operatorname{sym}P\\|^{2}+\mu L_{c}^{2}\,\\|\nabla(\operatorname{ sym}P)\\|^{2},\] (2.12) where $\varkappa^{+}$ is a dimensionless penalty coefficient, then Forest’s microstrain theory (3+6 parameter theory) [55] (see Subsection 4.2) is nothing else but a penalyzed strain gradient elasticity formulation. In the case of a strain gradient material both, the _local force stresses and the total force stresses_ are _symmetric_, see the discussion from Subsection 4.9. Therefore, the asymmetry of the force stress tensor in a continuum theory is not a consequence of the presence of microstructure in the body, it is rather a constitutive assumption [20]. Moreover, for an isotropic strain gradient material it is easy to see that both, the local force stresses and the nonlocal force stresses can be chosen symmetric, see Subsection 4.9. We will deal with the complete modeling issue in another contribution. ### The relaxed micromorphic continuum model The ultimate goal of science is the reduction to a minimum of necessary complexity in the description of nature. In the classical asymmetric micromorphic theory there are involved more than 1000 constitutive coefficients in the general anisotropic case, and even for isotropic materials the constitutive equations contain a great number of material constants (7+11 parameters in Mindlin’s and Eringen’s theory [49, 107, 50]). Unfortunately, this makes the general micromorphic model suitable for anything and nothing and has severely hindered the application of micromorphic models. We consider here a relaxed version of the classical micromorphic model with _symmetric Cauchy-stresses_$\sigma$ and drastically reduced numbers of constitutive coefficients. More precisely, our model is a subset of the classical model in which we allow the elasticity tensors $\widehat{\mathbb{C}}$ and $\widehat{\mathbb{L}}$ to become positive-semidefinite only. The proof of the well-posedness of this model [66] necessitates the _application of new mathematical tools_[135, 136, 137, 7, 5, 6, 88]. The curvature dependence is reduced to a dependence only on the _micro-dislocation tensor_$\alpha:=\operatorname{Curl}e=-\operatorname{Curl}P\in\mathbb{R}^{3\times 3}$ instead of $\gamma=\nabla P\in\mathbb{R}^{27}=\mathbb{R}^{3\times 3\times 3}$ and the local response is reduced to a dependence on the symmetric part of the elastic distortion (relative distortion) $\varepsilon_{e}=\operatorname{sym}e=\operatorname{sym}(\nabla u-P)$, while the _full kinematical degrees of freedom_ for $u$ and $P$ are kept, notably _rotation of the microstructure remains possible_. Our new set of _independent constitutive variables_ for the relaxed micromorphic model is thus \[\varepsilon_{e}=\operatorname{sym}(\nabla u-P),\quad\quad\quad \varepsilon_{p}=\operatorname{sym}P,\quad\quad\quad\alpha=-\operatorname{Curl}P.\] (2.13) The stretch strain tensor defined in (2.13)${}_{1}$ is symmetric. For simplicity, the following systems of partial differential equations are considered in a normalized form, i.e. the left hand sides of the equations are not multiplied with $\varrho$ or $\varrho\,I$, respectively. We consider the following system of partial differential equations which corresponds to this special linear anisotropic micromorphic continuum \[u_{,tt} =\textrm{Div}[\mathbb{C}.\operatorname{sym}(\nabla u-P)]+f\,,\] (2.14) \[P_{,tt} =-\textrm{Curl}[\mathbb{L}_{c}.\textrm{Curl}\,P]+\mathbb{C}. \operatorname{sym}(\nabla u-P)-\mathbb{H}.\operatorname{sym}P+M\,\ \ \ \text{ in}\ \ \ \Omega\times[0,T],\] where $f:\Omega\times[0,T]\rightarrow\mathbb{R}^{3}$ describes the body force and $M:\Omega\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ describes the external body moment, $\mathbb{C}\!:\!\Omega\to L(\mathbb{R}^{3\times 3},\mathbb{R}^{3\times 3})$, $\mathbb{L}_{c}\!:\!\Omega\to L(\mathbb{R}^{3\times 3},\mathbb{R}^{3 \times 3})$ and $\mathbb{H}\!:\!\Omega\to L(\mathbb{R}^{3\times 3},\mathbb{R}^{3\times 3})$ are fourth order elasticity tensors, positive definite and functions of class $C^{1}(\Omega)$. For the rest of the paper we assume that the constitutive coefficients have the following symmetries \[\mathbb{C}_{ijrs}=\mathbb{C}_{rsij}=\mathbb{C}_{jirs},\quad\quad \quad\quad\mathbb{H}_{ijrs}=\mathbb{H}_{rsij}=\mathbb{H}_{jirs},\quad\quad \quad\quad{(\mathbb{L}_{c})}_{ijrs}={(\mathbb{L}_{c})}_{rsij}\,.\] (2.15) The system (2.14) is derived from the following free energy \[\quad 2\,\mathcal{E}(\varepsilon_{e},\varepsilon_{p},\alpha) =\langle\mathbb{C}.\,\varepsilon_{e},\varepsilon_{e}\rangle+ \langle\mathbb{H}.\,\varepsilon_{p},\varepsilon_{p}\rangle+\langle\mathbb{L}_{ c}.\,\alpha,\alpha\rangle\] (2.16) \[=\underbrace{\langle\mathbb{C}.\,\operatorname{sym}(\nabla u-P), \operatorname{sym}(\nabla u-P)\rangle}_{\text{elastic energy}}+\underbrace{ \langle\mathbb{H}.\,\operatorname{sym}P,\operatorname{sym}P\rangle}_{\text{ microstrain self-energy}\lx@notemark{footnote}}+\underbrace{\langle\mathbb{L}_ {c}.\,\operatorname{Curl}P,\operatorname{Curl}P\rangle}_{\text{dislocation energy}},\] \[\sigma=D_{\varepsilon_{e}}\,\mathcal{E}(\varepsilon_{e}, \varepsilon_{p},\alpha)\in{\rm{Sym}}(3),\quad\quad s=D_{\varepsilon_{p}}\, \mathcal{E}(\varepsilon_{e},\varepsilon_{p},\alpha),\in{\rm{Sym}}(3),\quad \quad m=D_{\alpha}\,\mathcal{E}(\varepsilon_{e},\varepsilon_{p},\alpha)\in \mathbb{R}^{3\times 3}.\] Note again that in this theory, the elastic distortion $e=\nabla u-P$ may still be non-symmetric but the possible asymmetry of $e$ does not produce a related asymmetric stress contribution. The comparison with the classical Eringen’s equations (2.1)–(2.3) is achieved through observing again that \[\langle\widehat{\mathbb{C}}.X,X\rangle_{\mathbb{R}^{3\times 3}}:= \langle\mathbb{C}.\operatorname{sym}X,\operatorname{sym}X\rangle_{\mathbb{R}^{ 3\times 3}},\] (2.17) \[\langle\widehat{\mathbb{L}}.\nabla P,\nabla P\rangle_{\mathbb{R}^ {3\times 3\times 3}}:=\langle\mathbb{L}_{c}.\operatorname{Curl}P,\operatorname {Curl}P\rangle_{\mathbb{R}^{3\times 3}}\] define only _positive semi-definite tensors $\widehat{\mathbb{C}}$ and $\widehat{\mathbb{L}}$_ in terms of _positive definite tensors $\mathbb{C}$ and $\mathbb{L}_{c}$_ acting on linear subspaces of $\operatorname{\mathfrak{gl}}(3)\cong\mathbb{R}^{3\times 3}$. More precisely \[\widehat{\mathbb{C}}:\mathbb{R}^{3\times 3}\rightarrow\mathbb{R}^ {3\times 3},\quad\quad\quad\widehat{\mathbb{L}}:\mathbb{R}^{3\times 3\times 3} \rightarrow\mathbb{R}^{3\times 3\times 3},\] (2.18) while \[\mathbb{C}:{\rm{Sym}}(3)\rightarrow{\rm{Sym}}(3),\quad\quad\quad \mathbb{L}_{c}:\mathbb{R}^{3\times 3}\rightarrow\mathbb{R}^{3\times 3}.\] (2.19) We assume that the new fourth order elasticity tensors $\mathbb{C}$, $\mathbb{L}_{c}$ and $\mathbb{H}$ are positive definite. Then, there are positive numbers ${c_{M}}$, ${c_{m}}$ (the maximum and minimum elastic moduli for $\mathbb{C}$), ${(L_{c})}_{M}$ ,${(L_{c})}_{m}$ (the maximum and minimum moduli for $\mathbb{L}_{c}$) and $h_{M}$, $h_{m}$ (the maximum and minimum moduli for $\mathbb{H}$) such that \[{c_{m}}\\|X\\|^{2}\leq\langle\,\mathbb{C}.X,X\rangle\leq{c_{M}}\\|X \\|^{2} \text{for all }\ \ X\in{\rm{Sym}}(3),\] \[{(L_{c})}_{m}\\|X\\|^{2}\leq\langle\mathbb{L}_{c}.X,X\rangle\leq{(L _{c})}_{M}\\|X\\|^{2} \text{for all }\ \ X\in\mathbb{R}^{3\times 3},\] (2.20) \[h_{m}\\|X\\|^{2}\leq\langle\mathbb{H}.X,X\rangle\leq h_{M}\\|X\\|^{2} \text{for all }\ \ X\in{\rm{Sym}}(3),\] Further we assume, without loss of generality, that ${c_{M}}$, ${c_{m}}$, ${(L_{c})}_{M}$, $h_{M}$, $h_{m}$ and ${(L_{c})}_{m}$ are constants. Our new approach, in marked contrast to classical asymmetric micromorphic models, features a _symmetric Cauchy stress tensor_$\sigma=\mathbb{C}.\operatorname{sym}(\nabla u-P)$. Therefore, the linear Cosserat approach ([120]: $\mu_{c}>0$) is excluded here. Compared with Forest’s microstrain theory [55], the local force stress is similar, however, the micromorphic distortion $P$ in our new model is not necessarily symmetric but endowed with a weakest curvature response defined in terms of the _micro-dislocation_ tensor $\alpha=-\operatorname{Curl}P$. The skew symmetric part is uniquely determined by the solution $P$ of the boundary value problem. The relaxed formulation proposed in the present paper still shows size effects and smaller samples are relatively stiffer. It is clear to us that for this reduced model of relaxed micromorphic elasticity _unphysical effects of singular stiffening behaviour for small sample sizes_ (”bounded stiffness”, see [81]) _cannot appear_. In case of the isotropic Cosserat model this is only true for a reduced curvature energy depending only on $\\|\operatorname{dev}\operatorname{sym}\operatorname{Curl}P\\|^{2}$, see the discussion in [81, 130]. Remarkably, the necessary property of bounded stiffness is impossible to obtain for the indeterminate couple stress model (elastic energy $\sim\\|\operatorname{sym}\nabla u\\|^{2}+\\|\nabla\mathop{\rm skew}\nabla u\\|^{2}$, [81]). Whether bounded stiffness is true for the general strain gradient model (elastic energy $\sim\\|\operatorname{sym}\nabla u\\|^{2}+\\|\nabla\operatorname{sym}\nabla u\\|^{2}$, [81]) or the general gradient elasticity model (elastic energy $\sim\\|\operatorname{sym}\nabla u\\|^{2}+\\|D^{2}u\\|^{2}$) is unclear. The model introduced by Teisseyre [162] for the study of seismic wave propagation due to earthquake processes [44, 117, 118, 128] is also taking a symmetric relative stress tensor (see [162], p. 204 and 208) and in fact it is a particular case of the micromorphic approach to dislocation theory proposed by Eringen and Claus [31, 48, 32]. However, Teisseyre [162, 44, 117, 118] fails in choosing a positive definite dislocation energy, see Subsection 4.6. To our system of partial differential equations we adjoin the weaker boundary conditions⁹ (compare with the conditions (2.6)) [FOOTNOTE:9][ENDFOOTNOTE] \[{u}({x},t)=0,\ \ \ \text{and the {\it tangential condition}}\quad {P}_{i}({x},t)\times\,n(x)=0,\ \ \ i=1,2,3,\ \ \ \ ({x},t)\in\partial\Omega \times[0,T],\] (2.21) where $\times$ denotes the vector product, $n$ is the unit outward normal vector at the surface $\partial\Omega$ , $P_{i}$, $i=1,2,3$ are the rows of $P$. The model is driven by nonzero initial conditions \[{u}({x},0)={u}_{0}(x),\quad\quad\quad\dot{u}({x},0)=\dot{u}_{0}(x ),\quad\quad\quad{P}({x},0)={P}_{0}(x),\quad\quad\quad\dot{P}({x},0)=\dot{P}_{ 0}(x),\ \ \text{\ \ }{x}\in\overline{\Omega},\] (2.22) where ${u}_{0},\dot{u}_{0},{P}_{0}$ and $\dot{P}_{0}$ are prescribed functions, satisfying $u_{0}(x)=0$ and $P_{0i}(x)\times n(x)=0$ on $\partial\Omega$. Remark 2.1: _Since $P$ is determined in ${\rm H}({\rm Curl}\,;\Omega)$ in our relaxed model the only possible description of boundary value is in terms of tangential traces $P.\tau$. This follows from the standard theory of the ${\rm H}({\rm Curl}\,;\Omega)$-space, see [67]._ In contrast with the 7+11 parameters isotropic Mindlin and Eringen model [107, 49, 50], we have altogether only seven parameters $\mu_{e},\lambda_{e},\mu_{h},\lambda_{h},\alpha_{1},\alpha_{2}$, $\alpha_{3}$. For isotropic materials, our system reads \[u_{,tt} =\textrm{Div}\,\sigma+f\,,\] (2.23) \[P_{,tt} =-\operatorname{Curl}m+\sigma-s+M\,\ \ \ \text{in}\ \ \ \Omega \times[0,T].\] where \[\sigma =2\mu_{e}\operatorname{sym}(\nabla u-P)+\lambda_{e}\textrm{tr}( \nabla u-P){\,\!\cdot\!\,}1\!\!1,\] \[m =\alpha_{1}\operatorname{dev}\operatorname{sym}\operatorname{Curl }P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+\alpha_{3}\,\textrm{tr}( \operatorname{Curl}P){\,\!\cdot\!\,}1\!\!1,\] (2.24) \[s =2\mu_{h}\operatorname{sym}P+\lambda_{h}\textrm{tr}(P){\,\!\cdot \!\,}1\!\!1\,.\] Thus, we obtain the complete system of linear partial differential equations in terms of the kinematical unknowns $u$ and $P$ \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}(\nabla u-P)+\lambda_{e} \textrm{tr}(\nabla u-P){\,\!\cdot\!\,}1\!\!1]+f\,,\] (2.25) \[P_{,tt} =-\operatorname{Curl}[\alpha_{1}\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+ \alpha_{3}\,\textrm{tr}(\operatorname{Curl}P){\,\!\cdot\!\,}1\!\!1]\] \[\quad\ +2\mu_{e}\operatorname{sym}(\nabla u-P)+\lambda_{e}\textrm {tr}(\nabla u-P){\,\!\cdot\!\,}1\!\!1-2\mu_{h}\operatorname{sym}P-\lambda_{h} \textrm{tr}(P){\,\!\cdot\!\,}1\!\!1+M\,\ \ \ \text{in}\ \ \ \Omega\times[0,T].\] In this model, _the asymmetric parts of $P$_ are entirely due only to _moment stresses_ and _applied body moments_ ! In this sense, the _macroscopic_ and _microscopic scales_ are neatly _separated_. The positive definiteness required for the tensors $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{L}_{c}$ implies for isotropic materials the following restriction upon the parameters $\mu_{e},\lambda_{e},\mu_{h},\lambda_{h},\alpha_{1},\alpha_{2}$ and $\alpha_{3}$ \[\mu_{e}>0,\quad\quad 2\mu_{e}+3\lambda_{e}>0,\quad\quad\mu_{h}>0, \quad\quad 2\mu_{h}+3\lambda_{h}>0,\quad\quad\alpha_{1}>0,\quad\quad\alpha_{2} >0,\quad\quad\alpha_{3}>0.\] (2.26) Therefore, positive definiteness for our isotropic model does not involve extra nonlinear side conditions [47, 158]. In our relaxed model, exclusively, the material parameters $\mu_{e},\lambda_{e},\mu_{h},\lambda_{h}$ can even be uniquely determined from homogenization theory, see [119, 127, 79]: considering very large samples of an assumed heterogeneous structure, i.e. the characteristic length tends to zero, we must have [119, 127] \[\mu_{e}=\frac{\mu_{h}\,\mu}{\mu_{h}-\mu},\quad\quad\quad\quad 2 \mu_{e}+3\lambda_{e}=\frac{(2\mu_{h}+3\lambda_{h})\,(2\mu+3\lambda)}{(2\mu_{h} +3\lambda_{h})-(2\mu+3\lambda)}\,,\] (2.27) where $\lambda$, $\mu$ are the unique _macroscopic Lamé moduli_ obtained in classical experiments for large samples and $\lambda_{e},\mu_{e}$ are _isotropic scale transition parameters_ that control the interaction between the macro and the micro deformation. Thus, the macroscopic Lamé moduli $\lambda$ and $\mu$ should be always smaller than the _microstructural Lamé constants_$\mu_{h}$ and $\lambda_{h}$ related to the response of a representative volume element of the substructure. If, by neglect of our guiding assumption, we add the anti-symmetric term $2\mu_{c}\mathop{\rm skew}(\nabla u-P)$ in the expression of the Cauchy stress tensor $\sigma$, where $\mu_{c}\geq 0$ is the _Cosserat couple modulus¹⁰_, then our analysis also works for $\mu_{c}\geq 0$. The model in which $\mu_{c}>0$ is the isotropic Eringen-Claus model for dislocation dynamics [31, 48, 32] (see also the Subsection 4.4 and 4.3) and it is derived from the following free energy [FOOTNOTE:10][ENDFOOTNOTE] \[\mathcal{E}(e,\varepsilon_{p},\alpha) =\mu_{e}\\|\operatorname{sym}(\nabla u-P)\\|^{2}+\mu_{c}\\|\mathop{ \rm skew}(\nabla u-P)\\|^{2}+\frac{\lambda_{e}}{2}\,[\textrm{tr}(\nabla u-P)]^{ 2}+\mu_{h}\\|\operatorname{sym}P\\|^{2}+\frac{\lambda_{h}}{2}[\textrm{tr}\,(P)]^ {2}\] \[\quad\quad+\frac{\alpha_{1}}{2}\\|\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P\\|^{2}+\frac{\alpha_{2}}{2}\\|\mathop{\rm skew} \operatorname{Curl}P\\|+\frac{\alpha_{3}}{2}\,\textrm{tr}(\operatorname{Curl}P) ^{2}.\] (2.28) For $\mu_{c}>0$ and if the other inequalities (2.26) are satisfied, the existence and uniqueness follow along the classical lines. There is no need for any new integral inequalities. By means of a suitable decomposition (2.2) of the Mindlin-Eringen strain energy density, we are able to attribute to the unique parameter $\mu_{c}$ the asymmetry of the stress tensor in the isotropic case [120]. We believe that this unique parameter plays a fundamental role in the description of wave band-gaps in artificial metamaterials such as phononic crystals. Since the particular decomposition of the Mindlin-Eringen deformation energy for isotropic micromorphic media which we introduce in Eq. (2.2) allows for isolating few additional constitutive parameters with respect to standard Cauchy continuum theory, we may think to associate to each of these additional parameters a particular effect on wave propagation. Indeed, the search for wave solutions of the set of governing equations associated to the introduced micromorphic energy density may help to attribute a specific role to each of these parameters. An exhaustive treatment of wave propagation in relaxed Mindlin-Eringen media will be given in a forthcoming paper. Here, we limit ourselves to show the most characteristic features of the different elastic parameters introduced in this paper for the isotropic case. To do so, we summarize the basic role of the most important micromorphic parameters with respect to wave propagation: * The parameter $\mu_{h}$ (associated to the microstrain energy $\left\\|\mathrm{\text{\,}sym}\,{P}\,\right\\|^{2}$ in the energy density) regulates the propagation of acoustic waves inside the considered medium. More particularly, when setting $\mu_{h}=0$ it can be observed that no acoustic waves can propagate in the considered relaxed medium and hence only optic waves can propagate. This is sensible, since when considering the limit case $\mathrm{\text{\,}sym}\,{P}=\mathrm{\text{\,}sym}\,\nabla{u}$ our relaxed model reduces to a second gradient continuum in which $\mu_{h}$ is the only first gradient elastic parameter. It is indeed known that only acoustic waves can propagate in second gradient continua (see e.g. [38]). * When studying wave propagation phenomena in isotropic micromorphic media, the fact of accounting for the curvature dependence only via the parameters $\alpha_{1},\alpha_{2},\alpha_{3}$ (the terms involved in the energy density, multiplying $\left\\|\operatorname{dev}\operatorname{sym}\operatorname{Curl}{P}\,\right\\|^{2}$, $\left\\|\mathop{\rm skew}\operatorname{Curl}{P}\,\right\\|^{2}$ and $[\textrm{tr}(\operatorname{Curl}{P})]^{2}$, respectively) gives rise to dispersion curves (curves in the frequency/wavenumber plane) which have fixed concavity. This could, to some extent, make more difficult the fitting of the proposed relaxed model with some very particular classes of possible material behaviours. * It can be shown that the parameters $\alpha_{1},\alpha_{2},\alpha_{3}$ are related to the propagation of some particular optic waves. More particularly, when setting $\alpha_{1}=0,\,\alpha_{2}=0,\,\alpha_{3}=0$ in the considered relaxed model, no propagation is associated to the microdisplacement field ${P}$, which becomes an internal variable. Nevertheless, the global propagation inside the considered relaxed medium is not affected by the presence of $\alpha_{1},\alpha_{2},\alpha_{3}$, since macroscopic optic and acoustic waves can always propagate for all frequency ranges. * As far as a nonvanishing Cosserat couple modulus $\mu_{c}>0$ (which is associated to $\left\\|\mathrm{\text{\,}skew}\,\left(\nabla{u}-{P}\right)\,\right\\|^{2}$ in the energy density) is considered in the presented relaxed model, the micromorphic continuum starts exhibiting exotic properties which may be of use to describe the mechanical behavior of very particular metamaterials as lattice structures and phonon crystals. Indeed, when setting $\mu_{c}\neq 0$, the existence of frequency band-gaps is predicted by the considered micromorphic model. More particularly, when switching on the parameter $\mu_{c}$, there exist some frequency ranges in which neither acoustic nor optic waves can propagate. This means that, in these frequency ranges, only standing waves can exist which continue oscillating without propagating, thus keeping the energy trapped in the same region. We can conclude that the modeling of such exotic behavior is indeed directly related to the asymmetry of the stress tensor, at least for what concerns the linearized case. In the light of the aforementioned remarks, it is clear that the decomposition (2.2) of the strain energy density for the considered micromorphic media allows for a very effective identification of the elastic parameters and it may help in the identification of their physical meaning. Our model can also be compared with the model considered by Lazar and Anastassiadis [95]. In fact, in [95, 92] a simplified static version of the isotropic Eringen-Claus model for dislocation dynamics [31] has been investigated with $\mathbb{H}=0$ and $\mu_{c}>0$, with a focus on the gauge theory of dislocations (see Subsection 4.5). However, the dynamical theory of Lazar [94, 93] cannot be deduced from Mindlins dynamic theory, since in [94] there appears an additional gauge field which has no counterpart in Mindlins model. The theory proposed by Teisseyre in [162] is also using a symmetric force stress force and is a fully symmetric theory (see the assumption from [162], p. 204) which means that $\mu_{c}=0$[120, 128], see Subsection 4.6. However, for the mathematical treatment there arises the need for new integral type inequalities which we present in the next section. In the energy density given by Teisseyre, there exists a dislocation energy whose sign is not obvious. This is the reason why he did not take into account the influence of this energy. Using the new results established by Neff, Pauly and Witsch [135, 136, 137] and by Bauer, Neff, Pauly and Starke [7, 5, 6] we are now able to manage also energies depending on the dislocation energy and having symmetric Cauchy stresses [66]. ### Mathematical analysis In this subsection, for conciseness, we state only the obtained well-posedness results. The full proof of these mathematical results are included in [66]. The boundary–initial value problem defined by the equations (2.14), the boundary conditions (2.21) and the initial conditions (2.22) will be denoted by $(\mathcal{P})$. In order to establish an existence theorem for the solution of the problem $(\mathcal{P})$ we use the results of the semigroup theory of linear operators. First, we will rewrite the initial boundary value problem $({\mathcal{P}})$ as an abstract Cauchy problem in a Hilbert space [143, 167]. Let us define the space \[\mathcal{X}\,{=}\,\big{\{}\,w=(u,v,P,K)\,\|\,\ u{\in}{H}^{1}_{0}(\Omega),\quad v \in L^{2}(\Omega),\quad P\,{\in}\,H_{0}(\operatorname{Curl};\Omega),\quad K\in L ^{2}(\Omega)\big{\}}.\] Further, we introduce the operators $ A_{1}\,w=v,\quad A_{2}\,{w}=\textrm{Div}[\mathbb{C}. \operatorname{sym}(\nabla u-P)],\quad A_{3}\,{w}=K,$$A_{4}\,{w}=-\textrm{Curl}[\mathbb{L}_{c}.\textrm{Curl}\,P]+\mathbb{C}. \operatorname{sym}(\nabla u-P)-\mathbb{H}.\operatorname{sym}P\,,$ where all the derivatives of the functions are understood in the sense of distributions. Let $\mathcal{A}$ be the operator $\mathcal{A}=(A_{1},A_{2},A_{3},A_{4})$ with domain \[\mathcal{D}(\mathcal{A})=\{{w}=(u,v,P,K)\in\mathcal{X}\ \|\ \mathcal{A}{w}\in \mathcal{X}\}.\] With the above definitions, the problem $({\mathcal{P}})$ can be transformed into the following abstract equation in the Hilbert space $\mathcal{X}$ \[\frac{d{w}}{dt}(t)=\mathcal{A}{w}(t)+{\mathcal{F}}(t),\ \ {w}(0)={w}_{0},\] (2.29) where ${\mathcal{F}}(t)=\left({0},f,{0},M\right)$ and ${w}_{0}=(u_{0},\dot{u}_{0},P_{0},\dot{P}_{0}).$ Theorem 2.1: (Existence and uniqueness of the solution) _Assume that $f,M\,\in C^{1}([0,t_{1});L^{2}(\Omega))$, ${w}_{0}\in\mathcal{D}(\mathcal{A})$ and the fourth order elasticity tensors $\mathbb{C}$, $\mathbb{L}_{c}$ and $\mathbb{H}$ are symmetric and positive definite. Then, there exists a unique solution ${w}\!\in\!{C}^{1}((0,t_{1});\mathcal{X})\cap{C}^{0}([0,t_{1});\mathcal{D}( \mathcal{A}))$ of the Cauchy problem_ (2.29)._$\Box$_ Corollary 2.2: (Continuous dependence) _In the hypothesis of Theorem 2.1 we have the following estimate_ \[\begin{array}[]{crl}&&\\|{w}(t)\\|_{\mathcal{X}}\leq\\|{w}_{0}(t)\\|_{\mathcal{X}} +C\int_{0}^{t}\left(\\|f(s)\\|_{{L}^{2}(\Omega)}+\\|M(s)\\|_{{L}^{2}( \Omega)}\right)ds,\end{array}\] _where $C$ is a positive constant. $\Box$_ ## 3 Another further relaxed problem In this section, we weaken our energy expression further in the following model, where the corresponding elastic energy depends now only on the set of _independent constitutive variables_ \[\varepsilon_{e}=\operatorname{sym}(\nabla u-P),\quad\quad\quad \operatorname{dev}\varepsilon_{p}=\operatorname{dev}\operatorname{sym}P,\quad \quad\quad\operatorname{dev}\alpha=-\operatorname{dev}\operatorname{Curl}P.\] (3.1) In this model, it is neither implied that $P$ remains symmetric, nor that $P$ is trace-free, but only the trace free symmetric part of the micro-distortion $P$ and the trace-free part of the micro-dislocation tensor $\alpha$ contribute to the stored energy. ### Formulation of the problem The model in its general anisotropic form is: \[u_{,tt} ={\textrm{Div}}[\mathbb{C}.\operatorname{sym}(\nabla u-P)]+f\,,\] (3.2) \[P_{,tt} =-{\operatorname{Curl}}[\operatorname{dev}[\mathbb{L}_{c}. \operatorname{dev}\operatorname{Curl}P]]+\mathbb{C}.\operatorname{sym}(\nabla u -P)-\mathbb{H}.\operatorname{dev}\operatorname{sym}P+M\,\ \ \ \text{in}\ \ \ \Omega\times[0,T].\] In the isotropic case the model becomes \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}(\nabla u-P)+\lambda_{e} \textrm{tr}(\nabla u-P){\,\!\cdot\!\,}1\!\!1]+f\,,\] (3.3) \[P_{,tt} =-\operatorname{Curl}[\alpha_{1}\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P]\] \[\quad\ +2\mu_{e}\operatorname{sym}(\nabla u-P)+\lambda_{e}\textrm {tr}(\nabla u-P){\,\!\cdot\!\,}1\!\!1-2\mu_{h}\operatorname{dev}\operatorname{ sym}P+M\,\ \ \ \text{in}\ \ \ \Omega\times[0,T].\] To the system of partial differential equations of this model we adjoin the weaker boundary conditions \[{u}({x},t)=0,\ \ \ \quad\quad{P}_{i}({x},t)\times n(x)=0,\ \ \ i= 1,2,3,\ \ \ \ ({x},t)\in\partial\Omega\times[0,T],\] (3.4) and the nonzero initial conditions \[{u}({x},0)={u}_{0}(x),\quad\quad\quad\dot{u}({x},0)=\dot{u}_{0}(x ),\quad\quad\quad{P}({x},0)={P}_{0}(x),\quad\quad\quad\dot{P}({x},0)=\dot{P}_{ 0}(x),\ \ \text{\ \ }{x}\in\bar{\Omega},\] (3.5) where ${u}_{0},\dot{u}_{0},{P}_{0}$ and $\dot{P}_{0}$ are prescribed functions, satisfying $u_{0}(x)=0$ and $P_{0i}(x)\times n(x)=0$ on $\partial\Omega$. We remark again that $P$ is not trace-free in this formulation and no projection is performed, compare with Subsection 4.4 and 4.2. We denote the new problem defined by the above equations, the boundary conditions (3.4) and the initial conditions (3.5) by $(\widetilde{\mathcal{P}})$. ### Mathematical analysis The study of problem $(\widetilde{\mathcal{P}})$ follows along the same lines as in Subsection 2.3. We consider the operators $\widetilde{A}_{1}\,w=v,\quad\widetilde{A}_{2}\,{w}=\textrm{Div}[ \mathbb{C}.\operatorname{sym}(\nabla u-P)],\quad\widetilde{A}_{3}\,{w}=K,\quad \widetilde{A}_{4}\,{w}=-\textrm{Curl}[\operatorname{dev}[\mathbb{L}_{c}. \operatorname{dev}\textrm{Curl}\,P]]+\mathbb{C}.\operatorname{sym}(\nabla u-P)$$-\mathbb{H}.\operatorname{dev}\operatorname{sym}P\,,$ where all the derivatives of the functions are understood in the sense of distributions, and the operator $\widetilde{\mathcal{A}}=(\widetilde{A}_{1},\widetilde{A}_{2},\widetilde{A}_{3} ,\widetilde{A}_{4})$ with the domain $\mathcal{D}(\widetilde{\mathcal{A}})=\{{w}=(u,v,P,K)\in\mathcal{X}\ \|\ \widetilde{\mathcal{A}}{w}\in\mathcal{X}\}.$ Theorem 3.1: _Assume that $f,M\,\in C^{1}([0,t_{1});L^{2}(\Omega))$, ${w}_{0}\in\mathcal{D}(\widetilde{\mathcal{A}})$ and the fourth order elasticity tensors $\mathbb{C}$, $\mathbb{L}_{c}$ and $\mathbb{H}$ are symmetric and positive definite. Then, there exists a unique solution ${w}\!\in\!{C}^{1}((0,t_{1});\mathcal{X})\cap{C}^{0}([0,t_{1});\mathcal{D}( \widetilde{\mathcal{A}}))$ of the following Cauchy problem $\frac{d{w}}{dt}(t)=\widetilde{\mathcal{A}}{w}(t)+{\mathcal{F}}(t) ,\ \ {w}(0)={w}_{0},$ where ${\mathcal{F}}(t)=\left({0},f,{0},M\right)$ and ${w}_{0}=(u_{0},\dot{u}_{0},P_{0},\dot{P}_{0}).$ Moreover, we have the estimate_ \[\\|{w}(t)\\|_{\mathcal{X}}\leq\\|{w}_{0}(t)\\|_{\mathcal{X}}+C\int_{0 }^{t}\left(\\|f(s)\\|_{{L}^{2}(\Omega)}+\\|M(s)\\|_{{L}^{2}(\Omega)}\right)ds,\] _where $C$ is a positive constant. $\Box$_ ## 4 New and/or existing relaxed models In this section we propose a review of some existing relaxed models and we underline the possible connections between these models and the new relaxed models which we have proposed in this paper. ### Kröner’s view 4.1.1 Kröner’s discussion of a dislocated body and the Cosserat continuum: symmetric versus asymmetric force stresses Beginning from mid 1950 Kröner tried to link the theory of static dislocations to the Cosserat model with asymmetric force stresses. However, since 1964 it was clear to Kröner that the force stress $\sigma$ in the dislocation theory is always symmetric¹¹. [FOOTNOTE:11][ENDFOOTNOTE] We reproduce here the old, but nevertheless refreshing and clear comments of Kröner ([86, p. 1059-1060]) regarding the papers by Eringen and Claus [48], and Fox [58]. Kröner remarks: ”I would like to make clear why the skew symmetric stress does not appear in dislocated bodies. Assume particles which are little crystalline domains, for instance little cubes which build up a perfect crystal. Now imagine two of these particles to be isolated from the rest and be rotated through the same angle (Fig. 1(a)). By this operation the atomic structure is not disturbed and the state of the crystal along the interface between particles is not changed. So there is no static response to this kind of deformation and that is why the skew symmetric part of the ordinary stresses vanishes in dislocation theory. It does not vanish in Cosserat type theories where one considers oriented point particles which do not possess a crystalline structure (Fig. 1(b)). Such bodies could be, for instance, non-primitive crystal lattices where atoms in a cell are so tightly bound that the deformation of a cell can be disregarded whereas the bonds between the cells are weak. In this example the cells are the particles of the Cosserat continuum; they possess the usual translational and rotational degrees of freedom. Now rotate these particles through the same angle and the body is in a different state. So you expect a response. <figure><img src="content_image/1308.3219/x1.png"><figcaption>Figure 1: (a) Two adjacent ”particles” of a crystalline body before and aftera rotation through the same angle. This kind of rotation implies the slip of adislocation along the interface. It does not change the state of the crystal.(b) Four adjacent “particles” of a Cosserat type material before and after arotation through the same angle. This kind of rotation does change the stateof the body.</figcaption></figure> I call the body described firstly a _dislocated body_ and the other a _Cosserat continuum_. In the dislocated body one observes the occurrence of slip because the above described rotation of the two crystalline domains implies the slip of a dislocation along the interface between them. Slip has no meaning in the usual Cosserat continuum.” The comments of Teodosiu¹²[163, p. 1053-1054] regarding the papers by Eringen and Claus [48], and Fox [58] enforce the Kröner’s point of view. In order to defend their theory [48], Claus [169, p. 1054-1055] gave the following answered to Teodosiu’s comments: “If one includes extra degrees of freedom into the angular momentum equation, we claim that the equation lead to a non-symmetric stress tensor, whether it is a couple stress or a stress moment tensor. […] That is precisely the problem. Everybody these days is looking for situations in which the stress is non-symmetric. In continuum mechanics many people are trying to think along these lines. Some of the areas of promise to be pointed out are liquid crystal experiments where inherently there is a structure to the liquid which could conceivably lead to a non-symmetric stress tensor. Another area is in a body which contains a polarization, and the behavior of that body in an internal field. Many people are trying to look for asymmetries there. But I cannot quote an experimental paper where it has been demonstrated.[…] Concerning the question about elastic and plastic distortion, the interpretation here is that we have a body with dislocations that are deforming elastically so there is no slip in a lattice sense. There is no plastic deformation taking place; you put loads on the body and get only elastic reactions. Obviously what we are trying to construct is a plasticity theory, and we think we have the beginning of a mechanism to do that.” Moreover, Eringen said [169, p. 1054-1055]: “The ultimate goal of the present theory is to determine the motions and micromotions by solving an initial-boundary-value problem. Once they are determined, the dislocation density can be calculated in a straightforward manner. This point of view is, perhaps, in clash with the long-established traditions in other well-developed fields of continuum physics, I suggest that the continuum dislocation theory should offer a set of field equations subject to a set of well-posed initial and boundary conditions to predict the evolution of the motion and of the dislocations. Present practice in this field requires that the distribution of a second-oder tensor (the dislocation density) be given throughout the body at all times in order that we determine another second order tensor, namely, the stress tensor. This is not only unreasonable on logical grounds, but also not feasible experimentally. After all, why not ask for the stress tensor in the first place!”. [FOOTNOTE:12][ENDFOOTNOTE] 4.1.2 The Popov-Kröner dislocation model If we combine the dynamic model [150] of Popov with the Popov-Kröner static model of elastoplastic media with mesostructure [153, 151] we obtain the following equations \[{u}_{,tt} =\operatorname{Div}[{\mathbb{C}}.\,\operatorname{sym}(\nabla u-P) ]+f,\] (4.1) \[P_{,tt} =-\operatorname{Curl}(\alpha_{1}\,\operatorname{dev}\operatorname {sym}\operatorname{Curl}P+\alpha_{2}\,\mathop{\rm skew}\operatorname{Curl}P)+{ \mathbb{C}}.\,\operatorname{sym}(\nabla u-P)+M,\text{\ \ in \ \ }\Omega\times[ 0,T]\,,\] where $\alpha_{1},\alpha_{2}>0$. The Popov-Kröner model is derived from the internal free energy \[2\mathcal{E}(\varepsilon_{e},\alpha)=\langle\mathbb{C}.\, \operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P)\rangle+W_{ \operatorname{Curl}}(\alpha),\] (4.2) where \[W_{\operatorname{Curl}}(\alpha)=\mathfrak{a}_{1}\\|\alpha\\|^{2}+ \mathfrak{a}_{2}\langle\alpha,\alpha^{T}\rangle+\mathfrak{a}_{3}[\textrm{tr}( \alpha)]^{2},\] (4.3) and \[\mathfrak{a}_{1} =\frac{\mu(2d)^{2}}{24}\left(3+\frac{2\nu}{1-\nu}\right),\quad \quad\quad\mathfrak{a}_{2}=-\frac{\mu(2d)^{2}}{24}\frac{2\nu}{1-\nu},\quad \quad\quad\mathfrak{a}_{3}=-\frac{\mu(2d)^{2}}{24},\] (4.4) \[\nu =\frac{\lambda}{2(\mu+\lambda)}\quad\quad\quad\text{(the Poisson' s ratio)},\quad\quad\quad-1<\nu<\frac{1}{2}.\] The energy $W_{\operatorname{Curl}}$ can be expressed in terms of $\operatorname{dev}\operatorname{sym}\operatorname{Curl}P,\mathop{\rm skew} \operatorname{Curl}P$ and $\textrm{tr}(\operatorname{Curl}P)$ as in the following \[W_{\operatorname{Curl}}(\alpha) =(\mathfrak{a}_{1}+\mathfrak{a}_{2})\\|\operatorname{dev} \operatorname{sym}\operatorname{Curl}P\\|^{2}+(\mathfrak{a}_{1}-\mathfrak{a}_{2 })\\|\mathop{\rm skew}\operatorname{Curl}P\\|^{2}+\frac{\mathfrak{a}_{1}+ \mathfrak{a}_{2}+3\mathfrak{a}_{3}}{3}[\textrm{tr}(\operatorname{Curl}P)]^{2}\] \[=\frac{3\mu(2d)^{2}}{24}\\|\operatorname{dev}\operatorname{sym} \operatorname{Curl}P\\|^{2}+\frac{\mu(2d)^{2}}{24}\left(3+\frac{4\nu}{1-\nu} \right)\\|\mathop{\rm skew}\operatorname{Curl}P\\|^{2}.\] (4.5) We remark that $\textrm{tr}(\operatorname{Curl}P)$ is therefore, in fact, not present in the energy considered by Popov-Kröner [153, 150, 151] and in consequence it is also absent in the system of linear partial differential equations (4.1). Thus, the Popov-Kröner equations (4.1) coincide with our further relaxed model (3.3) in which $\mathbb{H}=0$ and \[\alpha_{1}=\frac{3\mu(2d)^{2}}{24},\quad\quad\ \alpha_{2}=\frac{ \mu(2d)^{2}}{24}\left(3+\frac{4\nu}{1-\nu}\right),\quad\quad\alpha_{3}=0.\] (4.6) In gradient plasticity models, there is another modeling issue at work: the microstrain $\operatorname{sym}P$ is not a state variable, therefore it should not appear in the free energy, as such $\mathbb{H}=0$. However, it may enter the equations through the notion of _equivalent plastic strain_, governing the isotropic linear hardening response [45]. Let us assume that $P$ is restricted to $\operatorname{\mathfrak{sl}}(3)$, which is the standard assumption in plasticity theory (plastic incompressibility, $\textrm{tr}(P)=0$). By subsequent orthogonal projection of the second equation (4.1)${}_{2}$ to the space of trace-free matrices, the full system of equations for the Popov-Kröner model [153, 150, 151] become \[{u}_{,tt} =\operatorname{Div}[{\mathbb{C}}.\,\operatorname{sym}(\nabla u- \operatorname{dev}P)]+f,\] (4.7) \[(\operatorname{dev}P)_{,tt} =-\operatorname{dev}[\operatorname{Curl}(\alpha_{1}\, \operatorname{dev}\operatorname{sym}\operatorname{Curl}\operatorname{dev}P+ \alpha_{2}\,\mathop{\rm skew}\operatorname{Curl}P)]+\operatorname{dev}[{ \mathbb{C}}.\,\operatorname{sym}(\nabla u-\operatorname{dev}P)]+\operatorname{ dev}M.\] The obtained model (4.7) is a 11 dof model, $(u,\operatorname{dev}P)$. In the isotropic case this is a 2+2 parameter model or a 2+1+2 parameter model if $\operatorname{dev}\operatorname{sym}P$ is taken into account as a constitutive variable. The constitutive variable $\operatorname{sym}P$ is not a state variable in this model. The model follows the line of argument given by Kröner¹³[85, p. 148]. [FOOTNOTE:13][ENDFOOTNOTE] ### Forest’s approach 4.2.1 Forest’s dynamic microstrain model In this Subsection we give a short description of the linear dynamic microstrain model [55]. The basic system of partial differential equation of this model can be obtained assuming that the micro-distortion $P$ is restricted to ${\rm{Sym}}(3)$ and by subsequent orthogonal projection of the second equation (2.1)${}_{2}$ (from the general Eringen’s micromorphic dynamics) to the space of symmetric-matrices. In addition, the ordinary elasticity tensor ${\mathbb{C}}:{\rm{Sym}}(3)\rightarrow{\rm{Sym}}(3)$ has to be taken, instead of Eringen’s elasticity tensor $\widehat{{\mathbb{C}}}$, since $\widehat{{\mathbb{C}}}$ does not map symmetric matrices into symmetric matrices. This leads to the system \[{u}_{,tt} =\operatorname{Div}[{\mathbb{C}}.\,\operatorname{sym}(\nabla u-P) ]+f,\] (4.8) \[(\operatorname{sym}{P})_{,tt} =\operatorname{sym}[\operatorname{Div}\widehat{\mathbb{L}}.\, \nabla(\operatorname{sym}P)]+{\mathbb{C}}.\,\operatorname{sym}(\nabla u-P)]+ \mathbb{H}.\,\operatorname{sym}P+\operatorname{sym}M,\text{\ \ in \ \ }\Omega \times[0,T]\,.\] The microstrain model is, however, incapable of describing rotation of the microstructure and features only $3+6$ degrees of freedom. For comparison we give the version without coupling terms. In fact, the mathematical problem for the microstrain model is to find functions $u\in H^{1}(\Omega)$ and $\varepsilon_{p}=\operatorname{sym}P\in H^{1}(\Omega)$ which satisfy the partial differential equations (4.13). A noteworthy feature of this model is a symmetric Cauchy-stress tensor $\sigma=\mathbb{C}.\operatorname{sym}(\nabla u-P)$. The curvature is only active on the gradient of microstrain, i.e. curvature depends only on $\nabla\varepsilon_{p}=\nabla(\operatorname{sym}P)$. Thus, the remaining set of _independent constitutive variables_ in the microstrain theory is \[\varepsilon_{e}=\operatorname{sym}(\nabla u-P),\quad\quad\quad \varepsilon_{p}=\operatorname{sym}P,\quad\quad\quad\nabla\varepsilon_{p}= \nabla\operatorname{sym}P.\] (4.9) The total energy corresponding to the microstrain micromorphic model is given by \[2\,\widehat{E}(t):=\int_{\Omega}\bigg{(}\\|u_{,t}\\|^{2} +\\|(\operatorname{sym}P)_{,t}\\|^{2}+\langle\mathbb{C}.\, \operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P)\rangle\] \[\ \ \ \ \quad\ +\langle\mathbb{H}.\,\operatorname{sym}P, \operatorname{sym}P\rangle+\langle\widehat{\mathbb{L}}.\,\nabla(\operatorname{ sym}P),\nabla(\operatorname{sym}P)\rangle\bigg{)}dv\,.\] (4.10) It is easy to obtain qualitative properties (uniqueness, continuous dependence, existence) of the microstrain micromorphic model because we only have to use the well known Korn’s inequality and the positive definiteness of $\mathbb{C},\mathbb{H},\widehat{\mathbb{L}}$[66], there is no need to specify Dirichlet boundary conditions on $\operatorname{sym}P$. 4.2.2 A microstrain-dislocation model without rotational degrees of freedom Let us consider now a new set of _independent constitutive variables_, i.e. \[\varepsilon_{e}=\operatorname{sym}(\nabla u-P),\quad\quad\quad \varepsilon_{p}=\operatorname{sym}P,\quad\quad\quad\operatorname{Curl} \varepsilon_{p}=\operatorname{Curl}\operatorname{sym}P,\] (4.11) and the corresponding total energy \[2\,\widehat{E}(t):=\int_{\Omega}\bigg{(}\\|u_{,t}\\|^{2} +\\|(\operatorname{sym}P)_{,t}\\|^{2}+\langle\mathbb{C}.\, \operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P)\rangle\] \[\ \ \ \ \quad\ +\langle\mathbb{H}.\,\operatorname{sym}P, \operatorname{sym}P\rangle+\langle{\widehat{\mathbb{L}}}_{c}.\,\operatorname{ Curl}\,\operatorname{sym}P,\operatorname{Curl}\,\operatorname{sym}P\rangle \bigg{)}dv\,.\] (4.12) Hence, the model equations are given by \[{u}_{,tt} =\operatorname{Div}[\,{\mathbb{C}}.\,\operatorname{sym}(\nabla u- P)]+f,\] (4.13) \[(\operatorname{sym}{P})_{,tt} =\operatorname{sym}[\operatorname{Curl}(\widehat{\mathbb{L}}_{c}. \,\operatorname{Curl}\,\operatorname{sym}P)]+{\mathbb{C}}.\,\operatorname{sym} (\nabla u-P)]+\mathbb{H}.\,\operatorname{sym}P+\operatorname{sym}M,\text{\ \ in \ \ }\Omega\times[0,T]\,.\] Existence and uniqueness follows along the lines given by our relaxed model, without the need for new inequalities. 4.2.3 The microcurl model The microcurl model is intended to furnish an approximation of a gradient plasticity model [33]. The free energy of the original system reads \[2\mathcal{E}_{\chi}(\varepsilon_{e},e_{p},\Gamma_{\chi})=\langle \mathbb{C}.\,\operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P) \rangle+\langle\mathbb{L}_{c}.\,\operatorname{Curl}P,\operatorname{Curl}P\rangle\] (4.14) and leads to some difficulties when one implements nonlinear pde-systems due to couplings with plasticity theory [138, 139, 140, 45, 126, 43]. The idea then is to introduce a new micromorphic-type variable $\chi_{p}$ and to couple it to elasto-plasticity. The _independent constitutive variables_ are the elastic strain tensor $\varepsilon_{e}=\operatorname{sym}(\nabla u-P)$, the relative plastic strain $e_{p}=P-\chi_{p}$ measuring the difference between plastic deformation and the plastic microvariable, and the dislocation density tensor $\Gamma_{\chi}=\operatorname{Curl}\chi_{p}$. The new free energy reads \[2\mathcal{E}_{\chi}(\varepsilon_{e},e_{p},\Gamma_{\chi})=\langle \mathbb{C}.\,\operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P) \rangle+\langle\mathbb{H}_{\chi}.\,(P-\chi_{p}),P-\chi_{p}\rangle+\langle \mathbb{L}_{\chi}.\,\operatorname{Curl}\chi_{p},\operatorname{Curl}\chi_{p} \rangle\,.\] (4.15) The quasistatic equations are \[0 =\textrm{Div}[\mathbb{C}.\operatorname{sym}(\nabla u-P)]\,,\] (4.16) \[0 =-\textrm{Curl}[\mathbb{L}_{c}.\textrm{Curl}\,\chi_{p}]+\mathbb{H }_{\chi}.(P-\chi_{p})\notin{\rm{Sym}}(3)\,\ \text{in general}\,,\] together with flow rules for the plastic variable $P$ (these are missing here). Since $\chi_{p}\in\mathbb{R}^{3\times 3}$, we have altogether 12 elastic degrees of freedom. Let us consider two alternative energies (with different coupling of $P$ and $\chi_{p}$) \[\mathcal{E}_{\chi}^{(1)} =\langle\mathbb{C}.\,\operatorname{sym}(\nabla u-P),\operatorname {sym}(\nabla u-P)\rangle+\langle\mathbb{L}_{\chi}.\operatorname{Curl}\chi_{p}, \operatorname{Curl}\chi_{p}\rangle+\langle\mathbb{H}_{\chi}.\,(P-\chi_{p}),P- \chi_{p}\rangle,\] (4.17) \[\mathcal{E}_{\chi}^{(2)} =\langle\mathbb{C}.\,\operatorname{sym}(\nabla u-P),\operatorname {sym}(\nabla u-P)\rangle+\langle\mathbb{L}_{\chi}.\operatorname{Curl}\chi_{p}, \operatorname{Curl}\chi_{p}\rangle+\langle\mathbb{H}_{\chi}.\,\operatorname{ sym}(P-\chi_{p}),\operatorname{sym}(P-\chi_{p})\rangle.\] The corresponding minimization problem in terms of the energy \[\left\{\begin{array}[]{l}\mathcal{E}_{\chi}^{(1)}\\ \mathcal{E}_{\chi}^{(2)}\end{array}\right.=\left\{\begin{array}[]{l}\text{has a unique solution }\chi_{p}\in{\rm H}(\operatorname{Curl},\Omega)\text{ for given }P\in L^{2}(\Omega),\\ \text{\quad\quad\quad\quad and there is no need for Dirichlet-boundary conditions (for uniqueness)},\\ \text{\quad\quad\quad\quad natural boundary conditions, determined by the variational formulation, suffice};\\ \text{has a solution }\chi_{p}\in{\rm H}(\operatorname{Curl},\Omega)\text{ for given }\operatorname{sym}P\in L^{2}(\Omega),\\ \text{\quad\quad\quad\quad uniqueness of }\chi_{p}\text{ requires tangential boundary conditions}.\end{array}\right.\] (4.18) ### The asymmetric isotropic Eringen-Claus model for dislocation dynamics This model is intended to describe a solid already containing dislocations undergoing elastic deformations: the dislocations bow out under the applied load, but do so reversibly. The system of equations derived by Eringen and Claus ([48], Eq. (3.39)) consists, as consequences of the balance laws of momentum and of the moment of momentum, of the following equations \[\varrho\,u_{l,tt} =\widetilde{\sigma}_{kl,k}+f_{l},\] (4.19) \[\varrho\,I{\,\!\cdot\!\,}P_{lm,tt} =\epsilon_{kmn}m_{nl,k}+\widetilde{\sigma}_{ml}-\widetilde{s}_{ml }+M_{lm},\] where (see the constitutive equations (3.32), (3.33) and (3.41) from [48] and the equations (36) and (37) from [161]) \[\widetilde{\sigma}_{kl} =(\overline{\lambda}+\tau)\,\varepsilon_{mm}\delta_{kl}+2( \overline{\mu}+\varsigma)\,\varepsilon_{kl}+\eta\,\overline{e}_{mm}\delta_{kl} +\overline{\nu}\,\overline{e}_{lk}+\kappa\,\overline{e}_{kl},\] \[\widetilde{s}_{kl} =(\overline{\lambda}+2\tau)\,\varepsilon_{mm}\delta_{kl}+2( \overline{\mu}+2\varsigma)\,\varepsilon_{kl}+(2\eta-\tau)\,\overline{e}_{mm} \delta_{kl}+(\overline{\nu}+\kappa-\varsigma)\,(\overline{e}_{kl}+\overline{e} _{lk}),\] (4.20) \[m_{kl} =-a_{3}\,\alpha_{mm}\delta_{kl}-a_{1}\,\alpha_{kl}+(a_{1}-a_{2}+a _{3})\,\alpha_{lk},\] and the set of _independent constitutive variables_ ([48], Eq. (1.7)) is \[\varepsilon=\operatorname{sym}\nabla u,\quad\quad\quad\overline{e }=\nabla u^{T}+P,\quad\quad\quad\alpha=-\operatorname{Curl}P.\] (4.21) The rest of the quantities have the same meaning as in Subsection 2.1. Let us remark that \[\varepsilon=\operatorname{sym}e+\varepsilon_{p},\quad\quad\quad \text{and}\quad\quad\quad\overline{e}=\operatorname{sym}e+2\varepsilon_{p}- \mathop{\rm skew}e\] (4.22) depend actually only on the _independent constitutive variables¹⁴_$e,\varepsilon_{p},\alpha$. [FOOTNOTE:14][ENDFOOTNOTE] We also remark that \[\epsilon_{kmn}m_{nl,k}=-\epsilon_{mkn}m_{nl,k}.\] (4.23) According with the definition of the $\operatorname{curl}$ operator, we have \[(\operatorname{curl}v)_{k}=\epsilon_{klm}v_{m,l},\ \ \text{for any vector}\ \ v=(v_{1},v_{2},v_{3})^{T}\in C^{1}(\Omega).\] (4.24) Hence, if we fix the indices $l$, then $\varepsilon_{ikn}m_{nl,k}$ gives the $i$-component of $\operatorname{curl}(m_{1l},m_{2l},m_{3l})$, i.e. \[(\epsilon_{kin}m_{nl,k})_{li}=-\big{(}\operatorname{Curl}(m^{T}) \big{)}_{li}.\] (4.25) Thus, written in terms of the operator $\operatorname{Curl}$, the constitutive equations (4.3)${}_{3}$ become \[m= a_{3}\,\textrm{tr}(\operatorname{Curl}P){\,\!\cdot\!\,}1\!\!1+2a _{1}\,\mathop{\rm skew}\operatorname{Curl}P+(a_{2}-a_{3})\,(\operatorname{Curl }P)^{T}.\] (4.26) In consequence, we deduce \[\operatorname{Curl}(m^{T}) =\operatorname{Curl}\bigg{[}(a_{2}-a_{3})\operatorname{Curl}P-2a_ {1}\,\mathop{\rm skew}\operatorname{Curl}P+a_{3}\,\textrm{tr}(\operatorname{ Curl}P)\!\cdot\!1\!\!1\bigg{]}\] (4.27) \[=\operatorname{Curl}\bigg{[}(a_{2}-a_{3})\,\operatorname{dev} \operatorname{sym}\operatorname{Curl}P+(a_{2}-a_{3}-2a_{1})\,\mathop{\rm skew} \operatorname{Curl}P+\frac{2a_{3}+a_{2}}{3}\textrm{tr}(\operatorname{Curl}P)\! \cdot\!1\!\!1\bigg{]}.\] We are thus able to identify the constitutive coefficients of the dislocation energy in the Eringen-Claus model [31, 48, 32] with the coefficients in our isotropic case, namely \[\alpha_{1}=a_{2}-a_{3},\quad\quad\ \alpha_{2}=a_{2}-a_{3}-2a_{1}, \quad\quad\ \ \alpha_{3}=\frac{2a_{3}+a_{2}}{3}.\] (4.28) Regarding the term $\widetilde{\sigma}_{ml}-\widetilde{s}_{ml}$ from the equations of motion (4.19), if we take $\overline{\nu}=\kappa$, then we have only the elastic strain tensor $\varepsilon_{e}=\operatorname{sym}e=\operatorname{sym}(\nabla u-P)$ and the micro-strain tensor $\varepsilon_{p}=\operatorname{sym}P$ taken into account. The condition $\overline{\nu}=\kappa$ is necessary and sufficient in order to have a symmetric force-stress tensor $\widetilde{\sigma}$ (see the discussion from Subsection 4.1), it corresponds to a vanishing Cosserat couple modulus $\mu_{c}=0$. Moreover, the force-stress tensor $\widetilde{\sigma}$ vanishes when $P=\nabla u$ if and only if $\overline{\mu}+\tau=-(\overline{\nu}+\kappa)$ and $\overline{\lambda}+\tau=-2\overline{\nu}$. However, Eringen and Claus strictly considered $\mu_{c}>0$, i.e. the asymmetry of the force stresses. ### The linear isotropic Cosserat model in terms of the dislocation density tensor In this subsection, we assume that the micro-distortion tensor is skew-symmetric, i.e. $P\in\operatorname{\mathfrak{so}}(3)$. For isotropic materials and with the asymmetric term $2\mu_{c}\mathop{\rm skew}(\nabla u-P)$ incorporated [16, 15], by orthogonal projection of the second equation (2.23)${}_{2}$ to the space of skew-symmetric matrices, the full system of equations for our reduced model is now \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}\nabla u+2\mu_{c}\mathop{ \rm skew}(\nabla u-(\mathop{\rm skew}P))+\lambda_{e}\textrm{tr}(\nabla u){\,\! \cdot\!\,}1\!\!1]+f\,,\] (4.29) \[(\mathop{\rm skew}P)_{,tt} =-\mathop{\rm skew}\operatorname{Curl}\bigg{[}\alpha_{1} \operatorname{dev}\operatorname{sym}\operatorname{Curl}(\mathop{\rm skew}P)+ \alpha_{2}\mathop{\rm skew}\operatorname{Curl}(\mathop{\rm skew}P)+\alpha_{3} \,\textrm{tr}(\operatorname{Curl}(\mathop{\rm skew}P)){\,\!\cdot\!\,}1\!\!1 \bigg{]}\] \[\ \ \ \ \ \ +2\mu_{c}\mathop{\rm skew}(\nabla u-(\mathop{\rm skew }P))+\mathop{\rm skew}M\,\ \ \ \text{in}\ \ \ \Omega\times[0,T].\] Then, switching to $A:=\mathop{\rm skew}P$, the equations (4.29) become \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}\nabla u+2\mu_{c}\mathop{ \rm skew}(\nabla u-A)+\lambda_{e}\textrm{tr}(\nabla u){\,\!\cdot\!\,}1\!\!1]+f\,,\] (4.30) \[A_{,tt} =-\mathop{\rm skew}\operatorname{Curl}\bigg{[}\alpha_{1} \operatorname{dev}\operatorname{sym}\operatorname{Curl}A+\alpha_{2}\mathop{\rm skew }\operatorname{Curl}A+\alpha_{3}\,\textrm{tr}(\operatorname{Curl}A){\,\!\cdot \!\,}1\!\!1\bigg{]}\] \[\ \ \ \ \ \ +2\mu_{c}\mathop{\rm skew}(\nabla u-A)+\mathop{\rm skew }M\,\ \ \ \text{in}\ \ \ \Omega\times[0,T].\] Moreover, for antisymmetric $A\in\operatorname{\mathfrak{so}}(3)$ the tangential boundary condition \[{A}_{i}({x},t)\cdot\tau(x)=0,\ \ \ i=1,2,3\ \ \ \ \ \ \text{ implies the strong anchoring condition }\ \ \ \ \ A=0\quad\text{on}\quad \partial\Omega.\] (4.31) We introduce the canonical identification of $\mathbb{R}^{3}$ with $\operatorname{\mathfrak{so}}(3)$. For \[A=\left(\begin{array}[]{ccc}0&-a_{3}&a_{2}\\ a_{3}&0&-a_{1}\\ -a_{2}&a_{1}&0\end{array}\right)\in\operatorname{\mathfrak{so}}(3)\] (4.32) we introduce the operators $\operatorname{axl}:\operatorname{\mathfrak{so}}(3)\rightarrow\mathbb{R}^{3}$ and $\operatorname{anti}:\mathbb{R}^{3}\rightarrow\operatorname{\mathfrak{so}}(3)$ through \[\operatorname{axl}\left(\begin{array}[]{ccc}0&-a_{3}&a_{2}\\ a_{3}&0&-a_{1}\\ -a_{2}&a_{1}&0\end{array}\right):=\left(\begin{array}[]{c}a_{1}\\ a_{2}\\ a_{3}\end{array}\right),\quad\quad A\cdot v=(\operatorname{axl}A)\times v, \quad\quad\forall v\in\mathbb{R}^{3},\] (4.33) \[\quad A_{ij}=\sum\limits_{k=1}^{3}-\epsilon_{ijk}(\operatorname{ axl}A)_{k}=:\operatorname{anti}(\operatorname{axl}A)_{ij},\quad\quad( \operatorname{axl}A)_{k}=\sum\limits_{i,j=1}^{3}-\frac{1}{2}\epsilon_{ijk}A_{ ij}\,,\] where $\epsilon_{ijk}$ is the totally antisymmetric third order permutation tensor. We also have the following identities (see [133], Nye’s formula [142] ) \[-\operatorname{Curl}A =(\nabla\operatorname{axl}A)^{T}-\textrm{tr}[(\nabla\operatorname {axl}A)^{T}]{\,\!\cdot\!\,}1\!\!1,\] (4.34) \[\nabla\operatorname{axl}A =-(\operatorname{Curl}A)^{T}+\frac{1}{2}\textrm{tr}[( \operatorname{Curl}A)^{T}]{\,\!\cdot\!\,}1\!\!1,\] (4.35) for all matrices $A\in\operatorname{\mathfrak{so}}(3)$. Using the above Curl-$\nabla\operatorname{axl}$ identity, it is simple to obtain \[\alpha_{1}\operatorname{dev}\operatorname{sym}\operatorname{Curl}A +\alpha_{2}\mathop{\rm skew}\operatorname{Curl}A+\alpha_{3}\, \textrm{tr}(\operatorname{Curl}A){\,\!\cdot\!\,}1\!\!1=\] (4.36) \[-\alpha_{1}\operatorname{dev}\operatorname{sym}(\nabla \operatorname{axl}A)^{T}-\alpha_{2}\mathop{\rm skew}(\nabla\operatorname{axl}A )^{T}-\alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A)^{T}{\,\!\cdot\!\,}1\! \!1+\] \[+\alpha_{1}\operatorname{dev}\operatorname{sym}\textrm{tr}[( \nabla\operatorname{axl}A)^{T}]{\,\!\cdot\!\,}1\!\!1+\alpha_{2}\mathop{\rm skew }\textrm{tr}[(\nabla\operatorname{axl}A)^{T}]{\,\!\cdot\!\,}1\!\!1+\alpha_{3} \,\textrm{tr}(\textrm{tr}[(\nabla\operatorname{axl}A)^{T}]{\,\!\cdot\!\,}1\!\! 1){\,\!\cdot\!\,}1\!\!1\] \[=-\alpha_{1}\operatorname{dev}\operatorname{sym}(\nabla \operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla\operatorname{axl}A)- \alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\!\cdot\!\,}1\!\!1+3 \alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\!\cdot\!\,}1\!\!1\] \[=-\alpha_{1}\operatorname{dev}\operatorname{sym}(\nabla \operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla\operatorname{axl}A)+2 \alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\!\cdot\!\,}1\!\!1.\] Hence, we have after multiplication with $A_{,t}$ \[\langle\operatorname{Curl}[\alpha_{1}\operatorname{dev} \operatorname{sym}\operatorname{Curl}A+\alpha_{2} \mathop{\rm skew}\operatorname{Curl}A+\alpha_{3}\,\textrm{tr}( \operatorname{Curl}A){\,\!\cdot\!\,}1\!\!1],A_{,t}\rangle\] (4.37) \[=\langle\mathop{\rm skew}\operatorname{Curl}[\alpha_{1} \operatorname{dev}\operatorname{sym} \operatorname{Curl}A+\alpha_{2}\mathop{\rm skew}\operatorname{ Curl}A+\alpha_{3}\,\textrm{tr}(\operatorname{Curl}A){\,\!\cdot\!\,}1\!\!1],A_{ ,t}\rangle\] \[=\langle\mathop{\rm skew}\operatorname{Curl}[-\alpha_{1} \operatorname{dev} \operatorname{sym}(\nabla\operatorname{axl}A)+\alpha_{2}\mathop{ \rm skew}(\nabla\operatorname{axl}A)+2\alpha_{3}\,\textrm{tr}(\nabla \operatorname{axl}A){\,\!\cdot\!\,}1\!\!1],A_{,t}\rangle\] \[=\langle\operatorname{Curl}[-\alpha_{1}\operatorname{dev} \operatorname{sym} (\nabla\operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla \operatorname{axl}A)+2\alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\! \cdot\!\,}1\!\!1],A_{,t}\rangle\] which, using the strong anchoring boundary conditions (4.31), implies \[\int_{\Omega} \langle\alpha_{1}\operatorname{dev}\operatorname{sym} \operatorname{Curl}A+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}A+\alpha_{3 }\,\textrm{tr}(\operatorname{Curl}A){\,\!\cdot\!\,}1\!\!1,\] \[\quad\quad\quad\operatorname{dev}\operatorname{sym}\operatorname{ Curl}A_{,t}+\mathop{\rm skew}\operatorname{Curl}A_{,t}+\frac{1}{3}\textrm{tr}( \operatorname{Curl}A_{,t})\rangle\,dv\] (4.38) \[=\int_{\Omega}\langle-\alpha_{1}\operatorname{dev}\operatorname{ sym}(\nabla\operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla \operatorname{axl}A)+2\alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\! \cdot\!\,}1\!\!1,\operatorname{Curl}A_{,t}\rangle\,dv.\] Moreover, we deduce that \[\frac{1}{2}\frac{d}{dt}\int_{\Omega}\bigg{(}\alpha_{1}\\| \operatorname{dev}\operatorname{sym}\operatorname{Curl}A\\|^{2}+\alpha_{2}\\| \mathop{\rm skew}\operatorname{Curl}A\\|+{\alpha_{3}}\,\textrm{tr}( \operatorname{Curl}A)^{2}\bigg{)}\,dv\] \[=\int_{\Omega}\langle-\alpha_{1}\operatorname{dev}\operatorname{ sym}(\nabla\operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla \operatorname{axl}A)+2\alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\! \cdot\!\,}1\!\!1,-(\nabla\operatorname{axl}A_{,t})^{T}+\textrm{tr}[(\nabla \operatorname{axl}A_{,t})^{T}]{\,\!\cdot\!\,}1\!\!1\rangle\,dv\] \[=\int_{\Omega}\langle-\alpha_{1}\operatorname{dev}\operatorname{ sym}(\nabla\operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla \operatorname{axl}A)+2\alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\! \cdot\!\,}1\!\!1,\] \[\ \ \ \ \ -\operatorname{dev}\operatorname{sym}(\nabla \operatorname{axl}A_{,t})^{T}-\mathop{\rm skew}(\nabla\operatorname{axl}A_{,t} )^{T}-\frac{1}{3}\textrm{tr}[(\nabla\operatorname{axl}A_{,t})^{T}]{\,\!\cdot\! \,}1\!\!1+\textrm{tr}[(\nabla\operatorname{axl}A_{,t})^{T}]{\,\!\cdot\!\,}1\! \!1\rangle\,dv\] \[=\int_{\Omega}\langle-\alpha_{1}\operatorname{dev}\operatorname{ sym}(\nabla\operatorname{axl}A)+\alpha_{2}\mathop{\rm skew}(\nabla \operatorname{axl}A)+2\alpha_{3}\,\textrm{tr}(\nabla\operatorname{axl}A){\,\! \cdot\!\,}1\!\!1,\] (4.39) \[\ \ \ \ \ -\operatorname{dev}\operatorname{sym}(\nabla \operatorname{axl}A_{,t})^{T}-\mathop{\rm skew}(\nabla\operatorname{axl}A_{,t} )^{T}+\frac{2}{3}\textrm{tr}[(\nabla\operatorname{axl}A_{,t})^{T}]{\,\!\cdot\! \,}1\!\!1\rangle\,dv\] \[=\frac{1}{2}\frac{d}{dt}\int_{\Omega}\bigg{(}\alpha_{1}\\| \operatorname{dev}\operatorname{sym}(\nabla\operatorname{axl}A)\\|^{2}+\alpha_{ 2}\\|\mathop{\rm skew}(\nabla\operatorname{axl}A)\\|^{2}+{4}\alpha_{3}\,[\textrm {tr}(\nabla\operatorname{axl}A)]^{2}\bigg{)}\,dv.\] Because $A$ is skew-symmetric, it is completely defined by its axial vector $\operatorname{axl}A$ and we have \[\operatorname{sym}A=0,\quad\quad\textrm{tr}(A)=0,\quad\quad\\|A\\|^ {2}=2\\|\operatorname{axl}A\\|^{2},\quad\quad\textrm{tr}(\operatorname{Curl}A)=2 \,\textrm{tr}(\nabla\operatorname{axl}A)\] \[\\|\mathop{\rm skew}(\nabla u-A)\\|^{2}=2\\|\operatorname{axl}( \mathop{\rm skew}\nabla u)-\operatorname{axl}A\\|^{2}=\frac{1}{2}\\| \operatorname{curl}\,u-2\operatorname{axl}A\\|^{2}.\] (4.40) Then, the total energies \[\mathcal{L}_{1} (u_{,t},A_{,t},\nabla u-A,\operatorname{sym}A,\operatorname{Curl}A)\] \[=\int_{\Omega}\bigg{(}\frac{1}{2}\\|u_{,t}\\|^{2}+\frac{1}{2}\\|A_{, t}\\|^{2}+\mu_{e}\\|\operatorname{sym}\nabla u\\|^{2}+\mu_{c}\\|\mathop{\rm skew}( \nabla u-A)\\|^{2}+\frac{\lambda_{e}}{2}\textrm{tr}(\nabla u)^{2}\] (4.41) \[\ \ \ \ \ \ \ \ \ \ \quad\quad+\frac{\alpha_{1}}{2}\\| \operatorname{dev}\operatorname{sym}\operatorname{Curl}A\\|^{2}+\frac{\alpha_{2 }}{2}\\|\mathop{\rm skew}\operatorname{Curl}A\\|+\frac{\alpha_{3}}{2}\,[\textrm{ tr}(\operatorname{Curl}A)]^{2}\bigg{)}\,dv,\] and \[\mathcal{L}_{2} (u_{,t},(\operatorname{axl}A)_{,t},\nabla u-A,\operatorname{axl}A)\] \[=\int_{\Omega}\bigg{(}\frac{1}{2}\\|u_{,t}\\|^{2}+\\|(\operatorname{ axl}A)_{,t}\\|^{2}+\mu_{e}\\|\operatorname{sym}\nabla u\\|^{2}+\mu_{c}\\|\mathop{ \rm skew}(\nabla u-A)\\|^{2}+\frac{\lambda_{e}}{2}[\textrm{tr}(\nabla u)]^{2}\] (4.42) \[\ \ \ \ \ \ \ \ \ \ \quad\quad+\frac{\alpha_{1}}{2}\\| \operatorname{dev}\operatorname{sym}(\nabla\operatorname{axl}A)\\|^{2}+\frac{ \alpha_{2}}{2}\\|\mathop{\rm skew}(\nabla\operatorname{axl}A)\\|^{2}+{2}\alpha_{ 3}\,[\textrm{tr}(\nabla\operatorname{axl}A)]^{2}\bigg{)}\,dv\] are equivalent and lead to equivalent Euler-Lagrange equations. The power function is given by \[\Pi(t) =\int_{\Omega}(\langle f,{u}_{t}\rangle+\langle M,A_ {t}\rangle)\,dv\,=\int_{\Omega}(\langle f,{u}_{t}\rangle+\langle \mathop{\rm skew}M,A_{t}\rangle)\,dv\] (4.43) \[=\int_{\Omega}(\langle f,{u}_{t}\rangle+2\langle \operatorname{axl}\mathop{\rm skew}M,\operatorname{axl}A_{t}\rangle)\,dv\,.\] In conclusion, in view of (4.4), the Euler-Lagrange equation gives us the following system of partial differential equations for $u$ and $A$ \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}\nabla u+2\mu_{c}\mathop{ \rm skew}(\nabla u-A)+\lambda_{e}\textrm{tr}(\nabla u){\,\!\cdot\!\,}1\!\!1]+f\,,\] \[\,(\operatorname{axl}A)_{,tt} =\operatorname{Div}\bigg{[}\frac{\alpha_{1}}{2}\operatorname{dev} \operatorname{sym}(\nabla\operatorname{axl}A)+\frac{\alpha_{2}}{2}\mathop{\rm skew }(\nabla\operatorname{axl}A)+{2}\alpha_{3}\,\textrm{tr}(\nabla\operatorname{ axl}A){\,\!\cdot\!\,}1\!\!1\bigg{]}\] \[\ \ \ \ \ \ +2\mu_{c}\operatorname{axl}(\mathop{\rm skew}\nabla u -A)+\operatorname{axl}\mathop{\rm skew}M\,\ \ \ \text{in}\ \ \ \Omega\times[0, T],\] which is completely equivalent with the system (4.29). In the case of the Cosserat theory we must put $\mu_{e}=\mu$ and $\lambda_{e}=\lambda$, where $\mu$ and $\lambda$ are the Lamé constants from classical elasticity¹⁵. The set of _independent constitutive variables_ for the Cosserat model is [FOOTNOTE:15][ENDFOOTNOTE] \[{e}=\nabla u-A,\quad\quad\quad{\alpha}=-\operatorname{Curl}A.\] (4.44) In terms of the microrotation vector $\vartheta=\operatorname{axl}A$, the above system turns into the classical format \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}\nabla u+2\mu_{c}(\mathop {\rm skew}\nabla u-\operatorname{anti}(\vartheta))+\lambda_{e}\textrm{tr}( \nabla u){\,\!\cdot\!\,}1\!\!1]+f\,,\] \[\,\vartheta_{,tt} =\operatorname{Div}\bigg{[}\frac{\alpha_{1}}{2}\operatorname{dev} \operatorname{sym}\nabla\vartheta+\frac{\alpha_{2}}{2}\mathop{\rm skew}\nabla \vartheta+{2}\alpha_{3}\,\textrm{tr}(\nabla\vartheta){\,\!\cdot\!\,}1\!\!1 \bigg{]}\] \[\ \ \ \ \ \ +2\mu_{c}\big{[}\operatorname{axl}(\mathop{\rm skew} \nabla u)-\vartheta\big{]}+\operatorname{axl}\mathop{\rm skew}M\,\ \ \ \text{ in}\ \ \ \Omega\times[0,T],\] We point out that for the static case and for $\mu_{c}>0$ in this model, existence and uniqueness can be shown for a very weak curvature energy, namely for $\alpha_{1}>0$, $\alpha_{2},\alpha_{3}\geq 0$, see [81]. For $\mu_{c}=0$ in the linear Cosserat model, the system uncouples. This is another artefact of the linear Cosserat model. Let us remark that if we relax the isotropic energy from the gradient elasticity formulation [41, 160, 113] \[\mathcal{E}(\nabla u,\nabla(\mathop{\rm skew}\nabla u))=\mu\,\\|\operatorname{ dev}\operatorname{sym}\nabla u\\|^{2}+\frac{2\mu+3\lambda}{6}\,[\textrm{tr}( \nabla u)]^{2}+\mu L_{c}^{2}\,\\|\nabla(\mathop{\rm skew}\nabla u)\\|^{2},\] (4.45) corresponding to the _indeterminate couple stress problem_, such that \[\mathcal{E}(\nabla u,A,\nabla A)=\mu\,\\|\operatorname{dev}\operatorname{sym} \nabla u\\|^{2}+\frac{2\mu+3\lambda}{6}\,[\textrm{tr}(\nabla u)]^{2}+\mu L_{c}^ {2}\,\\|\nabla A\\|^{2}+\varkappa^{+}\mu\,\\|\mathop{\rm skew}\nabla u-A\\|^{2},\] (4.46) where $\varkappa^{+}$ is a dimensionless penalty coefficient, then we obtain the isotropic Cosserat model. The coefficient $\varkappa^{+}\mu=\mu_{c}$ is the Cosserat couple modulus. We observe that no mixed terms appear. ### Lazar’s translational gauge theory of dislocations The static equations used by Lazar and Anastassiadis [95, 91] in the isotropic gauge theory of dislocations can be expressed as \[0= \operatorname{Div}[2{\mu}_{e}\operatorname{sym}( \nabla u-{P})+2{\mu_{c}}\mathop{\rm skew}(\nabla u-{P})+{\lambda}_{e}\,\textrm {tr}(\nabla u-{P}){\,\!\cdot\!\,}1\!\!1]+f,\] \[\sigma^{0}= -\operatorname{Curl}[\alpha_{1}\operatorname{dev} \operatorname{sym}(\operatorname{Curl}{P})+\alpha_{2}\mathop{\rm skew}( \operatorname{Curl}P)+{\alpha_{3}}\textrm{tr}(\operatorname{Curl}{P}){\,\! \cdot\!\,}1\!\!1]\] (4.47) \[+2{\mu}_{e}\operatorname{sym}(\nabla u-{P})+2{\mu_{c }}\mathop{\rm skew}(\nabla u-{P})+{\lambda}_{e}\,\textrm{tr}(\nabla u-{P}){\, \!\cdot\!\,}1\!\!1\,,\] where the coefficients $\alpha_{1},\,\alpha_{2},\,\alpha_{3}$ correspond to $a_{1},\,a_{2},\,\frac{a_{3}}{3}$ from the Lazar’s notations, $\sigma^{0}$ is a statically admissible background field (the body moment tensor $M$ in the Eringen-Claus model (4.19)), i.e. \[\operatorname{Div}\sigma_{0}+f=0,\quad\quad\quad\sigma_{0}.\,n\|_{ \partial\Omega\setminus\Gamma}=N\,,\] (4.48) with $N$ prescribed. Lazar and Anastassiadis have decomposed the dislocation tensor $\operatorname{Curl}P$ into its ${\rm SO}(3)$-irreducible pieces, “the axitor”, “the tentor” and “the trator” parts, i.e. \[\operatorname{Curl}P =\underbrace{\operatorname{dev}\operatorname{sym}(\operatorname{ Curl}{P})}_{\textrm{``tentor"}}+\underbrace{\mathop{\rm skew}(\operatorname{ Curl}P)}_{\textrm{``trator"}}+\underbrace{\frac{1}{3}\textrm{tr}(\operatorname {Curl}{P}){\,\!\cdot\!\,}1\!\!1}_{\textrm{``axitor"}}\,.\] (4.49) It is clear that the Lazar’s model [95] is a simplified static version of the asymmetric isotropic Eringen-Claus model for dislocation dynamics [31] (see the Subsection 4.3) with $\mathbb{H}=0$ and $\mu_{c}>0$. The tensor $\mathbb{H}$ is absent since the term $\langle\mathbb{H}.\,\operatorname{sym}P,\operatorname{sym}P\rangle$ is not translation gauge invariant. In [95] various special solutions to (4.5) for screw and edge dislocations are constructed. Abbreviating $\beta_{e}:=\nabla u-P\in\mathbb{R}^{3\times 3}$ the system is equivalent to the Euler-Lagrange equations of \[\int_{\Omega}\bigg{[} \mu_{e}\\|\operatorname{sym}\beta_{e}\\|^{2}+\mu_{c}\\|\mathop{\rm skew }\beta_{e}\\|^{2}+\frac{\lambda_{e}}{2}[\textrm{tr}(\beta_{e})]^{2}\] \[+\frac{\alpha_{1}}{2}\\|\operatorname{dev}\operatorname{sym} \operatorname{Curl}\beta_{e}\\|^{2}+\frac{\alpha_{2}}{2}\\|\operatorname{dev} \operatorname{sym}\operatorname{Curl}\beta_{e}\\|^{2}+\frac{\alpha_{3}}{2}[ \textrm{tr}(\operatorname{Curl}\beta_{e})]^{2}\] (4.50) \[+\langle\sigma_{0},\beta_{e}\rangle\bigg{]}dv\quad\to \quad\min.\ \beta_{e},\quad\quad\quad\quad\beta_{e}\cdot\tau=0\quad\text{ on } \quad\Gamma\subset\partial\Omega.\] In the variational formulation, the dislocation model can be seen as an elastic (reversible) description of a material, which may respond to external loads by an elastic distortion field $\beta_{e}$ which is not anymore a gradient (incompatible). This is not yet an irreversible plasticity formulation, since elasticity does not change the state of the body. The Euler-Lagrange equations turn out to be \[-f =\operatorname{Div}\sigma_{0}= \operatorname{Div}[2{\mu}_{e}\operatorname{sym}\beta_{e}+2{\mu_{c}}\mathop{\rm skew }\beta_{e}+{\lambda}_{e}\,\textrm{tr}(\beta_{e}){\,\!\cdot\!\,}1\!\!1],\] \[\sigma^{0} =\operatorname{Curl}[\alpha_{1}\operatorname{dev} \operatorname{sym}(\operatorname{Curl}{\beta_{e}})+\alpha_{2}\mathop{\rm skew} (\operatorname{Curl}\beta_{e})+{\alpha_{3}}\textrm{tr}(\operatorname{Curl}{ \beta_{e}}){\,\!\cdot\!\,}1\!\!1]\] (4.51) \[\quad+2{\mu}_{e}\operatorname{sym}\beta_{e}+2{\mu_{c }}\mathop{\rm skew}\beta_{e}+{\lambda}_{e}\,\textrm{tr}(\beta_{e}){\,\!\cdot\! \,}1\!\!1\,.\] We will deal with the well-posedness for the minimization problem (4.5) in another work. ### The symmetric earthquake structure model of Teisseyre Teisseyre [161, 162] followed closely the approach to micromorphic continuum theory developed by Suhubi and Eringen [49] and by Eringen and Claus [48]. In fact he used the equations of motion given by Eringen and Claus ([48], Eq. (3.39)) written in terms of the divergence operator (see Eqs. (1)-(4) from [161] and also [48]) \[\varrho\,u_{l,tt} =\widetilde{\sigma}_{kl,k}+f_{l},\] (4.52) \[P_{lm,tt} =\Lambda_{plm,p}+\widetilde{\sigma}_{ml}-\widetilde{s}_{ml}+M_{lm},\] where (see the constitutive equations (36)-(38) from [161]) \[\widetilde{\sigma}_{kl} =(\overline{\lambda}+\tau)\,\varepsilon_{mm}\delta_{kl}+2( \overline{\mu}+\varsigma)\,\varepsilon_{kl}+\eta\overline{e}_{mm}\delta_{kl}+ \overline{\nu}\,\overline{e}_{lk}+\kappa\,\overline{e}_{kl},\] \[\widetilde{s}_{kl} =(\overline{\lambda}+2\tau)\,\varepsilon_{mm}\delta_{kl}+2( \overline{\mu}+2\varsigma)\,\varepsilon_{kl}+(2\eta-\tau)\,\overline{e}_{mm} \delta_{kl}+(\overline{\nu}+\kappa-\varsigma)\,(\overline{e}_{kl}+\overline{e} _{lk}),\] (4.53) \[\Lambda_{plk} =a_{1}\,\alpha_{rn}(\epsilon_{prn}\delta_{kl}-\epsilon_{krn} \delta_{pl})+a_{2}\,\epsilon_{pkn}\alpha_{ln}+a_{3}\,(\epsilon_{pln}\alpha_{kn }-\epsilon_{kln}\alpha_{pn}),\] the _constitutive variables_$\varepsilon_{kl},\overline{e}_{kl}$ and $\alpha_{kl}$ are the same as in the Eringen-Claus theory (see Subsection 4.3) and the rest of the quantities have the same signification as in Subsection 2.1. For simplicity, the system (4.52) is considered in a normalized form. The constitutive equations and the equation of motion are the same as in the Eringen-Claus model [48]: in fact, using that $\Lambda_{klm}=-\Lambda_{mlk}$ from (4.6)${}_{3}$, Eringen and Claus considered the tensor \[m_{kl}=\frac{1}{2}\epsilon_{kmn}\Lambda_{mln}\] (4.54) and rewrote the equation (4.52)${}_{3}$ in the following format \[P_{lm,tt} =\epsilon_{kmn}m_{nl,k}+\widetilde{\sigma}_{ml}-\widetilde{s}_{ml }+M_{lm}\Leftrightarrow P_{,tt}=-\operatorname{Curl}(m^{T})+\widetilde{\sigma} ^{T}-\widetilde{s}^{T}+M\,.\] (4.55) Moreover, because the tensor $m_{kl}$ is given by \[m_{kl} =-a_{3}\alpha_{mm}\delta_{kl}-a_{1}\alpha_{kl}+(a_{1}-a_{2}+a_{3} )\alpha_{lk},\] (4.56) the equations of motion give us an energy whose form can be found following the Subsection 4.3. More precisely, the energy \[\mathcal{K}_{1}(P)=\langle\Lambda,\nabla P\rangle,\] (4.57) from [162] is in fact the energy \[\mathcal{K}_{2}(P) =\langle m,\operatorname{Curl}P\rangle\] \[=(a_{2}-a_{3})\\|\operatorname{dev}\operatorname{sym}( \operatorname{Curl}{P})\\|^{2}+(a_{2}-a_{3}-2a_{1})\\|\mathop{\rm skew}( \operatorname{Curl}P)\\|^{2}+\frac{2a_{3}+a_{2}}{3}\,[\textrm{tr}(\operatorname {Curl}{P})]^{2}\] \[=\alpha_{1}\\|\operatorname{dev}\operatorname{sym}(\operatorname{ Curl}{P})\\|^{2}+\alpha_{2}\\|\mathop{\rm skew}(\operatorname{Curl}P)\\|^{2}+ \alpha_{3}\,[\textrm{tr}(\operatorname{Curl}{P})]^{2}\,.\] (4.58) The first assumption of Teisseyre is that $\kappa=\overline{\nu}$ which implies that the force stress tensor $\widetilde{\sigma}$ is symmetric. Imposing the additional assumption that the moments of rotations have to vanish he also requires that the corresponding differences between the stress moment components and body couples appearing in the equation vanish. This is the reason why he assumed that \[\Lambda_{plk,p}=\Lambda_{pkl,p},\quad\quad\quad M_{lk}=M_{kl}\,.\] (4.59) If (4.59) are satisfied, we see immediately that (4.52) determines $P_{,tt}$ to be symmetric. Using the identity \[\Lambda_{klm}=\epsilon_{kmn}m_{nl}\] (4.60) the symmetry constraint $\Lambda_{plk,p}=\Lambda_{pkl,p}$, can be rewritten in terms of $m$, i.e. \[(\operatorname{Curl}(m^{T}))_{ml} =\epsilon_{lkn}m_{nm,k}=-\epsilon_{kln}m_{nm,k}=-\Lambda_{kml,k}\] (4.61) \[=-\Lambda_{klm,k}=-\epsilon_{kmn}m_{nl,k}=\epsilon_{mkn}m_{nl,k}= (\operatorname{Curl}(m^{T}))_{lm}\,.\] In other words, (4.59)${}_{1}$ demands that $m$ is such that \[\operatorname{Curl}(m^{T})\in{\rm{Sym}}(3).\] (4.62) Obviously, $P_{,tt}$ symmetric does not imply that $P$ must be symmetric. Hence, in view of (4.26) the constraint (4.62) means that \[\operatorname{Curl}[\alpha_{1}\operatorname{dev}\operatorname{sym }\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+\alpha_{ 3}\,\textrm{tr}(\operatorname{Curl}P){\,\!\cdot\!\,}1\!\!1]\in{\rm{Sym}}(3)\,,\] (4.63) and further that \[\operatorname{Curl}\{\alpha_{1}\operatorname{dev}\operatorname{ sym}(\operatorname{Curl}P)^{T}-\alpha_{2}\mathop{\rm skew}(\operatorname{Curl} P)^{T}+\alpha_{3}\,\textrm{tr}[(\operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1\!\! 1\}\in{\rm{Sym}}(3).\] (4.64) In order to satisfy (4.59)${}_{1}$, Teisseyre considered the following sufficient condition¹⁶ [FOOTNOTE:16][ENDFOOTNOTE] \[a_{2}=-a_{3},\quad\quad\quad a_{1}=-2a_{3}\,.\] (4.65) In terms of our notations these imply that \[\alpha_{1}=-6\alpha_{3},\quad\quad\quad\alpha_{2}=6\alpha_{3}\,.\] (4.66) The conditions (4.59) and (4.66) are the so-called _Einstein choice_ in three dimensions and they were used by Malyshev [98] and Lazar [89, 90] in order to investigate dislocations with symmetric force stress¹⁷. [FOOTNOTE:17][ENDFOOTNOTE] In addition, in another paper [162], Teisseyre assumed that $a_{3}=0$ which removes the effects of the _micro-dislocation_ tensor $\alpha=-\operatorname{Curl}P$ completely. In other words, the Einstein choice (4.66) leads to \[\operatorname{Curl}\{\alpha_{1} \operatorname{dev}\operatorname{sym}(\operatorname{Curl}P)^{T}- \alpha_{2}\mathop{\rm skew}(\operatorname{Curl}P)^{T}+\alpha_{3}\textrm{tr}[( \operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1\!\!1\}\] \[=\operatorname{Curl}\{-6\alpha_{3}\operatorname{dev}\operatorname {sym}(\operatorname{Curl}P)^{T}-6\alpha_{3}\mathop{\rm skew}(\operatorname{ Curl}P)^{T}+\alpha_{3}\textrm{tr}[(\operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1 \!\!1\}\] (4.67) \[=-6\alpha_{3}\operatorname{Curl}\{\operatorname{dev}\operatorname {sym}(\operatorname{Curl}P)^{T}+\mathop{\rm skew}(\operatorname{Curl}P)^{T}- \frac{1}{6}\textrm{tr}[(\operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1\!\!1\}\] \[=-6\alpha_{3}\operatorname{Curl}\{\operatorname{dev}\operatorname {sym}(\operatorname{Curl}P)^{T}+\mathop{\rm skew}(\operatorname{Curl}P)^{T}+ \frac{1}{3}\textrm{tr}[(\operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1\!\!1-\frac{ 1}{2}\textrm{tr}[(\operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1\!\!1\}.\] Using the decomposition (4.49) of the dislocation tensor $\operatorname{Curl}P$, (4.6) implies \[\operatorname{Curl}\{\alpha_{1} \operatorname{dev}\operatorname{sym}(\operatorname{Curl}P)^{T}- \alpha_{2}\mathop{\rm skew}(\operatorname{Curl}P)^{T}+\alpha_{3}\textrm{tr}[( \operatorname{Curl}P)^{T}]{\,\!\cdot\!\,}1\!\!1\}\] (4.68) \[=-6\alpha_{3}\operatorname{Curl}[(\operatorname{Curl}P)^{T}]+3 \alpha_{3}\operatorname{Curl}\{\textrm{tr}[(\operatorname{Curl}P)^{T}]{\,\! \cdot\!\,}1\!\!1\}.\] Let us remark that for all differentiable functions $\zeta:\mathbb{R}\rightarrow\mathbb{R}$ on $\Omega$ we have (4.69) On the other hand we have \[\operatorname{Curl}[(\operatorname{Curl}S)^{T}] \in{\rm{Sym}}(3)\,,\quad\quad\quad\text{for all}\quad\quad S\in{ \rm{Sym}}(3),\] \[\operatorname{Curl}[(\operatorname{Curl}A)^{T}] \in\operatorname{\mathfrak{so}}(3)\,,\quad\quad\quad\text{for all }\quad\quad A\in\operatorname{\mathfrak{so}}(3).\] (4.70) \[\textrm{tr}(\operatorname{Curl}S) =0\,,\quad\quad\quad\text{for all}\quad\quad S\in{\rm{Sym}}(3).\] Hence, from (4.62), (4.68), (4.6) and (4.69) we obtain \[\operatorname{Curl}(m^{T})\|_{\alpha_{1}=-6\alpha_{3},\alpha_{2}=6 \alpha_{3}}= -6\alpha_{3}\underbrace{\operatorname{Curl}\{[\operatorname{Curl} (\operatorname{sym}P)]^{T}\}}_{\in{\rm{Sym}}(3)}\] (4.71) \[-6\alpha_{3}\underbrace{\operatorname{Curl}\{[\operatorname{Curl} (\mathop{\rm skew}P)]^{T}\}}_{\in\operatorname{\mathfrak{so}}(3)}+3\alpha_{3} \underbrace{\operatorname{Curl}\{\textrm{tr}[\big{(}\operatorname{Curl}( \mathop{\rm skew}P)\big{)}^{T}]{\,\!\cdot\!\,}1\!\!1\}}_{\in\operatorname{ \mathfrak{so}}(3)}.\] Let us also remark that if we consider \[\mathop{\rm skew}P=\left(\begin{array}[]{ccc}0&-p_{3}&p_{2}\\ p_{3}&0&-p_{1}\\ -p_{2}&p_{1}&0\end{array}\right)\] (4.72) then we have \[\big{(}\operatorname{Curl}(\mathop{\rm skew}P)\big{)}^{T}=\left( \begin{array}[]{ccc}p_{3,3}+p_{2,2}&-p_{1,2}&-p_{1,3}\\ -p_{2,1}&p_{3,3}+p_{1,1}&-p_{2,3}\\ -p_{3,1}&-p_{3,2}&p_{2,2}+p_{1,1}\\ \end{array}\right).\] (4.73) Thus, we deduce \[\textrm{tr}[\big{(}\operatorname{Curl}(\mathop{\rm skew}P)\big{)} ^{T}]=2(p_{1,1}+p_{2,2}+p_{3,3})=2\,{\rm div}\,p\,,\] (4.74) where $p=\operatorname{axl}(\mathop{\rm skew}P)$. Moreover, in view of (4.69), (4.74) implies \[\operatorname{Curl}\{\textrm{tr}[\big{(}\operatorname{Curl}( \mathop{\rm skew}P)\big{)}^{T}]{\,\!\cdot\!\,}1\!\!1\}=2\left(\begin{array}[]{ ccc}0&\,{\rm div}\,p_{,3}&-{\rm div}\,p_{,2}\\ -{\rm div}\,p_{,3}&0&{\rm div}\,p_{,1}\\ {\rm div}\,p_{,2}&-{\rm div}\,p_{,1}&0\end{array}\right)=-2\operatorname{anti} \nabla({\rm div}\,p).\] (4.75) On the other hand, we have \[\operatorname{Curl}\{[\operatorname{Curl}(\mathop{\rm skew}P)]^{T }\}=\left(\begin{array}[]{ccc}0&\,{\rm div}\,p_{,3}&-{\rm div}\,p_{,2}\\ -{\rm div}\,p_{,3}&0&{\rm div}\,p_{,1}\\ {\rm div}\,p_{,2}&-{\rm div}\,p_{,1}&0\end{array}\right)=-\operatorname{anti} \nabla({\rm div}\,p).\] (4.76) From the above two identities, we deduce \[-2\operatorname{Curl}\{[\operatorname{Curl}(\mathop{\rm skew}P)]^ {T}\}+\operatorname{Curl}\{\textrm{tr}[\big{(}\operatorname{Curl}(\mathop{\rm skew }P)\big{)}^{T}]{\,\!\cdot\!\,}1\!\!1\}=0,\quad\quad\text{for all}\quad\quad P \in\mathbb{R}^{3\times 3}.\] (4.77) Thus, we obtain \[\operatorname{Curl}(m^{T})\|_{\alpha_{1}=-6\alpha_{3},\alpha_{2}=6 \alpha_{3}}= -6\alpha_{3}\,\operatorname{Curl}\{[\operatorname{Curl}( \operatorname{sym}P)]^{T}\}\in{\rm{Sym}}(3),\quad\quad\text{for all}\quad\quad P \in\mathbb{R}^{3\times 3}.\] (4.78) Summarizing, we have the following result which gives information about the symmetry of the model. Remark 4.1: * * _If_ $\alpha_{1}=-6\alpha_{3}$ _and_ $\alpha_{2}=6\alpha_{3}$_, then_ \[\operatorname{Curl}\{\alpha_{1}\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+{ \alpha_{3}}\ {\rm tr}(\operatorname{Curl}P)\!\cdot\!1\!\!1\}\in{\rm{Sym}}(3) \quad\text{ for all }\quad\quad P\in\mathbb{R}^{3\times 3}.\] (4.79) * _Given_ $P\in{\rm{Sym}}(3)$_, then we have_ \[\operatorname{Curl}\{\alpha_{1}\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+{ \alpha_{3}}\ {\rm tr}(\operatorname{Curl}P)\!\cdot\!1\!\!1\}\in{\rm{Sym}}(3).\] (4.80) _if and only if_ $\ \alpha_{1}=-\alpha_{2}$_._ We conclude that, in view of (4.71) and (4.77), the Einstein choice (4.66) implies that \[\operatorname{Curl}[m^{T}]\in{\rm{Sym}}(3)\,,\quad\quad\text{for all}\quad\quad P\in\mathbb{R}^{3\times 3}.\] (4.81) Thus, the condition (4.65) is in concordance, without any restriction and projection of the equation, with the assumption that $P_{,tt}$ remains symmetric since the right hand side of the equations (4.19) is now symmetric. Therefore Teisseyre’s model does have a symmetric stress tensor, it is based on the dislocation tensor $\alpha$, and determines nevertheless a symmetric micro-distortion such that $P_{,tt}$. In addition, if $P\in{\rm{Sym}}(3)$, then from (4.55) it follows that $\mathop{\rm skew}P$ is solution of the problem \[(\mathop{\rm skew}P)_{,tt}=0,\quad\quad\quad(\mathop{\rm skew}P)( x,0)=0,\quad\quad\quad(\mathop{\rm skew}P)_{,t}(x,0)=0.\] (4.82) The unique solution of the above problem is $\mathop{\rm skew}P=0$. Thus, we conclude that if $P(x,0)\in{\rm{Sym}}(3)$ and $P_{,t}(x,0)\in{\rm{Sym}}(3)$, then $P(x,t)\in{\rm{Sym}}(P)$ and in consequence the Teisseyre’s model is a fully “symmetric” micromorphic model. If we consider the supplementary conditions (4.65), then the energy $\mathcal{K}_{2}$ becomes \[\mathcal{K}_{2}(P)=-2a_{3}\,\\|\operatorname{dev}\operatorname{sym }(\operatorname{Curl}{P})\\|^{2}+4a_{3}\,\\|\mathop{\rm skew}(\operatorname{Curl }P)\\|^{2}+\frac{a_{3}}{3}\,[\textrm{tr}(\operatorname{Curl}{P})]^{2}\,,\] (4.83) which has no sign! The constraint (4.65), introduced for having $P_{,tt}\in{\rm{Sym}}(3)$ destroys, therefore, the positive definiteness of the dislocation energy (4.83). In fact, the most general form of the energy (4.83) considered by Teisseyre is \[\mathcal{K}_{2}(\operatorname{sym}P)=\langle\widehat{\mathbb{L}}_ {c}.\,\operatorname{Curl}\operatorname{sym}P,\operatorname{Curl}\operatorname{ sym}P\rangle,\] (4.84) where $\widehat{\mathbb{L}}_{c}$ is a _non-positive definite_ isotropic tensor. In view of (2.17), this energy is equivalent with \[\mathcal{K}_{T}(\operatorname{sym}P)=\langle\widehat{\mathbb{L}}_ {T}.\,\nabla\operatorname{sym}P,\nabla\operatorname{sym}P\rangle,\] (4.85) where \[\widehat{\mathbb{L}}_{T}:\mathbb{R}^{3\times 3\times 3} \rightarrow\mathbb{R}^{3\times 3\times 3}.\] (4.86) The energy $\mathcal{K}_{T}(\operatorname{sym}P)$ is similar with the energy from the gradient elasticity formulation [41] \[\mathcal{E}(\operatorname{sym}\nabla u)=\langle\widehat{\mathbb{L }}_{T}.\,\nabla(\operatorname{sym}\nabla u),\nabla(\operatorname{sym}\nabla u)\rangle.\] (4.87) If we extend the Teisseyre’s model to the anisotropic case, then the total energy is equivalent with the energy \[2\widehat{E}(t):=\int_{\Omega}\bigg{(}\\|u_{,t}\\|^{2} +\\|(\operatorname{sym}P)_{,t}\\|^{2}+\langle\mathbb{C}.\, \operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P)\rangle\] \[\ \ \ \ \quad\ +\langle\mathbb{H}.\,\operatorname{sym}P, \operatorname{sym}P\rangle+\langle\widehat{\mathbb{L}}.\,\nabla(\operatorname{ sym}P),\nabla(\operatorname{sym}P)\rangle\bigg{)}dv\,,\] (4.88) from the microstrain model [55] (see Subsection 4.2). In conclusion, the Teisseyre’s model is a special degenerate isotropic microstrain model and it is therefore incapable of describing rotation of the microstructure, hence the name “symmetric” micromorphic model. ### The asymmetric microstretch model in dislocation format It is well known that the theory of microstretch elastic materials is a special subclass of the class of micromorphic materials [47, 18, 14, 13]. In this Subsection we show that the microstretch model is already contained in our relaxed micromorphic model in dislocation format. To this aim, we assume that the micro-distortion tensor has the form $P=\zeta{\,\!\cdot\!\,}1\!\!1+A$, where $A\in\operatorname{\mathfrak{so}}(3)$ and $\zeta$ is a scalar function. It is easy to check that \[\operatorname{Curl}P=\operatorname{Curl}A+\operatorname{Curl}( \zeta{\,\!\cdot\!\,}1\!\!1)=\operatorname{Curl}A+\left(\begin{array}[]{ccc}0& \zeta_{,3}&-\zeta_{,2}\\ -\zeta_{,3}&0&\zeta_{,1}\\ \zeta_{,2}&-\zeta_{,1}&0\end{array}\right),\] (4.89) and \[\operatorname{Curl}(\operatorname{Curl}P)=\operatorname{Curl}( \operatorname{Curl}A)+\left(\begin{array}[]{ccc}-(\zeta_{,22}+\zeta_{,33})& \zeta_{,12}&\zeta_{,13}\\ \zeta_{,12}&-(\zeta_{,11}+\zeta_{,33})&\zeta_{,23}\\ \zeta_{,13}&\zeta_{,23}&-(\zeta_{,11}+\zeta_{,22})\end{array}\right)\,.\] (4.90) As in the construction of the linear Cosserat model in terms of the dislocation density tensor (Subsection 4.4), for isotropic materials and with the asymmetric factor $2\mu_{c}\mathop{\rm skew}(\nabla u-P)$ incorporated, the constitutive equations become \[\sigma =2\mu_{e}\,\operatorname{sym}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1 )+2\mu_{c}\,\mathop{\rm skew}(\nabla u-A)+\lambda_{e}\,\textrm{tr}(\nabla u- \zeta{\,\!\cdot\!\,}1\!\!1){\,\!\cdot\!\,}1\!\!1\,,\] \[m =\alpha_{1}\operatorname{dev}\operatorname{sym}\operatorname{Curl }A+\alpha_{2}\,\mathop{\rm skew}\operatorname{Curl}A+\alpha_{3}\,\textrm{tr}( \operatorname{Curl}A){\,\!\cdot\!\,}1\!\!1\] \[\quad+\alpha_{1}\,\operatorname{dev}\underbrace{\operatorname{sym }\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1)}_{=0}+\alpha_{2}\mathop{\rm skew }\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1)+\alpha_{3}\,\underbrace{ \textrm{tr}(\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1))}_{=0}{\,\!\cdot\! \,}1\!\!1\] (4.91) \[s =2\mu_{h}\operatorname{sym}(\zeta{\,\!\cdot\!\,}1\!\!1)+\lambda_{ h}\textrm{tr}(\zeta{\,\!\cdot\!\,}1\!\!1){\,\!\cdot\!\,}1\!\!1\,,\quad\quad \quad\ \ \ \text{in}\ \quad\quad\Omega\times[0,T].\] Observing that for all matrices $X\in\mathbb{R}^{3\times 3}$ we have the decomposition \[X-\operatorname{dev}\operatorname{sym}X=\mathop{\rm skew}X+\frac {1}{3}\textrm{tr}(X){\,\!\cdot\!\,}1\!\!1,\] (4.92) we obtain by restriction and projection the equations \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}(\nabla u-P)+2\mu_{c} \mathop{\rm skew}(\nabla u-P)+\lambda_{e}\textrm{tr}(\nabla u-P){\,\!\cdot\!\, }1\!\!1]+f\,,\] (4.93) \[P_{,tt} -(\operatorname{dev}\operatorname{sym}P)_{,tt}=-\operatorname{ Curl}\bigg{[}\alpha_{1}\operatorname{dev}\operatorname{sym}\operatorname{Curl} P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+\alpha_{3}\,\textrm{tr}( \operatorname{Curl}P){\,\!\cdot\!\,}1\!\!1\bigg{]}\] \[\quad+\operatorname{dev}\operatorname{sym}\operatorname{Curl} \bigg{[}\alpha_{1}\operatorname{dev}\operatorname{sym}\operatorname{Curl}P+ \alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+\alpha_{3}\,\textrm{tr}( \operatorname{Curl}P){\,\!\cdot\!\,}1\!\!1\bigg{]}\] \[\quad+2\mu_{e}\operatorname{sym}(\nabla u-P)+2\mu_{c}\mathop{\rm skew }(\nabla u-P)+\lambda_{e}\textrm{tr}(\nabla u-P){\,\!\cdot\!\,}1\!\!1\] \[\quad-2\mu_{e}\operatorname{dev}\operatorname{sym}(\nabla u-P)-2 \mu_{c}\operatorname{dev}\mathop{\rm skew}(\nabla u-P)+\lambda_{e} \operatorname{dev}\textrm{tr}(\nabla u-P){\,\!\cdot\!\,}1\!\!1\] \[\quad-2\mu_{h}\operatorname{sym}P+2\mu_{h}\operatorname{dev} \operatorname{sym}P-\lambda_{h}\textrm{tr}(P){\,\!\cdot\!\,}1\!\!1+\lambda_{h} \operatorname{dev}(\textrm{tr}(P){\,\!\cdot\!\,}1\!\!1)\] \[\quad+M-\operatorname{dev}\operatorname{sym}M\,\quad\quad\quad\ \ \ \text{in}\ \quad\quad\Omega\times[0,T].\] and further \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}(\nabla u-\zeta{\,\!\cdot \!\,}1\!\!1)+2\mu_{c}\mathop{\rm skew}(\nabla u-A)+\lambda_{e}\textrm{tr}( \nabla u-\zeta{\,\!\cdot\!\,}1\!\!1){\,\!\cdot\!\,}1\!\!1]+f\,,\] (4.94) \[(\zeta{\,\!\cdot\!\,}1\!\!1+A)_{,tt} -(\operatorname{dev}\operatorname{sym}(\zeta{\,\!\cdot\!\,}1\!\!1 +A))_{,tt}=-\operatorname{Curl}\bigg{[}\alpha_{1}\operatorname{dev} \operatorname{sym}\operatorname{Curl}A+\alpha_{2}\mathop{\rm skew} \operatorname{Curl}A+\alpha_{3}\,\textrm{tr}(\operatorname{Curl}A){\,\!\cdot\! \,}1\!\!1\bigg{]}\] \[\quad-\operatorname{dev}\operatorname{sym}\operatorname{Curl} \bigg{[}\alpha_{1}\operatorname{dev}\operatorname{sym}\operatorname{Curl}A+ \alpha_{2}\mathop{\rm skew}\operatorname{Curl}A+\alpha_{3}\,\textrm{tr}( \operatorname{Curl}A){\,\!\cdot\!\,}1\!\!1\bigg{]}\] \[\quad-\alpha_{2}\operatorname{Curl}\operatorname{Curl}(\zeta{\,\! \cdot\!\,}1\!\!1)-\alpha_{2}\operatorname{dev}\operatorname{sym}\operatorname{ Curl}\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1)\] \[\quad+2\mu_{e}\operatorname{sym}(\nabla u-\zeta{\,\!\cdot\!\,}1\! \!1)-2\mu_{e}\operatorname{dev}\operatorname{sym}(\nabla u-\zeta{\,\!\cdot\!\, }1\!\!1)\] \[\quad+2\mu_{c}\mathop{\rm skew}(\nabla u-A)-2\mu_{c}\operatorname {dev}\mathop{\rm skew}(\nabla u-A)\] \[\quad+\lambda_{e}\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1) {\,\!\cdot\!\,}1\!\!1-\lambda_{e}\operatorname{dev}(\textrm{tr}(\nabla u-\zeta {\,\!\cdot\!\,}1\!\!1){\,\!\cdot\!\,}1\!\!1)\] \[\quad-2\mu_{h}\operatorname{sym}(\zeta{\,\!\cdot\!\,}1\!\!1)+2\mu _{h}\operatorname{dev}\operatorname{sym}(\zeta{\,\!\cdot\!\,}1\!\!1)\] \[\quad-\lambda_{h}\textrm{tr}(\zeta{\,\!\cdot\!\,}1\!\!1){\,\! \cdot\!\,}1\!\!1+\lambda_{h}\operatorname{dev}(\textrm{tr}(\zeta{\,\!\cdot\!\, }1\!\!1){\,\!\cdot\!\,}1\!\!1)+M-\operatorname{dev}\operatorname{sym}M\,\quad \quad\quad\ \ \ \text{in}\ \quad\quad\Omega\times[0,T].\] By orthogonal projection of the second equation (4.94)${}_{2}$ to the space of skew-matrices and to the spherical part, respectively, and using the fact that $\textrm{tr}(\operatorname{Curl}S)=0$ for all $S\in{\rm{Sym}}(3)$ and $\textrm{tr}[\operatorname{Curl}(\mathop{\rm skew}\operatorname{Curl}A)]=0$ for all $A\in\operatorname{\mathfrak{so}}(3)$, the full system of equations for our reduced model is \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}(\nabla u-\zeta{\,\!\cdot \!\,}1\!\!1)+2\mu_{c}\mathop{\rm skew}(\nabla u-A)+\lambda_{e}\textrm{tr}( \nabla u-\zeta{\,\!\cdot\!\,}1\!\!1){\,\!\cdot\!\,}1\!\!1]+f\,,\] \[\mathop{\rm skew}A_{,tt} =-\mathop{\rm skew}\operatorname{Curl}\bigg{[}\alpha_{1} \operatorname{dev}\operatorname{sym}\operatorname{Curl}A+\alpha_{2}\mathop{\rm skew }\operatorname{Curl}A+\alpha_{3}\,\textrm{tr}(\operatorname{Curl}A){\,\!\cdot \!\,}1\!\!1\bigg{]}\] \[\ \ \ \ \ \ +2\mu_{c}\mathop{\rm skew}(\nabla u-A)+\mathop{\rm skew }M\,,\] (4.95) \[\textrm{tr}(\zeta_{,tt}{\,\!\cdot\!\,}1\!\!1) =-\alpha_{2}\textrm{tr}[\operatorname{Curl}\operatorname{Curl}( \zeta{\,\!\cdot\!\,}1\!\!1)]\] \[\quad\ +2\mu_{e}\textrm{tr}[\operatorname{sym}(\nabla u-\zeta{\, \!\cdot\!\,}1\!\!1)]+3\lambda_{e}\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\! \!1)\] \[\quad\ -2\mu_{h}\textrm{tr}\operatorname{sym}(\zeta{\,\!\cdot\!\, }1\!\!1)-3\lambda_{h}\textrm{tr}(\zeta{\,\!\cdot\!\,}1\!\!1)+\frac{1}{3} \textrm{tr}(M)\quad\quad\quad\ \ \ \text{in}\ \quad\quad\Omega\times[0,T].\] But \[\operatorname{Curl}\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1)=\left( \begin{array}[]{ccc}-(\zeta_{,22}+\zeta_{,33})&\zeta_{,12}&\zeta_{,13}\\ \zeta_{,12}&-(\zeta_{,11}+\zeta_{,33})&\zeta_{,23}\\ \zeta_{,13}&\zeta_{,23}&-(\zeta_{,11}+\zeta_{,22})\end{array}\right).\] Hence, we deduce $\textrm{tr}[\operatorname{Curl}\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1) ]=-2(\zeta_{,11}+\zeta_{,22}+\zeta_{,33})=-2\Delta\zeta.$ In terms of the microrotation vector $\vartheta=\operatorname{axl}A$, the above system becomes \[u_{,tt} =\textrm{Div}[2\mu_{e}\,\operatorname{sym}(\nabla u-\zeta{\,\! \cdot\!\,}1\!\!1)+2\mu_{c}\,(\mathop{\rm skew}\nabla u-\operatorname{anti}( \vartheta))+\lambda_{e}\,\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1){\,\! \cdot\!\,}1\!\!1]+f\,,\] \[\vartheta_{,tt} =\operatorname{Div}\bigg{[}\frac{\alpha_{1}}{2}\,\operatorname{ dev}\operatorname{sym}\nabla\vartheta+\frac{\alpha_{2}}{2}\,\mathop{\rm skew} \nabla\vartheta+{2}\alpha_{3}\,\textrm{tr}(\nabla\vartheta){\,\!\cdot\!\,}1\! \!1\bigg{]}\] \[\ \ \ \ \ \ +2\mu_{c}\,\big{[}\operatorname{axl}(\mathop{\rm skew }\nabla u)-\vartheta\big{]}+\operatorname{axl}\mathop{\rm skew}M\] (4.96) \[\zeta_{,tt} =\frac{2}{3}\alpha_{2}\,\Delta\zeta+\frac{2\mu_{e}+3\lambda_{e}}{ 3}\,\textrm{div}\,u-(2\mu_{e}+3\lambda_{e}+2\mu_{h}+3\lambda_{h})\,\zeta+\frac {1}{3}\,\textrm{tr}(M)\quad\quad\quad\ \ \ \text{in}\ \quad\quad\Omega\times[0 ,T].\] Using the micro-distortion tensor specific to the microstretch model, the tangential boundary condition (2.21) implies the strong anchoring condition¹⁸ [FOOTNOTE:18][ENDFOOTNOTE] \[A({x},t)=0\ \quad\ \text{and}\ \quad\ \ \ \ \zeta({x},t)=0\quad \quad\text{on}\quad\partial\Omega\times[0,T].\] (4.97) The above system is the microstretch model in dislocation format and the total energy for this model is given by \[\mathcal{L}_{3}=\int_{\Omega}\bigg{(} \frac{1}{2}\\|u_{,t}\\|^{2}+\\|\vartheta_{,t}\\|^{2}+\frac{3}{2}\\| \zeta_{,t}\\|^{2}\] \[+\mu_{e}\,\\|\operatorname{sym}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\! 1)\\|^{2}+\mu_{c}\,\\|\mathop{\rm skew}\nabla u-\operatorname{anti}(\vartheta)\\| ^{2}+\frac{\lambda_{e}}{2}\,\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1)^{2}\] (4.98) \[+\frac{\alpha_{1}}{2}\,\\|\operatorname{dev}\operatorname{sym} \nabla\vartheta\\|^{2}+\frac{\alpha_{2}}{2}\,\\|\mathop{\rm skew}\nabla\vartheta \\|^{2}+{2}\alpha_{3}\,[\,\textrm{tr}(\nabla\vartheta)]^{2}+\frac{3}{2}(2\mu_{h }+3\lambda_{h})\,\zeta^{2}+\alpha_{2}\,\\|\nabla\zeta\\|^{2}\bigg{)}\,dv.\] We may also obtain the model of microstretch materials if we replace the energy from the gradient elasticity formulation [41, 160, 113] \[\mathcal{E}(\nabla u,\nabla(\mathop{\rm skew}\nabla u),\nabla( \textrm{tr}(\nabla u)))= \mu\,\\|\operatorname{dev}\operatorname{sym}\nabla u\\|^{2}+\frac{2 \mu+3\lambda}{6}\,\,[\textrm{tr}(\nabla u)]^{2}\] (4.99) \[\quad+\mu L_{c_{1}}^{2}\,\\|\nabla(\textrm{tr}(\nabla u))\\|^{2}+ \mu L_{c_{2}}^{2}\,\\|\nabla(\mathop{\rm skew}\nabla u)\\|^{2},\] with \[\mathcal{E}(\nabla u,A,\nabla A,\zeta,\nabla\zeta)= \mu\,\\|\operatorname{dev}\operatorname{sym}\nabla u\\|^{2}+\frac{2 \mu+3\lambda}{6}\,[\textrm{tr}(\nabla u)]^{2}\] (4.100) \[\quad+\mu\,L_{c_{1}}^{2}\,\\|\nabla\zeta\\|^{2}+\mu\,L_{c_{2}}^{2} \,\\|\nabla A\\|^{2}+\varkappa_{1}^{+}\mu\,[\textrm{tr}(\nabla u-\zeta{\,\!\cdot \!\,}1\!\!1)]^{2}+\varkappa_{2}^{+}\mu\,\\|\mathop{\rm skew}\nabla u-A\\|^{2},\] where $\varkappa^{+}_{1}$ and $\varkappa^{+}_{2}$ are _dimensionless penalty coefficients_. The coefficient $\varkappa^{+}_{2}\mu=\mu_{c}$ is the _Cosserat couple modulus¹⁹_. Again, no mixed terms appear. [FOOTNOTE:19][ENDFOOTNOTE] For comparison, the classical linear microstretch formulation has the energy (see [47], p. 253) \[\mathcal{L}_{4}=\int_{\Omega}\bigg{(} \frac{1}{2}\\|u_{,t}\\|^{2}+\frac{1}{2}\\|\vartheta_{,t}\\|^{2}+\frac {1}{2}\\|\zeta_{,t}\\|^{2}\] \[+\mu_{e}\,\\|\operatorname{sym}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\! 1)\\|^{2}+\mu_{c}\,\\|\mathop{\rm skew}\nabla u-\operatorname{anti}(\vartheta)\\| ^{2}+\frac{\lambda_{e}}{2}\,\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1)^{2}\] (4.101) \[+\frac{\gamma_{1}}{2}\,\\|\operatorname{dev}\operatorname{sym} \nabla\vartheta\\|^{2}+\frac{\gamma_{2}}{2}\,\\|\mathop{\rm skew}\nabla\vartheta \\|^{2}+\frac{\gamma_{3}}{2}\,[\,\textrm{tr}(\nabla\vartheta)]^{2}+\frac{ \lambda_{1}}{2}\,\zeta^{2}+\frac{\gamma_{0}}{2}\,\\|\nabla\zeta\\|^{2}\] \[+\lambda_{0}\,\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1)\, \zeta+b_{0}\,\langle\operatorname{anti}(\nabla\zeta),\nabla\vartheta\rangle \bigg{)}\,dv.\] The microstretch model in dislocation format involves only three curvature coefficients, instead of four considered in the classical model [47]. ### The microvoids model in dislocation format It is well known that the theory of elasticity with voids is a subset of the micromorphic model [47, 23, 64, 63, 17, 12]. In this subsection we show that the microvoid model is a special case of our relaxed micromorphic model in dislocation format. Indeed this is a particular case of (2.25) in which we assume $P=\zeta{\,\!\cdot\!\,}1\!\!1$. Hence, using (2.25) we obtain \[u_{,tt} =\textrm{Div}[2\mu_{e}\operatorname{sym}(\nabla u-\zeta{\,\!\cdot \!\,}1\!\!1)+\lambda_{e}\textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1){\,\! \cdot\!\,}1\!\!1]+f\,,\] \[\zeta_{,tt} =\frac{2}{3}\alpha_{2}\Delta\zeta+\frac{2\mu_{e}+3\lambda_{e}}{3} \,\textrm{div}\,u-(2\mu_{e}+3\lambda_{e}+2\mu_{h}+3\lambda_{h})\zeta+\frac{1}{ 3}\textrm{tr}(M)\quad\quad\quad\ \ \ \text{in}\ \quad\quad\Omega\times[0,T].\] The boundary condition which follows from the tangential boundary condition (2.21) is the strong anchoring condition \[\zeta(x,t)=0,\quad\quad\text{on}\quad\partial\Omega\times[0,T],\] (4.102) which implies that on the boundary the volumes of the voids do not change. This microvoids model in dislocation format is related to the theory of “micro-voids” [141, 35] and corresponds to the following choice of the total energy \[\mathcal{L}_{5}=\int_{\Omega}\bigg{(}\frac{1}{2}\\|u_{,t}\\|^{2}+ \frac{3}{2}\\|\zeta_{,t}\\|^{2} +\mu_{e}\\|\operatorname{dev}\operatorname{sym}(\nabla u-\zeta{\, \!\cdot\!\,}1\!\!1)\\|^{2}+\frac{2\mu_{e}+3\lambda_{e}}{6}[\textrm{tr}(\nabla u -\zeta{\,\!\cdot\!\,}1\!\!1)]^{2}\] (4.103) \[+\frac{3}{2}(2\mu_{h}+3\lambda_{h})\,\zeta^{2}+\alpha_{2}\,\\| \nabla\zeta\\|^{2}\bigg{)}\,dv.\] From the above equations we remark that the parameter $\alpha_{2}$ describes the creation of micro-voids. This observation suggests to skip $\alpha_{2}$ when $\textrm{tr}(P)=0$. Cowin and Nunziato [141, 35] introduced the following symmetric stress tensor \[{\sigma}_{\mathrm{v}} =2\mu_{\mathrm{v}}\,\operatorname{sym}\nabla u+\lambda_{\mathrm{v }}\,\textrm{tr}(\nabla u){\,\!\cdot\!\,}1\!\!1+b_{\mathrm{v}}\,\zeta{\,\!\cdot \!\,}1\!\!1,\] (4.104) while the _“balance of equilibrated forces”_ is given by \[{\zeta}_{,tt}=\alpha_{\mathrm{v}}\,\Delta\zeta-b_{\mathrm{v}}\, \textrm{div}\,u-\xi_{\mathrm{v}}\,\zeta+\ell,\] (4.105) where $\lambda_{\mathrm{v}},\mu_{\mathrm{v}},b_{\mathrm{v}},\alpha_{\mathrm{v}}$ and $\xi_{\mathrm{v}}$ are constitutive constants and $\ell$ is called _“extrinsic equilibrated body force”_. Our symmetric Cauchy stress tensor is given by \[\sigma=2\mu_{e}\,\operatorname{sym}\nabla u+\lambda_{e}\,\textrm{ tr}(\nabla u){\,\!\cdot\!\,}1\!\!1-(2\mu_{e}+3\lambda_{e})\,\zeta{\,\!\cdot\! \,}1\!\!1.\] (4.106) A direct identification of the coefficients gives us that the coefficient of the Cowin-Nunziato theory can be expressed in terms of our constitutive coefficients \[\mu_{\mathrm{v}}\] (4.107) \[\xi_{\mathrm{v}} =2\mu_{e}+3\lambda_{e}+2\mu_{h}+3\lambda_{h}=-3b_{\mathrm{v}}+2 \mu_{h}+3\lambda_{h}.\] In our model we have only four parameters, because $2\mu_{h}+3\lambda_{h}$ can be regarded as a single parameter, instead of five considered by Cowin and Nunziato [35]. Moreover, we have all the terms considered in the microvoids theory but without having any mixed terms involving two constitutive variables. The positivity conditions for the Cowin-Nunziato theory with voids [35] are (4.108) while in our microvoids model in dislocation format the positivity conditions are obvious \[\mu_{e}>0,\quad\quad\quad 2\mu_{e}+3\lambda_{e}>0,\quad\quad\quad 2 \mu_{h}+3\lambda_{h}>0,\quad\quad\quad\alpha_{2}>0\,.\] (4.109) Let us consider the energy from the isotropic second gradient poromechanics model [37, 41] \[\mathcal{E}(\nabla u,\nabla(\textrm{tr}(\operatorname{sym}\nabla u)))=\mu\,\\| \operatorname{dev}\operatorname{sym}\nabla u\\|^{2}+\frac{2\mu+3\lambda}{6}\,[ \textrm{tr}(\operatorname{sym}\nabla u)]^{2}+\mu L_{c}^{2}\,\\|\nabla(\textrm{ tr}(\operatorname{sym}\nabla u)\\|^{2}.\] (4.110) If we rewrite this energy into a two-field formulation for $u$ and $\zeta$ \[\mathcal{E}(\nabla u,\zeta,\nabla\zeta)= \mu\,\\|\operatorname{dev}\operatorname{sym}\nabla u\\|^{2}+\frac{2 \mu+3\lambda}{6}\,[\textrm{tr}(\operatorname{sym}\nabla u)]^{2}\] (4.111) \[\quad+\mu L_{c}^{2}\,\\|\nabla\zeta\\|^{2}+\varkappa^{+}\mu\,[ \textrm{tr}(\nabla u-\zeta{\,\!\cdot\!\,}1\!\!1)]^{2},\] where $\varkappa^{+}$ is a dimensionless penalty coefficient, we obtain a 4-parameter microvoids model in which no mixed terms appear. We give below, only for comparison, the total energy of the classical linear elastic microvoid model (see [78, 65]) \[\mathcal{L}_{6}=\int_{\Omega}\bigg{(} \frac{1}{2}\\|u_{,t}\\|^{2}+\frac{1}{2}\\|\zeta_{,t}\\|^{2}+\mu_{ \mathrm{v}}\,\\|\operatorname{sym}\nabla u\\|^{2}+\frac{\lambda_{\mathrm{v}}}{2} \,\textrm{tr}(\nabla u)^{2}+\frac{\xi_{\mathrm{v}}}{2}\,\zeta^{2}+\frac{\alpha _{\mathrm{v}}}{2}\,\\|\nabla\zeta\\|^{2}+b_{\mathrm{v}}\,\textrm{tr}(\nabla u)\, \zeta\bigg{)}\,dv.\] According to Lakes [87], the Cowin-Nunziato theory of porous materials predicts that size effects will occur in bending of bars but not in torsion and not in tension in an isotropic material. However, size effects occur always both in bending and in torsion, which means that the void theory cannot adequately describe materials with microvoids. ### A glimpse on the isotropic strain gradient model Let us consider the general energy from the isotropic strain gradient model [61, 62, 37, 41, 160, 3, 113] \[\mathcal{E}(\nabla u,D^{2}u)=\mu\,\\|\operatorname{sym}\nabla u\\|^{2}+\frac{ \lambda}{2}\,[\textrm{tr}(\operatorname{sym}\nabla u)]^{2}+W_{\rm curv}(\nabla \operatorname{sym}\nabla u).\] (4.112) In general, the strain gradient models have the great advantage of simplicity and physical transparency. Due to isotropy, the curvature energy $W_{\rm curv}(\nabla\operatorname{sym}\nabla u)$ involves 5 additional constitutive constants. Taking free variations in the energy (4.112), we obtain \[\int_{\Omega}\bigg{[}2\mu\,\langle\operatorname{sym}\nabla u, \operatorname{sym}\nabla\delta u\rangle_{\mathbb{R}^{3\times 3}}+\lambda \textrm{tr}(\nabla u) \textrm{tr}(\nabla\delta u)\] (4.113) \[+\langle D\,W_{\rm curv}(\nabla\operatorname{sym}\nabla u),\nabla \operatorname{sym}\nabla\delta u\rangle_{\mathbb{R}^{27}}\bigg{]}dv=0,\quad \quad\quad\forall\,\delta u\in C_{0}^{\infty}(\Omega).\] But for all $\delta u\in C^{\infty}(\Omega)$, we have \[\int_{\Omega}\bigg{[}\langle D \,W_{\rm curv}(\nabla\operatorname{sym}\nabla u),\nabla \operatorname{sym}\nabla\delta u\rangle_{\mathbb{R}^{27}}\bigg{]}dv\] (4.114) \[=\int_{\Omega}\bigg{[}\textrm{Div}\big{[}\big{(}D\,W_{\rm curv}( \nabla\operatorname{sym}\nabla u)\big{)}^{T}.\,(\operatorname{sym}\nabla\delta u )\big{]}-\langle\textrm{Div}(D\,W_{\rm curv}(\nabla\operatorname{sym}\nabla u) ),\operatorname{sym}\nabla\delta u\rangle_{\mathbb{R}^{3\times 3}}\bigg{]}dv\] \[=\int_{\Omega}\bigg{[}\textrm{Div}\big{[}\big{(}D\,W_{\rm curv}( \nabla\operatorname{sym}\nabla u)\big{)}^{T}.\,(\operatorname{sym}\nabla\delta u )\big{]}-\langle\operatorname{sym}\textrm{Div}(D\,W_{\rm curv}(\nabla \operatorname{sym}\nabla u)),\nabla\delta u\rangle_{\mathbb{R}^{3\times 3}} \bigg{]}dv\] \[=\int_{\Omega}\bigg{[}\textrm{Div}\big{[}\big{(}D\,W_{\rm curv}( \nabla\operatorname{sym}\nabla u)\big{)}^{T}.\,(\operatorname{sym}\nabla\delta u )\big{]}-\textrm{Div}\big{[}\big{(}\operatorname{sym}\textrm{Div}\,D\,W_{\rm curv }(\nabla\operatorname{sym}\nabla u)\big{)}^{T}.\,\delta u\big{]}\] \[\quad\quad\quad\quad+\langle\textrm{Div}[\operatorname{sym} \textrm{Div}(D\,W_{\rm curv}(\nabla\operatorname{sym}\nabla u))],\delta u \rangle_{\mathbb{R}^{3\times 3}}\bigg{]}dv.\] Hence, the relation (4.113) leads to \[\int_{\Omega} \bigg{\{}\langle\textrm{Div}\big{\{}\underbrace{\underbrace{2\mu \,\operatorname{sym}\nabla u+\lambda\textrm{tr}(\nabla u)\!\cdot\!1\!\!1}_{ \text{``local force stress"}}-\underbrace{\operatorname{sym}\textrm{Div}( \underbrace{D\,W_{\rm curl}(\nabla\operatorname{sym}\nabla u)}_{\text{`` hyperstress"}})}_{\text{``nonlocal force stress"}}}_{\text{``total force stress"}}\big{\}},\delta u\rangle_{\mathbb{R}^{3}}\bigg{\}}\,dv\] (4.115) \[\quad-\int_{\partial\Omega}\bigg{\{}\langle\big{(}D\,W_{\rm curv} (\nabla\operatorname{sym}\nabla u)\big{)}.\,n,\operatorname{sym}\nabla\delta u \rangle+\langle\big{(}\operatorname{sym}\textrm{Div}\,D\,W_{\rm curv}(\nabla \operatorname{sym}\nabla u)\big{)}.\,n,\delta u\rangle\bigg{\}}\,da=0,\] for all $\delta u\in C^{\infty}(\Omega)$. In this representation, the local and nonlocal parts of the force stress tensor are both seen to be symmetric. This is in line with the observation that the generalized Cauchy stresses in a second grade elastic material can always be assumed in symmetric form if frame-indifference is satisfied [160, 113], see the footnote 6. ## 5 Conclusion Let us summarize the main thrust of the paper regarding the new relaxed micromorphic model. We * reconcile Kröner’s rejection of antisymmetric force stresses in dislocation theory with the dislocation model of Eringen and Claus and show that the concept of asymmetric force stress is not needed in order to describe rotations of the microstructure in non-polar materials. * preserve full kinematical freedom (12 degree of freedom) by reducing the model in order to obtain symmetric Cauchy stresses. The proposed relaxed model is still able to fully describe rotations of the microstructure and to fit a huge class of mechanical behaviours of materials with microstructure. * note that the possible non-symmetry of the micro-distortion $P$ is governed solely by moment stresses and applied body moments. The macroscopic and microscopic scales are separated, in this sense. * define a dependence of the free energy only on the elastic strain, microstrain and dislocation density tensor. * provide a standard set of tangential boundary conditions for the micro-distortion, i.e. $P.\,\tau=0$ ($P_{i}\times\,n=0$) on $\partial\Omega$ and not the usual strong anchoring condition $P=0$ on $\partial\Omega$. * obtain well-posedness results for the relaxed formulation regarding: existence, uniqueness and continuous dependence for tangential boundary conditions. * disclose as unnecessary the concept of asymmetric force stresses for a wide class of microstructured materials. * conclude that the linear Cosserat theory is a redundant model for dislocated bodies and for the description of a huge class of material behaviours. * remark that for the isotropic relaxed micromorphic model only 3 curvature parameters remain to be determined, which may eventually be reduced to 2 parameters, which are needed for fitting bending and torsion experiments. * allow in principle for non-smooth solutions and the possibility of fracture. The solution space for the elastic distortion and micro-distortion is only ${\rm H}(\operatorname{Curl};\Omega)$ and for the macroscopic displacement $u\in{\rm H}^{1}(\Omega)$. For non-smooth external data we expect slip lines. * introduce a suitable decomposition of the Mindlin-Eringen strain energy density for micromorphic media (see Eq. (2.2)) which allows to determine a unique parameter $\mu_{c}$ which is responsible for eventual asymmetry of the stress tensor. * observe that the Cosserat couple modulus $\mu_{c}\geq 0$ can be set to zero, the relaxed micromorphic model is still capable to describe rotations of the microstructure and to fit a large class of microstructured material behaviours. * understand that for $\mu_{c}=0$, the seemingly absent (local) control of the antisymmetric part of the elastic distortion is provided by the dislocation energy, the microstrain energy and the tangential boundary conditions. Thus, the skew-symmetric part of the distortion is fully determined by the boundary value problem. * note that the asymmetry of the Cauchy stress tensor may arise in theories where there are couple stresses due to a non-mechanical nature, e.g. in models with electromagnetic interactions and in polarizable media (piezoelectricity, ferroelectricity). As far as purely mechanical models are considered in the framework of linear elasticity, the need of introducing asymmetric stresses becomes rarer. Indeed, only some very special engineering materials like lattice structures and phononic crystals may be seen to need asymmetric stresses to disclose their complete mechanical behavior. In Figure 2 we indicate the place in the literature of our relaxed model and we point out the relations between the existing models. [FIGURE:S5.F2][ENDFIGURE] Moreover, Figure 3 gives the relations between our relaxed micromorphic, microstretch model, Cosserat model, microstrain model and microvoid models in dislocation format. [FIGURE:S5.F3][ENDFIGURE] The diagram from Figure 4 gives some new possible relaxed micromorphic models and, in view of the status of the mathematical background, we indicate the well-posedness of the dynamic and static problem. [FIGURE:S5.F4][ENDFIGURE] ## 6 Outlook The new concept of metamaterials is attracting more and more the interest of physicists and mechanicians. It is described and studied in many works: we refer here for instance to [46] or [172]. Metamaterials are obtained by suitably assembling multiple individual elements but usually arranged in (quasi-)periodic substructures in order to show very peculiar and especially designed mechanical properties. Indeed, the particular shape, geometry, size, orientation and arrangement of their constituting elements can affect e.g. the propagation of waves of light or sound in a not-already-observed manner, creating material properties which cannot be found in conventional materials. Particularly promising in the design of metamaterials are those micro-structures which present high-contrast in microscopic properties: these micro-structures, once homogenized, may produce generalized continuum models (see e.g. [21, 2, 144, 55, 54, 112]). The micro-structures of such metamaterials, although remaining quasi-periodical, are conceived so that some of the physical micro-properties characterizing their behavior diverge when the size of the REV tends to zero, while simultaneously some other properties are vanishing in the same limit. In the present paper we have mathematically studied a large class of evolution equations which are governing the propagation of linear waves in micromorphic or generalized continua (see e.g. [38, 149, 155]). The mathematical existence, uniqueness and continuous dependence theorems which we have obtained in [66] are the logical basis of the studies which will be developed in further investigations, where the manifold variety of propagating mechanical waves which may exist in micromorphic continua may unfold unexpected applications in the design of particularly tailored metamaterials, showing very useful and up-to-now unimagined features. Indeed, as already remarked, second gradient materials can be seen as a particular limit case of the micromorphic media introduced in this paper. Such materials can be obtained from micromorphic ones constraining the micromorphic strain tensor $\operatorname{sym}{P}$ to be equal to the classical strain tensor. More precisely, the elastic distortion $\nabla u-P$ is considered to be zero. In this sense, the study of wave propagation in micromorphic media intrinsically contains all the results which are valid for second gradient media. Previous results on wave propagation in second gradient elastic media have shown a wide variety of exotic phenomena basically related to screening or transmitting properties of interfaces embedded in such media. It has been shown that (see e.g. [38, 149, 155]) for waves at frequencies which are sufficiently high to interact with the underlying microstructure, the screening or transmitting properties of the interface can be sensibly enhanced. It is clear that materials which are able to show such exotic properties with respect to wave propagation could lead to the design of technologically relevant devices for example in the field of stealth technology or vibration and acoustic passive control. Some preliminary results on the study of wave propagation in micromorphic media indicate that, for particular sets of the constitutive parameters suggested by our mathematical analysis, propagation of some types of waves can be inhibited or waves which propagate without carrying energy can also be observed. Such frequency-dependent exotic properties are already observable in the bulk of the considered micromorphic medium without considering more complicated reflection and transmission phenomena at surfaces of discontinuity of material properties. This means that well-conceived micromorphic materials could be used as exotic waveguides which allows to filter and/or switch on and off some typical waves depending on the envisaged use. Recently the theory of material symmetry for the Cosserat continuum was extended in [147]. In [147] it is mentioned that some relaxed Cosserat models can be interpreted as micropolar liquid crystals. Although the theory of material symmetry for the relaxed micromorhic models similar to [147] is not elaborated into details, such relaxed micromorphic model can be also interpreted as liquid crystal in the sense of the material symmetry group. We remark that the theorems obtained in [66] can also be used to give a better grounded basis to many results which are already available in the literature (see e.g. [39, 40]). Acknowledgements. I.D. Ghiba acknowledges support from the Romanian National Authority for Scientific Research (CNCS-UEFISCDI), Project No. PN-II-ID-PCE-2011-3-0521. I.D. Ghiba would like to thank P. Neff for his kind hospitality during his visit at the Faculty of Mathematics, Universität Duisburg-Essen, Campus Essen. P. Neff is grateful to F. dell’Isola for making his visit to CISTERNA di LATINA (M&MoCS), in spring 2013, a wonderful scientific experience. A. Madeo thanks INSA-Lyon for the financial support assigned to the project BQR 2013-0054 “Matériaux Méso et Micro-Hétérogènes: Optimisation par Modèles de Second Gradient et Applications en Ingénierie . ## References * [1] R.A. Adams. _Sobolev Spaces._, volume 65 of _Pure and Applied Mathematics_. Academic Press, London, 1. edition, 1975. * [2] J.-J. Alibert, P. Seppecher, and F. dell’Isola. Truss modular beams with deformation energy depending on higher displacement gradients. _Math. Mech. Solids_, 8(1):51–73, 2003. * [3] H. Askes and E.C. Aifantis. Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. _Int. J. Solids Struct._, 48:1962–1990, 2011. * [4] N. Auffray, F. dell’Isola F., V. Eremeyev V., A. Madeo, and G. Rosi. Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids. _Math. Mech. Solids_, in press, 2013. * [5] S. Bauer, P. Neff, D. Pauly, and G. Starke. Dev-Div and DevSym-DevCurl inequalities for incompatible square tensor fields with mixed boundary conditions. _submitted_, 2013. * [6] S. Bauer, P. Neff, D. Pauly, and G. Starke. New Poincaré type inequalities. _submitted to C. R. Acad. Sci. Paris, Ser. I, 340_, 2013. * [7] S. Bauer, P. Neff, D. Pauly, and G. Starke. Some Poincaré type inequalities for quadratic matrix fields. _Proce. Appl. Math. Mech._, to appear, 2013. * [8] V.L. Berdichevsky. Continuum theory of dislocations revisited. _Cont. Mech. Thermo._, 18:195–222, 2006. * [9] K. Berglund. Investigation of a two-dimensional model of a micropolar continuum. _Archiwum Mechaniki Stosowanej_, 29:383–392, 1977. * [10] K. Berglund. Structural models of micropolar media. In O. Brulin and R.K.T. Hsieh, editors, _Mechanics of micropolar media._, volume 132 of _CISM-Lecture Notes_, pages 35–86. 1982. * [11] D. Bigoni and W.J. Drugan. Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. _J. Appl. Mech._, 74:741–753, 2007. * [12] M. Bîrsan. Saint-Venant’s problem for Cosserat shells with voids. _Int. J. Solids Struct._, 42:2033–2057, 2005. * [13] M. Bîrsan. On a thermodynamic theory of porous Cosserat elastic shells. _J. Thermal Stresses_, 29:879–899, 2006. * [14] M. Bîrsan. On the dynamic deformation of porous Cosserat linear-thermoelastic shells. _Z. Angew. Math. Mech._, 88:74–78, 2008. * [15] M. Bîrsan. On Saint-Venant’s problem for anisotropic, inhomogeneous, cylindrical Cosserat elastic shells. _Int. J. Eng. Sci._, 47:21–38, 2009. * [16] M. Bîrsan. Thermal stresses in cylindrical Cosserat elastic shells. _Eur. J. Mech. A/Solids_, 28:94–101, 2009. * [17] M. Bîrsan and H. Altenbach. On the theory of porous elastic rods. _Int. J. Solids Struct._, 48:910–924, 2011. * [18] M. Bîrsan and H. Altenbach. On the Cosserat model for thin rods made of thermoelastic materials with voids. _Discrete and Continuous Dynamical Systems - Series S_, 6(6):1473–1485, 2013. * [19] J.L. Bleustein. A note on the boundary conditions of toupin’s strain gradient-theory. _Int. J. Solids Struct._, 3:1053–1057, 1967. * [20] C.I. Borş. Deformable solids with microstructure having a symmetric stress tensor. _Analele Ştiinţifice ale Universitaţii Al. I. Cuza din Iaşi_, XXVII, s.Ia, f.1(f.1):177–184, 1981. * [21] C. Boutin, S. Hans, and C. Chesnais. Generalized beams and continua. Dynamics of reticulated structures. In G.A. Maugin and A.V. Metrikine, editors, _Mechanics of Generalized Continua_, pages 131–141. Springer, New York, 2011. * [22] P.M. Buechner and R.S. Lakes. Size effects in the elasticity and viscoelasticity of bone. _Biomech. Model. Mechanobio._, 1:295–301, 2003. * [23] E. Bulgariu and I.D. Ghiba. On the thermal stresses in anisotropic porous cylinders. _Discrete and Continuous Dynamical Systems - Series S_, 6:1539–1550, 2013. * [24] G. Capriz. _Continua with Microstructure._ Springer, Heidelberg, 1989. * [25] G. Capriz and P. M. Mariano. Symmetries and Hamiltonian formalism for complex materials. _J. Elasticity_, 72:57–90, 2003. * [26] Y. Chen and J.D. Lee. Connecting molecular dynamics to micromorphic theory. I. Instantaneous and averaged mechanical variables. _Physica A_, 322:359–376, 2003. * [27] Y. Chen, J.D. Lee, and A. Eskandarian. Atomistic viewpoint of the applicability of microcontinuum theories. _Int. J. Solids Struct._, 41:2085–2097, 2004. * [28] S. Chiriţă and I.D. Ghiba. Inhomogeneous plane waves in elastic materials with voids. _Wave Motion_, 47:333–342, 2010. * [29] S. Chiriţă and I.D. Ghiba. Rayleigh waves in Cosserat elastic materials. _Int. J. Eng. Sci._, 51:117–127, 2012. * [30] S. Chiriţă and I.D. Ghiba. Strong ellipticity and progressive waves in elastic materials with voids. _Proc. R. Soc. A_, 466:439–458, 2010. * [31] W.D. Claus and A.C. Eringen. Three dislocation concepts and micromorphic mechanics. In _Developments in Mechanics, Proceedings of the 12th Midwestern Mechanics Conference_, volume 6, pages 349–358. Midwestern, 1969. * [32] W.D. Claus and A.C. Eringen. Dislocation dispersion of elastic waves. _Int. J. Engng. Sci._, 9:605–610, 1971. * [33] N.M. Cordero, A. Gaubert, S. Forest, E.P. Busso, F. Gallerneau, and S. Kruch. Size effects in generalised continuum crystal plasticity for two-phase laminates. _J. Mech. Phys. Solids 58 (2010)_, 28:1963–1994, 2010. * [34] E. Cosserat and F. Cosserat. _Théorie des corps déformables._ Librairie Scientifique A. Hermann et Fils (engl. translation by D. Delphenich 2007, pdf available at http://www.uni-due.de/ hm0014/Cosserat_files/Cosserat09_eng.pdf), reprint 2009 by Hermann Librairie Scientifique, ISBN 978 27056 6920 1, Paris, 1909. * [35] S.C. Cowin and J.W. Nunziato. Linear elastic materials with voids. _J. Elasticity_, 13:125–147, 1983. * [36] C. de Fabritiis and P. M. Mariano. Geometry of interactions in complex bodies. _J. Geom. and Physics_, 54:301–323, 2005. * [37] F. dell’Isola, M. Guarascio, and K. Hutter. A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. _Arch. Appl. Mech._, 70(5):323–337, 2000. * [38] F. dell’Isola, A. Madeo, and L. Placidi. Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3d continua. _Z. Angew. Math. Mech._, 92(1):52–71, 2012. * [39] F. dell’Isola, L. Rosa, and Cz. Wozniak. Dynamics of solids with micro periodic nonconnected fluid inclusions. _Arch. Appl. Mech._, 67(4):215–228, 1997. * [40] F. dell’Isola, L. Rosa, and Cz. Wozniak. A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter. _Acta Mech._, 127(1-4):165–182, 1998. * [41] F. dell’Isola, G. Sciarra, and S. Vidoli. Generalized Hooke’s law for isotropic second gradient materials. _Proc. R. Soc. A_, 465:2177–2196, 2009. * [42] F. dell’Isola and P. Seppecher. The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power. _C.R. Acad. Sci. II, Mec. Phys. Chim. Astron._, 321:303–308, 1995. * [43] J.K. Djoko, F. Ebobisse, A.T. McBride, and B.D. Reddy. A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: Algorithms and numerical analysis. _Comp. Meth. Appl. Mech. Engrg._, 197(1):1–21, 2007. * [44] L. Dresen, J. Kozak, A. Spicac, L. Waniek, and R. Teisseyre. Wave propagation in physical models of micromorphic media. _Stud. Geoph. et Geod._, 28:272–285, 1984. * [45] F. Ebobisse and P. Neff. Rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin. _Math. Mech. Solids_, 15:691–703, 2010. * [46] N. Engheta and R.W. Ziolkowski. _Metamaterials: Physics and Engineering Explorations_. Wiley & Sons, 2006. * [47] A. C. Eringen. _Microcontinuum Field Theories._ Springer, Heidelberg, 1999. * [48] A.C. Eringen and W.D. Claus. A micromorphic approach to dislocation theory and its relation to several existing theories. In J.A. Simmons, R. de Wit, and R. Bullough, editors, _Fundamental Aspects of Dislocation Theory._, volume 1 of _Nat. Bur. Stand. (U.S.), Spec. Publ._, pages 1023–1040. Spec. Publ., 1970. * [49] A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple micro-elastic solids. I. _Int. J. Eng. Sci._, 2:189–203, 1964. * [50] A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids: II. _Int. J. Eng. Sci._, 2:389–404, 1964. * [51] M. Ferretti, M. Madeo A., F. dell’Isola, and P. Boisse. Modelling the onset of shear boundary layers in fibrous composite reinforcements by second gradient theory. _Z. Angew. Math. Phys._, accepted, 2013. * [52] S. Forest. Mechanics of generalized continua: construction by homogenization. _J. Phys. IV France_, 8:Pr4–39–Pr4–48, 1998. * [53] S. Forest. Homogenization methods and the mechanics of generalized continua - Part 2. _Theoret. Appl. Mech. (Belgrad)_, 28-29:113–143, 2002. * [54] S. Forest. Micromorphic approach for gradient elasticity, viscoplasticity, and damage. _J. Eng. Mech._, 135(3):117–131, 2009. * [55] S. Forest and R. Sievert. Nonlinear microstrain theories. _Int. J. Solids Struct._, 43:7224–7245, 2006. * [56] S. Forest, R. Sievert, and E.C. Aifantis. Strain gradient crystal plasticity: thermodynamical formulations and applications. _J. Mech. Beh. Mat._, 13:219–232, 2002. * [57] S. Forest and D.K. Trinh. Generalized continua and non-homogeneous boundary conditions in homogenisation methods. _Z. Angew. Math. Mech._, 91:90–109, 2011. * [58] N. Fox. On the continuum theory of dislocation. In J.A. Simmons, R. de Wit, and R. Bullough, editors, _Fundamental Aspects of Dislocation Theory._, volume 1 of _Nat. Bur. Stand. (U.S.), Spec. Publ._, pages 1041–1052. Spec. Publ., 1970. * [59] C. Galeş. Some results in micromorphic piezoelectricity. _Eur. J. Mech.-A/Solids_, 31:37–46, 2012. * [60] C. Galeş, I.D. Ghiba, and I. Ignatescu. Asymptotic partition of energy in micromorphic thermopiezoelectricity. _Journal of Thermal Stresses_, 34:1241–1249, 2011. * [61] P. Germain. La méthode des puissances virtuelles en mécanique des milieux continus-I: Théorie du second gradient. _J. Mécanique_, 12:235–274, 1973. * [62] P. Germain. The method of virtual power in continuum mechanics. Part 2: Microstructure. _SIAM J. Appl. Math._, 25:556–575, 1973. * [63] I.D. Ghiba. Semi-inverse solution for Saint-Venant’s problem in the theory of porous elastic materials. _Eur. J. Mech. A/Solids_, 27:1060–1074, 2008. * [64] I.D. Ghiba. On the deformation of transversely isotropic porous elastic circular cylinder. _Arch. Mech._, 61:407–421, 2009. * [65] I.D. Ghiba. On the temporal behaviour in the bending theory of porous thermoelastic plates. _Z. Angew. Math. Mech._, 93:284–296, 2013. * [66] I.D. Ghiba, P. Neff, A. Madeo, L. Placidi, and G. Rosi. The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics. _submitted_, 2013. * [67] V. Girault and P.A. Raviart. _Finite Element Approximation of the Navier-Stokes Equations._, volume 749 of _Lect. Notes Math._ Springer, Heidelberg, 1979. * [68] J.D. Goddard. From granular matter to generalized continuum. In G. Capriz, P. Giovine, and P.M. Mariano, editors, _Mathematical Models of Granular Matter, Lecture Notes in Applied Mathematics_, volume 1937, chapter 1, pages 1–20. Springer-Verlag, 2008. * [69] A.E. Green and R.S. Rivlin. Multipolar continuum mechanics. _Arch. Rat. Mech. Anal._, 17(2):113–147, 1964. * [70] A.E. Green and R.S. Rivlin. On Cauchy’s equations of motion. _Z. Angew. Math. Phys._, 15:290–292, 1964. * [71] A.E. Green and R.S. Rivlin. Simple force and stress multipoles. _Arch. Rat. Mech. Anal._, 16:325–353, 1964. * [72] A.E. Green and R.S. Rivlin. Multipolar continuum mechanics: functional theory. I. _Proc. R. Soc. A_, 284:303–324, 1965. * [73] E.F. Grekova and G.A. Maugin. Modelling of complex elastic crystals by means of multi-spin micromorphic media. _Int. J. Eng. Sci._, 43:494–519, 2005. * [74] I. Hlaváček and M. Hlaváček. On the existence and uniqueness of solutions and some variational principles in linear theories of elasticity with couple-stresses. I: Cosserat continuum. II: Mindlin’s elasticity with micro-structure and the first strain gradient. _J. Apl. Mat._, 14:387–426, 1969. * [75] D. Ieşan. A theory of thermoelastic materials with voids. _Acta Mech._, 60:67–89, 1986. * [76] D. Ieşan. Extremum principle and existence results in micromorphic elasticity. _Int. J. Eng. Sci._, 39:2051–2070, 2001. * [77] D. Ieşan. On the micromorphic thermoelasticity. _Int. J. Eng. Sci._, 40:549–567, 2002. * [78] D. Ieşan and M. Ciarletta. _Non-classical elastic solids_. Longman Scientific and Technical, Harlow, Essex, UK and John Wiley$\&$Sons, Inc., New York, 1993. * [79] R. Jänicke, S. Diebels, H.G. Sehlhorst, and A. Düster. Two-scale modelling of micromorphic continua. _Cont. Mech. Therm._, 21:297–315, 2009. * [80] R Jänicke and H. Steeb. Wave propagation in periodic microstructures by homogenisation of extended continua. _Comp. Mater. Sci._, 52:209–211, 2012. * [81] J. Jeong and P. Neff. Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. _Math. Mech. Solids_, 15(1):78–95, 2010. * [82] N. Kirchner and P. Steinmann. A unifying treatise on variational principles for gradient and micro-morphic continua. _Phil. Mag._, 85:3975–3995, 2005. * [83] A. Klawonn, P. Neff, O. Rheinbach, and S. Vanis. FETI-DP domain decomposition methods for elasticity with structural changes: $P$-elasticity. _ESAIM: Math. Mod. Num. Anal._, 45:563–602, 2011. * [84] E. Kröner. _Mechanics of Generalized Continua. (Proceedings of the IUTAM-Symposium on the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart, Germany, 1967)_. Springer, Berlin/Heidelberg/New York. * [85] E. Kröner. Das physikalische Problem der antisymmetrischen Spannungen und der sogenannten Momentenspannungen. In H. Görtler, editor, _Proceedings of 11th. Int. Congress Applied Mechanics., Munich, 1964_, pages 143–158. 1966. * [86] E. Kröner. Discussion on Papers by A.C. Eringen and W.D. Claus, Jr., and N. Fox. In J.A. Simmons, R. de Wit, and R. Bullough, editors, _Fundamental Aspects of Dislocation Theory._, volume 1 of _Nat. Bur. Stand. (U.S.), Spec. Publ._, pages 1054–1059. Spec. Publ., 1970. * [87] R.S. Lakes. Experimental microelasticity of two porous solids. _Int. J. Solids Struct._, 22:55–63, 1985. * [88] J. Lankeit, P. Neff, and D. Pauly. Uniqueness of integrable solutions to $\nabla\xi={G}\xi,$$\xi\|_{\gamma}=0$ for integrable tensor coefficients ${G}$ and applications to elasticity. _Z. Angew. Math. Phys._, DOI 10.1007/s00033-013-0314-4, 2013. * [89] M. Lazar. An elastoplastic theory of dislocations as a physical field theory with torsion. _J. Phys. A: Math. Gen._, 35:1983–2004, 2002. * [90] M. Lazar. Screw dislocations in the field theory of elastoplasticity. _Ann. Phys._, 11:635–649, 2002. * [91] M. Lazar. The gauge theory of dislocations: a uniformly moving screw dislocation. _Proc. R. Soc. A_, 465:2505 2520, 2009. * [92] M. Lazar. Dislocations in generalized continuum mechanics. In Metrikine A.V. Maugin, G.A., editor, _Mechanics of Generalized Continua. One hundred years after the Cosserats_, volume 21 of _Advances in Mechanics and Mathematics_, chapter 24, pages 223–232. Springer, 2010. * [93] M. Lazar. On the fundamentals of the three-dimensional translation gauge theory of dislocations. _Math. Mech. Solids_, 16(253-264), 2011. * [94] M. Lazar and C. Anastassiadis. The gauge theory of dislocations: conservation and balance laws. _Phil. Mag._, 88:1673–1699, 2008. * [95] M. Lazar and C. Anastassiadis. The gauge theory of dislocations: static solutions of screw and edge dislocations. _Phil. Mag._, 98:199–231, 2009. * [96] J.D. Lee and Y. Chen. Constitutive relations of micromorphic thermoplasticity. _Int. J. Engrg. Sci._, 41:387–399, 2002. * [97] R. Leis. _Initial Boundary Value problems in Mathematical Physics_. Teubner, Stuttgart, 1986. * [98] C. Malyshev. The ${T}(3)$-gauge model, the Einstein-like gauge equation, and Volterra dislocations with modified asymptotics. _Annals of Physics_, 286:249–277, 2000. * [99] P. M. Mariano. Representation of material elements and geometry of substructural interaction. _Quaderni di Matematica_, 2008. * [100] P. M. Mariano and G. Modica. Ground states in complex bodies. _ESAIM: COCV_, 15(2):377–402, 2009. * [101] P. M. Mariano and F.L. Stazi. Computational aspects of the mechanics of complex materials. _Arch. Comput. Meth. Engng._, 12:392–478, 2005. * [102] G.A. Maugin. Un principe variationnel pour des milieux micromorphiques non dissipatifs. _C. R. Acad. Sci. Paris, Serie II_, 271:807–810, 1970. * [103] G.A. Maugin. Electromagnetism and generalized continua. In H. Altenbach and V. Eremeyev, editors, _Generalized Continua from the Theory to Engineering Applications_, volume 541 of _CISM International Centre for Mechanical Sciences_, pages 301–360. Springer, 2013. * [104] G.A. Maugin. The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. In memory of Paul Germain (1920-2009). _Cont. Mech. Thermodyn._, pages 1–20, 2013. * [105] G.A. Maugin and V.A. Metrikine (eds.). Mechanics of generalized continua. One hundred years after the Cosserats. volume 21 of _Advances in Mechanics and Mathematics_. Springer, Berlin, 2010. * [106] R.D. Mindlin. Influence of couple-stresses on stress concentrations. _Experimental Mechanics_, 3(1):1–7, 1963. * [107] R.D. Mindlin. Micro-structure in linear elasticity. _Arch. Rat. Mech. Anal._, 16:51–77, 1964. * [108] R.D. Mindlin. Second gradient of strain and surface tension in linear elasticity. _Int. J. Solids Struct._, 1:417–438, 1965. * [109] R.D. Mindlin. Theories of elastic continua and crystal lattice theories. In E. Kröner, editor, _Mechanics of Generalized Continua. Proceedings of the IUTAM-Symposium on the generalized Cosserat continuum and the continuum theory of dislocations with applications in Freudenstadt, 1967_, pages 312–320. Springer, 1968. * [110] R.D. Mindlin and N.N. Eshel. On first strain-gradient theories in linear elasticity. _Int. J. Solids Struct._, 4:109–124, 1968. * [111] R.D. Mindlin and H.F. Tiersten. Effects of couple stresses in linear elasticity. _Arch. Rat. Mech. Anal._, 11:415–447, 1962. * [112] A. Misra and C.S. Chang. Effective elastic moduli of heterogeneous granular solids. _Int. J. Solids Struct._, 30:2547–2566, 1993. * [113] A. Morro and M. Vianello. Interstitial energy flux and stress-power for second-gradient elasticity. In _The 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM 2013)_, pages 1–7, Paper ID 702, Montréal, Canada, July 23-26 2013. McGill University. * [114] H.B. Mühlhaus and E.C. Aifantis. A variational principle for gradient plasticity. _Int. J. Solids Struct._, 28:845–857, 1991. * [115] I. Münch and P. Neff. A nonlinear micropolar continuum theory for initial plasticity. In A. Zigoni, editor, _Advances and Trends in Structural Engineering, Mechanics and Computation._, pages 269–272. CRC Press/Balkema, ISBN 978-0-415-58472-2, 2010. * [116] I. Münch, W. Wagner, and P. Neff. Transversely isotropic material: nonlinear Cosserat versus classical approach. _Cont. Mech. Thermod._, 23(1):27–34, 2011. * [117] H. Nagahama and R. Teisseyre. Micromorphic continuum and fractal fracturing in the lithosphere. _Pure Appl. Geophysics_, 157:559–574, 2000. * [118] H. Nagahama and R. Teisseyre. Micromorphic continuum and fractal fracturing in the lithosphere. _Int. Geophysics_, 76:425–440, 2001. * [119] P. Neff. On material constants for micromorphic continua. In Y. Wang and K. Hutter, editors, _Trends in Applications of Mathematics to Mechanics_, STAMM Proceedings, Seeheim 2004, pages 337–348. Shaker Verlag, Aachen, 2005. * [120] P. Neff. The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. _Z. Angew. Math. Mech._, 86:892–912, 2006. * [121] P. Neff. Existence of minimizers for a finite-strain micromorphic elastic solid. _Proc. Roy. Soc. Edinb. A_, 136:997–1012, 2006. * [122] P. Neff. Remarks on invariant modelling in finite strain gradient plasticity. _Technische Mechanik_, 28(1):13–21, 2008. * [123] P. Neff and K. Chełmiński. Infinitesimal elastic-plastic Cosserat micropolar theory. Modelling and global existence in the rate-independent case. _Proc. Roy. Soc. Edinb. A_, 135:1017–1039, 2005. * [124] P. Neff and K. Chełmiński. Well-posedness of dynamic Cosserat plasticity. _Appl. Math. Optim._, 56:19–35, 2007. * [125] P. Neff and K. Chełmiński. $H^{1}_{\rm loc}$-stress and strain regularity in Cosserat-Plasticity. _Z. Angew. Math. Mech._, 89(4):257–266, 2008. * [126] P. Neff, K. Chełmiński, and H.D. Alber. Notes on strain gradient plasticity. Finite strain covariant modelling and global existence in the infinitesimal rate-independent case. _Math. Mod. Meth. Appl. Sci. (M3AS)_, 19(2):1–40, 2009. * [127] P. Neff and S. Forest. A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. _J. Elasticity_, 87:239–276, 2007. * [128] P. Neff and J. Jeong. A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. _Z. Angew. Math. Mech._, 89(2):107–122, 2009. * [129] P. Neff, J. Jeong, I. Münch, and H. Ramezani. Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity. _Z. Angew. Math. Phys._, 3(60):479–497, 2009. * [130] P. Neff, J. Jeong, I. Münch, and H. Ramezani. Linear Cosserat Elasticity, Conformal Curvature and Bounded Stiffness. In G.A. Maugin and V.A. Metrikine, editors, _Mechanics of Generalized Continua. One hundred years after the Cosserats_, volume 21 of _Advances in Mechanics and Mathematics_, pages 55–63. Springer, Berlin, 2010. * [131] P. Neff, J. Jeong, and H. Ramezani. Subgrid interaction and micro-randomness - novel invariance requirements in infinitesimal gradient elasticity. _Int. J. Solids Struct._, 46(25-26):4261–4276, 2009. * [132] P. Neff and D. Knees. Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity. _SIAM J. Math. Anal._, 40(1):21–43, 2008. * [133] P. Neff and I. Münch. Curl bounds Grad on ${\rm SO}(3)$. _ESAIM: Control, Optimisation and Calculus of Variations_, 14(1):148–159, 2008. * [134] P. Neff and I. Münch. Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure. _Cont. Mech. Thermod._, 21(3):195–221, 2009. * [135] P. Neff, D. Pauly, and K.J. Witsch. Poincaré meets Korn via Maxwell: Extending Korn’s first inequality to incompatible tensor fields. _arXiv:1203.2744_, submitted. * [136] P. Neff, D. Pauly, and K.J. Witsch. A canonical extension of Korn’s first inequality to ${\rm H(Curl)}$ motivated by gradient plasticity with plastic spin. _C. R. Acad. Sci. Paris, Ser. I_, 349:1251–1254, 2011. * [137] P. Neff, D. Pauly, and K.J. Witsch. Maxwell meets Korn: a new coercive inequality for tensor fields in $\mathbb{R}^{N\times N}$ with square-integrable exterior derivative. _Math. Methods Appl. Sci._, 35:65–71, 2012. * [138] P. Neff, A. Sydow, and C. Wieners. Numerical approximation of incremental infinitesimal gradient plasticity. _Int. J. Num. Meth. Engrg._, 77(3):414–436, 2009. * [139] S. Nesenenko and P. Neff. Well-posedness for dislocation based gradient viscoplasticity I: Subdifferential case. _SIAM J. Math. Anal._, 44(3):1694–1712, 2012. * [140] S. Nesenenko and P. Neff. Well-posedness for dislocation based gradient visco-plasticity II: Monotone case. _MEMOCS: Mathematics and Mechanics of Complex Systems_, 1(2):149–176, 2013. * [141] J.W. Nunziato and S.C. Cowin. A nonlinear theory of elastic materials with voids. _Arch. Rat. Mech. Anal._, 72, 1979. * [142] J.F. Nye. Some geometrical relations in dislocated crystals. _Acta Metall._, 1:153–162, 1953. * [143] A. Pazy. _Semigroups of linear operators and applications to partial differential equations._ Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. * [144] C. Pideri and P. Seppecher. A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. _Cont. Mech. Thermo._, 9:241–257, 1997. * [145] W. Pietraszkiewicz and V.A. Eremeyev. On natural strain measures of the non-linear micropolar continuum. _Int. J. Solids Struct._, 46:774–787, 2009. * [146] W. Pietraszkiewicz and V.A. Eremeyev. On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. _Int. J. Solids Struct._, 46:2477–2480, 2009. * [147] W. Pietraszkiewicz and V.A. Eremeyev. Material symmetry group of the non-linear polar-elastic continuum. _Int. J. Solids Struct._, 49:1993–2005, 2012. * [148] G. Piola. Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, Modena, Tipi del R.D. Camera, 1846, translated by F. dell’Isola, U. Andreaus and L. Placidi. In U. Andreaus, F. dell’Isola, R. Esposito, S. Forest, G. Maier, and U. Perego, editors, _The complete works of Gabrio Piola_, volume I. Springer, in preparation. * [149] L. Placidi, G. Rosi, I. Giorgio, and A. Madeo. Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second gradient materials. _Math. Mech. Solids_, DOI: 10.1177/1081286512474016, 2013. * [150] V.L. Popov. Gauge theory of “plastically incompressible” medium without dissipation-I. Dispersion relations and propagation of perturbations without dissipation. _Int. J. Engng. Sci._, 30(3):329–334, 1992. * [151] V.L. Popov. Coupling of an elastoplastic continuum and a Cosserat continuum. _Russ. Phys. Journal_, 37:337–342, 1994. * [152] V.L. Popov. Dynamics of plastic rotations in a medium with dislocations and disclinations. _Tech. Phys. Letters_, 20:576, 1994. * [153] V.L. Popov and E. Kröner. Theory of elastoplastic media with mesostructure. _Theor. Appl. Fract. Mech._, pages 299–310, 2001. * [154] G. Rosi, I. Giorgio, and V.A. Eremeyev. Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. _ZAMM_, DOI: 10.1002/zamm.201200285, 2013. * [155] G. Rosi, A. Madeo, and J.-L. Guyader. Switch between fast and slow Biot compression waves induced by second gradient microstructure at material discontinuity surfaces in porous media. _Int. J. Solids Struct._, 50:1721–1746, 2013. * [156] C. Sansour. A unified concept of elastic-viscoplastic Cosserat and micromorphic continua. In A. Bertram and F. Sidoroff, editors, _Mechanics of Materials with Intrinsic Length Scale: Physics, Experiments, Modelling and Applications._, Journal Physique IV France 8, pages 341–348. EDP Sciences, France, 1998. * [157] G. Sciarra, F. dell’Isola, and O. Coussy. Second gradient poromechanics. _Int. J. Solids Struct._, 44:6607–6629, 2007. * [158] A.C. Smith. Inequalities between the constants of a linear micro-elastic solid. _Int. J. Eng. Sci._, 6:65–74, 1968. * [159] D.J. Steigmann. Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. _Int. J. Non-linear Mech._, 47(734-742), 2012. * [160] B. Svendsen, P. Neff, and A. Menzel. On constitutive and configurational aspects of models for gradient continua with microstructure. _Z. Angew. Math. Mech._, 89(8):687–697, 2009. * [161] R. Teisseyre. Earthquake process in a micromorphic continuum. _Pure Appl. Geophysics_, 102(1):15–28, 1973. * [162] R. Teisseyre. Symmetric micromorphic continuum: wave propagation, point source solutions and some applications to earthquake processes. In P. Thoft-Christensen, editor, _Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics_, volume 12 of _NATO Advanced Study Institute Series_. Springer, 1974. * [163] C. Teodosiu. Discussion on Papers by A.C. Eringen and W.D. Claus, Jr., and N. Fox. In J.A. Simmons, R. de Wit, and R. Bullough, editors, _Fundamental Aspects of Dislocation Theory._, volume 1 of _Nat. Bur. Stand. (U.S.), Spec. Publ._, pages 1054–1059. Spec. Publ., 1970. * [164] R.A. Toupin. Elastic materials with couple stresses. _Arch. Rat. Mech. Anal._, 11:385–413, 1962. * [165] R.A. Toupin. Theory of elasticity with couple stresses. _Arch. Rat. Mech. Anal._, 17:85–112, 1964. * [166] S. Vidoli and F. dell’Isola. Modal coupling in one-dimensional electromechanical structured continua. _Acta Mech._, 141:37–50, 2000. * [167] I. Vrabie. $C_{0}$_-Semigroups and applications, Ser. Mathematics Studies no. 191._ Elsevier, Amsterdam, 2003. * [168] X. Wang and J. Lee. Micromorphic theory: a gateway to nano world. _Int. J. of Smart and Nano Materials_, 1:115–135, 2010. * [169] Jr. W.D. Claus. Discussion on Papers by A.C. Eringen and W.D. Claus, Jr., and N. Fox. In J.A. Simmons, R. de Wit, and R. Bullough, editors, _Fundamental Aspects of Dislocation Theory._, volume 1 of _Nat. Bur. Stand. (U.S.), Spec. Publ._, pages 1054–1059. Spec. Publ., 1970. * [170] A. Yavari. Covariant balance laws in continua with microstructure. _Reports on Mathematical Physics_, 63:1–42, 2009. * [171] X. Zeng, Y. Chen, and J.D. Lee. Determining material constants in nonlocal micromorphic theory through phonon dispersion relations. _Int. J. Eng. Sci._, 44:1334–1345, 2006. * [172] S. Zouhdi, A. Sihvola, and A.P. Vinogradov. _Metamaterials and Plasmonics: Fundamentals, Modelling, Applications._ NATO Science for Peace and Security Series B: Physics and Biophysics. Springer-Verlag, New York, 2009. ## Appendix A Some useful identities In this Appendix we outline some identities which could be useful for the readers: * For all matrices $A\in\operatorname{\mathfrak{so}}(3)$ we have the Nye’s formula [142] \[-\operatorname{Curl}A =(\nabla\operatorname{axl}A)^{T}-\textrm{tr}[(\nabla\operatorname {axl}A)^{T}]{\,\!\cdot\!\,}1\!\!1,\] \[\nabla(\operatorname{axl}A) =-(\operatorname{Curl}A)^{T}+\frac{1}{2}\textrm{tr}[( \operatorname{Curl}A)^{T}]{\,\!\cdot\!\,}1\!\!1\,\quad\quad\text{ ``Nye's curvature tensor"}.\] * For all differentiable functions $\zeta:\mathbb{R}\rightarrow\mathbb{R}$ on $\Omega$ we have $\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1)=\left(\begin{array}[]{ccc}0& \zeta_{,3}&-\zeta_{,2}\\ -\zeta_{,3}&0&\zeta_{,1}\\ \zeta_{,2}&-\zeta_{,1}&0\end{array}\right)\in\operatorname{\mathfrak{so}}(3)$ and $\operatorname{Curl}\operatorname{Curl}(\zeta{\,\!\cdot\!\,}1\!\!1)=\left( \begin{array}[]{ccc}-(\zeta_{,22}+\zeta_{,33})&\zeta_{,12}&\zeta_{,13}\\ \zeta_{,12}&-(\zeta_{,11}+\zeta_{,33})&\zeta_{,23}\\ \zeta_{,13}&\zeta_{,23}&-(\zeta_{,11}+\zeta_{,22})\end{array}\right)\in{\rm{ Sym}}(3)$ * If $A=\left(\begin{array}[]{ccc}0&-x_{3}&x_{2}\\ x_{3}&0&-x_{1}\\ -x_{2}&x_{1}&0\end{array}\right),$ then $\operatorname{Curl}A=1\!\!1\in{\rm{Sym}}(3)$. * $\textrm{tr}(\operatorname{Curl}S)=0$ for all $S\in{\rm{Sym}}(3)$. * $\operatorname{Curl}[(\operatorname{Curl}S)^{T}]\in{\rm{Sym}}(3)\,,$ for all $S\in{\rm{Sym}}(3)$. * $\operatorname{Curl}[(\operatorname{Curl}A)^{T}]\in\operatorname{\mathfrak{so}} (3)\,,$ for all $A\in\operatorname{\mathfrak{so}}(3)$. * In view of b), e) and f) we have \[\operatorname{Curl}[(\operatorname{Curl}\operatorname{sym}P)^{T}]\in{\rm{Sym}} (3),\quad\quad\quad\operatorname{Curl}[(\operatorname{Curl}\mathop{\rm skew}P) ^{T}]\in\operatorname{\mathfrak{so}}(3)\quad\quad\forall\ P\in\mathbb{R}^{3 \times 3}.\] * $\textrm{tr}[\operatorname{Curl}(\mathop{\rm skew}\operatorname{Curl}A)]=0$ for all $A\in\operatorname{\mathfrak{so}}(3)$. * Saint-Venant compatibility conditions²⁰: if ${\mathbf{inc}}(S):=\operatorname{Curl}((\operatorname{Curl}S)^{T})=0$ and $S\in{\rm{Sym}}(3)$ then $S=\operatorname{sym}\nabla u$ in a simply connected domain. [FOOTNOTE:20][ENDFOOTNOTE] * $\nabla\operatorname{axl}(\mathop{\rm skew}\nabla u)=[\operatorname{Curl}( \operatorname{sym}\nabla u)]^{T}$ but $\nabla\operatorname{axl}(\mathop{\rm skew}P)\neq[\operatorname{Curl}( \operatorname{sym}P)]^{T}$ for general $P\in\mathbb{R}^{3\times 3}$. * $\mathfrak{a}_{1}\\|X\\|^{2}+\mathfrak{a}_{2}\langle X,X^{T}\rangle+\mathfrak{a}_ {3}[\textrm{tr}(X)]^{2}=(\mathfrak{a}_{1}+\mathfrak{a}_{2})\\|\operatorname{dev }\operatorname{sym}X\\|^{2}+(\mathfrak{a}_{1}-\mathfrak{a}_{2})\\|\mathop{\rm skew }X\\|^{2}+\frac{\mathfrak{a}_{1}+\mathfrak{a}_{2}+3\mathfrak{a}_{3}}{3}[\textrm {tr}(X)]^{2}$ for all $X\in\mathbb{R}^{3\times 3}$. * For all $P\in\mathbb{R}^{3\times 3}$ we have \[\textrm{tr}[\big{(}\operatorname{Curl}(\mathop{\rm skew}P)\big{)} ^{T}]=2\,{\rm div}\operatorname{axl}(\mathop{\rm skew}P)\] (A.1) \[\operatorname{Curl}\{\textrm{tr}[\big{(}\operatorname{Curl}( \mathop{\rm skew}\operatorname{axl}(\mathop{\rm skew}P))\big{)}^{T}]{\,\!\cdot \!\,}1\!\!1\}=-2\operatorname{anti}\nabla({\rm div}\operatorname{axl}(\mathop{ \rm skew}P)).\] * For all $P\in\mathbb{R}^{3\times 3}$ we have \[\operatorname{Curl}\{[\operatorname{Curl}(\mathop{\rm skew}P)]^{T }\}=\,-\operatorname{anti}\nabla({\rm div}\operatorname{axl}(\mathop{\rm skew} P))\,.\] (A.2) * We have the identity \[-2\operatorname{Curl}\{[\operatorname{Curl}(\mathop{\rm skew}P)]^ {T}\}+\operatorname{Curl}\{\textrm{tr}[\big{(}\operatorname{Curl}(\mathop{\rm skew }P)\big{)}^{T}]{\,\!\cdot\!\,}1\!\!1\}=0,\quad\quad\text{for all}\quad\quad P \in\mathbb{R}^{3\times 3}.\] (A.3) * If $\alpha_{1}=-6\alpha_{3}$ and $\alpha_{2}=6\alpha_{3}$, then \[\operatorname{Curl}\{\alpha_{1}\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+{ \alpha_{3}} \ \textrm{tr}(\operatorname{Curl}P)\!\cdot\!1\!\!1\}\] \[=-6\alpha_{3}\,\operatorname{Curl}\{[\operatorname{Curl}( \operatorname{sym}P)]^{T}\}\in{\rm{Sym}}(3)\] (A.4) for all $P\in\mathbb{R}^{3\times 3}$. * Given $P\in{\rm{Sym}}(3)$, then we have \[\operatorname{Curl}\{\alpha_{1}\operatorname{dev}\operatorname{ sym}\operatorname{Curl}P+\alpha_{2}\mathop{\rm skew}\operatorname{Curl}P+{ \alpha_{3}}\ \textrm{tr}(\operatorname{Curl}P)\!\cdot\!1\!\!1\}\in{\rm{Sym}}(3).\] (A.5) if and only if $\ \alpha_{1}=-\alpha_{2}$. * $\langle v,\operatorname{axl}(W)\rangle_{\mathbb{R}^{3}}=\frac{1}{2}\langle \operatorname{anti}(v),W\rangle_{\mathbb{R}^{3\times 3}}$ $\forall\ W\in\operatorname{\mathfrak{so}}(3)$. The adjoint of the operator $\operatorname{axl}:\operatorname{\mathfrak{so}}(3)\rightarrow\mathbb{R}^{3}$ is the mapping $\operatorname{axl}^{}:\mathbb{R}^{3}\rightarrow\operatorname{\mathfrak{so}}(3)$, $\operatorname{axl}^{}(\cdot)=\frac{1}{2}\,\operatorname{anti}(\cdot)$.
1604.08299	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 38689, "num_imgs": 0, "llama3_tokens_count": 12363 }	[]	# On the clique number of a strongly regular graph Gary R. W. Greaves Leonard H. Soicher Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan grwgrvs@gmail.com School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK L.H.Soicher@qmul.ac.uk ###### Abstract. We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs. Key words and phrases:Delsarte bound, Hoffman bound, block intersection polynomial, clique number, conference graphs, strongly regular graphs, clique adjacency bound. 2010 Mathematics Subject Classification: The first author was supported by JSPS KAKENHI; grant number: 26$\cdot$03903 ## 1. Introduction The clique number$\omega(\Gamma)$ of a graph $\Gamma$ is defined to be the cardinality of a clique of maximum size in $\Gamma$. For a $k$-regular strongly regular graph with smallest eigenvalue $s<0$, Delsarte [12, Section 3.3.2] proved that $\omega(\Gamma)\leqslant\lfloor 1-k/s\rfloor$; we refer to this bound as the Delsarte bound. Therefore, since one can write $s$ in terms of the parameters of $\Gamma$, one can determine the Delsarte bound knowing only the parameters $(v,k,\lambda,\mu)$ of $\Gamma$. In this paper we determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. Our bounds improve on the Delsarte bound infinitely often. Let $q=p^{k}$ be a power of a prime $p$ congruent to $1$ mod $4$. A Paley graph has vertex set equal to the finite field $\mathbb{F}_{q}$, and two vertices $a$ and $b$ are adjacent if and only if $a-b$ is a nonzero square. For a Paley graph $\Gamma$ on $q$ vertices with $k$ even, Blokhuis [4] showed that $\omega(\Gamma)=\sqrt{q}$; this corresponds to equality in the Delsarte bound. Bachoc et al. [1] recently considered the case when $\Gamma$ is a Paley graph on $q$ vertices with $k$ odd and, for certain such $q$, showed that $\omega(\Gamma)\leqslant\lfloor\sqrt{q}-1\rfloor$. This corresponds to an improvement to the Delsarte bound for these Paley graphs. Here, working much more generally, given a strongly regular graph $\Gamma$ with parameters $(v,k,\lambda,\mu)$, we provide inequalities in terms of the parameters of $\Gamma$ that, when satisfied, guarantee that the clique number of $\Gamma$ is strictly less than the Delsarte bound. We show that these inequalities are satisfied by infinitely many feasible parameters tuples for strongly regular graphs and, in particular, are satisfied by infinitely many parameter tuples that correspond to Paley graphs. Our inequalities are obtained using what we call the “clique adjacency bound” (see Section 4), a bound defined by the second author [17]. We also show that the clique adjacency bound is always at most the Delsarte bound when applied to strongly regular graphs. The paper is organised as follows. In Section 2 we state our main results and in Section 3 we state some standard identities that we will use in our proofs. Section 4 contains the proofs of our main results. In Section 5 we examine the strength of the clique adjacency bound and in Section 6 we provide an illustrative example comparing certain bounds for the clique number of an edge-regular graph that is not necessarily strongly regular. Finally, we give an appendix in which we describe our symbolic computations. ## 2. Definitions and main results A non-empty $k$-regular graph on $v$ vertices is called edge-regular if there exists a constant $\lambda$ such that every pair of adjacent vertices has precisely $\lambda$ common neighbours. The triple $(v,k,\lambda)$ is called the parameter tuple of such a graph. A strongly regular graph$\Gamma$ with parameter tuple$(v,k,\lambda,\mu)$ is defined to be a non-complete edge-regular graph with parameter tuple $(v,k,\lambda)$ such that every pair of non-adjacent vertices has precisely $\mu$ common neighbours. We refer to the elements of the parameter tuple as the parameters of $\Gamma$. We call the parameter tuple of a strongly regular graph feasible if its elements satisfy certain nonnegativity and divisibility constraints given by Brouwer [6, VII.11.5]). Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$. It is well-known that $\Gamma$ has at most three distinct eigenvalues, and moreover, the eigenvalues can be written in terms of the parameters of $\Gamma$ (see [14, Section 10.2]). In what follows we denote the eigenvalues of $\Gamma$ as $k>r\geqslant s$. Strongly regular graphs whose parameters satisfy $k=(v-1)/2$, $\lambda=(v-5)/4$, and $\mu=(v-1)/4$ are called type I or conference graphs. Strongly regular graphs all of whose eigenvalues are integers are called type II. Every strongly regular graph is either type I, type II, or both type I and type II (see Cameron and Van Lint [9, Chapter 2]). The fractional part of a real number $a\in\mbox{$\mathbb{R}$}$ is defined as $\operatorname{frac}\left(a\right):=a-\lfloor a\rfloor$. We are now ready to state our main results. Theorem 2.1.: _Let $\Gamma$ be a type-I strongly regular graph with $v$ vertices. Suppose that_ \[0<\operatorname{frac}\left(\sqrt{v}/2\right)<1/4+(\sqrt{v}-\sqrt{v+5/4})/2.\] _Then $\omega(\Gamma)\leqslant\lfloor\sqrt{v}-1\rfloor$._ Proof.: Follows from Theorem 4.1 together with Corollary 4.6 below. ∎ _Remark 2.2_.: For a prime $p$ satisfying $0<\operatorname{frac}\left(\sqrt{p}/2\right)<1/4+(\sqrt{p}-\sqrt{p+5/4})/2$, we have that $\lfloor\sqrt{p}\rfloor=2\lfloor\sqrt{p}/2\rfloor$ is even. Furthermore, for $n:=\lfloor\sqrt{p}\rfloor$, since $\lfloor\sqrt{p}\rfloor>\sqrt{p+5/4}-1/2$, we have $n^{2}+n-1>p$. Hence, if $\Gamma$ is a Paley graph on $p$ vertices then $\omega(\Gamma)\leqslant\lfloor\sqrt{p}-1\rfloor$ by Bachoc et al. [1, Theorem 2.1 (i)] (see [1, Remark 2.5]). Therefore, for type-I strongly regular graphs with $p$ (a prime) vertices, satisfying $0<\operatorname{frac}\left(\sqrt{p}/2\right)<1/4+(\sqrt{p}-\sqrt{p+5/4})/2$, Theorem 2.1 is a generalisation of Bachoc et al. [1, Theorem 2.1 (i)]. _Remark 2.3_.: Let $g$ be a positive integer. Then $(1+4g,2g,g-1,g)$ is a feasible parameter tuple for a type-I strongly regular graph on $v=1+4g$ vertices. Observe that $(\sqrt{v}-\sqrt{v+5/4})/2$ tends to $0$ as $v$ tends to infinity. Using Fejér’s theorem (see Kuipers and Niederreiter [15, page 13]), it is straightforward to show that the sequence $(\sqrt{1+4g}/2)_{g\in\mathbb{N}}$ is uniformly distributed modulo $1$. Therefore we can apply Theorem 2.1 to about a quarter of all feasible parameter tuples for type-I strongly regular graphs. Let $\mathcal{P}$ denote the set of all primes $p$ of the form $p=1+4g$ for some $g\in\mathbb{N}$. Then the sequence $(\sqrt{p}/2)_{p\in\mathcal{P}}$ is uniformly distributed modulo $1$ (see Balog [2, Theorem 1]). Therefore, since Paley graphs on $p$ vertices exist for all $p\in\mathcal{P}$, Theorem 2.1 is applicable to infinitely many strongly regular graphs. Note that the example in [17] with parameters $(65,32,15,16)$ is an example of a (potential) graph satisfying the hypothesis of Theorem 2.1. A graph is called co-connected if its complement is connected. We have the following: Theorem 2.4.: _Let $\Gamma$ be a co-connected type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Suppose that_ \[0<\operatorname{frac}\left(-k/s\right)<1-(r^{2}+r)/(v-2k+\lambda).\] _Then $\omega(\Gamma)\leqslant\lfloor-k/s\rfloor$._ Proof.: Follows from Theorem 4.1 together with Corollary 4.9 below. ∎ _Remark 2.5_.: Currently Brouwer [7] lists the feasible parameter tuples for connected and co-connected strongly regular graphs on up to $1300$ vertices. Of these, about $1/8$ of the parameter tuples of type-II strongly regular graphs satisfy the hypothesis of Theorem 2.4. By the remark following Corollary 4.9, it follows that Theorem 2.4 can be applied to about $1/4$ of the complementary pairs of type-II strongly regular graphs on Brouwer’s list. Note that the example in [17] of a strongly regular graph with parameter tuple $(144,39,6,12)$ is an example of a graph satisfying the hypothesis of Theorem 2.4; in fact, in this case, the conclusion of Theorem 2.4 is satisfied with equality. The parameter tuple $(88,27,6,9)$ is the first parameter tuple in Brouwer’s list to which we can apply Theorem 2.4 and whose corresponding graphs are not yet known to exist (or not exist). ## 3. Parameters of strongly regular graphs Here we state some well-known properties of strongly regular graphs and their parameters. The first two propositions are standard (see Brouwer and Haemers [8, Chapter 9] or Cameron and Van Lint [9, Chapter 2]). Proposition 3.1.: _Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Then_ \[(v-k-1)\mu =k(k-\lambda-1);\] \[\lambda-\mu =r+s;\] \[\mu-k =rs.\] Proposition 3.2.: _Let $\Gamma$ be a type-I strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r>s$. Then_ \[k=(v-1)/2;\quad\lambda=(v- 5)/4;\quad\mu=(v-1)/4;\] \[r=(\sqrt{v}-1)/2; \quad\quad s=-(\sqrt{v}+1)/2.\] The next proposition is a key observation. Proposition 3.3.: _Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$._ 1. _[(i)]_ 2. _If_ $\Gamma$ _is type I then_ $k/s-2\operatorname{frac}\left(\mu/s\right)$ _is an integer._ 3. _If_ $\Gamma$ _is type II then_ $k/s-\operatorname{frac}\left(\mu/s\right)$ _is an integer._ Proof.: If $\Gamma$ is type I then, by Proposition 3.2, we have $k-2\mu=0$. If $\Gamma$ is type II then, by Proposition 3.1, we have $k/s-\mu/s=-r$ and $r$ is an integer. ∎ Next, the complement $\overline{\Gamma}$ of a strongly regular graph $\Gamma$ is also a strongly regular graph. This is again a standard result (see Cameron and Van Lint [9, Chapter 2]). Proposition 3.4.: _Let $\Gamma$ be a connected and co-connected strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r>s$. Then $\overline{\Gamma}$ is strongly regular with parameters $(v,v-k-1,v-2k+\mu-2,v-2k+\lambda)$ and eigenvalues $v-k-1>-s-1>-r-1$._ Finally we state some straightforward bounds for the parameters of strongly regular graphs. Proposition 3.5.: _Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$. Then_ 1. _[(i)]_ 2. $v-2k+\lambda\geqslant 0$ _with equality if and only if_ $\Gamma$ _is complete multipartite;_ 3. $k-\lambda-1\geqslant 0$ _with equality if and only if_ $\overline{\Gamma}$ _is complete multipartite._ ## 4. The clique adjacency polynomial Now we define our main tool, the clique adjacency polynomial. Given an edge-regular graph $\Gamma$ with parameters $(v,k,\lambda)$, define the clique adjacency polynomial$C_{\Gamma}(x,y)$ as \[C_{\Gamma}(x,y):=x(x+1)(v-y)-2xy(k-y+1)+y(y-1)(\lambda-y+2).\] The utility of the clique adjacency polynomial follows from [16, Theorem 1.1] (see also [17, Theorem 3.1]), giving: Theorem 4.1.: _Let $\Gamma$ be an edge-regular graph with parameters $(v,k,\lambda)$. Suppose that $\Gamma$ has a clique of size $c\geqslant 2$. Then $C_{\Gamma}(b,c)\geqslant 0$ for all integers $b$._ As discussed in [16] and [17], Theorem 4.1 provides a way of bounding the clique number of an edge-regular graph using only its parameters. Indeed, by Theorem 4.1, for an edge-regular graph $\Gamma$ and some integer $c\geqslant 2$, if there exists an integer $b$ such that $C_{\Gamma}(b,c)<0$ then $\omega(\Gamma)\leqslant c-1$. Hence we define the clique adjacency bound (CAB) to be the least integer $c\geqslant 2$ such that $C_{\Gamma}(b,c+1)<0$ for some $b\in\mathbb{Z}$; note that such a $c$ always exists. We will show that, for a $k$-regular strongly regular graph $\Gamma$, the clique adjacency bound gives $\omega(\Gamma)\leqslant\lfloor 1-k/s\rfloor$. That is, the clique adjacency bound is always at least as good as the Delsarte bound when applied to strongly regular graphs. This follows from Theorem 4.1 together with Theorem 4.2 below. More interestingly, we will also show that the clique adjacency bound does better than the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs. In this section we consider the univariate polynomial $C_{\Gamma}(f(t),g(t))$ in the variable $t$, where $f(t)$ and $g(t)$ are linear polynomials in $t$. The main idea is to choose the linear polynomials $f$ and $g$ such that there exists $t\in\mbox{$\mathbb{R}$}$ such that $C_{\Gamma}(f(t),g(t))<0$, $f(t)\in\mathbb{Z}$, and $g(t)$ is an integer at least $2$. We begin by stating one of the main results of this paper. Theorem 4.2.: _Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Then,_ Proof.: Follows from Corollary 4.5 and Corollary 4.8 below. ∎ Observe that, together with Theorem 4.1, Theorem 4.2 shows that the clique adjacency bound always does as well as the Delsarte bound for strongly regular graphs. Now we can state our first polynomial identity, which shows that the clique adjacency polynomial is negative at a certain point. Lemma 4.3.: _Let $\Gamma$ be a connected strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r>s$. Then_ \[C_{\Gamma}(-\mu/s,2-k/s)={(2s-r)(r+1)}<0.\] Proof.: The equality follows from direct calculation (see Appendix A), using the equalities in Proposition 3.1. The right-hand side is negative since $s<0$ and $r\geqslant 0$. ∎ Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ such that both $\mu/s$ and $k/s$ are integers. Then by Lemma 4.3, together with Theorem 4.1, we recover the Delsarte bound, i.e., $\omega(\Gamma)\leqslant\left\lfloor 1-k/s\right\rfloor$. It remains for us to deal with the situation when $k/s$ and $\mu/s$ are not integers. In the remainder of this section, motivated by Lemma 4.3, we consider integral points $(x,y)\in\mathbb{Z}^{2}$ close to $(-\mu/s,2-k/s)$ such that $C_{\Gamma}(x,y)$ is negative. We deal with the type I and type II cases separately. ### Type-I strongly regular graphs Let $\Gamma$ be a type-I strongly regular graph (or conference graph) with $v$ vertices. By Proposition 3.2 we have $-\mu/s=r$ and $-k/s=2r$. Therefore, we consider integral points $(x,y)$ close to $(r,2+2r)$ at which to evaluate the clique adjacency polynomial. In view of Proposition 3.3, we evaluate $C_{\Gamma}(x,y)$ at points of the form $(r-t,a+2r-2t)$ for some $a\in\mathbb{N}$, thinking of $t$ as the fractional part of $r$. Lemma 4.4.: _Let $\Gamma$ be a type-I strongly regular graph with $v$ vertices and eigenvalues $k>r>s$. Then_ (1) \[C_{\Gamma}(r-t,3+2r-2t) =2(t-1)(t+s-2)(t+2s);\] (2) \[C_{\Gamma}(r-t,2+2r-2t) =(t+s)(2t^{2}+(4s-1)t-3s-1).\] Proof.: The equalities follow from direct calculation (see Appendix A), applying Proposition 3.1 and the definition of a type-I strongly regular graph. ∎ The right-hand side of Equation (2) is a cubic polynomial in the indeterminate $t$ with positive leading coefficient. Furthermore, since for a type-I strongly regular graph we have $s=-(\sqrt{v}+1)/2$, we observe that the smallest zero of the right-hand side of Equation (2) is equal to $3/4+(\sqrt{v}-\sqrt{v+5/4})/2$. Hence $C_{\Gamma}(r-t,2+2r-2t)$ is negative for $t<3/4+(\sqrt{v}-\sqrt{v+5/4})/2$. We use this observation in the next result, which can be used with Theorem 4.1 to obtain the Delsarte bound for conference graphs. Corollary 4.5.: _Let $\Gamma$ be a type-I strongly regular graph with $v$ vertices. Then_ \[C_{\Gamma}\left(\left\lfloor(\sqrt{v}-1)/2\right\rfloor,\left\lfloor\sqrt{v}+1 \right\rfloor\right)<0.\] Proof.: Let $t=\operatorname{frac}\left(r\right)$. If $t>1/2$ then \[C_{\Gamma}\left(\left\lfloor(\sqrt{v}-1)/2\right\rfloor,\left\lfloor\sqrt{v}+1 \right\rfloor\right)=C_{\Gamma}(r-t,3+2r-2t)\] and the right-hand side of Equation (1) is negative for $t<1$. Otherwise, if $t<1/2$ then \[C_{\Gamma}\left(\left\lfloor(\sqrt{v}-1)/2\right\rfloor,\left\lfloor\sqrt{v}+1 \right\rfloor\right)=C_{\Gamma}(r-t,2+2r-2t),\] which is negative since $t<1/2<3/4+(\sqrt{v}-\sqrt{v+5/4})/2$. Note that $t$ cannot be equal to $1/2$ since $r$ is an algebraic integer. ∎ The next corollary follows in a similar fashion. Corollary 4.6.: _Let $\Gamma$ be a type-I strongly regular graph with $v$ vertices. Suppose that_ \[0<\operatorname{frac}\left(\sqrt{v}/2\right)<1/4+(\sqrt{v}-\sqrt{v+5/4})/2.\] _Then $C_{\Gamma}\left(\left\lfloor(\sqrt{v}-1)/2\right\rfloor,\left\lfloor\sqrt{v} \right\rfloor\right)<0$._ Proof.: Let $t=\operatorname{frac}\left(r\right)=\operatorname{frac}\left((\sqrt{v}-1)/2\right)$. Then by our hypothesis $1/2<t<3/4+(\sqrt{v}-\sqrt{v+5/4})/2$. Therefore we have \[C_{\Gamma}\left(\left\lfloor(\sqrt{v}-1)/2\right\rfloor,\left\lfloor\sqrt{v} \right\rfloor\right)=C_{\Gamma}(r-t,2+2r-2t),\] which is negative since $t<3/4+(\sqrt{v}-\sqrt{v+5/4})/2$. ∎ ### Type-II strongly regular graphs Let $\Gamma$ be a type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$. Again, in view of Proposition 3.3, we evaluate $C_{\Gamma}(x,y)$ at points of the form $(-\mu/s-t,a-k/s-t)$ for some $a\in\mathbb{Z}$, thinking of $t$ as the fractional part of $-\mu/s$. Lemma 4.7.: _Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Then_ (3) \[C_{\Gamma}(-\mu/s-t,2-k/s-t) =(t-1)((v-2k+\lambda)t-(2s-r)(r+1));\] (4) \[C_{\Gamma}(-\mu/s-t,1-k/s-t) =t((v-2k+\lambda)(t-1)+r(r+1)).\] _Moreover, if $\Gamma$ is co-connected then these polynomials have positive leading coefficients._ Proof.: The equalities follow from direct calculation (see Appendix A), using the equalities in Proposition 3.1. By Proposition 3.5 if $\Gamma$ is co-connected then the polynomials have positive leading coefficients. ∎ Note that Lemma 4.3 is a special case of Lemma 4.7, for $t=0$. Corollary 4.8.: _Let $\Gamma$ be a type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Then_ \[C_{\Gamma}(\lfloor-\mu/s\rfloor,\lfloor 2-k/s\rfloor)<0.\] Proof.: If $\Gamma$ is disconnected then we have $\mu=0$ and $2-k/s=\lambda+3$. Whence $C_{\Gamma}(\lfloor-\mu/s\rfloor,\lfloor 2-k/s\rfloor)=C_{\Gamma}(0,\lambda+3)= -(\lambda+3)(\lambda+2)<0$, as required. Hence we can assume that $\Gamma$ is connected. Let $t=\operatorname{frac}\left(-\mu/s\right)$. Then, using Proposition 3.3 and Equation (3), we have \[C_{\Gamma}(\lfloor-\mu/s\rfloor,\lfloor 2-k/s\rfloor)=(t-1)((v-2k+\lambda)t-(2 s-r)(r+1)).\] Suppose first that $\Gamma$ is co-connected. The right-hand side of Equation (3) is negative on the open interval $(\eta,1)$, where $\eta=(2s-r)(r+1)/(v-2k+\lambda)$ is negative. Hence the corollary holds for $\Gamma$. On the other hand, for complete multipartite graphs we have $t=0$, in which case the right-hand side of Equation (3) is negative. ∎ The next corollary follows similarly, using the fact that the right-hand side of Equation (4) is negative on the open interval $(0,\eta)$, where $\eta=1-(r^{2}+r)/(v-2k+\lambda)$. Corollary 4.9.: _Let $\Gamma$ be a co-connected type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Suppose that_ \[0<\operatorname{frac}\left(-k/s\right)<1-(r^{2}+r)/(v-2k+\lambda).\] _Then $C_{\Gamma}(\lfloor-\mu/s\rfloor,\lfloor 1-k/s\rfloor)<0$._ _Remark 4.10_.: We remark that if a type-II strongly regular graph satisfies the hypothesis of Corollary 4.9 then its complement cannot also satisfy the hypothesis. Indeed, suppose that $\Gamma$ satisfies the hypothesis of Corollary 4.9. Since $\operatorname{frac}\left(-k/s\right)>0$ we have that $s\neq-1$ and hence $\Gamma$ is connected. Then, using Proposition 3.4, we see that the complement of $\Gamma$ also satisfies the hypothesis of Corollary 4.9 if \[0<\operatorname{frac}\left((v-k-1)/(r+1)\right)<1-(s^{2}+s)/\mu.\] In particular, for the corollary to hold for both $\Gamma$ and its complement, we must have both $(r^{2}+r)/(v-2k+\lambda)<1$ and $(s^{2}+s)/\mu<1$. But we find that $(r^{2}+r)/(v-2k+\lambda)<1$ if and only if $(s^{2}+s)/\mu>1$. One can see this by using the equalities in Proposition 3.1 (see Appendix A) to obtain the equality \[\mu(v-2k+\lambda)=(r^{2}+r)(s^{2}+s).\] ## 5. How sharp is the clique adjacency bound? In this section we show that the clique adjacency bound is sharp for strongly regular graphs in certain instances. We also comment on the sharpness of the clique adjacency bound for general strongly regular graphs. Theorem 5.1.: _Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r\geqslant s$. Suppose that $\lambda+1\leqslant-k/s$. Then the clique adjacency bound is equal to $\lambda+2$._ As the second author observed in [17], for an edge-regular graph with parameters $(v,k,\lambda)$ we have $C_{\Gamma}(0,y)=-y(y-1)(y-(\lambda+2))$, so for all $y>\lambda+2$, we have $C_{\Gamma}(0,y)<0$. Hence the clique adjacency bound is always at most the trivial bound of $\lambda+2$. Therefore, to prove Theorem 5.1, it suffices to show that, for strongly regular graph parameters satisfying $\lambda+1\leqslant-k/s$, the clique adjacency bound is at least $\lambda+2$. Lemma 5.2.: _Let $\Gamma$ be a connected type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k>r>s$. Suppose that $\lambda+1\leqslant-k/s$. Then $C_{\Gamma}(1,\lambda+2)\geqslant 0$ with equality if and only if $\lambda=-k/s-1$._ Proof.: First suppose $\lambda+1=-k/s$. Equivalently, since $\lambda+1=k+(r+1)(s+1)$ and $\mu=k+rs$, we have $-\mu/s=1$. In this case, $C_{\Gamma}(1,\lambda+2)=C_{\Gamma}(-\mu/s,1-k/s)$, which is zero by Equation (4). It remains to assume $\lambda+1<-k/s$. Using Proposition 3.1 (see Appendix A) we can write \[\frac{\mu}{2}C_{\Gamma}(1,\lambda+2)=k(k-(\mu+1)(\lambda+1))+\mu(\lambda+1)^{2}.\] To show this quantity is nonnegative, it suffices to show that $k-(\mu+1)(\lambda+1)$ is nonnegative. Using the inequality $\lambda+1<-k/s$, we have $k-(\mu+1)(\lambda+1)>k(1+(\mu+1)/s)$. It therefore suffices to show that $1+(\mu+1)/s\geqslant 0$. Since $\lambda=k+r+s+rs$, the inequality $\lambda+1<-k/s$ becomes \[k+(r+1)(s+1)<-k/s.\] Since $s<-1$, it follows that $r+1>-k/s$. Multiplying this inequality by $-s$ gives $-s(r+1)>k$. Since both $s$ and $r$ are integers, we have $-s(r+1)\geqslant k+1$ Now by rearranging and substituting $\mu=k+rs$, we obtain the inequality $1+(\mu+1)/s\geqslant 0$ as required. ∎ Lemma 5.3.: _Let $\Gamma$ be an edge-regular graph with parameters $(v,k,\lambda)$ such that $C_{\Gamma}(b,\lambda+2)\geqslant 0$ for all integers $b$. Then $C_{\Gamma}(b,c)\geqslant 0$ for all $c\in\{2,\ldots,\lambda+2\}$ and all integers $b$._ Proof.: Let $c\in\{2,\ldots,\lambda+2\}$ and let $b$ be an integer. If $b\leqslant 0$, then from the definition of the clique adjacency polynomial $C_{\Gamma}(x,y)$, we see that $C_{\Gamma}(b,c)\geqslant 0$, so we now assume that $b$ is positive. A calculation (see Appendix A) shows that \[C_{\Gamma}(b,c)-C_{\Gamma}(b,\lambda+2)=(\lambda+2-c)(b-c)(b-c+1)+2b(\lambda+2 -c)(k-\lambda-1).\] This quantity is nonnegative since $b$ and $\lambda+2-c$ are nonnegative integers, the product of two consecutive integers is nonnegative, and $k-\lambda-1$ is also nonnegative by Proposition 3.5. Hence \[C_{\Gamma}(b,c)\geqslant C_{\Gamma}(b,\lambda+2)\geqslant 0,\] as required. ∎ Now we prove Theorem 5.1. Proof of Theorem 5.1.: Firstly, if $\Gamma$ is disconnected then $\Gamma$ is the disjoint union of complete graphs and hence contains cliques of size $\lambda+2$. Therefore the clique adjacency bound is at least $\lambda+2$. Now we assume that $\Gamma$ is connected. By Lemma 5.3, the clique adjacency bound is less than $\lambda+2$ only if there exists some integer $b$ such that $C_{\Gamma}(b,\lambda+2)$ is less than zero. To ease notation set $f(x):=C_{\Gamma}(x,\lambda+2)$. Hence \[f(x)=x\left((v-\lambda-2)x+v+(2\lambda-2k+1)(\lambda+2)\right).\] It suffices to show that there does not exist any integer $b$ such that $f(b)<0$. Observe that the polynomial $f(x)$ is a quadratic polynomial in the variable $x$. Furthermore, the leading coefficient of $f(x)$ is $v-\lambda-2\geqslant 0$, and $f(0)=0$. Let $\xi$ be the other zero of $f(x)$. Now, $f(x)$ is negative if and only if $x$ is between $0$ and $\xi$. Hence, if $f(-1)$ and $f(1)$ are both nonnegative then there are no integers $b$ such that $f(b)<0$. As in the proof of the previous result $f(-1)$ is nonnegative. Therefore Lemma 5.2 completes the proof for type-II strongly regular graphs. The inequality $\lambda+1\leqslant-k/s$ only holds for type-I strongly regular graphs on $5$ vertices or $9$ vertices (where we have equality). One can explicitly compute the clique adjacency bound for these two cases: the unique $(5,2,0,1)$-strongly regular graph and the unique $(9,4,1,2)$-strongly regular graph. For each of these graphs the clique adjacency bound is equal to $\lambda+2$. ∎ Now we give a couple of remarks about Theorem 5.1. _Remark 5.4_.: For strongly regular graphs with $\lambda\leqslant 1$, it is easy to see that the clique number is $\lambda+2$. By Theorem 5.1, the clique adjacency bound is equal to the clique number for such graphs. Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$. By Proposition 3.1, we see that $k=-s(r+1)-r+\lambda$. Therefore, for strongly regular graphs with $\lambda=2$ and $r\geqslant 2$, we have $\lambda+1=3\leqslant-k/s$, and so Theorem 5.1 applies to such graphs. _Remark 5.5_.: We conjecture that if the clique adjacency bound is less than $-k/s$ then $\lambda+1\leqslant-k/s$. We have verified this conjecture for all feasible parameter tuples for strongly regular graphs on up to $1300$ vertices, making use of Brouwer’s website [7]. In Table 1, we list all the feasible parameter tuples for strongly regular graphs on at most $150$ vertices to which we can apply either Theorem 2.1 or Theorem 2.4. In other words, Table 1 displays the feasible parameters for strongly regular graphs on at most $150$ vertices for which the clique adjacency bound is strictly less than the Delsarte bound. In the column labelled ‘Exists’, if there exists a strongly regular graph with the appropriate parameters then we put ‘+’, or ‘!’ if the graph is known to be unique; otherwise, if the existence is unknown, we put ‘?’. In the final column of Table 1, we put ‘Y’ (resp. ‘N’) if there exists (resp. does not exist) a strongly regular graph with the corresponding parameters that has clique number equal to the clique adjacency bound, otherwise we put a ‘?’ if such existence is unknown. We refer to Brouwer’s website [7] for details on the existence of strongly regular graphs with given parameters. For the parameter tuples in Table 1, the Delsarte bound is equal to the clique adjacency bound plus $1$. As an example of a parameter tuple for which the clique adjacency bound differs from the Delsarte bound by $2$, we have $(378,52,1,8)$ for which there exists a corresponding graph [11]. For this graph the Delsarte bound is $5$, but the clique adjacency bound is $3$. Parameters \| Type \| CAB \| Exists \| Sharp ---\|---\|---\|---\|--- (17,8,3,4) \| I \| 3 \| ! \| Y (37,18,8,9) \| I \| 5 \| + \| Y (50,7,0,1) \| II \| 2 \| ! \| Y (56,10,0,2) \| II \| 2 \| ! \| Y (65,32,15,16) \| I \| 7 \| ? \| ? (77,16,0,4) \| II \| 2 \| ! \| Y (88,27,6,9) \| II \| 4 \| ? \| ? (99,14,1,2) \| II \| 3 \| ? \| Y (100,22,0,6) \| II \| 2 \| ! \| Y (101,50,24,25) \| I \| 9 \| + \| ? (105,32,4,12) \| II \| 3 \| ! \| Y (111,30,5,9) \| II \| 4 \| ? \| ? (115,18,1,3) \| II \| 3 \| ? \| Y (120,42,8,18) \| II \| 3 \| ! \| Y (121,36,7,12) \| II \| 4 \| ? \| ? (133,32,6,8) \| II \| 5 \| ? \| ? (144,39,6,12) \| II \| 4 \| + \| Y (144,52,16,20) \| II \| 6 \| ? \| ? (145,72,35,36) \| I \| 11 \| ? \| ? (149,74,36,37) \| I \| 11 \| + \| ? Table 1. Feasible parameter tuples for strongly regular graphs on at most 150 vertices to which we can apply either Theorem 2.1 or Theorem 2.4. Feasible parameters for which there does not exist a corresponding strongly regular graph whose clique number is equal to the clique adjacency bound include $(16,10,6,6)$ and $(27,16,10,8)$. However, we ask the following question. Do there exist strongly regular graphs with parameters $(v,k,\lambda,\mu)$, with $k<v/2$, such that every strongly regular graph having those parameters has clique number less than the clique adjacency bound? ## 6. Hoffman bound vs Delsarte bound vs clique adjacency bound Let $\Gamma$ be a connected non-complete regular graph with $v$ vertices, valency $k$, and second largest eigenvalue $r<k$. Then the complement $\overline{\Gamma}$ of $\Gamma$ is a regular graph with valency $\overline{k}=v-k-1$ and least eigenvalue $\overline{s}=-r-1<0$. We may obtain a bound for the clique number of $\Gamma$ by applying the Hoffman bound (also called the ratio bound) [13, Theorem 2.4.1] on the size of a largest independent set (coclique) of $\overline{\Gamma}$. This gives \[\omega(\Gamma)\leqslant\left\lfloor\frac{v}{1-\overline{k}/\overline{s}}\right\rfloor.\] If $\Gamma$ is strongly regular, then it is known (and follows from the relations of Proposition 3.1) that the Delsarte bound for $\omega(\Gamma)$ is the same as that given by the Hoffman bound above. Now the Delsarte bound applies not only to strongly regular graphs, but also to the graphs $\{\Gamma_{1},\ldots,\Gamma_{d}\}$ of the relations (other than equality) of any $d$-class symmetric association scheme (see [13, Corollary 3.7.2]). Thus, if $\Gamma$ is such a graph, having valency $k$ and least eigenvalue $s$, then $\omega(\Gamma)\leqslant\lfloor 1-k/s\rfloor$. Here is an interesting illustrative example. Let $\Delta$ be the edge graph (or line graph) of the incidence graph of the projective plane of order $2$. Then $\Delta$ is the unique distance-regular graph with intersection array $\{4,2,2;1,1,2\}$. Now let $\Delta_{3}$ be the graph on the vertices of $\Delta$, with two vertices joined by an edge if and only if they have distance $3$ in $\Delta$. Then $\Delta_{3}$ is the graph of a relation in the usual symmetric association scheme associated with a distance-regular graph, where two vertices are in relation $i$ precisely when they are at distance $i$ in the distance-regular graph. The graph $\Delta_{3}$ has diameter $2$ and is edge-regular (but not strongly regular) with parameters $(v,k,\lambda)=(21,8,3)$. The clique adjacency bound for $\Delta_{3}$ is $4$. The least eigenvalue of $\Delta_{3}$ is $-\sqrt{8}$, and the Delsarte bound gives $3$, and indeed, $\omega(\Delta_{3})=3$. However, the complement of $\Delta_{3}$ has least eigenvalue $-1-\sqrt{8}$, and the Hoffman bound for independent sets in the complement of $\Delta_{3}$ gives $5$. Thus, for $\Delta_{3}$, the Delsarte bound is better than the clique adjacency bound which is better than that obtained from the Hoffman bound. However, the three bounds are for different classes of graphs. For example, there may well be an edge-regular graph with parameters $(21,8,3)$ and clique number $4$. It would be interesting to find one. We conjecture that if $\Gamma$ is any connected non-complete edge-regular graph, then the clique adjacency bound for $\omega(\Gamma)$ is at most that obtained from the Hoffman bound for $\overline{\Gamma}$. ## Appendix A Algebraic computational verification of identities In this appendix we present the algebraic computations in Maple [3] that were used to verify certain identities employed in this paper. These identities were also checked independently using Magma [5]. We start up Maple (version 18) and assign to $C$ the clique adjacency polynomial. Ψ> C:=x(x+1)(v-y)-2xy(k-y+1)+y(y-1)(lambda-y+2): Ψ We then make a set of relators, obtained from Proposition 3.1, which evaluate to $0$ on the parameters $(v,k,\lambda,\mu)$ and eigenvalues $r,s$ (with $k>r\geqslant s$) of a strongly regular graph. Ψ> srg_rels:={(v-k-1)mu-k(k-lambda-1),(lambda-mu)-(r+s),(mu-k)-rs}: Ψ We make a further set of relators which evaluate to $0$ on the parameters and eigenvalues of a type-I strongly regular graph. Ψ> type1_rels:=srg_rels union {2k-(v-1),4lambda-(v-5),4mu-(v-1)}: Ψ Let $R=\mathbb{Q}[t,v,k,\lambda,\mu,r,s]$ be the ring of polynomials over $\mathbb{Q}$ in the indeterminates $t,v,k,\lambda,\mu,r,s$, let $S$ be the ideal of $R$ generated by srg_rels given above, and let $T$ be the ideal of $R$ generated by type1_rels. We use the Maple package Groebner to caclulate and employ Gröbner bases [10] to work in the factor rings $R/S$ and $R/T$. We set the monomial ordering for the Gröbner basis calculations to be the Maple tdeg ordering, more commonly called the grevlex ordering, with the indeterminates ordered as $t>v>k>\lambda>\mu>r>s$. Ψ> ordering:=tdeg(t,v,k,lambda,mu,r,s): Ψ Then we compute a Gröbner basis $G$ for $S$. Ψ> G:=Groebner[Basis](srg_rels,ordering): Ψ For the record, $G=[\lambda-\mu-r-s,rs+k-\mu,{k}^{2}-kr-ks-\mu\,v-k+\mu]$. Similarly, we compute a Gröbner basis $H$ for $T$. Ψ> H:=Groebner[Basis](type1_rels,ordering): Ψ Here, we obtain $H=[r+1+s,\lambda+1-\mu,k-2\,\mu,v-1-4\,\mu,{s}^{2}-\mu+s]$. We now verify that the identity of Lemma 4.3 holds, by checking that $s^{3}(C(-\mu/s,2-k/s)-(2s-r)(r+1))=0$ in $R/S$. Ψ> Groebner[NormalForm](expand(s^3(eval(C,[x=-mu/s,y=2-k/s]) Ψ> - (2s-r)(r+1))),G,ordering); Ψ 0 Ψ Similarly, we verify that the identities of Lemma 4.4 hold for type-I strongly regular graphs, by working in $R/T$. Ψ> Groebner[NormalForm](eval(C,[x=r-t,y=3+2r-2t]) Ψ> - 2(t-1)(t+s-2)(t+2s),H,ordering); Ψ 0 Ψ> Groebner[NormalForm](eval(C,[x=r-t,y=2+2r-2t]) Ψ> - (t+s)(2t^2+(4s-1)t-3s-1),H,ordering); Ψ 0 Ψ Next are the verifications of the identities of Lemma 4.7. Ψ> Groebner[NormalForm](expand(s^3(eval(C,[x=-mu/s-t,y=2-k/s-t]) Ψ> - (t-1)((v-2k+lambda)t-(2s-r)(r+1)))),G,ordering); Ψ 0 Ψ> Groebner[NormalForm](expand(s^3(eval(C,[x=-mu/s-t,y=1-k/s-t]) Ψ> - t((v-2k+lambda)(t-1)+r(r+1)))),G,ordering); Ψ 0 Ψ Here is confirmation of the identity used in Remark 4.10. Ψ> Groebner[NormalForm](mu(v-2k+lambda)-(r^2+r)(s^2+s),G,ordering); Ψ 0 Ψ Next is verification of the identity used in the proof of Lemma 5.2. ΨGroebner[NormalForm](mueval(C,[x=1,y=lambda+2])/2 Ψ - (k(k-(mu+1)(lambda+1))+mu(lambda+1)^2),G,ordering); Ψ 0 Ψ Finally, here is a confirmation of the polynomial equality used in the proof of Lemma 5.3. Ψ> expand((eval(C,[x=b,y=c])-eval(C,[x=b,y=lambda+2])) Ψ> - ((lambda+2-c)(b-c)(b-c+1)+2b(lambda+2-c)(k-lambda-1))); Ψ 0 Ψ We remark that the total CPU time for all these computations on a desktop Linux PC was only about 0.16 seconds, and the total store used by Maple was 2.4MB. ## Acknowledgement We thank Anton Betten for organising the Combinatorics and Computer Algebra 2015 conference, whose problem sessions brought us together to begin this work. ## References * [1] C. Bachoc, M. Matolcsi, and I. Z. Ruzsa. Squares and difference sets in finite fields. _Integers_, 13:A77, 2013. * [2] A. Balog. On the distribution of $p^{\theta}\,{\rm mod}\,1$. _Acta Math. Hungar._, 45(1-2):179–199, 1985. * [3] L. Bernardin, P. Chin, P. DeMarco, K. O. Geddes, D. E. G. Hare, K. M. Heal, G. Labahn, J. P. May, J. McCarron, M. B. Monagan, D. Ohashi, and S. M. Vorkoetter. _Maple Programming Guide_. Maplesoft, Waterloo ON, Canada, 2014. * [4] A. Blokhuis. On subsets of ${\rm GF}(q^{2})$ with square differences. _Nederl. Akad. Wetensch. Indag. Math._, 46(4):369–372, 1984. * [5] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. _J. Symbolic Comput._, 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993). * [6] A. E. Brouwer. Strongly regular graphs. In C. J. Colbourn and J. H. Dinitz, editors, _Handbook of Combinatorial Designs_, Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2007. * [7] A. E. Brouwer. http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html, 2016. * [8] A. E. Brouwer and W. H. Haemers. _Spectra of Graphs_. Universitext, Springer, New York (2012), 2012. * [9] P. J. Cameron and J. H. van Lint. _Designs, Graphs, Codes, and their Links_. Cambridge University Press, 1992. * [10] A. Cohen. Gröbner bases, an introduction. In A. Cohen, H. Cuypers, and H. Sterk, editors, _Some Tapas of Computer Algebra_, pages 1–33. Springer, Berlin-Heidelberg-New York,, 1999. * [11] A. Cossidente and T. Penttila. Hemisystems on the Hermitian surface. _J. London Math. Soc. (2)_, 72(3):731–741, 2005. * [12] P. Delsarte. _An algebraic approach to the association schemes of coding theory_. PhD thesis, Universite Catholique de Louvain., 1973. * [13] C. Godsil and K. Meagher. _Erdős—Ko—Rado Theorems: Algebraic Approaches_. Cambridge University Press, 2016. * [14] C. Godsil and G. Royle. _Algebraic Graph Theory_, volume 207 of _Graduate Texts in Mathematics_. Springer-Verlag, New York, 2001. * [15] L. Kuipers and H. Niederreiter. _Uniform Distribution of Sequences_. Wiley Interscience, New York, 1974. * [16] L. H. Soicher. More on block intersection polynomials and new applications to graphs and block designs. _J. Combin. Theory Ser. A_, 117(7):799–809, 2010. * [17] L. H. Soicher. On cliques in edge-regular graphs. _J. Algebra_, 421:260–267, 2015.
1801.07275	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 102290, "num_imgs": 19, "llama3_tokens_count": 27914 }	[ "content_image/1801.07275/x1.png", "content_image/1801.07275/x2.png", "content_image/1801.07275/ch4contoura1.jpg", "content_image/1801.07275/po2.png", "content_image/1801.07275/BifEnergyAction_abio.png", "content_image/1801.07275/BifEnergyRes_abio.png", "content_image/1801.07275/rot1.jpg", "content_image/1801.07275/rot_number1.jpg", "content_image/1801.07275/disI_ps_res_1.jpg", "content_image/1801.07275/disO_sos1.png", "content_image/1801.07275/sos_disA_1.png", "content_image/1801.07275/sos_disA_2.png", "content_image/1801.07275/sos_disA_2_5.png", "content_image/1801.07275/ConleyContour.jpg", "content_image/1801.07275/rlevelspolar0.jpg", "content_image/1801.07275/E25.jpg", "content_image/1801.07275/manif5.jpg", "content_image/1801.07275/slice2.jpg", "content_image/1801.07275/slice25.jpg" ]	# The phase space geometry underlying roaming reaction dynamics Vladimír Krajňák¹ and Holger Waalkens² [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] ###### Abstract Recent studies have found an unusual way of dissociation in formaldehyde. It can be characterized by a hydrogen atom that separates from the molecule, but instead of dissociating immediately it roams around the molecule for a considerable amount of time and extracts another hydrogen atom from the molecule prior to dissociation. This phenomenon has been coined roaming and has since been reported in the dissociation of a number of other molecules. In this paper we investigate roaming in Chesnavich’s CH${}_{4}^{+}$ model. During dissociation the free hydrogen must pass through three phase space bottleneck for the classical motion, that can be shown to exist due to unstable periodic orbits. None of these orbits is associated with saddle points of the potential energy surface and hence related to transition states in the usual sense. We explain how the intricate phase space geometry influences the shape and intersections of invariant manifolds that form separatrices, and establish the impact of these phase space structures on residence times and rotation numbers. Ultimately we use this knowledge to attribute the roaming phenomenon to particular heteroclinic intersections. Keywords: reaction dynamics, roaming, transition state theory ## 1 Introduction ### Roaming For a long time it was believed that dissociation of molecules can only happen in two ways. Firstly, the original molecule can dissociate into smaller molecules and this is sometimes referred to as dissociation via the molecular channel. In order to dissociate, the system has to pass over a potential barrier representing the energy needed to break existing bonds and form new ones. Quantitative results on dissociation rates (or reaction rates in general) can be obtained via transition state theory. Alternatively an individual atom, called free radical, can escape from a molecule without forming new bonds and thus without passing over a potential barrier [3]. This is sometimes referred to as dissociation via the radical channel. Dissociation via both channels is well understood. Recently, however, van Zee et al. [35] reported having experimentally observed dissociation of formaldehyde (H${}_{2}$CO) with CO in low rotational levels at an energy where dissociation through the molecular channel should have rather resulted in high rotational states of CO. The two proposed explanations for this behaviour are that either at least one of the vibrational modes of the transition state is quite anharmonic or there have to be two distinct molecular channels. Townsend et al. [33] discovered in their study of formaldehyde (H${}_{2}$CO) a new form of dissociation that appears not to be associated with the molecular or the radical channel. In the process, an H atom separates from the molecule following the radical channel, but instead of dissociating it spends a considerable amount of time near the molecule and eventually abstracting the remaining H atom from the molecule to form H${}_{2}$. This type of dissociation is called _roaming_ due to the nature of behaviour of the escaping H atom. No potential barrier or dynamical transition state is known to be involved in roaming. The discovery of roaming stimulated extensive studies of formaldehyde photodissociation and roaming has since been accepted as the cause for the phenomenon observed by van Zee et al. [35]. Bowman and Shelper [4] have studied the dynamics of H${}_{2}$CO and CH${}_{3}$CHO to find evidence that roaming is more connected to the radical rather than the molecular channel. At the same time, roaming was observed at energies below the radical threshold. ### Known results In recent years dynamical systems theory has made an impact in chemistry by providing the means to understand the classical phase space structures that underlie reaction type dynamics [38, 34, 36]. This concerns the phase space geometry that governs transport across a saddle equilibrium point referred to as the molecular channel above. These ideas make precise the notion of a _transition state_ which forms the basis of computing reaction rates from _Transition State Theory_ which can be considered to be the most important approach to compute reaction rates in chemistry [39]. It is shown that surfaces of constant energy contain for energies above the saddle an invariant manifold with the topology of a sphere. This sphere is unstable. More precisely it is a normally hyperbolic invariant manifold (NHIM) which is of codimension 2 in the energy surface. The NHIM can be identified with the transition state: it forms an unstable invariant subsystem located between reactants and products. What is crucial for the computation of reaction rates is that the NHIM is spanned by the hemispheres of another higher dimensional sphere which is of codimension 1 in the energy surface and which is referred to as a _dividing surface_. It divides the energy surface into a reactants region and a products region in such a way that trajectories extending from reactants to products have exactly one intersection with one of the hemispheres and trajectories extending from products to reactants have exactly one intersection with the other hemisphere. The construction of such a _recrossing-free_ dividing surface is crucial for Transition State Theory where reaction rates are computed from the flux through a dividing surface which is computationally much cheaper than sampling trajectories. For the global dynamics, the NHIM is significant because of its stable and unstable manifolds. The latter have the topology of spherical cylinders or ‘tubes’ which are of codimension 1 in the energy surface and hence have sufficient dimensionality to act as separatrices. In fact the stable and unstable manifolds separate the reactive trajectories from the non-reactive ones. The geometry of the stable and unstable manifolds and their location and intersections in the reactants and products regions carry the full information about the transition process including, e.g., state specific reactivity [9]. In the case of two degrees of freedom the NHIM is the Lyapunov periodic orbit associated with the saddle equilibrium point and the approach reduces the _periodic orbit dividing surface_ (PODS) introduced earlier by Pechukas and Pollak and others [29, 30]. Explaining the roaming phenomenon poses a new challenge to dynamical systems theory. The first attempts to use methods from dynamical systems theory related to ones underlying transition state theory to explain roaming can be found in the work of Mauguière et al. [21] in which they identified the region of the classical phase space where roaming occurs with the aid of numerous invariant phase space structures for Chesnavich’s CH${}_{4}^{+}$ dissociation model [5]. They also introduced a classification of trajectories present in the system and matched them with the experimentally observed behaviour. The definition of roaming was formulated using the number of turning points in the radial direction in the roaming region. In light of a gap time analysis of Mauguière et al. in [22], the definition was refined by means of the number of crossings of a dividing surface constructed in the roaming region. This refined definition is the best dynamical description of roaming to date. In [24], Mauguière et al. studied the model of formaldehyde to find unstable periodic orbits in the roaming region. The homoclinic tangle of one such orbit was shown to be responsible for transport between two potential wells in a process that is closely linked to roaming. The periodic orbit involved do not arise as the Lyapunov periodic orbits associated with a saddle equilibrium points and the situation is hence different from the usual setting of transition state theory built on a potential saddle. Yet a recrossing free dividing surface can be constructed from such a periodic orbit. Such dividing surfaces may be other than spherical. It was shown in [19] and [20] that a spherical dividing surface near an index-$1$ critical point may bifurcate into a torus. In [23] a toric dividing surface was constructed near an index-$2$ critical point. The authors of [25] found that the local geometry of the energy surface in an O${}_{3}$ model may be toric and constructed a toric dividing surface using two unstable periodic orbits. Even more recently, Huston et al. [14] report that they have found a correlation between the distribution of internal energies of CO and H${}_{2}$ with the molecular channel and roaming. Particularly at lower energies, roaming trajectories have significantly more energy in H${}_{2}$. With increasing energy the differences decrease. Their definition of roaming is slightly different though, it involves rotation of H${}_{2}$ around CO at a ‘slowly varying and elongated distance’. The precise definition involves technical conditions on H${}_{2}$ vibrational energy, time spent at a certain minimal distance from CO with low kinetic energy and large H-H bond length. ### Objectives and outline of this paper Because there is no single generally accepted definition of roaming, there is a clear need for a deeper understanding of the mechanisms behind dissociations. In this work we present a detailed study of dissociation in the CH${}_{4}^{+}$ model by [5]. We discuss all types of dynamics present in this model and explain their connection to the underlying phase space geometry and invariant structures. We construct various surfaces of section and from the dynamics on these surfaces we deduce the role of invariant manifolds in slow dissociation and ultimately show a certain structure of heteroclinic tangles that causes roaming. From the point of view of transition state theory we address two interesting problems. Firstly, it is not very well understood what happens in case reactants and products are divided by multiple transition states in series, which is a problem we address in this work. Secondly, we study the role of the local energy surface geometry in interactions of multiple transition states. In the study we employ surfaces of section, all of which satisfy the Birkhoff condition [2] of being bounded by invariant manifolds. Using the surfaces of section we can observe dynamical behaviour such as roaming, but to understand the role of the local energy surface geometry and its implications to roaming, we generalise the Conley-McGehee representation [7, 26, 17] and study the dynamics on the energy surface in full $3$ dimensions. The paper is organized as follows. In Sec. 2 we introduce the Chesnavich’s CH${}_{4}^{+}$ model. In Sec. 3 we discuss various periodic orbits and their role in setting up the problem of transport of phase space volumes between different phase space regions. In Sec. 4 we study the dynamics of the Chesnavich model by looking at trajectories from various perspectives. This section is followed by relating the dynamics to roaming in Sec. 5. The invariant manifolds that govern the dynamics and in particular roaming are discussed from a global perspective in Sec. 6. Conclusions are given in Sec. 7 ## 2 Set-up ### Chesnavich’s CH${}_{4}^{+}$ dissociation model Like [21] and [22], we use the model for the $\text{CH}_{4}^{+}\rightarrow\text{CH}_{3}^{+}+\text{H}$ dissociation introduced by Chesnavich [5]. The system is a $2$-degree-of-freedom “phenomenological model” that is intended for the study of multiple transition states. In this model, only one H atom is free and the CH${}_{3}^{+}$ molecule is considered to be a rigid complex. It is a planar system and we study it in a centre of mass frame in polar coordinates $(r,\theta_{1},\theta_{2},p_{r},p_{1},p_{2})$, where $r$ is the distance of the free H atom to the centre of mass (magnitude of the Jacobi vector), $\theta_{1}$ is the angle between a fixed axis through the centre of mass and the Jacobi vector between CH${}_{3}^{+}$ and H, and $\theta_{2}$ is the angle representing the orientation of CH${}_{3}^{+}$ with respect to the fixed axis. In these coordinates the kinetic energy has the form \[T=\frac{1}{2m}\left(p_{r}^{2}+\frac{1}{r^{2}}p_{2}^{2}\right)+\frac{1}{2I}p_{1 }^{2},\] where $m$ is the reduced mass of the system, and $I$ is the moment of inertia of the rigid body CH${}_{3}^{+}$. The system has a rotational $SO(2)$ symmetry, which can be reduced giving a family of systems parametrised by the (conserved) angular momentum. This reduction can be obtained from the following canonical transformation: \[\theta_{1}=\theta+\psi,\quad\theta_{2}=\psi,\quad p_{1}=p_{\theta},\quad p_{2} =p_{\psi}-p_{\theta}.\] Then $p_{\psi}=p_{1}+p_{2}=:\lambda$ is the total angular momentum and it is conserved. It follows that \[H(r,\theta,p_{r},p_{\theta};\lambda) =\frac{1}{2m}p_{r}^{2}+\frac{1}{2I}p_{\theta}^{2}+\frac{1}{2mr^{2 }}(p_{\theta}-\lambda)^{2}+U(r,\theta)\] \[=\frac{1}{2m}p_{r}^{2}+\frac{1}{2}\left(\frac{1}{I}+\frac{1}{mr^{ 2}}\right)p_{\theta}^{2}-\frac{\lambda}{mr^{2}}p_{\theta}+\frac{\lambda^{2}}{2 mr^{2}}+U(r,\theta),\] where $U(r,\theta)$ is the potential energy from [5] that we will discuss later in Section 2.3. In the last expression the term $\frac{\lambda}{mr^{2}}p_{\theta}$ gives rise to a Coriolis force in the equations of motion. ### General setting As explained by [3], systems exhibiting roaming have a potential well for a small radius, representing the stable molecule, and with increasing distance between the dissociated components converges monotonously to a certain base energy, which we can assume to be $0$. This is unlike the traditional bimolecular reactions that involve flux over a potential saddle. As shown in [25], under certain conditions the potential $U$ admits two unstable periodic orbits that are not associated with any potential which, however, form the transition state to dissociation. We will find these orbits and use them to construct a toric dividing surface from them. The argument for the existence of the periodic orbits is as follows. The dependence of the potential $U(r,\theta)$ on $\theta$ is due to the interaction between the anisotropic rigid molecule and the free atom. When $r$ is sufficiently large, the potential $U(r,\theta)$ is essentially independent of $\theta$. This is because for sufficiently large distances, the orientation of the CH${}_{3}^{+}$ molecule does essentially not influence the interaction with the free atom. Let us therefore assume for a moment that $r$ is sufficiently large, so that $U$ is rotationally symmetric and we can drop $\theta$ in the argument of $U$. The system reduced by the rotational symmetry then has the effective potential \[V_{red}(r;\lambda)=\frac{(p_{\theta}-\lambda)^{2}}{2mr^{2}}+U(r),\] where $p_{\theta}$ becomes a constant of motion. The reduced system admits a relative equilibrium, provided $U(r)$ is monotonous, $U(r)<0$ and $U\in o(r^{-2})$ as $r\rightarrow\infty$. Potentials of most chemical reactions, including $\text{CH}_{4}^{+}\rightarrow\text{CH}_{3}^{+}+\text{H}$, meet this condition. The relative equilibrium is given by $r=r_{p_{\theta}}$, $p_{r}=0$, where $r_{p_{\theta}}$ is the solution of \[\dot{p}_{r}=-\frac{\partial H}{\partial r}=\frac{1}{mr^{3}}(p_{\theta}-\lambda )^{2}-\frac{\mathrm{d}U}{\mathrm{d}r}=0.\] For the class of potentials $U=-cr^{-(2+\epsilon)}$ with $c,\epsilon>0$, the relative equilibrium is unstable (Fig. 1). This follows from the reduced $1$-degree-of-freedom Hamiltonian having a saddle at this equilibrium as can be seen from computing the Hessian matrix which is diagonal with the elements \[\frac{\partial^{2}H}{\partial p_{r}^{2}}=\frac{1}{m},\] and \[\frac{\partial^{2}H}{\partial r^{2}}=\frac{3}{mr^{4}}(p_{\theta}- \lambda)^{2}+\frac{\mathrm{d^{2}}U}{\mathrm{d}r^{2}}=\frac{3}{r}\frac{\mathrm{ d}U}{\mathrm{d}r}+\frac{\mathrm{d^{2}}U}{\mathrm{d}r^{2}}\\ =c\frac{3(2+\epsilon)}{r^{4+\epsilon}}-c\frac{(3+\epsilon)(2+ \epsilon)}{r^{4+\epsilon}}=-c\frac{\epsilon(2+\epsilon)}{r^{4+\epsilon}}.\] <figure><img src="content_image/1801.07275/x1.png"><figcaption>Figure 1: Schematic representation of the dominant long range potential andthe “centrifugal term” over r (left), and of r1 over pθ (right).</figcaption></figure> In the full system, the relative equilibrium is manifested as the unstable periodic orbits $r=r_{p_{\theta}}$, $p_{r}=0$ and $p^{\pm}_{\theta}$ such that $(p^{+}_{\theta}-\lambda)^{2}=(p^{-}_{\theta}-\lambda)^{2}$. Following general results on the persistence of normally hyperbolic invariant manifolds [10], these periodic orbit persist if the rotational symmetry is broken, provided the perturbation is not too big. Note that according to our assumptions these periodic orbits are not associated with a local maximum of $U$. The condition $U\in o(r^{-2})$ as $r\rightarrow\infty$ is reminiscent of the assumption made by the authors of [6]. However, they consider a growth restriction near the origin, namely that for all $\theta$ \[\left(\frac{\lambda^{2}}{2mr^{2}}+U(r,\theta)\right)\in o(r^{-2})\quad\text{as }\quad r\to 0,\] and additionally require $\left(\frac{\lambda^{2}}{2mr^{2}}+U(r,\theta)\right)$ to have at most one maximum for each $\theta$. We do not impose restrictions on $U$ near $r=0$ and admit several maxima. ### Potential energy The potential as suggested by Chesnavich [5] is the sum \[U(r,\theta)=U_{CH}(r)+U_{}(r,\theta),\] where $U_{CH}$ is a radial long range potential and $U_{}$ a short range “hindered rotor” potential that represents the anisotropy of the rigid molecule CH${}_{3}^{+}$ ([16], [6]). The long range potential is defined by \[U_{CH}(r)=\frac{D_{e}}{c_{1}-6}\left(2(3-c_{2})e^{c_{1}(1-x)}-\left(4c_{2}-c_{ 1}c_{2}+c_{1}\right)x^{-6}-(c_{1}-6)c_{2}x^{-4}\right),\] where $x=\frac{r}{r_{e}}$. The constants $D_{e}=47$ kcal/mol and $r_{e}=1.1$ Å represent the C-H dissociation energy and equilibrium bond length respectively. $c_{1}=7.37$ and $c_{2}=1.61$ result in a harmonic oscillator limit with stretching frequency $3000$ cm${}^{-1}$. A graph of $U_{CH}$, using Chesnavich’s choice of coefficients, can be found in Figure 2. As expected for long range interactions, it is meant to dominate the potential for large values of $r$ and not be subject to the orientation of CH${}_{3}^{+}$. Therefore $U_{CH}$ is independent of the angle and its leading term for large $r$ is $r^{-4}$. Since $U_{CH}$ also dominates the short range potential in the neighbourhood of $r=0$, Chesnavich suggest a cut-off at $r=0.9$. The cut-off is not near the region of interest in our study of roaming, nor does it have any significant implications. <figure><img src="content_image/1801.07275/x2.png"><figcaption>Figure 2: Left: Graph of UCH versus r. Right: Graph of U0 with a=1 versus r.</figcaption></figure> The short range potential has the form \[U_{}(r,\theta)=\frac{U_{0}(r)}{2}(1-\cos 2\theta),\] where \[U_{0}(r)=U_{e}e^{-a(r-r_{e})^{2}},\] is the rotor barrier, which is a smoothly decreasing function of the distance $r$, and $U_{e}=55$ kcal/mol is the barrier height, see Figure 2. The constant $a$ influences the value of $r$ at which the transition from vibration to rotation occurs. The transition is referred to as _early_ if it occurs at small $r$ and as _late_ otherwise. For comparison, see the late transition for $a=1$, which we will be using, and the early transition for $a=5$ in Figure 3. Note the different proportions of the potential well (dark blue) with respect to the high potential islands along the vertical axis. <figure><img src="content_image/1801.07275/ch4contoura1.jpg"><figcaption>Figure 3: Contour plots of potential for a=1, corresponding to a latetransition, and a=5, corresponding to an early transition.</figcaption></figure> Note that the angular dependence $(1-\cos 2\theta)$ in $U_{}$ is $\pi$-periodic and even. These properties induce a reflection symmetry of $U$ with respect to the $x$ and $y$ axes, because \[U(r,\theta)=U(r,-\theta),\] corresponds to the reflection about the $x$ axis and \[U(r,\theta)=U(r,-\theta+\pi),\] corresponds to the reflection about the $y$ axis. ## 3 Setting up the transport problem We follow [21] and set $a=1$ for a slow transition from vibration to rotation. In what follows we also assume $\lambda=0$, unless stated otherwise. This section introduces features of the potential relevant to finding periodic orbits, defining dividing surfaces and formulating roaming in terms of transport between regions on the energy surface. ### Energy levels and Hill regions Here we give details about the features of the potential relevant to the dynamics of the system. Being the most basic characteristic of the potential, we look at critical points of the potential that give valuable information about local dynamics and at level sets that tell us about the accessible area in configuration space. Due to the reflection symmetry of the potential about the $x$ and $y$ axes introduced above, critical points always come in pairs. We will denote them by $q_{i}^{\pm}$, where $i$ indicates the index of the critical point and the superscript $+$ stands for the upper half plane $\theta\in[0,\pi)$, while $-$ stands for the lower. Here we present a list of critical points: * $q_{0}^{\pm}$ - two wells at $(r,\theta)=(1.1,0)$ and $(1.1,\pi)$ with $U(q_{0}^{\pm})=E_{0}\approx-47$, * $q_{1}^{\pm}$ - two index-$1$ saddles at $(3.45,\frac{\pi}{2})$ and $(3.45,\frac{3\pi}{2})$ with $U(q_{1}^{\pm})=E_{1}\approx-0.63$, * $\widetilde{q}_{1}^{\pm}$ - two index-$1$ saddles at $(1.1,\frac{\pi}{2})$ and $(1.1,\frac{3\pi}{2})$ with $U(\widetilde{q}_{1}^{\pm})=\widetilde{E}_{1}\approx 8$, * $q_{2}^{\pm}$ - two index-$2$ saddles at $(1.63,\frac{\pi}{2})$ and $(1.63,\frac{3\pi}{2})$ with $U(q_{2}^{\pm})=E_{2}\approx 22.27$. The potential wells correspond to the two isomers of CH${}_{4}^{+}$ with the free H atom close to the CH${}_{3}^{+}$ molecule. All index-$1$ saddles are involved in isomerisation and the two index-$2$ saddles provide us with interesting geometries of the accessible regions in configuration space. For zero angular momentum ($\lambda=0$), the critical phase space points of $H$ are given by $z_{i}^{\pm}=(q_{i}^{\pm},0)$ and $\widetilde{z}_{1}^{\pm}=(\widetilde{q}_{1}^{\pm},0)$. The critical energies are ordered as \[E_{0}<E_{1}<0<\widetilde{E}_{1}<E_{2},\] and all critical points can be found in the contour plot in Figure 3. For a given fixed energy $E$, we are interested in the accessible region in the configuration space which, following the celestial mechanics literature, we refer to as Hill region [12], and the geometry of the energy surface. Since the system as defined in Section 2.1 is a natural mechanical system $H=T+U$ and the kinetic energy $T$ is always non-negative, the Hill region consists of all $(r,\theta)$ such that $U(r,\theta)\leq E$ and is bounded by the equipotential $U(r,\theta)=E$. To see what the Hill regions look like, we note that the two wells $q_{0}^{\pm}$ give rise to two topological discs that connect into an annulus at $E=E_{1}$ via $q_{1}^{\pm}$. With $E\to 0$ the annulus widens until at $E=0$ it loses compactness and covers the whole plane except for a disc near the origin. This cut-out disc decomposes at $E=\widetilde{E}_{1}$ into three, two areas of high potential around $q_{2}^{\pm}$ and the cut-off of the potential at $r=0.9$ mentioned earlier. Above $E=E_{2}$ only the cut-off at $r=0.9$ remains inaccessible. Topologically this is equivalent to the case with the energy $0<E<\widetilde{E}_{1}$. Hill regions are shown for various energies in Figure 3. For comparison, we also include Hill regions for the case $a=5$ in Figure 3, where the transition from vibration to rotation occurs earlier. Although energy levels remain topologically equivalent, note the larger potential well and the smaller energy interval where the boundary of Hill region consists of three circles. ### Relevant periodic orbits Next we study the invariant structures that can be found in the system at various energies. Critical points $z_{i}^{\pm}$ described in Section 3.1 are the most basic invariant structures at energies $E_{i}$. In the following we discuss (non-degenerate) periodic orbits on the $3$-dimensional energy surface. They create a backbone for the understanding the dynamical behaviour of our system. Similarly to critical points, periodic orbits also come in pairs because of the symmetry of the potential. The periodic orbits are then related by the discrete rotational symmetry \[(r,\theta,p_{r},p_{\theta})\mapsto(r,\theta+\pi,p_{r},p_{\theta}),\] or the discrete reflection symmetry \[(r,\theta,p_{r},p_{\theta})\mapsto(r,-\theta,p_{r},-p_{\theta}).\] In contrast to critical points, non-degenerate periodic orbits persist in energy intervals forming one-parameter families. As periodic orbits evolve with varying energy, they occasionally bifurcate with other families of periodic orbits. Based on the knowledge of Hill regions we gained in Section 3.1, we can formulate some expectations about periodic orbits in this system. For $E\leq E_{1}$, the system does not admit rotating periodic orbits, orbits that are periodic in $\theta$ and along which always $p_{\theta}>0$ or $p_{\theta}<0$. Rotating orbits project onto the configuration space as circles with the origin contained in their interior. Instead in the interval $E_{0}<E\leq E_{1}$ we can only expect vibrating, oscillator like, periodic orbits. Of special interest are periodic orbits that project onto a line with both ends on an equipotential. In the celestial mechanics literature these orbits are referred to as periodic _brake orbits_ (the name is due to Ruiz [32]). Here we present a list of the important families of periodic orbits together with a brief description of their evolution. Configuration space projections of the periodic orbits at $E=2$ are shown in Fig. 4. <figure><img src="content_image/1801.07275/po2.png"><figcaption>Figure 4: Configuration space projections of Γi± (blue), Γo± (black), Γa±(red) and one orbit of the family Γb (magenta) at energy E=2.</figcaption></figure> * $\Gamma^{i}$: The family of periodic orbits $\Gamma^{i}$ is born in a saddle-centre bifurcation at energy $E=-.29$. Until a host of bifurcations above $E=20$, $\Gamma^{i}$ consists of hyperbolic brake orbits. Around $E=21.47$ the orbits become inverse hyperbolic and at $E=22.27$ they become heteroclinic to $z_{2}^{\pm}$ and undergo a Morse bifurcation, similar to those described in [19]. At higher energies, $\Gamma^{i}$ consists of rotating orbits that undergo further bifurcations. The periodic orbits are by some authors referred to as inner or tight periodic orbits. We denote the individual orbits by $\Gamma^{i}_{+}$ and $\Gamma^{i}_{-}$. For $E\leq 22.27$$\Gamma^{i}_{\pm}$ is the brake orbit in the potential well associated with $z^{\pm}_{0}$ and for $E>22.27$ the subscript $\pm$ corresponds to the sign of $p_{\theta}$ along the rotating periodic orbit. * $\Gamma^{o}$: This family of unstable periodic orbits originates at $r=\infty$ at $E=0$. With increasing energy the orbits monotonously decrease in radius and remain unstable until a bifurcation with $\Gamma^{a}$ and $\Gamma^{b}$ at $E=6.13$, where $\Gamma^{a}$ and $\Gamma^{b}$ are described below. These periodic orbits are sometimes called outer or orbiting periodic orbits, because these are the periodic orbits with the largest radius at the energies where they exist. We denote the individual orbits with $p_{\theta}>0$ and $p_{\theta}<0$ by $\Gamma^{o}_{+}$ and $\Gamma^{o}_{-}$ respectively. * $\Gamma^{a}$: These periodic orbits are created in a saddle-centre bifurcation at $E=-.0602$ as stable, turn unstable at $E=-.009$ and remain unstable until a period doubling bifurcation at $2.72$. The family disappears in the aforementioned bifurcation with $\Gamma^{o}$ and $\Gamma^{b}$. At all energies, the configuration space projection of $\Gamma^{a}$ is located between those of $\Gamma^{i}$ and $\Gamma^{o}$ and we will refer to the orbits as the middle periodic orbits. We denote the individual orbits with $p_{\theta}>0$ and $p_{\theta}<0$ by $\Gamma^{a}_{+}$ and $\Gamma^{a}_{-}$ respectively. * $\Gamma^{b}$: The product of a saddle-centre bifurcation at $E=-.0023$ that quickly becomes inverse hyperbolic. Around $E=2.37$ the orbits become elliptic and undergo a reverse period doubling at $E=2.4025$. Note that the energetic gap between these two bifurcations is so small that they are almost indistinguishable in Figure 5. After that $\Gamma^{b}$ remains stable until it collides with $\Gamma^{a}$ and $\Gamma^{o}$. For $E<2.4025$ the family consists of four periodic orbits with twice the period compared to all the previously mentioned ones. The orbits related by discrete symmetries mentioned above. $\Gamma^{i}$ is important because its orbits lie in the potential well and have the largest radial coordinate $r$ of all periodic orbits in the well. $\Gamma^{o}$ are the outermost periodic orbits and trajectories with a larger radial coordinate $r$ and $p_{r}>0$ go to infinity in forward time, i.e. $r\rightarrow\infty$ as $t\rightarrow\infty$. As mentioned above, the configuration space projections of $\Gamma^{a}$ lie between $\Gamma^{i}$ and $\Gamma^{o}$. In fact there are no other periodic orbits with single period ($2\pi$-periodic in $\theta$) in this region of configuration space. We use orbits of the family $\Gamma^{a}$ in Section 3.4 to define dividing surfaces and divide phase space into regions. $\Gamma^{b}$ is needed for a complete description of the evolution of $\Gamma^{o}$ and $\Gamma^{a}$ and its bifurcations may hint at qualitative changes of structures formed by invariant manifolds. <figure><img src="content_image/1801.07275/BifEnergyAction_abio.png"><figcaption>Figure 5: Bifurcation diagrams showing Γi± (blue), Γo± (black), Γa± (red) andorbits of the family Γb (magenta) in the energy-action (E,S) plane.</figcaption></figure> There are various other periodic orbits, most notably ones corresponding to stable vibrations of the bound CH${}_{4}^{+}$ molecule, Lyapunov orbits associated with $z_{1}^{\pm}$ and $\widetilde{z}_{1}^{\pm}$ that play a role in isomerisation and periodic orbits involved in various bifurcations with the orbits mentioned above. All of these will not play a role in our further considerations. With non-zero angular momentum, periodic orbits of a family remain related by the discrete rotational symmetry, but not by the discrete reflection symmetry and some other properties are different too. The inner periodic orbits are no longer brake orbits for $\lambda\neq 0$ and their projections onto configuration space are topological circles instead of lines. Similarly rotating orbits of the same family do not have the same configuration space projection and bifurcate at different energies. With increasing $\|\lambda\|$ the differences become more pronounced. In Figure 5 we present the evolution of the orbits in above-mentioned families in the energy-action ($E,S$) plane. We will explain in Section 3.3 why flux through a dividing surface associated with a vibrating periodic orbit is equal to its action, while for rotating periodic orbits it is equal to twice its action. Figure 6 shows the evolution of the Greene residue of orbits in the families. The Greene residue, due to J. M. Greene [11] is a quantity characterizing the stability of the orbits. It is derived from the monodromy matrix, a matrix that describes the behaviour of solutions in the neighbourhood of a periodic orbit. <figure><img src="content_image/1801.07275/BifEnergyRes_abio.png"><figcaption>Figure 6: Bifurcation diagrams showing Γi± (blue), Γo± (black), Γa± (red) andorbits of the family Γb (magenta) in the energy-residue (E,R) plane.</figcaption></figure> The monodromy matrix and the Greene residue are defined as follows. For a periodic orbit $\Gamma$ with the parametrisation $\gamma(t)$ and period $T$, let $M(t)$ be the matrix satisfying the variational equation \[\dot{M}(t)=JD^{2}H(\gamma(t))M(t),\] where $J=\begin{pmatrix}0&Id\\ -Id&0\end{pmatrix}$, with the initial condition \[M(0)=Id.\] The monodromy matrix is defined by $M=M(T)$. It describes how an initial deviation $\delta$ from $\gamma(0)$ changes after a full period $T$. For $\delta$ sufficiently small the relationship is \[\Phi_{H}^{T}(\gamma(0)+\delta)=\gamma(T)+M\delta+O(\delta^{2}),\] where $\Phi_{H}^{t}$ is the Hamiltonian flow. If $\delta$ is an initial displacement along the periodic orbit $\delta\parallel J\nabla H$, then by periodicity $\delta$ is preserved after a full period $T$, i.e. $M\delta=\delta$. A similar argument holds for an initial displacement perpendicular to the energy surface $\delta\parallel\nabla H$. Consequently, two of the eigenvalues of $M$ are $\lambda_{1}=\lambda_{2}=1$. More details including a reduction of $M$ can be found in [8]. As the variational equation satisfied by $M(t)$ is Hamiltonian, the preservation of phase space volume following Liouville’s theorem implies that the determinant $\det M(t)=\det M(0)=1$ for all $t$. Therefore for the two remaining eigenvalues we have $\lambda_{3}\lambda_{4}=1$ and we can write them as $\lambda$ and $\frac{1}{\lambda}$. $\Gamma$ is hyperbolic if $\lambda>1$, it is elliptic if $\|\lambda\|=1$ and it is inverse hyperbolic if $\lambda<-1$. Definition 1: _The Greene residue of $\Gamma$ is defined as_ \[R=\frac{1}{4}(4-TrM),\] _where $M$ is the monodromy matrix corresponding to the periodic orbit $\Gamma$._ Knowing that $\lambda_{1}=\lambda_{2}=1$, we can write $R$ as \[R=\frac{1}{4}\left(2-\lambda-\frac{1}{\lambda}\right).\] By definition $R<0$ if $\Gamma$ is hyperbolic, $0<R<1$ if it is elliptic and $R>1$ if it is inverse hyperbolic. ### Transition states and dividing surfaces In this section we discuss dividing surfaces associated with transition states, the backbone of Transition State Theory. Following [19, 20] we define transition states more formally as follows. Definition 2 (Ts): _A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-$2$ submanifold of the energy surface that can be spanned by two surfaces of unidirectional flux, whose union divides the energy surface into two components and has no local recrossings._ The name transition state is due to the fact it is a structure found between areas of qualitatively different types of motion, a transition between two types of motion so to say. One can imagine the transition between types of motion corresponding to physical states like reactants and products or the transition between rotation and vibration. In a system with $2$ degrees of freedom, a TS consists of unstable periodic orbit. Generally a TS is a codimension-$2$ normally hyperbolic invariant manifold, a manifold on the energy surface invariant under the Hamiltonian flow, such that instabilities transversal to it dominate the instabilities tangential to it ([10], [13]). In general, a dividing surface (DS) is a surface that divides the energy surface into two disjoint components. By a DS associated with a TS we mean a union of the two surfaces of unidirectional flux that is constructed as follows. For a fixed energy $E$, let $(r_{\Gamma},\theta_{\Gamma})$ be the projection of the periodic orbit $\Gamma$ onto configuration space, then the DS is the surface $(r_{\Gamma},\theta_{\Gamma},p_{r},p_{\theta})$, where $(p_{r},p_{\theta})$ are given implicitly by the energy equation \[E=\frac{1}{2m}p_{r}^{2}+\frac{1}{2I}p_{\theta}^{2}+\frac{1}{2mr^{2}}(p_{\theta }-\lambda)^{2}+U(r,\theta).\] This construction also works for stable periodic orbits, but the resulting DS admits local recrossings. In the following a DS associated with a TS is always the surface constructed this way. We will refer to the DSs associated with $\Gamma^{i}$, $\Gamma^{o}$ and $\Gamma^{a}$ as inner, outer and middle, respectively. For our investigation, we do not need to distinguish between the DSs associated to $\Gamma^{i}_{+}$ and $\Gamma^{i}_{-}$, therefore we always refer to the former unless explicitly stated otherwise. We are mainly interested in the influence of local energy surface geometry on the geometry of DSs and in the dynamics on DSs under the corresponding return map. The geometry of the DSs is due to the form of the kinetic energy and the local geometry of the energy surface. It is well known that a DS associated to a brake periodic orbit is a sphere and the brake periodic orbit is an equator of this sphere, [36]. The equator divides the sphere into hemispheres, whereby the flux through the two hemispheres is equal in size and opposite in direction. Trajectories passing this sphere from reactants to products intersect one hemisphere and the other hemisphere is crossed on the way from products to reactants. The flux through a hemisphere is then by Stokes’ theorem equal to the action of the periodic orbit [17]. Rotating periodic orbits, on the other hand, such as $\Gamma^{o}$, give rise to a DS that is a torus. The two orbits of the same family with opposite orientation are circles on the torus and divide it into two annuli with properties identical to the hemispheres. Using Stokes’ theorem we find that the flux across each annulus is given by the sum of the actions of the two orbits, or simply twice the action of a single orbit [19]. Should it be necessary to distinguish the hemispheres or annuli of a DS by the direction of flux, the outward hemisphere or annulus is the one intersected by the prototypical dissociating trajectory defined by $\theta=0$, $p_{r}>0$, $p_{\theta}=0$ and/or $\theta=\pi$, $p_{r}>0$, $p_{\theta}=0$. The inward hemisphere or annulus is then intersected by $\theta=0$, $p_{r}<0$, $p_{\theta}=0$ and/or $\theta=\pi$, $p_{r}<0$, $p_{\theta}=0$ ### Division of energy surface Using the inner and outer DSs we can define regions on the energy surface and formulate roaming as a transport problem. The area bounded by the surface $r=0.9$ and the two inner DSs represents the two isomers of CH${}_{4}^{+}$. We denote the two regions by $B_{1}^{+}$ and $B_{1}^{-}$. The unbounded region beyond the outer DS, denoted $B_{3}$, represents the dissociated molecule. It is therefore in the interaction region between the inner and the outer DS, denoted $B_{2}$, where the transition between CH${}_{4}^{+}$ and CH${}_{3}^{+}+$H occurs. When in $B_{2}$, the H atom is no longer in the proximity of CH${}_{3}^{+}$, but still bound to the CH${}_{3}^{+}$ core. This is the region, where the system exhibits roaming. Contained in $B_{2}$ are $\Gamma^{a}$ and various other periodic orbits that may play a role in roaming. Dissociation can in this context be formulated as a problem of transport of energy surface volume from $B_{1}$ to $B_{3}$. Such volume contains trajectories that originate in the potential well, pass through the interaction region and never return after crossing the outer DS. Since each trajectory passing from $B_{1}$ to $B_{2}$ crosses the inner DS and leaves $B_{2}$ by crossing the outer DS, we may restrict the problem to the interaction region. Because roaming is a particular form of dissociation, it too has to be subject to transport from the inner DS to the outer DS. It is well known that transport to and from a neighbourhood of a unstable periodic orbits is governed by its stable and unstable invariant manifolds. The problem can be reformulated accordingly. This means, of course, by studying the structure of heteroclinic intersections of stable and unstable invariant manifolds of $\Gamma^{i}$ and $\Gamma^{o}$, as well as with $\Gamma^{a}$ that, as we will soon see, sits inside the homoclinic tangle of $\Gamma^{o}$. We will denote the invariant manifolds of $\Gamma^{i}_{+}$ by $W_{\Gamma^{i}_{+}}$. We will further use a superscript $s$ and $u$ to label the stable and unstable invariant manifolds and add an extra superscript $-$ and $+$ for the branches that leave the neighbourhood of $\Gamma^{i}_{+}$ to the CH${}_{4}^{+}$ side ($r$ smaller) or to the CH${}_{3}^{+}+$H side ($r$ larger), respectively. $W_{\Gamma^{i}_{+}}^{u+}$ therefore denotes the unstable branch of the invariant manifolds of $\Gamma^{i}_{+}$ that leaves the neighbourhood of $\Gamma^{i}_{+}$ to the CH${}_{3}^{+}+$H side. Invariant manifolds of other TSs will be denoted analogously. We remark that we may use TST to consider the evolution of periodic orbits in the energy-action plane shown in Figure 5 in the context of transport of energy surface volume from $B_{1}$ to $B_{3}$. Recall for Section 3.3 that the flux across the outer and middle DSs is twice the action of $\Gamma^{o}_{+}$ and $\Gamma^{a}_{+}$ respectively. The combined flux through both inner DSs is twice the action of $\Gamma^{i}_{+}$. We see that for $E\leq.32$, the outer DS has the lowest flux, while for higher energies it is the inner DS. ## 4 Dynamics of the Chesnavich model Before we proceed to the discussion of how invariant manifolds cause slow dissociations, let us describe some numerical observations of how the system behaves in certain phase space regions. The observations will later be explained using invariant manifolds. In the following, we offer insight into the amount of time needed to dissociate, the locations where dissociation is fast or slow and how these properties change with increasing energy. We use this knowledge to establish a link between invariant manifolds and slow dissociation on which we further elaborate in Section 5 in the context of roaming. ### Residence times and rotation numbers For various energies $0<E<6.13$ where $\Gamma^{o}$ exists, we investigate trajectories starting in $B_{1}^{+}$, $B_{2}$ and $B_{3}$ on the surface $\theta=0,p_{\theta}>0$. We study how long it takes trajectories to reach a terminal condition representing the dissociated state. In Section 3.4 we said that we consider the molecule dissociated as soon as the system enters $B_{3}$. Naturally, then the terminal condition should be that trajectories reach the outer DS. However using the outer DS raises uncertainty of whether a faster dissociation is a dynamical property or a result of the changing position of $\Gamma^{o}_{\pm}$ with energy. To prevent this uncertainty, we use a fixed terminal condition. Since for $E\to 0$, the radius of $\Gamma^{o}_{\pm}$ diverges, no fixed terminal condition can represent the dissociated state for all energies. We decided to define the terminal condition by $r=15$ that works well for $E\geq 0.4$ at the cost of losing the energy interval $E<0.4$. In the following we consider residence times and rotation numbers, i.e. time and change in angle needed for trajectories starting on $\theta=0,p_{\theta}>0$ to reach the surface $r=15$ in $B_{3}$. <figure><img src="content_image/1801.07275/rot1.jpg"><figcaption>Figure 7: From top left to bottom right the plots show residence times on thesurface of section θ=0, pθ>0 for energies E=1,2,2.5,5. The dots correspond tothe periodic orbits Γi+ (blue), Γa+ (red), orbits of the family Γb (magenta)and Γo+ (cyan). Invariant manifolds of Γi+ (green) and Γo+ (black) are alsoincluded.</figcaption></figure> Figure 7 shows rotation numbers for selected energies, with marked periodic orbits and invariant manifolds. As expected, initial conditions with $p_{r}>0$ large are the fastest ones to escape. The slowest ones are located near the periodic orbits and near $p_{r}=0$ ($p_{\theta}$ large). For $E\leq 2.5$, almost all initial conditions in $B_{1}^{+}$ were slow to escape. For higher energies, most of the slow dissociation occurs around $\Gamma^{a}_{+}$ and $\Gamma^{o}_{+}$, the slowly dissociating trajectories have a negative initial $p_{r}$ very close to zero. This observation is easily explained by noting that configuration space projections of these trajectories are almost circular and spend most of the time in the region where the potential is very flat and almost independent of $\theta$, thus $\dot{p}_{\theta}\approx 0$. Chaotic structures that can be seen in $B_{1}^{+}$ are the result of lengthy escape from a potential well. The only known structure responsible for fractal-like patterns and one closely linked to chaotic dynamics are invariant manifolds, in this case $W_{\Gamma^{i}_{+}}$. Note that at $E=5$, it seems that $W_{\Gamma^{o}_{+}}$ slows the dynamics down considerably more than $W_{\Gamma^{i}_{+}}$. <figure><img src="content_image/1801.07275/rot_number1.jpg"><figcaption>Figure 8: From top left to bottom right the plots show rotation numbers on thesurface of section θ=0, pθ>0 for energies E=1,2,2.5,5. The dots correspond tothe periodic orbits Γi+ (blue), Γa+ (red), orbits of the family Γb (magenta)and Γo+ (cyan). Invariant manifolds of Γi+ (green) and Γo+ (black) are alsoincluded.</figcaption></figure> Rotation numbers, i.e. number of completed full rotations upon dissociation, closely match residence times suggesting that slowly dissociating trajectories are ones that rotate in $B_{2}$ and $B_{3}$ for a long time. More pronounced, due to the discrete nature of the number of rotations, are structures inside $B_{1}^{+}$, just below $p_{r}=0$ and in the neighbourhood of $\Gamma^{o}_{+}$ and $W_{\Gamma^{o}_{+}}$. Note in Figure 8 that the fractal like structures recede with increasing energy and by $E=5$ most of them lie either in $B_{1}^{+}$, near $p_{r}=0$ as mentioned above and in the proximity of the homoclinic tangle of $\Gamma^{o}_{+}$. The homoclinic tangle seems to tend to a homoclinic loop as it disappears for $E\to 6.13$. It is also worth noting that fast and simple dissociation, i.e. low residence time and low rotation number, is not only becoming more dominant, but also speeding up, see Figures 7 and 8. Due to the increase in kinetic energy in the angular degree of freedom, the dissociating trajectories are naturally not becoming more direct with increasing energy. ### Residence times on the inner DS Similarly to the surface $\theta=0,p_{\theta}>0$, we can study residence times and rotation numbers for trajectories starting on a DS. In Section 3.4 we formulated our problem as a transport problem from the inner to the outer DS. According to Section 3.3, trajectories enter $B_{1}^{+}$ through one hemisphere of the inner DS and leave through the other. Naturally we are interested in the latter hemisphere. <figure><img src="content_image/1801.07275/disI_ps_res_1.jpg"><figcaption>Figure 9: Residence times for initial conditions on the inner DS with outwarddirection for energies E=1,2,2.5,5. Note that the scale for E=5 is different,because 9 is an upper bound for the residence time for initial conditions onthe inner DS.</figcaption></figure> Although it is not absolutely indispensable for qualitative purposes, we prefer to work on the DS in canonical coordinates. Due to the preservation of the canonical $2$-form by the Hamiltonian flow, if we use canonical coordinates, the map from one surface of section to another is area preserving. Consequently areas of initial conditions on the inner DS corresponding slow or fast dissociation can be directly compared to the areas on the surface of section $\theta=0$, $p_{\theta}>0$. Canonical coordinates are obtained by defining a new radial variable $\rho(r,\theta)=r-\bar{r}(\theta)$ that is constant along $\Gamma^{i}_{+}$, where the curve $\bar{r}(\theta)$ is the approximation of the configuration space projection of $\Gamma^{i}_{+}$, similarly to [15]. Due to the symmetry of the system, $\Gamma^{i}_{+}$ can be very well approximated by a quadratic polynomial for every energy. Next we use the generating function (type 2 in [1]) \[G(r,\theta,p_{\rho},p_{\sigma})=(r-\bar{r}(\theta))p_{\rho}+\theta p_{\sigma}.\] From that we obtain \[p_{r}=\frac{\partial G}{\partial r} = p_{\rho},\] \[p_{\theta}=\frac{\partial G}{\partial\theta} = p_{\sigma}-\bar{r}^{\prime}(\theta)p_{\rho},\] and therefore $p_{\sigma}=p_{\theta}+\bar{r}^{\prime}(\theta)p_{\rho}$. The surface of section is now defined by $\rho=0$, $\dot{\rho}>0$, i.e. the outward hemisphere of the inner DS corresponding to transport in the direction from $B_{1}^{+}$ to $B_{2}$. Figure 9 shows the distribution of residence times for initial conditions on the inner DS. We can see that slow dissociation is specific to two areas of the surface of section. Initial conditions on the rest of the surface leave $B_{2}$ quickly. Information from the two surfaces of section suggests that $W_{\Gamma^{o}_{+}}$, and eventually $W_{\Gamma^{a}_{+}}$, intersect the inner DS in the area with long dissociation times. The areas of slow dissociation are the most pronounced for low energies, at $E=2.5$ they almost disappear. At $E=5$ we see no sign of slow dissociation, the longest residence time found at the current resolution ($6000\times 6000$ initial conditions) was below $9$. This suggests that the structure responsible for roaming disappears at an energy below $2.5$. Note that even the slowest dissociation at $E=5$ takes as long as the fastest ones at $E=1$ or $E=2$. In summary we can say that the system exhibits various types of dissociation ranging from fast and direct, where the H atom escapes almost radially, to slow that involves H revolving a multitude of times around CH${}_{3}^{+}$. Long dissociations seem to occur in fractal-like structures that are caused by invariant manifolds, proof of which will be given in Section 4.3. ### Sections of manifolds Let us now have a closer look at manifolds on the two surfaces of section presented above and establish a link between invariant structures and slow dissociation. In Section 4.1 we already noted that the homoclinic tangle of $\Gamma^{i}_{+}$ is responsible for a fractal structure of slow dissociation of initial conditions in $B_{1}^{+}$. Furthermore the homoclinic tangle of $\Gamma^{o}_{+}$ (and $\Gamma^{o}_{-}$) is responsible for slow dissociation in the interaction region $B_{2}$, especially at the top half of the energy interval. It is important to say that the section $\theta=const$, $p_{\theta}>0$ is not very well suited for the study of invariant manifolds. This is mainly due to the transition from vibration to rotation. The invariant manifolds $W_{\Gamma^{i}_{+}}$ may be nicely visible, but during this transition the invariant manifolds are not barriers to transport of surface area on this surface of section. Because parts $W_{\Gamma^{i}_{+}}$ rotate with $p_{\theta}<0$ after the transition, they do not return to the surface of section. For the same reason there are trajectories that do not return to the surface of section. The return map associated with this surface of section is therefore not area preserving. This anomaly can be seen from odd shapes of invariant manifolds - heteroclinic points seem to be mapped to infinity. Apart from the transition of $W_{\Gamma^{i}_{+}}$ from vibration to rotation, invariant manifolds may enter $B_{1}^{\pm}$ and be captured therein for a significant amount of time. Upon leaving $B_{1}^{\pm}$ the direction of rotation is unpredictable and this is true for invariant manifolds of all TSs. That is all we can say about the section $\theta=const$, $p_{\theta}>0$. The section on the inner DS, just as all other DSs, does not suffer from these problems, because they do not depend on the direction of rotation. Moreover, these surfaces are almost everywhere transversal to the flow. In Figure 10 we present the intersection of $W_{\Gamma^{o}_{+}}$ with the inner DS at $E=1$ and $E=2$. Since slow dissociation fades away at higher energies, we do not present the section at higher energies. In fact, for $E\geq 2.5$ the manifolds $W_{\Gamma^{o}_{+}}$ do not intersect the inner DS and therefore $W_{\Gamma^{i}_{+}}$ and $W_{\Gamma^{o}_{+}}$ do not intersect at all. Clearly then, slow dissociation, and thereby roaming, is induced by the heteroclinic tangle of $W_{\Gamma^{i}_{+}}$ and $W_{\Gamma^{o}_{+}}$. This claim is further supported by what we see in Figures 9 and 10. When we compare Figures 9 and 10, we clearly see that longer residence times are prevalent in the same locations where $W_{\Gamma^{o}_{+}}^{s-}$ intersects the inner DS. At $E=1$ we can even recognize the structure of the of the intersection in both figures. As the manifolds $W_{\Gamma^{o}_{+}}^{s-}$ recede with increasing energy, the area of slow dissociation at $E=2.5$ remains as a relic of the intersection. Afterall, trajectories close to $W_{\Gamma^{o}_{+}}^{s-}$ follow the manifold and approach $\Gamma^{o}_{+}$ before dissociation is completed. <figure><img src="content_image/1801.07275/disO_sos1.png"><figcaption>Figure 10: Ws−Γo+ invariant manifolds on the inner DS for E=1 (left) and E=2(right). For energies E≥2.5 the manifolds W−Γo+ don’t reach the inner DS.</figcaption></figure> Note that there was no word of $W_{\Gamma^{o}_{+}}^{u-}$. This is mainly due to the fact that it influences the residence time in backward time, hence cannot be seen in forward time. Furthermore, $W_{\Gamma^{o}_{+}}^{u-}$ first intersects the other hemisphere of the inner DS, spends considerable time in $B_{1}^{+}$ and becomes heavily distorted before intersecting the outward hemisphere of the inner DS. In backward time, however, we expect a result symmetric to the one presented here due to time reversibility of the system. What really prevents us from making more fundamental conclusions at this point is the fact that $W_{\Gamma^{o}_{+}}^{s-}$ is heavily distorted when it reaches the inner DS. The reason is very simple - heteroclinic points. Here we not only mean trajectories on $W_{\Gamma^{o}_{+}}^{s-}$ that tend toward $\Gamma^{i}_{+}$, but also to $\Gamma^{a}_{+}$. Due to this fact, it is impossible to tell which area is enclosed by $W_{\Gamma^{o}_{+}}^{s-}$ and which is outside of it. For the majority of the area we note that $\sigma=0$, $p_{\rho}>0$, $p_{\sigma}=0$ (equivalent to $\theta=0$, $p_{r}>0$, $p_{\theta}=0$), the prototype of a fast dissociation, must lie inside $W_{\Gamma^{o}_{+}}^{s-}$ to quickly reach the outer DS. The tongues of $W_{\Gamma^{o}_{+}}^{s-}$ visible in Figure 10 therefore mostly contain trajectories that do not dissociate immediately. This problem is present on both the inner and outer DSs. Sections on both suffer from the fractal structure that is so characteristic for homoclinic and heteroclinic tangles. The ideal choice seems to be $\Gamma^{a}_{+}$ because $W_{\Gamma^{i}_{+}}^{+}$ and $W_{\Gamma^{o}_{+}}^{-}$ reach the middle DS very quickly. The first image of the manifolds under the Poincaré map associated with this surface does not display heteroclinic orbits, all manifolds are mapped to (topological) circles. Heteroclinic points become visible after applying the return map at least once, the resulting tongues wind around the previously mentioned circles. On the downside, $\Gamma^{a}_{+}$ is only hyperbolic until $2.72$, therefore for higher energies the middle DS allows local recrossings. It can be still used as a surface of section and we can expect to see fewer heteroclinic points that cause tongues, but we need to keep local recrossings in mind. In the next section we present a detailed view on the dynamics on the middle DS. ## 5 The observed dynamics and roaming In this section we recall possible definitions of roaming used in previous works. We then elaborate on the observations above and analyse invariant manifolds on the middle DS with the aim to thoroughly explain how exactly roaming is linked to the heteroclinic tangles. Based on the explanation, a natural definition of roaming follows. ### Roaming Roaming in the chemistry literature refers to a kind of dissociation that is longer or more complicated than the usual dissociation with a monotonically increasing reaction coordinate that involves a saddle type equilibrium. While there is a sufficient amount of observations and intuitive understanding of what roaming is, an exact definition has not yet been generally adopted. Mauguière et al. [21] proposed a classification of trajectories based on the number of turning points of trajectories in the interaction region $B_{2}$. Later the authors refine their definition in [22] based on the number of intersections of a trajectory with the middle DS. Dissociating trajectories need to cross the middle DS at least three times before they are classified as roaming. Huston et al. [14], on the other hand, set the criteria such that roaming trajectories have to spend a certain amount of time at a minimum radius, have low average kinetic energy and have on average a certain number of bonds over time. ### The mechanism of roaming Based on intersections of invariant manifolds, we would like to report on the types of trajectories in this dissociation problem and explain why the types exist. There is a general accord on the mechanism behind direct dissociation along the radical and molecular channel. The framework, that describes how codimension-$1$ invariant manifolds divide the energy surface in two and thereby separate reactive trajectories from non-reactive ones, is very well known in reaction dynamics, see [18], [28], [31], [27]. Due to the different local geometries of the energy surface, we need to be careful with the invariant manifolds at this point. TSs that are brake orbits give rise to spherical DS and their invariant manifolds are spherical cylinders. TSs that are rotating orbits, just like ones belonging to the families $\Gamma^{o}$ and $\Gamma^{a}$, give rise a toric DS that is based on two orbits instead of one. Therefore in the description of transport, invariant manifolds of both orbits have to make up a toric cylinder together. Invariant manifolds govern transport of energy surface volume as follows. In $\text{CH}_{4}^{+}\rightarrow\text{CH}_{3}^{+}+\text{H}$, we cannot discuss the molecular channel, but the radical channel and roaming is present. In general, if the H atom has enough kinetic energy to break bonds with $\text{CH}_{3}^{+}$, it escapes. Such a trajectory is contained in the interior of the invariant cylinder $W_{\Gamma^{i}_{+}}^{u+}$, because it leaves the inner DS to the $\text{CH}_{3}^{+}+\text{H}$ side. The same is true for $W_{\Gamma^{i}_{+}}^{u+}$. Since the trajectory corresponding to $\theta=0$, $p_{r}>0$, $p_{\theta}=0$ on the inner DS dissociates immediately, a part of $W_{\Gamma^{i}_{+}}^{u+}$ reaches the middle and outer DS without returning to the inner DS. A part of the interior of $W_{\Gamma^{i}_{+}}^{u+}$ must therefore be contained in the invariant toric cylinder made up of $W_{\Gamma^{a}_{+}}^{s-}$ and $W_{\Gamma^{a}_{-}}^{s-}$, that we will refer to as $W_{\Gamma^{a}}^{s-}$. Other invariant toric cylinders will be denoted analogously. Trajectories that have too little energy in the radial degree of freedom do not reach the middle DS and are therefore not contained in the invariant cylinder. It does not matter whether $p_{\theta}>0$ or $p_{\theta}<0$. Considering that invariant manifolds are of codimension-$1$ on the energy surface and that $W_{\Gamma^{a}_{+}}^{s-}$ and $W_{\Gamma^{a}_{-}}^{s-}$ never intersect, by the inside of the invariant toric cylinder $W_{\Gamma^{a}}^{s-}$ we mean the energy surface volume enclosed between $W_{\Gamma^{a}_{+}}^{s-}$ and $W_{\Gamma^{a}_{-}}^{s-}$. As we shall see, $W_{\Gamma^{i}_{+}}^{u+}$ is entirely contained in the invariant cylinder $W_{\Gamma^{a}}^{s-}$. After crossing the middle DS, the interior of the cylinder $W_{\Gamma^{a}}^{s-}$ is lead away from the surface by the cylinder consisting of $W_{\Gamma^{a}}^{u+}$. The trajectories that dissociate are further guided by $W_{\Gamma^{o}}^{s-}$ towards the outer DS and further away by $W_{\Gamma^{o}}^{u+}$ to complete dissociation. All directly dissociating trajectories will be contained in the interior all of the above mentioned invariant cylinders. Moreover, directly dissociating trajectories are not contained in the interior of any other invariant cylinder. As soon as a trajectory is contained in another cylinder, it is guided by that cylinder to cross the corresponding DS. Should a trajectory be contained in $W_{\Gamma^{i}_{+}}^{s+}$, it will come back to the inner DS. In this way isomerisation, i.e. transport of energy surface volume between $B_{1}^{+}$ and $B_{1}^{-}$, is possible via the intersection of the interiors of $W_{\Gamma^{i}_{+}}^{u+}$ and $W_{\Gamma^{i}_{-}}^{s+}$ or $W_{\Gamma^{i}_{-}}^{u+}$ and $W_{\Gamma^{i}_{+}}^{s+}$. The intersections of the interiors of $W_{\Gamma^{a}}^{s+}$ and $W_{\Gamma^{a}}^{u+}$ or $W_{\Gamma^{a}}^{s-}$ and $W_{\Gamma^{a}}^{u-}$, on the other hand, lead to the recrossing of the middle DS. In case a trajectory originating in $B_{1}^{\pm}$ dissociates after recrossing of the middle DS, by the definition of Mauguière et al. [22] it is a roaming trajectory. From the above it is clear that roaming trajectories are contained in intersection of the interiors of $W_{\Gamma^{i}_{\pm}}^{u+}$, $W_{\Gamma^{a}}^{s-}$, $W_{\Gamma^{a}}^{u+}$, $W_{\Gamma^{a}}^{s+}$, $W_{\Gamma^{a}}^{u-}$ and $W_{\Gamma^{o}}^{s-}$. It remains to express the order of intersections of the DSs by a roaming trajectory with the invariant cylinders above. In summary the arguments above enable us to say that, * directly dissociating trajectories are contained in $W_{\Gamma^{i}_{+}}^{u+}$ (or $W_{\Gamma^{i}_{-}}^{u+}$), $W_{\Gamma^{a}}^{s-}$, $W_{\Gamma^{a}}^{u+}$, $W_{\Gamma^{o}}^{s-}$ and no other, * isomerisation and non-dissociating trajectories are contained in $W_{\Gamma^{i}_{\pm}}^{u+}$ and $W_{\Gamma^{i}_{\mp}}^{s+}$, * roaming trajectories are contained in $W_{\Gamma^{i}_{\pm}}^{u+}$, $W_{\Gamma^{a}}^{s-}$, $W_{\Gamma^{a}}^{u+}$, $W_{\Gamma^{a}}^{s+}$, $W_{\Gamma^{a}}^{u-}$ and $W_{\Gamma^{o}}^{s-}$. Note that since a trajectory contained in the cylinder $W_{\Gamma^{a}}^{s-}$ is automatically conveyed to $W_{\Gamma^{a}}^{u+}$ after crossing the middle DS, we may omit mentioning one of the cylinders. A roaming trajectory could therefore be shortly characterized by $W_{\Gamma^{i}_{\pm}}^{u+}$, $W_{\Gamma^{a}}^{s+}$ and $W_{\Gamma^{o}}^{s-}$. The definition of Mauguière et al. admits nondissociating roaming trajectories. These are contained in $W_{\Gamma^{i}_{\pm}}^{u+}$ and $W_{\Gamma^{a}}^{s+}$, but not in $W_{\Gamma^{o}}^{s-}$. <figure><img src="content_image/1801.07275/sos_disA_1.png"><figcaption>Figure 11: First and last intersections of invariant manifolds with theoutward annulus of the middle DS for E=1. Wu+Γi+ (green) forms the boundary ofγu+i, Ws−Γo+ (red) and Ws−Γo− (orange) form the boundary of γs−o, Wu−Γo+ isblack and Wu−Γo− is grey. Wu−Γi+ copies the shape of Wu−Γo+ inside γu+i.Selected initial conditions for roaming with very long residence times aremarked with blue crosses.</figcaption></figure> ### Roaming on the middle DS As mentioned in Section 4.3, the middle DS seems to be better suited for the study of roaming than the inner and outer DSs. More precisely, we will study dynamics on the outward annulus of the middle DS, i.e. the annulus crossed by the prototypical dissociating trajectory $\theta=0$, $p_{r}>0$, $p_{\theta}=0$. We may introduce canonical coordinates on this annulus using a generating function in the same way as we did in Section 4.2, but for for the sake of simplicity we continue using the coordinates $(\theta,p_{\theta})$. In the following elaboration we need means to precisely express the order in which invariant cylinders intersect the outward annulus of the middle DS. Based on the arguments in Section 5.2, roaming involves the invariant cylinders $W_{\Gamma^{i}_{\pm}}^{u+}$, $W_{\Gamma^{a}}^{s+}$ and $W_{\Gamma^{o}}^{s-}$. Due to symmetry we have that every statement regarding $W_{\Gamma^{i}_{+}}^{u+}$ also holds for $W_{\Gamma^{i}_{-}}^{u+}$. The dynamics under the return map associated with the surface of section does not require $W_{\Gamma^{a}}^{s+}$ for a complete and detailed description of dynamics. The simple fact that a point on the surface is mapped by the return map to another point on the surface is enough to deduce that the corresponding trajectory is contained $W_{\Gamma^{a}}^{s+}$ and in fact, all other invariant cylinders made up of invariant manifolds of $\Gamma^{a}_{\pm}$. Consequently, for a description of roaming on the outward annulus of the middle DS we only need $W_{\Gamma^{i}_{+}}^{u+}$ and $W_{\Gamma^{o}}^{s-}$. Every branch of the invariant manifolds intersects the middle DS in a topological circle. Since it is possible that a branch of invariant manifold returns to the middle DS, by the first intersection of an unstable branch of invariant manifold with the outward annulus of the middle DS we mean that all points on the circle converge in backward time to the respective TS without reintersecting the outward annulus of the middle DS. Similarly we define the last intersection of a stable branch in forward time. <figure><img src="content_image/1801.07275/sos_disA_2.png"><figcaption>Figure 12: First and last intersections of invariant manifolds with theoutward annulus of the middle DS for E=2. Wu+Γi+ (green) forms the boundary ofγu+i, Ws−Γo+ (red) and Ws−Γo− (orange) form the boundary of γs−o, Wu−Γo+ isblack and Wu−Γo− is grey. Wu−Γi+ copies the shape of Wu−Γo+ inside γu+i.Selected initial conditions for roaming with very long residence times aremarked with blue crosses.</figcaption></figure> Denote the interior of the first/last intersection of the invariant cylinders $W_{\Gamma^{i}_{+}}^{u+}$ and $W_{\Gamma^{o}}^{s-}$ with the outward annulus of the middle DS by $\gamma^{u+}_{i}$ and $\gamma^{s-}_{o}$, respectively. Denote the Poincaré return map associated with the outward annulus of the middle DS by $P$. By our findings all trajectories originating in $B_{1}^{+}$ and all trajectories that cross the inward annulus of the outer DS reach the middle DS. By definition we have that \[\gamma^{u+}_{i}\cap\gamma^{s-}_{o},\] contains trajectories that dissociate quickly. This is due to the fact that $\gamma^{u+}_{i}$ contains trajectories that just escaped from $B_{1}^{+}$ and $\gamma^{s-}_{o}$ contains those that reach the outer DS and therefore never return to the middle DS. Therefore points in $\gamma^{u+}_{i}\cap\gamma^{s-}_{o}$ do not have an image under the return map $P$, in fact the whole of $\gamma^{s-}_{o}$ does not have an image. This is in accordance with the results on “reactive islands” by [28]. Figures 11, 12 and 13 show this intersection for various energies together with the first/last intersections of other invariant cylinders. Note that trajectories passing through $\gamma^{u+}_{i}\cap\gamma^{s-}_{o}$ reach the outer DS in varying amounts of time. We can expect the trajectory representing fast dissociation passing through $\theta=0$, $p_{r}>0$, $p_{\theta}=0$ to take significantly less time than trajectories in the proximity of $W_{\Gamma^{o}}^{s-}$, which may take arbitrarily long as they approach $\Gamma^{o}_{\pm}$. Therefore if roaming was to be only defined by time spent in $B_{2}$ or in the neighbourhood of a periodic orbit, we can always find a suitable trajectory in $\gamma^{u+}_{i}\cap\gamma^{s-}_{o}$ that is monotonous in $r$. Arguably, such a trajectory does not lead to an intramolecular hydrogen abstraction that has been reported in the context of roaming. <figure><img src="content_image/1801.07275/sos_disA_2_5.png"><figcaption>Figure 13: First and last intersections of invariant manifolds with theoutward annulus of the middle DS for E=2.5. Wu+Γi+ (green) forms the boundaryof γu+i, Ws−Γo+ (red) and Ws−Γo− (orange) form the boundary of γs−o, Wu−Γo+ isblack and Wu−Γo− is grey, Wu−Γi+ is cyan. Roaming is not present becauseγu+i⊂γs−o and Wu+Γi+ and Ws−Γo+ are disjoint.</figcaption></figure> It remains to explain what happens to $\gamma^{u+}_{i}\setminus\gamma^{s-}_{o}$. Since trajectories corresponding to points in $\gamma^{u+}_{i}\setminus\gamma^{s-}_{o}$ do not dissociate, they return to the outward annulus of the middle DS unless they are asymptotic to a periodic orbit. The set $\gamma^{u+}_{i}\setminus\gamma^{s-}_{o}$ has an image under the return map $P$ and it is $P\gamma^{u+}_{i}$. Note that the corresponding trajectories are guided to and from the middle DS by the invariant cylinders $W_{\Gamma^{a}}^{s-}$, $W_{\Gamma^{a}}^{u+}$, $W_{\Gamma^{a}}^{s+}$, $W_{\Gamma^{a}}^{u-}$. We remark that based on the understanding of lobe dynamics ([31], trajectories that cross the outward annulus of the middle DS and do not reach the outer DS are repelled by $\Gamma^{o}_{\pm}$ and necessarily pass through the homoclinic tangle of $\Gamma^{o}_{\pm}$. By the definition of Mauguière et al. [22], roaming trajectories cross the middle DS at least three times, which means crossing the outward annulus at least twice and the inward annulus at least once. Roaming trajectories must therefore contained in $P\gamma^{u+}_{i}$. In fact roaming trajectories that cross the outward annulus of the middle DS precisely $n$ times before dissociating pass through \[P^{n-1}\gamma^{u+}_{i}\cap\gamma^{s-}_{o}.\] Since recrossings of the middle DS are possible due to the homoclinic tangle of $\Gamma^{o}_{\pm}$, roaming requires that the invariant cylinder $W_{\Gamma^{i}_{+}}^{u+}$ conveys trajectories into the homoclinic tangle. Heteroclinic intersections are therefore necessary. Arguably, recrossings of the middle DS are inevitable to capture the process of intramolecular hydrogen abstraction, as reported by [4], where the free H atom has to return back to the $\text{CH}_{3}^{+}$ core. Isomerisation trajectories are also contained in $\gamma^{u+}_{i}\setminus\gamma^{s-}_{o}$ and are guided by $W_{\Gamma^{i}_{\pm}}^{s+}$ to $B_{1}^{\pm}$ and by $W_{\Gamma^{i}_{\pm}}^{u+}$ out of $B_{1}^{\pm}$. Trajectories that return to $B_{1}^{+}$ pass through the intersection of $P^{n}\gamma^{u+}_{i}\cap W_{\Gamma^{i}_{-}}^{u+}$, for some $n$. Trajectories corresponding isomerisation pass through the intersection of $P^{n}\gamma^{u+}_{i}$ and the last intersection of $W_{\Gamma^{i}_{-}}^{s+}$ with the outward annulus, for some $n$. It remains to discuss the first intersection of $W_{\Gamma^{o}_{+}}^{u-}$, $W_{\Gamma^{o}_{-}}^{u-}$ and $W_{\Gamma^{i}_{+}}^{u-}$ on the surface of section shown in Figures 11, 12 and 13. We shall denote the intersections according to the convention above by $\gamma^{u-}_{o}$ and $\gamma^{u-}_{i}$, respectively. The invariant cylinder $W_{\Gamma^{o}}^{u-}$ guides energy surface volume from the inward annulus of the outer DS into the interaction region. Clearly then $\gamma^{s-}_{o}\cap\gamma^{u-}_{o}$ corresponds to trajectories that intersect the surface of section only once. The part of the intersection in $\gamma^{u+}_{i}$ passes through $B_{1}^{+}$. $W_{\Gamma^{o}}^{u-}$ is guided from the inward hemisphere of the inner DS by $W_{\Gamma^{i}_{+}}^{u-}$ and its homoclinic intersections cause tongues. In the process, $W_{\Gamma^{o}}^{u-}$ and $W_{\Gamma^{i}_{+}}^{u-}$ are stretched and compressed causing only one to be visible in Figures 11 and 12. In Figure 13$W_{\Gamma^{o}}^{u-}$ does not enter $B_{1}^{+}$ and the two invariant cylinders are visible. It is important to point out that seemingly $W_{\Gamma^{o}}^{u-}$ and $W_{\Gamma^{i}_{+}}^{u-}$ intersect, which is impossible. Instead we observe a discontinuity caused by points on $W_{\Gamma^{o}}^{u-}$ heteroclinic to $\Gamma^{i}_{+}$. As mentioned above, a part of $W_{\Gamma^{o}}^{u-}$ passes through $B_{1}^{+}$ and is mapped by $P$ into $\gamma^{u+}_{i}$, the remainder visible on the surface of section stays in $B_{2}$ and is mapped outside of $\gamma^{u+}_{i}$. The points inbetween are not intersections between $W_{\Gamma^{o}}^{u-}$ and $W_{\Gamma^{i}_{+}}^{u-}$, but between $W_{\Gamma^{o}}^{u-}$ and $W_{\Gamma^{i}_{+}}^{s\pm}$ and do not have an image under $P$. Note that $\gamma^{u-}_{o}$ carries information about roaming in backward time. Since points in $\gamma^{u-}_{o}$ do not have a preimage under $P$, all points in $\gamma^{u+}_{i}\cap\gamma^{s-}_{o}$ that are not in $\gamma^{u-}_{o}$ must have a preimage under $P$. These points correspond to trajectories that qualify as roaming in backward time. With increasing energy it becomes difficult to study $\gamma^{u+}_{i}\setminus\gamma^{s-}_{o}$ due to the fractal structures of $W_{\Gamma^{o}}^{u-}$ caused by heteroclinic points. We can, however, expect proportionally fewer roaming trajectories to enter $B_{1}^{+}$ multiple times at $E=2$ than at $E=1$. Instead it is probable to find roaming trajectories spending the majority of their residence time in $B_{2}$ at $E=2$. ## 6 Global study of the invariant manifolds that govern the dynamics In this section we discuss an alternative way of studying dynamics on a $3$-dimensional energy surface using the so called Conley-McGehee representation [7], [26], [17], described along with other alternatives in [37]. This is a very useful way of studying dynamics in full $3$ dimensions, but to date has only been defined for subsets of energy surfaces that are locally a spherical shell. Since the Conley-McGehee representation does only works in $B_{1}^{\pm}$ of Chesnavich’s CH${}_{4}^{+}$ model studied here, we introduce an extension of the Conley-McGehee representation that enables us to study energy surfaces with other geometry than in the Conley-McGehee case. ### Conley-McGehee representation The dynamics on the energy surface can be visualized in many ways. Just as it was done above, it can be viewed on various surfaces of section, most notably ones constructed around TSs. It is also possible to study the system locally using normal form approximations. The Williamson normal form [40] of the Hamiltonian in the neighbourhood of an index-$1$ critical point is \[H_{2}(q_{1},p_{1},q_{2},p_{2})=\frac{1}{2}\lambda(p_{1}^{2}-q_{1}^{2})+\frac{1 }{2}\omega(p_{2}^{2}+q_{2}^{2}),\] for some $\lambda,\omega>0$. We found that for a fixed energy $H_{2}(q_{1},p_{1},q_{2},p_{2})=h_{2}$, the energy surface can be locally viewed as a continuum of spheres parametrized by $q_{1}$. In the Conley-McGehee representation [7], [26], is based on the spherical local geometry of an energy surface. While the normal form perspective above only applies locally, in the Conley-McGehee representation the whole energy surface is represented as a nested set of spheres parametrised in the radial direction by the reaction coordinate. Advantages are immediate - the representation gives a global model of the energy surface and by construction reveals the spherical structure of the energy surface. For $2$ degrees of freedom it enables us to study the $3$-dimensional energy surface in the full $3$ dimensions. Moreover, it enables to visualise the DSs as spheres that separate the energy surface into two disjoint components. It is also very natural that the flux through the hemispheres of the DSs is unidirectional and trajectories have to cross a particular hemisphere of the DS to pass from one component to the other. Apart from DSs the Conley-McGehee representation enables us to visualise and therefore study TSs and their invariant manifolds that are spherical cylinders in a natural environment. ### Toric extension of the Conley-McGehee representation The Conley-McGehee representation in its original form applies to spherical geometries. The energy surface of the CH${}_{4}^{+}$ model, on the other hand, has a partially spherical and partially toric geometry, where many periodic orbits come in pairs and several DSs are tori. We therefore adapt the Conley-McGehee representation for the energy surface as follows. The energy surface is defined by \[M_{E}=\Big{\{}(r,\theta,p_{r},p_{\theta})\in\mathbb{R}^{4}\Big{\|}H(r,\theta,p_ {r},p_{\theta})=\frac{1}{2m}p_{r}^{2}+\frac{1}{2}\left(\frac{1}{I}+\frac{1}{mr ^{2}}\right)p_{\theta}^{2}+U(r,\theta)=E\Big{\}}.\] For very high energies, $E>E_{2}$, where only $r<0.9$ is energetically inaccessible due to the cut-off of the potential and $E>U(r,\theta)$ for all $r\geq 0.9$, the whole energy surface has a toric local geometry. For any fixed radius $r_{0}$ and a fixed $\theta_{0}$, \[M_{E}(r_{0},\theta_{0})=\Big{\{}(p_{r},p_{\theta})\in\mathbb{R}^{2}\Big{\|} \frac{1}{2m}p_{r}^{2}+\frac{1}{2}\left(\frac{1}{I}+\frac{1}{mr_{0}^{2}}\right) p_{\theta}^{2}=E-U(r_{0},\theta_{0})\Big{\}},\] is a $\mathbb{S}^{1}$. If $\theta$ is not fixed, $M_{E}(r_{0},\theta)$ defines a $\mathbb{S}^{1}\times\mathbb{S}^{1}=\mathbb{T}^{2}$ and therefore the whole energy surface $M_{E}$ is a $\mathbb{T}^{2}\times\mathbb{R}^{+}$. The radii of the concentric circles $M_{E}(r_{0},\theta_{0})$ on the $(p_{r},p_{\theta})$-plane depend on $r_{0}$ and $\theta_{0}$ through $U(r_{0},\theta_{0})$. The potential energy is not monotonous in $r$ nor in $\theta$. Recall from Section 3.1 that * the two wells $q_{0}^{\pm}$ are located at $(1.1,0)$ and $(1.1,\pi)$ with $U(q_{0}^{\pm})=E_{0}\approx-47$, * the two index-$2$ saddles $q_{2}^{\pm}$ are located at $(1.63,\frac{\pi}{2})$ and $(1.63,\frac{3\pi}{2})$ with $U(q_{2}^{\pm})=E_{2}\approx 22.27$. Further recall from Section 2.2 that for $r$ sufficiently large $U(r,\theta)$ is essentially independent of $\theta$ and $r^{2}U(r,\theta)\to 0$ as $r\rightarrow\infty$ for all $\theta$. It follows that the tori corresponding to $r_{0}=1.1$ and $r_{0}=r_{large}$, for some $r_{large}$ sufficiently large, intersect. This is because the circle $M_{E}(1.1,0)$ has a larger radius than $M_{E}(r_{large},0)$, while $M_{E}(1.1,\frac{\pi}{2})$ has a smaller radius than $M_{E}(r_{large},\frac{\pi}{2})$. The tori will always intersect if the radius is not a monotonous in $r$. In order to extend the Conley-McGehee representation, we need to reparametrise these tori so that their radii are monotonous in $r$ for every $\theta$. Define \[P_{r}=\frac{r}{\sqrt{2m(E-U(r,\theta))}}p_{r},\] and \[P_{\theta}=\frac{r\sqrt{\frac{1}{I}+\frac{1}{mr^{2}}}}{\sqrt{2(E-U(r,\theta))} }p_{\theta}.\] Now we have \[P_{r}^{2}+P_{\theta}^{2}=r^{2}\frac{1}{E-U(r,\theta)}\Big{(}\frac{1}{2m}p_{r}^ {2}+\frac{1}{2}\left(\frac{1}{I}+\frac{1}{mr^{2}}\right)p_{\theta}^{2}\Big{)}= r^{2}.\] The radius of the tori is monotonous in $r$ and independent of $\theta$ and therefore the tori $P_{r}^{2}+P_{\theta}^{2}=r^{2}$ foliating the energy surface are, unlike before the reparametrisation, disjoint in $(\theta,P_{r},P_{\theta})$-space. Note that a section of the tori with a plane of section $\theta=\theta_{0}$ shows concentric circles, where the smallest one has the radius $r=0.9$ due to the cut-off of the system. This is due to the fact that the boundary of the energy surface corresponds to a torus. Should it be desirable to have the whole $(\theta,P_{r},P_{\theta})$-space foliated by tori, it can be done by replacing $r$ by $r-0.9$ in the definitions of $P_{r}$ and $P_{\theta}$. ### Extension to non-constant geometries We remark that the construction above relies on the fact that at $E>E_{2}$ the energy surface has a purely toric geometry. We can slightly amend the construction to work for lower energies $E\leq E_{2}$, where the energy surface is not purely toric. $E=U(r,\theta)$ does not pose a problem for the definition of $P_{r}$ and $P_{\theta}$ as it may seem on first sight. $P_{r}$ and $P_{\theta}$ are only normalized conjugate momenta and by definition \[\|P_{r}\|,\|P_{\theta}\|\leq r.\] The momenta are therefore well defined on the whole energy surface. Points on $E=U(r,\theta)$ are degenerate circles with radius $0$ on the energy surface, but due to normalization correspond to circles in $(P_{r},P_{\theta})$. Such a representation of the energy surface for lower energies is clearly flawed. For $r$ large, we still have tori, but for smaller $r$, e.g. near $\Gamma^{i}$, we do not see the spherical geometry we expect. To solve this issue, we introduce different momenta in which the radius $P_{r}^{2}+P_{\theta}^{2}\to 0$ as $U(r,\theta)\to E$. We remark that the following only works for $E\leq\widetilde{E}_{1}$. In the interval $\widetilde{E}_{1}<E<E_{2}$, the projection of the energy surface on configuration space is the whole plane minus three discs, see Figure 3. One is the potential energy cut-off and the other two are areas of high potential around index-$2$ critical points $q_{2}^{\pm}$. Spherical and toric geometry cannot accurately represent a genus $3$ surface. Since for energies $E<0$ the standard Conley-McGehee representation applies, we will restrict ourselves to the more interesting case $0\leq E\leq\widetilde{E}_{1}$. For the sake of simplicity, we retain the notation $P_{r}$ and $P_{\theta}$, making clear that we are discussing different momenta in a different energy interval than before. <figure><img src="content_image/1801.07275/ConleyContour.jpg"><figcaption>Figure 14: Contour plot of r6(E−U(r,θ)) for E=0. The black circle correspondsto the radius rE defined in the text.</figcaption></figure> Let \[P_{r}=\frac{r^{3}}{\sqrt{2m}}p_{r},\] and \[P_{\theta}=r^{3}\sqrt{\frac{1}{2}\left(\frac{1}{I}+\frac{1}{mr^{2}}\right)}p_{ \theta}.\] It follows that \[P_{r}^{2}+P_{\theta}^{2}=r^{6}\Big{(}\frac{1}{2m}p_{r}^{2}+\frac{1}{2}\left( \frac{1}{I}+\frac{1}{mr^{2}}\right)p_{\theta}^{2}\Big{)}=r^{6}(E-U(r,\theta)).\] While $E-U(r,\theta)$ makes sure that zero kinetic energy corresponds to $P_{r}=P_{\theta}=0$, the term $r^{6}$ seems perhaps less obvious. In the previous section we showed that it is important for $P_{r}^{2}+P_{\theta}^{2}$ to be monotonous in $r$ for every $\theta$. This is however not possible, because for some fixed angles $\theta=const$ the term $(E-U(r,\theta))$ vanishes for several values of $r$. This is only possible in coordinates in which $E=U(r,\theta)$ correspond to coordinate lines for all $E$. Let $r_{E}$ be the smallest $r$ such that $E=U(r,\theta)$ has at most one solution for every $\theta$ for $r_{E}\leq r$. The term $r^{6}$ is the smallest even power of $r$ such that $r^{6}(E-U(r,\theta))$ is monotonous in $r$ on $r_{E}\leq r$ for $0\leq E\leq\widetilde{E}_{1}$. As a consequence of restricting the radius, the representation omits a significant part of $B_{1}^{\pm}$. Since we formulated roaming as a transport problem from the inner DS to the outer DS, dynamics inside $B_{1}^{\pm}$ does not play a significant role in our study. All significant periodic orbits and DSs are well defined in the Conley-McGehee representation as presented here. Figure 14 shows the contour plot of $r^{6}(E-U(r,\theta))$ with a highlighted circle marking $r_{0}$, the boundary of the representation defined above. ### Consequences of the extension <figure><img src="content_image/1801.07275/rlevelspolar0.jpg"><figcaption>Figure 15: Comparison of energy surface geometry representation for E=0 in(θ,pr,pθ), left, and (θ,Pr,Pθ), right. The surfaces shown correspond toselected fixed values of r. The value r=r0=2.0267 corresponds to the spheres,r=2.802 defines the pinched torus and r=3.5 is a regular torus. Additionally,the right figure also shows r=4.</figcaption></figure> In the representation as defined above, the geometry of the energy surface is preserved. For a fixed radius $r_{0}$, we can see that the surfaces have the following topologies: * $\mathbb{T}^{2}$ if $U(r_{0},\theta)<E$ for all $\theta$, * a pinched torus if $U(r_{0},\theta)\leq E$ for all $\theta$ and $U(r_{0},\theta_{0})=E$ for some $\theta_{0}$, * $\mathbb{S}^{2}\cup\mathbb{S}^{2}$ if $U(r_{0},\theta_{0})>E$ for some $\theta_{0}$. For $E=0$, an example of each is shown in Figure 15 in the canonical phase space coordinates $(\theta,p_{r},p_{\theta})$ and in the proposed extension of Conley-McGehee representation $(\theta,P_{r},P_{\theta})$. Note indeed in the latter that the surfaces are disjoint and present part of a foliation of the energy surface. We added an additional value of $r=4$ to the extended Conley-McGehee representation to illustrate that the radius of the tori diverges for $r\rightarrow\infty$, whereas it converges in the canonical phase space coordinates. Due to the properties of the extended Conley-McGehee representation, we may study invariant structures and the aforementioned DSs globally on the energy surface. All techniques used to date either relied on surfaces of section or local approximations of the energy surface. In what follows, we study the structures on the energy surface in full three dimensions. In Figure 16 we present the periodic orbits $\Gamma^{i}_{+}$, $\Gamma^{a}_{\pm}$ and $\Gamma^{o}_{\pm}$, all of which are TSs, and the associated DSs at energy $E=2.5$. The construction of the DSs was discussed in Section 3.3. We remark that from a qualitative perspective Figure 16 can be thought to represent the whole energy interval $0<E<2.72$, where $\Gamma^{a}_{\pm}$ are unstable. The difference at higher energies is that $\Gamma^{a}_{\pm}$ is not a TS and the associated torus is not a DS. We have used some of the DSs mentioned above in previous sections to study residence times, rotation numbers, and most importantly, the intersections of stable and unstable invariant manifolds of TSs. For the sake of clarity, in the following we left out the $\Gamma^{i}_{-}$ and all associated structures, but everything said about $\Gamma^{i}_{+}$ also holds for $\Gamma^{i}_{-}$. In the figures one can easily imagine another sphere just like the inner DS but shifted by $\pi$ in the angular direction. Note that here we take full advantage of the proposed extension of the Conley-McGehee representation to present structures on the energy surface with different local geometries - the inner DS is a sphere whereas the middle and the outer DSs are tori. This has to our knowledge not been done before. <figure><img src="content_image/1801.07275/E25.jpg"><figcaption>Figure 16: TSs and DSs in the Conley-McGehee representation for E=2.5. Theinner DS is shown in green, middle DS in red, outer DS in blue. Γi+, Γa± andΓo± are the thick lines on the corresponding DSs shown in darker green, redand blue, respectively.</figcaption></figure> On the DSs we highlighted the respective TSs. Note that the inner DS is defined using one periodic orbit whereas the middle and the outer are defined using two. The individual orbits in the families $\Gamma^{a}$ and $\Gamma^{o}$ can be distinguished by the sign of $P_{\theta}$ as they run in opposite directions. The inner DS is divided by $\Gamma^{i}_{+}$ into two hemispheres and the surfaces of unidirectional flux from the definition of a TS. Flux from $B_{1}^{+}$ to $B_{2}$ crosses the hemisphere where predominantly $P_{r}>0$. The situation is similar for the annuli of the middle and outer DS. The two orbits divide the torus into two annuli, where the outward annulus is the one with larger $P_{r}$. <figure><img src="content_image/1801.07275/manif5.jpg"><figcaption>Figure 17: TSs, corresponding DSs and invariant manifolds in the Conley-McGehee representation at E=5. The inner DS is shown in green, middle DS inred and outer DS in blue. Γi+, Γa± and Γo± are the thick lines on thecorresponding DSs shown in darker green, red and blue respectively. Note thatΓa± are stable at this energy. The invariant manifolds WΓi+ (green) are only asketch based on the computed sections on the surface θ=0 shown in thick green.</figcaption></figure> The mechanism behind energy surface volume transport across DSs is governed by invariant manifolds of the corresponding TSs as discussed in the Sections 4.3 and 5. Here we present a new perspective for the study of invariant manifolds. In the following we use $E=5$, because at this energy the TSs are evenly spaced and the lack of heteroclinic intersections facilitates understanding of this new perspective. Everything we say is applicable to the whole interval $0<E<6.13$ relevant to roaming. Figure 17 displays the invariant manifolds of $\Gamma^{i}_{+}$ and $\Gamma^{o}_{\pm}$ in full $3$ dimensions. Note that the manifolds $W_{\Gamma^{i}_{+}}$ are only a sketch based on computed sections on the surface $\theta=0$. Computing the whole invariant manifold numerically is in this case relatively straight-forward, but for the sole purpose of illustration unnecessarily expensive. <figure><img src="content_image/1801.07275/slice2.jpg"><figcaption>Figure 18: A section of invariant manifolds for θ=0 at E=2 in the Conley-McGehee representation. The inner DS is shown in green, middle DS in red,outer DS in blue. Γi+, Γa± and Γo± are the thick lines on the correspondingDSs shown in darker green, red and blue respectively and the curves extendingfrom the TSs are their respective invariant manifolds. For clarity, theintersections of WΓi and WΓo are only indicated.</figcaption></figure> Clearly visible is the structure of the manifolds, spherical cylinders formed by $W_{\Gamma^{i}_{+}}$ and toric cylinders formed by $W_{\Gamma^{o}_{\pm}}$. This is how the sections of invariant manifolds in the extended Conley-McGehee representation near $\theta=0$ shown in Figures 18 and 19 should be interpreted. For clarity, the invariant cylinders are indicated by the section of invariant manifolds for $\theta=0$ in these figures, the scale and complexity of intersections of the invariant cylinders would make the figures incomprehensible. One can clearly see that at $E=5$, in fact in the whole energy interval $E\geq 2.5$, the intersection of $W_{\Gamma^{i}_{+}}^{u+}$ with the middle and the outer DSs produces a topological circle centred at $P_{\theta}=0$. This is purely the consequence of the spherical geometry induced by $\Gamma^{i}_{+}$. $W_{\Gamma^{o}_{\pm}}$ for the same reason intersects the middle DS in two lines that should be seen as circles concentric with $\Gamma^{a}_{\pm}$. This is in agreement with the sections of invariant manifolds on the outward annulus of the middle DS at $E=2.5$ shown in Figure 13. Note that $W_{\Gamma^{i}_{+}}^{u-}$ intersects the outward hemisphere of the middle DS at $E=2.5$ in a shape that cannot be identified as a circle. This is due to the fact that we study $W_{\Gamma^{i}_{+}}^{u-}$ that is asymptotic to $\Gamma^{i}$ in $B_{1}^{+}$ as it leaves from $B_{1}^{+}$ to $B_{2}$. Moreover, it is deformed in the proximity of $\Gamma^{a}$ that exhibits a different kind of dynamics than $\Gamma^{i}$, rotating as opposed to vibrating. According to the findings in Section 5.3, the non-existence of roaming at higher energies due to the lack of intersection of $W_{\Gamma^{i}_{\pm}}$ with $W_{\Gamma^{o}_{\pm}}$ is immediate from Figures 17 and 19. <figure><img src="content_image/1801.07275/slice25.jpg"><figcaption>Figure 19: A section of invariant manifolds for θ=0 at E=2.5 in the Conley-McGehee representation. The inner DS is shown in green, middle DS in red,outer DS in blue. Γi+, Γa± and Γo± are the thick lines on the correspondingDSs shown in darker green, red and blue respectively and the curves extendingfrom the TSs are their respective invariant manifolds. Note that WΓi and WΓodo not intersect.</figcaption></figure> The situation at $E<2.5$ represented by Figure 12 is very similar to the energy interval $E\geq 2.5$. The main difference here are the heteroclinic intersections that cause roaming. These intersections are visible on the middle DS as well as in the extended Conley-McGehee representation. We remark that in the extended Conley-McGehee representation roaming occurs in the thin stripes between the two invariant cylinders $W_{\Gamma^{o}}^{s-}$ and $W_{\Gamma^{o}}^{u-}$ around $P_{r}=0$. Clearly the majority of the energy surface is occupied by more direct dynamics. ## 7 Conclusion We have shown that numerical observations of long dissociation are caused by particular structures formed by invariant manifolds of TSs. These invariant manifolds are also responsible for multiple recrossings of the middle DS and consequently also for roaming. We have shown that roaming trajectories that originate in the potential wells are captured in the homoclinic tangles of the outer TS and the middle TS dissociating. The transition for the potential wells into the homoclinic tangles is only possible in case the invariant manifolds of the inner and outer DS intersect and create a heteroclinic tangle. In case of Chesnavich’s CH${}_{4}^{+}$ model, this heteroclinic intersection only exists for energies $E\leq 2.5$ and therefore the system does not admit roaming at higher energy levels. Our results can possible be directly extended to other chemical reactions, as the only significant assumption on the potential energy is that for all $\theta$ \[U(r,\theta)\in o(r^{-2})\text{ as }r\rightarrow\infty.\] This condition guarantees the existence of an outer TS thanks to which we may restrict roaming to a transport problem from potential wells representing a stable molecule to the DS associated with the outer TS. Furthermore, our findings support then dynamical definition of roaming as suggested in [22]. We studied the above-mentioned invariant manifolds on various dividing surfaces, some of which highlighted the difficulties posed by the unusual energy surface geometry. This was the case even hough all surfaces of section satisfy the Birkhoff condition [2] of being bounded by invariant manifolds. These difficulties motivated us to construct an extension of the Conley-McGehee representation. Using the extension we were able to study the unusual energy surface geometry, TSs, the associated invariant manifolds and DSs in full three dimensions. ## Acknowledgment We are grateful to Dayal Strub for many very useful discussions and remarks on the contents of this paper. ## References * [1] V. I. Arnold. _Les méthodes mathématiques de la mécanique classique_. Nauka, Moscow, Éditions Mir, 1976. * [2] G. D. Birkhoff. _Dynamical Systems_. AMS, 1927. * [3] J. M. Bowman. Roaming radicals. _Mol. Phys._, 112(19):2516–2528, 2014. doi:10.1080/00268976.2014.897395. * [4] J. M. Bowman and B. C. Shepler. Roaming radicals. _Annu. Rev. Phys. Chem._, 62:531–553, 2011. doi:10.1146/annurev-physchem-032210-103518. * [5] W. J. Chesnavich. Multiple transition states in unimolecular reactions. _J. Chem. Phys._, 84(5):2615–2619, 1986. doi:10.1063/1.450331. * [6] W. J. Chesnavich, T. Su, and M. T. Bowers. Collisions in a noncentral field: A variational and trajectory investigation of ion-dipole capture. _J. Chem. Phys._, 72(4):2641, 1980. doi:10.1063/1.439409. * [7] C. C. Conley. Low energy transit orbits in the restricted three-body problem. _SIAM J. Appl. Math._, 16:732–746, 1968. doi:10.1137/0116060. * [8] B. Eckhardt and D. Wintgen. Indices in classical mechanics. _J. Phys. A_, 24(18):4335, 1991. doi:10.1088/0305-4470/24/18/020. * [9] G. S. Ezra, H. Waalkens, and S. Wiggins. Microcanonical rates, gap times, and phase space dividing surfaces. _J. Chem. Phys._, 130(16):164118, 2009. URL: http://dx.doi.org/10.1063/1.3119365. * [10] N. Fenichel. Persistence and smoothness of invariant manifolds for flows. _Indiana Univ. Math. J._, 21:193–226, 1971. doi:10.1512/iumj.1972.21.21017. * [11] J. M. Greene. Two-dimensional measure-preserving mappings. _J. Math. Phys._, 9(5):760–768, 1968. doi:10.1063/1.1664639. * [12] G. W. Hill. _Collected Mathematical Works of G. W. Hill, vol. 1_. Washington:Carnegie Institute, 1905. * [13] M. W. Hirsch, C. C. Pugh, and M. Shub. _Invariant Manifolds_. Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg, 1977. * [14] P. L. Huston, R. Conte, and J. M. Bowman. Roaming under the microscope: Trajectory study of formaldehyde dissociation. _J. Phys. Chem. A_, 120:5103–5114, 2016. doi:10.1021/acs.jpca.6b00488. * [15] C. Jaffé, D. Farrelly, and T. Uzer. Transition state in atomic physics. _Phys. Rev. A_, 60(5):3833, 1999. doi:10.1103/PhysRevA.60.3833. * [16] M. J. T. Jordan, S. C. Smith, and R. G. Gilbert. Variational transition state theory: A simple model for dissociation and recombination reactions of small species. _J. Phys. Chem._, 95(22):8685–8694, 1991. doi:10.1021/j100175a050. * [17] R. S. MacKay. Flux over a saddle. _Phys. Lett. A_, 145(8,9):425–427, 1990. doi:10.1016/0375-9601(90)90306-9. * [18] R. S. MacKay, J. D. Meiss, and I. C. Percival. Transport in Hamiltonian systems. _Phys. D_, 13(1-2):55–81, 1984. doi:10.1016/0167-2789(84)90270-7. * [19] R. S. MacKay and D. C. Strub. Bifurcations of transition states: Morse bifurcations. _Nonlinearity_, 27(5):859–895, 2014. doi:10.1088/0951-7715/27/5/859. * [20] R. S. MacKay and D. C. Strub. Morse bifurcations of transition states in bimolecular reactions. _Nonlinearity_, 28(12):4303, 2015. doi:10.1088/0951-7715/28/12/4303. * [21] F. A. L. Mauguière, P. Collins, G. S. Ezra, S. C. Farantos, and S. Wiggins. Multiple transition states and roaming in ion–molecule reactions: A phase space perspective. _Chem. Phys. Lett._, 592:282–287, 2014. doi:10.1016/j.cplett.2013.12.051. * [22] F. A. L. Mauguière, P. Collins, G. S. Ezra, S. C. Farantos, and S. Wiggins. Roaming dynamics in ion-molecule reactions: Phase space reaction pathways and geometrical interpretation. _J. Chem. Phys._, 140(13):134112, 2014. doi:10.1063/1.4870060. * [23] F. A. L. Mauguière, P. Collins, G. S. Ezra, and S. Wiggins. Bond breaking in a Morse chain under tension: Fragmentation patterns, higher index saddles, and bond healing. _J. Chem. Phys._, 138(13):134118, 2013. doi:10.1063/1.4798641. * [24] F. A. L. Mauguière, P. Collins, Z. C. Kramer, B. K. Carpenter, G. S. Ezra, S. C. Farantos, and S. Wiggins. Phase space structures explain hydrogen atom roaming in formaldehyde decomposition. _J. Phys. Chem. Lett._, 6(20):4123–4128, 2015. doi:10.1021/acs.jpclett.5b01930. * [25] F. A. L. Mauguière, P. Collins, Z. C. Kramer, B. K. Carpenter, G. S. Ezra, S. C. Farantos, and S. Wiggins. Phase space barriers and dividing surfaces in the absence of critical points of the potential energy: Application to roaming in ozone. _J. Chem. Phys._, 144(5):54107, 2016. doi:10.1063/1.4940798. * [26] R. P. McGehee. Some homoclinic orbits in the restricted three-body problem. _PhD thesis_, 1969. * [27] J. D. Meiss. Thirty years of turnstiles and transport. _Chaos_, 25(9):097602, 2015. doi:10.1063/1.4915831. * [28] A. M. Ozorio de Almeida, N. de Leon, M. A. Mehta, and C. C. Marston. Geometry and dynamics of stable and unstable cylinders in hamiltonian systems. _Phys. D_, 46(2):265–285, 1990. doi:10.1016/0167-2789(90)90040-V. * [29] P. Pechukas and F. J. McLafferty. On transition-state theory and the classical mechanics of collinear collisions. _J. Chem. Phys._, 58:1622–1625, 1973. doi:10.1063/1.1679404. * [30] P. Pechukas and E. Pollak. Transition states, trapped trajectories, and classical bound states embedded in the continuum. _J. Chem. Phys._, 69:1218–1226, 1978. doi:10.1063/1.436658. * [31] V. Rom-Kedar and S. Wiggins. Transport in two-dimensional maps. _Arch. Ration. Mech. Anal._, 109(3):239–298, 1990. doi:10.1007/BF00375090. * [32] O. R. Ruiz. Existence of brake-orbits in Finsler mechanical systems. _PhD thesis_, 1975. * [33] D. Townsend, A. Lahankar, S. K. Lee, S. D. Chambreau, A. G. Suits, X. Zhang, J. Rheinecker, L. B. Harding, and J. M. Bowman. The roaming atom: Straying from the reaction path in formaldehyde decomposition. _Science_, 306(5699):1158–1161, 2004. doi:10.1126/science.1104386. * [34] T. Uzer, C. Jaffé, J. Palacián, P. Yanguas, and S. Wiggins. The geometry of reaction dynamics. _Nonlinearity_, 15:957–992, 2002. doi:10.1088/0951-7715/15/4/301. * [35] R. D. van Zee, M. F. Foltz, and C. B. Moore. Evidence for a second molecular channel in the fragmentation of formaldehyde. _J. Chem. Phys._, 99(3):1664–1673, 1993. doi:10.1063/1.465335. * [36] H. Waalkens and S. Wiggins. Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed. _J. Phys. A_, 37:L435, 2004. doi:10.1088/0305-4470/37/35/L02. * [37] H. Waalkens and S. Wiggins. Geometric models of the phase space structures governing reaction dynamics. _Regul. Chaotic Dyn._, 15:1–39, 2010. doi:10.1134/S1560354710010016. * [38] S. Wiggins, L. Wiesenfeld, C. Jaffé, and T. Uzer. Impenetrable barriers in phase-space. _Phys. Rev. Lett._, 86:5478–5481, 2001. doi:10.1103/PhysRevLett.86.5478. * [39] E. Wigner. Calculation of the rate of elementary association reactions. _J. Chem. Phys._, 5:720–725, 1937. doi:10.1063/1.1750107. * [40] J. Williamson. On the algebraic problem concerning the normal forms of linear dynamical systems. _Amer. J. Math._, 58(1):141–163, 1936. doi:10.2307/2371062.
1911.01574	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 48100, "num_imgs": 6, "llama3_tokens_count": 12726 }	[ "content_image/1911.01574/diffest-gray.jpg", "content_image/1911.01574/wallrelaxspheresim-gray.png", "content_image/1911.01574/FEM_vs_Freeman.png", "content_image/1911.01574/RbGeometry.png", "content_image/1911.01574/110ccfulltrans.png", "content_image/1911.01574/110cellsteadystatexepol.png" ]	# A Novel, Finite-Element Model for Spin-Exchange Optical Pumping Using an Open-Source Code G.M. Schrank February 20, 2024 ###### Abstract A new model is presented for spin-exchange optical pumping using an open-source code, ElmerFEM-CSC. The model builds on previous models by adding the effects of alkali-vapor heterogeneity in optical pumping cells and by modeling the effects of hyperpolarized-gas wall-relaxation using a diffusion model. The code supports full, three-dimensional solutions to optical-pumping models, and solves for (1) laser absorption, (2) alkali vapor concentration, (3) fluid flow parameters, (4) thermal effects due to the pumping laser, and (5) noble gas polarization. The source code for the model is available for researchers to utilize and modify. pacs: Valid PACS appear here † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction Spin-exchange optical pumping (SEOP) is a technique whereby the ensemble nuclear spin-angular momentum of certain noble gasses can be increased to of order 10%. The technique is currently most notably used clinically and pre-clinically in lung imaging using MRI Oros and Shah (2004), but it has also been used in the NMR characterization of porous media Terskikh et al. (2002) and protein dynamics Schröder (2013). The physics of SEOP are described comprehensively in other places Walker (2011); Appelt et al. (1998). Briefly, the technique involves two steps: (1) optical pumping of an alkali-metal vapor and (2) spin-exchange from the alkali-metal vapor to a noble-gas nuclei. In the first step, optical pumping, a beam of circularly-polarized light is directed onto a transparent cell containing a macroscopic amount of alkali metal. The cell is heated, usually to between 100-200 ${}^{\circ}$C, in order to vaporize some amount of alkali metal. The laser interacts with the metal vapor to create close to 100% spin polarization of the alkali vapor. In the second step, spin-exchange, the alkali vapor transfers spin-angular momentum to the noble gas nuclei. A gas mixture is introduced into the cell containing a noble gas and some other inert gasses, and through collisional interactions, the metal vapor transfers its spin-polarization to the noble gas. The alkali metal vapor atoms become depolarized in this interaction, but because of optical pumping, the alkali atoms are quickly repolarized. One popular method of SEOP involves the hyperpolarization of xenon-129 (${}^{129}$Xe) using rubidium (Rb) vapor. Hyperpolarized ${}^{129}$Xe (HP ${}^{129}$Xe) gas is typically produced in a continuous manner using a flow-through polarizer Driehuys,B. et al. (1996); Ruset et al. (2006); Schrank et al. (2009). A flow-through polarizer operates by flowing a ${}^{129}$Xe gas mixture through an optical pumping cell containing Rb vapor. The ${}^{129}$Xe spin-exchange interaction occurs on a short enough timescale that considerable ${}^{129}$Xe polarizations can be achieved during the short transit through the optical pumping cell. Currently, there are no SEOP models that attempt to account for the full three-dimensional flow-dynamics of optical pumping cell geometries. Computational models for SEOP can be broken into two groups: finite difference models and finite element models (FEM). The finite difference models appear to have been the first in use, but detailed descriptions of the models do not appear extensively in the literature. Models of this type approximate flow through the cell by either a one-dimensional, plug-flow model (such as used in Ref. Freeman et al. (2014)) or two-dimensional, laminar-flow model(such as Ref. Ruset (2005)). The first computational model of spin-exchange optical pumping to be extensively described in the literature was done so by Ref. Fink et al. (2005). It originally described a model approximating an optical pumping cell geometry with only a half-cylinder. The model included the effects of laser heating, fluid dynamics, and heat transfer. The original model only investigated a static optical pumping cell in which the gas was not flowing. This was later extended in Ref. Fink and Brunner (2007) to analyze two geometries with gas flow: a half cylinder and a simplified geometry of the optical pumping cell used by Ref. Ruset et al. (2006). Recently, Ref. Burant (2018) has described an FEM fluid model of an optical pumping cell. However, it does not appear the model included SEOP, and it only included fluid flow, heat transfer, and diffusion of Rb metal vapor. Here, we present an open-source FEM code that attempts to model the dynamics of SEOP using the full three-dimensional geometry of a common optical-pumping-cell design. The model incorporates fluid flow, Rb vapor diffusion, thermal transfer, laser absorption, and ${}^{129}$Xe polarization. ## II Model Description The FEM model presented here utilizes the finite-element method to numerically solve five differential equations. The author used an open-source code, ElmerFEM-CSC, to implement the model. The model simulates (1) fluid flow through the cell, (2) diffusion and transport of the Rb vapor, (3) heat transfer through the gas stream, (4) absorption of the laser by the Rb and subsequent polarization, and (5) spin-transfer between the Rb vapor and the ${}^{129}$Xe. Of the five modeled quantities, ElmerFEM-CSC contained modules for solving the first three. Descriptions of these three modules can be found in Ref. Råback et al. (2015). The last two equations, for laser absorption and ${}^{129}$Xe polarization, were obtained by modifying existing ElmerFEM-CSC modules to accommodate the expressions that describe the dynamics of those processes. That this is an open-source model is a notable difference from the other models described in section I. The other FEM models were produced using commercial codes that are not easily accessible to many researchers. Other finite-difference codes have not been formally published in an open-access environment. The specifics of the expressions used to describe laser absorption and ${}^{129}$Xe polarization very closely followed Ref. Fink et al. (2005). Major differences between the current model and other models will be highlighted in the following sections. ### Laser Absorption Model Laser absorption is one of the key features of an SEOP model. The expression for modeling laser absorption by the Rb-metal vapor is Wagshul and Chupp (1989): \[\frac{\partial\psi(\nu,z)}{\partial z}=-n_{Rb}(z)\sigma_{s}(\nu)\frac{\Gamma_{ SD}(z)}{\gamma_{opt}(z)+\Gamma_{SD}(z)}\psi(\nu,z)\] (1) where $\psi$ is the photon flux density, $z$ is the azimuthal spacial coordinate, $\nu$ is the frequency of the light, $n_{Rb}$ is the number-density of the Rb, $\sigma_{s}$ is the cross-section for absorption by unpolarized Rb, $\Gamma_{SD}$ is the spin-destruction rate of the Rb, and $\gamma_{opt}=\int_{0}^{\infty}\psi\sigma_{s}\partial\nu$ is the optical pumping rate. The expression is easily solved using finite-difference methods, and it was used in many previous models. However, the presence of the integral expression is challenging for standard finite-element methods. Instead, the FEM model presented here uses the method described by Ref. Fink et al. (2005): \[\frac{\partial\gamma_{opt}}{\partial z}=-\beta\gamma_{opt}n_{Rb}\left(1-\frac{ \gamma_{opt}}{\gamma_{opt}+\Gamma_{SD}}\right)\] (2) where \[\beta=2\sqrt{\pi\textrm{ln}(2)}\frac{r_{e}f\lambda_{l}^{2}w^{\prime}\left(i \sqrt{ln(2)}(r+is)\right)}{\delta\lambda}.\] (3) Here, $r_{e}$ is the classical radius of the electron, $f$ is the oscillator strength of the Rb D-1 line, $\lambda_{l}$ is the laser wavelength, and $\delta\lambda_{l}$ is the laser-line width. $w^{\prime}(Z)$ is the real part of the Faddeeva function; with $s$ denoting the ratio of the laser-to-Rb-line-frequency difference and the laser line width, and $r$ denoting the ratio of the Rb-absorption-line width and the laser-line width. Equation (2) solves explicitly for $\gamma_{opt}$ with the assumption that the beam’s spectral profile is Gaussian throughout the optical pumping cell. This is notably different from finite-difference models, in which the shape of the spectral profile changes during passage through the optical pumping cell. The laser absorption solver used the ElmerFEM-CSC Advection-Reaction module as its template. The equation solved by the Advection-Reaction Module is Råback et al. (2015): \[\frac{\partial c}{\partial t}+\vec{v}\cdot\vec{\nabla}c+\Gamma c=S\] (4) The following modifications were made to eq. (4). First, $\vec{v}$ can be constrained to be a unit vector, $\hat{n}$, pointing in the direction of the laser beam propagation. The spatial derivative of eq. (2) can be rewritten as $\hat{n}\cdot\vec{\nabla}\gamma_{opt}=\frac{\partial\gamma_{opt}}{\partial z}$. Second, $\Gamma$ from eq. (4) can be set equal to: \[\Gamma=-\beta n_{Rb}\left(1-\frac{\gamma_{opt}}{\gamma_{opt}+\Gamma_{SD}} \right).\] (5) The non-linear portion of the equation is solved iteratively by the Picard method Råback et al. (2015). The source term, $S$, is set to 0. ### Diffusion of the Alkali Metal In many previous models, the Rb-metal vapor distribution was assumed to be uniform. In this FEM model, a diffusion model of the Rb-metal vapor is implemented using the diffusion module supplied with ElmerFEM-CSC. Simulated geometries include temperature-dependent sources and sinks, which use the Hertz-Knudsen equation for the boundary condition of the source/sink Fink and Brunner (2007): \[j_{Rb}=\alpha_{Rb}\frac{p_{sat}(T)-p}{\sqrt{2\pi M_{Rb}k_{B}T}}\] (6) where $p_{sat}(T)$ is the saturation partial pressure for a given temperature, $p$ is the instantaneous partial pressure, $M_{Rb}$ is the molecular mass of the Rb, $k_{B}$ is Boltzmann’s constant, $T$ is the absolute temperature, and $\alpha_{Rb}$ is the evaporation coefficient of Rb. The saturation partial pressure $p_{sat}(T)$ for Rb is calculated by the Killian equation Killian (1926). The evaporation rate $j_{Rb}$ can be positive (sources) or negative (sinks) depending on the local values of $p_{sat}(T)$ and $p$. Although $\alpha_{Rb}$ has not been measured for Rb, Ref. Fink and Brunner (2007) notes that the ideal value of $\alpha_{Rb}=1$ is expected. For all the simulations presented here, this value of $\alpha_{Rb}$ was used. For the current model, the saturation and instantaneous partial pressures were converted to the absolute mass concentration used as default in ElmerFEM-CSC diffusion module Råback et al. (2015). ### Wall-Relaxation of HP ${}^{129}$Xe In most previous simulations, wall-relaxation is modeled as a constant term in the HP ${}^{129}$Xe spin-relaxation term. In this simulation, the diffusion-based model of HP ${}^{129}$Xe wall-relaxation is a refinement of the expression presented in Ref. Fink et al. (2005). In that model, the wall boundary-conditions were modeled as completely depolarizing HP ${}^{129}$Xe spin-polarization at the walls. The authors offered an alternative model with the depolarization set to 1% rather than 100%. However, they noted that the lack of experimental data hindered more precise estimates. The present model attempts to refine this approximation and connect the boundary condition at the walls to the wall-relaxation time. The wall-relaxation time (at room temperature) can easily be measured for a given cell by filling the cell with HP ${}^{129}$Xe and monitoring the amplitude of the HP ${}^{129}$Xe NMR as a function of time. It is known that this relaxation time is typically 10s of minutes Freeman et al. (2014). <figure><img src="content_image/1911.01574/diffest-gray.jpg"><figcaption>Figure 1: A plot of the decay curve (eq. (9)). The modeled sphere had a radiusof R=0.03 m and a diffusion coefficient of DXe=1∗10−5m2s. The mass transfercoefficient of αRW=3.5∗10−5ms was calculated using eq. (11) for a wall-relaxation time of τwall=300 sec. The overall decay curve (solid blue)overlaps with the first-order term (dashed red). The second-order term doesnot contribute more than 1 part per thousand to the overall decay curve, andit decays to less than 10−8 after less than 100 seconds. The higher-orderterms contribute less and decay even more quickly. This wall-relaxation timeis even shorter than discussed in the text, and illustrates the robustness ofthe assumption at reasonable wall-relaxation times. Compare the overall decaycurve with the decay curve from Figure 2</figcaption></figure> <figure><img src="content_image/1911.01574/wallrelaxspheresim-gray.png"><figcaption>Figure 2: The visualization of an FEM diffusion model using the sameparameters as those in used in Figure 1. The left graphic depicts theconcentration of the species (normalized to one) at τwall=300 sec. The graphon the right is the averaged concentration as a function of time. Note thatthe curve is qualitatively identical to the theoretical curve in Figure 1.</figcaption></figure> The present model uses the solution to the diffusion equation for surface-evaporation in a sphere. Like with the evaporation of the Rb, we wish to have an expression that relates the flux of polarization to the surface: \[j=-D_{Xe}\frac{\partial P}{\partial r}=\alpha_{RW}P,\] (7) where $\alpha_{RW}$ is the so-called mass transfer coefficient. The solution to the diffusion equation for a spherical geometry with this boundary condition is Crank (1975): \[\frac{P(r)}{P_{i}}=\frac{2LR}{r}\sum_{n=1}^{\infty}\frac{\textrm{exp}(-D_{Xe} \beta^{2}_{n}t/R^{2})}{\beta^{2}_{n}+L(L-1)}\frac{\textrm{sin}(\beta_{n}r/R)}{ \textrm{sin}\beta_{n}}\] (8) where the $\beta_{n}$’s are the roots of $\beta_{n}\textrm{cot}\left(\beta_{n}\right)+L-1=0$, $L=\frac{R\alpha_{RW}}{D_{Xe}}$, $P$ is the polarization after time $t$, $P_{i}$ is the initial polarization, and $R$ is the radius of the sphere. The total polarization, $P_{tot}$, decay curve is given by Crank (1975): \[\frac{P_{tot}}{P_{i}}=\sum_{n=1}^{\infty}\frac{6L^{2}\textrm{exp}(-D_{Xe}\beta ^{2}_{n}t/R^{2})}{\beta^{2}_{n}\left[\beta^{2}_{n}+L\left(L-1\right)\right]}.\] (9) The wall-relaxation time, $\tau_{wall}$, is usually found by fitting the decaying HP${}^{129}$Xe NMR amplitude to an exponential function: \[\frac{P_{tot}}{P_{i}}=\textrm{exp}\left(\frac{-t}{\tau_{wall}}\right).\] (10) In the limit of $\beta_{n}>>\beta_{1}\textrm{;}\forall n\neq 1$, the larger $\beta_{n}$s can be ignored, and only the $\beta_{1}$ term will significantly contribute at long timescales. In this case, we can compare the time-dependent portion of just the first term of eq. (9) to eq. (10), and we find that $\tau_{wall}=\frac{R^{2}}{D_{Xe}\beta_{1}^{2}}$. It turns out that for typical values found in SEOP systems (i.e. $R>5$ cm, $D_{Xe}\approx 0.1\frac{\textrm{cm}^{2}}{\textrm{s}}$ and, $\tau_{wall}>10$ min.), $\beta_{1}\approx 1$, in which case $L<0.5$, the ratio $\frac{\beta_{2}}{\beta_{1}}\gtrapprox 4$, and all other $\beta_{n}$s are much larger. For wall-relaxation times longer than 10 min., the ratio between the $\beta_{n}$s and $\beta_{1}$ is even greater. Using this approximation, we can solve for $\alpha_{RW}$ and find that: \[\alpha_{RW}=\frac{D_{Xe}}{R}\left(1-\sqrt{\frac{R^{2}}{D_{Xe}\tau_{wall}}} \textrm{cot}\left[\sqrt{\frac{R^{2}}{D_{Xe}\tau_{wall}}}\right]\right).\] (11) Equation (11) is a poorly behaved equation and needs to be applied with care. In particular, for a given $\tau_{wall}$, the expression has asymptotes at $R=\pi n\sqrt{D_{Xe}\tau_{wall}}$; $n=1,2,3,...$. The spacing of the asymptotes is proportional to multiples of the diffusion length. Physically, this indicates that the HP${}^{129}$Xe cannot undergo wall relaxation faster than it takes for a polarized ${}^{129}$Xe atom at the center of the sphere to travel to the wall. Therefore, eq. (11) is only valid for $\tau_{wall}>\frac{R^{2}}{\pi^{2}D_{Xe}}$. As an example of the validity of the assumption that the higher-order terms in eq. (8) and (9) do not significantly contribute, Figure 1 shows the theoretical contributions to the decrease in concentration of the polarized species as a function of time for a sphere. The first-order term almost precisely overlays the overall decay. The higher-order terms quickly decay and do not significantly contribute to the decay curve. The expression from eq. (11) was used to model the wall-relaxation of HP${}^{129}$Xe that diffused to the wall. An FEM model using this expression was checked computationally in a simple spherical model to give the correct transient relaxation time and decay curve (see Figure 2). ### Thermal Transfer through Cell Walls The present model attempts to capture the nuances of thermal transfer through the optical pumping cell walls. The model assumes the optical pumping cell is in a forced-air oven where the temperature of the air is held constant. Heat transfer in the walls of the cell is solved by imposing the boundary condition: \[-k\frac{\partial T}{\partial n}=\alpha_{T}\left(T-T_{ext}\right)\] (12) where $k$ is the heat conductivity of the gas mixture in the cell, $T$ is the temperature at the boundary, $T_{ext}$ is the temperature at which the external flowing air is held, and $\alpha_{T}$ is the heat transfer coefficient. The heat transfer coefficient is calculated by combining the affects of the optical pumping wall (usually glass; thermal conductivity $k_{wall}=1.005\frac{\textrm{W}}{\textrm{m}\cdot\textbf{K}}$) and that of forced-air convection (heat transfer coefficient $\alpha_{air}\approx 35\frac{\textrm{W}}{\textrm{m}^{2}\cdot\textbf{K}}$). These two terms can be combined into a single heat transfer coefficient boundary condition by using the expression Bird et al. (2007): \[\frac{1}{\alpha_{T}}=\frac{t_{wall}}{k_{wall}}+\frac{1}{\alpha_{air}}\] (13) where $t_{wall}$ is the thickness of the wall. The expression assumes that the contact area of the different boundaries are approximately equal, which is true when $t_{wall}$ is small compared with the other linear dimensions of the optical-pumping cell. ### Other Considerations Other quantities, such as viscosity and heat capacity, were calculated using standard expressions (see Table 1). These quantities were used as inputs for the standard modules in ElmerFEM-CSC. For more information, see appendix A. ## III Verification Although the model contains the relevant SEOP physics, it is necessary to verify that the model is providing an adequate computational solution to those expressions. In order to confirm that the FEM model conformed to accepted results, a comparison study was run with the model described in Ref. Freeman et al. (2014), the Freeman model. The Freeman model differs from the model described here in several important ways. The Freeman model uses a finite-difference method rather than a finite-element method. The Freeman model is a one-dimensional simulation. The fluid velocity, temperature, and the Rb-density distribution are approximated as uniform in the Freeman model. The laser spectral profile used in the Freeman model is not constrained to be Gaussian; rather the spectral profile is discretized and calculated at each spacial node in the model. Finally, in the Freeman model, the ${}^{129}$Xe polarization is calculated using the average Rb polarization rather than Rb polarization at each spacial node. It was possible to reproduce aspects of this model in the infrastructure of the FEM model described here. The geometry of the FEM model was drawn as a cylinder. The Navier-Stokes fluid model, heat transfer model, and Rb-diffusion model were disabled. They were replaced with uniform axial flow, a uniform temperature, and a homogeneous Rb number-density. In addition, for this comparison, the wall-relaxation described in section II.3 was replaced by an additional term in the HP${}^{129}$Xe relaxation expression. The outlet ${}^{129}$Xe polarization of this FEM model was compared with the predicted ${}^{129}$Xe polarization from the Freeman model. The results of this comparison are shown in Figure 3. The two models predicted very close to the same polarization at low temperature for all flow velocities. The discrepancy between the models increases with both the temperature and the flow velocity. The Freeman model nearly always predicts a lower polarization. The largest absolute and relative discrepancies are 9 percentage-points (16%) and 19% (8 percentage-points), respectively. <figure><img src="content_image/1911.01574/FEM_vs_Freeman.png"><figcaption>Figure 3: The predicted 129Xe polarization from the Freeman model and the FEMmodel as a function of temperature and flow velocity using substantiallysimilar assumptions regarding flow, temperature distribution, and Rb number-density distribution. Each curve represents a different flow velocity (0.2,0.5, 1.0, and 1.5 SLM) from a different model. The dotted lines are thepredictions of the Freeman model, and the solid lines are the predictions ofthe FEM model. The Freeman model almost always predicts a lower polarizationthan the FEM model. This is thought to be because the FEM model includes acombination of a diffusion model for the 129Xe polarization solver and aheterogeneous Rb polarization spatial-distribution.</figcaption></figure> A couple of explanations for this discrepancy were considered and rejected. One explanation for this discrepancy is that the calculated laser absorption differs in the two models. The Freeman model does not enforce the assumption of a Gaussian spectral profile of the laser, while the FEM model described here does. The assumption of a Gaussian spectral profile might influence the calculated Rb polarization, and thus the final ${}^{129}$Xe polarization. However, when a comparison of the 0.2 SLM flow-rate series was conducted, it was both found that the average calculated Rb polarization differed by no more than 0.2 percentage-points (0.2%) and that the FEM model usually predicted a _lower_ Rb polarization. Thus, it seems unlikely that the Gaussian spectral profile, which is enforced by the FEM model, would result in a higher prediction for the ${}^{129}$Xe polarization. Another possible cause for the discrepancy between the two models was the Rb polarization spacial-distribution that existed in FEM model. Although the Freeman model does calculate a spacial-distribution for the Rb polarization, the final ${}^{129}$Xe polarization calculation uses a spatially-averaged value. In order to test if this was the cause of the discrepancy, the flow direction in the FEM model was reversed so that it was directed parallel to the direction of laser propagation instead of anti-parallel. The Rb polarization in the FEM model is highest where the laser first enters the cell geometry. Therefore, reversing the direction of the flow in the model changes the spatial-distribution of the Rb polarization for the simulated ${}^{129}$Xe. In the anti-parallel flow configuration, the gas will first encounter Rb with a lower-than-average Rb polarization and progress to a region of higher-than-average Rb polarization. In the parallel flow configuration, this order is reversed. Although the reversed-flow FEM model did result in a lower prediction of the ${}^{129}$Xe polarization, this affect alone was not sufficient to account for the discrepancy between the Freeman model and the FEM model. The predicted ${}^{129}$Xe polarization at 150 ${}^{\circ}$C from the FEM model decreased by only $\sim$1 percentage point when the flow was reversed. A final possible cause, which was considered, was that the FEM model incorporates diffusion of the ${}^{129}$Xe polarization while the Freeman model does not. This difference, coupled with a Rb polarization spatial-distribution discussed above, will cause the FEM model to predict higher ${}^{129}$Xe polarizations than the Freeman model by significant amounts. The solution to the one-dimensional _diffusion-reaction equation_, assuming a linear Rb polarization as a function of axial position, causes the final predicted ${}^{129}$Xe polarization to fall off linearly as a function of gas velocity. Solving the _advection-reaction equation_, which is used in the Freeman model, with a Rb polarization independent of axial position causes the final predicted ${}^{129}$Xe polarization to fall off exponentially with gas velocity. The details of this derivation are provided in appendix B. This final combined effect from both the diffusion-reaction equation and the Rb-spatial distribution is likely the cause of the discrepancy between the predictions of the Freeman model and the FEM model. <figure><img src="content_image/1911.01574/RbGeometry.png"><figcaption>Figure 4: A visualization of the geometry chosen on which to test the FEMcode. The geometry is a 100-cc standard optical pumping cell. The entiregeometry (optical pumping region, inlet, and outlet) was drawn using OnshapeCAD software, and a mesh was created from this geometry using Salome 9.2.1.The Rb source was modelled as a thin, cylindrical film that encircled themiddle of the optical pumping region of the cell (black section in figure). Anadditional Rb sink was prescribed in the entire outlet tube for the geometry(not pictured).</figcaption></figure> ## IV Test on a Standard SEOP Geometry <figure><img src="content_image/1911.01574/110ccfulltrans.png"><figcaption>Figure 5: A visualization of the 100-cc cell simulation. The image on the leftshows the predicted flow-lines for the gas as it moves through the cell. Theflow lines are color-coded by the predicted temperature of the gas at eachpoint. The slice of the cell is color-coded by the predicted Rb number-densityin the cell. The graph on the right shows three of the simulated quantitiesaveraged over the optical-pumping region of the cell: (1) Rb polarization, (2)absorbed laser power, and (3) temperature. The Rb polarization and absorbedlaser power are both normalized to one. The temperature is normalized to thefraction above the set-point temperature, in this case, 110 ∘C.</figcaption></figure> After verification, limited tests were conducted on a standard, SEOP-cell geometry to assure that all modules would work in concert. A 100-cc-cell geometry with Rb in the optical-pumping body was chosen as the first geometry to test (see Figure 4). The test was conducted with the following boundary conditions: * Laser Power: 75 W * Laser Spectral Width: 0.3 nm * Laser Beam Radius: 1 mm smaller than optical pumping region radius * Wall Temperature: 110 ${}^{\circ}$C * Flow Rate: 1.5 SLM * Wall-Relaxation Time: 56 min. * Fraction ${}^{129}$Xe: 1% * Fraction N${}_{2}$: 10% * Fraction He: 89% * Cell Pressure: 73 psig (assumed to be at sea level) Initially, a steady-state solution for the model was attempted. However, when this failed to converge, the model was solved as a transient problem with 0.1 sec. per time-step. A total of 611 time-steps were simulated. However, due to a file-corruption error, the last 6 time-steps were discarded. Due to possible instability with the module, the ${}^{129}$Xe polarization module was not used during the transient simulation. Instead, after the transient simulation reached steady-state in (1) temperature, (2) Rb polarization, and (3) laser power absorbed, the last transient solution was used as the initial conditions for a steady-state calculation with the HP${}^{129}$Xe polarization module enabled. Steady-state was defined as less than a 0.05% change in all of the three other metrics. The transient simulation shows convergence to a steady-state solution after the 611 time-steps (Figure 5). The predicted ${}^{129}$Xe polarization is comparable to the observed ${}^{129}$Xe polarization, made in Ref. Freeman et al. (2014), from a 100-cc cell similar to the geometry simulated in the FEM model. In the Ref. Freeman et al. (2014) study, a 100-cc cell was observed to absorb 30% of the incident light and produce HP${}^{129}$Xe polarized to $\sim$15% at 90 ${}^{\circ}$C. The FEM model predicted that the simulated 100-cc cell would absorb 7.9% of the incident light and produce HP${}^{129}$Xe polarized to $\sim$4.7% at 110 ${}^{\circ}$C. The HP${}^{129}$Xe polarization for the FEM simulation was taken as the average polarization across a slice 5 cm from the edge of the outlet of the model’s geometry. The discrepancy between the observed light absorption of the 100-cc optical pumping cell and the simulated cell may be from a couple sources. First, as previously mentioned, the FEM model enforces a Gaussian spectral profile throughout the optical pumping cell. This assumption may affect the calculated total absorption of the light. The affects of the imposed Gaussian distribution are not entirely understood, however, as stated in section III, the Gaussian spectral distribution was found to not significantly alter the average Rb polarization in the FEM model when compared with a model that does not enforce a Gaussian spectral distribution. Second, the Rb-source distribution in the model may not accurately describe the distribution in the optical pumping cell tested in Ref. Freeman et al. (2014). The authors do not describe the Rb-metal distribution in their 100-cc cell (e.g. a Rb droplet, a thin layer of Rb on the side of the cell, etc.). The distribution and location of the Rb-metal in the cell may affect the details of the dynamics in the optical-pumping region. The discrepancy between the observed HP${}^{129}$Xe polarization may be similarly explained by the Rb-source distribution in the model. Because the Rb-source distribution may not reflect the conditions in the actual cell that was tested, the Rb number-density calculated in the model may be lower than the number-density that was realized in the actual cell. This would result in a lower spin-exchange rate than what was realized in the actual cell, and thus, a lower predicted HP${}^{129}$Xe polarization. <figure><img src="content_image/1911.01574/110cellsteadystatexepol.png"><figcaption>Figure 6: A visualization of the steady state solution to the HP129Xepolarization. The visualization on the left shows the polarization in theoptical pumping section of the cell. The maximum polarization is ∼6%. Thevisualization on the right is the polarization 5 cm before the edge of theoutlet of the cell. The total polarization averaged over this section is 4.7%.</figcaption></figure> ## V Summary and Future Work A new open-source, FEM-based SEOP model has been coded, tested against an existing model, and tested with a three-dimensional geometry. The model provides predictions of HP${}^{129}$Xe polarizations that are comparable to existing, accepted models when the conditions are limited to the scope of those existing models. The three-dimensional model predicts HP${}^{129}$Xe polarizations that are comparable to existing observations. The new FEM model provides the ability to visualize important SEOP phenomena such as laser heating, Rb-vapor distribution, and gas flow. The model can compute solutions for complicated geometries including current designs for optical pumping cells. The most immediate improvement that can be made is to add the ability to multi-thread computations. The lack of multi-threading limits the speed at which solutions can be computed, which in turn, limits transient studies to simulate only the first several hundred seconds after initialization. The three-dimensional simulation described in section IV required $\sim$22,000 cpu-min., and it only simulated the first $\sim$60 sec. of the cell after initialization. Important, long-term behavior cannot be thoroughly investigated. Multi-threading capabilities with appropriate increases in other computational resources could potentially decrease the amount of time required to calculate a single time-step and open up the possibility of exploring the long-term behavior of these systems. Better approximations to spin-exchange parameters and other model parameters may increase the fidelity of the model’s predictions. However, it is suspected that a major source of error in the model is likely due to the choice of Rb source distribution. Because all the parameters are correct to an order of magnitude, it is unlikely substantial gains will be realized by pursuing more accurate approximations to these parameters unless it can be shown that the current estimates of the parameters are incorrect by large margins. Further simulations on various optical pumping cell geometries will be reported in a follow-up report. ###### Acknowledgements. The author gratefully acknowledges J. Cook for useful conversations regarding the coding of the simulations. Also, the author is grateful for insightful discussions with and feedback on initial drafts from B.J. Anderson, T. Barthel, B. Driehuys, and B. Saam. Finally, much of the computation time on Amazon’s AWS Cloud Computing Service was supported by backers of the project from Experiments.com. The author gratefully acknowledges the financial support of all the backers of the project. ## Appendix A Model Details Term \| Model \| Expressions \| Reference ---\|---\|---\|--- \| Gas Mixture --- Density \| Navier-Stokes --- Heat \| ρ=PTR --- R=Cpγ−1γ Råback et al. (2015) \| Gas Mixture --- Viscosity Navier-Stokes \| \| μ=μ′(TT′)32T′+ST+S --- μtot=∑ixi(μi)13 \| Heat Capacity --- at Constant Pressure \| Navier-Stokes --- Heat Rb Diffusion Cp,mix=∑ixiCp,i \| Linstrom and Mallard (2018) \| Specific Heat --- Ratio \| Navier-Stoke --- Heat Rb Diffusion γ=∑ixiγi \| \| Alkali Evaporation --- Rate Rb Diffusion \| jRb=αpsat−p√2πMRbkBT \| Fink and Brunner (2007) Diffusion Constant \| \| Rb Diffusion --- 129Xe Polarization D1,2=1.8583×10−7T3/2√1M1+1M2pσ21,2Ω \| Bird et al. (2007) \| Optical Pumping --- Rate Laser Absorption \| ∂γp∂z=−βγpnRb(1−γpγp+ΓSD) \| Fink et al. (2005) \| Alkali Spin --- Destruction Rate Laser Absorption \| \| ΓSD=ΓRb+ΓXe+ΓN2+ΓHe+ΓVW --- ΓRb=κRb[Rb] ΓN2=170(1+T−90∘C194.36∘C)[N2] ΓHe=24.6(1+T−90∘C96.4∘C)[He] ΓXe=κXe[Xe] ΓVW=6469fXe+1.1fN2+3.2fHe \| Ruset (2005) --- Nelson (2001) Walter et al. (2002) Ben-Amar Baranga et al. (1998) \| Xenon Spin- --- Exchange Rate 129Xe Polarization \| γSE=κSE[Rb] \| Fink et al. (2005) \| Xenon Spin- --- Relaxation Rate 129Xe Polarization \| \| ΓSR=ΓB+ΓVW --- Γb=κXe[Xe] ΓvdW=ΓXevdW1+r[B][Xe] Chann et al. (2002) Wall-Relaxation \| 129Xe Polarization \| α=DXeR(1−√R2DXeτcot[√R2DXeτ]) \| Crank (1975) Thermal Conductivity \| Heat \| \| k=1.9881×10−4√T/Mσ2Ω --- kmix=∑αxαkα∑βxβϕβ Bird et al. (2007) \| Heat Transfer --- Coefficient Heat \| 1U=1h0+∑nj=1xj−xj−1kj−1,j−1hn \| Bird et al. (2007) Laser Heating \| Heat \| Q=hνlnRbγpΓSDγp+ΓSD \| Fink et al. (2005) Table 1: The table lists all of the expressions that are used to calculate various parameters in the model. The first column lists the term in the model. The second column lists in which model the term is specifically used. The third column lists the expression used. Note that because the model is coded in SI units, the actual implementation of some of the expression may have been multiplied by a constant to make all the units consistent. The final column lists the reference from which each expression is derived. In all cases, the notation used by the reference has been kept. For the meaning of the particular notation in the expressions, the reader should refer to the particular reference. Note: A previous version of this pre-print incorrectly listed the expressions used for specific heat ratio and viscosity. Section II was meant to highlight the important differences between the FEM model and previous models. However, readers that are interested in using the model may be interested in some more details of that model. The most recent version of the model can be found at https://github.com/drschrank/elmerfem and is free for use under the GNU license requirement found in the source code. Although the model can be used in a steady-state formulation of the differential equations, transient simulations are frequently useful because initial simulations on some optical pumping geometries fail to converge when solving the steady-state equations. During testing, the transient time-step for the simulation was 0.1 sec/step. Two different linear solvers with different convergence limits were used for the modules. The generalized minimal residual method (GMRES) and biconjugate gradient stabilized method (BiCGSTAB) were both used for the Navier-Stokes equation. The BiCGSTAB was used exclusively for all the other modules. Non-linear and linear convergence limits were all less than $10^{-4}$ and were usually $10^{-6}$. As described in section II, the model consists of five modules which are calculated in sequence. The first module which is calculated is the solution to the Navier-Stokes equation. It was provided with ElmerFEM-CSC and was used without modification. ElmerFEM-CSC provides two applicable compressiblity models for gas flow: the “Perfect Gas” compressibility model, which solves the full Navier-Stokes equation, and the “Incompressible” compressibility model, which solves the Navier-Stokes equation with the assumption that $\vec{\nabla}\cdot j=0$. The “Incompressible”compressiblity model requires less computational resources, and it is appropriate when the Mach number of the fluid flow is low, which is the case for all reasonable SEOP models. When using the “Incompressible” model, the density can be calculated as a function of temperature at each time-step. The Navier-Stokes calculation is dependent on the solution to the heat equation, which is discussed later. The second module which is calculated is Rb diffusion. This module uses the Advection-Diffusion module. This module was also provided with ElmerFEM-CSC and was used without modification. This module’s solution is only dependent on the resulting gas flow calculated by the Navier-Stokes equation. The third module is the laser absorption module. This module was based on the Advection-Reaction (eq. (4)) module, which was provided by ElmerFEM-CSC. The module was modified to use the Picard method to approximate the non-linear solution by a series of iterative steps (see Ref. Råback et al. (2015) section on linearization of the Navier-Stokes equation for an example). The specific equation solved using this method is described in section II.1. The solution to this module is only dependent on the solution to the previous module, the Rb diffusion module. The time-dependent portion of eq. (4) is ignored by ElmerFEM-CSC solvers when the model is used to solve steady-state problems. However, for the transient simulations, the time-dependent term had to be included. To handle this, the time-dependent term was multiplied by a coefficient that is much smaller than the characteristic time-steps used in the model. This modification effectively causes each time-step to be a steady-state solution of eq. (2). The fourth module is the ${}^{129}$Xe polarization module. This module is based on the Advection-Diffusion module which was provided by ElmerFEM-CSC. The method by which the equation was solved was not changed. The only changes made were the assignments to the various constant parameters of the Advection-Diffusion equation, and the solver was forced to always use the “absolute mass” setting because the equation in this form is easily adapted to eq. (14). The solution to this module is only dependent on the laser absorption module, and the solution does not effect any of the other modules. This allows for the possibility of running simulations with only the other four modules active and then using a final, steady-state solution of the other four modules to calculate the solution to the ${}^{129}$Xe polarization. The final module is the heat equation module. This module was provided by ElmerFEM-CSC without any modification. The solution is dependent on both the laser absorption solution and the Navier-Stokes solution. Parameters for the various modules are listed in table 1. The values of particular constants used in the equations in the table can be found in the references listed in the table. ## Appendix B Details of the Derivation of the Difference Between the Freeman Model and the FEM Model In section III, it is stated without proof that the assumptions of the Freeman model give rise to an exponential dependence on flow rate when a uniform Rb polarization is assumed, while the assumptions of the FEM model give rise to a linear dependence on flow rate when a Rb polarization gradient is assumed. In this appendix, the derivation of that result will be given. The FEM model uses the advection-diffusion equation to model ${}^{129}$Xe polarization. The form of that equation in one dimension is: \[D_{Xe}\frac{\partial^{2}P_{Xe}(x)}{\partial x^{2}}+v\frac{ \partial P_{Xe}(x)}{\partial x}+\left(\sigma+\Gamma\right)P_{Xe}(x)=\sigma P_{ Rb}(x).\] (14) The Rb polarization is assumed to have a linear gradient given by: \[P_{Rb}(x)=\frac{P_{L}-P_{0}}{L}x+P_{0}\] (15) where $P_{L}$ is the highest polarization of the Rb at the point $x=L$, and $P_{0}$ is the lowest Rb polarization at the point $x=0$. From Ref. Walker (2011), it is clear in the limit as $L\to\infty$ that $P_{Xe}\to\frac{\sigma}{\sigma+\Gamma}P_{L}$. The solution to eq. (14) is given by: \[P_{Xe}(x)=K_{1}e^{x\frac{v+\sqrt{v^{2}-4D_{Xe}(\sigma+\Gamma)}}{ 2D_{Xe}}}+K_{2}e^{x\frac{v-\sqrt{v^{2}-4D_{Xe}(\sigma+\Gamma)}}{2D_{Xe}}}-\\ \frac{v\sigma(P_{L}-P_{0})}{L(\sigma+\Gamma)^{2}}+\frac{\sigma(P_ {L}-P_{0})x}{L(\sigma+\Gamma)}+\frac{\sigma P_{0}}{(\sigma+\Gamma)}.\] (16) For $v^{2}>>4D_{Xe}(\sigma+\Gamma)$, this can be simplified to: \[P_{Xe}(x)=K_{1}e^{-\frac{vx}{D_{Xe}}}+K_{2}+\\ \frac{\sigma}{\sigma+\Gamma}\left(P_{Rb}(x)-\frac{v(P_{L}-P_{0}}{ L(\sigma+\Gamma)}\right).\] (17) If $\frac{vL}{D_{Xe}}>>1$, then \[P_{Xe}(L)=K_{2}+\frac{\sigma}{\sigma+\Gamma}\left(P_{L}-\frac{v( P_{L}-P_{0}}{L(\sigma+\Gamma)}\right).\] (18) For $L\to\infty$, we get: \[P_{Xe}(L\to\infty)\to K_{2}+\frac{\sigma}{\sigma+\Gamma}P_{L}=\frac{\sigma}{ \sigma+\Gamma}P_{L},\] (19) which implies: \[K_{2}=0.\] (20) Therefore, solving the one-dimensional advection-diffusion equation for the polarization of ${}^{129}$Xe at $x=L$ assuming a linear Rb polarization gradient gives: \[P_{Xe}(L)=\frac{\sigma}{\sigma+\Gamma}\left(P_{L}-\frac{v(P_{L}-P_{0}}{L( \sigma+\Gamma)}\right),\] (21) which linearly decreases as the flow rate, $v$, increases. The Freeman model uses eq. (14) with the diffusion term absent and an average Rb polarization, $\bar{P}_{ave}=\frac{P_{L}+P_{0}}{2}$. The solution to this equation can be trivially shown to be: \[P_{Xe}(L)=\frac{\bar{P}_{ave}\sigma}{\sigma+\Gamma}\left(1-e^{-\frac{(\sigma+ \Gamma)L}{v}}\right),\] (22) which exponentially decreases as a function of $v$. ## References S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer (1998)Theory of spin-exchange optical pumping of ${}^{3}\mathrm{He}$ and ${}^{129}\mathrm{Xe}$. Phys. Rev. A58, pp. 1412–1439. External Links: Document, Link Cited by: §I. * A. Ben-Amar Baranga, S. Appelt, M. V. Romalis, C. J. Erickson, A. R. Young, G. D. Cates, and W. Happer (1998)Polarization of ${}^{3}\mathrm{He}$ by spin exchange with optically pumped rb and k vapors. Phys. Rev. Lett.80, pp. 2801–2804. External Links: Document, Link Cited by: Table 1. * R. B. (. B. Bird, W. E. Stewart, and E. N. Lightfoot (2007)Transport phenomena. J. Wiley. External Links: Link, ISBN 9780470115398 Cited by: Table 1, §II.4. * A. Burant (2018)Characterizing Hyperpolarized ${}^{129}$Xe Depolarization Mechanisms During Continuous-Flow Spin Exchange Optical Pumping and as a Source of Image Contrast. Ph.D. Thesis, University of North Carolina, Chapel Hill. Cited by: §I. * B. Chann, I. A. Nelson, L. W. Anderson, B. Driehuys, and T. G. Walker (2002)${}^{129}Xe-\mathrm{Xe}$ Molecular spin relaxation. Phys. Rev. Lett.88, pp. 113201. External Links: Document, Link Cited by: Table 1. * J. Crank (1975)The Mathematics of Diffusion. 2 edition, Oxford University Press, London. Cited by: Table 1, §II.3. * Driehuys,B., Cates,G. D., Miron,E., Sauer,K., Walter,D. K., and Happer,W. (1996)High‐volume production of laser‐polarized 129xe. Applied Physics Letters69 (12), pp. 1668–1670. External Links: Document, Link Cited by: §I. * A. Fink, D. Baumer, and E. Brunner (2005)Production of hyperpolarized xenon in a static pump cell: numerical simulations and experiments. Phys. Rev. A72, pp. 053411. External Links: Document, Link Cited by: Table 1, §I, §II.1, §II.3, §II. * A. Fink and E. Brunner (2007)Optimization of continuous flow pump cells used for the production of hyperpolarized 129Xe: A theoretical study. Appl. Phys. B89 (1), pp. 65–71. External Links: Link, ISBN 0946-2171, Document Cited by: Table 1, §I, §II.2, §II.2. * M. S. Freeman, K. Emami, and B. Driehuys (2014)Characterizing and modeling the efficiency limits in large-scale production of hyperpolarized ${}^{129}\mathrm{Xe}$. Phys. Rev. A90, pp. 023406. External Links: Document, Link Cited by: §I, §II.3, §III, §IV, §IV. * T. J. Killian (1926)Thermionic phenomena caused by vapors of rubidium and potassium. Phys. Rev.27, pp. 578–587. External Links: Document, Link Cited by: §II.2. * L.J. Linstrom and W.G. Mallard (Eds.) (2018)NIST Chemistry WebBook NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg. External Links: Link, Document Cited by: Table 1. * I. A. Nelson (2001)Physics of Practical Spin-Exchange Optical Pumping. Ph.D. Thesis, University of Wisconsin-Madison. Cited by: Table 1. * A. Oros and N. J. Shah (2004)Hyperpolarized xenon in NMR and MRI. Physics in Medicine and Biology49 (20), pp. R105–R153. External Links: Document, Link Cited by: §I. * P. Råback, M. Malinen, J. Ruokolainen, A. Pursula, and T. Zwinger (2015)Elmer Models Manual. CSC – IT Center for Science. External Links: Link Cited by: Table 1, Appendix A, §II.1, §II.1, §II.2, §II. * I. C. Ruset, S. Ketel, and F. W. Hersman (2006)Optical pumping system design for large production of hyperpolarized ${}^{129}\mathrm{Xe}$. Phys. Rev. Lett.96, pp. 053002. External Links: Document, Link Cited by: §I, §I. * I. C. Ruset (2005)Hyperpolarized ${}^{129}$Xe Production and Applications. Ph.D. Thesis, University of New Hampshire. Cited by: Table 1, §I. * G. Schrank, Z. Ma, A. Schoeck, and B. Saam (2009)Characterization of a low-pressure high-capacity ${}^{129}\text{X}\text{e}$ flow-through polarizer. Phys. Rev. A80, pp. 063424. External Links: Document, Link Cited by: §I. * L. Schröder (2013)Xenon for nmr biosensing – inert but alert. Physica Medica29 (1), pp. 3 – 16. External Links: ISSN 1120-1797, Document, Link Cited by: §I. * V. V. Terskikh, I. L. Moudrakovski, S. R. Breeze, S. Lang, C. I. Ratcliffe, J. A. Ripmeester, and A. Sayari (2002)A general correlation for the 129xe nmr chemical shift−pore size relationship in porous silica-based materials. Langmuir18 (15), pp. 5653–5656. External Links: Document, Link Cited by: §I. * M. E. Wagshul and T. E. Chupp (1989)Optical pumping of high-density rb with a broadband dye laser and gaalas diode laser arrays: application to ${}^{3}\mathrm{He}$ polarization. Phys. Rev. A40, pp. 4447–4454. External Links: Document, Link Cited by: §II.1. * T. G. Walker (2011)Fundamentals of spin-exchange optical pumping. Journal of Physics: Conference Series294, pp. 012001. External Links: Document, Link Cited by: Appendix B, §I. * D. K. Walter, W. M. Griffith, and W. Happer (2002)Magnetic slowing down of spin relaxation due to binary collisions of alkali-metal atoms with buffer-gas atoms. Phys. Rev. Lett.88, pp. 093004. External Links: Document, Link Cited by: Table 1.
1710.00007	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 96453, "num_imgs": 13, "llama3_tokens_count": 24380 }	[ "content_image/1710.00007/x1.png", "content_image/1710.00007/x2.png", "content_image/1710.00007/x3.png", "content_image/1710.00007/x4.png", "content_image/1710.00007/x5.png", "content_image/1710.00007/x6.png", "content_image/1710.00007/x7.png", "content_image/1710.00007/x8.png", "content_image/1710.00007/x9.png", "content_image/1710.00007/x10.png", "content_image/1710.00007/x11.png", "content_image/1710.00007/x12.png", "content_image/1710.00007/x13.png" ]	# A Bayesian Hierarchical Approach to Galaxy-Galaxy Lensing Alessandro Sonnenfeld,${}^{1}$ Alexie Leauthaud,${}^{1,2}$ ${}^{1}$Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan ${}^{2}$ Department of Astronomy and Astrophysics, UCO/Lick Observatory, University of California, 1156 High Street, Santa Cruz, CA 95064, USA E-mail:alessandro.sonnenfeld@ipmu.jp ###### Abstract We present a Bayesian hierarchical inference formalism to study the relation between the properties of dark matter halos and those of their central galaxies using weak gravitational lensing. Unlike traditional methods, this technique does not resort to stacking the weak lensing signal in bins, and thus allows for a more efficient use of the information content in the data. Our method is particularly useful for constraining scaling relations between two or more galaxy properties and dark matter halo mass, and can also be used to constrain the intrinsic scatter in these scaling relations. We show that, if observational scatter is not properly accounted for, the traditional stacking method can produce biased results when exploring correlations between multiple galaxy properties and halo mass. For example, this bias can affect studies of the joint correlation between galaxy mass, halo mass, and galaxy size, or galaxy colour. In contrast, our method easily and efficiently handles the intrinsic and observational scatter in multiple galaxy properties and halo mass. We test our method on mocks with varying degrees of complexity. We find that we can recover the mean halo mass and concentration, each with a $0.1$ dex accuracy, and the intrinsic scatter in halo mass with a $0.05$ dex accuracy. In its current version, our method will be most useful for studying the weak lensing signal around central galaxies in groups and clusters, as well as massive galaxies samples with $\log{M_{}}>11$, which have low satellite fractions. keywords: gravitational lensing: weak – methods: statistical – cosmology: dark matter ## 1 Introduction The relation between the stellar mass of a galaxy and the mass of its host halo is a fundamental ingredient in our understanding of galaxy formation. At the high mass end of the galaxy population, weak lensing observations have helped constrain this relation up to redshift $z\sim 1$(e.g. Mandelbaum et al., 22; Leauthaud et al., 2012; Coupon et al., 2015; Zu & Mandelbaum, 2015). In addition to the well-known stellar-to-halo mass relation (SHMR), the colours of central galaxies are also found to depend on halo mass at fixed stellar mass (see e.g. Hoekstra et al., 2005; Mandelbaum et al., 22; Tinker et al., 36; Mandelbaum et al., 2016). This dependence is interpreted by Zu & Mandelbaum (2016) as evidence for halo-driven quenching of star formation. Because dark matter halos play an important role in galaxy evolution, we expect additional correlations between the properties of galaxies and those of their host halos. For example, at fixed stellar mass, we might expect correlations between halo mass and galaxy size, age of the stellar population, central black hole mass, to list a few. Weak gravitational lensing is one of the most direct ways of measuring halos masses, but the signal-to-noise ratio for weak lensing-based halo mass measurements around individual galaxies is typically low. As a result, it is in general necessary to statistically combine weak lensing measurements around a large number of lenses in order to obtain precise halo mass measurements for population ensembles. Traditionally, two methods have been used to carry out galaxy-galaxy lensing measurements. The first approach adopts a maximum-likelihood framework and consists of modeling the contribution from the dark matter halo of each individual lens galaxy to the shear signal, assuming simple parametrized halo density profiles. In order to make the problem computationally tractable, scatter-free scaling relations between galaxy and halo properties are typically assumed (e.g. Schneider & Rix, 1997; Hudson et al., 1998; Hoekstra et al., 2004; Limousin et al., 2005; Han et al., 2015). The second, somewhat more popular approach, commonly referred to as “stacking”, consists of the derivation of the average weak lensing signal in radial bins around a set of galaxies that have similar observed properties, such as stellar mass (e.g. Hoekstra et al., 2001; Parker et al., 2005; Mandelbaum et al., 23; Leauthaud et al., 2012; Velander et al., 2014; Brouwer et al., 2016). In stacking analyses, the SHMR is constrained by forward modeling the population of galaxies and halos, and comparing it with the stacked weak lensing signal, assuming a parametrized form for the halo density profile. A significant advantage of stacking over maximum-likelihood methods is the possibility of forward modeling the contribution of satellite galaxies to the weak lensing signal. One common limitation to maximum-likelihood and stacking methods, is the need to impose a strict self-similarity relation between galaxies, at some point in the analysis chain. This in turn results in loss of information, partly limiting the ability of inferring the properties of the SHMR, especially the intrinsic scatter around the mean relation. In maximum-likelihood studies, the scatter between stellar and halo mass is usually set to zero, and is not inferred directly. In stacking, when shape measurements around galaxies in a given bin are combined, we are essentially assuming that all objects in the bin are either identical or scaled up versions of one another, depending on the details of the analysis. Although some information on the intrinsic scatter can be obtained by forward modeling a halo population distribution and comparing it with the stacked signal, the process of stacking erases differences between halos in a given bin, making the resulting data less sensitive to variations in the scatter. Another limitation of traditional galaxy galaxy lensing methods is the difficulty in exploring dependences of the halo mass on more than one parameter. For example, let us consider the problem of simultaneously measuring the dependence of halo mass on stellar mass and galaxy size. If we wish to address this problem with a stacking approach, we need to make bins in the 2-dimensional space defined by stellar mass and size. This can increase dramatically the number of bins necessary for the analysis, making it difficult to find a number of galaxies sufficient to measure a weak lensing signal in every bin. Stacking is still effective when the secondary variable can be treated as a binary quantity. This is the case, for example, in the study of the relation between halo mass and colour: galaxies can be divided in two colour bins, red and blue, with a clear physical meaning. However, stacking can be rather cumbersome for studying the dependence of halo mass on variables with a continuous distribution (although see Zu & Mandelbaum, 2017, for a successful 2-dimensional stacked weak lensing analysis). In this work, we present a new method for inferring the distribution of halo masses and its dependence on galaxy properties using weak lensing measurements. The method is based on a Bayesian hierarchical inference formalism. It consists in forward modeling the distribution of galaxies in a multi-dimensional space defined by halo mass, stellar mass, and other properties of interest, and fitting it directly to individual shape measurements. Similarly to maximum-likelihood methods, we assume scaling relations between galaxy properties, such as stellar mass, and halo mass. However, a significant difference in our treatment is that we also allow for the presence of intrinsic and observational scatter, which we infer directly from the data. As we will show later, modeling the scatter is crucial for making accurate inferences in multi-dimensional problems. A Bayesian hierarchical method for the inference of the mass-concentration relation of galaxy groups and clusters was recently developed and applied to CFHTLenS data by Lieu et al. (2017). In this work, we describe how we can use a similar approach to determine the SHMR, its scatter, and to measure the correlation between halo mass and galaxy size at fixed stellar mass. The goal of this work is to present a method and demonstrate its effectiveness. We do this with a rather simplistic model for the description of the SHMR, but our approach can easily be generalized to the more sophisticated models commonly found in the literature. This paper is structured as follows. In Section 2 we briefly review the problem of inferring the SHMR, and its generalization to higher dimensions, highlighting the challenges of this task. In Section 3 we describe our Bayesian hierarchical inference formalism. In Section 4 we describe the mock galaxy catalog and mock weak lensing data used to test our method. In Section 5 we apply our inference method to the mock data, test our ability to infer a dependence of halo mass on galaxy size at fixed stellar mass, and perform various tests to asses the robustness of our analysis. We discuss the results and conclude in Section 6. Throughout our analysis we assume a flat cosmology with $\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$ and $h=0.7$. Stellar and halo masses are expressed in Solar units. Halo masses are defined as the mass enclosed within a sphere with average density equal to 200 times the critical density of the Universe (usually referred to as $M_{200c}$). The notation $\log{}$ refers to the base 10 logarithm. ## 2 The SHMR and its generalization In this Section we introduce the problem of inferring the SHMR with weak lensing data. In particular, we will examine how this problem is typically addressed in stacked weak lensing studies, pointing out possible shortcomings with this approach. ### Measuring the SHMR The SHMR can be described in two, complementary, ways: by specifying the distribution of halo masses at fixed stellar mass, ${\rm P}(M_{h}\|M_{})$, or, vice versa, by describing the distribution in $M_{}$ at fixed $M_{h}$, ${\rm P}(M_{}\|M_{h})$. The two descriptions are related as follows: \[{\rm P}(M_{}){\rm P}(M_{h}\|M_{})={\rm P}(M_{h}){\rm P}(M_{}\|M_{h})={\rm P}( M_{},M_{h}),\] (1) where ${\rm P}(M_{})$ is the stellar mass function, ${\rm P}(M_{h})$ is the halo mass function, and ${\rm P}(M_{},M_{h})$ is the joint distribution in the 2-dimensional space defined by stellar and halo mass. Observationally, when working with a sample selected in stellar mass, ${\rm P}(M_{h}\|M_{})$ is easier to obtain. However, in the comparison with theoretical models, a description in terms of ${\rm P}(M_{}\|M_{h})$ is usually preferred, because galaxy properties are thought to depend on their dark matter environments and hence $M_{h}$ is assumed to be the more fundamental property. Furthermore, ${\rm P}(M_{}\|M_{h})$ is more stable with respect to the observational scatter on $M_{}$, compared to ${\rm P}(M_{h}\|M_{})$ (we will discuss this point later in this Section). Stacked weak lensing provides measurements of the radial profile of the excess surface mass density, $\Delta\Sigma$, around galaxies binned in observed stellar mass. This is defined as \[\Delta\Sigma(R)=\bar{\Sigma}(<R)-\Sigma(R),\] (2) where $\Sigma(R)$ is the surface mass density at projected distance $R$ from a given galaxy, and $\bar{\Sigma}(<R)$ is the average surface mass density within the circle of radius $R$. The surface mass density around a galaxy is determined by contributions from its stellar component, its dark matter halo, and from neighboring galaxies and halos. In order to convert stacked measurements into measurements of halo mass, parametrized halo models are usually fit to $\Delta\Sigma$ profiles. The most straightforward way of interpreting a stacked measurement is by fitting it with a model consisting of a single halo. This is an appropriate choice when the sample of galaxies consists of central galaxies, and when the radial range of the measurements does not extend far beyond the virial radius of the main halo (see e.g. Mandelbaum et al., 2016). The inferred halo mass is close to, but not exactly equal to, the mean mass of the halos in the bin (Mandelbaum et al., 2005). This is because the excess surface mass density, which is the observable quantity, does not scale linearly with halo mass, and so the mean of the $\Delta\Sigma$ distribution is not equal to the mean of the $M_{h}$ distribution. However, the mean halo mass of the galaxies in the bin can be recovered by applying corrections calibrated with numerical simulations (Mandelbaum et al., 2016). With this approach, it is then possible to measure the mean halo mass of galaxies in different stellar mass bins. By making further assumptions on the scatter in halo mass around the mean relation, these measurements can be used to constrain ${\rm P}(M_{h}\|M_{})$. However, one caveat is that bins are, by necessity, defined in terms of the _observed_ stellar mass. Due to Eddington bias, the mean true stellar mass of galaxies in a bin is in general lower than the mean of the observed values. The inferred SHMR is then a relation between observed stellar mass and true halo mass. Although this distinction might seem pedantic, it can have important implications, especially when looking for secondary correlations between halo mass and galaxy properties at fixed stellar mass, as we will show later. A more complex approach consists of forward modeling the stacked weak lensing signal created by a population of galaxies and halos with a halo occupation distribution (HOD) model (e.g. Leauthaud et al., 2012; Tinker et al., 37; Zu & Mandelbaum, 2015). Given a theoretically motivated halo mass function and a functional form for ${\rm P}(M_{}\|M_{h})$, one can predict the stellar mass function ${\rm P}(M_{})$ and the stacked weak lensing signal in different bins. Comparing these predictions with the observed stellar mass function and the measured weak lensing signal allows to constrain the form of the SHMR. Through such a procedure, it is possible to account for the effect of the environment and for the presence of satellite galaxies. Moreover, forward modeling constrains the scatter around the SHMR, as changing the scatter modifies the distribution of halo masses within each stellar mass bin. However, this method needs to use information on the stellar mass function as an additional constraint that is independent from the weak lensing. Hence, in order to apply this method, it is necessary to work with complete samples, or at least, samples that are complete within redshift slices (Zu & Mandelbaum, 2015). In all practical applications of HOD forward model methods, observational errors in $M_{}$ measurements are, more or less explicitly, treated as an additional source of scatter. ${\rm P}(M_{}\|M_{h})$ is typically described as a Gaussian in $\log{M_{}}$ with dispersion $\sigma_{\log{M_{}}}$. Both intrinsic scatter and observational scatter contribute to this dispersion. In principle, the two sources of scatter can be deconvolved. In practice, this is usually not done carefully, for simplicity and due to difficulties estimating accurate stellar mass uncertainties (see discussion in subsection 4.2 of Leauthaud et al., 2012). As a result, the inferred SHMR is, also in this case, an observed stellar-to-halo mass relation (OSHMR), ${\rm P}(M_{}^{\mathrm{(obs)}}\|M_{h})$, which is the convolution between the distribution in true stellar mass and observational errors: \[{\rm P}(M_{}^{\mathrm{(obs)}}\|M_{h})\propto\int dM_{}{\rm P}(M_{}^{\mathrm{ (obs)}}\|M_{}){\rm P}(M_{}\|M_{h})\] (3) If the intrinsic scatter is Gaussian in $\log{M_{}}$ and observational errors on $\log{M_{}}$ are drawn from a Gaussian with zero mean and scatter $\sigma_{,err}$ (i.e. ${\rm P}(M_{}^{\mathrm{(obs)}}\|M_{})$ is a Gaussian in $\log{M_{}^{\mathrm{(obs)}}}$ centred on $\log{M_{}}$ and with dispersion $\sigma_{,err}$), then ${\rm P}(M_{}^{\mathrm{(obs)}}\|M_{h})$ is itself a Gaussian with the same mean as ${\rm P}(M_{}\|M_{h})$ and with variance given by \[\sigma_{\log{M_{}}}^{2}=\sigma_{,int}^{2}+\sigma_{,err}^{2},\] (4) where $\sigma_{,int}$ is the intrinsic scatter. This means that the inferred scatter is the quadrature sum of the intrinsic and the observational scatter, and that the inferred mean stellar mass at fixed halo mass is unbiased. Although intrinsic and observational scatter affect the OSHMR in the same way, the physical interpretation of the two sources of scatter is drastically different: the former is set by processes determining the buildup of stellar mass in dark matter halos, the latter is purely artificial and goes to zero in the limit of perfect observations. ### Correlations with two or more variables As we are about to show, the distinction between observed and intrinsic quantities is crucial when measuring correlations between halo mass and two or more galaxy properties simultaneously. Let us consider the problem of determining the dependence of halo mass on galaxy size, at fixed stellar mass. Naively, one could proceed as follows: 1) select a sample of galaxies in a bin of stellar mass, 2) sub-divide this sample according to whether a galaxy lies above or below the mass-size relation, 3) measure the stacked weak lensing signals for the two samples with different sizes, and 4) compare the inferred surface mass density profile. Such an approach produces a biased result. The reason is that, observationally, we have no access to the true stellar mass of a galaxy, but only to a noisy estimate of it. At fixed _observed_ stellar mass, objects with smaller size are more likely to be intrinsically less massive galaxies that have scattered in the bin due to observational uncertainty. These galaxies will on average live in less massive halos. Therefore, such a measurement would appear to show a correlation between size and halo mass even if such a correlation is not present. A toy representation of the problem is illustrated in Figure 1, where we plot the effect of observational scatter in stellar mass on a mock mass-size relation. We make a bin in observed stellar mass, and divide it in two subsamples, based on whether the observed data points lie above or below the mass-size relation. As Figure 1 shows, objects scatter into the bin from higher and lower true stellar masses. Most of the objects entering the bin from the lower mass side have smaller sizes than the average (blue circles), while most of the objects scattered from the higher stellar mass region have larger sizes (red circles). As a result, the average true stellar mass of the larger size subsample is actually larger than that of the smaller size subsample, even though the two subsamples share the same observed stellar mass. <figure><img src="content_image/1710.00007/x1.png"><figcaption>Figure 1: Illustration of the bias introduced when splitting a sample ofgalaxies by size at fixed observed stellar mass. The plot shows stellar massvs. effective radius for a mock sample of galaxies. True values are shown asfilled circles, while observed values, generated by adding a Gaussian scatterin the logarithm of the stellar mass, are marked by empty circles. The twovertical lines mark the bounds of a bin in observed stellar mass. Objectsmarked in blue (red) are galaxies that belong to the bin in observed stellarmass, and that have a smaller (larger) size compared to the average mass-sizerelation for galaxies of the same observed stellar mass. Horizontal linesconnect true and observed stellar masses for the objects in the observedstellar mass bin. Most of the objects scattered into the bin from lower(higher) stellar masses are marked in blue (red), indicating that they havesmaller (higher) sizes compared to the average for their observed stellarmass. To enhance the effect of the bias, we have assumed, when making thismock, that sizes are proportional to the square of the stellar mass, acorrelation four times stronger than observed for massive early-type galaxies.</figcaption></figure> The importance of this effect is higher the stronger the correlation between mass and size, and the larger the observational scatter in stellar mass. For the example in Figure 1, we have artificially steepened the mass-size relation for a better visualization of the bias. Nevertheless, in order to robustly determine a correlation between size (or any other quantity) and halo mass at fixed stellar mass, the underlying correlation between mass and size and the effects of observational uncertainties must be properly modeled and taken into account. This bias does not pose a fundamental problem to stacking: in fact, it can be avoided by forward modeling the effects of observational scatter, making a clear distinction between true and observed quantities. However, this bias is also naturally avoided with the inference method presented in the next Section. ## 3 The model In this Section we describe the statistical inference formalism we use for our analysis (in subsection 3.1), the model used to describe the relation between galaxies and their halos (subsection 3.2) and the model adopted to fit weak lensing data (subsection 3.3). Subsection 3.4 will be dedicated to technical aspects of the computations required to perform the fit. For the sake of clarity, we begin by describing these for a simplified version of the problem, adding more complexity step by step. ### The inference problem Let us consider a sample of galaxies selected in observed stellar mass. We would like to infer the distribution of host halo masses, the intrinsic stellar masses of the sample, and the relation between the two quantities. This is equivalent to determining the distribution of galaxies in the 2-dimensional space defined by true stellar and halo mass. Let us assume that this distribution can be described analytically by a set of parameters $\boldsymbol{\eta}$. We call these the _hyper-parameters_ of the model and refer to the distribution of stellar and halo masses, given by the hyper-parameters, as ${\rm P}(M_{},M_{h}\|\boldsymbol{\eta})$. We wish to infer the posterior probability distribution of the hyper-parameters given weak lensing and stellar mass data $\mathbf{d}$, ${\rm P}(\boldsymbol{\eta}\|\mathbf{d})$. From Bayes theorem, \[{\rm P}(\boldsymbol{\eta}\|\mathbf{d})\propto{\rm P}(\boldsymbol{\eta}){\rm P}( \mathbf{d}\|\boldsymbol{\eta}),\] (5) where ${\rm P}(\boldsymbol{\eta})$ is the prior probability distribution on the hyper-parameters and ${\rm P}(\mathbf{d}\|\boldsymbol{\eta})$ is the likelihood of observing the data given the model. The latter term expands as follows: \[{\rm P}(\mathbf{d}\|\boldsymbol{\eta})=\idotsint dM_{,1}dM_{h,1} \ldots dM_{,N}dM_{h,N}\times\\ {\rm P}(\mathbf{d}\|M_{,1},M_{h,1},\ldots,M_{,N},M_{h,N})\times \\ {\rm P}(M_{,1},M_{h,1},\ldots,M_{,N},M_{h,N}\|\boldsymbol{\eta}).\] (6) In order to calculate the likelihood of observing the data given the hyper-parameters, we need to marginalize over all possible values of the individual stellar and halo masses, resulting in a $2\times N$-dimensional integral, where $N$ is the number of lenses in the sample. The data consists of stellar mass measurements of all lenses and weak lensing shape measurements on background sources. In principle, each background source is lensed by all lenses, therefore the above integral cannot be simplified without further assumptions. In maximum-likelihood weak lensing methods, the problem is simplified by assuming no scatter in the SHMR: the term ${\rm P}(M_{,1},M_{h,1},\ldots,M_{,N},M_{h,N}\|\boldsymbol{\eta})$ reduces to a product of delta functions, and Equation 6 results in a trivial integral. Here we wish to model the scatter, therefore we make a different assumption: we assume that lenses are isolated from each other, and that each background source is only lensed by one lens galaxy. The likelihood term then simplifies to \[{\rm P}(\mathbf{d}\|\boldsymbol{\eta})=\prod_{i}{\rm P}(\mathbf{d}_{i}\| \boldsymbol{\eta}),\] (7) with \[{\rm P}(\mathbf{d}_{i}\|\boldsymbol{\eta})=\iint dM_{,i}dM_{h,i}{\rm P}( \mathbf{d}_{i}\|M_{,i},M_{h,i}){\rm P}(M_{,i},M_{h,i}\|\boldsymbol{\eta}).\] (8) Here $\mathbf{d}_{i}$ indicates the data relative to the $i$-th lens only. This consists of a stellar mass measurement of the lens galaxy, and shape measurements of background sources located around the lens. Section 5 will describe how source galaxies are assigned to lenses. In short, we will only consider the lensing effect on sources located roughly within the expected virial radius of each lens, because the assumption of isolated lenses will break down at large distances where the effects of neighboring halos become significant. The effects of the isolated lens assumption will be tested in subsection 5.7. Equation 8 relates the distribution of stellar and halo masses ${\rm P}(M_{,i},M_{h,i}\|\boldsymbol{\eta})$, which we wish to infer, to the data. The first term in the integral in the right hand side of Equation 8 is the likelihood of observing the data $\mathbf{d}_{i}$ given the values of the stellar and halo mass of the $i$-th galaxy. To calculate this term, we must specify a model describing the galaxy+halo system and predict the expected values of the shear and stellar mass measurements given $M_{}$ and $M_{h}$. Our choice for the model halo density profile will be discussed in 3.3. Equation 8 shows the hierarchical nature of the problem: the likelihood of the data is given by the values of parameters describing individual objects, $M_{,i}$ and $M_{h,i}$. The probability distribution for these parameters ${\rm P}(M_{,i},M_{h,i}\|\boldsymbol{\eta})$ is in turn specified by the hyper-parameters $\boldsymbol{\eta}$. ${\rm P}(M_{,i},M_{h,i}\|\boldsymbol{\eta})$ can be thought of as a prior on stellar and halo mass, where the parameters describing this prior are free, and have priors of their own. Provided that our model allows us to compute the likelihood term ${\rm P}(\mathbf{d}_{i}\|M_{,i},M_{h,i})$, we can then use Equations 5 and 7 to calculate the posterior probability distribution of the hyper-parameters, through the integrals given by Equation 8. The Bayesian hierarchical inference formalism described above is very general and can be applied to any problem in which _i)_ measurements are carried out on a family of objects that can be described by a finite number of parameters and _ii)_ these parameters can be described as being drawn from a single distribution for the whole sample. From here on we will make assumptions specific to the weak lensing problem at hand. ### The stellar-to-halo mass relation We now describe the distribution in halo mass at fixed stellar mass, ${\rm P}(M_{h}\|M_{})$. To do so, we assume the following form for the distribution in stellar and halo masses: \[{\rm P}(M_{,i},M_{h,i}\|\boldsymbol{\eta})=\mathcal{S}(M_{,i}\|\boldsymbol{ \eta})\mathcal{H}(M_{h,i}\|\boldsymbol{\eta},M_{,i}).\] (9) We describe the halo mass term $\mathcal{H}(M_{h}\|\boldsymbol{\eta},M_{})$ as a Gaussian distribution \[\mathcal{H}(M_{h}\|\boldsymbol{\eta},M_{})=\frac{1}{\sqrt{2\pi}\sigma_{h}}\exp {\left\{-\frac{(\log{M_{h}}-\mu_{h}(M_{}))^{2}}{2\sigma_{h}^{2}}\right\}},\] (10) with a mean scaling with stellar mass as \[\mu_{h}(M_{})=\mu_{h,0}+\beta_{h}(\log{M_{}}-\log{M_{}^{\mathrm{piv}}})\] (11) and dispersion $\sigma_{h}$, with $M_{}^{\mathrm{piv}}$ being an arbitrary pivot point that can be left constant in the analysis. Equation 11 corresponds to a power-law relation between stellar and halo mass. We make this assumption to simplify the presentation of the method, but the model can be trivially generalized to more complex descriptions of the distribution of halo masses at fixed stellar mass commonly found in the literature. In subsection 5.5 we will consider a modified version of this model, in which the relation between stellar and halo mass is treated as a broken power-law. We model the stellar mass term, $\mathcal{S}$, as a skewed Gaussian distribution in the logarithm of the stellar mass, (12) with \[\Phi(\log{M_{}})=1+\mathrm{erf}\left(\alpha_{}\frac{\log{M_{}}-\mu_{}}{ \sqrt{2}\sigma_{}}\right).\] (13) The choice of a skewed Gaussian is motivated by the need to model a sample selected in stellar mass. For example, for a sample obtained by selecting galaxies above a given value of observed stellar mass, $M_{,{\mathrm{min}}}$, we expect the distribution in true stellar mass to have a sharp (yet continuous, due to observational errors) drop for $M_{}<M_{,{\mathrm{min}}}$, which can be captured by large values of the parameter $\alpha_{}$ in Equation 13. The hyper-parameters introduced so far are \[\boldsymbol{\eta}\equiv\left\{\mu_{h,0},\sigma_{h},\beta_{h},\mu_{},\sigma_{ },\alpha_{}\right\}.\] (14) A brief description of each parameter is provided in Table 1. ### The likelihood term and halo profile As discussed in Subsection 3.1, in order to infer the posterior probability distribution of the hyper-parameters we need to calculate the likelihood of the data given the values of stellar and halo mass of each object. For each lens galaxy, the data consists of an observed stellar mass $M_{}^{\mathrm{(obs)}}$ and weak lensing shape measurements $\left\{\rm{WL}\right\}$. The likelihood can then be separated as \[{\rm P}(\mathbf{d}\|M_{},M_{h})={\rm P}(M_{}^{\mathrm{(obs)}}\|M_{}){\rm P}( \left\{\rm{WL}\right\}\|M_{},M_{h}).\] (15) Assuming known Gaussian uncertainties on $\log{M_{}}$, $\sigma_{,{\mathrm{err}}}$, the stellar mass term in the right-hand side becomes \[{\rm P}(M_{}^{\mathrm{(obs)}}\|M_{})=\frac{A}{\sqrt{2\pi}\sigma_{,{\mathrm{ err}}}}\exp{\left\{-\frac{(\log{M_{}}-\log{M_{}^{\mathrm{(obs)}}})^{2}}{2 \sigma_{,{\mathrm{err}}}^{2}}\right\}},\] (16) where $A$ is a normalization constant such as the integral over all possible values of the observed stellar mass, set by the lower limit $M_{,{\mathrm{min}}}$, is unity: \[\int_{M_{,{\mathrm{min}}}}^{\infty}d\log{M_{}^{\mathrm{(obs)}}}{\rm P}(M_{} ^{\mathrm{(obs)}}\|M_{})=1.\] (17) In other terms, Equation 17 states that the probability of a lens galaxy having an observed value of $M_{}$ between $M_{,{\mathrm{min}}}$ and infinity, given the fact that it is part of the selected sample, is one. Calculating the weak lensing term of the likelihood requires making assumptions on the density profile of the dark matter halo. _We assume a spherical Navarro Frenk and White profile for the dark matter distribution_(NFW, Navarro et al., 1997): \[\rho(r)=\frac{\rho_{0}}{r/r_{s}(1+r/r_{s})^{2}}.\] (18) An NFW profile can be fully described by two parameters. We choose halo mass and concentration, defined as the ratio between the virial radius and the scale radius $r_{s}$. Consistently with our definition of halo mass, $M_{h}=M_{200c}$, we define the virial radius as the radius of the sphere enclosing an average density equal to 200 times the critical density of the Universe, $r_{200c}$. The concentration is then \[c_{h}\equiv\frac{r_{200c}}{r_{s}}.\] (19) We assume the following mass-concentration relation for the halos: \[{\rm P}(c_{h}\|M_{h})=\frac{1}{\sqrt{2\pi}\sigma_{c}}\exp{\left\{-\frac{(\log{c _{h}}-\mu_{c}(M_{h}))^{2}}{2\sigma_{c}^{2}}\right\}},\] (20) with \[\mu_{c}(M_{h})=\mu_{c,0}+\beta_{c}(\log{M_{h}}-\log{M_{h}}^{\mathrm{piv}}).\] (21) The parameters $\mu_{c,0}$, $\beta_{c}$ and the scatter $\sigma_{c}$ are to be inferred from the data, while $M_{h}^{\mathrm{piv}}$ is an arbitrary pivot point. Having introduced an additional parameter, the concentration, the likelihood term (Equation 8) becomes \[{\rm P}(\mathbf{d}_{i}\|\boldsymbol{\eta})=\iiint d\log{M_{,i}}d \log{M_{h,i}}d\log{c_{h,i}}\times\\ {\rm P}(\mathbf{d}_{i}\|M_{,i},M_{h,i},c_{h,i}){\rm P}(c_{h,i}\|M_ {h,i}){\rm P}(M_{,i},M_{h,i}\|\boldsymbol{\eta})\] (22) Note that, since we expressed all our probability distributions in terms of the logarithm of the parameters $M_{}$, $M_{h}$ and $c_{h}$, the integration variables have been changed accordingly. The contribution from the baryons in the lens galaxy to the weak lensing signal is modeled with a circular de Vaucouleurs profile (de Vaucouleurs, 1948), with half-light radius inferred from the data. We also assume that the measurements of the stellar mass, $M_{}^{\mathrm{(obs)}}$, provide an _accurate_ estimate of the baryonic mass. This will not necessarily be the case in practical applications of this method: typical stellar mass measurements are based on a set of assumptions on the stellar population, which could bias the inference. Particularly important is the assumption of a stellar initial mass function (IMF): if the true IMF is different from the one assumed to obtain $M_{}^{\mathrm{(obs)}}$, then the stellar mass estimate will be biased. For simplicity, we ignore possible variations of the stellar IMF in this theoretical study. As we will show, the contribution of the stellar mass to the lensing signal is in any case very small. Finally, we assume that the redshift of the lens galaxy is known exactly, and that the dark matter halo is centred at the galaxy position. Given our mass model, we can calculate the predicted reduced-shear signal of the lens at a given position $\boldsymbol{\theta}$ and source redshift $z_{s}$, $g(\boldsymbol{\theta},z_{s})$, defined as \[g(\boldsymbol{\theta},z_{s})=\frac{\gamma(\boldsymbol{\theta},z_{s})}{1-\kappa (\boldsymbol{\theta},z_{s})}.\] (23) $\kappa(\boldsymbol{\theta},z_{s})$ and $\gamma(\boldsymbol{,}z_{s})$ are respectively the dimensionless surface mass density and the complex shear generated by the lens at image position $\boldsymbol{\theta}$ for a source at redshift $z_{s}$. We refer to Bartelmann (1996); Wright & Brainerd (2000) for details on the lensing properties of NFW halos. The reduced-shear is a complex quantity. The tangential component of it, $g_{t}$, is equal to the tangential ellipticity induced by the lens on a circular source at $z_{s}$. The likelihood of an individual shape measurement $\epsilon_{t}^{(\mathrm{obs})}$ from a source at _known_ redshift $z_{s}$ given the model is \[{\rm P}(\epsilon_{t}^{(\mathrm{obs})}\|M_{},M_{h},c_{h})=\frac{1}{\sqrt{2\pi}{ \sigma_{\epsilon}}}\exp{\left\{-\frac{(g(\boldsymbol{\theta},z_{s})-\epsilon_{ t}^{(\mathrm{obs})})^{2}}{2\sigma_{\epsilon}^{2}}\right\}},\] (24) where $\sigma_{\epsilon}$ is the observational uncertainty on the tangential shear, typically dominated by the intrinsic dispersion in galaxy shapes. In cases when the estimate of the source redshift is noisy, i.e. most practical applications of weak lensing, the uncertainty on the source redshift, $\Delta z_{s}$, introduces an additional source of error on the tangential shear. This error is due to the dependence of the reduced shear $g$ on the lensing critical surface mass density $\Sigma_{cr}$, and, in the limit of small $\Delta z_{s}$, is given by \[\left\|\frac{\partial g}{\partial z_{s}}\right\|\Delta z_{s}=\left\|\frac{ \partial g}{\partial\Sigma_{cr}}\right\|\left\|\frac{\partial\Sigma_{cr}}{ \partial z_{s}}\right\|\Delta z_{s}=\frac{\|\gamma\|}{(1-\kappa)^{2}}\left\|\frac{ \partial\ln{\Sigma_{cr}}}{\partial z_{s}}\right\|\Delta z_{s}.\] (25) Under the assumption that the uncertainties on the source shape and redshift are independent, the term above can be added in quadrature to $\sigma_{\epsilon}$ to obtain the final uncertainty on the observed tangential shear. This additional source of uncertainty is typically much smaller than the intrinsic shape noise, since it is on the order of the magnitude of the shear. However, Equation 25 is an approximation: since the critical density is not linear in $z_{s}$, we cannot simply propagate an uncertainty on the redshift into one in $\Sigma_{cr}$. In principle, the likelihood term 24 should be calculated by obtaining the model tangential shear for all possible values of the source redshift and marginalizing over $z_{s}$. Ignoring the non-linearity of $\Sigma_{cr}$ with respect to $z_{s}$ can in principle introduce a systematic bias. However, we will show that this bias is very small. The likelihood of the weak lensing measurements given the halo mass, concentration, and stellar mass, is given by the following product over all shape measurements: \[{\rm P}(\left\{\rm{WL}\right\}\|M_{h},c_{h},M_{})=\prod_{j}{\rm P}(\epsilon_{t ,j}^{(\mathrm{obs})}\|M_{},M_{h},c_{h})\] (26) At this point we have described all ingredients necessary for the inference of the hyper-parameters of the model in the context of the assumptions made so far. In Figure 2 we display a probabilistic graphical model of the inference problem, showing its hierarchical structure. Hyper-parameters, represented as circles on the upper and left-hand side of the figure, feed into lower level variables, the individual stellar and halo mass and the concentration, which in turn determine the observed quantities: the observed stellar mass and source shapes. <figure><img src="content_image/1710.00007/x2.png"><figcaption>Figure 2: Probabilistic graphical model of the SHMR Bayesian hierarchicalinference problem with weak lensing data described in Subsections 3.1 through3.3. Circles indicate probability distributions, while dots indicate fixedquantities. Shaded regions refer to observed quantities, while unshaded onesare free parameters. Arrows indicate the dependence between the parameters.Tables indicate ensembles of parameters. The hyper-parameters, shown asunshaded circles on the left-hand side of the figure, determine thedistributions for the individual stellar and halo masses. Halo massesdetermine halo concentrations, together with the hyper-parameters describingthe mass-concentration relation. Halo mass, concentration and stellar massdetermine the tangential ellipticity of background sources. This is asimplified version of the full problem treated in this paper.</figcaption></figure> ### Sampling the posterior probability distribution We would like to measure the posterior probability distribution of the hyper-parameters using a Markov Chain Monte Carlo (MCMC). In order to evaluate the posterior at each step of the chain we must calculate the three-dimensional integral in Equation 22 for each lens galaxy. This is a computationally expensive task because the likelihood is not an analytical function of halo mass, concentration and stellar mass. Following Schneider et al. (2015) we use Monte Carlo integration together with importance sampling to evaluate these integrals: we sample the likelihood for each object beforehand and then use these samples to do the marginalization over the individual parameters for each drawn value of the hyper-parameters, as explained below. To simplify the notation, we refer to the individual lens parameters collectively as $\boldsymbol{\psi}_{i}\equiv\{M_{,i},M_{h,i},c_{h,i}\}$. We introduce a prior on the distribution in $\boldsymbol{\psi}_{i}$. This prior, which we call _interim prior_ and label $I$, should be broad enough to cover the region of parameter space where the likelihood is nonzero (our choice for the interim prior will be discussed later). We consider the posterior probability distribution of $\boldsymbol{\psi}_{i}$ given the data and the interim prior, ${\rm P}(\boldsymbol{\psi}_{i}\|\mathbf{d}_{i},I)$. This can be written as, \[{\rm P}(\boldsymbol{\psi}_{i}\|\mathbf{d}_{i},I)\propto{\rm P}(\boldsymbol{\psi }_{i}\|I){\rm P}(\mathbf{d}_{i}\|\boldsymbol{\psi}_{i}).\] (27) Using the above we can write Equation 22 as \[{\rm P}(\mathbf{d}_{i}\|\boldsymbol{\eta})=\int d\boldsymbol{\psi}_{i}\frac{{ \rm P}(\boldsymbol{\psi}_{i}\|\mathbf{d}_{i},I)}{{\rm P}(\boldsymbol{\psi}_{i}\| I)}{\rm P}(\boldsymbol{\psi}_{i}\|\boldsymbol{\eta})\] (28) up to a multiplicative constant irrelevant for the problem. At this point we draw samples $\{\boldsymbol{\psi}_{i}^{(k)}\}$ of sufficiently large size $N$ from ${\rm P}(\boldsymbol{\psi}_{i}\|\mathbf{d}_{i},I)$ with an MCMC and approximate the above integral with the sum over the samples \[{\rm P}(\mathbf{d}_{i}\|\boldsymbol{\eta})\approx\frac{1}{N}\sum_{k}\frac{1}{{ \rm P}(\boldsymbol{\psi}_{i}^{(k)}\|I)}{\rm P}(\boldsymbol{\psi}_{i}^{(k)}\| \boldsymbol{\eta}).\] (29) The advantages of this approximation are that ${\rm P}(\boldsymbol{\psi}_{i}\|\mathbf{d}_{i})$ only needs to be sampled once at the beginning of the inference, and that the triple integral reduces to a sum. We choose a uniform distribution in the range $(10,13)$ as a prior on $\log{M_{,i}}$, a Gaussian distribution with mean $13.0$ and dispersion $0.5$ on $\log{M_{h,i}}$, and a Gaussian distribution with mean $0.8$ and dispersion $0.3$ on $\log{c_{h}}$. The exact choice of the interim prior does not affect the results of the inference, as it is divided out in Equation 29. ### The full problem We can now generalize the method to include a secondary dependence of halo mass on a property of its central galaxy. We choose galaxy size to be this parameter. We introduce the effective radius variable $R_{\mathrm{e}}$ and update the model to account for the dependence of halo and stellar mass on $R_{\mathrm{e}}$. We assume that stellar mass, halo mass and effective radius are now drawn from a distribution ${\rm P}(M_{},M_{h},R_{\mathrm{e}}\|\boldsymbol{\eta})$ with the following form: \[{\rm P}(M_{},M_{h},R_{\mathrm{e}}\|\boldsymbol{\eta})=\mathcal{S}(M_{}\| \boldsymbol{\eta})\mathcal{R}(R_{\mathrm{e}}\|M_{},\boldsymbol{\eta})\mathcal{ H}(M_{h}\|M_{},R_{\mathrm{e}},\boldsymbol{\eta}).\] (30) Here $\mathcal{S}(M_{}\|\boldsymbol{\eta})$ is the same skewed Gaussian as in Equation 12. $\mathcal{R}(R_{\mathrm{e}}\|M_{},\boldsymbol{\eta})$ is the following Gaussian \[\mathcal{R}(R_{\mathrm{e}}\|M_{},\boldsymbol{\eta})=\frac{1}{\sqrt{2\pi}\sigma _{R}}\exp{\left\{-\frac{(\log{R_{\mathrm{e}}}-\mu_{R}(M_{}))^{2}}{2\sigma_{R} ^{2}}\right\}}\] (31) with mean \[\mu_{R}(M_{})=\mu_{R,0}+\beta_{R}(\log{M_{}}-\log{M_{}^{\mathrm{piv}}})\] (32) and dispersion $\sigma_{R}$. $\mathcal{H}(M_{h}\|M_{},R_{\mathrm{e}},\boldsymbol{\eta})$ has the same form as the Gaussian in Equation 10, but its mean is updated as follows to include a dependence on stellar mass density \[\mu_{h}(M_{},R_{\mathrm{e}})=\mu_{h,0}+\beta_{h}(\log{M_{}}-\log{M_{}^{ \mathrm{piv}}})+\xi_{h}\log{(\Sigma_{}/\Sigma_{,0})},\] (33) where $\Sigma_{}=M_{}/(2\pi R_{\mathrm{e}}^{2})$ and $\log{\Sigma_{,0}}=\log{M_{}^{\mathrm{piv}}}-\log{2\pi}-2\mu_{R,0}$ (the stellar mass density of a galaxy with $M_{}=M_{}^{\mathrm{piv}}$ and average size for its mass). We choose to parametrize the model with a dependence of halo mass on stellar mass density rather then directly on effective radius to minimize correlations between the model parameters. Stellar mass and size are correlated, and it is difficult with weak lensing data alone to determine which of the two is the fundamental parameter on which halo mass depends. On the other hand, since $R_{\mathrm{e}}$ scales with a power of $M_{}$ close to $0.5$, massive galaxies occupy roughly a region of constant stellar mass density as a function of mass. Using $M_{}$ and $\Sigma_{}$ in Equation 33 allows us to decouple the dependence on mass from that on size, helping in the interpretation of the results. The full list of hyper-parameters is summarized in Table 1, among with a short description of each parameter. \| Hyper-parameter \| Description \| Prior ---\|---\|---\|--- \| μh,0 \| Mean logMh at logM∗=11.2 \| Uniform(11,15) \| σh \| Intrinsic scatter in logMh \| Uniform(0,2) \| βh \| Power-law index of Mh-M∗ correlation \| Uniform(−3,3) Simplified model \| μ∗ \| Mean of Gaussian component of stellar mass distribution \| Uniform(10,12) (subsection 3.2) \| σ∗ \| Dispersion of Gaussian component of stellar mass distribution \| Uniform(0,2) \| α∗ \| Skewness parameter of stellar mass distribution \| Uniform in log (−1,1) \| μc,0 \| Mean logch at logMh=13 \| Uniform(0,2) \| σc \| Intrinsic scatter in logch \| Uniform(0,1) \| βc \| Power-law index of ch-Mh correlation \| Uniform(−1,1) \| ξh \| Power-law index of Mh-Σ∗ correlation \| Uniform(−2,2) (Full model only) \| μR,0 \| Mean logRe at logM∗=11.2 \| Uniform(−1,2) \| σR \| Intrinsic scatter in logRe \| Uniform(0,1) \| βR \| Power-law index of Re-M∗ correlation \| Uniform(−1,2) Table 1: Hyper-parameters of the model. The horizontal line divides the hyper- parameters between those relative to the simplified model, described in subsections 3.2 and 3.3, and the ones appearing only in the full version of the model, introduced in subsection 3.5. The likelihood must also be updated to account for the observed effective radius. Equation 15 becomes \[{\rm P}(\mathbf{d}\|M_{},M_{h},c_{h},R_{\mathrm{e}})={\rm P}(M_{* }^{\mathrm{(obs)}}\|M_{}){\rm P}(R_{\mathrm{e}}^{\mathrm{(obs)}}\|R_{\mathrm{e} })\times\\ {\rm P}(\left\{\rm{WL}\right\}\|M_{},R_{\mathrm{e}},M_{h},c_{h}).\] (34) Here we have assumed that measurements of stellar mass and effective radius are independent. This, however, is not a critical assumption. A probabilistic graphical model of the problem is shown in Figure 3 . <figure><img src="content_image/1710.00007/x3.png"><figcaption>Figure 3: Probabilistic graphical model of the SHMR Bayesian hierarchicalinference problem described in subsection 3.5.</figcaption></figure> ### Possible model extensions In the model considered so far, the average halo mass scales with stellar mass and stellar mass density following a power-law relation. Observational studies of the SHMR, however, indicate that the relation between stellar and halo mass is not a single power-law at all mass scales: the dependence of $M_{h}$ on $M_{}$ is steeper at the high mass end, compared to the low mass regime, the transition occurring around $\log{M_{}}\sim 10.5$(see e.g. Leauthaud et al., 2012). Our model can be modified to allow for an SHMR with different slopes in different mass ranges: we will show one such example in subsection 5.5, where we describe the relation between stellar and halo mass as a broken power-law. Another implicit assumption in our model is that the SHMR does not depend evolve with time. This approximation is reasonable if the sample of lenses covers a thin slice in redshift, but breaks down over a broad redshift range (see e.g. Moster et al., 2010). Nevertheless, we can in principle account for evolution by adding a redshift dependence term to the expression for the average halo mass, Equation 33. In fact, our method can allow us to measure the time evolution of the SHMR without resorting to dividing the sample of lens galaxies in redshift bins. Finally, we have not discussed the possibility of a dependence on galaxy type of the SHMR. Current constraints from weak lensing (Mandelbaum et al., 2016) and satellite kinematics (More et al., 2011) suggest that blue central galaxies live in less massive halos compared to red centrals of the same stellar mass (although see Section 6 for a possible issue with measuring such a correlation with binning and stacking). While a dependence of the SHMR on galaxy type can be included in our model, we assume for simplicity a single SHMR for all galaxies. Strictly speaking, then, the results of the tests presented in this work will only be relevant for situations in which the sample of lens galaxies is constructed by selecting only objects of a given type. ## 4 Mock observations We follow a semi-empirical approach to create $5$ different sets of mock weak lensing observations. For each set, we use different ingredients to create the population of lens galaxies. We generate mocks of varying complexity to test the impact that different approximations have on the inference. In the next two subsections we describe the most simple and the most complex models among these mock lens populations. We further create an additional intermediate set of three mocks that will be described in Section 5 – these will be used in Section 5. Table 2 summarizes the characteristics of each mock sample. Label \| SHMR \| Miscent. \| Sat. \| 2-halo ---\|---\|---\|---\|--- A \| Power-law \| N \| N \| N B \| Power-law \| Y \| N \| N C \| B13 \| Y \| N \| N D \| B13 \| Y \| Y \| N E \| B13 \| Y \| Y \| Y Table 2: Mock lens samples. The second column indicates which form of the SHMR has been used. Columns 3 to 5 indicate whether miscentering, the presence of satellite galaxies and the effects of neighboring halos (the 2-halo term) are included in the model. ### A toy model (model A) The most simple model we consider, labeled “mock A”, is generated by drawing stellar and halo masses from from a very similar distribution to the one assumed in subsection 3.2. We take a Gaussian distribution in $\log{M_{}}$ with mean $11.0$ and dispersion $0.4$. We draw a large number of objects from this distribution, apply a $0.15$ dex observational uncertainty to the stellar mass, apply a cut selecting only objects with $\log{M_{}^{\mathrm{(obs)}}}>11$, and finally draw a subsample of 5,000 objects. We then generate halo masses from a Gaussian distribution, with a mean that scales as a power of the (true) stellar mass according to Equation 11, with parameters $\mu_{h,0}=13$, $\beta_{h}=1.5$, $\log{M_{}^{\mathrm{piv}}}=11.2$, and with dispersion $\sigma_{h}=0.4$. Essentially, we are using the same form as Equation 10 to describe the SHMR. We distribute the galaxies in redshift using a Gaussian distribution centred at $z=0.2$, with dispersion $0.1$ and truncated at a minimum redshift $z_{\mathrm{min}}=0.1$ to ensure that each galaxy contributes with an appreciable lensing signal. We assign a size to each galaxy, drawn from a mass-size relation of the same form as Equation 31 and Equation 32, with $\mu_{R,0}=0.80$, $\beta_{R}=0.57$ and $\sigma_{R}=0.16$, as measured by Newman et al. (2012) using SDSS data, corrected to a Chabrier IMF. Note that we do not assume any additional dependence of size on halo mass. We model the mass distribution of each system with a spherical NFW profile describing the halo and a circularly symmetric de Vaucouleurs profile describing the stars. To set the scale radius of each halo, we use a mass-concentration relation of the form given by Equation 20, with $\mu_{c,0}=0.72$, $\beta_{c}=-0.098$, and $\sigma_{c}=0.1$, as suggested by dark matter only simulations (see e.g. Macciò et al., 2008). Finally, we assume that each galaxy is exactly at the center of its dark matter halo, ignoring the possibility of miscentering, and that each halo is at a virtually infinite projected distance from any other halo, neglecting the contribution from neighboring halos to the lensing signal. ### The most complex model (model E) For the most complex of our mock lens realizations, we make use of the package Halotools (Hearin et al., 2016). We draw a sample of galaxies and host halos using the Behroozi et al. (2010, B10 from here on) SHMR and halo catalogs from the Bolshoi simulation at $z=0$. We take half of the simulation box, preserving the size along line-of-sight direction, then, as for mock A, we apply a $0.15$ dex observational uncertainty to the stellar masses, and apply a stellar mass cut by selecting only objects with $\log{M_{}^{\mathrm{(obs)}}}>11$. The final sample results in $\sim 4,500$ galaxies, 16% of which are satellites. Galaxy sizes are generated in the same way as mock A. With this simulation we wish to test the effects of the environment on our inference, therefore we drop the approximation of galaxies being at infinite distance from each other, and use the full halo position information from the Bolshoi simulation. However, for simplicity, we still model halos and galaxies with analytical density profiles, similarly to mock A. In order for the simulation to be realistic, we need halos to have a finite mass. We then model the mass distribution of each halo with a smoothly truncated spherical NFW profile, following Baltz et al. (2009): \[\rho(r)=\frac{\rho_{0}r_{s}}{r/r_{s}(1+r/r_{s})^{2}}\left(\frac{r_{t}^{2}}{r^{ 2}+r_{t}^{2}}\right)^{2}.\] (35) We set the truncation radius $r_{t}$ to be the same as the value of $r_{200c}$ for each halo, and assume the same mass-concentration relation used for mock A. The distribution of stellar mass is modeled as a de Vaucouleurs profile. In addition to the halos of the galaxies in the sample, we model the contribution to the lensing signal of all halos with $M_{h}>10^{12.5}M_{\odot}$. We allow for miscentering between galaxy and halos. This is added as a random shift in projection between the centers of the two mass components, drawn from a Gaussian distribution with zero mean and $10\,\rm{kpc}$ dispersion. In the literature, the term miscentering is sometimes used in a broader sense, to indicate situations in which the galaxy at the center of the halo is not the most massive among those bound to the main halo. The B10 model used for this mock already includes such cases. We treat the sample of galaxies and halos as a thin screen of lenses at $z=0.3$. The Bolshoi box is $250\,\rm{Mpc}/h$ on the side. For our fiducial cosmology, this corresponds to an area of 243 square degrees at $z=0.3$. For the sake of computational time, we only use half of this area, leaving us with a $\sim 120$ square degrees simulation. This is similar to the area covered by the first-year shear catalog of the Hyper Suprime-Cam (HSC) survey (Aihara et al., 2017; Mandelbaum et al., 2017). The absence of a lens redshift distribution is the only element that is more simple compared to mock A. Otherwise, this mock contains a variety of features that are present in the real Universe and could bias our inference method: satellite galaxies, the effects of neighboring halos, and miscentering. ### Mock source sample We generate a mock source sample with a uniform distribution in the image plane of $20\,\rm{arcmin}^{-2}$ and a Gaussian redshift distribution centred at $z=1.0$ and with dispersion $\sigma_{z}=0.5$. The two components of the complex ellipticity of each source are drawn from a Gaussian centred in zero and with dispersion $\sigma_{\epsilon}=0.27$. We then add lensing distortion. For each source, we calculate the reduced-shear resulting from the contribution of all halos within a physical projected distance in the image plane smaller than ten times their virial radius. In mock E, all lenses are at the same redshift and the reduced-shear can be calculated using Equation 23, where the surface mass density $\kappa$ and the shear $\gamma$ are just the sum of the individual values of $\kappa$ and $\gamma$ of each contributing lens. Although we also consider mocks with a distribution in lens redshifts, in all such cases lenses are assumed to be at infinite distance from each other, so that Equation 23 can be used directly, with $\kappa$ and $\gamma$ being the surface mass density and complex shear of the only lens affecting the source. Given the value of the reduced shear $g$, and given a value for the intrinsic complex ellipticity of the source, $\epsilon_{s}$, the observed complex ellipticity is given by (Seitz & Schneider, 1997) \[\epsilon=\left\{\begin{array}[]{ll}\dfrac{\epsilon-g}{1-g^{}\epsilon}&\rm{for }\,\|g\|<1\\ &\\ \dfrac{1-g\epsilon^{}}{\epsilon^{}-g^{8}}&\rm{for}\,\|g\|>1\end{array}\right..\] (36) Finally, we add a photo-z measurement error of $\Delta z_{s}=0.1$. The parameters used to simulate weak lensing measurements are set to resemble those of the HSC survey (Mandelbaum et al., 2017; Tanaka et al., 2017). ## 5 Results ### Mock A: source photo-z uncertainty We fit the model introduced in 3.5 to the data from mock A (weak lensing, stellar masses, and effective radii, see subsections 4.1 and 4.3). We only use sources located within a cone of angular radius $\theta_{max}$ centred on each lens. This corresponds to $300\,\rm{kpc}$ in projected physical distance at the redshift of our lenses. This choice approximates what we would do when dealing with actual data, in order to stay in a regime where our isolated lens approximation is valid. Although no effects from the environment are included in mock A, we still apply this cut in projected source distance to check whether we can still make a meaningful inference on the SHMR with weak lensing data not extending too far from the lens. In Figure 4 we plot the inferred posterior probability distribution on the hyper-parameters describing the distribution of halo masses and concentrations, along with the true values used to generate the mock. The inference is accurate. The inference on the coefficient $\xi$ is consistent with zero indicating no correlation between halo mass and stellar mass density at fixed galaxy mass, consistent with the input model. This is perhaps not surprising, since the model we are fitting is identical to the one used to generate the mock. The average halo mass is recovered with a precision of $0.03$ dex. This value is somewhat arbitrary, as it is essentially set by the size of the mock sample and the number density of sources, which we picked to be 5,000 lens galaxies and $20\,\rm{arcmin}^{-2}$ respectively, but it sets the sensitivity of our experiments to possible systematic effects. With mock A, we are testing for biases introduced by our treatment of the source redshift uncertainty, which we propagate directly onto the uncertainty in the tangential shear. The fact that we obtain an accurate answer implies that any bias related to this procedure is too small to be detected with our mock (i.e. smaller than $0.03$ dex in halo mass). <figure><img src="content_image/1710.00007/x4.png"><figcaption>Figure 4: Posterior probability distribution for the hyper-parametersdescribing the distribution in halo mass and halo concentration, obtained fromfits of different models to sets of mock observations A and B. Blue contoursshow the inference obtained by fitting the model to mock A. Black Solid linescorrespond to the inference made on mock A by fitting a model that ignores thecontribution of the stars to the lensing signal. Red contours correspond tothe inference made by fitting the model to mock B, which differs from mock Aby the presence of miscentering between galaxies and halos. Contours delimit68%, 95% and 99.7% enclosed probability regions. Black dots and dotted linesshow the true values of the hyper-parameters used to create the mocks.</figcaption></figure> In addition to the hyper-parameters describing the halo mass distribution, it is also interesting to check whether the model is able to recover the intrinsic distribution in stellar mass of the population. In our model, the stellar mass distribution is described as a skewed Gaussian, meant to recover the drop at low masses generated by the sharp cut in $M_{}^{\mathrm{(obs)}}$ applied to define the sample. In Figure 5 we plot the distribution in true stellar mass of the sample, together with the inferred distribution, as given by the maximum likelihood inference on the skewed Gaussian parameters $\mu_{}$, $\alpha_{}$, $\sigma_{}$. There is good agreement between the two distributions: the model correctly infers the presence of a tail of galaxies with true mass below the $\log{M_{}}=11$ observational cut. This is a key feature of our method, that allows for an unbiased estimate of the correlation between halo mass and galaxy properties. <figure><img src="content_image/1710.00007/x5.png"><figcaption>Figure 5: Histogram: Distribution in stellar mass of the mock A sample. Line:Maximum likelihood stellar mass distribution, described by Equation 12,inferred by fitting the model in subsection 3.5 to the mock data. Thecorresponding parameter values are μ∗=10.99, α∗=2.63, σ∗=0.40.</figcaption></figure> Mock A was generated assuming that halo masses are independent of galaxy sizes, at fixed stellar mass. Our inference recovers this feature ($\xi=0$). However, we also wish to show that our method can also recover a positive or negative correlation with size, if this is present in the data. For this purpose, we generate two new mock populations, similar to mock A, but with an added dependence of halo mass on size: values of $\log{M_{h}}$ are still drawn from a Gaussian distribution with mean given by Equation 33, but the value of parameter $\xi$ is set to $-0.5$ and $0.5$. The two cases correspond to a positive linear dependence of halo mass on galaxy size, at fixed stellar mass, and a negative linear dependence, respectively. Our model still provides an accurate inference on the parameter $\xi$, as shown in Figure 6. <figure><img src="content_image/1710.00007/x6.png"><figcaption>Figure 6: Recovered values of the parameter ξ defined in Equation 33,describing a dependence of halo mass on stellar mass density at fixed stellarmass, as a function of the input value, for three different variations of themock A simulation. The inference is consistent with the input truth in allcases.</figcaption></figure> ### Mock A: stellar mass contribution to $\Delta\Sigma$ Before applying our method to a more complex simulation, we can use mock A to study the sensitivity of our inference to the presence of the central galaxy. We would like to know what would the inferred halo mass distribution be if we were to neglect the contribution of the stellar mass to the weak lensing signal. We then fit a simpler version of our model, in which galaxies are treated as massless, to the mock A data. The resulting inference on the hyper-parameters describing the dark matter distribution is plotted in Figure 4. The inferred average halo mass at $M_{}^{\mathrm{piv}}$, parameter $\mu_{h,0}$, is consistent with the true value, and does not shift significantly with respect to the case in which the stellar mass is included. However, the inferred average concentration, $\mu_{c,0}$ is overestimated by about $0.1$ dex. This makes sense, as the model compensates the lack of a central component, present in the data, with a more concentrated halo. ### Mock B: the effect of miscentering The second set of simulated weak lensing observations, mock B, is a variation of mock A in which we allow for miscentering between central galaxies and their halos. We do this by adding a random shift in the projected distance between the two mass components, drawn from a Gaussian distribution with dispersion $10\,\rm{kpc}$, as described in subsection 4.2. The inferred hyper-parameters are plotted in Figure 4. There is only a $0.06$ dex shift in the inference on the average halo mass, moving towards smaller values. Although the shift would become larger for increasingly higher amplitude of the simulated miscentering, we do not expect large ($>10\,\rm{kpc}$) displacements between central galaxies and halos to be a common occurrence in the real Universe. Therefore we conclude that miscentering has a small impact on studies of this kind. Miscentering has a somewhat larger impact on the inferred concentration distribution. For instance, the average concentration is underestimated by about $0.10$ dex for this sample. ### Mock C: a different SHMR We now fit our model to a simulation created using a more complex SHMR compared to the simple power-law scaling between stellar and halo mass that we assume in our model. Mock C is obtained from the more complex mock E, described in Section 4.2, which is based on the B10 SHMR. However, to isolate the effect of the form of the SHMR from other effects included in mock E, we set the lenses at infinite distance from each other. Moreover, we eliminate satellite galaxies from the sample. In mock E, and therefore also in mock C, dark matter halos are no longer described by a pure NFW profile, but by a smoothly truncated NFW profile, with density profiles described by Equation 35 and truncation radii equal to the virial radius $r_{200c}$. In the real Universe, the radius at which the dark matter profile drops sharply can in general be different from the virial radius (see Diemer & Kravtsov, 2014, for a discussion). The truncation radius then introduces an additional parameter, which should in principle be inferred from the data. However, for simplicity, we assume that the ratio between truncation radius and virial radius is known perfectly. Therefore, the model used to fit this mock, as well as the following mocks, is modified to a smoothly truncated NFW halo with $r_{t}=r_{200c}$. The results are shown in Figure 7. Because the SHMR of this mock is no longer a power-law with constant scatter, there is no obvious definition of the true values of $\mu_{h,0}$, $\sigma_{h}$ and $\beta_{h}$ for this sample. We fit a power-law relation to the distribution of $M_{h}$ as a function of $M_{}$, with a minimum least squares method, and use the slope and intercept at $\log{M_{}^{\mathrm{piv}}}=11.2$ as proxies for the true values of $\mu_{h,0}$ and $\beta_{h}$. We then take the standard deviation in $\log{M_{h}}$ around the best fit relation as a proxy for $\sigma_{h}$. The true values defined with this procedure are plotted in Figure 7 as black dots. Additionally, we plot in Figure 8 the inferred average halo mass as a function of stellar mass, given by Equation 33. <figure><img src="content_image/1710.00007/x7.png"><figcaption>Figure 7: Posterior probability distribution for the hyper-parametersdescribing the distribution in halo mass and halo concentration, obtained fromfits of different models to sets of mock observations C, D and E. Bluecontours show the inference obtained by fitting the model to mock C (a samplewith only central galaxies, each at infinite distance from each other). Greencontours show the inference obtained by fitting the model to mock D (a samplewith both central and satellite galaxies, but no effects from theenvironment). Red contours correspond to the inference made by fitting themodel to mock E (a mock that includes the lensing effect created byneighboring halos). Contours delimit 68%, 95% and 99.7% enclosed probabilityregions. Black dots and dotted lines show the true values of the hyper-parameters used to create the mocks.</figcaption></figure> <figure><img src="content_image/1710.00007/x8.png"><figcaption>Figure 8: Stellar vs. halo mass for mock galaxies generated with Halotoolsusing the B10 model applied to the halo catalog of the Bolshoi simulation atz=0, on which mock C, D and E are based. Gray circles: central galaxies. Blacksolid line: median halo mass as a function of stellar mass, for centralgalaxies only. Blue, green and red dashed lines delimit the 68% confidenceregion of the inference on the SHMR obtained by fitting our model to mock C, Dand E respectively.</figcaption></figure> The recovered average halo mass is in very good agreement with the underlying truth, as can be seen from Figure 7. Although the true SHMR departs from a pure power-law relation, the inferred model is still a good description of the SHMR, as can be seen by comparing the dashed blue lines in Figure 8, which delimit the 68% confidence region of the inference, with the solid black line, corresponding to the median halo mass as a function of stellar mass of the mock. It is also important to check how the inferred scatter in halo mass compares with that of the mock. This is shown in Figure 9, where we plot the distribution in halo masses for central galaxies with $11.15<\log{M_{}}<11.25$, together with the maximum-likelihood inferred distribution at $\log{M_{}}=11.2$. Our model assumes a Gaussian distribution at fixed stellar mass. This appears to be a good description of the actual distribution. The inferred scatter, however, is $\sim 0.1$ dex larger than the actual dispersion in halo mass, as can be read from Figure 7. This is the result of the true SHMR being different from a pure power-law. We expect this small bias to be reduced if we were to use a more accurate model. <figure><img src="content_image/1710.00007/x9.png"><figcaption>Figure 9: Histogram: Distribution in dark matter mass of central galaxies inthe stellar mass bin logM∗∈(11.15,11.25). Line: Maximum likelihood halo massdistribution at logM∗=11.2, described by Equation 10, inferred by fitting themodel in subsection 3.5 to the mock data. The corresponding parameter valuesare μh,0=13.34, σh=0.46.</figcaption></figure> As for mock B, the inferred concentration is $\sim 0.1$ dex lower than the truth. We understand this to be the result of miscentering. ### Mock C: measuring the change in slope of the SHMR We now fit the data from the mock C sample with a more complex model for the SHMR. Instead of assuming a power-law relation between the average halo mass and the stellar mass, we allow for an SHMR with a different slope at the low and high mass end, modifying Equation 33 as follows: \[\mu_{h}(M_{},R_{\mathrm{e}})=\mu_{h,0}+\xi_{h}\log{(\Sigma_{}/ \Sigma_{,0})}+\\ \left\{\begin{array}[]{ll}\beta_{h,1}(\log{M_{}}-\log{M_{}^{ \mathrm{piv}}})&\rm{if}\,\,M_{}<M_{}^{\mathrm{piv}}\\ \beta_{h,2}(\log{M_{}}-\log{M_{}^{\mathrm{piv}}})&\rm{if}\,\,M_{}>M_{}^{ \mathrm{piv}}\end{array}\right.\] (37) The equation above corresponds to a broken power-law relation between stellar and halo mass at fixed stellar mass density, with a change in the slope of the SHMR occurring at $M_{}^{\mathrm{piv}}$. In principle, $M_{}^{\mathrm{piv}}$ could be left as a free parameter to be inferred from the data. For simplicity, however, we keep its value fixed to $\log{M_{}^{\mathrm{piv}}}=11.2$. In Figure 10, we plot the inference on the parameters $\beta_{h,1}$ and $\beta_{h,2}$, describing the correlation between halo mass and stellar mass below and above $M_{}^{\mathrm{piv}}$, respectively. The true relation between halo mass and stellar mass is steeper at the high mass end ($\beta_{h,2}>\beta_{h,1}$). Our broken power-law model recovers this result, although with a large uncertainty. In order to make a more precise inference on the change in slope of the SHMR, a larger sample and/or a larger dynamic range in mass is needed. Solutions with $\beta_{h,1}=\beta_{h,2}$, corresponding to the pure power-law model considered previously and marked as a dotted line in Figure 10, are within the inferred $1\sigma$ confidence region. This means that models with a broken power-law SHMR do not provide a significantly better description of the data used in our simulation, compared to a single power-law model. Therefore, from here on, we will focus exclusively on the simpler model with a single value of $\beta_{h}$ over the whole mass range. <figure><img src="content_image/1710.00007/x10.png"><figcaption>Figure 10: Posterior probability distribution obtained by fitting a brokenpower-law model for the SHMR to the data from the mock C sample, projected onthe space defined by the parameters βh,1 and βh,2 defined in Equation 37. Theblack dot indicates the true values of the parameters, obtained by fittingpower-law relations to the distribution of halo mass as a function of stellarmass, above and below Mpiv∗. The dotted line corresponds to models with asingle SHMR slope at all masses: βh,1=βh,2.</figcaption></figure> ### Mock D: the effect of satellites We take one step further in complexity, and add satellite galaxies to the sample. For these objects, the lensing effect of the main halo, offset from the galaxy, is included. Lens galaxies are still assumed to be at infinite distance from each other, so that the number of halos distorting sources behind any given galaxy is at most two (one for centrals). In our model, galaxies are treated as isolated, therefore the presence of a more massive halo offset from it could in principle bias our inference. We first fit our model to a sample consisting only of satellite galaxies. The inference is shown as solid curves in Figure 7. Since satellites account for only 16% of the galaxies in the simulation, the uncertainty on the hyper-parameters is broadened due to the smaller sample size. The main difference with respect to the inference based on mock C, is the value of the mean concentration: satellite galaxies push the parameter $\mu_{c,0}$ towards values as low as allowed by the prior. We then use the full sample of galaxies, both centrals and satellites. In this case, the inference does not change significantly with respect to the case with centrals only. This can be seen by comparing the green and blue contours and lines in Figure 7 and Figure 8. We conclude that our method is robust to the presence of a small fraction (16% in this case) of satellite galaxies, at least in the mass regime probed in our test. ### Mock E: the full simulation We can finally apply our method to the full mock realization of the weak lensing observations, consisting of shape measurements over a contiguous area of 120 square degrees. This simulation offers an additional challenge with respect to the tests carried out so far. Up until this point, lenses were at an infinite distance from each other, therefore the assumption of isolated lenses was true by construction. The model was fit to sources located within an angle $\theta_{max}$ in projection from each galaxy ($300\,\rm{kpc}$ in physical distance), and there was no ambiguity over which foreground galaxy lensed any given source. In mock E, and in the real Universe, instead, each source is lensed by every galaxy and halo in its foreground. In order to apply the isolated lens approximation, on which our method is based, we must arbitrarily assign background sources to each lens in our sample. This can be a problem, especially in cases where pairs or multiplets of lens galaxies are found at close projected distances from each other. We cannot build sets of weak lensing observations within cones of radius $\theta_{max}$ around every lens galaxy in the sample without using the same data twice. A possible solution could be using a more complex subdivision of sources among foreground lenses, for example using adaptive boundaries. Alternatively, we could arbitrarily decide to remove some lens galaxies from the sample until the regions within $\theta_{max}$ are no longer overlapping. None of these solutions is ideal. It would be best to explicitly model the contribution of multiple lenses to each background source. This, however, would be computationally very challenging, as it would require, for each draw of a set of hyper-parameters in the MCMC, to marginalize over $3\times N_{lens}$ parameters, where $N_{lens}$ is the number of lenses in the sample, and the factor of $3$ is the number of free parameters for each lens: $M_{}$, $M_{h}$ and $c_{h}$. In other words, it would require the evaluation of a $3\times N_{lens}$-dimensional integral. With the isolated lens assumption, the problem reduces to the much more tractable calculation of $N_{lens}$$3$-dimensional integrals (Equation 22). For our test, we adopt the second of the two solutions discussed above: whenever two lens galaxies are located at a projected distance smaller than $2\theta_{max}$, the least massive of the two, according to the observed stellar mass, is removed from the sample. The rationale for this choice is that we expect the lensing signal to be dominated by the most massive halo, which we expect to correspond to the most massive galaxy. With this procedure, we remove 14% of the lens galaxies. Many of the excluded galaxies are satellites: the residual satellite fraction is 10%, from the initial 16%. The inferred hyper-parameters are plotted in Figure 7, and the corresponding SHMR is shown in Figure 8. Remarkably, the inference is accurate to better than $0.1$ dex in halo mass over a decade in stellar mass. ### A biased approach As a last test, we examine what would happen if we were to adopt a more traditional approach to search for a correlation between halo mass and galaxy size at fixed stellar mass, consisting in making a bin in observed stellar mass, splitting it according to size, and measuring a stacked weak lensing signal around galaxies in each bin. Let us take mock E, and make a bin in observed stellar mass with $11.0<\log{M_{}^{\mathrm{(obs)}}}<11.2$. Let us split this mass bin in two sub-bins: galaxies lying above the mass-size relation at their observed stellar mass go into a ‘larger size’ bin, while galaxies with smaller size compared to the average at their $M_{}^{\mathrm{(obs)}}$ go into a ‘smaller size’ bin. In Table 3 we report the mean values of the observed stellar mass, true stellar mass, effective radius and halo mass for the full bin and the two sub-bins. M(obs)∗ Bin \| \| Full bin \| Larger \| Smaller ---\|---\|---\|---\|--- [11.0,11.2] \| logM(obs)∗ \| 11.08 \| 11.08 \| 11.09 \| logM∗ \| 10.98 \| 11.04 \| 10.94 \| logRe \| 0.66 \| 0.85 \| 0.56 \| logMh \| 12.81 \| 12.92 \| 12.75 [11.2,11.4] \| logM(obs)∗ \| 11.28 \| 11.28 \| 11.28 \| logM∗ \| 11.14 \| 11.21 \| 11.11 \| logRe \| 0.78 \| 0.97 \| 0.67 \| logMh \| 13.12 \| 13.25 \| 13.05 >11.4 \| logM(obs)∗ \| 11.51 \| 11.50 \| 11.51 \| logM∗ \| 11.32 \| 11.39 \| 11.30 \| logRe \| 0.86 \| 1.08 \| 0.79 \| logMh \| 13.46 \| 13.71 \| 13.38 Table 3: Mean properties of each M(obs)∗ bin and relative sub-bins. The two sub-bins have the same mean observed stellar mass. However, their average true stellar mass is different, with the larger size bin containing galaxies that are more massive by $0.06$ dex on average. The mean halo mass of the two sub-samples is correspondingly different, by an even larger amount: $0.10$ dex. This is the effect of observational scatter, as we discussed in Section 2. Figure 11 shows a graphical representation of the problem. In the bottom panel, we plot the size of our mock galaxies as a function of their stellar mass. This plot is analogous to that of Figure 1, but obtained with a realistic distribution of masses and sizes. Two different biases are acting. One is the well-known Eddington bias. Galaxies get scattered into the mass bin from lower and higher intrinsic masses due to observational errors. Since the galaxy stellar mass distribution is rapidly declining with increasing mass, there are more galaxies with intrinsically lower stellar mass populating the bin. This explains why the average true stellar mass of our bins is smaller than the average over the observed values. The second source of bias is the one described in subsection 2.2. Due to the correlation between mass and size, objects scattering into the bin from higher values of their intrinsic mass have on average larger sizes compared to objects with smaller intrinsic mass. Therefore, when we split the sample in two size bins, the larger size bin tends to be populated with objects with larger intrinsic stellar mass. Although the amount of the bias in stellar mass between the two sub-bins is relatively small, the corresponding bias in halo mass gets amplified due to the steep relation between stellar and halo mass at the high mass end. Since the difference in average size between the two sub-samples is $0.3$ dex, and the corresponding difference in the mean halo mass is $0.16$ dex, failure to take this bias into account would lead us to believe that halo mass correlates with the $1/2$ power of $R_{\mathrm{e}}$ at fixed stellar mass. <figure><img src="content_image/1710.00007/x11.png"><figcaption>Figure 11: Mass-size relation of mock galaxies. Bottom: effective radius as afunction of stellar mass for a subset of 1,000 objects in the mock sample.Filled circles mark the true stellar mass, while empty circles mark theobserved values. Objects in blue (red) correspond to galaxies in thelogM(obs)∗∈[11.0,11.2] bin with smaller (larger) sizes compared to the mass-size relation. The dashed line shows the mass-size relation used to create themock. Top: Distribution in true stellar mass (filled histogram) and observedstellar mass (empty histogram) of the two subsamples. Vertical dashed (dotted)lines mark the median of the true (observed) stellar mass of each sub-bin.</figcaption></figure> Note that weak lensing did not enter at all the above argument: this is a general bias that applies to any situation in which one tries to infer separate correlations between a quantity and two different variables that are correlated with each other. Let us now obtain a stacked weak lensing signal around galaxies in each sub-bin, and compare them. For this purpose, we use the software Swot(Coupon et al., 2012; Coupon et al., 2017). We consider two more bins: one covering the range $11.2<\log{M_{}^{\mathrm{(obs)}}}<11.4$, and one with $\log{M_{}^{\mathrm{(obs)}}}>11.4$. The excess surface mass density for the larger and smaller sized galaxies in the three stellar mass bins is plotted in Figure 12. In each bin, the signal obtained for the larger size subsample is significantly higher compared to the smaller size subsample. Since the two curves have been obtained on samples of galaxies of the same observed stellar mass, a naive interpretation of this plot would lead to the conclusion that halo mass correlates with size at fixed stellar mass. However, such a correlation is not present in the simulation used to produce this plot, therefore this would be a wrong conclusion. <figure><img src="content_image/1710.00007/x12.png"><figcaption>Figure 12: Excess surface mass density in different radial bins around twosamples of lens galaxies from mock E, selected by having their observedstellar mass in three different bins (with mass range indicated in eachpanel), and split according to their position with respect to the mean mass-size relation. The signal is obtained by stacking weak lensing measurements,using the software Swot (Coupon et al., 2012; Coupon et al., 2017). In eachstellar mass bin, the larger size sample appears to have a higher excesssurface mass density compared to the smaller size sample. This is just theresult of the two subsamples having different values of their true stellarmass.</figcaption></figure> The effect of Eddington bias, and its differential effect on the two subsamples of galaxies is further illustrated in Figure 13. In the upper panel, we plot the average halo mass as a function of the mean observed stellar mass in each bin, for large and small galaxies. This is the OSHMR: the distribution of halo mass as a function of observed stellar mass. Since the mean observed stellar mass in each bin is larger than the true value, the OSHMR is lower than the truth, for both large and small galaxies. In addition, large and small galaxies are affected differently from Eddington bias, resulting in a different OSHMR for the two subsamples. We stress out that this definition of large or small is only in relation to the _observed_ stellar mass of each galaxy. <figure><img src="content_image/1710.00007/x13.png"><figcaption>Figure 13: Top: average halo mass as a function of stellar mass of mock E(black dashed line), compared with the values inferred with our method (cyanshaded region). The blue and red circles show the average halo mass ofgalaxies in each stellar mass bin, as a function of the average value oflogM(obs)∗ in each bin. The arrows show the effect of Eddington bias on thetwo subsamples of galaxies. Eddington bias has a stronger effect on the sampleof smaller galaxies, leading to an apparent difference in halo mass. Bottom:correlation between halo mass and stellar mass density at fixed stellar mass.The dashed line shows the true value in mock E (no correlation), the cyan bandshows the 68% confidence region of the inference obtained with our method. Themagenta dots show the values one would obtain by combining perfect halo massmeasurements with the observed values of the average stellar mass and size ineach bin.</figcaption></figure> In the bottom panel of Figure 13, we plot the values of $\xi_{h}$ (the correlation between halo mass and stellar mass density at fixed stellar mass) one would infer by interpreting the difference in OSHMR between larger and smaller galaxies as real. As expected, the result is biased. Our method, on the other hand, is able to recover the correct answer. ## 6 Discussion and summary We introduced a Bayesian hierarchical inference method to infer the SHMR of a population of massive galaxies, generalized by allowing for a secondary dependence of halo mass on stellar mass density. Our method is similar in spirit to existing maximum-likelihood approaches, but it differs from them in two fundamental aspects. Firstly, it explicitly models the intrinsic scatter in the SHMR. The second, more subtle, difference, is that it makes a clear distinction between true and observed quantities. As shown in Section 2, making such a distinction and carefully modeling the effects of observational uncertainties is crucial in order to make accurate inferences. We tested the method on mock observations generated with semi-analytic models with increasing degrees of complexity, and showed that it can recover accurate halo masses to within $\sim 0.1$ dex over a decade in stellar mass. Although our mocks are somewhat simplistic in their nature (we have not allowed for departures from spherical symmetry in the mass distribution of our galaxies, for instance), our tests have allowed us to gauge the role of two important systematic effects: the role of satellites and that of the environment, which are shown to be small in the mass regime considered ($\log{M_{}}\gtrsim 11$). Our model is also able to infer the mass-concentration relation of the halos. Although miscentering between galaxies and their halos can introduce a $\sim 20\%$ bias on the average concentration, the correlation between concentration and halo mass appears to be robust with respect to the potential sources of systematic uncertainty explored in our work. This opens up interesting possibilities for the exploration of correlation between concentration and other galaxy properties at fixed halo mass. The concentration of a halo is believed to be tightly linked to its formation time (Navarro et al., 1997; Wechsler et al., 2002; Zhao et al., 2003), and is in principle sensitive to baryonic physics effects, such as adiabatic contraction. Measuring correlations between galaxy properties and halo mass then has ample potential for discovery. This is one of the possible uses of our inference method. Nevertheless, we point out that, like all methods based purely on weak lensing data, our model suffers from the mass-sheet degeneracy, which sets a fundamental limit to the ability to robustly determine the lens density profile. With our approach, we are artificially breaking the degeneracy by asserting a specific form for the density profile of the lenses. However, in order to robustly break this degeneracy, complementary information, such as magnification measurements, is needed. Our method offers various advantages over more traditional approaches, such as stacking and maximum-likelihood methods. It allows for the unbiased exploration of secondary dependences of halo mass on quantities other than stellar mass, such as size, as shown in Section 5. The same is not true if a simple stacked weak lensing method is used, as our test in subsection 5.8 clearly shows. Recently, Charlton et al. (2017) claimed a detection of a positive correlation between halo mass and galaxy size at fixed stellar mass, $M_{h}\propto R_{\mathrm{e}}^{\eta}$, with $\eta=0.42\pm 0.12$, obtained by comparing the stacked weak lensing signal around galaxies samples split by size at fixed observed stellar mass. It is possible that this correlation might in part be the result of the bias described qualitatively in subsection 2.2 and, quantitatively, in 5.8. In fact, the value of the correlation measured by Charlton et al. (2017) is very similar to the value found in our simulation when ignoring the effects of observational scatter in $M_{}$ (see bottom panel of Figure 13). Stacked weak lensing studies of the correlation between halo mass and galaxy colour at fixed stellar mass are also susceptible to the same bias. Red galaxies are on average more massive than blue galaxies: for instance, only a small fraction of galaxies at $\log{M_{}^{\mathrm{(obs)}}}>11$ are blue. This means that, when binning in observed stellar mass, Eddington bias affects red and blue galaxies differently: for the same value of $M_{}^{\mathrm{(obs)}}$, red galaxies have, on average, an intrinsically larger stellar mass compared to blue galaxies. Therefore, the measured difference in the average halo mass between red and blue galaxies at fixed observed stellar mass (Mandelbaum et al., 2016) might also be, in part, the result of the bias described in subsection 2.2. Moster et al. (2017) argued that the observed difference is a result of scatter. However, they did not specify whether they referred to intrinsic or observational scatter. As we explained in Section 2, this distinction is instead very important, as it leads to drastically different physical interpretations. Another advantage of our method, although not exploited in this work, is that it can be extended to allow for the inclusion of datasets other than weak lensing, such as X-ray emission from the diffuse halo gas, or galaxy kinematics. This is simply done by adding multiplicative terms to the likelihood, as long as it is possible to predict the corresponding observables from the model. The method is based on a crucial approximation: the assumption that all lenses are isolated, or, in other words, that for a given background source there is only one lens contributing to its lensing distortion. Although this approximation breaks down for satellite galaxies and for pairs of halos in close proximity, our tests on mocks show that we can still accurately recover the SHMR, if we restrict the analysis to lens galaxies more massive than $\sim 10^{11}M_{\odot}$, and sources located within $300\,\rm{kpc}$ in projection. We stress out that the necessity for the isolated lens assumption is purely technical, since it arises from the difficulty of exploring the highly-dimensional parameter space of the full problem, in which each source is lensed by every lens in the sample. We do not exclude the possibility that this difficulty could be overcome with the use of more sophisticated sampling techniques. A possible step forward can be made, with the tools already at hand, by abandoning the isolated lens assumption in favor of a less strong _isolated group_ assumption. When a few galaxies are found to lie in close projected distance from each other, it is possible to explore the parameter space of all the lenses in the “group” (which does not need to correspond to a physically bound association) self-consistently, in a finite computational time. This could, on the one hand, reduce the systematic uncertainties due to neglecting the contribution of neighboring halos to the lensing signal around each group galaxy, and, on the other hand, would get rid of the need to eliminate lenses from the sample in case of overlapping regions of influence. We leave the exploration of such models for future work. At the moment, our method is most suited to samples of massive ($\log{M_{}}\gtrsim 11$) galaxies, probing the group and cluster regime, for which the satellite fraction is small and galaxies are well separated in projected distance. ## acknowledgments We thank Surhud More and Jiaxin Han for helpful discussion. This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and the National Science Foundation under Grant No. NSF PHY17-48958. AS is partly supported by KAKENHI Grant Number JP17K14250. ## References * Aihara et al. (2017) Aihara H., et al., 2017, preprint, (arXiv:1704.05858) * Baltz et al. (2009) Baltz E. A., Marshall P., Oguri M., 2009, J. Cosmology Astropart. Phys., 1, 015 * Bartelmann (1996) Bartelmann M., 1996, A&A, 313, 697 * Behroozi et al. (2010) Behroozi P. S., Conroy C., Wechsler R. H., 2010, ApJ, 717, 379 * Brouwer et al. (2016) Brouwer M. M., et al., 2016, MNRAS, 462, 4451 * Charlton et al. (2017) Charlton P. J. L., Hudson M. J., Balogh M. L., Khatri S., 2017, preprint, (arXiv:1707.04924) * Coupon et al. (2012) Coupon J., et al., 2012, A&A, 542, A5 * Coupon et al. (2015) Coupon J., et al., 2015, MNRAS, 449, 1352 * Coupon et al. (2017) Coupon J., Leauthaud A., Kilbinger M., Medezinski E., 2017, swot: Super W Of Theta, Astrophysics Source Code Library (ascl:1707.007) * Diemer & Kravtsov (2014) Diemer B., Kravtsov A. V., 2014, ApJ, 789, 1 * Han et al. (2015) Han J., et al., 2015, MNRAS, 446, 1356 * Hearin et al. (2016) Hearin A., et al., 2016, preprint, (arXiv:1606.04106) * Hoekstra et al. (2001) Hoekstra H., et al., 2001, ApJ, 548, L5 * Hoekstra et al. (2004) Hoekstra H., Yee H. K. C., Gladders M. D., 2004, ApJ, 606, 67 * Hoekstra et al. (2005) Hoekstra H., Hsieh B. C., Yee H. K. C., Lin H., Gladders M. D., 2005, ApJ, 635, 73 * Hudson et al. (1998) Hudson M. J., Gwyn S. D. J., Dahle H., Kaiser N., 1998, ApJ, 503, 531 * Leauthaud et al. (2012) Leauthaud A., et al., 2012, ApJ, 744, 159 * Lieu et al. (2017) Lieu M., Farr W. M., Betancourt M., Smith G. P., Sereno M., McCarthy I. G., 2017, MNRAS, 468, 4872 * Limousin et al. (2005) Limousin M., Kneib J.-P., Natarajan P., 2005, MNRAS, 356, 309 * Macciò et al. (2008) Macciò A. V., Dutton A. A., van den Bosch F. C., 2008, MNRAS, 391, 1940 * Mandelbaum et al. (2005) Mandelbaum R., Tasitsiomi A., Seljak U., Kravtsov A. V., Wechsler R. H., 2005, MNRAS, 362, 1451 * Mandelbaum et al. (2006a) Mandelbaum R., Seljak U., Kauffmann G., Hirata C. M., Brinkmann J., 2006a, MNRAS, 368, 715 * Mandelbaum et al. (2006b) Mandelbaum R., Seljak U., Cool R. J., Blanton M., Hirata C. M., Brinkmann J., 2006b, MNRAS, 372, 758 * Mandelbaum et al. (2016) Mandelbaum R., Wang W., Zu Y., White S., Henriques B., More S., 2016, MNRAS, 457, 3200 * Mandelbaum et al. (2017) Mandelbaum R., et al., 2017, preprint, (arXiv:1705.06745) * More et al. (2011) More S., van den Bosch F. C., Cacciato M., Skibba R., Mo H. J., Yang X., 2011, MNRAS, 410, 210 * Moster et al. (2010) Moster B. P., Somerville R. S., Maulbetsch C., van den Bosch F. C., Macciò A. V., Naab T., Oser L., 2010, ApJ, 710, 903 * Moster et al. (2017) Moster B. P., Naab T., White S. D. M., 2017, preprint, (arXiv:1705.05373) * Navarro et al. (1997) Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493 * Newman et al. (2012) Newman A. B., Ellis R. S., Bundy K., Treu T., 2012, ApJ, 746, 162 * Parker et al. (2005) Parker L. C., Hudson M. J., Carlberg R. G., Hoekstra H., 2005, ApJ, 634, 806 * Schneider & Rix (1997) Schneider P., Rix H.-W., 1997, ApJ, 474, 25 * Schneider et al. (2015) Schneider M. D., Hogg D. W., Marshall P. J., Dawson W. A., Meyers J., Bard D. J., Lang D., 2015, ApJ, 807, 87 * Seitz & Schneider (1997) Seitz C., Schneider P., 1997, A&A, 318, 687 * Tanaka et al. (2017) Tanaka M., et al., 2017, preprint, (arXiv:1704.05988) * Tinker et al. (2013a) Tinker J. L., Leauthaud A., Bundy K., George M. R., Behroozi P., Massey R., Rhodes J., Wechsler R. H., 2013a, ApJ, 778, 93 * Tinker et al. (2013b) Tinker J. L., Leauthaud A., Bundy K., George M. R., Behroozi P., Massey R., Rhodes J., Wechsler R. H., 2013b, ApJ, 778, 93 * Velander et al. (2014) Velander M., et al., 2014, MNRAS, 437, 2111 * Wechsler et al. (2002) Wechsler R. H., Bullock J. S., Primack J. R., Kravtsov A. V., Dekel A., 2002, ApJ, 568, 52 * Wright & Brainerd (2000) Wright C. O., Brainerd T. G., 2000, ApJ, 534, 34 * Zhao et al. (2003) Zhao D. H., Jing Y. P., Mo H. J., Börner G., 2003, ApJ, 597, L9 * Zu & Mandelbaum (2015) Zu Y., Mandelbaum R., 2015, MNRAS, 454, 1161 * Zu & Mandelbaum (2016) Zu Y., Mandelbaum R., 2016, MNRAS, 457, 4360 * Zu & Mandelbaum (2017) Zu Y., Mandelbaum R., 2017, preprint, (arXiv:1703.09219) * de Vaucouleurs (1948) de Vaucouleurs G., 1948, Annales d’Astrophysique, 11, 247
0808.0743	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 28696, "num_imgs": 2, "llama3_tokens_count": 9044 }	[ "content_image/0808.0743/x1.png", "content_image/0808.0743/x2.png" ]	# Kerr nonlinearities and nonclassical states with superconducting qubits and nanomechanical resonators F. L. Semião Departamento de Física, Universidade Estadual de Ponta Grossa - Campus Uvaranas, 84030-900 Ponta Grossa, Paraná, Brazil K. Furuya Institute of Physics “Gleb Wataghin”, P.O. Box 6165, University of Campinas - UNICAMP, 13083-970 Campinas, SP, Brazil G. J. Milburn School of Physical Sciences - University of Queensland, Brisbane, Queensland 4072, Australia ###### Abstract We propose the use of a superconducting charge qubit capacitively coupled to two resonant nanomechanical resonators to generate Yurke-Stoler states, i.e. quantum superpositions of pairs of distinguishable coherent states 180${}^{\circ}$ out of phase with each other. This is achieved by effectively implementing Kerr nonlinearities induced through application of a strong external driving field in one of the resonators. A simple study of the effect of dissipation on our scheme is also presented, and lower bounds of fidelity and purity of the generated state are calculated. Our procedure to implement a Kerr nonlinearity in this system may be used for high precision measurements in nanomechanical resonators. Suggested keywords: nano-eletromechanical systems, charge qubits, nonclassical states, Kerr Hamiltonian pacs: 85.85.+j,42.50.Dv ## I Introduction Quantum nonlinear dynamics is an important topic in physics. In quantum optics, nonlinear interactions have been widely used to generate nonclassical field states, such as squeezed or sub-Poissonian light [1]. A special class of optical nonlinearity results in an intensity dependent phase shift, commonly known as the Kerr effect. In the single mode case, the time evolution of an initial coherent state, under the influence of such a Kerr medium and very low loss, will evolve into a quantum superposition of two coherent states 180${}^{\circ}$ out of phase with each other. This was first discovered by Yurke and Stoler [2], and since then such states have been called Yurke-Stoler states. A single-mode Kerr medium preserves the photon statistics but modifies the quadrature uncertainties generally leading to squeezing [1]. There is great interest in observing this quantum nonlinear couplings in solid state systems. This would allows us to deepen our current understanding of the classical-quantum frontier by studying how long can superpositions of mesoscopically distinct states survive in such systems. Some interesting proposals involving nanomechanical resonators have been published during the last years. In one scheme [3], the use of a time dependent drive in a Cooper pair box coupled to a nanomechanical resonator is shown to generate a number of nonlinear Hamiltonians for the latter. By parametrically driving a nanomechanical resonator capacitively coupled to a superconducting coplanar waveguide one can generate interesting nonlinear Hamiltonians suitable for generation and detection of squeezed states as proposed in [4]. In this system, entangled states in temperatures up to tens of milliKelvin may be achieved as discussed in [5]. Nanomechanical oscillators have also been shown to be feasible for coupling to other important physical systems besides Cooper pair boxes or microwave fields of coplanar wave guides. Nanomechanical resonators may, for instance, be coupled to Bose-Einstein condensates [6], trapped ions [7; 8] or spin degrees of freedom of a sample of neutral atoms in the gas phase [9]. In this paper, we propose a theoretical scheme to engineer Kerr Hamiltonians using a system composed of a Cooper pair box capacitively coupled to two resonant nanomechanical resonators. We show in Sec. II that such nonlinear Hamiltonians can be achieved in a dispersive regime by appropriately choosing the system’s parameters and by using a properly tuned strong classical field in one of the resonators. The integration of superconducting qubits with nanoresonators is an important topic and has been previously considered in [10; 11; 14; 15; 12; 13]. We start from a well known Hamiltonian describing the interaction between a charge qubit and two resonant nanomechanical resonators in a quantum regime [14; 16; 17; 18], and we then include an external driving in one of the oscillators. By considering the regime of intense driving, we show that a nonlinear Kerr-type effective Hamiltonian may be obtained. This Hamiltonian is induced by the common coupling of the resonators with the qubit and intense external driving. This is the central result of this paper, and as an application, we show in Sec. III how to generate the Yurke-Stoler state in the normal modes of the nanomechanical resonators. We also discuss the zero temperature decoherence in a particular regime of relaxation, and evaluate both the fidelity and the purity of the generated superposition state. Finally, we would like to point out that the ability to implement Kerr nonlinearities in nanomechanical resonators has recently been shown to find applications also in high precision measurements. In a recent paper [19], Woolley et al. have proposed a new protocol for high precision measurement in a nanomechanical resonator that makes explicit use of such nonlinearities. This might be a potential application for the results presented in this paper. In Sec. IV we draw some conclusions. ## II The model and the Kerr type interactions The simplest charge qubit, the Cooper pair box (CPB), consists of a small superconducting island with an excess number, $n$, of Cooper-pairs, connected by a tunnel junction (capacitance $C_{J}$ and Josephson coupling $E_{J}$) to a superconducting electrode. External control is achieved by the application of a voltage gate $V_{g}$ coupled to the CPB via a gate capacitor with capacitance $C_{g}$. More details can be found in the review [20]. For specific qubit proposals and decoherence analysis see [21]. In our study, we will assume that the CPB is coupled capacitively to two nano-eletromechanical systems (NEMS) [18], as depicted in Fig.(1). <figure><img src="content_image/0808.0743/x1.png"><figcaption>Figure 1: Two nanomechanical resonators (with lowering operators a and b) arecapacitively coupled to a Cooper pair box (CPB). One of the oscillators isdriven by a classical force.</figcaption></figure> In the two level approximation for the CPB, the capacitive coupling between the qubit and two NEMS is described by the Hamiltonian [18] \[H = \hbar\omega a^{\dagger}a+\hbar\omega b^{\dagger}b+\frac{\hbar \omega_{0}}{2}\bar{\sigma}_{z}+\frac{\hbar\bar{\Delta}}{2}\bar{\sigma}_{x}+ \hbar\lambda_{1}(a+a^{\dagger})\bar{\sigma}_{z}\] (1) \[+\hbar\lambda_{2}(b+b^{\dagger})\bar{\sigma}_{z}+\hbar g(ae^{i \omega_{e}t}+a^{\dagger}e^{-i\omega_{e}t}),\] where $a,a^{\dagger}$ are the raising and lowering operators for the driven NEMS [22], $b,b^{\dagger}$ are the raising and lowering operators for the other NEMS with the same resonance frequency $\omega$, and $g$ represents the amplitude of the external nanomechanical drive (frequency $\omega_{\rm{e}}$). The parameters appearing in (1) are given by \[\hbar\omega_{\rm{0}} = -4E_{c}(1-2n_{g}),\] (2) \[\hbar\bar{\Delta} = -2E_{J}\cos(\pi\phi/\phi_{0}),\] (3) \[\hbar\lambda_{i} = e\frac{V_{g}C_{g,i}}{C_{\Sigma}d_{i}}\sqrt{\frac{\hbar}{2m\omega }},\] (4) where $C_{g,i}$ is the capacitance between the CPB and $i$-th nanomechanical bias gate, $C_{\Sigma}$ is the total capacitance, $d_{i}$ is the distance between the $i$-th nanomechanical bias gate and the CPB, and $m$ is the mass of the NEMS. The couplings can be made different by varying, for example, the distances $d_{i}$ or applying DC voltages to the resonators [23]. Our goal now is to show how the application of the external driving field may be used to engineer nonclassical states. We first make a rotation of the qubit to new variables $\bar{\sigma}_{\alpha}\rightarrow\sigma_{\alpha}$: \[H = \hbar\omega a^{\dagger}a+\hbar\omega b^{\dagger}b+\hbar g(ae^{i \omega_{\rm{e}}t}+a^{\dagger}e^{-i\omega_{\rm{e}}t})+\] (5) \[\frac{\hbar\bar{\Omega}}{2}\sigma_{z}+\hbar\lambda_{1}(a+a^{ \dagger})(\cos\theta\sigma_{z}-\sin\theta\sigma_{x})\] \[+\hbar\lambda_{2}(b+b^{\dagger})(\cos\theta\sigma_{z}-\sin\theta \sigma_{x}),\] where \[\cos\theta = \frac{\omega_{\rm{0}}}{\bar{\Omega}},\] (6) \[\sin\theta = \frac{\bar{\Delta}}{\bar{\Omega}},\] (7) \[\bar{\Omega} = (\omega_{\rm{0}}^{2}+\bar{\Delta}^{2})^{1/2}.\] (8) Now, moving to a rotating frame with frequency $\omega_{\rm{e}}$, and setting $\bar{\Omega}=\omega_{\rm{e}}$ and $\delta=\omega-\omega_{\rm{e}}$, we get \[H = \hbar\delta a^{\dagger}a+\hbar\delta b^{\dagger}b+\hbar g(a+a^{ \dagger})+\hbar\lambda_{1}(ae^{-i\omega_{e}t}+\] (9) \[a^{\dagger}e^{i\omega_{e}t})[\cos\theta\sigma_{z}-\sin\theta( \sigma_{+}e^{i\omega_{e}t}+\sigma_{-}e^{-i\omega_{e}t})]+\] \[\hbar\lambda_{2}(be^{-i\omega_{e}t}+b^{\dagger}e^{i\omega_{e}t})[ \cos\theta\sigma_{z}-\sin\theta(\sigma_{+}e^{i\omega_{e}t}\] \[+\sigma_{-}e^{-i\omega_{e}t})].\] We can now make the rotating wave approximation, to get the interaction picture Hamiltonian, \[H = \hbar g(ae^{-i\delta t}+a^{\dagger}e^{i\delta t})-\hbar\lambda_{1 }\sin\theta(a\sigma_{+}e^{-i\delta t}+a^{\dagger}\sigma_{-}e^{i\delta t})\] (10) \[\mbox{}-\hbar\lambda_{2}\sin\theta(b\sigma_{+}e^{-i\delta t}+b^{ \dagger}\sigma_{-}e^{i\delta t})\] An interesting situation appears when one takes the dispersive approximation ($\|\delta\|\gg\lambda_{1},\lambda_{2},g$) for the above Hamiltonian (applying similar methods to those described in [24]). In this regime, the Hamiltonian (10) may be approximated by \[H=\hbar\Omega a^{\dagger}a\sigma_{z}+\hbar\chi b^{\dagger}b \sigma_{z}+\hbar\Delta\sigma_{x}+\hbar r(a^{\dagger}b+ab^{\dagger})\sigma_{z},\] (11) where \[\Omega = -\frac{\lambda_{1}^{2}}{\delta}\sin^{2}\theta,\] (12) \[\chi = -\frac{\lambda_{2}^{2}}{\delta}\sin^{2}\theta,\] (13) \[\Delta = \frac{g\lambda_{1}}{\delta}\sin\theta,\] (14) \[r = -\frac{\lambda_{1}\lambda_{2}}{\delta}\sin^{2}\theta.\] (15) The Hamiltonian (11) can be diagonalized by using new bosonic composite operators $a_{1}=(\cos{\frac{\gamma}{2}}a+\sin{\frac{\gamma}{2}}b)$ and $a_{2}=(-\sin{\frac{\gamma}{2}}a+\cos{\frac{\gamma}{2}}b)$ with appropriate choice for $\gamma$. We set from now on $\Omega=\chi$, i.e. $\lambda_{1}=\pm\lambda_{2}$, since for this case the simple choice $\gamma=\frac{\pi}{2}$ solves the problem. In terms of the new operators $a_{1}=2^{-1/2}(a+b)$ and $a_{2}=2^{-1/2}(a-b)$, $H$ is written (setting $\hbar=1$) as \[\tilde{H}_{+(-)}=\xi a^{\dagger}_{1(2)}a_{1(2)}\sigma_{z}+\Delta\sigma_{x}\] (16) where $\zeta=-\frac{2\lambda_{1}^{2}}{\delta}\sin^{2}\theta$. We will now show that in the regime $\|\Delta\|\gg\|\zeta\|$, a Kerr type Hamiltonian can be generated. From $\zeta=-\frac{2\lambda_{1}^{2}}{\delta}\sin^{2}\theta$, we see that $\|\Delta\|\gg\|\zeta\|$ implies that we must have $g\gg 2\lambda_{i}\sin\theta$ ($i=1\,{\rm{or}}\,2$), i.e a strong driving $(g\gg\lambda_{i})$. To make this clear, lets us assume $\lambda_{1}=\lambda_{2}$ and $\Delta>0$. By transforming $H_{+}$ to an interaction picture with respect to $\Delta\sigma_{x}$, one obtains \[\tilde{\mathcal{V}}_{+}(t)=\frac{\zeta}{2}\{a_{1}^{\dagger}a_{1}[ (\sigma_{z}-i\sigma_{y})e^{2i\Delta t}+(\sigma_{z}+i\sigma_{y})e^{-2i\Delta t} ]\}.\] Now, if one defines the operator $A=a_{1}^{\dagger}a_{1}(\sigma_{z}-i\sigma_{y})$ and the constant $\lambda=\frac{\zeta}{2}$, the above Hamiltonian will read $\tilde{\mathcal{V}}_{+}(t)=\lambda(Ae^{i2\Delta t}+A^{\dagger}e^{-i2\Delta t})$. It can be shown [24] that for $\Delta>>\lambda$, the effective Hamiltonian $\tilde{\mathcal{V}}_{+}^{\,{\rm{eff}}}=\hbar\frac{\lambda^{2}}{2\Delta}[A,A^{ \dagger}]$ can be used. By evaluating this commutator, one finds \[\tilde{\mathcal{V}}_{+}^{\,{\rm{eff}}}=\mu(a_{1}^{\dagger}a_{1})^ {2}\sigma_{x}\] (18) where $\mu=\zeta^{2}/2\Delta$. Remarkably, this Hamiltonian mimics the single mode Kerr effect. If the CPB is prepared in an eigenstate of $\sigma_{x}$, the bosonic mode will follow a decoupled evolution under the nonlinear Hamiltonian $\mu(a_{1}^{\dagger}a_{1})^{2}$. Going back to the definitions, one can see that the magnitude of the nonlinearity $\mu$ is in fact controlled by the system parameters $\lambda_{1}$ (coupling constant for the interaction of resonator $a$ with the qubit), $g$ (related to the amplitude of the classical driving), and $\delta$ (detuning between driving field and nanoresonators). Thus, it is possible to control the the intensity of the present Kerr type effect, which is always important in the applications. ## III Yurke-Stoler state and inclusion of dissipation in the NEMS Consider now the initial preparation, $\|\psi(0)\rangle=\|\alpha\rangle_{a}\|\alpha\rangle_{b}\|+\rangle_{x}$, i.e both resonators in coherent states with the same amplitude $\alpha$, and the CBP in an eigenstate of $\sigma_{x}$ with eigenvalue equal to one. In the transformed space of the composite modes $a_{1}$ and $a_{2}$, this initial state becomes $\|\tilde{\psi}(0)\rangle=\|\sqrt{2}\alpha\rangle_{1}\|0\rangle_{2}\|+\rangle_{x}$. It means that the composite mode-$1$ is initially in a coherent state $\|\alpha_{1}=\sqrt{2}\alpha\rangle_{1}$, mode-$2$ in the vacuum state $\|\alpha_{2}=0\rangle_{2}$, and the qubit in the eigenstate of $\sigma_{x}$ corresponding to the eigenvalue $1$. For this initial condition, Hamiltonian (18) leads to the following time evolved state: \[\|\tilde{\psi}_{I}(t)\rangle=\left[e^{-\|\alpha_{1}\|^{2}/2}\sum_{n= 0}^{\infty}\frac{(\alpha_{1})^{n}}{\sqrt{n!}}e^{-it\mu n^{2}}\|n\rangle_{1} \right]\|0\rangle_{2}\|+\rangle_{x},\] with $\alpha_{1}=\sqrt{2}\alpha$. For an interaction time $t_{I}$ such that $\mu t_{I}=\pi/2$, the state (III) evolves to $\|\tilde{\psi}_{I}(t_{I})\rangle=\|{\rm{YS}}\rangle_{1}\|0\rangle_{2}\|+\rangle,$ where \[\|{\rm{YS}}\rangle_{1}=\frac{\|\alpha_{1}\rangle_{1}+i\|-\alpha_{1} \rangle_{1}}{\sqrt{2}}\] (20) is the Yurke-Stoler state. We remark that no measurement whatsoever was needed to generate this state, so this scheme is deterministic. If initially one prepares $\|\psi(0)\rangle=\|\alpha\rangle_{a}\|-\alpha\rangle_{b}\|+\rangle_{x}$ and choose $\lambda_{1}=-\lambda_{2}$, a Yurke-Stoler state is generated in the mode-$2$. Many applications for superpositions of coherent states have been suggested in the quantum optics and quantum information literature [25], along with a considerable variety of generation protocols [26]. Since we have performed a perturbation approach of the problem (effective Hamiltonians), it is now important to make a brief discussion about the experimental values of the parameters and the feasibility of the regimes we used. From (9) to (10), we have realized a rotating wave approximation, and this is justified when $\lambda_{1},\lambda_{2}\ll\omega,\bar{\Omega},g$. According to experimental reference [17], it is currently possible to achieve $\omega/2\pi=1.0$ GHz. For charge qubits, ordinary values for $\bar{\Omega}$ are also about a few gigahertz [27]. In principle, the external driving $g$ may also be of the same order or even stronger than $\omega$ and $\bar{\Delta}$. We have also demanded $\bar{\Omega}=\omega_{e}$, and this means that the frequency of the drive field is also of a few gigahertz. Taking all these into account, we see that the coupling constants $\lambda_{1}$ and $\lambda_{2}$ must be at most around a few megahertz for the rotating wave approximation to be valid. This seems not be a problem since such coupling constants may be tuned by changing the distance between the CPB and the nanoresonators or through additional DC voltages on the resonator. When going from (10) to (11), we took the dispersive regime that demands $\lambda_{1},\lambda_{2}\ll\delta$. Again, this might not be a problem since $\lambda_{i}$ depends on $d_{i}$. Finally, our last approximation corresponds to the regime of strong driving $\lambda_{1},\lambda_{2}\ll g$. This seems to be easy to achieve since $g$ is externally controlled via a driving gate and do not depend on the fabrication features of the CPB or the resonators. It is well known that superposition states of this kind are easily corrupted in noisy or dissipative environment. For this reason, it is important to find a way to evaluate, at least approximately, how our generation protocol is affected by such irreversible effects. A complete treatment of the problem would involve modeling the qubit decoherence and relaxation as well as different dissipative effects in the nanomechanical resonators. It is not our intention here to account for all these noise mechanisms. Instead, we will present one simple situation which allows of a very illustrative _exact solution_. We consider the case in which both NEMS (with lowering operators $a$ and $b$) lose energy to their surrounding with decay rates $\kappa_{a}$ and $\kappa_{b}$, respectively. For simplicity, we will not include the qubit decoherence and relaxation. This is justified if the qubit decoherence times are longer compared to the resonators ones. At present, the charge qubits are notably more robust against decoherence and relaxation than the nanomechanical resonators. In this situation, and considering $\lambda_{1}=\lambda_{2}$, the system master equation at zero temperature, when expressed in terms of the mode operators, is written as \[\frac{\partial\tilde{\rho}_{I}}{\partial t} = -i[\mu(a_{1}^{\dagger}a_{1})^{2}\sigma_{x},\tilde{\rho}_{I}]\] (21) \[+\frac{\kappa_{a}+\kappa_{b}}{4}(2a_{1}\tilde{\rho}_{I}a_{1}^{{ \dagger}}-a_{1}^{{\dagger}}\hat{a_{1}}\tilde{\rho}_{I}-\tilde{\rho}_{I}a_{1}^{ {\dagger}}a_{1})\] \[+\frac{\kappa_{a}+\kappa_{b}}{4}(2a_{2}\tilde{\rho}_{I}a_{2}^{{ \dagger}}-a_{2}^{{\dagger}}\hat{a_{2}}\tilde{\rho}_{I}-\tilde{\rho}_{I}a_{2}^{ {\dagger}}a_{2})-\] \[\frac{\kappa_{a}-\kappa_{b}}{4}[(a_{1}^{\dagger}a_{2}+a_{2}^{ \dagger}a_{1})\tilde{\rho}_{I}+\tilde{\rho}_{I}(a_{1}^{\dagger}a_{2}+a_{2}^{ \dagger}a_{1})]\] \[+\frac{\kappa_{a}-\kappa_{b}}{4}[2(a_{1}+a_{2})\tilde{\rho}_{I}(a _{1}^{\dagger}+a_{2}^{\dagger})].\] We can see that the master equation contains extra terms due to the transformation to normal modes. The full treatment for arbitrary $\kappa_{a}$ and $\kappa_{b}$ makes analytical progress quite difficult [28] and a numerical calculation may be presented elsewhere. However, the simple regime in which both resonators decay with similar rates can be readily investigated. In this case, $\|\kappa_{a}+\kappa_{b}\|\gg\|\kappa_{a}-\kappa_{b}\|$, and we can drop the terms proportional to $(\kappa_{a}-\kappa_{b})$. This assumption is realistic here since both resonators are assumed to be identical (same mass and natural frequencies). Even when this is not exactly the case, the quantities calculated below under the assumption of $(\kappa_{a}\approx\kappa_{b})$ will, at least, serve as an upper bound to the case in which the dissipation rates are disparate. We calculate here the degree of purity and the fidelity of the state, generated in such a noisy environment, as compared to the Yurke-Stoler state obtained in the ideal unitary case. Therefore, we need to take the same initial preparation used in the ideal case i.e. $\|\psi(0)\rangle=\|\alpha\rangle_{a}\|\alpha\rangle_{b}\|+\rangle_{x}$. As this implies that mode-$2$ will be in the vacuum state, we need to consider only the terms in (21) that contain operators for mode-$1$. The master equation (21) reduces to, \[\frac{\partial\tilde{\rho}_{I}}{\partial t} = -i[\mu(a_{1}^{\dagger}a_{1})^{2},\tilde{\rho}_{I}]+\kappa(2a_{1} \tilde{\rho}_{I}a_{1}^{{\dagger}}-a_{1}^{{\dagger}}\hat{a_{1}}\tilde{\rho}_{I}\] (22) \[-\tilde{\rho}_{I}a_{1}^{{\dagger}}a_{1}),\] where $\kappa=(\kappa_{a}+\kappa_{b})/4$. An exact solution for (22) using the Q-function approach is presented in [29], but we will use a recent solution obtained directly for the density operator [30] which allows us to readily obtain the purity $P={\rm{Tr}}[\rho^{2}]$, and fidelity $F=\langle{\rm{YS}}\|\rho\|{\rm{YS}}\rangle$. According to [30], the solution of (22) is \[\tilde{\rho}_{I}(t) = \left\{\sum_{k,n,m=0}^{\infty}\tilde{\rho}_{n+k,m+k}(0)e^{-i\mu t (n^{2}-m^{2})-\kappa t(n+m)}\sqrt{\frac{(n+k)!(m+k)!}{n!m!}}\left[\frac{1-e^{- 2i\mu t(n-m)-2\kappa t}}{2i\mu(n-m)+2\kappa}\right]^{k}\frac{(2\kappa)^{k}}{k! }\|n\rangle_{1}{}_{1}\langle m\|\right\}\] (23) \[\otimes\|0\rangle_{2}{}_{2}\langle 0\|\otimes\|+\rangle_{x}{}_{x} \langle+\|,\] where $\tilde{\rho}_{n,m}(0)$ are the (Fock) matrix elements of the initial density matrix of mode-1. In Fig.(2), we show the decay of fidelity (solid) and purity (dotted), as a function of the dimensionless parameter $\Gamma=\kappa/\mu$ for $\alpha=2$, at the time for which the Yurke-Stoler state arises, i.e., $\mu t_{I}=\pi/2$. We can see that the fidelity is quite high ($F>0.99$) for $\Gamma\leq 10^{-3}$. As expected, Fig.2 reveals that the purity is more affected with increasing $\Gamma$ than is the fidelity. However, it also presents satisfactory values for $\Gamma\approx 10^{-3}$ ($P\approx 0.99$). For more realistic values such as $\Gamma\approx 10^{-2}$, we find $F>0.95$ and $P>0.90$. <figure><img src="content_image/0808.0743/x2.png"><figcaption>Figure 2: Fidelity (solid) and purity (dotted) in function of thedimensionless parameter Γ=κ/μ for α=2, at the moment the Yurke-Stoler statewould be perfectly generated in the ideal lossless case.</figcaption></figure> Finally, a few words about detection of superpositions of pairs of distinguishable coherent states is in order. This is an important topic, and several methods for detecting these states have already been proposed in the literature [31; 32; 33]. Among them, it seems that the most suitable method for the system treated here is the one presented in [32], whereby motional states of a single trapped ion have been experimentally determined. This method relies upon implementation of displacement operators and Jaynes-Cummings interactions to determine both the density matrix in the number state basis and the Wigner function. Thi ion techniques could be an alternative to detect the Yurke-Stoler state proposed in this paper, but it should be remarked that a CPB coupled to two NEMS has not yet been operated in strong coupling regime. ## IV Conclusions To summarize, we have proposed a theoretical scheme to engineer a nonlinear Kerr Hamiltonian using superconducting charge qubits and nanoresonators. We have shown how such systems may be used to mimic a Kerr Hamiltonian. The formation of the Yurke-Stoler states in the composite mode of both resonators occurs naturally at an appropriate interaction time without needing to make a measurement on the system. For the case in which both resonators have equal decay rates, a simple exact expression for the total density matrix was derived. The present treatment, while not complete (more complex models of dissipation and noise could be considered), serves as an upper bound for the case in which the qubit decoherence can be neglected. In this context, we have shown that the fidelity of the generated state can high for moderate values of the decay constants. As a final remark, recently Woolley _et al._[19] have proposed a new protocol for high precision measurement in a nanomechanical resonator that makes explicit use of a Kerr nonlinearity. The method of the present paper could enable the use of linear nanomechanical resonators for such measurements instead of the intrinsically nonlinear nanomechanical resonators assumed in [19]. _Acknowledgments:_ FLS wishes to thanks F. Brito for helpful discussions and KF wishes to thank the Australian Research Council Centre for Quantum Computer Technology at The University of Queensland for hosting her visit to Brisbane. GJM would like to acknowledge the support of the Australian Research Council, FLS to FAPESP (Brazil) and CNPq (Brazil), and KF to CNPq (Brazil). ## References * (1) R. Loudon and P. L. Knight, J. Mod. Optics. 34, 709 (1987); A. D. Wilson-Gordon, V. Bužek, and P. L. Knight, Phys. Rev. A 44, 7647 (1991). * (2)B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986). * (3) K. Jacobs, Phys. Rev. Lett. 99, 117203 (2007). * (4) M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, Phys. Rev. A 78, 062303 (2008) * (5) D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, Phys. Rev. A 76, 042336 (2007). * (6) P. Treutlein, D. Hunger, S. Camerer, T. W. Hänsch, and J. Reichel, Phys. Rev. Lett. 99, 140403 (2007). * (7) L. Tian and P. Zoller, Phys. Rev. Lett 93, 266403 (2004). * (8) W. K. Hensinger, D. W. Utami, H.-S. Goan, K. Schwab, C. Monroe, and G. J. Milburn, Phys. Rev. A 72, 041405(R) (2005). * (9) Y. -J. Wang, M. Eardley, S. Knappe, J. Moreland, L. Hollberg, and John Kitching, Phys. Rev. Lett. 97, 227602 (2006). * (10)A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004); M. R. Geller and A. N. Cleland, Phys. Rev. A 71, 032311 (2005). * (11) A. D. Armour, M. P. Blencowe, K. C. Schwab, Phys. Rev. Lett. 88, 148301 (2002). * (12) Y. Hu, Y. -F. Xiao, Z. -W. Zhou, and G. -C. Guo, Phys. Rev. A 75, 012314 (2007). * (13) M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm and E. Solano, Phys. Rev. B 78, 104508 (2008). * (14) E. K. Irish and K. Schwab, Phys. Rev. B 68, 155311 (2003). * (15) E. K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Phys. Rev. B 72, 195410 (2005). * (16) K. C. Schwab and M. L. Roukes, Phys. Today 58, 36 (2005). * (17) X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes, Nature 421 496 (2003). * (18) S. Bose and G. S. Agarwal, New J. Phys. 8 (2006) 34. * (19) M. J. Woolley, G. J. Milburn, C. M. Caves, New J. Phys. 10, 125018 (2008). * (20)Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). * (21)J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, S. Lloyd, Science 285, 1036 (1999); Y. Nakamura, Yu. A. Pashkin, J. S. Tsai, Nature 398, 786 (1999); J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens, Nature 406, 43 (2000); R. Koch, J. Rozen, G. Keefe, F. Milliken, C. Tsuei, J. Kirtley, and D. DiVincenzo, Phys. Rev. B 72, 092512 (2005); G. Burkard and F. Brito, Phys. Rev. B 72, 054528 (2005). * (22) R. Ruskov, K. Schwab, and A. N. Korotkov, IEEE Trans. Nanothec.,4, 132 (2005). * (23) A. Naik, O. Buu, M. D. LaHaye, A. D. Clerk, M. P. Blencowe, K. C. Schwab, Nature 443 (2006) 193. * (24) W. P. Schleich, _Quantum Optics in Phase Space_ (Wiley-VCH, Weinhein, 2001); C. Gerry and P. Knight, _Introductory Quantum Optics_ (Cambridge University Press, Cambridge, 2005; F. L. Semião, J. Opt. B: At. Mol. Opt. Phys. 41, 081004 (2008). * (25) B. C. Sanders, Phys. Rev. A 45, 6811 (1992); S. J. van Enk and O. Hirota, Phys. Rev. A 64, 022313 (2001); X. Wang, Phys. Rev. A 64, 022302 (2001); H. Jeong, M. S. Kim, and J. Lee, Phys. Rev. A 64, 052308 (2001); H. Jeong and M. S. Kim, Phys. Rev. A 65, 042305 (2002). * (26) J. F. Poyatos, J. I. Cirac, R. Blatt, and P. Zoller, Phys. Rev. A 54, 1532 (1996); C. C. Gerry, Phys. Rev. A 59, 4095 (1999); H. Jeong, M. S. Kim, T. C. Ralph, and B. S. Ham, Phys. Rev. A 70, 061801(R) (2004); F. L. Semião and A. Vidiella-Barranco, Phys. Rev. A 71, 065802 (2005); A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, Nature 448, 784 (2007); M. Takeoka and M. Sasaki, Phys. Rev. A 75, 064302 (2007). * (27) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. -S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 163 (2004). * (28) A. R. Bosco de Magalhães, S. G. Mokarzel, M. C. Nemes, M. O. Terra Cunha, Physica A, 341, 234 (2004). * (29) G. J. Milburn and C. A. Holmes, Phys. Rev. Lett. 56, 2237 (1986); D. J. Daniel and G. J. Milburn, Phys. Rev. A 39, 4628 (1989). * (30) H. Moya-Cessa, Phys. Rep. 432, 1 (2006); R. Juárez-Amaro, J. M. Vargas-Martínez and H. Moya-Cessa, Laser Physics 18, 344 (2008). * (31) M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. A 45, 5193 (1992). * (32) D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996). * (33) D. Vitali, P. Tombesi, Ph. Grangier, Appl. Phys. B 64, 249 (1997); Y. P. Huang and M. G. Moore, Phys. Rev. A. 73, 023606 (2006).
1001.4994	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 101439, "num_imgs": 0, "llama3_tokens_count": 39213 }	[]	# Vector and Axial anomaly in the Thirring-Wess model Pierluigi Falco Institute for Advanced Study, Princeton, New Jersey 08540 ###### Abstract We study the 2D Vector Meson model introduced by Thirring and Wess, that is to say the Schwinger model with massive photon and massless fermion. We prove, with a renormalization group approach, that the vector and axial Ward identities are broken by the Adler-Bell-Jackiw anomaly; and we rigorously establish three widely believed consequences: a) the interacting meson-meson correlation equals a free boson propagator, though the mass is additively renormalized by the anomaly; b) the anomaly is quadratic in the charge, in agreement with the Adler-Bardeen formula; c) the fermion-fermion correlation has an anomalous long-distance decay. ## 1 Introduction Since early days of Quantum Field Theory, (QFT), 1+1 dimensional models have been widely investigated as example of relativistic fields with local interaction: the Thirring and the Schwinger models, [40], [36], are probably the most celebrated cases. Although these systems are so simplified to have an exact solution, they nonetheless suggest ideas and mathematical tools to approach realistic theories in four space-time dimensions. One of the aspects that are certain relevant also in higher dimensions is the role played by two Ward Identities (Wi) related to the invariance of the Lagrangian under vector and axial transformations. In agreement with the general Adler-Bell-Jackiw mechanism, [2], [5], the vector and the axial symmetries are _broken_ at quantum level by the Wi _anomaly_. Many salient features of QFT are related to such an anomaly; let’s consider some of them. In any dimensions, by the Adler-Bardeen (AB) argument, [3], the anomaly is expected to be _linear_ in the bare coupling, i.e. not renormalized in loop perturbation theory at any order bigger than one; besides, it is a _topological quantity_, i.e. it doesn’t depend on the choice of the cutoff (to some extent). The anomaly is also responsible for the _anomalous dimension_ in the distance decay of the fermion correlations, [26]. And, finally, when a fermion field and a gauge photon interact through a _minimal coupling_, as for instance in the Schwinger model, the anomaly also represents a dynamically-generated physical mass for the photon field, [36]. In two dimensions the anomaly is said “mild”: although the formal Wi are broken, the anomaly has the only consequence of changing the normalization of the vector and axial currents, that remain conserved. Therefore using the Wi it was possible to find _formal exact solutions_ of the Schwinger-Dyson equation, (SDe), of the Thirring, the Schwinger and related models, in this way computing all the correlations: see [26], [15], [39], [24], [27] and [29]. An alternative approach - still related to the mildness of the anomaly - is the _bosonization_, i.e. the equivalence of the the fermion currents with boson free fields. This fact is behind the solution of the Thirring and the Schwinger model, [22], [20], in the path integral formalism. The above analysis - and much more, see chapters X and XII of [1] - has been based on formal methods and sound assumptions only. Rigorous results are few; though, at this stage, the reader might have few interest left for them. It is worth explaining, then, why the matter is still very tangled. In fact, earlier formal solutions of the Thirring and the Schwinger models were incomplete or incorrect, as pointed out in [24], [42], [1]. And, after all, the distinction among _formally correct_, _incorrect_ or _incomplete_ solutions may be quite faint. Wightman, [42], to put order in the confusion of the results, took the approach of considering any set of correlation functions, no matter how they were obtained, as trial theories; and then of promoting them to QFT if they satisfied certain axioms. This viewpoint has been moderately prolific (see for example [19], [18], [17]). There is at least one clear issue with it: being based of some sort of exact solution of the correlation functions, it is limited to few special models. A massive fermion field, for example, or an additional interactions that, in the renormalization group language, is irrelevant, would represent a severe obstruction to the method. The recent approach to the Thirring model in [6], [7] is different. We derived the correlations from the Lagrangian, so that in the massless case we obtained the exact solution, while in the massive case, where no exact solution is known, we can still prove the axioms. But, most of all, the major advantage of using the Lagrangian (as opposed to correlations) as starting point, is that it keeps track of the relationship between a special class of statistical mechanical problems - such as the Eight-Vertex models and the XYZ quantum chain - and their scaling limit, that turns out to be the Thirring model; in this way we were able prove some scaling formulas for non solvable models, [8], [13]. The heart of the technique is the control of the vanishing of the “beta function” for the effective coupling: this route, that have been useful also for other statistical models, was opened in [9] by exploiting the exact solution of the Luttinger model; and reached its final form, based on Wi and independent from any exact solution, in [11], [12]. The extension of the above techniques to the Schwinger model poses some serious issues because the infrared divergences related with the massless photon; therefore we take up a problem of intermediate difficulty, the theory of the Vector Meson, [41], i.e. a Schwinger model in which a photon mass is added by hand. This model still preserves some interests, because we can prove that the bare photon mass is renormalized into the physical mass just by an additive constant, exactly as expected in the Schwinger model; besides, we can establish the anomalous dimension of the fermion field. Finally, we can prove the AB-formula in a genuine example of QFT, i.e. removing the all the cutoffs of the theory: in this sense this paper is a completion of the objective of [30], [31]. Although the present technique allows us to treat other aspects of the theory, we shall not verify the Osterwalder-Schrader axioms, nor we shall discuss the case with massive fermion or the bosonization of the final result; this is because details would be largely similar to [6], [7]. We shall rather focus on the novelties with respect to those papers. To conclude, we mention that other rigorous results on the Wi anomalies, but in different regimes and with different techniques, were established in [38], [35]. Whereas for the study the true Schwinger model, i.e. the case with null bare photon mass, a change of viewpoint seems needed: boson and fermion propagators should be treated on the same ground along the Renormalization Group (RG) flow; perhaps that is possible in the approach of [32], [16]. ## 2 Definitions and main results Let’s begin with the definition of the formal path integral formula in Euclidean formulation; afterwords we will introduce _infrared and ultraviolet cutoffs_ to evaluate the correlations. The Vector Meson model is made of one fermion field, $(\bar{\psi}_{\bf x},\psi_{\bf x})$, for ${\bf x}=(x_{0},x_{1})\in\hbox{\msytw R}^{2}$, and one vector boson field, $(A^{0}_{\bf x},A^{1}_{\bf x})$, interacting through minimal coupling. The free meson with mass $\mu$ (a _gauge field_ if $\mu=0$) is described by the action \[{1\over 4}\int\!d{\bf x}\;(F_{\bf x}^{\mu\nu})^{2}+{\mu^{2}\over 2}\int\!d{\bf x }\;(A^{\mu}_{\bf x})^{2}\] for $F^{\mu\nu}_{\bf x}=\partial^{\mu}A^{\nu}_{\bf x}-\partial^{\nu}A^{\mu}_{\bf x}$. The quantization of a gauge theory (namely the case $\mu=0$) requires a _gauge fixing term_ which makes convergent the integration along the orbits of gauge transformation: for $\alpha>0$ \[{\alpha\over 2}\int\!d{\bf x}\;(\partial^{\mu}_{\bf x}A^{\mu}_{\bf x})^{2}\;.\] In our case $\mu\neq 0$ and the gauge fixing would not be required to make sense of the theory. Nevertheless, the Vector Meson model defined by Thirring and Wess had the purpose to be equivalent to the Schwinger model in the limit of vanishing $\mu$; therefore, with them, we shall consider $\alpha>0$ only, that will make the interaction _superrinormalizable_. For notational convenience, we also introduce a further non-local term for the vector field \[{\sigma\over 2}\int\!d{\bf x}d{\bf y}\;(\partial^{\mu}A^{\mu}_{\bf x})\Delta^{ -1}({\bf x}-{\bf y})(\partial^{\mu}A^{\mu}_{\bf y})\] where $\Delta^{-1}$ is the inverse of the Laplacian and $\sigma$ is a real parameter with the dimension of the square of a mass. The three terms we have introduced so far are collected together into the following term \[{1\over 2}\int\!d{\bf x}d{\bf x}\;A^{\mu}_{\bf x}D^{\mu\nu}({\bf x }-{\bf y})A^{\nu}_{\bf y}\] (1) (2) \[{\buildrel\;{\rm def.}\;\over{=}}{1\over 2}\int\!{d{\bf k}\over(2 \pi)^{2}}\;{\widehat{A}}^{\mu}_{\bf k}\left[({\bf k}^{2}+\mu^{2})\delta^{\mu \nu}-\left(1-\alpha+{\sigma\over{\bf k}^{2}}\right){\bf k}^{\mu}{\bf k}^{\nu} \right]{\widehat{A}}^{\nu}_{-{\bf k}}\;.\] Sometimes we will write $D^{\mu\nu}$ for the operator with kernel $D^{\mu\nu}({\bf x})$. The massless electron, with charge $q$, interacts with the photon through a _minimal coupling_ \[\int\!d{\bf x}\;\bar{\psi}_{\bf x}\gamma^{\mu}\left(i\partial^{\mu}_{\bf x}+qA ^{\mu}_{\bf x}\right)\psi_{\bf x}\] where $\gamma^{0}$ and $\gamma^{1}$ are generators of the Euclidean Clifford algebra. The total Euclidean action of the Vector Meson is \[{1\over 2}\int\!d{\bf x}d{\bf y}\ A^{\mu}_{\bf x}D^{\mu\nu}({\bf x}-{\bf y})A^ {\nu}_{\bf y}+\int\!d{\bf x}\;\bar{\psi}_{\bf x}\gamma^{\mu}\left(i\partial^{ \mu}_{\bf x}+qA^{\mu}_{\bf x}\right)\psi_{\bf x}\;.\] Finally, putting together all the above terms, we define the _generating functional_ of the truncated correlations of the the Vector Meson model, ${\cal K}(J,\eta)$, as follows: \[e^{{\cal K}(J,\eta)}{\buildrel\;{\rm def.}\;\over{=}}\int\!dP(\psi)dP(A)\;\exp \left\{\int\!d{\bf x}\;\left[-qA^{\mu}_{\bf x}(\bar{\psi}_{\bf x}\gamma^{\mu} \psi_{\bf x})+J^{\mu}_{\bf x}A^{\mu}_{\bf x}+\bar{\eta}_{\bf x}\psi_{\bf x}+ \bar{\psi}_{\bf x}\eta_{\bf x}\right]\right\}\] (3) where $J$ is a real external field; $\eta$, $\bar{\eta}$ are Grassmann external fields; $dP(\psi)$ is a Gaussian measure on Grassmann variables $(\psi_{{\bf x}},\bar{\psi}_{{\bf x}})_{{\bf x}}$ with zero covariances, but for $\int\!dP(\psi)\;\psi_{\bf x}\bar{\psi}_{\bf y}$ that equals \[S_{0}({\bf x}-{\bf y}){\buildrel\;{\rm def.}\;\over{=}}i\gamma^{\mu}\int\!{d{ \bf k}\over(2\pi)^{2}}\;{e^{-i{\bf k}({\bf x}-{\bf y})}\over{{\bf k}^{2}}}{\bf k }^{\mu}\;;\] and $dP(A)$ is a Gaussian measure on real variables $(A^{0}_{\bf x},A^{1}_{\bf x})_{\bf x}$ with covariances $\int\!dP(A)\;A^{\mu}_{\bf x}A^{\nu}_{\bf y}$ equal to \[G_{0}^{\mu\nu}({\bf x}-{\bf y};\mu^{2},\sigma){\buildrel\;{\rm def.}\;\over{=} }\int\!{d{\bf k}\over(2\pi)^{2}}\;e^{-i{\bf k}({\bf x}-{\bf y})}\left[{\delta^ {\mu\nu}\over{\bf k}^{2}+\mu^{2}}-\left({1\over{\bf k}^{2}+\mu^{2}}-{1\over \alpha{\bf k}^{2}+\mu^{2}-\sigma}\right){{\bf k}^{\mu}{\bf k}^{\nu}\over{\bf k }^{2}}\right]\] (we will abridge the notation of $G_{0}^{\mu\nu}({\bf x};\mu^{2},\sigma)$ into $G_{0}^{\mu\nu}({\bf x})$, sometimes). These covariances are also called _free propagators_ as To make sense of (3) we have to introduce a cutoff function. For a fixed $\gamma>1$, let ${\widehat{\chi}}(t)$ be a smooth function, positive in $[0,\gamma)$ and \[{\widehat{\chi}}(t)=\left\{\matrix{1\hfill&\hfill{\rm if\ }0\leq t\leq 1\cr 0 \hfill&\hfill{\rm if\ }t\geq\gamma.}\right.\;;\] (4) then, given two integers $h,h^{\prime}$, define \[{\widehat{\chi}}_{h,h^{\prime}}({\bf k})={\widehat{\chi}}\left(\gamma^{-h}\|{ \bf k}\|\right)-{\widehat{\chi}}\left(\gamma^{-h^{\prime}+1}\|{\bf k}\|\right)\;.\] and in correspondence, define two Gaussian measure $dP_{h,h^{\prime}}(\psi)$ and $dP_{h,h^{\prime}}(A)$, determined by the covariances: \[S_{0,h,h^{\prime}}({\bf x}){\buildrel\;{\rm def.}\;\over{=}}\gamma^{\mu}\int\! {d{\bf k}\over(2\pi)^{2}}\;{\widehat{\chi}}_{h,h^{\prime}}({\bf k}){e^{-i{\bf k }{\bf x}}\over{{\bf k}^{2}}}{\bf k}^{\mu}\;,\] \[G_{0,h,h^{\prime}}^{\mu\nu}({\bf x}){\buildrel\;{\rm def.}\;\over{=}}\int\!{d{ \bf k}\over(2\pi)^{2}}\;{\widehat{\chi}}_{h,h^{\prime}}({\bf k})e^{-i{\bf k}{ \bf x}}\left[{\delta^{\mu\nu}\over{\bf k}^{2}+\mu^{2}}-\left({1\over{\bf k}^{2 }+\mu^{2}}-{1\over\alpha{\bf k}^{2}+\mu^{2}-\sigma}\right){{\bf k}^{\mu}{\bf k }^{\nu}\over{\bf k}^{2}}\right]\;.\] Given the integers $-l,N>0$, the _regularized functional integral_, ${\cal K}_{l,N}(J,\eta)$, is given by (3), replacing $dP(\psi)$ and $dP(A)$ with $dP_{l,N}(\psi)$ and $dP_{l,N}(A)$. Finally, let $S({\bf x}-{\bf y})$ and $G^{\mu\nu}({\bf x}-{\bf y})$ be the _interacting propagators_, namely the correlations \[S({\bf x}-{\bf y}){\buildrel\;{\rm def.}\;\over{=}}\lim_{-l,N\to\infty}{ \partial^{2}{\cal K}_{l,N}\over\partial\bar{\eta}_{\bf x}\partial\eta_{\bf y}} (0,0)\] (5) \[G^{\mu\nu}({\bf x}-{\bf y};\mu^{2},\sigma){\buildrel\;{\rm def.}\;\over{=}} \lim_{-l,N\to\infty}{\partial^{2}{\cal K}_{l,N}\over\partial J^{\mu}_{\bf x} \partial J^{\nu}_{\bf y}}(0,0)\] (6) where the derivatives in $\eta$ are taken from the right; and the limit in $N$ is taken before the limit in $l$. Define \[F({\bf z};\mu^{2},\sigma)=\int\!{d{\bf k}\over(2\pi)^{2}}\;{e^{i {\bf k}{\bf z}}-1\over[\alpha{\bf k}^{2}+\mu^{2}-\sigma]{\bf k}^{2}}\] (7) (8) \[F_{5}({\bf z};\mu^{2})=\int\!{d{\bf k}\over(2\pi)^{2}}\;{e^{i{ \bf k}{\bf z}}-1\over[{\bf k}^{2}+\mu^{2}]{\bf k}^{2}}\;,\] (9) and note that, for $\mu^{2}>0$, $\sigma<\mu^{2}$, $\alpha>0$, we have the following large $\|{\bf z}\|$ asymptotic: \[F({\bf z};\mu^{2},\sigma)\sim-{1\over 2\pi(\mu^{2}-\sigma)}\ln\|{\bf z}\|\;, \qquad F_{5}({\bf z};\mu^{2},\sigma)\sim-{1\over 2\pi\mu^{2}}\ln\|{\bf z}\|\;.\] Theorem 2.1: _Given the meson mass $\mu^{2}>0$, for $\sigma<\mu^{2}$, $\alpha\geq\alpha_{0}>0$ and $\|q\|$ small enough, the explicit expression of the interacting propagators are:_ \[S({\bf x})=e^{q^{2}\Big{[}F({\bf x};\mu^{2}-\nu_{5},\sigma+\nu-\nu_{5})-F_{5}( {\bf x};\mu^{2}-\nu_{5})\Big{]}}S_{0}({\bf x})\] (10) \[G^{\mu\nu}({\bf x};\mu^{2},\sigma)=G_{0}^{\mu\nu}({\bf x};\mu^{2}-\nu_{5}, \sigma+\nu-\nu_{5})\] (11) _with $\nu=-\nu_{5}=q^{2}/(2\pi)$._ This result is uniform in $\alpha>a_{0}$: Feynman-’t Hooft’s and Landau’s gauges, for example, are recovered for $\alpha=1$ and $\alpha\to\infty$, respectively; the two point correlation of $F^{\mu\nu}$ is $\alpha$-independent. Although (11) means that the interaction changes the free meson correlation of an additional mass term only, the theory is not free; indeed, in (10) we can see that the large distance decay of the fermion correlation has an anomalous dimension $\eta$: \[S({\bf x})\sim{1\over\|{\bf x}\|^{1+\eta}}\qquad\eta={q^{2}\over 2\pi}\left[{1 \over\mu^{2}-\nu^{5}}-{1\over\mu^{2}-\sigma-\nu}\right]\] (12) We shall see that $\nu$ and $\nu_{5}$ are the anomalies of the vector and axial Wi’s, respectively. To clarify the relation of this result with the literature, it is worth mentioning the (unproven) _uncertainty principle_ of the anomalies, [24],[25], [20], [14], namely the fact that the most general numerical values are \[\nu={q^{2}\over\pi}(1-\xi)\qquad\nu_{5}=-{q^{2}\over\pi}\xi\] for $\xi$ a real parameter fixed by the kind of regularization of the functional integral. Hence the meson mass, $\nu_{5}$, is regularization dependent (this is not an issue for the meaning of the model, because $q$ is the _bare_, not the _physical_ charge). Our result is in agreement with [41], where $\sigma=0$, $\xi=1$ and $\alpha=1$; and with [20], where $\xi$ is any, $\sigma=0$, $\alpha=\infty$. Solutions of the Vector Meson model for $\alpha=0$ (a case that this paper doesn’t cover) are in [39], [15] and [25], for $\xi=1/2$, $\xi=1$ and any $\xi$, respectively. Those results are in agreement with our theorem only formally: in that case $F({\bf x},\mu^{2},\nu)$ is not defined and from $S({\bf x})$ one has to divide out a divergent factor. This is not a surprise: when $\alpha=0$ the large momentum asymptotic of the free meson propagator is \[{\widehat{G}}_{0}^{\mu\nu}({\bf k})\sim{\bf k}^{\mu}{\bf k}^{\nu}/{\bf k}^{2}\] that makes the interaction _renormalizable_ as in the case of the gradient coupling model, [34], [4]. The correct approach for this case would be the one in [6], with vanishing beta function and field renormalization. Anyways that rises a question: the AB-formula is not valid in [6], where radiative corrections do change the numerical value of the anomaly; is this the case also for the $\alpha=0$ Vector Meson model? We will discuss this issue in a possible forthcoming paper. As mentioned in the Introduction, were $\mu=0$, we would read (11) as the _dynamical mass generation_ of the Schwinger model; but unfortunately we are not able to cover that case. ## 3 Idea of the proof. Ward Identities. Anomalies Firstly, we have to prove that there exist the limits (5) and (6). To do that, we use some Lesniewski’s ideas, [28]. Define the functional integral \[{\cal W}_{l,N}(J,\eta)=\ln\int\!dP_{l,N}(\psi)\;e^{{\cal V}(\psi,J,\eta)}\] where ${\cal V}(\psi,J,\eta)$ is the interaction (self-interaction plus coupling with external fields) of a non-local version of the Thirring model: \[{\cal V}(\psi,J,\eta)={\lambda\over 4}\int\!d{\bf x}d{\bf y}\;( \bar{\psi}_{\bf x}\gamma^{\mu}\psi_{\bf x})\;G_{0}^{\mu\nu}({\bf x}-{\bf y})\; (\bar{\psi}_{\bf y}\gamma^{\nu}\psi_{\bf y})\] (13) (14) \[+\int\!d{\bf x}\;J^{\mu}_{\bf x}\;(\bar{\psi}_{\bf x}\gamma^{\mu} \psi_{\bf x})+\int\!d{\bf x}\;(\bar{\eta}_{\bf x}\psi_{\bf x}+\bar{\psi}_{\bf x }\eta_{\bf x})\] (15) Now take the integral over the vector field $A^{\mu}$ in (3), and obtain an identity between the functional integrals of the Thirring and the Vector meson models, for coupling $\lambda=2q^{2}$ \[{\cal K}(DJ,\eta)={\cal W}(-qJ,\eta)+{1\over 2}\int\!d{\bf x}d{\bf y}\;J^{\mu} _{\bf x}D^{\mu\nu}({\bf x}-{\bf y})J^{\nu}_{\bf y}\;.\] (16) If we can control the field and the current correlation derived from ${\cal W}$, we can also construct the limits (5) and (6). To bound the Feynman graphs, one has to use that, above the scale of the photon mass, the interaction in ${\cal K}$ is superrinormalizable; but, to see that in ${\cal W}$, we will need some identities among Feynman graphs, as in [28], [30] and [21]. This point is discussed in Section 4.1. Then, with a classical bound for fermion determinant, [28], the convergence of the perturbation theory is proved. Secondly, in Section 4.2, we shall prove that the correlations generated from ${\cal W}$ satisfy two anomalous Wi’s. From the (formal) invariance under the vector transformation $\psi\to e^{i\vartheta}\psi$, $\bar{\psi}\to\bar{\psi}e^{-i\vartheta}$ we obtain the _vector Ward identity_, \[\;i\partial^{\mu}_{\bf x}\int\!d{\bf z}\;\left[\delta^{\mu\nu} \delta({\bf x}-{\bf z})-\nu G_{0}^{\mu\nu}({\bf x}-{\bf z})\right]{\partial{ \cal W}(J,\eta)\over\partial J^{\nu}_{\bf z}}-2\Theta i\partial_{\bf x}^{\mu}J ^{\mu}_{\bf x}\] (17) (18) \[={\partial{\cal W}(J,\eta)\over\partial\eta_{\bf x}}\eta_{\bf x}- \bar{\eta}_{\bf x}{\partial{\cal W}(J,\eta)\over\partial\bar{\eta}_{\bf x}}\;;\] (19) and from the (formal) invariance under (Euclidean) axial-vector transformation $\psi\to e^{\gamma^{5}\vartheta}\psi$, $\bar{\psi}\to\bar{\psi}e^{\gamma^{5}\vartheta}$, with $\gamma^{5}=-i\gamma^{0}\gamma^{1}$, using $\gamma^{\mu}\gamma^{5}=-i\varepsilon^{\mu\nu}\gamma^{\nu}$, we obtain the _axial Ward identity_ \[\;\varepsilon^{\mu\rho}i\partial^{\mu}_{\bf x}\int\!d{\bf z}\; \left[\delta^{\rho\nu}\delta({\bf x}-{\bf z})-\nu_{5}G_{0}^{\rho\nu}({\bf x}-{ \bf z})\right]{\partial{\cal W}(J,\eta)\over\partial J^{\nu}_{\bf z}}-2\Theta_ {5}\varepsilon^{\rho\mu}i\partial_{\bf x}^{\rho}J^{\mu}_{\bf x}\] (20) (21) \[=-{\partial{\cal W}(J,\eta)\over\partial\eta_{\bf x}}i\gamma^{5} \eta_{\bf x}-\bar{\eta}_{\bf x}i\gamma^{5}{\partial{\cal W}(J,\eta)\over \partial\bar{\eta}_{\bf x}}\;.\] (22) $\nu$, $\nu_{5}$, $\Theta$ and $\Theta_{5}$ are the _Adler-Bell-Jackiw anomalies_. Assuming the validity of (17) and (20), the proof of (11) is just a computation in which the AB-formula plays a crucial role. Before showing that, let’s pause for some technical comments. At $J=0$ (i.e. without the terms proportional to $\Theta$ and $\Theta_{5}$), (17) and (20) were proved in [6] for local self interaction of the fermion field, i.e. $G_{0}^{\mu\nu}({\bf x})=\delta^{\mu\nu}\delta({\bf x})$; and later on they were proved in [30] for $G_{0}^{\mu\nu}({\bf x})=\delta^{\mu\nu}v({\bf x})$ where $v({\bf x})$ is a short-range, bounded self interaction, i.e. without removing IR and UV cutoffs in $v({\bf x})$. Of course the latter case is technically simpler; nonetheless it is remarkable for, as opposed to the former, it gives an example in which the AB-formula is valid. The main task of the present paper is to extend the proof of the Wi’s to the case $J\neq 0$ and for the given $G_{0}^{\mu\nu}$, that is a symmetric matrix with short-ranged but unbounded entries. To continue the computation, use (16) to turn (17) and (20) into identities for derivatives of ${\cal K}$; since \[\partial^{\mu}_{\bf x}D^{\mu\nu}_{\bf x}=\left[-\alpha\Delta_{\bf x}+\mu^{2}- \sigma\right]\partial^{\nu}_{\bf x}\;,\qquad\varepsilon^{\mu\rho}\partial^{\mu }_{\bf x}D^{\rho\nu}_{\bf x}=\left[-\Delta_{\bf x}+\mu^{2}\right]\varepsilon^{ \rho\nu}\partial^{\rho}_{\bf x}\;\] take a further derivative in $J^{\sigma}_{\bf y}$, at $\eta=\bar{\eta}=J=0$ and obtain: \[\;\left[-\alpha\Delta_{\bf x}+\mu^{2}-\sigma-\nu\right]i\partial^ {\mu}_{\bf x}G^{\mu\nu}({\bf x}-{\bf y})=i\partial^{\nu}_{\bf x}\delta({\bf x} -{\bf y})+\] (23) (24) \[+\left(2q^{2}\Theta-\nu\right)i\partial_{\bf x}^{\mu}G_{0}^{\mu \nu}({\bf x}-{\bf y})\] (25) \[\;\left[-\Delta_{\bf x}+\mu^{2}-\nu_{5}\right]\varepsilon^{\rho \mu}i\partial^{\rho}_{\bf x}G^{\mu\nu}({\bf x}-{\bf y})=\varepsilon^{\rho\nu}i \partial^{\rho}_{\bf x}\delta({\bf x}-{\bf y})+\] (26) (27) \[+\left(2q^{2}\Theta_{5}-\nu_{5}\right)\varepsilon^{\rho\mu}i \partial_{\bf x}^{\rho}G_{0}^{\mu\nu}({\bf x}-{\bf y})\] (28) Here comes the crucial point: in establishing the validity of the Wi and the presence of the anomaly, we will also verify the AB-formula, i.e. the fact the anomaly is given by the first order perturbation theory, without higher order corrections. Therefore we can explicitly evaluate $\nu=\lambda\Theta=-\nu_{5}=-\lambda\Theta_{5}={\lambda\over 4\pi}$. Since $\lambda=2q^{2}$, the second line in both equations is zero; and the theorem is proved by explicit solution of (23) and (26). We stress that in formal versions of this computation, [33], the quantum anomaly is just added into the classical equations by hand where it is expected, and so the terms proportional to $2q^{2}\Theta-\nu$ and $2q^{2}\Theta_{5}-\nu_{5}$ never appear at all. To prove (10), we need the SDe, i.e. the field equations written in terms of the correlations. In the functional integral approach that is nothing but the Wick theorem for the Gaussian measure. For the fermion fermion correlation we have: \[\;i\gamma^{\mu}\partial^{\mu}_{\bf x}{\partial^{2}{\cal W}\over\partial\bar{ \eta}_{\bf x}\partial\eta_{\bf y}}(0,0)=\delta({\bf x}-{\bf y})+{\lambda\over 2 }\gamma^{\mu}\int\!d{\bf z}\;G^{\mu\nu}_{0}({\bf x}-{\bf z}){\partial^{3}{\cal W }\over\partial J^{\nu}_{\bf z}\partial\bar{\eta}_{\bf x}\partial\eta_{\bf y}}( 0,0)\] (29) That is not a closed equation, though we can use the Wi to close it. Take derivatives in (17) and (20) w.r.t. $\bar{\eta}_{\bf x}$ and $\eta_{\bf y}$ at $J=\bar{\eta}=\eta=0$: we obtain two equations that are equivalent to (30) (31) \[+\varepsilon^{\rho\mu}i\partial^{\rho}_{\bf x}\left[F_{5}({\bf x} -{\bf y})-F_{5}({\bf x}-{\bf w})\right]\Big{\|}_{\mu^{2}-\nu_{5}\atop\sigma+\nu -\nu_{5}}i\gamma^{5}S({\bf w}-{\bf y})\] (32) (we have abridged the notation of the mass terms in $F$ and $F_{5}$). Plug (30) into (29), and obtain the closed equation \[\;i\gamma^{\mu}\partial^{\mu}_{\bf x}S({\bf x}-{\bf y})=\delta({\bf x}-{\bf y} )+{\lambda\over 2}i\gamma^{\mu}\left[\partial^{\mu}_{\bf x}F({\bf x}-{\bf y})- \partial^{\mu}_{\bf x}F_{5}({\bf x}-{\bf y})\right]\Big{\|}_{\mu^{2}-\nu_{5} \atop\sigma+\nu-\nu_{5}}S({\bf x}-{\bf y})\] (33) that is solved by (10). There is a subtle point, though. (30) can be plugged into (33) after the limit of removed cutoff has been removed only if one proves that the limit is continuous at ${\bf w}={\bf x}$. In Section 4.3 we will prove (33) in a slightly different way: we will plug the Wi into the SDe _before_ removing the cutoffs, and we will show that the the limit of the remainder is vanishing (as opposed to the case in [6], where the limit of the remainder gives rise to a further anomaly in the closed equation). Summarizing, in Section 4.1 we will study the limit of removed cutoffs; in Section 4.2 we will prove (17) and (20); in Section (4.3) we will prove (33). ## 4 Renormalization group approach As mentioned, from the viewpoint of the formal power series in $q$, the RG description of (3) is quite simple. Above the meson mass scale, i.e. in the UV regime, the coupling of a fermion current with a boson field is superrenormalizable; below the meson mass scale, i.e. the IR regime, the interaction is renormalizable, and the RG flow equals, up to irrelevant terms, the flow of the Thirring model. A qualitative explanation is the following. At energy $E>\mu$, boson and fermion propagators have typical sizes $E^{0}$ and $E^{1}$, respectively; then, the energy of a graph with $p$ vertexes, $2m$ external fermion legs and $n$ external boson legs is $E^{d(p,n,2m)}$, for $d(p,n,2m)=2-m-p$: the only relevant graphs are $(p,2m)=(1,1)$, the uncontracted vertex, and $(p,2m)=(2,0)$, which is zero by symmetry; so no renormalization of coupling constants is required. At $E<\mu$, the size of the boson propagator becomes $(E/\mu)^{2}$; then a graph size is $\mu^{n-p}E^{d^{\prime}(p,n,2m)}$, for $d^{\prime}(p,n,2m)=2-m-n$, that is the same _power counting_ of the Thirring model. Still qualitatively, the limit $\mu\to 0$ gives the Schwinger model; the limit $\mu\to\infty$ gives the free boson field; whereas replacing $G_{0}^{\mu\nu}$ with $\mu^{2}G_{0}^{\mu\nu}$, and taking the limit $\mu\to\infty$ give the Thirring model. In this paper we consider a fixed $\mu>0$. The issue with the above argument is that we are not able to prove the convergence of the perturbation theory with both boson and fermion fields. To overcome that we integrate the boson field before the RG analysis so that the fermion-boson interaction is turned into a fermion-fermion quartic interaction. Now the theory looks marginal at any scale. To recover the superrenormalizability of the UV scales we use identities among the Feynman graphs: the identities of this paper are the same as in [21] and [8]; as opposed to approach in [30], [31], they permit to take advantage of $L_{p}$ inequalities, which is the key to control an unbounded $G_{0}$. This part is largely inspired to [28]. Before discussing technical details, we set up some more notations. The explicit choice of generators of the Euclidean Clifford algebra is \[\gamma^{0}=\pmatrix{0&1\cr 1&0\cr}\;,\qquad\gamma^{1}=\pmatrix{0&-i\cr i&0\cr}\;;\] then we call $\eta=(\eta^{-}_{{\bf x},-},\eta^{-}_{{\bf x},+})$, $\bar{\eta}=(\eta^{+}_{{\bf x},+},\eta^{+}_{{\bf x},-})$, $\psi_{\bf x}=(\psi^{-}_{{\bf x},+},\psi^{-}_{{\bf x},-})$ and $\bar{\psi}_{{\bf x}}=(\psi^{+}_{{\bf x},-},\psi^{+}_{{\bf x},+})$ - note the opposite notation for the components of the spinors $\bar{\eta}$, $\eta$ and $\bar{\psi}$, $\psi$ - so that, for $\partial_{{\bf x},\omega}=i\partial^{0}-\omega\partial^{1}$, \[\psi^{+}_{{\bf x},\omega}\psi^{-}_{{\bf x},\omega}=\bar{\psi}_{\bf x}{\gamma^{ 0}-i\omega\gamma^{1}\over 2}\psi_{\bf x}\;,\qquad\sum_{\omega}\psi^{+}_{{\bf x },\omega}\partial_{\omega}\psi^{-}_{{\bf x},\omega}=i\bar{\psi}_{\bf x}\gamma^ {\mu}\partial^{\mu}\psi_{\bf x}\;.\] The interaction becomes \[v_{\omega,\omega^{\prime}}({\bf x})={1\over 2}\left[i\omega G_{0}^{10}({\bf x} )+i\omega^{\prime}G_{0}^{01}({\bf x})+G_{0}^{00}({\bf x})-\omega\omega^{\prime }G_{0}^{11}({\bf x})\right]\;;\] and since $G_{0}^{01}({\bf x})=G_{0}^{10}({\bf x})$, also $v_{\omega,\omega^{\prime}}({\bf x})=v_{\omega^{\prime},\omega}({\bf x})$. Finally, for $J_{{\bf x},\omega}=J^{0}_{\bf x}+i\omega J^{1}_{\bf x}$, the functional integral formula for the non-local Thirring model is \[e^{{\cal W}_{l,N}(J,\eta)}=\int\!dP_{l,N}(\psi)\;\exp\left\{{ \lambda\over 2}\sum_{\omega,\omega^{\prime}}\int\!d{\bf x}d{\bf y}\;\psi^{+}_{ {\bf x},\omega}\psi^{-}_{{\bf x},\omega}\;v_{\omega,\omega^{\prime}}({\bf x}-{ \bf y})\;\psi^{+}_{{\bf x},\omega^{\prime}}\psi^{-}_{{\bf x},\omega^{\prime}}\right\}\] (34) (35) \[\exp\left\{\sum_{\omega}\int\!d{\bf x}\;J_{{\bf x},\omega}\psi^{+ }_{{\bf x},\omega}\psi^{-}_{{\bf x},\omega}+\sum_{\omega}\int\!d{\bf x}\;(\eta ^{+}_{{\bf x},\omega}\psi^{-}_{{\bf x},\omega}+\psi^{+}_{{\bf x},\omega}\eta^{ -}_{{\bf x},\omega})\right\}\] (36) for $dP_{l,N}(\psi)$ determined by the covariance: \[\int\!dP_{l,N}(\psi)\;\psi^{-}_{{\bf x},\omega}\psi^{+}_{{\bf x},\omega}=g^{[l ,N]}_{\omega}({\bf x})=\int\!{d{\bf k}\over(2\pi)^{2}}\;{e^{-i{\bf k}{\bf x}} \over D_{\omega}({\bf k})}{\widehat{\chi}}_{l,N}({\bf k})\;,\qquad D({\bf k})= k^{0}+i\omega k^{1}\;.\] Finally, we stress two points. Firstly, we are assuming that the limit of removed cutoff in the propagator $v_{\omega,\omega^{\prime}}$ is already taken: this is not an abuse, since, otherwise, the estimates that will follow would be anyways uniform in the $l,N$ of $v_{\omega,\omega^{\prime}}$. Secondly, all the claims about ${\cal W}(J,\eta)$ (and the same for other functional integrals that we are about to define), must actually be understood in terms of the correlations that it generates, i.e. for a finite number of derivatives w.r.t the external fields, at $J=\eta=0$. (In fact there would be no need to prove that ${\cal W}(J,\eta)$ is a convergent power series of the external fields even if we wanted to verify the Osterwalder-Schrader axioms, see [6].) ### Correlations. In evaluating ${\cal W}_{l,N}(J,\eta)$, to have bounds that are uniform in $l$ and $N$, we have to slice the range of allowed momenta into scales. We use the decomposition \[{\widehat{\chi}}_{h^{\prime},h}({\bf k})=\sum_{k=h^{\prime}+1}^{h}f_{k}({\bf k})\] where $f_{k}({\bf k})={\widehat{\chi}}_{k-1,k}({\bf k})$; in correspondence we have the factorization of the Gaussian measure \[\psi=\psi^{(h^{\prime})}+\psi^{(h^{\prime}+1)}+\cdots+\psi^{(h)}\] (37) (38) \[dP_{h^{\prime},h}(\psi)=dP_{h^{\prime}}(\psi^{(h^{\prime})})dP_{ h^{\prime}+1}(\psi^{(h^{\prime}+1)})\cdots dP_{h}(\psi^{(h)})\] (39) and $dP_{k}(\psi^{(k)})$ is determined by the covariance \[g_{\omega}^{[k]}({\bf x})=\int\!{d{\bf k}\over(2\pi)^{2}}\ {e^{-i{\bf k}{\bf x }}\over D_{\omega}({\bf k})}f_{k}({\bf k})\;.\] We integrate iteratively the fields with smaller and smaller momentum. After the integration of $\psi=\psi^{(N)},\psi^{(N-1)},\ldots,\psi^{(h+1)}$ we have the effective potential on scale $h$, ${\cal V}^{(h)}$, such that \[\;e^{{\cal W}_{l,N}(J,\eta)}=\int\!dP_{l,h}(\psi)\ e^{{\cal V}^{(h)}(\psi,J, \eta)}\;.\] (40) Consider the case $\bar{\eta}=\eta=0$. Assume by induction that, for any scale $h=k+1$, the effective potential is a polynomial in the fields $(J_{{\bf z},\omega})$, $(\psi^{+}_{\omega,{\bf x}})$ and $(\psi^{-}_{\omega,{\bf y}})$. We call kernels on scale $h$ the coefficients of the monomials of ${\cal V}^{(h)}$: for ${\underline{{\bf z}}}=({\bf z}_{1},\ldots,{\bf z}_{n})$, ${\underline{{\bf x}}}=({\bf x}_{1},\ldots,{\bf x}_{m})$, ${\underline{{\bf y}}}=({\bf y}_{1},\ldots,{\bf y}_{m})$, ${\underline{\omega}}^{\prime}=(\omega_{1},\ldots\omega_{n})$ and ${\underline{\omega}}=(\omega_{1},\ldots\omega_{m})$, \[W^{(n;2m)(h)}_{{\underline{\omega}}^{\prime},{\underline{\omega}}}({\underline {{\bf z}}};{\underline{{\bf x}}},{\underline{{\bf y}}}){\buildrel\;{\rm def.} \;\over{=}}\left.\prod_{j=1}^{n}{\partial\over\partial J_{{\bf z}_{i},\omega^{ \prime}_{i}}}\prod_{i=1}^{m}{\partial\over\partial\psi^{+}_{{\bf x}_{i},\omega _{i}}}{\partial\over\partial\psi^{-}_{{\bf y}_{i},\omega_{i}}}{\cal V}^{(h)}( \psi,J,0)\right\|_{J=\psi=0}\;\] (41) (where the derivatives in $\partial\psi^{-}_{{\bf y}_{i},\omega_{i}}$ are taken from the right). To evaluate ${\cal V}^{(k)}(\psi,J,0)$, use the formula for the truncated expectations: \[{\cal V}^{(k)}(\psi,J,0)=\ln\int\!dP_{k+1}(\zeta)\ e^{{\cal V}^{( k+1)}(\psi+\zeta,J,0)}\] (42) (43) \[=\sum_{p\geq 0}{1\over p!}E^{T}_{k+1}\Big{[}\underbrace{{\cal V}^ {(k+1)}(\psi+\zeta,J,0);\cdots;{\cal V}^{(k+1)}(\psi+\zeta,J,0)}_{p{\rm\ times }}\Big{]}\] (44) where $E^{T}_{k+1}$ is by definition the truncated expectation w.r.t. the Gaussian random variables $(\zeta^{\varepsilon}_{{\bf x},\omega})$ with covariances $(g_{\omega}^{[k+1]}({\bf x}))$. Accordingly, (42) gives through (41) the kernels $W^{(n;2m)(k)}_{{\underline{\omega}}^{\prime},{\underline{\omega}}}$. Formula (42) gives also the well known interpretation of each kernels as sum of Feynman graphs belonging to a given class, that is determined by the “external legs”. Later on, we will take advantage of the following two identities on the structure of the graph expansion of the kernels. Lemma 4.1: _The derivatives of the effective potential satisfy two identities:_ \[{\partial{\cal V}^{(k)}\over\partial\psi^{+}_{{\bf x},\omega}}( \psi,J,0)=J_{{\bf x},\omega}\psi_{{\bf x},\omega}^{-}+J_{{\bf x},\omega}\int\! d{\bf u}\ g_{\omega}({\bf x}-{\bf u}){\partial{\cal V}^{(k)}\over\partial\psi_ {{\bf u},\omega}^{+}}(\psi,J,0)\] (45) (46) \[+\lambda\sum_{\omega,\omega^{\prime}}\int\!d{\bf w}d{\bf u}\ v_{ \omega,\omega^{\prime}}({\bf x}-{\bf w})g_{\omega}({\bf x}-{\bf u})\left[{ \partial^{2}{\cal V}^{(k)}\over\partial J_{{\bf w},\omega^{\prime}}\partial \psi^{+}_{{\bf u},\omega}}+{\partial{\cal V}^{(k)}\over\partial J_{{\bf w}, \omega^{\prime}}}{\partial{\cal V}^{(k)}\over\partial\psi^{+}_{{\bf u},\omega} }\right](\psi,J,0)\] (47) (48) \[+\lambda\sum_{\omega,\omega^{\prime}}\int\!d{\bf w}\ v_{\omega, \omega^{\prime}}({\bf x}-{\bf w})\psi_{{\bf x},\omega}^{-}{\partial{\cal V}^{( k)}\over\partial J_{{\bf w},\omega^{\prime}}}(\psi,J,0)\;,\] (49) (50) (51) \[{\partial{\cal V}^{(k)}\over\partial J_{{\bf x},\omega}}(\psi,J, \eta)=\psi^{+}_{{\bf x},\omega}\psi^{-}_{{\bf x},\omega}+\int\!d{\bf u}\ g_{ \omega}({\bf x}-{\bf u})\left[\psi^{+}_{{\bf x},\omega}{\partial{\cal V}^{(k)} \over\partial\psi^{+}_{{\bf u},\omega}}-{\partial{\cal V}^{(k)}\over\partial \psi^{-}_{{\bf u},\omega}}\psi^{-}_{{\bf x},\omega}\right](\psi,J,\eta)\] (52) (53) \[+\int\!d{\bf u}d{\bf u}^{\prime}\ g_{\omega}({\bf x}-{\bf u})g_{ \omega}({\bf x}-{\bf u}^{\prime})\left[{\partial^{2}{\cal V}^{(k)}\over \partial\psi^{+}_{{\bf u},\omega}\partial\psi^{-}_{{\bf u}^{\prime},\omega}}+{ \partial{\cal V}^{(k)}\over\partial\psi^{+}_{{\bf u},\omega}}{\partial{\cal V} ^{(k)}\over\partial\psi^{-}_{{\bf u}^{\prime},\omega}}\right](\psi,J,\eta)\;.\] (54) These identities are clear from graphical interpretation of the multiscale integration; in appendix A we will prove them from the definition of ${\cal V}^{(k)}$ and ${\cal V}$. We introduce the following $L_{1}$ norm \[\\|W^{(n;2m)(k)}_{{\underline{\omega}}^{\prime},{\underline{\omega }}}\\|=\int\!d{\underline{{\bf x}}}d{\underline{{\bf y}}}d{\underline{{\bf z}}} _{2}\;\left\|W^{(n;2m)(k)}_{{\underline{\omega}}^{\prime},{\underline{\omega}}} ({\underline{{\bf z}}};{\underline{{\bf x}}},{\underline{{\bf y}}})\right\|\] (55) for $d{\underline{{\bf z}}}_{2}=({\bf z}_{2},\ldots,{\bf z}_{m})$; namely in (55) we are integrating all but one variable; by translation invariance, the norm does not depend upon ${\bf z}_{1}$. Since $W^{(n;2m)(k)}_{{\underline{\omega}}^{\prime},{\underline{\omega}}}$ may contain delta-distributions, we extend the definition of $L_{1}$ norm by considering them as positive functions. Let $\mu^{2}=\gamma^{2M}$. We will use the following straightforward bounds, for $c,c_{p},c^{\prime},B,B_{p}>1$: \[\\|g_{\omega}^{(h)}\\|_{L_{\infty}}{\buildrel\;{\rm def.}\;\over{=} }\sup_{{\bf x}}\|g_{\omega}^{(h)}({\bf x})\|\leq c\gamma^{h}\;,\] (56) (57) \[\\|g_{\omega}^{(h)}\\|_{L_{p}}{\buildrel\;{\rm def.}\;\over{=}} \left[\int\!d{\bf x}\;\|g_{\omega}^{(h)}({\bf x})\|^{p}\right]^{1/p}\leq c_{p} \gamma^{\left(1-{2\over p}\right)h}\] (58) (59) \[\\|g_{\omega}^{(h)}\\|_{L_{1}(w)}{\buildrel\;{\rm def.}\;\over{=}} \int\!d{\bf x}\;\|x_{j}\|\|g_{\omega}^{(h)}({\bf x})\|\leq c^{\prime}\gamma^{-2h}\] (60) and, since $\alpha{\bf k}^{2}+\mu^{2}-\sigma\geq\alpha_{0}[{\bf k}^{2}+\alpha_{0}^{-1}(\mu ^{2}-\sigma)]$, uniformly in $\alpha\geq\alpha_{0}$ \[\\|v_{\omega,\omega^{\prime}}\\|_{L_{p}}{\buildrel\;{\rm def.}\; \over{=}}\left[\int\!d{\bf x}\|v_{\omega,\omega^{\prime}}({\bf x})\|^{p}\right]^ {1/p}\leq B_{p}\gamma^{2\left(1-{1\over p}\right)M}\] (61) (62) \[\\|\partial_{j}v_{\omega,\omega^{\prime}}\\|_{L_{1}}{\buildrel\;{ \rm def.}\;\over{=}}\int\!d{\bf x}\;\|(\partial_{j}v_{\omega,\omega^{\prime}})( {\bf x})\|\leq B\gamma^{M}\;.\] (63) Let’s consider separately the two different regimes: the UV one, that corresponds to the scales $k:M\leq k\leq N$, and the IR, for $k:l\leq k\leq M-1$. Define \[w^{(1;2)}_{\omega^{\prime},\omega}({\bf z},{\bf x},{\bf y})= \delta({\bf z}-{\bf x})\delta({\bf z}-{\bf y})\delta_{\omega,\omega^{\prime}}\] (64) (65) \[w^{(0;4)}_{\omega^{\prime},\omega}({\bf x},{\bf y},{\bf u},{\bf v })=\delta({\bf x}-{\bf y})v_{\omega^{\prime},\omega}({\bf x}-{\bf u})\delta({ \bf u}-{\bf v})\] (66) [FIGURE:S4.F1][ENDFIGURE] and note that at $h=N$ we have $W^{(0;2)(N)}_{\omega}({\bf x},{\bf y})=0$, $W^{(1;2)(N)}_{\omega^{\prime},\omega}({\bf z},{\bf x},{\bf y})=w^{(1;2)}_{ \omega^{\prime},\omega}({\bf z},{\bf x},{\bf y})$ and $W^{(0;4)(N)}_{\omega^{\prime},\omega}({\bf x},{\bf y},{\bf u},{\bf v})=\lambda w ^{(0;4)}_{\omega^{\prime},\omega}({\bf x},{\bf y},{\bf u},{\bf v})$; all the other $W^{(n;2m)(N)}_{{\underline{\omega}}^{\prime},{\underline{\omega}}}$ are zero. Theorem 4.2: _For $\|\lambda\|$ small enough, there exist constants $C_{0},C>1$ such that, for $M\leq h\leq N$_ \[\\|W^{(0;2)(h)}_{\omega}\\|\leq C\|\lambda\|\gamma^{h}\;,\] (67) (68) \[\\|W^{(1;2)(h)}_{\omega^{\prime},\omega}-w^{(1;2)}_{\omega^{\prime },\omega}\\|\leq C\|\lambda\|\;,\] (69) (70) \[\\|W^{(0;4)(h)}_{\omega^{\prime},\omega}-\lambda w^{(0;4)}_{\omega ^{\prime},\omega}\\|\leq C\|\lambda\|\;;\] (71) _and, for any other $(n;2m)$_ \[\\|W^{(n;2m)(h)}_{{\underline{\omega}}^{\prime},{\underline{\omega}}}\\|\leq C_{ 0}^{n+d_{n,2m}}(C\|\lambda\|)^{d_{n,2m}}\gamma^{h(2-n-m)}\] (72) _where $d_{0,2}=1$, $d_{n,0}=0$, otherwise $d_{n,2m}=m-1$._ The point in the bounds is that $C$, $C_{0}$ are $N-h$ independent. The proof of (72) for $h=k$, assumed iteratively (67) and (72) for $h\geq k+1$, is standard. We shall focus, therefore, on (67) that improves (72) in the cases of _marginal and relevant graphs_, i.e. $(n;2m)=(0;2),(0;4),(1;2)$. Proof. To shorten the notation, in this proof we define $\zeta{\buildrel\;{\rm def.}\;\over{=}}\psi^{(k+1)}+\psi^{(k+2)}+\cdots+\psi^{( N)}$ and $g_{\omega}{\buildrel\;{\rm def.}\;\over{=}}g_{\omega}^{[k+1,N]}$. The derivatives in $\psi^{-}$, $\eta^{-}$ and $\zeta^{-}$ are taken from the right. The proof is for $C$ large enough with respect to $C_{0},c,c_{p},c^{\prime},B,B_{p}$. _1. Improved bound for_ $(0;2)$_._ By symmetry we have $W^{(1;0)(k)}_{-\omega}({\bf w})\equiv 0$; hence from (45) and (54) we expand the two-points kernel as in Fig.2 [FIGURE:S4.F2][ENDFIGURE] \[W^{(0;2)(k)}_{\omega}({\bf x},{\bf y})=\lambda\sum_{\omega^{ \prime}}\int\!d{\bf w}d{\bf w}^{\prime}\;v_{\omega,\omega^{\prime}}({\bf x}-{ \bf w})g_{\omega}({\bf x}-{\bf w}^{\prime})W^{(1;2)(k)}_{\omega^{\prime}; \omega}({\bf w};{\bf w}^{\prime},{\bf y})\;,\] (73) so that, from $\\|w_{\omega^{\prime};\omega}^{(1;2)}\\|\leq 1$ and from (72) for $(n;2m)=(1;2)$, we obtain, for $C$ large enough, \[\\|W^{(0;2)(k)}_{\omega}\\|\leq\|\lambda\|(1+C_{0})\sum_{\omega^{ \prime}}\\|v_{\omega,\omega^{\prime}}\\|_{L_{3}}\sum_{j=k}^{N}\\|g^{(j)}_{\omega} \\|_{L_{3/2}}\leq{C\over r_{1}}\|\lambda\|\gamma^{k}\gamma^{-{4\over 3}(k-M)}\;,\] (74) that proves the first of (67). The factor $r_{1}>1$ will be useful for later. _2. Improved bound for_ $(1;2)$_._ By (45), the kernel $W^{(1;2)(k)}_{\omega^{\prime};\omega}$ can be rewritten as in Fig.3. [FIGURE:S4.F3][ENDFIGURE] Graph (a) in Fig.3 is given by: \[W^{(1;2)(k)}_{(a)\omega^{\prime};\omega}({\bf z};{\bf x},{\bf y} ){\buildrel\;{\rm def.}\;\over{=}}\lambda\sum_{\omega^{\prime\prime}}\int\!d{ \bf w}d{\bf u}\ v_{\omega,\omega^{\prime\prime}}({\bf x}-{\bf w})g_{\omega}({ \bf x}-{\bf u})W^{(2;2)(k)}_{\omega^{\prime},\omega^{\prime\prime};\omega}({ \bf z},{\bf w};{\bf u},{\bf y})\] (75) From (72) for $(n;2m)=(2;2)$, we obtain \[\\|W^{(1;2)(k)}_{(a);\omega^{\prime};\omega}\\|\leq\|\lambda\|C_{0}^{ 2}\gamma^{-k}\sum_{\omega^{\prime\prime}}\\|v_{\omega,\omega^{\prime\prime}}\\|_ {L_{3}}\sum_{j=k}^{N}\\|g^{(j)}_{\omega}\\|_{L_{3/2}}\leq{C\over 4r_{2}}\|\lambda \|\gamma^{-{4\over 3}(k-M)}\] (76) where a large enough constant $r_{2}>1$ will be used later. For graphs (c) and (d) we use the just improved bound for $W^{(0;2)(k)}_{\omega}$: for instance, graph (d) is given by \[W^{(1;2)(k)}_{(d)\omega^{\prime};\omega}({\bf z};{\bf x},{\bf y} ){\buildrel\;{\rm def.}\;\over{=}}\delta_{\omega,\omega^{\prime}}\delta({\bf x }-{\bf z})\int\!d{\bf u}\ g_{\omega}({\bf x}-{\bf u})W^{(0;2)(k)}_{\omega}({ \bf u},{\bf y})\] (77) and using (74) for a $r_{1}$ large enough to compensate other constants, we get \[\\|W^{(1;2)(k)}_{(d)\omega^{\prime};\omega}({\bf z};{\bf x},{\bf y })\\|\leq\\|W^{(0;2)(k)}_{\omega}\\|\sum_{j=k}^{N}\\|g^{(j)}_{\omega}\\|_{L_{1}} \leq{C\over 4r_{2}}\|\lambda\|\gamma^{-{4\over 3}(k-M)}\;.\] (78) In order to obtain an improved bound also for the graphs (b) of Fig.3, we need to further expand $W^{(2;0)(k)}_{\omega^{\prime};-\omega}$. Using (54), we find \[W^{(2;0)(k)}_{\omega^{\prime},-\omega}({\bf z},{\bf w})=\int\!d{ \bf u}^{\prime}d{\bf u}\ g_{\omega}({\bf w}-{\bf u})g_{\omega}({\bf w}-{\bf u} ^{\prime})W^{(1;2)(k)}_{\omega^{\prime};-\omega}({\bf z};{\bf u}^{\prime},{\bf u})\] (79) and then, replacing the expansion for $W^{(1;2)(k)}_{\omega^{\prime};-\omega}({\bf z};{\bf u}^{\prime},{\bf u})$ in the graph (79) we find for (b) what is depicted in Fig.4. Graphs (b4) and (b5) have been obtained also using the expansion (73). [FIGURE:S4.F4][ENDFIGURE] A bound for (b2) and (b3) can be found in the same way. Consider, for instance, the expression for (b2): \[W^{(1;2)(k)}_{(b2)\omega^{\prime};\omega}({\bf z};{\bf x},{\bf y }){\buildrel\;{\rm def.}\;\over{=}}\lambda\delta({\bf x}-{\bf y})\int\!d{\bf w }\ v_{\omega^{\prime},\omega}({\bf x}-{\bf w})g^{2}_{\omega^{\prime}}({\bf w}- {\bf z})\] (80) We want to use the cancellation $\int\!d{\bf u}\ g^{2}_{\omega}({\bf u})=0$ that is a consequence of the symmetry under rotation of the model. In order to to that, expand \[v_{\omega,\omega}({\bf x}-{\bf w})=v_{\omega,\omega^{\prime}}({\bf x}-{\bf z}) +\sum_{j=0,1}(z_{j}-w_{j})\int_{0}^{1}\!\!d\tau\ \big{(}\partial_{j}v_{\omega, \omega^{\prime}}\big{)}\big{(}{\bf x}-{\bf z}+\tau({\bf z}-{\bf w})\big{)}\] (81) and plug (81) into (80): one term in zero; the other can be bounded as follows: \[\\|W^{(1;2)(k)}_{(b2)\omega^{\prime};\omega}\\|\leq 4\|\lambda\|\\| \partial_{j}v_{\omega,\omega^{\prime}}\\|_{L_{1}}\sum_{i=k}^{N}\sum_{j=k}^{i}\\| g^{(j)}_{\omega^{\prime}}\\|_{L_{\infty}}\\|g^{(i)}_{\omega}\\|_{L_{1}(w)}\leq\| \lambda\|{C\over 20r_{2}}\gamma^{-(k-M)}\] (82) Now consider (b1) \[W^{(1;2)(k)}_{(b1)\omega^{\prime};\omega}({\bf z};{\bf x},{\bf y }){\buildrel\;{\rm def.}\;\over{=}}\lambda\delta({\bf x}-{\bf y})\sum_{\sigma, \sigma^{\prime}}\int\!d{\bf w}d{\bf u}^{\prime}d{\bf z}^{\prime}\ v_{\omega, \sigma}({\bf x}-{\bf w})v_{\sigma,\sigma^{\prime}}({\bf u}^{\prime}-{\bf z}^{ \prime})\cdot\] (83) (84) \[\cdot\int\!d{\bf u}d{\bf w}^{\prime}\ g_{\sigma}({\bf w}-{\bf u}) g_{\sigma}({\bf w}-{\bf u}^{\prime})g_{\sigma}({\bf u}^{\prime}-{\bf w}^{ \prime})W^{(2;2)(k)}_{\omega^{\prime},\sigma^{\prime};\sigma}({\bf z},{\bf z}^ {\prime};{\bf w}^{\prime},{\bf u})\;;\] (85) therefore \[\\|W^{(1;2)(k)}_{(b1)\omega^{\prime};\omega}\\|\leq\|\lambda\|\\|v_{ \omega,\sigma}\\|_{L_{1}}\int\!d{\bf w}^{\prime}\;d{\bf u}\;d{\bf z}^{\prime}\ \ \|W^{(2;2)(k)}_{\omega^{\prime},\sigma^{\prime};\sigma}(0,{\bf z}^{\prime};{ \bf w}^{\prime},{\bf u})\|\cdot\] (86) (87) \[\cdot\int\!d{\bf u}^{\prime}\;d{\bf w}\;\;\|v_{\sigma,\sigma^{ \prime}}({\bf u}^{\prime}-{\bf z}^{\prime})g_{\sigma}({\bf w}-{\bf u})g_{ \sigma}({\bf w}-{\bf u}^{\prime})g_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime}) \|\;.\] (88) We have to find a bound for the second line that is uniform in $N-k$. For that, it is convenient to decompose the three fermion propagators into scales, $\sum_{j,q,p=k}^{N}g^{(j)}_{\sigma}g^{(q)}_{\sigma}g^{(p)}_{\sigma}$ and then, for each realization of $j,q,p$, we take the $\\|\cdot\\|_{L_{1}}$ on the fermion propagator with lowest scale. This is always possible: for $p\leq q,j$ \[\int\!d{\bf w}\;d{\bf u}^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({ \bf u}^{\prime}-{\bf z}^{\prime})g^{(j)}_{\sigma}({\bf u}^{\prime}-{\bf w})g^{ (q)}_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime})g^{(p)}_{\sigma}({\bf w}-{\bf u })\|\] (89) (90) \[=\int\!d{\bf u}^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({\bf u}^{ \prime}-{\bf z}^{\prime})g^{(q)}_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime})\| \int\!d{\bf w}\;\|g^{(j)}_{\sigma}({\bf u}^{\prime}-{\bf w})g^{(p)}_{\sigma}({ \bf w}-{\bf u})\|\] (91) (92) \[\leq\\|v_{\sigma,\sigma^{\prime}}\\|_{L_{3}}\\|g^{(q)}_{\omega}\\|_{L _{3/2}}\\|g^{(j)}_{\sigma}\\|_{L_{1}}\\|g^{(p)}_{\sigma}\\|_{L_{\infty}}\leq C_{3} \gamma^{{4\over 3}M}\gamma^{-{q\over 3}}\gamma^{-j}\gamma^{p}\] (93) for $q\leq p,j$ \[\int\!d{\bf w}\;d{\bf u}^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({ \bf u}^{\prime}-{\bf z}^{\prime})g^{(j)}_{\sigma}({\bf u}^{\prime}-{\bf w})g^{ (q)}_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime})g^{(p)}_{\sigma}({\bf w}-{\bf u })\|\] (94) (95) \[=\int\!d{\bf w}\;\|g^{(p)}_{\sigma}({\bf w}-{\bf u})\|\int\!d{\bf u }^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({\bf u}^{\prime}-{\bf z}^{\prime})g^{( q)}_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime})g^{(j)}_{\sigma}({\bf u}^{ \prime}-{\bf w})\|\] (96) (97) \[\leq\\|g^{(p)}_{\sigma}\\|_{L_{1}}\\|v_{\sigma,\sigma^{\prime}}\\|_{L _{3}}\\|g^{(j)}_{\sigma}\\|_{L_{3/2}}\\|g^{(q)}_{\sigma}\\|_{L_{\infty}}\leq C_{3} \gamma^{{4\over 3}M}\gamma^{-p}\gamma^{-{j\over 3}}\gamma^{q}\] (98) and finally, for $j\leq p,q$ \[\int\!d{\bf w}\;d{\bf u}^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({ \bf u}^{\prime}-{\bf z}^{\prime})g^{(j)}_{\sigma}({\bf u}^{\prime}-{\bf w})g^{ (q)}_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime})g^{(p)}_{\sigma}({\bf w}-{\bf u })\|\] (99) (100) \[=\int\!d{\bf w}\;\|g^{(p)}_{\sigma}({\bf w}-{\bf u})\|\int\!d{\bf u }^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({\bf u}^{\prime}-{\bf z}^{\prime})g^{( q)}_{\sigma}({\bf u}^{\prime}-{\bf w}^{\prime})g^{(j)}_{\sigma}({\bf u}^{ \prime}-{\bf w})\|\] (101) (102) \[\leq\\|g^{(p)}_{\sigma}\\|_{L_{1}}\\|v_{\sigma,\sigma^{\prime}}\\|_{L _{3}}\\|g^{(q)}_{\sigma}\\|_{L_{3/2}}\\|g^{(j)}_{\sigma}\\|_{L_{\infty}}\leq C_{3} \gamma^{{4\over 3}M}\gamma^{-p}\gamma^{-{q\over 3}}\gamma^{j}\] (103) so that, summing over the scales $q,p,j$, we obtain \[\int\!d{\bf w}\;d{\bf u}^{\prime}\;\|v_{\sigma,\sigma^{\prime}}({ \bf u}^{\prime}-{\bf z}^{\prime})g_{\sigma}({\bf u}^{\prime}-{\bf w})g_{\sigma }({\bf u}^{\prime}-{\bf w}^{\prime})g_{\sigma}({\bf w}-{\bf u})\|\leq C_{4} \gamma^{{4\over 3}M}\gamma^{-{k\over 3}}\] (104) From (104) and (86), we obtain: \[\\|W^{(1;2)(k)}_{(b1)\omega^{\prime};\omega}\\|\leq{C\over 20}\| \lambda\|\gamma^{-{4\over 3}(k-M)}\] (105) Finally, the latter argument applies also to the bounds of (b4) and (b5). For instance, the expression for (b4) is \[W^{(1;2)(k)}_{(b4)\omega^{\prime};\omega}({\bf z};{\bf x},{\bf y }){\buildrel\;{\rm def.}\;\over{=}}\delta({\bf x}-{\bf y})\lambda^{2}\int\!d{ \bf z}^{\prime}d{\bf w}\ v_{\omega,\omega^{\prime}}({\bf x}-{\bf w})g_{\omega^ {\prime}}({\bf w}-{\bf z})\cdot\] (106) (107) \[\cdot\sum_{\sigma}\int\!d{\bf w}^{\prime}d{\bf u}^{\prime}d{\bf u }\ g_{\omega^{\prime}}({\bf w}-{\bf w}^{\prime})g_{\omega^{\prime}}({\bf w}^{ \prime}-{\bf u})v_{\omega^{\prime},\sigma}({\bf w}^{\prime}-{\bf u}^{\prime})\cdot\] (108) (109) \[\cdot W^{(1;2)(k)}_{\sigma;\omega^{\prime}}({\bf u}^{\prime};{\bf u },{\bf z}^{\prime})g_{\omega^{\prime}}({\bf z}^{\prime}-{\bf z})\] (110) Hence, the bound for such a kernel is: \[\\|W^{(1;2)(k)}_{(b4)\sigma^{\prime};\sigma}\\|\leq 2\|\lambda\|^{2} \\|v_{\omega,\omega^{\prime}}\\|_{L_{1}}\int\!d{\bf z}^{\prime}\;d{\bf u}^{ \prime}\;d{\bf u}\;\|W^{(1;2)(k)}_{-\omega;\omega}({\bf u}^{\prime};{\bf u},{ \bf z}^{\prime})g_{\omega}({\bf z}^{\prime})\|\] (111) (112) \[\cdot\int\!d{\bf w}\;d{\bf w}^{\prime}\;\|g_{\omega}({\bf w}-{\bf z })g_{\omega}({\bf w}-{\bf w}^{\prime})g_{\omega}({\bf w}^{\prime}-{\bf u})v_{ \omega^{\prime},\sigma}({\bf w}^{\prime}-{\bf u}^{\prime})\|\] (113) that by (104) becomes \[\\|W^{(1;2)(k)}_{(b4);\omega^{\prime};\omega}\\|\leq{C\over 20r_{2} }\|\lambda\|\gamma^{-{4\over 3}(k-M)}\] (114) Therefore we have proved \[\\|W^{(1;2)(k)}_{\omega;\omega^{\prime}}-w^{(1;2)}_{\omega;\omega^{\prime}}\\| \leq{C\over r_{2}}\|\lambda\|\gamma^{-{4\over 3}k}\] (115) that is the second (67). _3. Improved bound for_ $(0;4)$_._ By (45) we obtain the identity in Fig.5. [FIGURE:S4.F5][ENDFIGURE] The bound for the sum of the graphs (a), (b), (d), and (e), all together, can be easily obtained from (115): for $r_{2}$ large enough \[\|\lambda\|\\|v_{\omega,\omega^{\prime}}\\|_{L_{1}}\\|W^{(1;2)(k)}_{ \omega;\omega^{\prime}}-w^{(1;2)}_{\omega;\omega^{\prime}}\\|\left(1+\\|g_{ \omega}\\|_{L_{1}}\\|W^{(0;2)(k)}_{\omega}\\|\right)\leq\|\lambda\|{C\over 2}\gamma ^{-{4\over 3}(k-M)}\;.\] (116) To bound (c), use (72) for $(n;2m)=(1;4)$, to get, for $\|\lambda\|$ small enough, \[\\|W^{(0;4)(k)}_{(a),\omega;\omega,\omega^{\prime}}\\|\leq\|\lambda\| \\|v_{\omega,\omega^{\prime}}\\|_{L_{3}}C_{0}^{2}C\|\lambda\|\\|g_{\omega}\\|_{L_{3/ 2}}\leq{C\over 2}\|\lambda\|\gamma^{-{4\over 3}(k-M)}\] (117) The proof of theorem 4.2 is complete. The analysis for $\eta^{+}=\eta^{-}\neq 0$ is not different, because the monomials in the effective potential that are proportional to at least one field $\eta^{+}$ or $\eta^{-}$ multiply a kernel that doesn’t need any power counting improvement. This is important for pointwise estimations on correlations: w.r.t. the $L_{1}$ bounds of the kernels the pointwise estimates have some missing integrations; but they never involve $W^{(0;2)}$, $W^{(1;2)}$, $W^{(0;4)}$, where, as we showed, missing integrations would spoil the bound. On scales $k\leq M$ the above arguments do not give a power counting improvement: the factors of type $\gamma^{-\vartheta(k-M)}$ that we have seen in the estimates so far are unbounded. Indeed at IR regime the interaction $v_{\omega,\omega^{\prime}}$ is effectively local, namely the system is effectively like a Thirring model. The RG approach to use is the one in [6]: with respect to that paper, the UV scale now is replace by $M$; and to the interaction, that there is purely quartic, now has a further term, that is irrelevant, generated by the integrations of the scales $[M,N]$. We do not repeat all the details of [6]. We just stress that $W^{(0;2)}$, $W^{(1;2)}$, $W^{(0;4)}$ must be _localized_ to extract the relevant part of the interaction. In this way, $W^{(0;2)}$ causes the flow of the field renormalization, which is responsible of the anomalous dimension of the large distance decay of fermion correlations. $W^{(1;2)}$ causes the flow of the current renormalization, that is responsible for the anomalous dimension of current correlations. Finally $W^{(0;4)}$ changes scale by scale the effective coupling of the quartic self-interaction: the related flow stays bounded thanks to the vanishing of the beta function, asymptotically for $M-k\to\infty$; this crucial property is not modified by the additional irrelevant interaction on scale $M$, see [6]. The final result of the multiscale integration is that there exists the limit of removed cutoff of the two point correlations; and for large $\|{\bf x}-{\bf y}\|$, \[{\partial^{2}{\cal W}\over\partial\eta^{+}_{{\bf x},\omega}\partial\eta^{-}_{{ \bf y},\omega}}(0,0)\sim{C\over\|{\bf x}-{\bf y}\|^{1+\eta}}\;,\qquad{\partial^{ 2}{\cal W}\over\partial J_{{\bf x},\omega}\partial J_{{\bf y},\omega}}(0,0) \sim{C\over\|{\bf x}-{\bf y}\|^{2+y}}\;.\] At this stage we do not know yet the explicit expressions (10), (11) of the two above correlation; nor we know the formula for $\eta$ and the fact that $y=0$. We only have a convergent power series for them. Explicit evaluations come from Wi and SDe. ### Ward Identities By definition of $J_{{\bf x},\omega}$, we have ${\partial\over\partial J_{{\bf x},\omega}}={1\over 2}\left[{\partial\over \partial J^{0}_{{\bf x}}}-i\omega{\partial\over\partial J^{1}_{{\bf x}}}\right]$. The analysis of this section will be done in Fourier transform: these are the conventions \[\psi^{\varepsilon}_{{\bf x},\omega}=\int{d{\bf k}\over(2\pi)^{2}}\;e^{i \varepsilon{\bf k}{\bf x}}\;{\widehat{\psi}}^{\varepsilon}_{{\bf k},\omega}\;, \qquad\eta^{\varepsilon}_{{\bf x},\omega}=\int{d{\bf k}\over(2\pi)^{2}}\;e^{i \varepsilon{\bf k}{\bf x}}\;{\widehat{\eta}}^{\varepsilon}_{{\bf k},\omega}\;,\] while $J_{{\bf x},\omega}$ and $v_{\omega,\omega^{\prime}}({\bf x})$ follow the same convention of $\psi^{-}_{{\bf x},\omega}$ and $\psi^{+}_{{\bf x},\omega}$, respectively. Setting \[\nu_{\omega,\omega^{\prime}}({\bf p})={\lambda\over 4\pi}\widehat{v}_{\omega, \omega^{\prime}}({\bf p})\;,\qquad B_{\omega}({\bf p})=\int\!{d{\bf k}\over(2 \pi)^{2}}\left[{\widehat{\eta}}^{+}_{{\bf k}+{\bf p},\omega}{\partial{\cal W} \over\partial{\widehat{\eta}}^{+}_{{\bf k},\omega}}-{\partial{\cal W}\over \partial{\widehat{\eta}}^{-}_{{\bf k}+{\bf p},\omega}}{\widehat{\eta}}^{-}_{{ \bf k},\omega}\right]\;,\] we want to prove that the correlation functions satisfy, in the limit of removed cutoffs, identities generated by the following equation for ${\cal W}$ \[D_{\omega}({\bf p}){\partial{\cal W}\over\partial{\widehat{J}}_{{\bf p},\omega }}-D_{-\omega}({\bf p})\sum_{\omega^{\prime}}\nu_{\omega,\omega^{\prime}}({\bf p }){\partial{\cal W}\over\partial{\widehat{J}}_{{\bf p},\omega^{\prime}}}-{1 \over 4\pi}D_{-\omega}({\bf p}){\widehat{J}}_{-{\bf p},\omega}=B_{{\bf p}, \omega}(J,\eta)\;.\] (118) Since $\nu_{\omega,\omega^{\prime}}({\bf p})=\nu_{\omega^{\prime},\omega}({\bf p})$, summing (118) over $\omega$ we find \[\sum_{\omega,\omega^{\prime}}D_{\omega}({\bf p})\left[\delta_{\omega,\omega^{ \prime}}-\nu_{-\omega,\omega^{\prime}}({\bf p})\right]{\partial{\cal W}\over \partial{\widehat{J}}_{{\bf p},\omega^{\prime}}}-{1\over 4\pi}\sum_{\omega}D_{ \omega}({\bf p}){\widehat{J}}_{-{\bf p},-\omega}=\sum_{\omega}B_{{\bf p}, \omega}(J,\eta)\;,\] that is (17) for $\nu=\lambda\Theta={\lambda\over 4\pi}$. Whereas multiplying (118) times $\omega$ and summing over $\omega$ we find \[\sum_{\omega,\omega^{\prime}}\omega D_{\omega}({\bf p})\left[\delta_{\omega, \omega^{\prime}}+\nu_{-\omega,\omega^{\prime}}({\bf p})\right]{\partial{\cal W }\over\partial{\widehat{J}}_{{\bf p},\omega^{\prime}}}+{1\over 4\pi}\sum_{ \omega}\omega D_{\omega}({\bf p}){\widehat{J}}_{-{\bf p},-\omega}=\sum_{\omega }\omega B_{{\bf p},\omega}(J,\eta)\;,\] that is (20) for $\nu_{5}=\lambda\Theta_{5}=-{\lambda\over 4\pi}$. In order to prove (118), use a general combination of the vector and axial vector transformations: for a real ${\widehat{\alpha}}_{{\bf p},\omega}$ (with the same Fourier transform convention as ${\widehat{J}}_{{\bf p},\omega}$) and transform the fields in ${\cal W}_{l,N}(\eta,J)$ as follows \[{\widehat{\psi}}^{\varepsilon}_{{\bf k},\omega}\to{\widehat{\psi}}^{ \varepsilon}_{{\bf k},\omega}+\varepsilon\int\!{d{\bf p}\over(2\pi)^{2}}{ \widehat{\alpha}}_{{\bf p},\omega}{\widehat{\psi}}^{\varepsilon}_{{\bf k}+ \varepsilon{\bf p},\omega}\;;\] that gives the identity \[D_{\omega}({\bf p}){\partial{\cal W}_{l,N}\over\partial{\widehat{J}}_{{\bf p}, \omega}}(J,\eta)=B_{{\bf p},\omega}(J,\eta)+R_{\omega,l,N}({\bf p};J,\eta)\] (119) where $R_{\omega,l,N}({\bf p};J,\eta)$ is a remainder w.r.t. the formal Wi: the presence in the free measure of the cutoff function $\chi_{l,N}({\bf k})$ explicitly breaks the vector and axial-vector symmetries. To study $R_{\omega,l,N}({\bf p};J,\eta)$ we introduce a new functional integral, \[e^{{\cal H}_{l,N}(\alpha,J,\eta)}=\int\!dP_{l,N}(\psi)\;e^{{\cal V }(\psi,\eta,J)+{\cal A}_{0}(\psi,\alpha)-{\cal A}_{-}(\psi,\alpha)}\] where \[{\cal A}_{0}(\alpha,\psi)=\sum_{\omega=\pm}\int\!{d{\bf q}\;d{\bf p }\over(2\pi)^{4}}\ C_{\omega}({\bf q}+{\bf p},{\bf q}){\widehat{\alpha}}_{{\bf p },\omega}{\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\omega}{\widehat{\psi}}^{-}_{{ \bf q},\omega}\;,\] (120) (121) \[{\cal A}_{-}(\alpha,\psi)=\sum_{\omega,\omega^{\prime}=\pm}\int\! {d{\bf q}\;d{\bf p}\over(2\pi)^{4}}\ D_{-\omega}({\bf p})\nu_{\omega,\omega^{ \prime}}({\bf p}){\widehat{\alpha}}_{{\bf p},\omega}{\widehat{\psi}}^{+}_{{\bf q }+{\bf p},\omega^{\prime}}{\widehat{\psi}}^{-}_{{\bf q},\omega^{\prime}}\] and, for ${\bf p},{\bf q}\in(\gamma^{l-1},\gamma^{N+1})$, \[C_{\omega}({\bf q},{\bf p})=[\chi_{l,N}^{-1}({\bf p})-1]D_{\omega}({\bf p})-[ \chi_{l,N}^{-1}({\bf q})-1]D_{\omega}({\bf q})\;.\] By explicit computation one can check \[{\partial{\cal H}_{l,N}\over\partial{\widehat{\alpha}}_{{\bf p},\omega}}(0,J, \eta)=R_{\omega,l,N}({\bf p};J,\eta)-D_{-\omega}({\bf p})\sum_{\omega^{\prime} }\nu_{\omega,\omega^{\prime}}({\bf p}){\partial{\cal W}_{l,N}\over\partial{ \widehat{J}}_{{\bf p},\omega^{\prime}}}(J,\eta)\] The fundamental issue behind the Adler-Bell-Jackiw anomaly corresponds, in the viewpoint of our RG scheme, to the following fact: although $C_{\omega}({\bf q},{\bf p})$ is zero for ${\bf p},{\bf q}\in[\gamma^{l},\gamma^{N}]$ and point-wise vanishing in $(\gamma^{l-1},\gamma^{N+1})$ in the limit of removed cutoffs, its insertion in the graphs of the perturbation theory, i.e. $R_{\omega,l,N}({\bf p};J,\eta)$, is not vanishing at all. Our result is that the remainder, in the limit of removed cutoffs, can anyways be computed: \[\lim_{-l,N\to\infty}{\partial{\cal H}_{l,N}\over\partial{\widehat{\alpha}}_{{ \bf p},\omega}}(J,\eta,0)={1\over 4\pi}D_{-\omega}({\bf p}){\widehat{J}}_{-{ \bf p},\omega}\;.\] (122) (Recall that (122) must be understood as generator of identities for correlations). This formula gives (118). In order to prove it, we need a multiscale integration of ${\cal H}_{l,N}$. Define the effective potential on scale $h$, ${\cal A}^{(h)}$, such that \[\;e^{{\cal H}_{l,N}(\alpha,J,\eta)}=\int\!dP_{l,h}(\psi)\ e^{{\cal V}^{(h)}( \psi,J,\eta)+{\cal A}^{(h)}(\alpha,J,\eta,\psi)}\;,\] (123) (so that ${\cal A}^{(h)}(0,J,\eta,\psi)=0$) and, correspondingly, the kernels of the monomials of ${\cal A}^{(h)}$ that are linear in $\alpha$ \[H^{(1;n;2m)(h)}_{\omega;{\underline{\omega}}^{\prime};{\underline{\omega}}}({ \bf z};{\underline{{\bf w}}};{\underline{{\bf x}}},{\underline{{\bf y}}}){ \buildrel\;{\rm def.}\;\over{=}}\prod_{i=1}^{n}{\partial\over\partial J_{{\bf w }_{i},\omega^{\prime}_{i}}}\prod_{i=1}^{m}{\partial\over\partial\psi^{+}_{{\bf x }_{i},\omega_{i}}}{\partial\over\partial\psi^{-}_{{\bf y}_{i},\omega_{i}}}{ \partial{\cal A}^{(h)}\over\partial\alpha_{{\bf z},\omega}}(0,0,0,0)\;.\] (124) For the results in this paper, we only need $n=0,1$. Because of the definitions of ${\cal A}_{0}$ and ${\cal A}_{-}$, we can have quite an explicit formula for the Fourier transforms of $H^{(1;n;2m)(h)}_{\omega;{\underline{\omega}}^{\prime};{\underline{\omega}}}$. Consider the identity, at $\alpha=\eta=0$, \[{\partial{\cal A}^{(h)}\over\partial{\widehat{\alpha}}_{{\bf p}, \omega}}=\int{d{\bf q}\over(2\pi)^{2}}C_{\omega}({\bf q}+{\bf p},{\bf q}){ \widehat{\psi}}^{+}_{{\bf q},\omega}{\widehat{\psi}}^{-}_{{\bf p}+{\bf q},\omega}\] (125) (126) (127) \[+\int{d{\bf q}\over(2\pi)^{2}}C_{\omega}({\bf q}+{\bf p},{\bf q}) \left[{\widehat{\psi}}^{-}_{{\bf q}+{\bf p},\omega}{\partial{\cal V}^{(h)} \over\partial{\widehat{\psi}}^{-}_{{\bf q},\omega}}{\widehat{g}}_{\omega}({\bf q })-{\widehat{g}}_{\omega}({\bf q}+{\bf p}){\partial{\cal V}^{(h)}\over\partial {\widehat{\psi}}^{+}_{{\bf p}+{\bf q},\omega}}{\widehat{\psi}}^{+}_{{\bf q}, \omega}\right]\] (128) (129) (130) \[+\sum_{i,j=h}^{N}\int{d{\bf q}\over(2\pi)^{2}}{\widehat{U}}^{(i,j )}_{\omega}({\bf q}+{\bf p},{\bf q}){\partial{\cal V}^{(h)}\over\partial{ \widehat{\psi}}^{+}_{{\bf p}+{\bf q},\omega}}{\partial{\cal V}^{(h)}\over \partial{\widehat{\psi}}^{-}_{{\bf q},\omega}}\] (131) (132) (133) \[+\sum_{i,j=h}^{N}\int{d{\bf q}\over(2\pi)^{2}}{\widehat{U}}^{(i,j )}_{\omega}({\bf q}+{\bf p},{\bf q}){\partial^{2}{\cal V}^{(h)}\over\partial{ \widehat{\psi}}^{+}_{{\bf p}+{\bf q},\omega}\partial{\widehat{\psi}}^{-}_{{\bf q },\omega}}\] (134) (135) (136) \[-\sum_{\omega^{\prime\prime}}D_{-\omega}({\bf p})\nu_{\omega, \omega^{\prime\prime}}({\bf p}){\partial{\cal V}^{(h)}\over\partial{\widehat{J }}_{{\bf p},\omega^{\prime\prime}}}\;,\] (137) for \[{\widehat{U}}^{(i,j)}_{\omega}({\bf q}+{\bf p},{\bf q}){\buildrel\;{\rm def.} \;\over{=}}C_{\omega}({\bf q}+{\bf p},{\bf q}){\widehat{g}}^{(i)}_{\omega}({ \bf q}+{\bf p}){\widehat{g}}^{(j)}_{\omega}({\bf q})\;.\] That suggests the following decomposition of the kernels: \[{\widehat{H}}^{(1;n;2m)(h)}_{\omega;{\underline{\omega}}^{\prime} ;{\underline{\omega}}}({\bf p};{\underline{{\bf k}}};{\underline{{\bf q}}})={ \widehat{H}}^{(1;n;2m)(h)}_{0,\omega;{\underline{\omega}}^{\prime};{\underline {\omega}}}({\bf p};{\underline{{\bf k}}};{\underline{{\bf q}}})+\sum_{\sigma= \pm}D_{\sigma\omega}({\bf p}){\widehat{H}}^{(1;n;2m)(h)}_{\sigma,\omega;{ \underline{\omega}}^{\prime};{\underline{\omega}}}({\bf p};{\underline{{\bf k} }};{\underline{{\bf q}}})\] (138) (139) \[+\sum_{\sigma=\pm}D_{\sigma\omega}({\bf p}){\widehat{K}}^{(1;n;2m )(h)}_{\sigma,\omega;{\underline{\omega}}^{\prime};{\underline{\omega}}}({\bf p };{\underline{{\bf k}}};{\underline{{\bf q}}})\;.\] (140) The first line of (125) corresponds to uncontracted ${\cal A}_{0}$, therefore is not included in the kernels. In ${\widehat{H}}^{(1;n;2m)(h)}_{0,\omega;{\underline{\omega}}^{\prime};{ \underline{\omega}}}$ there are the terms generated by the second line of (125), i.e. graphs in which only one between $\psi^{-}_{{\bf q},\omega}$ and $\psi^{+}_{{\bf q}+{\bf p},\omega}$ is contracted; ${\widehat{H}}^{(1;n;2m)(h)}_{\sigma,\omega;{\underline{\omega}}^{\prime};{ \underline{\omega}}}$ comes from the third line of (125), i.e. when $\psi^{-}_{{\bf q},\omega}$ and $\psi^{+}_{{\bf q}+{\bf p},\omega}$ are both contracted but to different graphs. Fourth and fifth line of (125) are kept together (because we want to exploit a partial cancellation among them) and generate ${\widehat{K}}^{(1;n;2m)(h)}_{\sigma,\omega;{\underline{\omega}}^{\prime};{ \underline{\omega}}}$. To explain the sum over $\sigma$, we define ${\widehat{S}}_{\omega^{\prime},\omega}^{(i,j)}$ such that \[\chi_{l,N}({\bf p}/2){\widehat{U}}_{\omega}^{(i,j)}({\bf q}+{\bf p},{\bf q})= \sum_{\sigma=\pm}D_{\sigma\omega}({\bf p}){\widehat{S}}_{\sigma\omega,\omega}^ {(i,j)}({\bf q}+{\bf p},{\bf q})\;\] (141) (we freely add the factor $\chi_{l,N}({\bf p}/2)$ because we are only interested in the case of fixed ${\bf p}\neq 0$), so that, for example, for the Fourier transform of ${\widehat{K}}^{(1;n;2m)(h)}_{\sigma,\omega;{\underline{\omega}}^{\prime};{ \underline{\omega}}}$ we find \[\;K^{(1;n;2m)(h)}_{\sigma,\omega;{\underline{\omega}}^{\prime};{ \underline{\omega}}}({\bf z};{\underline{{\bf x}}};{\underline{{\bf y}}})=\sum _{i,j=h}^{N}\int\!d{\bf u}d{\bf w}\;S^{(i,j)}_{\sigma\omega,\omega}({\bf z};{ \bf u},{\bf w})W^{(n;2+2m)(h)}_{{\underline{\omega}}^{\prime};\omega,{ \underline{\omega}}}({\underline{{\bf x}}};{\bf u},{\bf w},{\underline{{\bf y} }})\] (142) (143) \[-\delta_{\sigma,-1}\sum_{\omega^{\prime\prime}}\int\!d{\bf w}\; \nu_{\omega,\omega^{\prime\prime}}({\bf z}-{\bf w})W^{(1+n;2m)(h)}_{\omega^{ \prime\prime},{\underline{\omega}}^{\prime};{\underline{\omega}}}({\bf w};{ \underline{{\bf x}}},{\underline{{\bf y}}})\] (144) This kernel for $(n,2m)=(0,2)$ and $(1,0)$ is depicted in the the first line of Fig.6 and Fig.7, respectively. Theorem 4.3: _For $\|\lambda\|$ small enough and fixed ${\bf p}\neq 0$, there exists a $C>1$ and a $\vartheta:0<\vartheta<1$ such that, for any $k:M\leq k\leq N$ and $\sigma=\pm$,_ \[\|{\widehat{K}}^{(1;0;2)(k)}_{\sigma,\omega;\omega^{\prime}}({\bf p };{\bf k})\|\leq C\|\lambda\|\gamma^{-\vartheta(N-k)}\] (145) (146) \[\|{\widehat{K}}^{(1;1;0)(k)}_{\sigma,\omega;\omega^{\prime}}({\bf p })-\delta_{\omega,\omega^{\prime}}\delta_{\sigma,-1}{\delta({\bf p})\over 4\pi }\|\leq C\gamma^{-\vartheta(N-k)}\] (147) _and, for any other integer $(n,2m)$,_ \[\|{\widehat{K}}^{(1;n;2m)(k)}_{\sigma,\omega;{\underline{\omega}}} ({\bf p};{\underline{{\bf q}}},{\underline{{\bf k}}})\|\leq(C\|\lambda\|)^{n} \gamma^{(1-m-n)k}\gamma^{-\vartheta(N-k)}\] (148) \[\|{\widehat{H}}^{(1;n;2m)(k)}_{\sigma,\omega;{\underline{\omega}}} ({\bf p};{\underline{{\bf q}}},{\underline{{\bf k}}})\|\leq(C\|\lambda\|)^{n} \gamma^{(1-m-n)k}\gamma^{-\vartheta(N-k)}\] (149) Proof. Note that ${\widehat{U}}^{(i,j)}_{\omega}({\bf q}+{\bf p},{\bf q})$ is zero if none of $i$ and $j$ is $N$ or $l$. Besides, in the appendix of [21] there is the proof of the following bound: for any $q$ positive integer, there exists a constant $C_{q}>1$ such that \[\|S_{\bar{\omega},\omega}^{(i,j)}({\bf z};{\bf x},{\bf y})\|\leq C_ {q}{\gamma^{i}\over 1+[\gamma^{i}\|{\bf x}-{\bf z}\|]^{q}}{\gamma^{j}\over 1+[ \gamma^{j}\|{\bf y}-{\bf z}\|]^{q}}\;.\] (150) Also, we need the result of [6], \[\lim_{-l,N\to\infty}\sum_{i,j=l+1}^{N}\int\!{d{\bf p}\over(2\pi)^{2}}\ { \widehat{S}}_{-\omega,\omega}^{(i,j)}({\bf p},{\bf p})={1\over 4\pi}\;.\] (151) We bound $\|{\widehat{K}}^{(1;n;2m)(k)}\|$ and $\|{\widehat{H}}^{(1;n;2m)(k)}\|$ with the $L_{1}$ norm of $K^{(1;n;2m)(k)}$ and $H^{(1;n;2m)(k)}$, respectively. The estimate (149) is a straightforward consequence of (67), (72) and (150) for $i$ or $j$ equal to $N$; indeed, in the graphs expansion if ${\widehat{H}}^{(1;n;2m)(k)}_{\sigma,\omega;{\underline{\omega}}}$, there is no loop to worry about other than those ones already in the kernels $W^{(n;2m)(k)}$. The estimate (148), for $(n,2m)\neq(0,2),(1,0)$, is simple, because it derives from (145) by standard methods. For the estimate of the relevant and marginal kernels, (145), we have to take advantage of partial cancellations. Using the expansion for $W^{(0;4)(h)}_{\omega,\omega^{\prime}}$ in Fig.5, we can expand (142) for $(n,2m)=(0,2)$ according to Fig.6 (we have also used, for the class of graphs (d), the decomposition in Fig.2). Consider the case $\sigma=-$. [FIGURE:S4.F6][ENDFIGURE] Fixed the integer $q$ and calling $b_{j}({\bf x}){\buildrel\;{\rm def.}\;\over{=}}C_{q}\gamma^{j}/(1+[\gamma^{j}\| {\bf x}\|]^{q})$, we bound the r.h.s. member in the same spirit as in Section 4.1; though this time we also want to find the exponential small factor $\gamma^{-\vartheta(N-k)}$. Let’s first consider graphs (a) and (b) together: \[\lambda\sum_{\omega^{\prime\prime}}\int\!d{\bf u}\;\left[\sum_{i, j=k}^{N}S^{(i,j)}_{-\omega,\omega}({\bf z};{\bf u},{\bf u})-{\delta({\bf z}-{ \bf u})\over 4\pi}\right]\int\!d{\bf w}\;v_{\omega,\omega^{\prime\prime}}({\bf u }-{\bf w})W^{(1;2)(k)}_{\omega^{\prime\prime};\omega^{\prime}}({\bf w};{\bf x} ,{\bf y})\] (152) Using the identity (81), for graph (a) we have \[\lambda\sum_{\omega^{\prime\prime}}\sum_{i,j=k}^{N}\int\!d{\bf u} d{\bf w}\;S^{(i,j)}_{-\omega,\omega}({\bf z};{\bf u},{\bf u})v_{\omega,\omega^ {\prime\prime}}({\bf u}-{\bf w})W^{(1;2),(k)}_{\omega^{\prime\prime};\omega^{ \prime}}({\bf w};{\bf x},{\bf y})\] (153) (154) \[=\lambda\sum_{\omega^{\prime\prime}}\int\!d{\bf w}\;v_{\omega, \omega^{\prime\prime}}({\bf z}-{\bf w})W^{(1;2),(k)}_{\omega^{\prime\prime}; \omega^{\prime}}({\bf w};{\bf x},{\bf y})\sum_{i,j=k}^{N}\int\!d{\bf u}\;S^{(i ,j)}_{-\omega,\omega}({\bf z};{\bf u},{\bf u})\] (155) (156) \[+\lambda\sum_{p=0,1}\sum_{\omega^{\prime\prime}}\sum_{i,j=k}^{N} \int\!d{\bf u}\;S^{(i,j)}_{-\omega,\omega}({\bf z};{\bf u},{\bf u})(u_{p}-z_{p })\cdot\] (157) (158) \[\cdot\int_{0}^{1}\!d\tau\;\int\!d{\bf w}\;(\partial_{p}v _{\omega,\omega^{\prime\prime}})({\bf z}-{\bf w}+\tau({\bf u}-{\bf z}))W^{(1;2 ),(k)}_{\omega^{\prime\prime};\omega^{\prime}}({\bf w};{\bf x},{\bf y})\] (159) The latter term of the r.h.s. member of (153) has the wanted estimate: using that one between $i$ and $j$ is on scale $N$, a bound for its norm is \[8\|\lambda\|\\|W^{(1;2),(k)}_{\omega^{\prime\prime};\omega^{\prime}}\\|\\|\partial v _{\omega,\omega^{\prime\prime}}\\|_{L_{1}}\\|b_{N}\\|_{L_{1}(w)}\sum_{i=k}^{N}\\|b _{i}\\|_{L_{\infty}}\leq\|\lambda\|C_{5}\gamma^{-(k-M)}\gamma^{-(N-k)}\;.\] (160) The former term of the r.h.s. member of (153) - as opposed to what happened for (b3) of Fig4 - is not zero, but is compensated by (b): \[\sum_{i,j=k}^{N}\int\!d{\bf u}\;S^{(i,j)}_{-\omega,\omega}({\bf z };{\bf u},{\bf u})-{1\over 4\pi}=2\sum_{j\leq k-1}\int\!d{\bf u}\;S^{(N,j)}_{- \omega,\omega}({\bf z};{\bf u},{\bf u})\] (161) and hence the bound for such a difference is $C_{5}\gamma^{-(N-k)}$. The global bound for (a) and (b) together is therefore $\|\lambda\|C_{6}\gamma^{-(N-k)}$. Graph (c) corresponds to \[\lambda\sum_{i,j=k}^{N}\sum_{\omega^{\prime\prime}}\int\!d{\bf u} d{\bf u}^{\prime}d{\bf w}d{\bf w}^{\prime}\;S^{(i,j)}_{-\omega,\omega}({\bf z} ;{\bf u},{\bf w})g_{\omega}({\bf u}-{\bf u}^{\prime})v_{\omega,\omega^{\prime \prime}}({\bf u}-{\bf w}^{\prime})\cdot\] (162) (163) \[\cdot W^{(1;4)(k)}_{\omega^{\prime\prime};\omega,\omega^{\prime}} ({\bf w}^{\prime};{\bf u}^{\prime},{\bf w},{\bf x},{\bf y})\;.\] (164) Since either $i$ or $j$ has to be $N$, and because of the bound (150), the norm of (c) is bounded by \[\|\lambda\|\sum_{m=k}^{N}\sum_{i,j=k}^{N}\int\!d{\bf x}d{\bf u}^{ \prime}d{\bf w}d{\bf w}^{\prime}\;\|W^{(1;4)(k)}_{\omega^{\prime\prime};\omega, \omega^{\prime}}({\bf w}^{\prime};{\bf u}^{\prime},{\bf w},{\bf x},{\bf y})\|\] (165) \[\cdot\int\!d{\bf z}d{\bf u}\;b_{i}({\bf z}-{\bf w})b_{j}({\bf z}- {\bf u})\|v_{\omega,\omega^{\prime\prime}}({\bf u}-{\bf w}^{\prime})g^{(m)}_{ \omega}({\bf u}-{\bf u}^{\prime})\|\] (166) where $N$ reminds that at least one between $i$ and $j$ has to be $N$. As in the previous section, we bound the second line as follows: \[\\|b_{N}\\|_{L_{1}}\ \\|b_{j}\\|_{L_{3/2}}\ \\|v_{\omega,\omega^{ \prime}}\\|_{L_{3}}\ \\|g^{(m)}_{\omega}\\|_{L_{\infty}}\quad{\rm for}\quad i=N, \ m\leq j\] (167) (168) \[\\|b_{N}\\|_{L_{1}}\ \\|b_{j}\\|_{L_{\infty}}\ \\|v_{\omega,\omega^{ \prime}}\\|_{L_{3}}\ \\|g^{(m)}_{\omega}\\|_{L_{3/2}}\quad{\rm for}\quad i=N,\ j<m\] (169) (170) \[\\|b_{i}\\|_{L_{1}}\ \\|b_{N}\\|_{L_{3/2}}\ \\|v_{\omega,\omega^{ \prime}}\\|_{L_{3}}\ \\|g^{(m)}_{\omega}\\|_{L_{\infty}}\quad{\rm for}\quad j=N, \ m\leq i\] (171) (172) \[\\|b_{i}\\|_{L_{\infty}}\ \\|b_{N}\\|_{L_{1}}\ \\|v_{\omega,\omega^{ \prime}}\\|_{L_{3}}\ \\|g^{(m)}_{\omega}\\|_{L_{3/2}}\quad{\rm for}\quad j=N,\ i<m\] (173) and hence we get the bound, for $0\leq\vartheta\leq 1/3$, $C_{3}>1$ \[\int\!d{\bf z}d{\bf u}\;b_{i}({\bf z}-{\bf w})b_{j}({\bf z}-{\bf u})\|v_{\omega ,\omega^{\prime\prime}}({\bf u}-{\bf w}^{\prime})g^{(m)}_{\omega}({\bf u}-{\bf u }^{\prime})\|\leq C_{3}\gamma^{-{4\over 3}(k-M)}\gamma^{-\vartheta(N-k)}\gamma^ {k}\] (174) (with $C_{3}\to\infty$ if $\vartheta\to 1/3$). Using (72) for $W^{(1;4)(k)}_{\omega^{\prime\prime};\omega,\omega^{\prime}}$ we have for graph (c) the bound $\|\lambda\|C_{6}\gamma^{-{4\over 3}(k-M)}\gamma^{-\vartheta(N-k)}$. Graph (d) is bounded in the same way as (c) (in fact (d) was distinguished from (c) only for enumeration reasons, whereas topologically they are the same). We find \[\sum_{\sigma}\int\!d{\bf x}d{\bf u}^{\prime}d{\bf w}d{\bf w}^{ \prime}\;\|W^{(1;2)(k)}_{\omega^{\prime\prime};\omega}({\bf u};{\bf w}^{\prime} ,{\bf u}^{\prime})g_{\omega}({\bf v}-{\bf w})v_{\omega,\sigma}({\bf w}-{\bf v} ^{\prime})W^{(1;2)(k)}_{\sigma;\omega^{\prime}}({\bf v}^{\prime};{\bf x},{\bf y })\|\] (175) \[\cdot\|\lambda\|^{2}\sum_{m=k}^{N}\sum_{i,j=k}^{N}\int\!d{\bf z}d{ \bf u}\;b_{i}({\bf z}-{\bf w})b_{j}({\bf z}-{\bf u})\|v_{\omega,\omega^{\prime \prime}}({\bf u}-{\bf w}^{\prime})g^{(m)}_{\omega}({\bf u}-{\bf u}^{\prime})\|\;;\] then, with the aid of (67) and (174), we find a bound of the type $\|\lambda\|C_{3}\gamma^{-{4\over 3}(k-M)}\gamma^{-\vartheta(N-k)}$. For (e) and (f), by a simple argument, we have the bound \[4\|\lambda\|\\|W^{(1;2)(k)}_{\omega^{\prime\prime};\omega^{\prime}}\\|\left[1+\\|g_ {\omega}\\|_{L_{1}}\\|W^{(0;2)(k)}_{\omega}\\|\right]\\|v_{\omega,\omega^{\prime}} \\|_{L_{3}}\sum_{i,j=k}^{N}\\|b_{i}\\|_{L_{1}}\\|b_{j}\\|_{L_{3/2}}\;,\] that less than $\|\lambda\|C_{3}\gamma^{-{4\over 3}(k-M)}\gamma^{-\vartheta(N-k)}$. Now consider the case $\sigma=+$. The graph expansion of $K^{(1;0;2)(k)}_{+,\omega;\omega^{\prime}}$ is given again by Fig.6; the only differences is that the graph (b) is missing (that because of the $\delta_{\sigma,-1}$ in (142)). Hence a bound can be obtained with the same above argument, with only one important difference: the contribution that in the previous analysis were compensated by (b) now are zero by symmetries. Indeed, calling ${\bf k}^{}$ the rotation of ${\bf k}$ of $\pi/2$ and since ${\widehat{S}}^{(i,j)}_{\bar{\omega},\omega}({\bf k}^{},{\bf p}^{})=-\omega \bar{\omega}{\widehat{S}}^{(i,j)}_{\bar{\omega},\omega}({\bf k},{\bf p})$, in place of (161), in this case we have: \[\sum_{i,j=k}^{N}\int\!d{\bf u}\;S^{(i,j)}_{\omega,\omega}({\bf z};{\bf u},{\bf u })=\sum_{i,j=k}^{N}\int\!{d{\bf k}\over(2\pi)^{2}}\;{\widehat{S}}^{(i,j)}_{ \omega,\omega}({\bf k},-{\bf k})=0\] (176) The proof of the first of (145) is completed. We now consider the second. Expand $W^{(1;2)(h)}_{\omega^{\prime};\omega}$ as in Fig.3, and obtain the decomposition of Fig.7. In particular, class (e) comes from the kernel $w^{(1;2)}_{\omega,\omega^{\prime}}$ that is is darker bubble of Fig.3; while for (d) and (f) we also used the identity in Fig.2 to extract a further wiggly line. It is also worth stressing that (e) is not included in (a), because by construction $W^{(2;0)(h)}_{\omega,\omega^{\prime}}({\bf w},{\bf x})$ does not contain $\delta_{\omega,\omega^{\prime}}\delta({\bf x}-{\bf w})$. [FIGURE:S4.F7][ENDFIGURE] It is evident that graphs of classes (a), (b), (c) and (d) can be bounded as the graphs in homonym classes in Fig.(6): the only difference is one external wiggly line in place of two external fermion lines, but that does not change the power counting, nor the topology of the graph. A bound for graph (f) is: \[\|\lambda\|\int\!d{\bf u}^{\prime}d{\bf w}^{\prime}d{\bf w}\;\|W^{(1 ;2)(k)}_{\omega^{\prime\prime},\omega}({\bf w}^{\prime};{\bf u}^{\prime},{\bf x })g_{\omega}({\bf w}-{\bf x})\|\] (177) (178) \[\sum_{i,j=k}^{N}\int\!d{\bf z}d{\bf u}\;b_{i}({\bf z}-{\bf u})b_ {j}({\bf z}-{\bf x})\|v_{\omega,\omega^{\prime\prime}}({\bf u}-{\bf w}^{\prime} )g_{\omega}({\bf u}-{\bf u}^{\prime})\|\;.\] (179) By (174) we have the bound $\|\lambda\|C_{3}\gamma^{-\vartheta(N-k)}$. The only graph that is not bounded by the exponential small factor is (e). In fact, this kernel is finite; and to cancel it we need the to subtract $\delta_{\sigma,-1}/(4\pi)$, see Fig.8. [FIGURE:S4.F8][ENDFIGURE] The difference equals (161) for $\sigma=-1$, and (176) for $\sigma=+1$. Therefore also the second of (72) is proved. We still have to consider a last kind of kernels, ${\widehat{H}}^{(1;n;2m)(k)}_{0,\omega;{\underline{\omega}}}$. They can easily bounded, supposing ${\bf p}\neq 0$ and finite. Anyways to extract the small factor one might need the IR integration also: indeed, either the small factor comes from a contraction of ${\cal A}_{0}$ on scale $N$, or on scale $l$; the (simple) details are in [6]. As consequence of the analysis in this section, by the argument in [6], in the limit of removed cutoffs, the correlations generated by ${\cal H}$ satisfy the identities generated by (122). ### Closed Equation From the Wick theorem, see (232), we find the SDe equation \[{\partial^{2}{\cal W}_{l,N}\over\partial{\widehat{\eta}}^{+}_{{\bf k},\omega} \partial{\widehat{\eta}}^{-}_{{\bf k},\omega}}(0,0)={\widehat{g}}^{[l,N]}_{ \omega}({\bf k})\left[1+\lambda\sum_{\omega^{\prime}}\int\!{d{\bf p}\over(2\pi )^{2}}{\widehat{v}}_{\omega,\omega^{\prime}}({\bf p}){\partial^{3}{\cal W}_{l, N}\over\ \partial{\widehat{J}}_{{\bf p},\omega^{\prime}}\partial{\widehat{\eta }}^{+}_{{\bf k}+{\bf p},\omega}\partial{\widehat{\eta}}^{-}_{{\bf k},\omega}}( 0,0)\right]\] (180) that, in the limit of removed cutoffs, would be equivalent to (29); we do not take the limit now, though. Using (119) and (4.2), we find \[\sum_{\omega^{\prime\prime},\omega^{\prime}}\Big{[}D_{\omega}({ \bf p})({\widehat{v}}^{-1}({\bf p}))_{\omega,\omega^{\prime\prime}}-D_{-\omega }({\bf p}){\delta_{\omega,\omega^{\prime\prime}}\over 4\pi}\Big{]}{\widehat{v} }_{\omega^{\prime\prime},\omega^{\prime}}({\bf p}){\partial{\cal W}_{l,N}\over \partial{\widehat{J}}_{{\bf p},\omega^{\prime}}}(0,\eta)\] (181) (182) \[\hskip 30.0pt=B_{{\bf p},\omega}(0,\eta)+{\partial{\cal H}_{l,N} \over\partial{\widehat{\alpha}}_{{\bf p},\omega}}(0,0,\eta)\] (183) where the inverse of the matrix ${\widehat{v}}_{\omega,\omega^{\prime}({\bf p})}$ is \[({\widehat{v}}^{-1}({\bf p}))_{\omega,\omega^{\prime}}=({\bf p}^{2}+\mu^{2}) \delta_{\omega,-\omega}-{1\over 2}\left(1-\alpha+{\sigma\over{\bf p}^{2}} \right)\left[({\bf p}^{0})^{2}-\omega\omega^{\prime}({\bf p}^{1})^{2}-i(\omega +\omega^{\prime}){\bf p}^{0}{\bf p}^{1}\right]\;.\] As done to prove (23) and (26), use that \[\sum_{\omega}D_{\omega}({\bf p})({\widehat{v}}^{-1}({\bf p}))_{\omega,\omega^{ \prime}}=\left(\alpha{\bf p}^{2}+\mu^{2}-\sigma\right)D_{-\omega^{\prime}}({ \bf p})\] \[\sum_{\omega}\omega D_{\omega}({\bf p})({\widehat{v}}^{-1}({\bf p}))_{\omega, \omega^{\prime}}=-\omega^{\prime}\left({\bf p}^{2}+\mu^{2}\right)D_{-\omega^{ \prime}}({\bf p})\] to obtain a more explicit form of (181) \[\sum_{\omega^{\prime}}{\widehat{v}}_{\omega,\omega^{\prime}}({\bf p }){\partial{\cal W}_{l,N}\over\partial{\widehat{J}}_{{\bf p},\omega^{\prime}}} (0,\eta)=\sum_{\omega^{\prime}}M_{\omega,\omega^{\prime}}({\bf p})\left[B_{{ \bf p},\omega^{\prime}}(0,\eta)+{\partial{\cal H}_{l,N}\over\partial{\widehat{ \alpha}}_{{\bf p},\omega^{\prime}}}(0,\eta)\right]\] (184) for $M_{\omega,\omega^{\prime}}({\bf p})$ the Fourier transform of $-\partial_{\omega}F({\bf x})+\omega^{\prime}\partial_{\omega}F_{5}({\bf x})$. Plug (184) into (180) and obtain an equation that, in the limit of removed cutoff, equals (33), but for a remainder term, \[\sum_{\omega^{\prime}}\int{d{\bf p}\over(2\pi)^{2}}M_{\omega,\omega^{\prime}}( {\bf p}){\partial^{2}{\cal H}_{l,N}\over\partial{\widehat{\alpha}}_{{\bf p}, \omega^{\prime}}\partial{\widehat{\eta}}^{+}_{{\bf k}+{\bf p},\omega}}(0,\eta)\;.\] (185) We have to prove that, in the limit of removed cutoffs, (185) is vanishing. To this purpose, as in the previous section, we introduce a new functional integral \[e^{{\cal T}_{\varepsilon,l,N}(\beta,\eta)}=\int\!dP_{l,N}(\psi)e ^{{\cal V}(\psi,0,\eta)+{\cal B}_{\varepsilon,0}(\psi,\beta)-{\cal B}_{ \varepsilon,-}(\psi,\beta)}\] (186) for \[{\cal B}_{\varepsilon,0}(\psi,\beta)=\sum_{\omega}\int\!{d{\bf k} \;d{\bf p}\;d{\bf q}\over(2\pi)^{6}}\;M_{\omega,\varepsilon\omega}({\bf p})C_{ \varepsilon\omega}({\bf q}+{\bf p},{\bf q})\;{\widehat{\beta}}_{{\bf k},\omega }{\widehat{\psi}}^{-}_{{\bf k}+{\bf p},\omega}{\widehat{\psi}}^{+}_{{\bf q}+{ \bf p},\varepsilon\omega}{\widehat{\psi}}^{-}_{{\bf q},\varepsilon\omega}\;,\] (187) \[{\cal B}_{\varepsilon,-}(\psi,\beta)=\sum_{\omega,\omega^{\prime}}\int\!{d{\bf k }\;d{\bf p}\;d{\bf q}\over(2\pi)^{6}}\;M_{\omega,\omega^{\prime}}({\bf p})D_{- \omega^{\prime}}({\bf p})\nu_{\omega^{\prime},\varepsilon\omega}({\bf p})\;{ \widehat{\beta}}_{{\bf k},\omega}{\widehat{\psi}}^{-}_{{\bf k}+{\bf p},\omega} {\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\varepsilon\omega}{\widehat{\psi}}^{-}_{ {\bf q},\varepsilon\omega}\;,\] (188) Therefore we find: \[{\partial{\cal T}_{\varepsilon,l,N}\over\partial\beta_{{\bf k},\omega}}(0,\eta )=\int{d{\bf p}\over(2\pi)^{2}}M_{\omega,\varepsilon\omega}({\bf p}){\partial^ {2}{\cal H}_{l,N}\over\partial{\widehat{\alpha}}_{{\bf p},\varepsilon\omega} \partial{\widehat{\eta}}^{+}_{{\bf k}+{\bf p},\omega}}(0,\eta)\;.\] We now perform a multiscale integration of ${\cal T}_{\varepsilon,l,N}(\beta,0)$. Define ${\cal B}^{(h)}(\beta,\eta,\psi)$, the effective potential on scale $h$, to be such that \[\;e^{{\cal T}_{\varepsilon,l,N}(\beta,\eta)}=\int\!dP_{l,h}(\psi)\ e^{{\cal V} ^{(h)}(\psi,0,\eta)+{\cal B}_{\varepsilon}^{(h)}(\beta,\eta,\psi)}\;,\] (189) and correspondingly, the kernels of the monomials of ${\cal T}_{\varepsilon}^{(h)}$ that are linear in $\beta$: \[T^{(2m;2)(h)}_{\varepsilon;{\underline{\omega}},\omega}({\underline{{\bf x}}}, {\underline{{\bf y}}};{\bf u},{\bf v}){\buildrel\;{\rm def.}\;\over{=}}\prod_{ i=1}^{m}{\partial\over\partial\psi^{+}_{{\bf x}_{i},\omega_{i}}}{\partial\over \partial\psi^{-}_{{\bf y}_{i},\omega_{i}}}{\partial^{2}{\cal B}_{\varepsilon}^ {(h)}\over\partial\beta_{{\bf u},\omega}\partial\psi^{-}_{{\bf v},\omega}}(0,0 ,0)\;.\] (190) To make more explicit the kernels $T^{(2m;2)(h)}_{\varepsilon;{\underline{\omega}},\omega}$ we use the following identity at $\beta=\eta=0$: \[{\partial{\cal B}^{(h)}_{\varepsilon}\over\partial\beta_{{\bf k}, \omega}}=\sum_{i,j=h}^{N}\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega, \varepsilon\omega}({\bf p}){\widehat{U}}^{(i,j)}_{\varepsilon\omega}({\bf q}+{ \bf p},{\bf q}){\partial^{2}{\cal V}^{(h)}\over\partial{\widehat{\psi}}^{+}_{{ \bf q},\varepsilon\omega}\partial{\widehat{\psi}}^{-}_{{\bf q}+{\bf p}, \varepsilon\omega}}{\partial{\cal V}^{(h)}\over\partial{\widehat{\eta}}^{+}_{{ \bf k}+{\bf p},\omega}}\] (191) (192) (193) \[+\sum_{\omega^{\prime}}\int{d{\bf p}\over(2\pi)^{2}}M_{\omega, \omega^{\prime}}({\bf p})D_{-\omega^{\prime}}({\bf p})\nu_{\omega^{\prime}, \varepsilon\omega}({\bf p}){\partial{\cal V}^{(h)}\over\partial{\widehat{J}}_{ {\bf p},\varepsilon\omega}}{\partial{\cal V}^{(h)}\over\partial{\widehat{\eta} }^{+}_{{\bf k}+{\bf p},\omega}}\] (194) (195) (196) \[+\sum_{i,j=h}^{N}\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega, \varepsilon\omega}({\bf p}){\widehat{U}}^{(i,j)}_{\varepsilon\omega}({\bf q}+{ \bf p},{\bf q}){\widehat{g}}_{\omega}({\bf k}+{\bf p}){\partial^{3}{\cal V}^{( h)}\over\partial{\widehat{\psi}}^{+}_{{\bf k}+{\bf p},\omega}\partial{\widehat {\psi}}^{+}_{{\bf q},\varepsilon\omega}\partial{\widehat{\psi}}^{-}_{{\bf q}+{ \bf p},\varepsilon\omega}}\] (197) (198) (199) \[+\sum_{\omega^{\prime}}\int{d{\bf p}\over(2\pi)^{2}}M_{\omega, \omega^{\prime}}({\bf p})D_{-\omega^{\prime}}({\bf p})\nu_{\omega^{\prime}, \varepsilon\omega}({\bf p}){\widehat{g}}_{\omega}({\bf k}+{\bf p}){\partial^{2 }{\cal V}^{(h)}\over\partial{\widehat{\psi}}^{+}_{{\bf k}+{\bf p},\omega} \partial{\widehat{J}}_{{\bf p},\varepsilon\omega}}\] (200) (201) (202) \[-\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega,\varepsilon\omega }({\bf p})C_{\varepsilon\omega}({\bf q}+{\bf p},{\bf q})\ e^{-{\cal V}^{(h)}}{ \partial\over\partial{\widehat{\eta}}^{+}_{{\bf k}+{\bf p},\omega}}\left[e^{{ \cal V}^{(h)}}{\partial{\cal V}^{(h)}\over\partial{\widehat{\eta}}^{+}_{{\bf q },\varepsilon\omega}}{\partial{\cal V}^{(h)}\over\partial{\widehat{\eta}}^{-}_ {{\bf q}+{\bf p},\varepsilon\omega}}\right]\] (203) We decompose the kernels as follows: \[{\widehat{T}}^{(2m;2)(h)}_{\varepsilon;{\underline{\omega}},\omega}({ \underline{{\bf q}}};{\bf k})={\widehat{T}}^{(2m;2)(h)}_{0,\varepsilon;{ \underline{\omega}},\omega}({\underline{{\bf q}}};{\bf k})+{\widehat{T}}^{(2m; 2)(h)}_{1,\varepsilon;{\underline{\omega}},\omega}({\underline{{\bf q}}};{\bf k })+{\widehat{T}}^{(2m;2)(h)}_{2,\varepsilon;{\underline{\omega}},\omega}({ \underline{{\bf q}}};{\bf k})\;;\] in $T^{(2m;2)(h)}_{0,\varepsilon;{\underline{\omega}},\omega}({\underline{{\bf x}} };{\bf u},{\bf v})$ we collect the term generated by the last line of (194); in $T^{(2m;2)(h)}_{1,\varepsilon;{\underline{\omega}},\omega}({\underline{{\bf x}} };{\bf u},{\bf v})$ we included the terms generated by the third and fourth line of (194); finally, $T^{(2m;2)(h)}_{2,\varepsilon;{\underline{\omega}},\omega}({\underline{{\bf x}} };{\bf u},{\bf v})$ is related to the first two lines of (194). Theorem 4.4: _For $\|\lambda\|$ small enough, there exists a $C>1$ and a $\vartheta:0<\vartheta<1$ such that, for any $k:M\leq k\leq N$, $r=1,2$ and $\varepsilon=\pm$_ \[\|{\widehat{T}}^{(2m;2)(k)}_{r,\varepsilon;{\underline{\omega}},\omega}({ \underline{{\bf q}}},{\bf k})\|\leq C^{m}\|\lambda\|\gamma^{(1-m)k}e^{-\vartheta( N-k)}\] (204) Proof. Again we use the $L_{1}$ norm of $T^{(2m;2)(k)}$ as upper bound of $\|T^{(2m;2)(k)}\|$. In the cases $r=1,2$ the proof of (204) is a direct consequence of (145), (148). Indeed, for $r=2$, there is not new loop in the graph expansion of ${\widehat{T}}^{(2m;2)(k)}_{r,\varepsilon;{\underline{\omega}},\omega}$ w.r.t. the graph expansion of ${\widehat{K}}^{(1;0;2m)(k)}_{\sigma,\omega;{\underline{\omega}}}$ for $\sigma=\pm$. Whereas for $r=1$ there is only one loop more, that can be easily bounded: for $u_{\omega,\varepsilon\omega}^{(\sigma)}({\bf x})$ the Fourier transform of $M_{\omega,\varepsilon\omega}({\bf p})D_{\sigma\varepsilon\omega}({\bf p})$ \[T^{(2m;2)(k)}_{1,\varepsilon;\omega}({\underline{{\bf w}}};{\bf x},{\bf y})= \sum_{\sigma}\int\!d{\bf z}d{\bf u}\;u^{(\sigma)}_{\omega,\varepsilon\omega}({ \bf x}-{\bf z})g_{\omega}({\bf x}-{\bf u})K^{(1;0;2m+2)(k)}_{\sigma; \varepsilon\omega,\omega^{\prime}}({\bf z};{\underline{{\bf w}}},{\bf u},{\bf y})\] [FIGURE:S4.F9][ENDFIGURE] Note that the bound for $\\|u_{\omega,\varepsilon\omega}^{(\sigma)}\\|_{L_{p}}$ is essentially the same of $\\|v_{\omega,\omega^{\prime}}\\|_{L_{p}}$; therefore, using (145), we obtain the bound \[2\\|u^{(\sigma)}_{\omega,\varepsilon\omega}\\|_{L_{3}}\sum_{j=k}^{N}\\|g_{\omega} ^{(j)}\\|_{L_{3/2}}\\|K^{(1;0;2m+2)(k)}_{\sigma;\varepsilon\omega,\omega^{\prime }}\\|\leq C\|\lambda\|\gamma^{k}\gamma^{-{4\over 3}(k-M)}\gamma^{-\vartheta(N-k)}\;.\] (205) That completes the proof of the theorem. We shall now analyze the last kind of kernel left, $T^{(2m;2)(k)}_{0,\varepsilon;\omega}({\underline{{\bf q}}};{\bf k})$. We further expand the last line of (194) \[\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega,\varepsilon\omega} ({\bf p})C_{\varepsilon\omega}({\bf q}+{\bf p},{\bf q}){\partial{\cal V}^{(h)} \over\partial{\widehat{\eta}}^{+}_{{\bf k}+{\bf p},\omega}}{\partial{\cal V}^{ (h)}\over\partial{\widehat{\eta}}^{+}_{{\bf q},\varepsilon\omega}}{\partial{ \cal V}^{(h)}\over\partial{\widehat{\eta}}^{-}_{{\bf q}+{\bf p},\varepsilon \omega}}\] (206) (207) (208) \[-\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega,\varepsilon\omega }({\bf p})C_{\varepsilon\omega}({\bf q}+{\bf p},{\bf q}){\widehat{g}}_{ \varepsilon\omega}({\bf q}){\partial^{2}{\cal V}^{(h)}\over\partial{\widehat{ \eta}}^{+}_{{\bf k}+{\bf p},\omega}\partial{\widehat{\psi}}^{+}_{{\bf q}, \varepsilon\omega}}{\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\varepsilon\omega}\] (209) (210) (211) \[+\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega,\varepsilon\omega }({\bf p})C_{\varepsilon\omega}({\bf q}+{\bf p},{\bf q}){\widehat{g}}_{ \varepsilon\omega}({\bf q}+{\bf p}){\partial{\cal V}^{(h)}\over\partial{ \widehat{\eta}}^{+}_{{\bf k}+{\bf p},\omega}}{\widehat{\psi}}^{-}_{{\bf q}, \varepsilon\omega}{\partial{\cal V}^{(h)}\over\partial{\widehat{\psi}}^{-}_{{ \bf q}+{\bf p},\varepsilon\omega}}\] (212) (213) (214) \[+\int{d{\bf p}d{\bf q}\over(2\pi)^{4}}M_{\omega,\varepsilon\omega }({\bf p}){\widehat{U}}_{\varepsilon\omega}({\bf q}+{\bf p},{\bf q}){\widehat{ g}}_{\omega}({\bf k}+{\bf p}){\partial\over\partial{\widehat{\psi}}^{+}_{{\bf k }+{\bf p},\omega}}\left[{\partial{\cal V}^{(h)}\over\partial{\widehat{\psi}}^{ +}_{{\bf q},\varepsilon\omega}}{\partial{\cal V}^{(h)}\over\partial{\widehat{ \psi}}^{-}_{{\bf q}+{\bf p},\varepsilon\omega}}\right]\] (215) In the terms generated by the first line each of ${\widehat{\psi}}^{-}_{{\bf k}+{\bf p}}$, ${\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\varepsilon\omega}$ and ${\widehat{\psi}}^{-}_{{\bf q},\varepsilon\omega}$ is contracted with a different kernel ${\widehat{W}}^{(0,2m)(k)}$ (if any); therefore this term is bounded with (67), (72); the small factor can be extracted only if at least one between ${\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\varepsilon\omega}$ and ${\widehat{\psi}}^{-}_{{\bf q},\varepsilon\omega}$ is contracted; otherwise it comes from the IR integration, as in [6]. In the terms generated by the second and third line, one between ${\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\varepsilon\omega}$ and ${\widehat{\psi}}^{-}_{{\bf q},\varepsilon\omega}$ is not contracted; anyways in the graphical representation there is a loop that is not included in the kernels ${\widehat{W}}^{(0;2m)(k)}$, with momentum ${\bf p}$. [FIGURE:S4.F10][ENDFIGURE] Its explicit expression is \[\int{d{\bf p}\over(2\pi)^{2}}\;M_{\omega,\varepsilon\omega}({\bf p })\Big{[}\big{(}1-f_{N}({\bf k}_{1}-{\bf p})\big{)}-\big{(}\chi_{l,N}^{-1}({ \bf k}_{1})-1\big{)}{\widehat{g}}_{\omega}({\bf k}_{1}-{\bf p})\Big{]}\] (216) (217) \[\cdot{\widehat{g}}_{\omega}({\bf k}+{\bf p}){\widehat{W}}^{(0;2m+ 2)(k)}_{\omega,\varepsilon\omega{\underline{\omega}}}({\bf k}+{\bf p},{\bf k}_ {1}-{\bf p},{\underline{{\bf k}}})\] (218) The addend proportional to $\big{(}1-f_{N}({\bf k}_{1}-{\bf p})\big{)}$ has the constraint that $\|{\bf p}\|\geq c\gamma^{N}$; then $\|M({\bf p})\|\leq c\gamma^{-3N}$, and the bound is $C\gamma^{-N}\\|g_{\omega}\\|_{L_{1}}\\|W^{(0;2m+2)(k)}\\|\leq C_{M}\gamma^{-(N-k)} \gamma^{-2(k-M)}\gamma^{-(1-m)k}$; whereas the addend proportional to $\big{(}\chi_{l,N}^{-1}({\bf k}_{1})-1\big{)}$ will be contracted on scale $l$ (otherwise is zero), so obtaining the exponentially small factor, see [6]. Finally, from the fourth line we obtain terms in which ${\widehat{\psi}}^{-}_{{\bf k}+{\bf p},\omega}$ is contracted; and also both ${\widehat{\psi}}^{+}_{{\bf q}+{\bf p},\varepsilon\omega}$ and ${\widehat{\psi}}^{-}_{{\bf q},\varepsilon\omega}$ are contracted, but one of them is linked to the same kernel as ${\widehat{\psi}}^{-}_{{\bf k}+{\bf p},\omega}$. We find \[\sum_{\sigma}\sum_{i,j=k}^{N}\int\!d{\bf z}d{\bf w}d{\bf w}^{ \prime}d{\bf u}d{\bf u}^{\prime}\;u^{\sigma}_{\omega,\varepsilon\omega}({\bf z }-{\bf w})S^{(i,j)}_{\sigma\omega,\omega}({\bf w}-{\bf u},{\bf w}-{\bf u}^{ \prime})g_{\omega}({\bf z}-{\bf w}^{\prime})\] (219) (220) \[W^{(0;2m_{1})(k)}({\bf u}^{\prime},{\underline{{\bf x}}})W^{(0;2 m_{2}+2)(k)}({\bf w}^{\prime},{\bf u},{\underline{{\bf x}}}^{\prime})\] (221) that is bounded by \[4\\|u^{\sigma}_{\omega,\varepsilon\omega}\\|_{L_{3}}\\|g_{\omega}\\| _{L_{3/2}}\\|b_{N}\\|_{L_{1}}\sum_{j=k}^{N}\\|b_{j}\\|_{L_{1}}\\|W^{(0;2m_{1})(k)} \\|\\|W^{(0;2m_{2}+2)(k)}\\|\] (222) (223) \[\leq C\gamma^{(2-m_{1}-m_{2})}\gamma^{-(N-k)}\gamma^{-{4\over 3}( k-M)}\;.\] (224) The consequence of the analysis in this section is that, using the argument in [6], the correlations generated by (185) are vanishing when cutoffs are removed. Acknowledgments. I thank V.Mastropietro for suggesting the problem. _This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation._ ## Appendix A Proof of the graphical identities We remind that the derivative in the Grassmann variables $\psi^{+}$, $\eta^{+}$ and $\zeta^{+}$ are taken from the left, while the derivatives in $\psi^{-}$, $\eta^{-}$ and $\zeta^{-}$ are taken from the right. The definition of ${\cal V}^{(k)}$ is given by (40). Accordingly, the relation between ${\cal V}^{(k)}$ and ${\cal V}$ is \[\;{\cal V}^{(k)}(\psi,J,0)=\ln\int\!dP_{k+1,N}(\zeta)\ e^{{\cal V}(\psi+\zeta, J,0)}\;.\] (225) and we have the two identities: \[{\partial e^{{\cal V}^{(k)}}\over\partial\psi^{+}_{{\bf x},\omega }}(\psi,J,0)=J_{{\bf x},\omega}{\partial e^{{\cal V}^{(k)}}\over\partial\eta^{ +}_{{\bf x},\omega}}(\psi,J,0)\] (226) (227) \[\phantom{*******}+\lambda\sum_{\omega^{\prime}}\int\!d{\bf w}\ v _{\omega,\omega^{\prime}}({\bf x}-{\bf w}){\partial^{2}e^{{\cal V}^{(k)}}\over \partial J_{{\bf w},\omega^{\prime}}\partial\eta^{+}_{{\bf x},\omega}}(\psi,J, 0)\;.\] (228) (229) (230) \[{\partial{\cal V}^{(k)}\over\partial J_{{\bf x},\omega}}(\psi,J, \eta)=-e^{-{\cal V}^{(k)}(\psi,J,\eta)}{\partial^{2}e^{{\cal V}^{(k)}}\over \partial\eta^{+}_{{\bf x},\omega}\partial\eta^{-}_{{\bf x},\omega}}(\psi,J,\eta)\] (231) Moreover the _Wick theorem_ for $dP(\zeta)$ Gaussian mean values gives \[\int\!dP_{k+1,N}(\zeta)\ \zeta_{{\bf x},\omega}^{-\varepsilon}F( \zeta)=\varepsilon\int\!d{\bf u}\ g^{[k+1,N]}_{\omega}({\bf x}-{\bf u})\int\! dP_{k+1,N}(\zeta)\ {\partial F(\zeta)\over\partial\zeta_{{\bf u},\omega}^{ \varepsilon}}\] (232) this identity is straightforward for $F=\exp\{\sum_{\bf x}\zeta^{+}_{{\bf x},\omega}\eta^{-}_{{\bf x},\omega}+\sum_{ \bf x}\eta^{+}_{{\bf x},\omega}\zeta^{-}_{{\bf x},\omega}\}$ and so is also for any formal power series with even numbers of fields. Therefore \[{\partial e^{{\cal V}^{(k)}}\over\partial\eta^{\varepsilon}_{{\bf x },\omega}}(\psi,J,\eta)=\int\!dP_{k+1,N}(\zeta)\ \left(\psi_{{\bf x},\omega}^{ -\varepsilon}+\zeta_{{\bf x},\omega}^{-\varepsilon}\right)e^{{\cal V}(\psi+ \zeta,J,\eta)}\] (233) (234) \[=\psi_{{\bf x},\omega}^{-\varepsilon}e^{{\cal V}^{(k)}}+ \varepsilon\int\!d{\bf u}\ g^{[k+1,N]}_{\omega}({\bf x}-{\bf u}){\partial e^{{ \cal V}^{(k)}}\over\partial\psi_{{\bf u},\omega}^{\varepsilon}}(\psi,J,\eta)\;.\] (235) We plug the identity for $\varepsilon=+$ into (226) and get (45). Also, since $g_{\omega}^{[k+1,N]}(0)=0$, from (233) and (231) we obtain: \[{\partial e^{{\cal V}^{(k)}}\over\partial J_{{\bf x},\omega}}( \psi,J,\eta)=-\psi^{-}_{{\bf x},\omega}{\partial e^{{\cal V}^{(k)}}\over \partial\eta^{-}_{{\bf x},\omega}}(\psi,J,\eta)\] (236) (237) \[-\int\!d{\bf u}\ g^{[k+1,N]}_{\omega}({\bf x}-{\bf u}){\partial^{ 2}e^{{\cal V}^{(k)}}\over\partial\psi_{{\bf u},\omega}^{+}\partial\eta^{-}_{{ \bf x},\omega}}(\psi,J,\eta)\] (238) that gives (54). ## References [1] Abdalla E., Abdalla M.C.B., Rothe K.D.: _Non-perturbative methods in 2 dimensional quantum field theory._ World Scientific. (1991). * [2] Adler S.: _Axial-Vector Vertex in Spinor Electrodynamics._ Phys.Rev. 177 2426-2438 (1969). * [3] Adler S.L., Bardeen W.A: _Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation._ Phys.Rev. 182 1517–1536 (1969). * [4] Banerjee R.: _A Study of Two-Dimensional Models._ Z.Phys.C 25 251-258 (1984). * [5] Bell J.S., Jackiw R.: _A PCAC Puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$ model._N.Cim.A 51 47-61 (1969). * [6] Benfatto G., Falco P., Mastropietro V.: _Functional Integral Construction of the Massive Thirring model: Verification of Axioms and Massless Limit._ Comm.Math.Phys. 273 67–118 (2007). * [7] Benfatto G., Falco P., Mastropietro V.: _Massless Sine-Gordon and Massive Thirring Models: proof of the Coleman’s equivalence._ Comm.Math.Phys. 285 713-762 (2009). * [8] Benfatto G., Falco P., Mastropietro V.: _Extended scaling relations for planar lattice models._ Comm.Math.Phys. 292 569-605 (2009). * [9] Benfatto G., Gallavotti G., Procacci A., Scoppola B.: _Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the Fermi surface._ Comm.Math.Phys. 160 93-171 (1994). * [10] Benfatto G., Mastropietro V.: _Renormalization Group, hidden symmetries and approximated Ward identities in the XYZ model._Rev.Math.Phys. 13 1323-1435 (2001). * [11] Benfatto G., Mastropietro V.: _On the Density-Density Critical Indices in Interacting Fermi Systems._ Comm.Math.Phys. 231 97-134 (2002). * [12] Benfatto G., Mastropietro V.: _Ward Identities and Chiral Anomaly in the Luttinger Liquid._ Comm.Math.Phys. 258 609-655 (2005). * [13] Benfatto G., Mastropietro V.: _Universality Relations in Non-solvable Quantum Spin Chains._ J.Stat.Phys. to be published (2010). * [14] Bertlmann RA: _Anomalies in Quantum Field Theory_. Clarendon Press, Oxford.(1996) * [15] Brown L.S.: _Gauge Invariance and Mass in a Two-Dimensional Model._ N.Cim. 29 617-643 (1963). * [16] Brydges D., Slade, G. in preparation. * [17] Carey A.L., Ruijsenaars S.N.M., Wrigth J.D.: _The massless Thirring model: Positivity of Klaiber’s n-point functions._ Comm.Math.Phys. 99 347-364 (1985). * [18] Dell’Antonio G.: _Two-dimensional exactly solvable models. Some remarks._ N.Cim.A 25 303-330 (1975). * [19] Dell’Antonio G., Frishman Y., Zwanziger D.: _Thirring Model in Terms of Currents: Solution and Ligth–Cone Expansions._ Phys.Rev.D 6 988-1007 (1972). * [20] Das A., Mathur V.S.: _Path-integral solubility of two-dimensional models._ Phys.Rev.D 33 489–495 (1985). * [21] Falco P., Mastropietro V.: _Renormalization Group and Asymptotic Spin–Charge Separation for Chiral Luttinger Liquid._ J.Stat.Phys. 131 79-116 (2008). * [22] Gamboa Saravì R.E., Schaposnik F.A., Solomin J.E.: _Path-integral formulation of two-dimensional gauge theories with massless fermions._ N.Phys.B 185 239-253 (1981). * [23] Georgi H., Kats Y.: _Unparticle Example in 2D._ Phys.Rev.Lett. 101 131603 (2008). * [24] Hagen C.R.: _New solutions of the Thirring model._ N.Cim.B 51 169-186 (1967). * [25] Hagen C.R.: _Current Definition and Mass Renormalization in a Model Field Theory._ N.Cim.A 51 1033-1052 (1967). * [26] Johnson K.: _Solution of the equations for the Green functions of a two dimensional relativistic field theory._ N.Cim. 4 773-790 (1961). * [27] Klaiber B.: _The Thirring model_. In: Quantum theory and statistical physics, Vol X A, Barut A.O. and Brittin. W.F. * [28] Lesniewski A.: _Effective action for the Yukawa${}_{2}$ quantum field theory._ Comm.Math.Phys. 108 437-467 (1987). * [29]Lowenstein J.H., Swieca J.A.: _Quantum electrodynamics in two dimensions._ Ann.Phys. 68 172-195 (1971). * [30] Mastropietro V.: _Nonperturbative Adler-Bardeen theorem._ J.Math.Phys. 48 022302 (2007). * [31] Mastropietro V.: _Non-perturbative aspects of chiral anomalies._ J.Phys.A40 10349-10365 (2007). * [32] Mitter P., Scoppola B.: _The global renormalization group trajectory in a critical supersymmetric field theory on the lattice $Z_{3}$._ J.Stat.Phys. 133 921–1011 (2008). * [33] Radozycki T., Namyslowski J.M.: _Four-point Green functions in the Schwinger model._ Phys.Rev.D 59 065010 (1999). * [34] Rothe K.D., Stamatescu I.O.: _Study of a two-dimensional model with axial-current-pseudoscalar derivative interaction._ Ann.Phys 95 202-226 (1985). * [35] Salmhofer M., Seiler E.: _Proof of chiral symmetry breaking in strongly coupled lattice gauge theory._ Comm.Math.Phys. 139 395-431 (1991). * [36] Schwinger J.: _Gauge Invariance and Mass. II._ Phys.Rev. 128 2425–2429 (1962). * [37] Seiler E.: _Some remarks on the fermion determinants in gauge theories._ Phys.Rev.D 22 2412–2415 (1980). * [38] Seiler E.: _Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics._ Lecture Notes in Physics 159. Springer * [39] Sommerfield C.: _On the Definition of Currents and the Action Principle in Field Theories of One Spatial Dimension._ Ann.Phys. 26 1-43 (1964). * [40] Thirring W.E: _A soluble relativistic field theory._ Ann.Phys. 3 91-112 (1958). * [41] Thirring W.E., Wess J.E.: _Solution of a field theoretical model in one space-one time dimension._ Ann.Phys 27 331-337 (1964). * [42] Wightman, A.S.: _Introduction to some aspects of quantized fields_ Cargese lectures, 1964, New York: Gorden and Beach.
0704.1381	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 21207, "num_imgs": 0, "llama3_tokens_count": 5443 }	[]	# Mpemba effect and phase transitions in the adiabatic cooling of water before freezing S. Esposito¹ R. De Risi L. Somma Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Complesso Universitario di Monte S. Angelo, Via Cinthia, I-80126 Napoli, Italy [FOOTNOTE:1][ENDFOOTNOTE] ###### Abstract An accurate experimental investigation on the Mpemba effect (that is, the freezing of initially hot water before cold one) is carried out, showing that in the adiabatic cooling of water a relevant role is played by supercooling as well as by phase transitions taking place at $6\pm 1^{\rm o}$C, $3.5\pm 0.5^{\rm o}$C and $1.3\pm 0.6^{\rm o}$C, respectively. The last transition, occurring with a non negligible probability of 0.21, has not been detected earlier. Supported by the experimental results achieved, a thorough theoretical analysis of supercooling and such phase transitions, which are interpreted in terms of different ordering of clusters of molecules in water, is given. A well-known phenomenon such as that of the freezing of water has attracted much interest in recent times due to some counter-intuitive experimental results [1] and the apparent lacking of a generally accepted physical interpretation of them [2], [4], [3], [5]. These results consist in the fact that, _many times_, initially hot water freezes more quickly than initially cold one, a phenomenon which is now referred to as the Mpemba effect (for a short historical and scientific survey see the references in [3]). The observations sound counter-intuitive when adopting the naive, simple view according to which initially hot water has first to cool down to the temperature of the initially cold one, and then closely follow the cooling curve of the last one. The effect takes place even for not pure water, with solutions or different liquids (the original Mpemba observation occurred when he tried to make an ice cream). Several possible physical phenomena, aimed to explain such observations, have been proposed in the literature, mainly pointing out that some change in water should occur when heated [2][5]. However, such explanations cannot be applied if some precautions are taken during the experiments (whilst the Mpemba effect has been observed even in these cases) and, in any case, calculations do not seem to support quantitatively the appearance of the effect (see references in [3]). Some novel light has been introduced in the discussion, in our opinion, in Ref. [3], where the Mpemba effect has been related to the occurrence of supercooling both in preheated and in non-preheated water. Initially hot water seems to supercool to a higher local temperature than cold water, thus spontaneously freezing earlier. As a consequence, such a scenario, apparently supported by experimental investigations, points toward a statistical explanation of the effect, neither the time elapsed nor the effective freezing temperature being predictable. Here, we prefer to face the problem by starting from what is known about the _freezing_ process, rather than the _cooling_ one. In general it is known that, for given values of the thermodynamic quantities (for example the volume and the energy), a physical system may exist in a state in which it is not homogeneous, but it breaks into two or more homogeneous parts in mutual equilibrium between them. This happens when stability conditions are not fulfilled, so that a phase transition occurs; it is, for example, just the case of water that, at the pressure $p$ of 1 atm and at temperature $T$ of $0^{\rm o}$C, becomes unstable. When liquid water is cooled, the average velocities of its molecules decreases but, even if the temperature goes down to $0^{\rm o}$C (the fixed temperature where liquid and solid phases coexist) or lower, this is not a sufficient condition for freezing to start. In fact, in order that ice begins to form, first of all some molecules of the liquid water should arrange in a well-defined order to form a minimum crystal and this, in the liquid state, may happen only randomly. Second, such starting _nucleus_ has to attract further molecules in the characteristic locations of the crystalline structure of ice, by means of the interaction forces of the nucleus with the non-ordered molecules in the liquid. Nucleation and crystal growth processes are both favored at temperatures lower than $0^{\rm o}$C, so that supercooling of liquid water is generally required before its effective freezing. In fact, in pure water, only molecules in the liquid with statistically lower velocities can arrange the initial nucleus and, furthermore, only slow moving molecules are able to join that cluster and put their kinetic energies into potential energy of bond formation. When ice begins to form, these molecules are removed from those attaining to the given Maxwell distribution for the liquid water, so that the average speed becomes larger, and the temperature of the system rises to $0^{\rm o}$C (obviously, the temperature is set at the value where the continuing exchange of molecules is equal in terms of those joining and those leaving the formed crystal surface). Thus supercooling is, _de facto_, a key ingredient in the freezing process, although supercooled water exists in a state of precarious equilibrium (water is in a metastable state). Minor perturbations such as impurities or other can trigger the sudden appearance of the stable crystalline phase for the whole liquid mass, again with the release of the entire crystallization heat (melting heat) which increases the temperature of the freezing liquid to the normal $0^{\rm o}$C one. In general, when a system is in a metastable state, sooner or later it will pass to another stable state. In water, density and entropy fluctuations favor the formation of crystallization nuclei but, if the liquid constitutes a stable state, such nuclei are always unstable and will disappear with time being. However, since the fluctuations become more pronounced the lower the temperature, if water is supercooled, for sufficiently large nuclei they will result to be stable and grew with time, becoming freezing centers. The starting of the phase transition is thus _determined_ by the probability of appearance of those nuclei, and the reported Mpemba effect could be simply a manifestation of this process. We have calculated just this probability $P$ as function of the absolute temperature $T$ of the metastable phase (the one at which the nucleus is in equilibrium with the liquid), obtaining the following result ²: [FOOTNOTE:2][ENDFOOTNOTE] \[P=\frac{\alpha}{T_{\ast}}\,{\rm exp}\left\{-\beta\,\frac{T_{\ast}^{2}}{(T-T_{ \ast})^{2}}\right\}.\] (1) Here $T_{\ast}$ is the equilibrium temperature of the liquid-solid phase, $\alpha$ is a dimensionless normalization factor and $\beta$ is a constant whose expression is given by \[\beta=\frac{16\pi\tau^{3}v^{2}}{3Q^{2}kT_{\ast}},\] (2) where $\tau$ is the surface tension, $v$ the molecular volume of the crystallization nucleus, $Q$ the molecular heat of the transition from the metastable phase to the nucleus phase, and $k$ is the Boltzmann constant. Just to give an idea of the macroscopic value of the constant $\beta$, let us note that $\tau^{3}v^{2}=W_{\rm surf}^{3}$ is the cube of the work done by the surface forces, and by assuming that $Q\sim kT_{\ast}$ we may write: \[\beta\simeq\frac{16\pi}{3}\left(\frac{W_{\rm surf}}{Q}\right)^{3},\] (3) that is the constant $\beta$ is ruled by the ratio $W_{\rm surf}/Q$. The probability $P$ has a minimum at the liquid-solid equilibrium temperature $T_{\ast}$ and increases for decreasing temperature, as expected. From the formulae above it is clear that the probability for nucleation, and thus the onset of the freezing process as well, is enhanced if the work done by the surface forces (or the surface tension itself) is lowered in some way. In normal daily conditions when a commercial refrigerator is employed, this is easily induced in two simple ways: either by the presence of impurities, when solutions (such as an ice cream solution, as in the Mpemba case) are used as the freezing liquid instead of pure water, or by fluctuations of the external pressure or temperature, caused in the commercial refrigerator itself. This explains why no appreciable supercooling is observed in normal situations. Obviously, the most direct way to induce freezing in supercooled water is to introduce an external body in it, in order to directly lower the surface tension. We have thus performed an accurate experimental investigation, accounting for a total of about one hundred runs, aimed to clarify the phenomenology of the Mpemba effect and its interpretation. In the first part of our experiments we have tested all the above qualitative predictions about supercooling, by studying the cooling and freezing of tens of cm${}^{3}$ of normal water in a commercial refrigerator, in daily operation conditions. The key point, in fact, is not to obtain the most favorable physical conditions, employing sophisticated setups, but rather to reproduce the Mpemba conditions, that is adiabatic cooling (with commercial refrigerators) of not extremely small quantities of water. We have used an Onofri refrigerator for the cooling of double distilled water and a NiCr-Ni thermocouple as a temperature sensor (Leybold 666193), interfaced with a Cassy Lab software for data acquisition. For fixed temperatures of the cryostat we have indeed observed supercooling in our samples, with the freezing occurring just along the lines predicted above. In particular, during the supercooling phase we have induced a number of small perturbations in our samples, namely, variations of external pressure or temperature, mechanical perturbations or introduction of an external macroscopic body (a glass thermometer held at the same temperature of the sample). In _all_ these cases we have registered the sudden interruption of the supercooling phase and a practically instantaneous increase of the temperature to the value of $0^{\rm o}$C, denoting the starting of the freezing process. Conversely, if no perturbation is induced (or takes places) the water reached an equilibrium with the cryostat at temperatures up to about $-30^{\rm o}$C (lasting also for several thousands of seconds). We have then verified that when the freezing process started from the supercooling phase, the Mpemba effect took place with a probability in agreement with that reported in Ref. [3]. \| V= 20 cm3 \| V= 50 cm3 \| V= 65 cm3 \| V= 80 cm3 ---\|---\|---\|---\|--- PSC \| 1 \| 0.28 \| 0 \| 0.46 \| Tc=−8oC \| Tc=−14oC \| Tc=−22oC \| Tc=−26oC PSC \| 0.75 \| 0.50 \| 0 \| 0.11 Table 1: Probabilities for the occurrence of supercooling for different volumes V of the sample and different temperatures of the cryostat Tc. In about half (with a total probability of 0.47) of the runs performed we have detected a supercooling phase. In Table 1 we report the observed probability $P_{SC}$ for the occurrence of supercooling for different volumes $V$ of the water sample and for different temperatures $T_{c}$ of the cryostat. We find the data to be fitted by a straight line, denoting (in the range considered) a linear decreasing of $P_{SC}$ for decreasing temperatures of the cryostat and for increasing volumes of the samples, this probability reaching the maximum $P_{SC}=1$ for $T_{c}=0^{\rm o}$C (and $V=0$). An interesting feature of what we have observed is the sensible appearance of iced water in our samples. In fact, when supercooling did not occur, the ice started to form around the walls of the beaker, while the inner parts were still in a liquid form, as usually expected. Instead the immediate freezing of supercooled water involved the _whole_ sample, this showing a very peculiar symmetric form. We have used cylindrical beakers with the temperature sensor in their periphery, near the walls; the observed structure was a pure radial (planar) one, with no liquid water and radial filaments of ice from the center of the beakers to the walls (in one case we have been also able to take a low resolution picture of this, before its destruction outside the refrigerator). However, although supercooling plays a relevant role in the manifestation of the Mpemba effect, the things are made more complicated by the occurrence of other statistical effects before the temperature of the water reaches the value of $0^{\rm o}$C. This comes out when an accurate measurement of the cooling curves is performed (some examples of what we have obtained during the second part of our experiments are reported in Fig. 1). [FIGURE:S0.F1][ENDFIGURE] According to a simple naive model, the heat exchange from the water sample (at initial temperature $T_{0}$) to the cryostat (at fixed temperature $T_{c}$) is described by the equation \[C\,{\rm d}T\;=\;\delta\left(T_{c}-T_{0}\right){\rm d}t,\] (4) where $C$ and $\delta$ are the thermal capacity and the heat conductivity of the water, respectively. Thus by solving the differential equation in (4), the following expression for the temperature as function of time $t$ is obtained: \[T\;=\;T_{c}-\left(T_{c}-T_{0}\right){\rm e}^{-t/\tau},\] (5) where $\tau=C/\delta$ is a time constant measuring the cooling rate of the sample. However, although the overall dependence of $T$ on time is that expressed by Eq. (5), our experimental data clearly reveal the presence of three transition points before freezing (or supercooling), where $\tau$ changes its value. This transitions occur at temperatures $T_{1}=6\pm 1^{\rm o}$C, $T_{2}=3.5\pm 0.5^{\rm o}$C and $T_{3}=1.3\pm 0.6^{\rm o}$C with a probability of $P_{1}=0.11$, $P_{2}=0.84$ and $P_{3}=0.21$, respectively. The time duration $\Delta t$ of each phase transition, during which the temperature keeps practically constant ³, depends on the volume of the sample and on the temperature of the cryostat. The data we have collected are summarized in Table 2. For the phase transition at $T_{2}$ these data show a linear increase of $\Delta t_{2}$ with $T_{c}$ and a quadratic one with $V$; in Fig. 1 we give the fitting curves corresponding to best fit function $\Delta t_{2}=(a+bT_{c})V^{2}$. Instead, for the other two phase transitions no sufficient data are available in order to draw any definite conclusion on the dependence on $V$ and $T_{c}$, though $\Delta t_{1}$ and $\Delta t_{3}$ appear to be shorter than $\Delta t_{2}$. [FOOTNOTE:3][ENDFOOTNOTE] Tc=−8±2oC --- \| V= 20 cm3 \| V= 50 cm3 \| V= 65 cm3 \| V= 80 cm3 Δt1 (s) \| 7±1 \| \| \| Δt2 (s) \| 11±6 \| 220±100 \| 500±170 \| 630±160 Δt3 (s) \| 12±6 \| \| 70±30 \| Tc=−14±2oC --- \| V= 20 cm3 \| V= 50 cm3 \| V= 65 cm3 \| V= 80 cm3 Δt1 (s) \| \| \| \| 37±1 Δt2 (s) \| 8±3 \| 130±80 \| 480±160 \| 500±60 Δt3 (s) \| 7±4 \| \| \| Tc=−22±1oC --- \| V= 20 cm3 \| V= 50 cm3 \| V= 65 cm3 \| V= 80 cm3 Δt1 (s) \| \| 63±1 \| \| 7±1 Δt2 (s) \| \| 170±100 \| \| 130±70 Δt3 (s) \| \| \| \| Tc=−26±1oC --- \| V= 20 cm3 \| V= 50 cm3 \| V= 65 cm3 \| V= 80 cm3 Δt1 (s) \| \| 3.5±0.7 \| \| Δt2 (s) \| \| 3±1 \| 320±70 \| 210±170 Δt3 (s) \| \| 200±70 \| 1±1 \| Table 2: Time duration of the phase transitions at 6oC (Δt1), 3.5oC (Δt2) and 1.3oC (Δt3) for different volumes V of the sample and different temperatures of the cryostat Tc. The occurrence of these phase transitions is likely related to the formation of more or less ordered structures in water, resulting from the competition between long-range density ordering and local bond ordering maximizing the number of local bonds [8]. The anomalous density maximum at about $4^{\rm o}$C (which we observe here at $T_{2}=3.5\pm 0.5^{\rm o}$C) is, for example, explained just in term of this: as water is cooled, the local specific volume increases due to the progressive increase in tetrahedral order, so that the entropy, that always decreases upon cooling, at $4^{\rm o}$C becomes anticorrelated with the volume, resulting in an inversion (from positive to negative) of the thermal expansion coefficient and a corresponding density maximum [9]. Similar explanations in terms of different ordering could apply also to the other two transitions we have observed, but an exhaustive discussion of them, which would require more experimental data, is beyond the scope of this Letter. We only note that, while the first transition at $T_{1}=6\pm 1^{\rm o}$C seems related to the effect observed in Ref. [10] at $8^{\rm o}$C, to the best of our knowledge no other author has reported the one at $T_{3}=1.3\pm 0.6^{\rm o}$C (which, as mentioned, occurs with an appreciable probability of 0.21). \| Tc=−8oC \| Tc=−14oC \| Tc=−22oC \| Tc=−26oC ---\|---\|---\|---\|--- τ1 (s) \| 600±110 \| 680±100 \| 1000±110 \| 950±190 τ2 (s) \| 1080±260 \| 1060±170 \| 530±90 \| 570±3 τ3 (s) \| 1590±930 \| 1520±730 \| 270±50 \| 220±80 τ4 (s) \| 620±480 \| 500±180 \| 150±30 \| 640±490 Table 3: Time constants τ1 (T<T1), τ2 (T1<T<T2), τ3 (T2<T<T3), τ4 (T>T3) of the cooling curves before and after the three phase transitions detected, for different temperatures of the cryostat Tc. The observed mean values of the four time constants of the cooling curves, before and after the three phase transitions, are reported in Table 3 for different values of $T_{c}$. All the time constants are approximately _independent_ on the volume $V,$ in disagreement with the naive model discussed above which predicts an increase of $\tau$ with the thermal capacity. Instead they depend linearly on $T_{c}$, showing a negative slope for $\tau_{1}$ and positive ones for $\tau_{2},\tau_{3},\tau_{4}$ and a finite value for $T_{c}=0^{\rm o}$C. Note that (in the naive model) the ratios of the different time constants, at fixed volumes, give the (inverse) ratios of the heat conductivities in the different ordered phases (all these ratios decrease with the cryostat temperature), which are directly related to microscopic quantities like the size and average velocity of the ordered clusters of molecules in water. Coming back to the Mpemba effect, it is easy to see that Eq. (5) predicts that, for constant $\tau$, initially hot water reaches the freezing point _later_ than initially cold water. However, from what just discussed, in general this could be no longer true if the time constant changes its value during the cooling process (the slope of the cooling curves changes), or phase transitions before freezing occur (with time durations sufficiently long/short). In addition to these effects, the reaching of the freezing point does not automatically guarantees the effective starting of the freezing process, since relevant supercooling may take place, thus statistically causing the freezing of initially hot water _before_ cold one. From the data we have collected we have verified that, for given $V$ and $T_{c}$, in many cases no inversion between the cooling curves happens before the freezing point, irrespective of the change in the value of $\tau$ or the time duration of the phase transitions. Nevertheless we have as well realized that this is mainly due to the not very large difference between the initial temperatures of the samples, and in few cases (among those studied by ourselves) it cannot be applied, the largest effect causing the inversion being the phase transition at $T_{2}$. In conclusion our experimental results, and their interpretation reported here, clearly point out the statistical nature of the Mpemba effect (as already realized in [3]), whose explanation is given in terms of transitions between differently ordered phases in water and supercooling. The very detection of such phenomena seems to require the cooling to be adiabatic (as fulfilled in our experiment, as well as in those performed by other authors [3]), since for non adiabatic processes (for example, in fused salt) the coexistence of local solid nuclei in the liquid phase has been observed ⁴. [FOOTNOTE:4][ENDFOOTNOTE] An unexpected novel transition at $T_{3}=1.3\pm 0.6^{\rm o}$C has been as well detected with a non negligible probability, calling for further accurate investigation in order to achieve a more complete understanding of the unique properties of water. _Acknowledgements:_ Interesting discussions with G. Salesi and M. Villa are kindly acknowledged. ## References * (1) E.B. Mpemba, Cool. Phys. Educ. 4, 172 (1969). * (2) G.S. Kell, Am. J. Phys. 37, 564 (1969) * (3) D. Auerbach, Am. J. Phys. 63, 882 (1995). * (4) B. Wojciechowski, I. Owczarek and G. Bednarz, Crystatl. Res. Tech. 23, 843 (1988). * (5) J.I. Katz, preprint arXiv:physics/0604224. * (6) See, for example, H.B. Callen, Thermodynamics (Wiley, New York, 1960). * (7) L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1980). * (8) H. Tanaka, Phys. Rev. Lett. 80, 5750 (1998). * (9) P.G. Debenedetti and H.E. Stanley, Physics Today, June 2003, 40. * (10) K. Kotera, T. Saito and T. Yamanaka, Phys. Lett. A 345, 184 (2005).
1207.1946	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 34089, "num_imgs": 2, "llama3_tokens_count": 10618 }	[ "content_image/1207.1946/x1.png", "content_image/1207.1946/x2.png" ]	# Macroscopic superpositions via nested interferometry: finite temperature and decoherence considerations Brian Pepper,${}^{1}$ Evan Jeffrey,${}^{2}$ Roohollah Ghobadi,${}^{3,4}$ Christoph Simon,${}^{3}$ and Dirk Bouwmeester${}^{1,2}$ ${}^{1}$Department of Physics, University of California, Santa Barbara, California 93106, USA ${}^{2}$Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands ${}^{3}$Institute for Quantum Information Science and Department of Physics and Astronomy, University of Calgary, Calgary T2N 1N4, Alberta, Canada ${}^{4}$Department of Physics, Sharif University of Technology, Tehran, Iran bpepper@physics.ucsb.edu ###### Abstract Recently there has been much interest in optomechanical devices for the production of macroscopic quantum states. Here we focus on a proposed scheme for achieving macroscopic superpositions via nested interferometry. We consider the effects of finite temperature on the superposition produced. We also investigate in detail the scheme’s feasibility for probing various novel decoherence mechanisms. pacs: 42.50.Wk, 03.67.Bg, 03.65.Ta ## 1 Introduction Optomechanical systems have long been investigated as a means of probing the quantum-to-classical transition in macroscopic devices [1, 2, 3, 4, 5, 6, 7]. However, it has generally proven difficult to meet all necessary conditions for such experiments. Firstly, a sideband-resolved device is required, allowing ground state cooling [8, 9, 10, 11, 12, 13]. Secondly, the device’s coupling rate must be faster than the mechanical frequency [2, 8], in order to create a distinguishable state displaced by more than the device’s zero point motion. Finally, the device must meet the strong coupling criterion, ensuring that single photons remain in the cavity long enough to cause significant effects [2, 14, 15]. In practice it is very difficult to meet all of these competing requirements simultaneously. The authors have recently proposed a method to create quantum superpositions in weakly coupled systems via postselected nested interferometry [16]. This method greatly relaxes the above requirements, allowing the creation of quantum superpositions with devices more easily in reach of current technology [13], as well as possible tests of novel decoherence mechanisms [16]. Here we consider the experimental requirements of the proposed nested interferometry scheme, investigating in detail its tolerance of finite temperature in the resonator and finite temperature in the surrounding environment. We also analyze in detail the time scale on which decoherence mechanisms operate, including both traditional environmentally induced decoherence [17] and proposed novel decoherence mechanisms [18, 19, 20, 21, 22, 23]. ## 2 Nested interferometry Optomechanical systems evolve under the following Hamiltonian [24]: \[\hat{\mathcal{H}}=\hbar\omega_{o}\hat{a}^{{\dagger}}\hat{a}+\hbar \omega_{m}\hat{c}^{{\dagger}}\hat{c}-\hbar g\hat{a}^{{\dagger}}\hat{a}\left( \hat{c}+\hat{c}^{{\dagger}}\right),\] (1) with $\hbar$ defined as the reduced Planck’s constant, $\omega_{o}$ the optical angular frequency, $\omega_{m}$ the mechanical angular frequency, the optomechanical coupling rate $g=\omega_{o}x_{0}/L$, with the zero point motion $x_{0}=\sqrt{\hbar/(2m\omega_{m})}$, $\hat{a}$ the optical annihilation operator, and $\hat{c}$ the mechanical annihilation operator. If a single photon is input to the cavity and the mechanical state begins in coherent state $\left\|\gamma\right>_{m}$, then the mechanical state will evolve as follows [1]: \[\left\|\psi(t)\right>_{m} = \rme^{i\phi(t)}\left\|\gamma(t)+\alpha(t)\right>_{m}\] (2) \[\gamma(t) \equiv \gamma\rme^{-i\omega_{m}t}\] \[\alpha(t) \equiv \kappa(1-\rme^{-i\omega_{m}t})\] \[\phi(t) \equiv \kappa^{2}(\omega_{m}t-\sin\omega_{m}t),\] with $\kappa=g/\omega_{m}$. Here we define the set of coherent states $\left\|\gamma\right>$ as well as the single quantum-added coherent states $\left\|\gamma,1\right>$[25]: \[\left\|\gamma\right> \equiv \rme^{-\|\gamma\|^{2}/2}\sum_{n=0}^{\infty}\frac{\gamma^{n}}{\sqrt{ n!}}\left\|n\right>\] (3) \[\left\|\gamma,1\right> \equiv \frac{\hat{c}^{{\dagger}}\left\|\gamma\right>}{\sqrt{\left<\gamma \right\|\hat{c}\hat{c}^{{\dagger}}\left\|\gamma\right>}}=\frac{\exp(-\|\gamma\|^{2 }/2)}{\sqrt{\|\gamma\|^{2}+1}}\sum_{n=1}^{\infty}\frac{\gamma^{n-1}\sqrt{n}}{ \sqrt{(n-1)!}}\left\|n\right>.\] (4) <figure><img src="content_image/1207.1946/x1.png"><figcaption>Figure 1: The inner interferometer is a Mach-Zehnder interferometer. The upperpath contains Cavity A which has a weak optomechanical coupling to aresonator. In the absence of optomechanical interaction the interferometer isbalanced and all light exits via the bright port. Postselecting only thephotons which exit the normally dark port prepares the resonator in itsexcited state. Cavity B is used to match the spectrum and time delay of cavityA, and has no optomechanical interaction.</figcaption></figure> As detailed in [16] the postselection is accomplished by means of an inner Mach-Zehnder interferometer (Fig. 1). The single photon is input and split into both cavities by a beam splitter, creating state $1/\sqrt{2}(\left\|1\right>_{a}\left\|0\right>_{b}+\left\|0\right>_{a}\left\|1 \right>_{b})$. After weakly interacting ($\kappa\ll 1$, $\alpha(t)\ll 1$) with the optomechanical resonator for time t, the state will be: \[\left\|\psi(t)\right> = \frac{1}{\sqrt{2}}\left[\rme^{i\phi(t)}\left\|1\right>_{a}\left\|0 \right>_{b}\left\|\gamma(t)+\alpha(t)\right>_{m}+\left\|0\right>_{a}\left\|1 \right>_{b}\left\|\gamma(t)\right>_{m}\right].\] (5) By postselecting for photons which exit the dark port, we select the $1/\sqrt{2}(\left\|1\right>_{a}\left\|0\right>_{b}-\left\|0\right>_{a}\left\|1 \right>_{b})$ component, and compute it to lowest order in $\kappa$: \[\left\|\psi_{\mathrm{ps}}(t)\right>_{m} = \frac{1}{2}\left[\rme^{i\phi(t)}\left\|\gamma(t)+\alpha(t)\right>_ {m}-\left\|\gamma(t)\right>_{m}\right]\] (6) \[\approx \frac{1}{2}\left[\rme^{-i\kappa\gamma\sin(\omega_{m}t)}\hat{D}( \alpha(t))-1\right]\left\|\gamma(t)\right>_{m}\] \[\approx \frac{1}{2}\left[(1-i\kappa\gamma\sin(\omega_{m}t))(1+\alpha(t) \hat{c}^{\dagger}-\alpha^{}(t)\hat{c})-1\right]\left\|\gamma(t)\right>_{m}\] \[\approx \frac{1}{2}\left[\kappa\gamma(1-\cos(\omega_{m}t))\left\|\gamma(t) \right>_{m}+\alpha(t)\sqrt{\|\gamma\|^{2}+1}\left\|\gamma(t),1\right>_{m}\right]\] with $\hat{D}(\eta)$ defined as the displacement operator. In the $\gamma=0$ case, where the resonator has been cooled to its ground state, the above simplifies to a postselected state of: \[\left\|\psi_{\mathrm{ps}}(t)\right>_{m}=\frac{\alpha(t)}{2}\left\|1 \right>_{m}.\] (7) Thus in this case, the resonator is placed into the first excited state with probability $\|\alpha(t)\|^{2}/4$. The weak interaction between the photon and the device is probabilistically amplified. ### Finite device temperature However, for a device of finite temperature, $\gamma\neq 0$. Consider a mechanical resonator initially in a thermal state, a statistical mixture of coherent states: \[\hat{\rho}_{\mathrm{th}}=\frac{1}{\pi\bar{n}_{\mathrm{th}}}\int \rme^{-\|\gamma\|^{2}/\bar{n}_{\mathrm{th}}}(\left\|\gamma\right>\left<\gamma \right\|)\rmd^{2}\gamma\] (8) where $\bar{n}_{\mathrm{th}}$ is the average number of phonons: \[\bar{n}_{\mathrm{th}}\equiv\frac{1}{\rme^{\hbar\omega_{m}/k_{ \mathrm{B}}T}-1},\] (9) and where $k_{\mathrm{B}}$ represents the Boltzmann constant. Note that $\bar{n}_{\mathrm{th}}$ is also the value of $\|\gamma\|^{2}$ averaged over the thermal distribution, Eqn. 8. Note that in this subsection we will deal only with mechanical states and will thus drop the $m$ subscript. For an initial coherent state, we will have created $\left\|\psi_{\mathrm{ps}}(t)\right>$ from Eqn. 6, a superposition between a small early component with mechanical state $\left\|\psi_{\mathrm{ps}}(t)\right>$ and a large late component still in $\left\|\gamma(t)\right>$. To lowest order in $\kappa$, the overall probability of successful postselection for an initial coherent state will be: \[\left<\psi_{\mathrm{ps}}(t)\|\psi_{\mathrm{ps}}(t)\right>\approx \frac{1}{2}\left[\kappa^{2}(1-\cos\omega_{m}t)+\frac{1}{2}\kappa^{2}\|\gamma\|^{ 2}\sin^{2}\omega_{m}t\right]\] (10) Note that $\|\left<\gamma(t)\|\psi_{\mathrm{ps}}(t)\right>\|^{2}\approx(1/4)\kappa^{2}\| \gamma\|^{2}\sin^{2}\omega_{m}t$, precisely the second term of Eqn. 10. Thus the first term represents our signal, while the second term represents a background noise of dark port events due to finite temperature rather than successfully conveying a phonon to the device. Averaging Eqn. 10 over the thermal distribution, Eqn. 8, we arrive at: \[\left<\left<\psi_{\mathrm{ps}}(t)\|\psi_{\mathrm{ps}}(t)\right> \right>_{\mathrm{th}}\approx\left[\kappa^{2}\sin^{2}\frac{\omega_{m}t}{2}+ \frac{1}{4}\kappa^{2}\bar{n}_{\mathrm{th}}\sin^{2}\omega_{m}t\right]\] (11) So for the signal to be larger than the noise, we must have $\bar{n}_{\mathrm{th}}\ll 4\left[\sin(\omega_{m}t/2)/\sin\omega_{m}t\right]^{2} =\sec^{2}(\omega_{m}t/2)$. This implies that the nested interferometry proposal will only be successful if $\bar{n}\ll 1$, that is $T\ll\hbar\omega_{m}/k_{B}$. Thus, ground state cooling is essential for the success of this scheme. For a sideband-resolved device, this can be accomplished by driving the red (anti-Stokes) sideband of the cavity with a coherent beam [9, 10, 11, 12]. ### Nested interferometry The nested interferometry proposal [16] aims to use this amplification to create macroscopic superposition states, doing so by means of the extended optical setup pictured in Fig. 2. The postselection interferometer of Fig. 1 is nested in a larger interferometer with both an early and a late path. <figure><img src="content_image/1207.1946/x2.png"><figcaption>Figure 2: The outer interferometer measures the coherence of superpositionstates by the use of matched time delays. The input pulse is split bypolarizing beam splitters (PBS) into an early and late component each of whichtraverses the inner interferometer (see Fig. 1). The early and late componentsare brought back together with a second delay line and the interferencevisibility is measured by varying the phase shift ϕ. During the intervalbetween the early and late components the resonator will be in a(postselected) superposition of excited and not-excited, and any decoherenceduring that time will reduce the final measured visibility.</figcaption></figure> An experiment begins with the optomechanical device being cooled to its ground state by standard optomechanical cooling techniques [9, 10]. Single photons are input to the outer interferometer and are split into an early component and a late component. The late component enters the first delay line. The early component immediately enters the inner interferometer where it interacts with the device, and only the $\left\|1\right>_{m}$ component is passed through the dark port, entering a second equal length delay line. At this point, the late component is associated with mechanical component $\left\|0\right>_{m}$ while the early component is associated with $\left\|1\right>_{m}$. These components are left to evolve freely for the length of the delay lines, which can, in principle, be arbitrarily long. During this time they may experience decoherence from either traditional environmentally induced decoherence [17] or one of many proposed novel decoherence mechanisms [18, 19, 20, 21, 22, 23]. Finally, the components exit the delay lines and the late component enters the interferometer. As before only the $\left\|1\right>_{m}$ component passes out of the dark port and we are left with both components in the $\left\|1\right>_{m}$ state, assuming no decoherence has occurred. At this point both components are interfered to check for visibility, allowing us to measure whether decoherence has taken place. This scheme has two advantages over previous schemes. First, it allows weakly coupled devices to be placed in superpositions by a single photon. Second, in principle, it allows observation of decoherence on an arbitrary time scale, as the delay lines can be varied. Previous schemes [2, 8, 5] were limited in the time scales by both the mechanical period of oscillation and the cavity lifetime. This would require new devices to measure at different time scales. Though it may be difficult to determine the cause of the decoherence beyond any doubt, it will be possible to vary the temperature and the characteristics of the device, such as mass, frequency, mechanical quality factor, and optical finesse, allowing parameter dependence to be established. ## 3 Decoherence Here we will review the various decoherence mechanisms to be considered in this paper. The devices to be considered are hypothetical optomechanical trampoline resonators [13, 16], optimized for the nested interference scheme (Tab. 1). Device \| m \| fm \| L \| F \| Qm \| TEID \| κ \| ωm/Γc ---\|---\|---\|---\|---\|---\|---\|---\|--- Tramp. #1 [13] \| 60 \| 158 \| \| 5 \| \| 38,000 \| 43,000 \| 0.3 \| 0 \| .000034 \| 2 \| .0 Tramp. #2 [13] \| 110 \| 9 \| .71 \| 5 \| \| 29,000 \| 940,000 \| 0.4 \| 0 \| .0016 \| 0 \| .09 Proposed #1 [16] \| 1 \| 300 \| \| 0 \| .5 \| 300,000 \| 20,000 \| 0.3 \| 0 \| .001 \| 3 \| .0 Proposed #2 [16] \| 100 \| 4 \| .5 \| 5 \| \| 2,000,000 \| 2,000,000 \| 0.4 \| 0 \| .005 \| 3 \| .0 Table 1: We include parameters for two trampoline resonators [13] close to being able to implement the scheme and two devices proposed in [16] that should allow the scheme to be implemented. The parameters are effective mass of the mechanical mode (ng), mechanical mode frequency (kHz), cavity length (cm), optical finesse of cavity, mechanical quality factor, environmentally induced decoherence temperature (K), κ=g/ωm, and sideband-resolution measure ωm/Γc. Proposed device no. 2 may be capable of observing novel decoherence mechanisms [20, 26, 19, 18]. ### Environmentally induced decoherence Most devices proposed for ground state cooling [11, 12, 13] require that the device be optically cooled below the temperature $T_{\mathrm{env}}$ that the surrounding environment can reach by conventional cooling (there is one notable exception [27]). This is also true of the devices proposed in Tab. 1. In this situation, the mechanical resonator is modeled as coupled to an infinite bath of harmonic oscillators [17, 8]. In the limit of $k_{\mathrm{B}}T_{\mathrm{env}}\gg\hbar\omega_{m}$, mechanical quality factor $Q_{m}\gg 1$, and a Markovian regime with no memory effects in the bath, the bath degrees of freedom can be eliminated and the system can be described by the master equation for the reduced density matrix $\hat{\rho}$[17, 8, 28]: \[\frac{\rmd}{\rmd t}\hat{\rho}=\frac{i}{\hbar}\left[\hat{\rho}, \hat{\mathcal{H}}_{\mathrm{ren}}\right]-\frac{i\gamma_{m}}{\hbar}\left[\hat{x} ,\left\{\hat{p},\hat{\rho}\right\}\right]-\frac{D}{\hbar^{2}}\left[\hat{x}, \left[\hat{x},\hat{\rho}\right]\right],\] (12) with $\hat{\mathcal{H}}_{\mathrm{ren}}$ the Hamiltonian from Eqn. 1 renormalized by the interaction of the device and the bath, the damping coefficient $\gamma_{m}=\omega_{m}/Q_{m}$, and the diffusion coefficient $D=2m\gamma_{m}k_{\mathrm{B}}T_{\mathrm{env}}$. The first term represents the unitary evolution of the system under the Hamiltonian from Eqn. 1, while the second term represents the damping and the third term represents the diffusion. In the macroscopic regime the diffusion term proportional to $D/\hbar^{2}$ dominates Eqn. 12[17, 8]. Thus the resulting time scale for decoherence is: \[\tau_{\mathrm{EID}}\approx\frac{\hbar^{2}}{D(\Delta x)^{2}}=\frac {\hbar Q_{m}}{2k_{\mathrm{B}}T_{\mathrm{env}}},\] (13) with the superposition size $\Delta x=x_{0}$. It is helpful at this point to define an environmentally induced decoherence temperature [8]: \[T_{\mathrm{EID}}=\frac{\hbar\omega_{m}Q_{m}}{k_{B}}.\] (14) We note that the inverse of the decoherence time scale is $\tau_{\mathrm{EID}}^{-1}=2\omega_{m}(T_{\mathrm{env}}/T_{\mathrm{EID}})$. Thus for the environmentally induced decoherence to act on a time scale slower than the mechanical frequency it is necessary that $T_{\mathrm{env}}\ll T_{\mathrm{EID}}$. We will consider EID with a base temperature of $T_{\mathrm{env}}=1$ mK, obtainable with a dilution refrigerator. For this case, for the $300$ kHz device, $\tau_{\mathrm{EID}}\approx 150$$\mu$s. For the $4.5$ kHz device, $\tau_{\mathrm{EID}}\approx 15$ ms. ### Gravitationally induced decoherence Gravitationally induced decoherence, proposed independently by Diósi [18] and Penrose [19], is a type of decoherence caused by an object in superposition’s perturbation of spacetime. The time scale for such decoherence is: \[\tau_{\mathrm{P}}=\hbar/\Delta_{\mathrm{P}}\] (15) with the $\Delta_{\mathrm{P}}$ defined as follows: \[\Delta_{\mathrm{P}}=4\pi G\int\!\!\!\int\frac{(\rho_{1}(\vec{x})- \rho_{2}(\vec{x}))(\rho_{1}(\vec{y})-\rho_{2}(\vec{y}))}{\|\vec{x}-\vec{y}\|} \rmd^{3}x\rmd^{3}y,\] (16) with $\rho_{1}(\vec{x})$ and $\rho_{2}(\vec{x})$ the mass distributions of the two superposed states. As in [8], we model the system as set of spheres representing nuclei. The Penrose energy for one sphere is given by $\Delta_{\mathrm{P}}^{0}=4\pi(E_{1,2}^{0}+E_{2,1}^{0}-E_{1,1}^{0}-E_{2,2}^{0})$, with $E_{m,n}^{0}=-G\int\!\!\!\int\rho_{m}(\vec{x})\rho_{n}(\vec{x})/\|\vec{x}-\vec{y }\|\rmd^{3}x\rmd^{3}y$. The spheres considered are far enough apart and displaced little enough that their most significant interaction is with themselves, and not neighboring spheres. This means that we can merely multiply by the number of spheres, $M/m$, to get the total energy $\Delta_{\mathrm{P}}=(M/m)\Delta_{\mathrm{P}}^{0}=4\pi(E_{1,2}+E_{2,1}-E_{1,1}- E_{2,2})$, with $E_{m,n}=(M/m)E_{m,n}^{0}$ For all cases, we will consider two spherical mass distributions with radii $a$ equal to the size of the specific mass distribution that will be chosen, separated by $\Delta x=x_{0}=\sqrt{\hbar/(2m\omega)}$, the zero point motion of the resonator. Note that this is mathematically equivalent to the model of one sphere at $x=0$ for $\left\|0\right>_{m}$, and two half-mass spheres at $x=\pm x_{0}$ for $\left\|1\right>_{m}$. As the radius of the two spheres will be greater than $x_{0}$ regardless of mass distribution used, there will always be significant overlap in the distributions. This will greatly complicate evaluation of Eqn. 16. This has no effect on the self-energy terms but does affect the interaction terms. The $1/r$ potential between overlapping spheres has been evaluated previously [29]: \[E_{1,2}=\cases{-GMm/\Delta x&if $\Delta x>2a$,\\ -GMm\left[\frac{12a^{2}-5\Delta x^{2}}{10a^{3}}-\frac{\Delta x^{5}-30\Delta x^ {3}a^{2}}{160a^{6}}\right]&if $0\leq\Delta x\leq 2a$.}\] (17) For the $E_{1,1}$ and $E_{2,2}$ terms, we can just plug $\Delta x=0$ into Eqn. 17. This gives $E_{1,1}=E_{2,2}=-\frac{6GMm}{5a}$. There is considerable theoretical disagreement about the proper mass distribution to use for gravitationally induced decoherence [2, 30, 8, 31, 32]. Previous papers have used the zero point motion of the resonator itself, the nuclear radius of the nuclei making up the resonator, the zero point motion of the nuclei making up the resonator, and a completely homogeneous mass with no nuclear granularity. At this point, we will define the mass distributions to be considered in this paper. #### 3.2.1 Zero point motion of resonator Zero point motion is defined as: \[a=x_{0}=\sqrt{\frac{\hbar}{2m\omega}}\] (18) For the $300$ kHz device, $a=5.3$ fm. For the $4.5$ kHz device, $a=4.3$ fm. For this case, for the $300$ kHz device, $\tau_{\mathrm{P}}\approx 3.5$ ms. For the $4.5$ kHz device, $\tau_{\mathrm{P}}\approx 28$$\mu$s. This type of decoherence might potentially be testable in the $4.5$ kHz device, as it is faster than EID. #### 3.2.2 Radius of tantalum The atomic nucleus has a size of approximately [33]: \[a=r_{0}A^{1/3},\] (19) with $r_{0}=1.25$ fm and A the atomic mass number. Since the largest component of the mass of a Ta${}_{2}$O${}_{5}$/SiO${}_{2}$ dielectric mirror will be tantalum, we will make the simplifying assumption that the mirrors are composed of tantalum. For tantalum, $A=181$, so $a\approx 7$ fm. For this case, for the $300$ kHz device, $\tau_{\mathrm{P}}\approx 7.1$ ms. For the $4.5$ kHz device, $\tau_{\mathrm{P}}\approx 100$$\mu$s. This type of decoherence might potentially be testable in the $4.5$ kHz device, as it is faster than EID. #### 3.2.3 Zero point motion of nuclei In the Debye model, the zero point motion of nuclei in a lattice is given (Eqn. 12.3.10 in [34]): \[a=x_{0\mathrm{,nuc}}=\frac{3\hbar}{2\sqrt{k_{B}\Theta_{D}M}}.\] (20) with $\Theta_{D}$ the Debye temperature and $M$ the atomic mass. Since the largest component of the mass of a Ta${}_{2}$O${}_{5}$/SiO${}_{2}$ dielectric mirror will be tantalum, we will make the simplifying assumption that the mirrors are composed of tantalum. The Debye temperature of tantalum is $\Theta_{D}=240$ K [35], and the atomic mass $M=181$ amu. Thus $a\approx 5$ pm. For this case, for the $300$ kHz device, $\tau_{\mathrm{P}}\approx 1.8\times 10^{6}$ s. For the $4.5$ kHz device, $\tau_{\mathrm{P}}\approx 28\times 10^{3}$ s. This type of decoherence would not be testable, as it is slower than EID in both devices. #### 3.2.4 Homogeneous mass Some have even proposed modeling the resonator as a perfectly homogeneous mass with no nuclear granularity [31, 32]. In general this sets an extremely high bar for the decoherence times, but we will compute it for completeness. In this case we will model the mass as a single sphere of radius $a=30$$\mu$m (compared to a $60$$\mu$m diameter cylinder) with mass $60$ ng. It is as though the mirror is composed of one very large nucleus. Though the shape is not correct, this model will suffice for an order of magnitude estimate. This can be represented by setting the nuclear mass $m$ equal to the resonator mass $M$ in Eqn. 17. For this case, for the $300$ kHz device, $\tau_{\mathrm{P}}\approx 12\times 10^{9}$ s. For the $4.5$ kHz device, $\tau_{\mathrm{P}}\approx 1.8\times 10^{12}$ s. This type of decoherence would not be testable, as it is slower than EID in both devices. ### Continuous Spontaneous Localization Continuous spontaneous localization is a proposed position-localized decoherence mechanism in which a nonlinear stochastic classical field interacts with objects causing collapse of macroscopic superpositions. Proposed by Ghirardi, Rimini, Weber and Pearle [22, 23], the master equation and decay rate for position-localized decoherence have the following form [32, 19, 22, 23, 18, 21, 26, 20]: \[\frac{\rmd}{\rmd t}\left<x\right\|\hat{\rho}\left\|x^{\prime}\right> = \frac{i}{\hbar}\left<x\right\|[\hat{\rho},\hat{\mathcal{H}}]\left\| x^{\prime}\right>-\Gamma(x-x^{\prime})\left<x\right\|\hat{\rho}\left\|x^{\prime}\right>\] (21) \[\Gamma(x) \equiv \gamma\left[1-\exp\left(-\frac{x^{2}}{4a^{2}}\right)\right]\] (22) \[\approx \cases{\Lambda x^{2}&if $x\ll 2a$,\\ \gamma&if $x\gg 2a$.},\] (23) with $\Gamma(x)$ the decay rate, $\Lambda=\gamma/(4a^{2})$ the localization parameter, $\gamma$ the localization strength, and $a$ the localization distance. In all cases, the trampoline resonators considered are in the $x\ll 2a$ limit. For the single nucleon case, the continuous spontaneous localization model [23] gives values $a_{\mathrm{CSL}}=100$ nm and $\gamma_{\mathrm{CSL}}^{0}=10^{-16}$ Hz based on phenomenological arguments. Following [36, 32], the value of the localization parameter $\Lambda_{\mathrm{CSL}}$ can be shown to be: \[\Lambda_{\mathrm{CSL}}=\frac{M^{2}}{m_{0}^{2}}\frac{\gamma_{ \mathrm{CSL}}^{0}}{4a_{\mathrm{CSL}}^{2}}f(R,b,a)\] (24) with $M$ the resonator mass, $m_{0}$ the nucleon mass, $R$ the radius of the sphere and $f(R,b,a)$ a parameter depending on the geometry of the device. Disk geometry was considered in [36]. For motion perpendicular to the disk face $f$ is evaluated (see [36], Sec. 5.2, App. A, and Eqn. A.11): \[f(R,b,a)=4\left(\frac{2a}{R}\right)^{4}\left(\frac{2a}{b}\right) ^{2}[1-e^{-b^{2}/4a^{2}}]\int_{0}^{R/2a}x\rmd x\int_{0}^{R/2a}x^{\prime}\rmd x ^{\prime}e^{-(x^{2}+x^{\prime 2})}I_{0}(2xx^{\prime})\] (25) with $R$ the disk radius, $b$ the disk thickness, $I_{0}(x)$ the $n=0$ modified Bessel function of the first kind, and $a$ the localization distance (for CSL, $a_{\mathrm{CSL}}=100$ nm). In the $(R/2a)^{2}\gg 1$ and $(b/2a)^{2}\gg 1$ limits, applicable in this case, $f\approx(2a/R)^{2}(2a/b)^{2}$. Thus, for the $300$ kHz device, using a thickness of $\sim 5$ $\mu$m and a radius of $\sim 4$ $\mu$m (values consistent with the proposed finesse and mass), we obtain a decoherence time of order $\tau_{\mathrm{CSL}}=10^{7}$ s. For the $4.5$ kHz device, using a thickness of $\sim 5$ $\mu$m and a radius of $\sim 40$ $\mu$m, we obtain a decoherence time of order $\tau_{\mathrm{CSL}}=1.5\times 10^{5}$ s. This type of decoherence would not be testable, as it is slower than EID in both devices. ### Quantum gravity It has been proposed that quantum gravity might cause a form of position-localized decoherence due to coupling of the system to spacetime foam. This was first proposed by Ellis, Nanopoulos, Hagelin, and Srednicki [20] and subsequently elaborated [21, 26] with others. Notably, this model is phenomenologically equivalent to the CSL model with altered values for the constants [32]: $a_{\mathrm{QG}}=\hbar m_{\mathrm{P}}/2cm_{0}^{2}$ with $m_{\mathrm{P}}=\sqrt{\hbar c/G}$ the Planck mass, and $\gamma_{\mathrm{QG}}^{0}=4a_{\mathrm{QG}}^{2}c^{4}m_{0}^{6}/\hbar^{3}m_{P}^{3}$. This gives us: \[\Lambda_{\mathrm{QG}}=\frac{M^{2}}{m_{0}^{2}}\frac{\gamma_{ \mathrm{QG}}^{0}}{4a_{\mathrm{QG}}^{2}}f(R,b,a)=\frac{c^{4}M^{2}m_{0}^{4}}{ \hbar^{3}m_{P}^{3}}f(R,b,a)\] (26) with $f(R,b,a)$ as in Eqn. 25. However, since $R\ll a_{\mathrm{QG}}$ and $b\ll a_{\mathrm{QG}}$, we can set $f$ to 1 [36]: \[\Lambda_{\mathrm{QG}}\approx\frac{c^{4}M^{2}m_{0}^{4}}{\hbar^{3}m _{P}^{3}}\] (27) Thus, for the $300$ kHz device, using a thickness of $\sim 5$ $\mu$m and a radius of $\sim 4$ $\mu$m, we get a decoherence time of order $\tau_{\mathrm{QG}}=7.1$ s. For the $4.5$ kHz device, using a thickness of $\sim 5$ $\mu$m and a radius of $\sim 40$ $\mu$m, we get a decoherence time of order $\tau_{\mathrm{QG}}=1.1$ ms. This type of decoherence might potentially be testable in the $4.5$ kHz device, as it is faster than EID. ## 4 Conclusion In conclusion, we have presented an analysis of the experimental requirements of the nested interferometry scheme [16]. The scheme allows for the creation of macroscopic superpositions in weakly coupled systems, and allows for investigation of their decoherence on arbitrary time scales limited only by external delay lines. In particular, we investigate the temperature dependence of the scheme and find that ground state cooling is necessary for implementation. We also investigate the time scales on which proposed novel decoherence mechanisms would be expected to operate. We conclude that two proposed versions of gravitationally induced decoherence [18, 19] are testable, and that quantum gravitational decoherence [20, 21] is testable by this scheme. ## Acknowledgments The authors gratefully acknowledge support by the National Science Foundation grant PHY-0804177, Marie-Curie EXT-CT-2006-042580, European Commission Project MINOS, NWO VICI grant 680-47-604, an AITF New Faculty Award, and an NSERC Discovery Grant. ## References [1] S. Bose, K. Jacobs, and P. L. Knight. Scheme to probe the decoherence of a macroscopic object. _Phys. Rev. A_, 59:3204–3210, May 1999. * [2] William Marshall, Christoph Simon, Roger Penrose, and Dik Bouwmeester. Towards quantum superpositions of a mirror. _Phys. Rev. Lett._, 91(13):130401, Sep 2003. * [3] D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer. Optomechanical entanglement between a movable mirror and a cavity field. _Phys. Rev. Lett._, 98(3):030405, Jan 2007. * [4] Dustin Kleckner, William T. M. Irvine, Sumant S. R. Oemrawsingh, and Dirk Bouwmeester. Diffraction-limited high-finesse optical cavities. _Phys. Rev. A_, 81(4):043814, Apr 2010. * [5] U Akram, N Kiesel, M Aspelmeyer, and G J Milburn. Single-photon opto-mechanics in the strong coupling regime. _New J. Phys._, 12(8):083030, 2010. * [6] Oriol Romero-Isart, Mathieu L Juan, Romain Quidant, and J Ignacio Cirac. Toward quantum superposition of living organisms. _New J. Phys._, 12(3):033015, 2010. * [7] O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac. Large quantum superpositions and interference of massive nanometer-sized objects. _Phys. Rev. Lett._, 107:020405, Jul 2011. * [8] D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. van den Brink, and D. Bouwmeester. Creating and verifying a quantum superposition in a micro-optomechanical system. _New J. Phys._, 10(9):095020–+, September 2008. * [9] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg. Theory of ground state cooling of a mechanical oscillator using dynamical backaction. _Phys. Rev. Lett._, 99(9):093901, Aug 2007. * [10] Florian Marquardt, Joe P. Chen, A. A. Clerk, and S. M. Girvin. Quantum theory of cavity-assisted sideband cooling of mechanical motion. _Phys. Rev. Lett._, 99(9):093902, Aug 2007. * [11] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds. Sideband cooling of micromechanical motion to the quantum ground state. _Nature_, 475:359–363, July 2011. * [12] Jasper Chan, T. P. Mayer Alegre, Amir H. Safavi-Naeini, Jeff T. Hill, Alex Krause, Simon Groblacher, Markus Aspelmeyer, and Oskar Painter. Laser cooling of a nanomechanical oscillator into its quantum ground state. _Nature_, 478(7367):89–92, 10 2011. * [13] Dustin Kleckner, Brian Pepper, Evan Jeffrey, Petro Sonin, Susanna M. Thon, and Dirk Bouwmeester. Optomechanical trampoline resonators. _Opt. Express_, 19(20):19708–19716, Sep 2011. * [14] J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris. Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. _Nature_, 452:72–75, March 2008. * [15] S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer. Observation of strong coupling between a micromechanical resonator and an optical cavity field. _Nature_, 460:724, 2009. * [16] Brian Pepper, Roohollah Ghobadi, Evan Jeffrey, Christoph Simon, and Dirk Bouwmeester. Optomechanical superpositions via nested interferometry. Phys. Rev. Lett. (to be published). * [17] Wojciech Hubert Zurek. Decoherence, einselection, and the quantum origins of the classical. _Rev. Mod. Phys._, 75(3):715–775, May 2003. * [18] L. Diósi. Models for universal reduction of macroscopic quantum fluctuations. _Phys. Rev. A_, 40:1165–1174, August 1989. * [19] R. Penrose. On Gravity’s role in Quantum State Reduction. _Gen. Relativ. Gravit._, 28:581–600, May 1996. * [20] John Ellis, John S. Hagelin, D.V. Nanopoulos, and M. Srednicki. Search for violations of quantum mechanics. _Nucl. Phys. B_, 241(2):381 – 405, 1984. * [21] John Ellis, Subhendra Mohanty, and Dimitri V. Nanopoulos. Quantum gravity and the collapse of the wavefunction. _Phys. Lett. B_, 221(2):113 – 119, 1989. * [22] G. C. Ghirardi, A. Rimini, and T. Weber. Unified dynamics for microscopic and macroscopic systems. _Phys. Rev. D_, 34:470–491, Jul 1986. * [23] Gian Carlo Ghirardi, Philip Pearle, and Alberto Rimini. Markov processes in hilbert space and continuous spontaneous localization of systems of identical particles. _Phys. Rev. A_, 42:78–89, Jul 1990. * [24] C. K. Law. Interaction between a moving mirror and radiation pressure: A hamiltonian formulation. _Phys. Rev. A_, 51(3):2537–2541, Mar 1995. * [25] G. S. Agarwal and K. Tara. Nonclassical properties of states generated by the excitations on a coherent state. _Phys. Rev. A_, 43:492–497, Jan 1991. * [26] John Ellis, N.E. Mavromatos, and D.V. Nanopoulos. String theory modifies quantum mechanics. _Phys. Lett. B_, 293(1-2):37 – 48, 1992. * [27] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland. Quantum ground state and single-phonon control of a mechanical resonator. _Nature_, 464:697–703, April 2010. * [28] A.O. Caldeira and A.J. Leggett. Path integral approach to quantum brownian motion. _Physica A: Statistical Mechanics and its Applications_, 121(3):587 – 616, 1983. * [29] M W Kermode, M M Mustafa, and N Rowley. Coulomb potential between two uniformly charged heavy ions. _Journal of Physics G: Nuclear and Particle Physics_, 16(12):L299, 1990. * [30] Lajos Diósi. Notes on certain newton gravity mechanisms of wavefunction localization and decoherence. _J. Phys. A_, 40(12):2989, 2007. * [31] Filippo Maimone, Giovanni Scelza, Adele Naddeo, and Vinicio Pelino. Quantum superpositions of a mirror for experimental tests for nonunitary newtonian gravity. _Phys. Rev. A_, 83:062124, Jun 2011. * [32] Oriol Romero-Isart. Quantum superposition of massive objects and collapse models. _Phys. Rev. A_, 84:052121, Nov 2011. * [33] Kenneth S. Krane. _Introductory Nuclear Physics_. Wiley, 1987. * [34] Jenő Sólyom. _Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics_. Springer, 2007. * [35] Charles Kittel. _Introduction to Solid State Physics_. Wiley, 2004. * [36] Brian Collett and Philip Pearle. Wavefunction collapse and random walk. _Found. Phys._, 33:1495–1541, 2003.
1803.01701	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 125088, "num_imgs": 2, "llama3_tokens_count": 47798 }	[ "content_image/1803.01701/4ptGraphs.jpg", "content_image/1803.01701/5ptGraphs.jpg" ]	# Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes Yi-Jian Du c,d , Yong Zhang yijian.du@whu.edu.cn, yongzhang@itp.ac.cn Center for Theoretical Physics, School of Physics and Technology, Wuhan University, No.299 Bayi Road, Wuhan 430072, ChinaSuzhou Institute of Wuhan University, No.377 Linquan Street, Suzhou, 215123, ChinaCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, ChinaUniversity of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China February 29, 2024 ###### Abstract In this paper, we derive generalized Bern-Carrasco-Johansson (BCJ) relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM, which are derived from Feynman rules, Abelian Z-theory and Cachazo-He-Yuan (CHY) formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other. Keywords:Amplitude Relation, BCJ Numerator, Gauge Invariance ## 1 Introduction Color-kinematic duality (BCJ duality), which was suggested by Bern Carrasco and Johansson [1; 2], provides a deep insight into the study of scattering amplitudes. According to BCJ duality, full color-dressed Yang-Mills amplitudes are expressed by summing over trivalent (Feynman-like) diagrams, each of which is associated with a color factor and a kinematic factor (BCJ numerator) sharing the same algebraic properties (_i.e._, antisymmetry and Jacobi identity). Once the color factors are replaced by BCJ numerators of another copy Yang-Mills amplitude, we obtain a gravity amplitude. A significant consequence of BCJ duality is that tree-level color-ordered Yang-Mills amplitudes satisfy BCJ relations where the coefficients for amplitudes are functions of Mandelstam variables. Together with the earlier proposed Kleiss-Kuijf [3] (KK) relations, BCJ relations reduce the number of independent color-ordered Yang-Mills amplitudes to $(n-3)!$ (see the field theory proofs [4; 5] and string theory approaches [6; 7]). Though BCJ relations are first discovered in Yang-Mills theory, they actually hold for amplitudes in many other theories including: bi-scalar theory, NLSM [8], which can be uniformly described in the framework of CHY formulation [9; 10; 11; 12]. It was pointed out that fundamental BCJ relation can be regarded as the most elementary one since the minimal basis [4] and a set of more general BCJ relations [6; 5] are generated by them [13]. Nevertheless, in some situations, one may encounter BCJ relations which have much more complicated forms than knowns ones. Such relations can be neither directly understood as a result of fundamental relations nor straightforwardly proved by Britto-Cachazo-Feng-Witten [14; 15] recursion or CHY formula. Therefore, a new approach to nontrivial BCJ relations is required. Apart from the BCJ relations for amplitudes, the construction of BCJ numerators in various theories is also an important direction. In NLSM, there are three distinct constructions of BCJ numerators, all of which are polynomial functions of Mandelstam variables. (i) A construction based on off-shell extended BCJ relation (see [8]) was suggested by Fu and one of the current authors [16] (DF). In DF approach, the set of half-ladder numerators with the first and the last lines fixed (which serves as a basis of BCJ numerators) are expressed by proper combinations of momentum kernels [17; 18; 19; 20; 21; 22]. Since the off-shell extended BCJ relation [8] was proved by the use of Berends-Giele recursion (Feyman rules), the DF type BCJ numerators can be essentially regarded as a result of Feyman rules. (ii) A much more compact construction of BCJ numerators in NLSM, which was based on Abelian Z theory, was provided by Carrasco, Mafra and Schlotterer (CMS) [23]. A half ladder numerator of CMS type is elegantly expressed by only one momentum kernel. (iii) In a more recent work [24], a graphic approach to polynomial BCJ numerators (DT type numerator) in NLSM, which was based on CHY formula was proposed by Teng and one current author. All the three distinct constructions given above must produce the same scattering amplitudes in NLSM, but this equivalence is still not proven explicitly. In this paper, we derive highly nontrivial generalized BCJ relations (gauge invariance induced relations) by imposing gauge invariance conditions and CHY-inspired dimensional reduction on the recent discovered graphic expansion of color-ordered Einstein-Yang-Mills (EYM) amplitudes [24]. Expansion of EYM amplitudes was first proposed in [25] and further studied in [26; 27; 28; 29; 30; 31; 24; 32]. In the series work [29; 31; 24; 32], general recursive expansion for all tree-level EYM amplitudes and the graphic expansion of EYM amplitudes in terms of pure Yang-Mills ones were established. When gauge invariance condition for the so-called fiducial graviton is imposed, the recursive expansion of EYM amplitudes induces relations between those amplitudes with fewer gravitons. Equivalently, when the graphic expansion [24] is considered, such gauge invariance induced relation implies a relation between color ordered Yang-Mills amplitudes whose coefficients are functions of both momenta and polarizations. To induce amplitude relations where all coefficients are functions of Mandelstam variables, one should convert all polarizations in the coefficients into momenta. In the current paper, we propose gauge invariance induced relations based on the following two crucial observations: (i) One can impose the gauge invariance conditions for several gravitons simultaneously. (ii) The gauge invariance conditions are independent of dimensions. With these two critical observations in hand and inspired by the dimensional reduction in CHY formula [12], we define $(d+d)$-dimensional polarizations and momenta whose nonzero components are expressed by only $d$-dimensional momenta. Imposing the gauge invariance in $(d+d)$ dimensions on the graphic expansion [24] of single-trace EYM amplitudes, we naturally induce nontrivial amplitude relations where all coefficients are polynomials of Mandelstam variables (in $d$ dimensions). In the framework of CHY formula, such relations become nontrivial relations between Parke-Taylor factors. As a consequence, the gauge invariance induced relations hold for not only color-ordered Yang-Mills amplitudes but also color-ordered amplitudes in other theories such as bi-scalar theory and NLSM. An interesting application of our gauge invariance induced relation is the proof of equivalence between different approaches to NLSM amplitudes. Full color-dressed NLSM amplitudes can be spanned in terms of bi-scalar amplitudes via dual Del Duca-Dixon-Maltoni (DDM) [33] decomposition (The dual DDM decomposition for Yang-Mills amplitudes are given in [34; 22; 35; 36; 37; 38; 11; 39; 40; 41], for NLSM amplitudes are provided in [8; 16; 23; 24]), in which the coefficients are half-ladder BCJ numerators with fixing the first and the last lines. Although the three distinct approaches: Feyman rules, Abelian Z theory and CHY formula provide different types of half-ladder BCJ numerators, they must produce the same NLSM amplitudes through the dual DDM decomposition. This equivalence condition then requires nontrivial relations between color-ordered bi-scalar amplitudes. By using the gauge invariance induced relations and defining partial momentum kernel, we prove that the three distinct constructions of BCJ numerators produce the same NLSM amplitudes precisely. In other words, the equivalence between the three different approaches to NLSM amplitudes is explicitly proven. The relation between main results of this paper is provided as \[\begin{array}[]{c}\text{gauge invariance}\\ +\\ \text{dimensional reduction}\\ \end{array}\Rightarrow\text{generalized BCJ \eqref{Eq:NewGaugeIDAmp1}} \Rightarrow\begin{array}[]{ccc}&\text{relation \eqref{Eq:GenEquiv1}}& \Rightarrow\text{equivalence between CMS $\&$ DT}\\ \nearrow&&\\ \searrow&&\\ &\text{relation \eqref{Eq:GenEquiv2}}&\Rightarrow\text{equivalence between DF $\&$ CMS}\\ \end{array}.\] The structure of this paper is given as follows. In section 2, we provide a review of the background knowledge including CHY formula, the recursive expansion and the graphic expansion of EYM amplitudes. In section 3, we induce generalized BCJ relations by combining gauge invariance conditions and dimensional reduction. Partial momentum kernel, which is important for the discussions in this paper, is introduced in section 3. A review of the three distinct constructions of BCJ numerators in NLSM is provided in section 4. In section 5, we prove the equivalence between CMS type and DT type numerators by inducing identities expressed by partial momentum kernel. The proof of equivalence between DF type and CMS type numerators is given in section 6. We summarized this paper in section 7. Complicated graphs and proofs are included by appendices. ## 2 A review of CHY formula and the expansion of EYM amplitudes In this section, we review the CHY formula [9; 10; 11; 42; 12] for various theories and the recursive/graphic expansion of EYM amplitudes which will be used in the coming sections. ### CHY formula CHY formula expresses a tree level on-shell amplitude with $n$ massless particles by integration over $n$ scattering variables $z_{i}$ \[A=\int d\Omega_{\text{CHY}}\mathcal{I}_{L}\mathcal{I}_{R},\] (1) where $d\Omega_{\text{CHY}}$ is Möbius invariant measure which contains the condition that scattering variables satisfy the following scattering equations \[\sum\limits_{j\neq i}{k_{i}\cdot k_{j}\over z_{i}-z_{j}}=0,~{}~{} ~{}\text{(}i=1,\dots,n).\] (2) Here $k_{i}$ denotes the momenta of the particle $i$. The integrand $\mathcal{I}_{L}\mathcal{I}_{R}$ in (1) relies on theories. An important feature is that the CHY formula is independent of dimensions. #### The CHY integrand for BS, CS, YM, EYM and GR amplitudes The CHY integrands for color-ordered bi-scalar (BS), Yang-Mills (YM), single-trace EYM amplitudes (EYM) as well as gravity (GR) amplitudes are given by¹ [FOOTNOTE:1][ENDFOOTNOTE] \[\mathcal{I}^{\text{BS}}_{L}(\boldsymbol{\sigma}_{1,n}) = (-1)^{{(n+1)(n+2)\over 2}}\text{PT}(\boldsymbol{\sigma}_{1,n}),~{ }~{}~{}~{}~{}~{}~{}~{}\mathcal{I}^{\text{BS}}_{R}(\boldsymbol{\rho}_{1,n})=(-1 )^{{(n+1)(n+2)\over 2}}\text{PT}(\boldsymbol{\rho}_{1,n})\] (3) \[\mathcal{I}^{\text{YM}}_{L}(\boldsymbol{\sigma}_{1,n}) = (-1)^{{(n+1)(n+2)\over 2}}\text{PT}(\boldsymbol{\sigma}_{1,n}),~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{I}^{ \text{YM}}_{R}\,={\mbox{Pf}}\,^{\prime}[\Psi]\] (4) \[\mathcal{I}^{\text{EYM}}_{L}(\boldsymbol{\sigma}_{1,r}) = (-1)^{{(n+1)(n+2)+s(s+1)\over 2}}\text{PT}(\boldsymbol{\sigma}_{1 ,r}){\mbox{Pf}}[\Psi_{\mathsf{H}}],~{}~{}~{}~{}\mathcal{I}^{\text{EYM}}_{R}={ \mbox{Pf}}\,^{\prime}[\Psi]\] (5) \[\mathcal{I}^{\text{GR}}_{R} = {\mbox{Pf}}\,^{\prime}[\Psi],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{} ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{} ~{}~{}~{}~{}~{}\mathcal{I}^{\text{GR}}_{R}\,={\mbox{Pf}}\,^{\prime}[\Psi].\] (6) In (3) and (4), the boldface Greek letters $\boldsymbol{\sigma}_{1,n}$ and $\boldsymbol{\rho}_{1,n}$ denote permutations of all $n$ external particles $1,2,\dots,n$. The Parke-Taylor factor $\text{PT}(\boldsymbol{\sigma}_{1,n})$ is defined by \[\text{PT}(\boldsymbol{\sigma}_{1,n})={1\over z_{\sigma(1)\sigma(2 )}z_{\sigma(2)\sigma(3)}\dots z_{\sigma(n)\sigma(1)}},~{}~{}~{}~{}z_{ij}\equiv z _{i}-z_{j}.\] (7) The reduced Pfaffian ${\mbox{Pf}}\,^{\prime}[\Psi]$ in (4), (5) and (6) is given by \[{\mbox{Pf}}\,^{\prime}\left[\Psi\right]\equiv{(-1)^{i+j}\over z_{ ij}}{\mbox{Pf}}\left[\Psi^{i,j}_{i,j}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{} ~{}~{}\Psi=\left(\begin{array}[]{cc}{A}&-{C}^{T}\\ {C}&{B}\end{array}\right)\,,\] (8) where $\Psi^{i,j}_{i,j}$ means that the $i$, $j$-th ($1\leq i,j\leq n$) rows and columns are removed. Building blocks of the $2n\times 2n$-skew matrix $\Psi$ are (9) in which $k_{a}$ and $\epsilon_{a}$ are momentum and polarization of the particle $a$. In the CHY expression of single-trace EYM amplitude (5), $\text{PT}(\boldsymbol{\sigma}_{1,r})$ denotes the Parke-Taylor factor for $r$ gluons with the order $\sigma(1),\sigma(2),\dots,\sigma(r)$. The matrix $\Psi_{\mathsf{H}}$ is the one obtained by removing those rows and columns with respect to gluons in $\Psi$. #### The CHY integrand for NLSM amplitudes The CHY integrands for color-ordered NLSM amplitudes are obtained by dimensional reduction strategy [12]. In particular, $\mathcal{I}_{L}^{\text{NLSM}}$ has the same expression with $\mathcal{I}_{L}^{\text{YM}}$, while $\mathcal{I}_{R}^{\text{NLSM}}$ is obtained by extending $\mathcal{I}_{R}^{\text{YM}}$ to $(d+d+d)$-dimensions and defining momenta and polarizations as follows: \[\mathcal{K}_{a}=(k_{a};0;0)\] (10) The matrix $\Psi^{(d+d+d)}$ is thus written as \[\Psi^{(d+d+d)}=\left(\begin{array}[]{cc}\mathbb{A}&-\mathbb{C}^{T}\\ \mathbb{C}&\mathbb{B}\end{array}\right)\,,\] (11) where the $\mathbb{A}$, $\mathbb{B}$, $\mathbb{C}$ are defined via replacing the polarizations and momenta in (9) by the $(d+d+d)$-dimensional ones $\mathcal{E}$ and $\mathcal{K}$ correspondingly. With the explicit components given in (10), we immediately arrive $\mathbb{C}=0$, $\mathbb{A}=A$ and $\mathbb{B}=B$. As a consequence, the reduced Pfaffian ${\mbox{Pf}}\,^{\prime}\left[\Psi^{(d+d+d)}\right]$ is factorized into: \[\text{Pf}\,^{\prime}\big{[}\Psi^{(d+d+d)}\big{]}=\text{Pf}\,^{\prime}(A){\mbox {Pf}}({B})=\frac{(-1)^{n+1}}{\sigma_{1n}}\frac{\epsilon_{1}\cdot\epsilon_{n}}{ {k_{1}\cdot k_{n}}}\,\text{Pf}\,^{\prime}(A)\,{\mbox{Pf}}(B_{1,n}^{1,n})\,.\] (12) By a further replacement $\epsilon_{a}\to k_{a}$, we reduce $\text{Pf}\,^{\prime}\big{[}\Psi^{(d+d+d)}\big{]}$ to the final expression of the NLSM integrand $\mathcal{I}_{R}^{\text{NLSM}}$ \[\left.\text{Pf}\,^{\prime}\big{[}\Psi^{(d+d+d)}\big{]}\right\|_{\epsilon_{a} \to k_{a}}=\left[\text{Pf}\,^{\prime}(A)\right]^{2}=\mathcal{I}_{R}^{ \text{NLSM}}\,.\] (13) _To sum up, NLSM amplitudes are obtained by performing the following replacements on Yang-Mills amplitudes_ \[\epsilon_{a}\cdot k_{b}\;\rightarrow\;0\] \[\epsilon_{a}\cdot\epsilon_{b}\;\rightarrow\;\left\{\begin{array}[ ]{>{}l @{\hspace{1.5em}} >{}l} k_{a} \cdot k_{b}\hskip 15.0pt&\{a,b\}\subset\{2\ldots n-1\}\\ 1\hskip 15.0pt&\{a,b\}=\{1,n\}\\ 0\hskip 15.0pt& a\in\{1,n\}\text{ and }b\in\{2 \ldots n-1\}\,\text{, or vice versa}\end{array}\right.\] (14) ### Expansions of EYM amplitudes Tree level color-ordered EYM amplitude can be expressed recursively by ones with fewer gravitons and/or fewer traces. One can repeat this expansion until all amplitudes become pure Yang-Mills ones, then the expansion coefficients are constructed by graphic rules. Now we review the expansions of single-trace EYM amplitudes. The expansions of multi-trace amplitudes can be found in [32]. #### The recursive expansion of single-trace EYM amplitudes Single-trace EYM amplitude $A(1,2,\dots,r\\|\,\mathsf{H})$ with $r$ gluons and $s$ gravitons was shown to satisfy the following recursive expansion [29] \[A(1,2,\dots,r\\|\,\mathsf{H}) = \sum\limits_{\boldsymbol{h}\|\,\widetilde{\mathsf{h}}}C_{h_{i}}( \boldsymbol{h})A(1,\{2,\dots,r-1\}\shuffle\{\boldsymbol{h},h_{i}\},r\\|\, \widetilde{\mathsf{h}}).\] (15) In the above equation, we choose a fiducial graviton $h_{i}\in\mathsf{H}$. The summation notation stands for the sum over all possible splittings of the graviton set $\mathsf{H}\setminus{h_{i}}\to\boldsymbol{h}\|\,\widetilde{\mathsf{h}}$ and sum over all permutations of elements in $\boldsymbol{h}$ for a given splitting. For example, if we have three gravitons $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$ and choose $h_{3}$ as the fiducial graviton, then $\boldsymbol{h}\|\,\widetilde{\mathsf{h}}$ implies the following five terms \[~{}~{}\mathsf{H}\setminus\{h_{3}\} \to \emptyset\,\|\,\{h_{1},h_{2}\};\] \[~{}~{}\mathsf{H}\setminus\{h_{3}\} \to \{h_{1}\}\,\|\,\{h_{2}\};~{}~{}\mathsf{H}\setminus\{h_{3}\}\to\{h_ {2}\}\,\|\,\{h_{1}\};\] \[~{}~{}\mathsf{H}\setminus\{h_{3}\} \to \{h_{1},h_{2}\}\,\|\,\emptyset;~{}~{}\mathsf{H}\setminus\{h_{3}\} \to\{h_{2},h_{1}\}\,\|\,\emptyset.\] (16) Assuming the permutation of elements of given $\boldsymbol{h}$ is $\{i_{1},i_{2},\dots,i_{j}\}$, the coefficient $C_{h_{i}}(\boldsymbol{h})$ is defined by \[C_{h_{i}}(\boldsymbol{h}_{1})\equiv\epsilon_{h_{i}}\cdot F_{i_{j }}\cdot F_{i_{j-1}}\cdot\dots\cdot F_{i_{1}}\cdot Y_{i_{1}},\] (17) where $F_{a}^{\mu\nu}$ is the linearized field strength of particle $a$ \[F_{a}^{\mu\nu}\equiv k_{a}^{\mu}\epsilon_{a}^{\nu}-\epsilon_{a}^ {\mu}k_{a}^{\nu}\] (18) and $Y_{i_{1}}$ denotes the sum of all momenta of gluons in the original gluon set which appear on the left hand side of $i_{1}$. An explicit example is given by the expansion of the single-trace EYM amplitude $A(1,2,\dots,r\\|\,h_{1},h_{2},h_{3})$ with $r$ gluons and three gravitons. By choosing $h_{3}$ as the fiducial graviton and summing over the five terms in (16), we finally express the single-trace EYM amplitude with three gravitons by those amplitudes with two, one and no graviton: \[A(1,2,\dots,r\\|\,h_{1},h_{2},h_{3}) = (\epsilon_{h_{3}}\cdot Y_{h_{3}})A(1,\{2,\dots,r-1\}\shuffle\{h_{ 3}\},r\\|\,h_{1},h_{2})\] (19) \[+ (\epsilon_{h_{3}}\cdot F_{h_{1}}\cdot Y_{h_{1}})A(1,\{2,\dots,r-1 \}\shuffle\{h_{1},h_{3}\},r\\|\,h_{2})\] \[+ (\epsilon_{h_{3}}\cdot F_{h_{2}}\cdot Y_{h_{2}})A(1,\{2,\dots,r-1 \}\shuffle\{h_{2},h_{3}\},r\\|\,h_{1})\] \[+ (\epsilon_{h_{3}}\cdot F_{h_{1}}\cdot F_{h_{2}}\cdot Y_{h_{2}})A( 1,\{2,\dots,r-1\}\shuffle\{h_{2},h_{1},h_{3}\},r)\] \[+ (\epsilon_{h_{3}}\cdot F_{h_{2}}\cdot F_{h_{1}}\cdot Y_{h_{1}})A( 1,\{2,\dots,r-1\}\shuffle\{h_{1},h_{2},h_{3}\},r).\] #### Graphic rule for the pure Yang-Mills expansion of single-trace EYM amplitudes Applying the recursive expansion (15) repeatedly until there is no graviton remaining in the graviton set, we finally expand the single-trace EYM amplitude in terms of color-ordered Yang-Mills amplitudes \[A(1,2,\dots,r\\|{\mathsf{H}}) = \sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,{\mathsf{H}}}\mathcal{C}(1,\boldsymbol{\sigma},r)A(1,\boldsymbol{ \sigma},r).\] (20) Here, we summed over all possible permutations obtained by merging together the original gluon set $\{2,\dots,r-1\}$ and the set of gluons (‘half gravitons’) which come from the graviton set $\mathsf{H}$. The relative order of gluons should be preserved, while the ‘perms’ under the summation notation means that all possible relative orders of elements in $\mathsf{H}$ should be considered. Given order $\boldsymbol{\sigma}$, the full coefficient $\mathcal{C}(1,\boldsymbol{\sigma},r)$ can be determined by the following graphic rule². [FOOTNOTE:2][ENDFOOTNOTE] _Graphic rule for the expansion of EYM amplitudes:_ * Define a reference order $\boldsymbol{\rho}$ of gravitons, then all gravitons are arranged into an ordered set \[\mathsf{R}=\{h_{\rho(1)},h_{\rho(2)},\dots,h_{\rho(s)}\}.\] (21) * Pick the last graviton $h_{\rho(s)}$ in the ordered set $\mathsf{R}$, an arbitrary gluon $l\in\{1,2,\dots,r-1\}$ (noting that the gluon $r$ is not considered here) as well as gravitons $h_{i_{1}},h_{i_{2}},\dots,h_{i_{j}}\in\mathsf{H}$ s.t. the relative order of them in $\boldsymbol{\sigma}$ satisfies³$\sigma^{-1}(l)<\sigma^{-1}(h_{i_{1}})<\sigma^{-1}(h_{i_{2}})<\dots\sigma^{-1}( h_{i_{j}})<\sigma^{-1}(h_{\rho(s)})$. Now consider each particle in the set $\{l,h_{i_{1}},h_{i_{2}},\dots,h_{i_{j}},h_{\rho(s)}\}$ as a node, we define a _chain_ starting from the node $h_{\rho(s)}$ and ending at the node $l$. The graviton $h_{\rho(s)}$ here is mentioned as a _the starting point of this chain_, while the gluon $l$ is mentioned as a _root_. All other gravitons on this chain are mentioned as _internal nodes of this chain_. The factor associated to this chain is [FOOTNOTE:3][ENDFOOTNOTE] \[\epsilon_{h_{\rho(s)}}\cdot F_{h_{i_{j}}}\cdot F_{h_{i_{j-1}}} \cdot\dots\cdot F_{h_{i_{1}}}\cdot k_{l}.\] (22) Remove $h_{i_{1}}$, $h_{i_{2}}$, …, $h_{i_{j}}$, $h_{\rho(s)}$ from the ordered set $\mathsf{R}$ and redefine $\mathsf{R}$ \[\mathsf{R}\to\mathsf{R}\,^{\prime}=\mathsf{R}\setminus\{h_{i_{1}} ,h_{i_{2}},...,h_{i_{j}},h_{\rho(s)}\}.\] (23) * Picking $l^{\prime}\in\{1,2,\dots,r-1\}\cup\{h_{i_{1}},h_{i_{2}},...,h_{i_{j}},h_{\rho( s)}\}$, the last element $h_{\rho^{\prime}(s^{\prime})}$ in $\mathsf{R}\,^{\prime}$ as well as gravitons $h_{i^{\prime}_{1}}$, $h_{i^{\prime}_{2}}$, …, $h_{i^{\prime}_{j^{\prime}}}$ in $\mathsf{R}\,^{\prime}$ s.t., $\sigma^{-1}(l^{\prime})<\sigma^{-1}(h_{i_{1}^{\prime}})<\sigma^{-1}(h_{i_{2}^{ \prime}})<\dots<\sigma^{-1}(h_{i_{j^{\prime}}^{\prime}})<\sigma^{-1}(h_{\rho^{ \prime}(s^{\prime})})$, we define a chain $\{l^{\prime},h_{i_{1}^{\prime}},h_{i_{2}^{\prime}},\dots,h_{i_{j^{\prime}}^{ \prime}},h_{\rho(s^{\prime})}\}$ starting from $h_{\rho(s^{\prime})}$ and ending at $l^{\prime}$. This chain is associated with a factor \[\epsilon_{h_{\rho^{\prime}(s^{\prime})}}\cdot F_{h_{i^{\prime}_{j ^{\prime}}}}\cdot F_{h_{i^{\prime}_{{j^{\prime}-1}}}}\cdot\dots\cdot F_{h_{i_{ 1}^{\prime}}}\cdot k_{l^{\prime}}.\] (24) Remove $h_{i_{1}^{\prime}}$,$h_{i_{2}^{\prime}}$, …, $h_{i_{j^{\prime}}^{\prime}}$, $h_{\rho^{\prime}(s^{\prime})}$ from $\mathsf{R}\,^{\prime}$ and redefine $\mathsf{R}\to\mathsf{R}\,^{\prime\prime}=\mathsf{R}^{\prime}\setminus\{h_{i_{1 }^{\prime}},h_{i_{2}^{\prime}},\dots,h_{i_{j^{\prime}}^{\prime}},h_{\rho^{ \prime}(s^{\prime})}\}$. * Repeating the above steps until the ordered set $\mathsf{R}$ becomes empty, we get a graph (‘forest’) with gluons as roots of trees ⁴. For a given graph $\mathcal{F}$, the product of the factors accompanied to all chains produces a term $\mathcal{C}^{[\mathcal{F}]}(\boldsymbol{\sigma})$ in the coefficient $\mathcal{C}(1,\boldsymbol{\sigma},r)$ in (20). Thus the final expression of $\mathcal{C}(1,\boldsymbol{\sigma},r)$ is given by summing over all possible graphs defined above [FOOTNOTE:4][ENDFOOTNOTE] \[\mathcal{C}(1,\boldsymbol{\sigma},r)=\sum\limits_{\mathcal{F}\in \{\text{Graphs}\}}\mathcal{C}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},r).\] (25) #### The expansions of Pfaffians in the CHY formula of single-trace EYM amplitudes It is worth closing this section by translating the expansions (15), (20) of EYM amplitudes into the language of CHY formulation (see [31]). In CHY formulation, the recursive expansion (15) reflects \[(-1)^{s(s+1)\over 2}\text{PT}(1,2,\dots,r){\mbox{Pf}}\left[\Psi_{ \mathsf{H}}\right] = \sum\limits_{\boldsymbol{h}\|\,\widetilde{\mathsf{h}}}(-1)^{\| \widetilde{\mathsf{h}}\|(\|\widetilde{\mathsf{h}}\|+1)\over 2}C_{h_{1}}( \boldsymbol{h})\text{PT}(1,\{2,\dots,r-1\}\shuffle\{\boldsymbol{h},h_{1}\},r){ \mbox{Pf}}\left[\Psi_{\widetilde{\mathsf{h}}}\right],\] where $r$ and $s$ are the numbers of gluons and gravitons respectively, $\|\widetilde{\mathsf{h}}\|$ denotes the number of elements in the set $\widetilde{\mathsf{h}}$. The pure Yang-Mills expansion (20) implies \[(-1)^{s(s+1)\over 2}\text{PT}(1,2,\dots,r){\mbox{Pf}}\left[\Psi_{ \mathsf{H}}\right]=\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle \,\text{perms}\,{\mathsf{H}}}\mathcal{C}(1,\boldsymbol{\sigma},r)\text{PT}(1, \boldsymbol{\sigma},r).\] (27) The expansion coefficients $C_{h_{1}}(\boldsymbol{h})$ and $\mathcal{C}(1,\boldsymbol{\sigma},r)$ in (2.2) and (27) are given by (17) and (25) respectively. We emphasize that the relations (2.2) and (27) hold for arbitrary dimensions. ## 3 Gauge invariance induced relations In this section, we induce nontrivial generalized BCJ relations for color-ordered Yang-Mills amplitudes (also bi-scalar amplitudes and color-ordered NLSM amplitudes) by combining gauge invariance conditions with CHY inspired dimensional reductions. The coefficients of amplitudes in the gauge invariance induced relations are polynomials of Mandelstam variables. ### Inducing generalized BCJ relations by gauge invariance and dimensional reduction In the pure Yang-Mills expansion (20) of EYM amplitude $A(1,2,\dots,r\\|\,\mathsf{H})$, each term $\mathcal{C}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},r)$ (see (25)) of the expansion coefficient $\mathcal{C}(1,\boldsymbol{\sigma},r)$ is expressed as a product of Lorentz invariants $\epsilon\cdot k$, $\epsilon\cdot\epsilon$ and $k\cdot k$ and constructed by the grapic rule in section 2.2. The gauge invariance states that the amplitude $A(1,2,\dots,r\\|\,\mathsf{H})$ has to vanish under the replacement $\epsilon_{h}\to k_{h}$ for any given graviton $h\in\mathsf{H}$. Hence, a relation for pure Yang-Mills amplitudes [29] follows \[0=\sum\limits_{\sigma\in\{2,\dots,r-1\}\shuffle\,\text{perms}\,{ \mathsf{H}}}\mathcal{C}(1,\boldsymbol{\sigma},r)\Big{\|}_{\epsilon_{h}\to k_{h} }A(1,\boldsymbol{\sigma},r).\] (28) For a given graph in the expansion of $\mathcal{C}(1,\boldsymbol{\sigma},r)$, the graviton $h$ can be either an _internal node_ or a _starting point of a chain_. In the former case, the gauge invariance condition is naturally encoded by $F_{h}^{\mu\nu}\|_{\epsilon_{h}\to k_{h}}=0$, thus this contribution has to vanish. The only nontrivial contributions are those graphs in which the graviton $h$ plays as the starting point of a chain. The gauge invariance condition is then reduced to \[0=\sum\limits_{\sigma\in\{2,\dots,r-1\}\shuffle\,\text{perms}\,{ \mathsf{H}}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\boldsymbol{\sigma}}_{ \mathsf{H}}[h]}}\mathcal{C}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},r)\Big{\|}_{ \scriptsize{\epsilon_{h}\to k_{h}}}A(1,\boldsymbol{\sigma},r),\] (29) where $\mathcal{G}^{\boldsymbol{\sigma}}_{\mathsf{H}}[h]$ denotes the set of graphs for permutation $\boldsymbol{\sigma}$, where $h$ plays as starting point of a chain. As shown by examples in [29; 32] ( similar discussions on the gauge invariance relations can be found in [43; 25; 26; 44; 30; 45]), (29) is generated by known BCJ relations, thus it is not new relation beyond known BCJ relations. Nevertheless, a systematical study on the connection between (29) and the standard KK and BCJ relations still deserves future work. Coefficients in the relation (29) still contain polarizations. To induce a relation where coefficients are only functions of Mandelstam variables $s_{ij}=k_{i}\cdot k_{j}$, we should _‘turn’ all polarizations in the expansion of coefficients to momenta._ One reasonable approach to realize this point is combining gauge invariance conditions with dimensional reduction inspired by CHY formulation. Our discussion is based on the following crucial observations: * _Gauge invariance conditions for more than one graviton can be imposed simultaneously._ This can be understood from two different aspects. (i) Since the pure Yang-Mills expansion (20) is obtained by applying the recursive expansion (15) repeatedly, we can take gauge invariance condition for (15) instead. If we replace $\epsilon_{h_{a}}$ by $k_{h_{a}}$ for more than one graviton $h_{a}\in\mathsf{A}\subseteq\mathsf{H}$ ($\mathsf{A}$ consists of at least two gravitons) on the RHS of (15), there is at most one graviton plays as the fiducial one. The polarizations of the rest of the gravitons belonging to $\mathsf{A}$ are contained by either $F^{\mu\nu}$ or an EYM amplitude with fewer gravitons. When replacing $\epsilon_{h_{a}}$ by $k_{h_{a}}$ for all $h_{a}\in\mathsf{A}$ on the RHS of (15), every term has to vanish due to the antisymmetry of $F^{\mu\nu}$ or/and the gauge invariance condition for EYM amplitudes with fewer gravitons (as an inductive assumption). (ii) In the language of CHY formula (1), polarizations are packaged into (reduced) Pffafians. When the replacement $\epsilon_{h}\to k_{h}$ for a given graviton $h\in\mathsf{H}$ is imposed, the $\Psi_{\mathsf{H}}$ matrix becomes degenerate because two rows/columns coincide with each other (Noting the diagonal entry $C_{h_{a}h_{a}}$ for $C$ matrix vanishes due to scattering equation (2)) as shown by the left matrix in the following \[\left(\begin{array}[]{ccc\|ccc}\cdots&\cdots&\cdots&\cdots&\cdots& \cdots\\ \cdots&{k_{h_{a}}\cdot k_{h_{b}}\over z_{h_{a}h_{b}}}&\cdots&\cdots&{k_{h_{a}} \cdot\epsilon_{h_{b}}\over z_{h_{a}h_{b}}}&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \hline\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&{k_{h_{a}}\cdot k_{h_{b}}\over z_{h_{a}h_{b}}}&\cdots&\cdots&{k_{h_{a}} \cdot\epsilon_{h_{b}}\over z_{h_{a}h_{b}}}&\dots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \end{array}\right)~{}~{}~{}~{}~{}\to~{}~{}~{}~{}~{}\left(\begin{array}[]{ccc\| ccc}\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&{k_{h_{a}}\cdot k_{h_{b}}\over z_{h_{a}h_{b}}}&\cdots&\cdots&{k_{h_{a}} \cdot k_{h_{b}}\over z_{h_{a}h_{b}}}&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \hline\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&{k_{h_{a}}\cdot k_{h_{b}}\over z_{h_{a}h_{b}}}&\cdots&\cdots&{k_{h_{a}} \cdot k_{h_{b}}\over z_{h_{a}h_{b}}}&\dots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \end{array}\right).\] (30) If we take gauge invariance conditions for more than one graviton, e.g. $h_{a}$ and $h_{b}$, the matrix $\Psi$ is also degenerate for the same reason (see the right matrix in (30)), thus the Pfaffian has to vanish. * _The gauge invariance conditions are independent of dimensions._ This is because the statements (i) and (ii) in (1) hold for arbitrary dimension space. Having (1) and (2), we can conveniently carry on our discussions in the framework of CHY formula. The recursive and graphic expansions for amplitude reflect corresponding relations for Pfaffians (2.2) and (27). Since CHY formula does not depend on the dimension of space, we can extend the Pfaffian ${\mbox{Pf}}\left[\Psi_{\mathsf{H}}\right]$ in the graphic expansion (27) to $(d+d)$-dimensions by defining $(d+d)$-dimensional polarizations $\mathcal{E}_{h_{a}}$ (all $h_{a}\in\mathsf{H}$) and $(d+d)$-dimensional momenta $\mathcal{K}_{i}$ for all external particles, so that \[\mathcal{E}_{h_{a}}\cdot\mathcal{K}_{h_{a}}=0,~{}~{}~{}(\text{for all~{}}h_{a}\in\mathsf{H});~{}~{}~{}\mathcal{K}_{i}\cdot\mathcal{K}_{i}=0~{}~ {}~{}(\text{for all particles $i$});~{}~{}~{}\sum\limits_{i=1}^{r+s}\mathcal{K }_{i}=0\] (31) are satisfied. According to our observations (1) and (2), the Pfaffian ${\mbox{Pf}}\left[\Psi_{\mathsf{H}}\right]$ in $(d+d)$ dimensions on the LHS of (27) must vanish under the replacement $\mathcal{E}_{h_{a}}\to\mathcal{K}_{h_{a}}$ for all $h_{a}\in\mathsf{A}$ where $\mathsf{A}$ is a nonempty subset of $\mathsf{H}$. Consequently, the RHS of the graphic expansion (27) in $d+d$ dimensions has to vanish when $\mathcal{E}_{h_{a}}$ are replaced by $\mathcal{K}_{h_{a}}$ for all $h_{a}\in\mathsf{A}\subseteq\mathsf{H}$: \[0=\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\, \text{perms}\,{\mathsf{H}}}\mathcal{C}(1,\boldsymbol{\sigma},r)\Big{\|}_{ \begin{subarray}{c}\mathcal{E}_{h_{a}}\to\mathcal{K}_{h_{a}}\\ \text{for all~{}}h_{a}\in\mathsf{A}\end{subarray}}\text{PT}(1,\boldsymbol{ \sigma},r).\] (32) Once the coefficients $\mathcal{C}(1,\boldsymbol{\sigma},r)$ in the above equation are expressed by graphs (see eq. (25)) and the gauge invariance conditions are imposed, a chain in which any $h_{a}\in\mathsf{A}\subseteq\mathsf{H}$ plays as an internal node vanishes due to the antisymmetry of the $(d+d)$ dimensional strength tensor $\mathbf{F}_{h_{a}}^{UV}\equiv\mathcal{K}_{h_{a}}^{U}\mathcal{E}_{h_{a}}^{V}- \mathcal{K}_{h_{a}}^{V}\mathcal{E}_{h_{a}}^{U}$. Thus only those graphs where all $h_{a}\in\mathsf{A}$ play as starting points of chains survive. The relation (32) then turns to \[0=\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\, \text{perms}\,{\mathsf{H}}}\biggl{[}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{ \boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]}}\mathcal{C}^{[\mathcal{F}]}(1, \boldsymbol{\sigma},r)\Big{\|}_{\begin{subarray}{c}\scriptsize{\mathcal{E}_{h_{ a}}\to\mathcal{K}_{h_{a}}}\\ \text{for all~{}}h_{a}\in\mathsf{A}\end{subarray}}\biggr{]}\text{PT}(1, \boldsymbol{\sigma},r).\] (33) Here, $\mathcal{G}^{\boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]$ denotes the set of graphs corresponding to the permutation $\boldsymbol{\sigma}$, where all elements in the nonempty subset $\mathsf{A}$ play as starting points of chains (Note that other elements in $\mathsf{H}$ may also be starting points of chains). The equation (33) does not rely on details of $(d+d)$-dimensional polarizations $\mathcal{E}$ and momenta $\mathcal{K}$, only the conditions (31) are required. Thus, we can assign details of polarizations and momenta in $(d+d)$ dimensions appropriately s.t. (31) is satisfied. A reasonable definition inspired by the dimensional reduction strategy (see (10)) in the CHY formula is \[\mathcal{K}_{i}=(k_{i};0),~{}~{}~{}(\text{for all external particles}); \mathcal{E}_{h_{a}}=(0;k_{h_{a}}),~{}~{}~{}~{}{h_{a}\in\mathsf{H}}\] (34) which apparently satisfies (31). With this assignment, the coefficients in the gauge invariance condition (33) become polynomial functions of Mandelstam variables. When the coefficients $\mathcal{C}(1,\boldsymbol{\sigma},r)$ in $(d+d)$ dimensions are expressed by the graphic rules and $\mathcal{E}_{h_{a}}$ in $\mathcal{C}(1,\boldsymbol{\sigma},r)$ are replaced by $\mathcal{K}_{h_{a}}$ ($h_{a}\in\mathsf{A}\subseteq\mathsf{H}$), chains in the graphs are classified into two types: * _Type-1_ Chains started by $(d+d)$-dimensional polarizations $\mathcal{E}_{a}$ ($a\in\mathsf{H}\setminus\mathsf{A}$) have the general form \[\mathcal{E}_{a}\cdot\mathbf{F}_{h_{i_{j}}}\cdot\mathbf{F}_{h_{i_{ j-1}}}\dots\mathbf{F}_{h_{i_{1}}}\cdot\mathcal{K}_{b}.\] (35) A chain of this type has to vanish if its length is odd, because we cannot avoid a factor of the form $\mathcal{E}_{i}\cdot\mathcal{K}_{j}$ which is zero in the definition (34). Thus the length of nonvanishing type-1 chains must be even. When plugging the components (34) into an even-length chain of type-1, we get a chain expressed by $d$-dimensional Mandelstam variables \[s_{ah_{i_{j}}}s_{h_{i_{j}}h_{i_{j-1}}}\dots s_{h_{i_{1}}b}\] (36) associated with a factor $(-1)^{j+1\over 2}$, where $j$ is odd. Since the length $L$ of this chain is $j+1$, the prefactor can be given by $(-1)^{L\over 2}$. * _Type-2_ Chains started by $(d+d)$-dimensional momenta $\mathcal{K}_{a}$ have the general form \[\mathcal{K}_{a}\cdot\mathbf{F}_{h_{i_{j}}}\cdot\mathbf{F}_{h_{i_{ j-1}}}\dots\mathbf{F}_{h_{i_{1}}}\cdot\mathcal{K}_{b}.\] (37) A chain of this type vanishes if its length is even, for an even-length type-2 chain must contain a vanishing factor of the form $\mathcal{E}_{i}\cdot\mathcal{K}_{j}$. Thus the length of nonvanishing type-2 chains are odd. Inserting the choice of $(d+d)$-dimensional polarizations and momenta (34) into an odd-length chain of this type, we arrive \[s_{ah_{i_{j}}}s_{h_{i_{j}}h_{i_{j-1}}}\dots s_{h_{i_{1}}b}\] (38) associated with a factor $(-1)^{j\over 2}$, where $j$ is even. The prefactor for this chain is further expressed by the length $L$ of the chain as $(-1)^{L-1\over 2}$. Collecting all nonzero chains together, we induce the following relation for PT factors in $d$ dimensions from the $(d+d)$-dimensional gauge invariance condition (32): \[0=\sum\limits_{\sigma\in\{2,\dots,r-1\}\shuffle\,\text{perms}\,{ \mathsf{H}}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\,\boldsymbol{ \sigma}}_{\mathsf{H}}[\mathsf{A}]}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{ \sigma},r)\text{PT}(1,\boldsymbol{\sigma},r).\] (39) Here, $\mathcal{G}^{\prime\,\boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]$ denotes the set of graphs (constructed by the same rule in section 2.2) where _all elements in $\mathsf{A}\subseteq\mathsf{H}$ ($\mathsf{A}\neq\emptyset$) play as starting points of all odd-length chains._ Possible chains of even length must be started by elements in $\mathsf{H}\setminus\mathsf{A}$. For a given permutation $\boldsymbol{\sigma}$ and a given graph $\mathcal{F}\in\mathcal{G}^{\prime\boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]$, $\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},r)$ is obtained by associating chains with factors of the form \[s_{ah_{i_{j}}}s_{h_{i_{j}}h_{i_{j-1}}}\dots s_{h_{i_{1}}b},\] (40) in which $a$ and $b$ are the starting points and ending points of a chain, while $h_{i_{1}}$, …, $h_{i_{j}}$ are internal nodes of this chain. Note that the prefactors of all chains in any given graph in $\mathcal{G}^{\prime\,\boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]$ together produce a same total factor $(-1)^{{s\over 2}-{1\over 2}{N_{o}}}$, where $s$ is the number of elements in the set $\mathsf{H}$ and equal to the total length of all chains, $N_{o}$ denotes the number of odd-length chains and is equal to the order of the set $\mathsf{A}$. The total factor thus does not appear in the equation (39). To translate the gauge invariance induced relation (39) for Parke-Taylor factors into amplitude relation, we consider the expression \[\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,{\mathsf{H}}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\, \boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]}}(-1)^{{(n+1)(n+2)\over 2}}\int d \Omega_{\text{CHY}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},r)\text{ PT}(1,\boldsymbol{\sigma},r)\mathcal{I}_{R},\] (41) where $\mathcal{I}_{R}$ can be $\mathcal{I}^{\text{BS}}_{R}$, $\mathcal{I}^{\text{YM}}_{R}$ or $\mathcal{I}^{\text{NLSM}}_{R}$ in (3), (4) or (13) correspondingly. Since the coefficients $\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},r)$ are independent of the scattering variables, it can be moved outside the integration. The relation for Parke-Taylor factors (39) then gives the following gauge invariance induced amplitude relations \[\boxed{0=\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\} \shuffle\,\text{perms}\,{\mathsf{H}}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{ \prime\,\boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{A}]}}\mathcal{D}^{[\mathcal{ F}]}(1,\boldsymbol{\sigma},r)A(1,\boldsymbol{\sigma},r)},\] (42) for any nonempty $\mathsf{A}$ ($\mathsf{A}\subseteq\mathsf{H}$). ### Examples for the gauge invariance induced relation (42) Now let us present several examples of the gauge invariance induced amplitude relation (42). <figure><img src="content_image/1803.01701/4ptGraphs.jpg"><figcaption>Figure 1: All possible graphs with H={h1,h2} and reference order R={h1,h2}.Graphs (a) and (b) correspond to the permutations {2,…,r−1}\shuffle{h1,h2},while graphs (a) and (b) correspond to the permutations{2,…,r−1}\shuffle{h2,h1}.</figcaption></figure> #### 3.2.1 $\mathsf{H}=\{h_{1},h_{2}\}$ The first example is given by $\mathsf{H}=\{h_{1},h_{2}\}$. If the reference order is fixed as $\mathsf{R}=\{h_{1},h_{2}\}$, all graphs given by the graphic rule in section 2.2 are displayed in figure 1. The graphs $(a)$, $(b)$ in figure 1 contribute to permutations $\{2,\dots,r-1\}\shuffle\{h_{1},h_{2}\}$, while $(c)$, $(d)$ contribute to the relative order $\{2,\dots,r-1\}\shuffle\{h_{2},h_{1}\}$. In the gauge invariance induced relation (42), the nonempty subset $\mathsf{A}$ cannot contain only one element because the total length of all chains is an even number $2$. If $\mathsf{A}$ contains for example $h_{1}$, _i.e._, there is an odd-length chain started by $h_{1}$, we must have another odd-length chain started by $h_{2}$ so that the total length of all chains is even. Thus the nonempty subset $\mathsf{A}$ of $\mathsf{H}$ can only be chosen as $\{h_{1},h_{2}\}$ while $h_{1}$ and $h_{2}$ are starting points of two length-1 chains in this example. The graph $(b)$ which contains a length-2 chain does not appear in our gauge invariance induced relation. The relation (42) for $\mathsf{A}=\{h_{1},h_{2}\}$ reads \[0 = \sum\limits_{\boldsymbol{\sigma}}s_{h_{2}X_{h_{2}}}s_{h_{1}X_{h_{ 1}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{1},h_{2}\},r)\] (43) \[+\sum\limits_{\boldsymbol{\sigma}}s_{h_{2}X_{h_{2}}}(s_{h_{1}X_{h _{1}}}+s_{h_{1}h_{2}})A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{2} ,h_{1}\},r),\] where $s_{aX_{a}}\equiv\sum\limits_{\scriptsize\begin{subarray}{c}i\in\{1,2,\dots,r-1 \}\\ \text{s.t.}\sigma^{-1}(i)<\sigma^{-1}(a)\end{subarray}}s_{ai}$. This relation is in agreement with a fundamental BCJ relation. #### 3.2.2 $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$ We consider the examples with $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$. For the reference order $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$, all possible graphs constructed by the graphic rules are provided by figure 2 in appendix A. For any graph, the total length of all chains must be $3$. As a result, the nonempty subset $\mathsf{A}$ in the relation (42) can only contain odd number of elements, _i.e._, $\mathsf{A}$ can be $\{h_{1}\}$, $\{h_{2}\}$, $\{h_{3}\}$ or $\{h_{1},h_{2},h_{3}\}$. #### $\mathsf{A}=\{h_{1}\}$ If $\mathsf{A}$ contains only one element $h_{1}$. Then $h_{1}$ must leads to a length-1 chain while $h_{3}$ must leads to a length-2 chain $s_{h_{3}h_{2}}s_{h_{2}a}$ with an internal node $h_{2}$. Among the graphs in figure 2, only $(a5)$ (for the relative order $\{h_{1},h_{2},h_{3}\}$), $(c3)$, $(c4)$ (for the relative order $\{h_{2},h_{1},h_{3}\}$) and $(d2)$, $(d4)$, $(d6)$ (for the relative order $\{h_{2},h_{3},h_{1}\}$) contribute. Hence the relation for $\mathsf{A}=\{h_{1}\}$ is \[0 = \sum\limits_{\boldsymbol{\sigma}}s_{h_{1}X_{h_{1}}}s_{h_{3}h_{2}} s_{h_{2}X_{h_{2}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{1},h_{ 2},h_{3}\},r)\] (44) \[+ \sum\limits_{\boldsymbol{\sigma}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{2 }})s_{h_{3}h_{2}}s_{h_{2}X_{h_{2}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\} \shuffle\{h_{2},h_{1},h_{3}\},r)\] \[+ \sum\limits_{\boldsymbol{\sigma}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{2 }}+s_{h_{1}h_{3}})s_{h_{3}h_{2}}s_{h_{2}X_{h_{2}}}A(1,\boldsymbol{\sigma}\in\{ 2,\dots,r-1\}\shuffle\{h_{2},h_{3},h_{1}\},r).\] This relation is consistent with a fundamental BCJ relation. #### $\mathsf{A}=\{h_{2}\}$ If $\mathsf{A}=\{h_{2}\}$, $h_{2}$ must be the starting point of a length-1 chain under the choice of reference order $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$, while $h_{3}$ must start a length-2 chain with the internal node $h_{1}$. The graphs $(a3)$, $(a4)$, $(b2)$, $(b4)$, $(b6)$ and $(c5)$ have nonvanishing contributions and the relation (42) gives \[0 = \sum\limits_{\boldsymbol{\sigma}}s_{h_{2}X_{h_{2}}}s_{h_{3}h_{1}} s_{h_{1}X_{h_{1}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{2},h_{ 1},h_{3}\},r)\] (45) \[+ \sum\limits_{\boldsymbol{\sigma}}(s_{h_{2}X_{h_{2}}}+s_{h_{2}h_{1 }})s_{h_{3}h_{1}}s_{h_{1}X_{h_{1}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\} \shuffle\{h_{1},h_{2},h_{3}\},r)\] \[+ \sum\limits_{\boldsymbol{\sigma}}(s_{h_{2}X_{h_{2}}}+s_{h_{2}h_{1 }}+s_{h_{2}h_{3}})s_{h_{3}h_{1}}s_{h_{1}X_{h_{1}}}A(1,\boldsymbol{\sigma}\in\{ 2,\dots,r-1\}\shuffle\{h_{1},h_{3},h_{2}\},r).\] Again, the vanish of RHS can be considered as a result of fundamental BCJ relation. #### $\mathsf{A}=\{h_{3}\}$ If $\mathsf{A}=\{h_{3}\}$, the element $h_{3}$ can start either a length-$3$ chain or a length-$1$ chain. In the former case, both $h_{1}$ and $h_{2}$ must be internal nodes of the length-3 chain ($(a6)$ and $(c6)$ in figure 2), while in the latter case $h_{2}$ must start a length-$2$ chain with $h_{1}$ as the internal node ((a2), (b3), (e5) and (e6) in figure 2). All together, the relation (42) turns to \[0 = \sum\limits_{\boldsymbol{\sigma}}(s_{h_{3}h_{2}}s_{h_{2}h_{1}}s_{ h_{1}X_{h_{1}}}+s_{h_{3}X_{h_{3}}}s_{h_{2}h_{1}}s_{h_{1}X_{h_{1}}})A(1, \boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{1},h_{2},h_{3}\},r)\] (46) \[+\sum\limits_{\boldsymbol{\sigma}}s_{h_{3}h_{1}}s_{h_{1}h_{2}}s_{ h_{2}X_{h_{2}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{2},h_{1}, h_{3}\},r)\] \[+\sum\limits_{\boldsymbol{\sigma}}s_{h_{3}X_{h_{3}}}s_{h_{2}h_{1} }s_{h_{1}X_{h_{1}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{1},h_ {3},h_{2}\},r)\] \[+\sum\limits_{\boldsymbol{\sigma}}s_{h_{3}X_{h_{3}}}s_{h_{2}h_{1} }(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{3}})A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\} \shuffle\{h_{3},h_{1},h_{2}\},r),\] which is not as trivial as previous examples. One can check this identity by expanding all amplitudes in terms of BCJ basis amplitudes. #### $\mathsf{A}=\{h_{1},h_{2},h_{3}\}$ Now we consider the case $\mathsf{A}=\{h_{1},h_{2},h_{3}\}$, for which all elements in $\mathsf{H}$ play as starting points of odd-length chains. The only possibility is that all chains are of length $1$. The relation (42) then gives rise \[0 = \sum\limits_{\boldsymbol{\sigma}}s_{h_{1}X_{h_{1}}}s_{h_{2}X_{h_{ 2}}}s_{h_{3}X_{h_{3}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{1} ,h_{2},h_{3}\},r)\] (47) \[+\sum\limits_{\boldsymbol{\sigma}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{ 2}})s_{h_{2}X_{h_{2}}}s_{h_{3}X_{h_{3}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r- 1\}\shuffle\{h_{2},h_{1},h_{3}\},r)\] \[+\sum\limits_{\boldsymbol{\sigma}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{ 2}}+s_{h_{1}h_{3}})s_{h_{2}X_{h_{2}}}s_{h_{3}X_{h_{3}}}A(1,\boldsymbol{\sigma} \in\{2,\dots,r-1\}\shuffle\{h_{2},h_{3},h_{1}\},r)\] \[+\sum\limits_{\boldsymbol{\sigma}}s_{h_{1}X_{h_{1}}}(s_{h_{2}X_{h _{2}}}+s_{h_{2}h_{3}})s_{h_{3}X_{h_{3}}}A(1,\boldsymbol{\sigma}\in\{2,\dots,r- 1\}\shuffle\{h_{1},h_{3},h_{2}\},r)\] \[+\sum\limits_{\boldsymbol{\sigma}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{ 3}})(s_{h_{2}X_{h_{2}}}+s_{h_{2}h_{3}})s_{h_{3}X_{h_{3}}}A(1,\boldsymbol{ \sigma}\in\{2,\dots,r-1\}\shuffle\{h_{3},h_{1},h_{2}\},r)\] \[+\sum\limits_{\boldsymbol{\sigma}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{ 3}}+s_{h_{1}h_{2}})(s_{h_{2}X_{h_{2}}}+s_{h_{2}h_{3}})s_{h_{3}X_{h_{3}}}A(1, \boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\{h_{3},h_{2},h_{1}\},r).\] The RHS of the above relation gets contributions from eighteen graphs $(a1)$, $(b1)$, $(b5)$, $(c1)$, $(c2)$, $(d1)$, $(d3)$, $(d5)$, $(e1)$, $(e2)$, $(e3)$, $(e4)$, $(f1)$, $(f2)$, $(f3)$, $(f4)$, $(f5)$ and $(f6)$. Both the sum of the first three rows and the sum of the last three rows vanish due to fundamental BCJ relation. #### 3.2.3 $\mathsf{H}=\{h_{1},h_{2},h_{3},h_{4}\}$ We consider a much more nontrivial case with $\mathsf{H}=\{h_{1},h_{2},h_{3},h_{4}\}$ as the last example. The nonempty subset in (42) is chosen as $\mathsf{A}=\{h_{3},h_{4}\}$ and the reference order is chosen as $\mathsf{R}=\{h_{1},h_{2},h_{3},h_{4}\}$. If $h_{4}$ ($h_{3}$) is starting point of a length-$3$ chain, $h_{3}$ ($h_{4}$) must be starting point of a length-$1$ chain. Such graphs contain only two chains; If both $h_{4}$ and $h_{3}$ are starting points of length-$1$ chains, we must also have an length-$2$ chain of the form $s_{h_{2}h_{1}}s_{h_{1}Y_{h_{1}}}$. The coefficients for all possible permutations are displayed as follows ($\{h_{1}h_{2}h_{3}h_{4}\}$ is used to denote the permutation $1,\{2,\dots,r-1\}\shuffle\{h_{1},h_{2},h_{3},h_{4}\},r$ for short) \[\{h_{3}h_{1}h_{2}h_{4}\}:s_{h_{4}h_{2}}s_{h_{2}h_{1}}s_{h_{1}X_{h _{1}}}s_{h_{3}X_{h_{3}}}+s_{h_{4}X_{h_{4}}}s_{h_{3}X_{h_{3}}}s_{h_{2}h_{1}}(s_ {h_{1}X_{h_{1}}}+s_{h_{1}h_{3}}),\] \[\{h_{1}h_{3}h_{2}h_{4}\}:s_{h_{4}h_{2}}s_{h_{2}h_{1}}s_{h_{1}X_{h _{1}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{1}})+s_{h_{4}X_{h_{4}}}s_{h_{3}X_{h_{3}}} s_{h_{2}h_{1}}s_{h_{1}X_{h_{1}}},\] \[\{h_{1}h_{2}h_{3}h_{4}\}:s_{h_{4}h_{2}}s_{h_{2}h_{1}}s_{h_{1}X_{h _{1}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{1}}+s_{h_{3}h_{2}})+s_{h_{4}X_{h_{4}}}(s_ {h_{3}X_{h_{3}}}+s_{h_{3}h_{2}})s_{h_{2}h_{1}}s_{h_{1}X_{h_{1}}},\] \[\{h_{1}h_{2}h_{4}h_{3}\}:s_{h_{4}h_{2}}s_{h_{2}h_{1}}s_{h_{1}X_{h _{1}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{1}}+s_{h_{3}h_{2}}+s_{h_{3}h_{4}})+s_{h_{ 4}X_{h_{4}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{2}}+s_{h_{3}h_{4}})s_{h_{2}h_{1}}s_ {h_{1}X_{h_{1}}},\] \[\{h_{3}h_{1}h_{4}h_{2}\}:s_{h_{4}X_{h_{4}}}s_{h_{3}X_{h_{3}}}s_{h _{2}h_{1}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{3}}),~{}~{}~{}~{}\{h_{1}h_{3}h_{4}h_{ 2}\}:s_{h_{4}X_{h_{4}}}s_{h_{3}X_{h_{3}}}s_{h_{2}h_{1}}s_{h_{1}X_{h_{1}}},\] \[\{h_{1}h_{4}h_{3}h_{2}\}:s_{h_{4}X_{h_{4}}}(s_{h_{3}X_{h_{3}}}+s_ {h_{3}h_{4}})s_{h_{2}h_{1}}s_{h_{1}X_{h_{1}}},~{}~{}~{}~{}\{h_{1}h_{4}h_{2}h_{ 3}\}:s_{h_{4}X_{h_{4}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{4}}+s_{h_{3}h_{2}})s_{h_ {2}h_{1}}s_{h_{1}X_{h_{1}}},\] \[\{h_{3}h_{4}h_{1}h_{2}\}:s_{h_{4}X_{h_{4}}}s_{h_{3}X_{h_{3}}}s_{h _{2}h_{1}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{3}}+s_{h_{1}h_{4}}),\] \[\{h_{4}h_{3}h_{1}h_{2}\}:s_{h_{4}X_{h_{4}}}(s_{h_{3}X_{h_{3}}}+s_ {h_{3}h_{4}})s_{h_{2}h_{1}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{3}}+s_{h_{1}h_{4}}),\] \[\{h_{4}h_{1}h_{3}h_{2}\}:s_{h_{4}X_{h_{4}}}(s_{h_{3}X_{h_{3}}}+s_ {h_{3}h_{4}})s_{h_{2}h_{1}}(s_{h_{1}X_{h_{1}}}+s_{h_{1}h_{4}}),\] \[\{h_{4}h_{1}h_{2}h_{3}\}:s_{h_{4}X_{h_{4}}}(s_{h_{3}X_{h_{3}}}+s_ {h_{3}h_{4}}+s_{h_{3}h_{2}})s_{h_{2}h_{1}}s_{h_{1}X_{h_{1}}},\] \[\{h_{3}h_{2}h_{1}h_{4}\}:s_{h_{4}h_{1}}s_{h_{1}h_{2}}s_{h_{2}X_{h _{2}}}s_{h_{3}X_{h_{3}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }\{h_{2}h_{3}h_{1}h_{4}\}:s_{h_{4}h_{1}}s_{h_{1}h_{2}}s_{h_{2}X_{h_{2}}}(s_{h_ {3}X_{h_{3}}}+s_{h_{3}h_{2}}),\] \[\{h_{2}h_{1}h_{3}h_{4}\}:s_{h_{4}h_{1}}s_{h_{1}h_{2}}s_{h_{2}X_{h _{2}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{2}}+s_{h_{3}h_{1}})+s_{h_{4}X_{h_{4}}}s_{ h_{3}h_{1}}s_{h_{1}h_{2}}s_{h_{2}X_{h_{2}}},\] \[\{h_{2}h_{1}h_{4}h_{3}\}:s_{h_{4}h_{1}}s_{h_{1}h_{2}}s_{h_{2}X_{h _{2}}}(s_{h_{3}X_{h_{3}}}+s_{h_{3}h_{2}}+s_{h_{3}h_{1}}+s_{h_{3}h_{4}})+s_{h_{ 4}X_{h_{4}}}s_{h_{3}h_{1}}s_{h_{1}h_{2}}s_{h_{2}X_{h_{2}}}\] \[\{h_{2}h_{4}h_{1}h_{3}\}:s_{h_{4}X_{h_{4}}}s_{h_{3}h_{1}}s_{h_{1} h_{2}}s_{h_{2}X_{h_{2}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }\{h_{4}h_{2}h_{1}h_{3}\}:s_{h_{4}X_{h_{4}}}s_{h_{3}h_{1}}s_{h_{1}h_{2}}(s_{h_ {2}X_{h_{2}}}+s_{h_{2}h_{4}}).\] (48) ### The boundary case $\mathsf{A}=\mathsf{H}$ and partial momentum kernel When we set $\mathsf{A}=\mathsf{H}$, every graph in the gauge invariance induced relation (42) only contains length-1 chains (as shown by examples (43) and (47)). Then the relation (42) becomes \[0=\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle\, \text{perms}\,{\mathsf{H}}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime \boldsymbol{\sigma}}_{\mathsf{H}}[\mathsf{H}]}}\mathcal{D}^{[\mathcal{F}]}(1, \boldsymbol{\sigma},r)A(1,\boldsymbol{\sigma},r).\] (49) Assuming that the reference order is $\mathsf{R}=\left\{h_{\rho(1)},h_{\rho(2)},\dots,h_{\rho(s)}\right\}$, let us analyze the coefficients in the above equation in more detail. A length-1 chain started by $h_{\rho(s)}$ can end at any gluon $l_{s}\in\{1,\dots,r-1\}$ s.t. $\sigma^{-1}(l_{s})<\sigma^{-1}(h_{\rho(s)})$ and is associated with a factor $s_{h_{\rho(s)}l_{s}}$. A length-1 chain started by $h_{\rho(s-1)}$ can end at any element $l_{s-1}\in\{1,\dots,r-1\}\cup\{h_{\rho(s)}\}$ s.t., $\sigma^{-1}(l_{s-1})<\sigma^{-1}(h_{\rho(s-1)})$ and is associated with a factor $s_{h_{\rho(s-1)}l_{s-1}}$. This observation can be extended to arbitrary case: a length-1 chain started by $h_{\rho(i)}$ in (49) can end at any $l_{i}\in\{1,\dots,r-1\}\cup\{h_{\rho(i+1)},\dots,h_{\rho(s)}\}$ s.t., $\sigma^{-1}(l_{i})<\sigma^{-1}(h_{\rho(i)})$. The coefficient for given permutation $\boldsymbol{\sigma}$ then reads \[\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\sigma }}_{\mathsf{H}}[\mathsf{H}]}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\sigma} ,r) = \sum\limits_{\begin{subarray}{c}l_{i}\in\{1,2,\dots,r-1\}\cup\{h_ {\rho(i+1)},\dots,h_{\rho(s)}\}\\ \text{s.t.\,}\sigma^{-1}(l_{i})<\sigma^{-1}(h_{\rho(i)})\,\text{for all}\,i=1, \dots,s\end{subarray}}s_{h_{\rho(1)}l_{1}}s_{h_{\rho(2)}l_{2}}\dots s_{h_{\rho (s)}l_{s}}.\] (50) An interesting observation is that we can reexpress the coefficient (50) by defining ‘partial momentum kernel’. Given two permutations $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$ of elements in $\{2,\dots,m\}$ and a nonempty subset $\mathsf{H}$ of $\{2,\dots,m\}$, the partial momentum kernel $\widetilde{S}_{\mathsf{H}}[\boldsymbol{\sigma}\|\boldsymbol{\rho}]$ is defined by \[\widetilde{S}_{\mathsf{H}}[\boldsymbol{\sigma}\|\boldsymbol{\rho}] \equiv\prod\limits_{a\in\mathsf{H}}\biggl{[}s_{a1}+\sum\limits_{l\in\{2,\dots, m\}}\theta(\sigma^{-1}(a)-\sigma^{-1}(l))\theta(\rho^{-1}(a)-\rho^{-1}(l))s_{ al}\biggr{]},\] (51) where $\sigma^{-1}(a)$ and $\rho^{-1}(a)$ denote the positions of $a$ in the permutations $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$ respectively . Given $a\in\mathsf{H}$ and $l\in\{2,\dots,m\}$, the product of two step functions in (51) is $1$ if both $\sigma^{-1}(a)>\sigma^{-1}(l)$ and $\rho^{-1}(a)>\rho^{-1}(l)$ are satisfied, otherwise $0$. Explicit examples of the partial momentum kernel are given as \[\widetilde{S}_{\{2\}}[2345\|2543] = s_{21},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{} ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\widetilde{S}_{\{3\}}[2345\|5423]=s _{31}+s_{32},\] \[\widetilde{S}_{\{2,5\}}[2345\|4235] = s_{21}(s_{51}+s_{52}+s_{53}+s_{54}),~{}\widetilde{S}_{\{2,3,4\}} [2345\|3542]=s_{21}s_{31}(s_{41}+s_{43}).\] (52) There are many useful properties satisfied by partial momentum kernels: * Partial momentum kernel $\widetilde{S}_{\mathsf{H}}[\boldsymbol{\sigma}\|\boldsymbol{\rho}]$ is symmetric under exchanging of permutations $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$, _i.e._, \[\widetilde{S}_{\mathsf{H}}[\boldsymbol{\sigma}\|\boldsymbol{\rho}] =\widetilde{S}_{\mathsf{H}}[\boldsymbol{\rho}\|\boldsymbol{\sigma}].\] (53) * If the subset $\mathsf{H}$ is chosen as the full set $\{2,\dots,m\}$, we arrive the usual momentum kernel \[\widetilde{S}_{\{2,\dots,m\}}[\boldsymbol{\sigma}_{2,m}\| \boldsymbol{\rho}_{2,m}]=S[\boldsymbol{\sigma}_{2,m}\|\boldsymbol{\rho}_{2,m}].\] (54) * Assuming that $\boldsymbol{\rho}_{\mathsf{B}}$ and $\boldsymbol{\rho}^{\prime}_{\mathsf{B}}$ are two permutations of elements of a set $\mathsf{B}$, while $\boldsymbol{\rho}_{\mathsf{C}}$ is a permutation of elements of $\mathsf{C}$, the partial momentum kernel $\widetilde{S}_{\mathsf{C}}\left[\boldsymbol{\rho}_{\mathsf{B}},\boldsymbol{ \rho}_{\mathsf{C}}\|\boldsymbol{\sigma}_{\mathsf{B}}\shuffle\boldsymbol{\sigma} _{\mathsf{C}}\right]$ satisfies \[\widetilde{S}_{\mathsf{C}}\left[\boldsymbol{\rho}_{\mathsf{B}}, \boldsymbol{\rho}_{\mathsf{C}}\|\boldsymbol{\sigma}_{\mathsf{B}}\shuffle \boldsymbol{\sigma}_{\mathsf{C}}\right]=\widetilde{S}_{\mathsf{C}}\left[ \boldsymbol{\rho}^{\prime}_{\mathsf{B}},\boldsymbol{\rho}_{\mathsf{C}}\| \boldsymbol{\sigma}_{\mathsf{B}}\shuffle\boldsymbol{\sigma}_{\mathsf{C}}\right].\] (55) * The following property which relates usual momentum kernel and partial momentum kernel will be useful in the coming sections: (56) Having defined the partial momentum kernel (51) and choosing the the reference order as $\mathsf{R}=\{h_{\rho(1)},h_{\rho(2)},\dots,h_{\rho(s)}\}$, we naturally write the coefficient (50) as \[\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\sigma }}_{\mathsf{H}}[\mathsf{H}]}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\sigma} ,r)=\widetilde{S}_{\mathsf{H}}[\boldsymbol{\sigma}\|2,\dots,r-1,h_{\rho(s)},h_{ \rho(s-1)},\dots,h_{\rho(1)}].\] (57) The relation (49) for $\mathsf{A}=\mathsf{H}$ is then conveniently given by \[0=\sum\limits_{\boldsymbol{\sigma}\in\{2,\dots,r-1\}\shuffle \text{perms~{}}\mathsf{H}}\widetilde{S}_{\mathsf{H}}[\boldsymbol{\sigma}\|2, \dots,r-1,h_{\rho(s)},h_{\rho(s-1)},\dots,h_{\rho(1)}]A(1,\boldsymbol{\sigma}, r).\] (58) For the cases with $\mathsf{H}=\{h_{1},h_{2}\}$ and $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$, (58) returns to the examples (43) and (47) respectively. In fact, the relation (58) is consistent with the following fundamental BCJ relation for given permutation $\boldsymbol{\eta}\in\{2,\dots,r-1\}\shuffle\text{perms\,}\{\mathsf{H}\setminus \{h_{\rho(s)}\}\}$ \[0 = s_{h_{\rho(s)}1}A(1,h_{\rho(s)},\eta(1),\eta(2),\dots,\eta(r+s-2 ),r)\] (59) \[+(s_{h_{\rho(s)}1}+s_{h_{\rho(s)}\eta(1)})A(1,\eta(1),h_{\rho(s)} ,\eta(2),\dots,\eta(r+s-2),r)\] \[+\dots+(s_{h_{\rho(s)}1}+s_{h_{\rho(s)}\eta(1)}+\dots+s_{h_{\rho( s)}\eta(r+s-2)})A(1,\eta(1),\eta(2),\dots,\eta(r+s-2),h_{\rho(s)},r).\] ## 4 Three types of BCJ numerators in NLSM As an application of the gauge invariance induced relation (42), we will prove the equivalence between distinct approaches to scattering amplitudes in NLSM: (i) traditional Feynman diagrams, (ii) the CHY formula and (iii) the Abelian Z theory in the remaining sections. The starting point of our proof is the fact that all three approaches result _dual DDM formula_ \[M(1,\dots,n)=\sum\limits_{\boldsymbol{\sigma}\in S_{n-2}}n_{1\| \boldsymbol{\sigma}\|n}A(1,\boldsymbol{\sigma},n),~{}~{}~{}(\text{$n$ is even})\] (60) with distinct (DF, CMS and DT) expressions of BCJ numerators $n_{1\|\boldsymbol{\sigma}\|n}$ (as polynomial functions of Mandelstam variables). The $A(1,\boldsymbol{\sigma},n)$ in (60) are bi-scalar amplitudes. Thus the three approaches are equivalent to each other if and only if the following relations for bi-scalar amplitudes are satisfied: \[\sum\limits_{\boldsymbol{\sigma}\in S_{n-2}}n^{\text{DF}}_{1\| \boldsymbol{\sigma}\|n}A(1,\boldsymbol{\sigma},n)=\sum\limits_{\boldsymbol{ \sigma}\in S_{n-2}}n^{\text{CMS}}_{1\|\boldsymbol{\sigma}\|n}A(1,\boldsymbol{ \sigma},n)=\sum\limits_{\boldsymbol{\sigma}\in S_{n-2}}n^{\text{DT}}_{1\| \boldsymbol{\sigma}\|n}A(1,\boldsymbol{\sigma},n).\] (61) We will review the three types of BCJ numerators in this section and prove the equivalence condition (61) by using (42) in sections 5 and 6. ### Three distinct constructions of BCJ numerators in NLSM Now let us review the DF, CMS and DT types of BCJ numerators which correspond to the Feynman diagram approach, Abelian Z theory and CHY formula. #### The DF type numerators The DF type BCJ numerator was derived by applying off-shell extended BCJ relation [8; 16], which is based on Berends-Giele recursion (thus Feynman diagrams). The explicit expression of DF type BCJ numerator is given by a proper combination of momentum kernel:⁵ [FOOTNOTE:5][ENDFOOTNOTE] \[n^{\text{DF}}_{1\|\boldsymbol{\sigma}\|n}=(-1)\sum\limits_{ \boldsymbol{\rho}\in\mathsf{\Gamma}}S[\boldsymbol{\sigma}\|\boldsymbol{\rho}],\] (62) where we summed over permutations $\boldsymbol{\rho}$ in $\mathsf{\Gamma}$ which is defined as the collection of permutations satisfying the following conditions. For any $a\in\{2,\dots,n-1\}$, we assume $b$ ($c$) is the nearest element on the LHS (RHS) of $a$ in the permutation $\boldsymbol{\rho}$, which satisfies $\sigma^{-1}(b)>\sigma^{-1}(a)$ ($\sigma^{-1}(c)>\sigma^{-1}(a)$) ⁶ . The permutations $\boldsymbol{\rho}$ in the DF type numerator (62) are those satisfying either of the following two conditions: (i) There are odd number of elements between $a$, $b$ as well as $a$, $c$ in the permutation $\boldsymbol{\rho}$. (ii) There is no element between both $a$, $b$ and $a$, $c$ in the permutation $\boldsymbol{\rho}$. Explicit examples are given as [FOOTNOTE:6][ENDFOOTNOTE] \[n^{\text{DF}}_{1\|23\|4} = S[23\|32]=-s_{21}s_{31},\] (63) \[n^{\text{DF}}_{1\|2345\|6} = -(S[2345\|5243]+S[2345\|5342]+S[2345\|4352]+S[2345\|4253]+S[2345\|3254])\] (64) \[= (-1)\Bigl{[}s_{51}\left(s_{41}+s_{42}\right)(s_{31}+s_{32})s_{21} +s_{51}\left(s_{41}+s_{43}\right)s_{31}s_{21}\] \[+(s_{51}+s_{54}+s_{53})s_{41}s_{31}s_{21}+(s_{51}+s_{54}+s_{52})s _{41}(s_{31}+s_{32})s_{21}\] \[+(s_{51}+s_{52}+s_{53})(s_{41}+s_{42}+s_{43})s_{31}s_{21}\Bigr{]}.\] #### The CMS type numerators The CMS type BCJ numerator, which comes from Abelian Z theory [23], expresses each numerator in dual DDM decomposition by only one momentum kernel: \[n^{\text{CMS}}_{1\|\boldsymbol{\sigma}\|n}=(-1)^{n\over 2}S[\sigma (2),\sigma(3),\dots,\sigma(n-1)\|\sigma(2),\sigma(3),\dots,\sigma(n-1)].\] (65) Explicit expressions for four- and six-point cases are \[n^{\text{CMS}}_{1\|23\|4} = S[23\|23]=s_{21}(s_{31}+s_{32})\] \[n^{\text{CMS}}_{1\|2345\|6} = S[2345\|2345]=s_{21}(s_{31}+s_{32})(s_{41}+s_{42}+s_{43})(s_{51}+ s_{52}+s_{53}+s_{54}).\] (66) It is worthy emphasizing that both DF and CMS types BCJ numerators manifest the relabeling symmetry of $n-2$ elements, _i.e._, $n_{1\|\sigma(2),\dots,\sigma(n-1)\|n}$ can be obtained from $n_{1\|2,\dots,n-1\|n}$ by the replacement $2,3,\dots,n-1\to\sigma(2),\sigma(3),\dots,\sigma(n-1)$. #### The DT type numerators Being different from the previous two constructions, the DT type numerator which is based on the graphic expansion of amplitudes and the dimensional reduction in CHY formula is not a symmetric form. This type of BCJ numerators are expanded by graphic rule instead of momentum kernels. The construction of $n^{\text{DT}}_{1\|\boldsymbol{\sigma}\|n}$ is given by * Consider $1$ as the root of a tree and define a reference order of elements in $\{2,\dots,n-1\}$, say $\mathsf{R}\equiv\{\rho(1),\dots,\rho(s=n-2)\}$. * Pick $\rho(s)$ in $\{\sigma(2),\dots,\sigma(n-1)\}$. Construct a chain $\mathbb{C}[1]\equiv\{l=1,i_{1},\dots,i_{j},\rho(s)\}$ of even length started by $\rho(s)$ towards $1$ with internal nodes $i_{1},i_{2},\dots,i_{j}$ ($j$ is odd) s.t. $\sigma^{-1}(l=1)<\sigma^{-1}({i_{1}})<\sigma^{-1}({i_{2}})<\dots<\sigma^{-1}(i _{j})<\sigma^{-1}(\rho(s))$. This chain is associated with a factor \[s_{\rho(s)i_{j}}s_{i_{j}i_{j-1}}\dots s_{i_{2}i_{1}}s_{i_{1}1}.\] (67) Remove this chain from the ordered set $\mathsf{R}\to\mathsf{R}^{\prime}=\mathsf{R}\setminus\{i_{1},i_{2},\dots,i_{j}, \rho(s)\}\equiv\{\rho^{\prime}(1),\dots,\rho^{\prime}(s^{\prime})\}$. * Repeat the previous step: Pick $\rho^{\prime}(s^{\prime})\in\mathsf{R}^{\prime}$ and construct a chain $\mathbb{C}[2]\equiv\{l^{\prime},i^{\prime}_{1},\dots,i^{\prime}_{j^{\prime}}, \rho^{\prime}({s^{\prime}})\}$ of even length ($j^{\prime}$ is odd), which starts from $\rho^{\prime}(s^{\prime})$ towards a node $l^{\prime}$ on $\mathbb{C}[1]$ and satisfies $\sigma^{-1}(l^{\prime})<\sigma^{-1}({i^{\prime}_{1}})<\dots<\sigma^{-1}(i^{ \prime}_{j^{\prime}})<\sigma^{-1}(\rho^{\prime}(s^{\prime}))$. The new chain $\mathbb{C}[2]$ is associated with a factor \[s_{\rho^{\prime}(s^{\prime})i^{\prime}_{j^{\prime}}}s_{i^{\prime }_{j^{\prime}}i^{\prime}_{j^{\prime}-1}}\dots s_{i^{\prime}_{2}i^{\prime}_{1}} s_{i^{\prime}_{1}l^{\prime}}.\] (68) Remove this chain from the ordered set $\mathsf{R}\to\mathsf{R}^{\prime\prime}=\mathsf{R}^{\prime}\setminus\{i^{\prime }_{1},i^{\prime}_{2},\dots,i^{\prime}_{j},\rho^{\prime}(s^{\prime})\}\equiv\{ \rho^{\prime\prime}(1),\dots,\rho^{\prime\prime}(s^{\prime\prime})\}$. * Repeat the above steps until the ordered set $\mathsf{R}$ becomes empty. Each new even-length chain is attached to nodes which have been used and associated with a factor. Collecting the factors corresponding to all chains in a graph and summing over all possible graphs (noting that the total phase factor is $(-1)^{{n\over 2}-1}$), we finally get the BCJ numerator $n^{\text{DT}}_{1\|\boldsymbol{\sigma}\|n}$. By means of the conventions of notations established for the gauge invariance induced relation (42), we can write the numerators of DT type as⁷ [FOOTNOTE:7][ENDFOOTNOTE] \[n^{\text{DT}}_{1\|\boldsymbol{\sigma}\|n}=(-1)^{n\over 2}\sum \limits_{\mathcal{F}\in{\mathcal{G}^{\prime\,\boldsymbol{\sigma}}_{\{2,\dots,n -1\}}[\emptyset]},}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\sigma},n)\] (69) where the $\mathsf{H}$ set, whose elements serve as starting points or internal nodes of trees, is chosen as $\{2,\dots,n-1\}$. The empty set $\emptyset$ in $\mathcal{G}^{\prime\,\boldsymbol{\sigma}}_{\{2,\dots,n-1\}}[\emptyset]$ means that all chains are of even length. The explicit expressions for four-point numerators $n^{\text{DT}}_{1\|23\|4}$ and $n^{\text{DT}}_{1\|32\|4}$ are given by \[n^{\text{DT}}_{1\|23\|4}=-s_{32}s_{21},~{}~{}~{}~{}~{}~{}n^{\text{ DT}}_{1\|32\|4}=0,\] (70) where the reference order is chosen as $\mathsf{R}=\{2,3\}$. ## 5 The equivalence between DT and CMS constructions of NLSM amplitudes The DT and the CMS types of numerators produce the same amplitude if and only if the second equality in (61) holds. Substituting (69) and (65) into (61), we arrive the following relation for bi-scalar amplitudes $A(1,\boldsymbol{\sigma},n)$ (71) To prove the equivalence condition (71), we carry on our discussions in a more generic framework: * The momentum kernel $S[\boldsymbol{\sigma}\|\boldsymbol{\sigma}]$ is generalized to the partial momentum kernel \[\widetilde{S}_{\mathsf{H}}\left[\{2,\dots,r-1\}\shuffle \boldsymbol{\sigma}_{\mathsf{H}}\|2,\dots,r-1,\boldsymbol{\sigma}_{\mathsf{H}}\right]\] (72) where $\mathsf{H}$ is an arbitrary nonempty set with $s$ elements. When setting $\{2,\dots,r-1\}=\emptyset$ and $\mathsf{H}=\{2,\dots,n-1\}$, we return to the original momentum kernel $S[\boldsymbol{\sigma}\|\boldsymbol{\sigma}]$ ($\boldsymbol{\sigma}\in S_{n-2}$). * The number of external particles is not limited to be even. Amplitudes with odd number external particles are also under consideration. * The amplitude $A(1,\boldsymbol{\sigma},n)$ can be color-ordered Yang-Mills, bi-scalar or color-ordered NLSM amplitudes. Having the above generalizations, we will prove the following two relations and (74) corresponding to whether the number of elements in the set $\mathsf{H}$ is even or odd. Coefficients of amplitudes therein are expressed by partial momentum kernels, while the summation $\sum\limits_{\boldsymbol{\sigma}_{\mathsf{H}}}$ means that we sum over all possible permutations of elements in $\mathsf{H}$. Consequences of the relations (5) and (74) are deduced: * When we we set $r=n$ (for even $n$), $\mathsf{H}=\{2,\dots,n-1\}$ and $\{2,\dots,r-1\}\to\emptyset$, the relation (5) naturally returns to the equivalence condition (71) for even $n$. Thus the equivalence condition (71) between DT and CMS constructions is proven. * When we set $\{2,\dots,r-1\}\to\{\rho(2),\dots,\rho(r-1)\}$ in the partial momentum kernel $\widetilde{S}_{\mathsf{H}}$ in (74) and apply the property (55) and (53), the relation (74) then becomes \[\sum\limits_{\boldsymbol{\sigma}_{\mathsf{H}}}\sum\limits_{ \boldsymbol{\alpha}\,\in\,\boldsymbol{\rho}\,\shuffle\,\boldsymbol{\sigma}_{ \mathsf{H}}}\widetilde{S}_{\mathsf{H}}\left[\boldsymbol{\alpha}\|2,\dots,r-1, \boldsymbol{\sigma}_{\mathsf{H}}\right]A(1,\boldsymbol{\alpha},r)=0\,(\text{ for odd $s$}).\] (75) Multiplying a momentum kernel $S[\rho(2),\rho(3),\dots,\rho(r-1)\|2,3,\dots,r-1]$ to both sides of the above relation and applying the relation (56) between usual momentum kernel and partial momentum kernel, we arrive an amplitude relation expressed by usual momentum kernels \[\boxed{\sum\limits_{\boldsymbol{\sigma}_{\mathsf{H}}}\sum\limits_ {\begin{subarray}{c}\boldsymbol{\alpha}\,\in\,\boldsymbol{\rho}\,\shuffle\, \boldsymbol{\sigma}_{\mathsf{H}}\end{subarray}}S\left[\boldsymbol{\alpha}\|2, \dots,r-1,\boldsymbol{\sigma}_{\mathsf{H}}\right]A(1,\boldsymbol{\alpha},r)=0 \,(\text{for odd $s$})},\] (76) where $\boldsymbol{\rho}$ is an arbitrary permutation of elements in $\{2,\dots,r-1\}$. The boundary case with $\mathsf{H}=\{2,\dots,n-1\}$, $\{1,\dots,r\}\to\{1,n\}$ shows very interesting relation for amplitudes with odd number of external particles \[\boxed{\sum\limits_{\boldsymbol{\sigma}\in S_{n-2}}S\left[ \boldsymbol{\sigma}\|\boldsymbol{\sigma}\right]A(1,\boldsymbol{\sigma},n)=0\,( \text{for odd $n$})}.\] (77) Although the relation (74) for odd $s$ is not used in the proof of the equivalence condition (71) between the DT and the CMS constructions of NLSM amplitudes, the relation (76) as a result of (74), plays a crucial role in the proof of the equivalence between the DF and CMS constructions in the next section. In the remaining discussions of this section, we establish the graphic expansion of the partial momentum kernel and prove the relations (5) and (74). ### Expressing partial momentum kernel by graphs The partial momentum kernel $\widetilde{S}_{\mathsf{H}}[\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle \boldsymbol{\sigma}_{\mathsf{H}}\|2,\dots,r-1,\boldsymbol{\sigma}_{\mathsf{H}}]$ can be conveniently expanded by the graphic rule in section (2.2), when replacing the factors $\epsilon_{h_{a}}\cdot F_{h_{i_{1}}}\cdot\dots\cdot F_{h_{i_{j}}}\cdot k_{b}$ for each chain by $s_{h_{a}h_{i_{1}}}s_{h_{i_{1}}h_{i_{2}}}\dots s_{h_{i_{j}}b}$. The reference order $\mathsf{R}=\{h_{\rho(1)},h_{\rho(2)},\dots,h_{\rho(s)}\}$ is chosen arbitrarily. We demonstrate this expansion by examples first. #### Example-1: $\mathsf{H}=\{h_{1},h_{2}\}$ The $\sigma_{\mathsf{H}}$ in the partial momentum kernel $\widetilde{S}_{\{h_{1},h_{2}\}}[\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle \sigma_{\mathsf{H}}\|2,\dots,r-1,\sigma_{\mathsf{H}}]$ can be either $\{h_{1},h_{2}\}$ or $\{h_{2},h_{1}\}$. If we define reference order $\mathsf{R}=\{h_{1},h_{2}\}$, the partial momentum kernel with $\sigma_{\mathsf{H}}=\{h_{1},h_{2}\}$ is expressed by the sum of $(a)$ and $(b)$ in figure 1, while the partial momentum kernel with $\sigma_{\mathsf{H}}=\{h_{2},h_{1}\}$ is expressed by the sum of $(c)$ and $(d)$ in figure 1. If we change the reference order to $\mathsf{R}=\{h_{2},h_{1}\}$, graphs contributing to $\boldsymbol{\sigma}_{\mathsf{H}}=\{h_{1},h_{2}\}$ ($\boldsymbol{\sigma}_{\mathsf{H}}=\{h_{2},h_{1}\}$) become the graphs $(c)$ and $(d)$ ($(a)$ and $(b)$) in figure 1 with exchanging $h_{1}$ and $h_{2}$. Though the chain structures are different for different choices of reference order, the expression of each partial momentum kernel $\widetilde{S}_{\{h_{1},h_{2}\}}[\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle \sigma_{\mathsf{H}}\|2,\dots,r-1,\sigma_{\mathsf{H}}]$ is not changed. #### Example-2: $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$ We now consider the partial momentum kernel \[\widetilde{S}_{\{h_{1},h_{2},h_{3}\}}[\boldsymbol{\alpha}\in\{2, \dots,r-1\}\shuffle\{h_{1},h_{3},h_{2}\}\|2,\dots,r-1,\{h_{1},h_{3},h_{2}\}]\] (78) where $\mathsf{H}$ contains three elements and $\boldsymbol{\sigma}_{\mathsf{H}}$ in this example is chosen as $\boldsymbol{\sigma}_{\mathsf{H}}=\{h_{1},h_{3},h_{2}\}$. From the definition (51), (78) is given by the product of three factors \[\Biggl{[}s_{h_{1}1}+\sum\limits_{\small\begin{subarray}{c}i\in\{2 ,\dots,r-1\}\\ \alpha^{-1}(i)<\alpha^{-1}(h_{1})\end{subarray}}s_{h_{1}i}\Biggr{]}\Biggl{[}s_ {h_{3}1}+s_{h_{3}h_{1}}+\sum\limits_{\small\begin{subarray}{c}i\in\{2,\dots,r- 1\}\\ \alpha^{-1}(i)<\alpha^{-1}(h_{3})\end{subarray}}s_{h_{3}i}\Biggr{]}\Biggl{[}s_ {h_{2}1}+s_{h_{2}h_{1}}+s_{h_{2}h_{3}}+\sum\limits_{\small\begin{subarray}{c}i \in\{2,\dots,r-1\}\\ \alpha^{-1}(i)<\alpha^{-1}(h_{2})\end{subarray}}s_{h_{2}i}\Biggr{]}.\] This partial momentum kernel can be obtained as follows: * Define a reference order of elements in $\mathsf{H}$, e.g., $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$. * Pick the last element $h_{3}$ in the ordered set $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$ and pick a term from the factor corresponding to $h_{3}$ in (5.1). Such a term has the form $s_{h_{3}j}$, where $j$ can be any element in $\{h_{1}\}\cup\{1,2,\dots,r-1\}$ s.t., $\alpha^{-1}(j)<\alpha^{-1}(h_{3})$. If $j$ is an element in $\{1,2,\dots,r-1\}$, we get a length-1 chain started from $h_{3}$ towards $\{1,2,\dots,r-1\}$. Else, if $j=h_{1}$, we further pick a factor $s_{h_{1}k}$ for $k\in\{1,2,\dots,r-1\}$ satisfying $\alpha^{-1}(k)<\alpha^{-1}(h_{1})$, then a chain $s_{h_{3}h_{1}}s_{h_{1}k}$ started from $h_{3}$ towards $k$ have been constructed. We take the $j=h_{1}$ case for instance and continue our discussion. * Remove the starting node $h_{3}$ and the internal node $h_{1}$ of the chain which have been already constructed, from the ordered set $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$ and redefine $\mathsf{R}$ as $\mathsf{R}\to\mathsf{R}^{\prime}=\{h_{2}\}$. Construct a chain started from the element $h_{2}$ in $\mathsf{R}^{\prime}$ towards $l\in\{h_{1},h_{3}\}\cup\{1,2,\dots,r-1\}$. Then we have a factor $s_{h_{2}l}$. For example, we choose $l=h_{1}$. * Remove $h_{2}$ from $\mathsf{R}^{\prime}$, then the set $\mathsf{R}^{\prime}$ becomes empty. Putting the chains obtained together, we arrive a term $s_{h_{3}h_{1}}s_{h_{1}k}s_{h_{2}h_{1}}$ corresponding to the graph $(b4)$ of figure 2. * The full partial momentum kernel in this example is obtained by summing over all possible graphs constructed by the above steps (displayed by the graphs $(b1)\sim(b6)$ in figure 2). Again, we emphasize that the reference order $\mathsf{R}$ can be chosen arbitrarily. If we change the reference order, only the chains are changed, the structure of graphs and the final expression of partial momentum kernel are not changed. Now we extend our discussions to the graphic expansion of any partial momentum kernel with the form: \[\widetilde{S}_{\mathsf{H}}[\{2,\dots,r-1\}\shuffle\boldsymbol{ \sigma}_{\mathsf{H}}\|2,\dots,r-1,\boldsymbol{\sigma}_{\mathsf{H}}]\] (80) \[= \Biggl{[}s_{{\sigma}_{\mathsf{H}}(1)1}+\sum\limits_{\small \begin{subarray}{c}i\in\{2,\dots,r-1\}\\ \alpha^{-1}(i)<\alpha^{-1}(\sigma_{{\mathsf{H}}(1)})\end{subarray}}s_{{\sigma} _{\mathsf{H}}(1)i}\Biggr{]}\Biggl{[}s_{{\sigma}_{\mathsf{H}}(2)1}+s_{{\sigma}_ {\mathsf{H}}(2)\sigma_{\mathsf{H}}(1)}+\sum\limits_{\small\begin{subarray}{c}i \in\{2,\dots,r-1\}\\ \alpha^{-1}(i)<\alpha^{-1}(\sigma_{{\mathsf{H}}(2)})\end{subarray}}s_{{\sigma} _{\mathsf{H}}(2)i}\Biggr{]}\] \[\times\dots\times\Biggl{[}s_{{\sigma}_{\mathsf{H}}(s)1}+s_{{ \sigma}_{\mathsf{H}}(s)\sigma_{\mathsf{H}}(1)}+\dots s_{{\sigma}_{\mathsf{H}}( s)\sigma_{\mathsf{H}}(s-1)}+\sum\limits_{\small\begin{subarray}{c}i\in\{2, \dots,r-1\}\\ \alpha^{-1}(i)<\alpha^{-1}(\sigma_{{\mathsf{H}}(s)})\end{subarray}}s_{{\sigma} _{\mathsf{H}}(s)i}\Biggr{]}.\] * Define a reference order $\mathsf{R}=\{h_{\rho(1)},h_{\rho(2)},\dots,h_{\rho(s)}\}$ for elements in the set $\mathsf{H}$ (assume there are $s$ elements in the set $\mathsf{H}$). Pick $h_{\rho(s)}$ and an arbitrary term $s_{h_{\rho(s)h_{i_{j}}}}$ ($\sigma^{-1}(h_{i_{j}})<\sigma^{-1}(h_{\rho(s)})$) from the factor corresponding to $h_{\rho(s)}$. Then pick an arbitrary term $s_{h_{i_{j}}h_{i_{j-1}}}$ ($\sigma^{-1}(h_{i_{j-1}})<\sigma^{-1}(h_{i_{j}})$) from the factor corresponding to $h_{i_{j}}$. Next, pick a term of the form $s_{h_{i_{j-1}}h_{i_{j-2}}}$ ($\sigma^{-1}(h_{i_{j-2}})<\sigma^{-1}(h_{i_{j-1}})$) from the factor corresponding to $h_{i_{j-1}}$, and so on. This procedure is terminated at a factor $s_{h_{i_{1}}l}$ where $l$ belongs to the set $\{1,2,\dots,r-1\}$. Putting all factors together, we get a chain $s_{h_{\rho(s)h_{i_{j}}}}s_{h_{i_{j}}h_{i_{j-1}}}\dots s_{h_{i_{1}}l}$. Redefine $\mathsf{R}$ by removing the internal nodes and the starting point of the chain which was already constructed: $\mathsf{R}\to\mathsf{R}^{\prime}=\mathsf{R}\setminus\{h_{i_{1}},\dots,h_{i_{j} },h_{\rho(s)}\}\equiv\{h_{\rho^{\prime}(1)},h_{\rho^{\prime}(2)},\dots,h_{\rho ^{\prime}(s^{\prime})}\}$. * We construct a chain from $h_{\rho^{\prime}(s^{\prime})}$ towards an element $l^{\prime}\in\{1,2,\dots,r\}\cup\{h_{i_{1}},\dots,h_{i_{j}},h_{\rho(s)}\}$ by picking $s_{h_{\rho^{\prime}(s^{\prime})}h_{i^{\prime}_{j^{\prime}}}}$, $s_{h_{i^{\prime}_{j^{\prime}}}h_{i^{\prime}_{j^{\prime}-1}}}$, …, $s_{h_{i^{\prime}_{1}}l^{\prime}}$ ($\sigma^{-1}(l^{\prime})<\sigma^{-1}(h_{i^{\prime}_{1}})<\dots<\sigma^{-1}(h_{i ^{\prime}_{j^{\prime}}})<\sigma^{-1}(h_{\rho(s^{\prime})})$) from the factors corresponding to $h_{\rho(s^{\prime})}$, $h_{i^{\prime}_{j^{\prime}}}$, …, $h_{i^{\prime}_{1}}$ in the partial momentum kernel (80). The we get another chain $s_{h_{\rho^{\prime}(s^{\prime})}h_{i^{\prime}_{j^{\prime}}}}s_{h_{i^{\prime}_{ j^{\prime}}}h_{i^{\prime}_{j^{\prime}-1}}}\dots s_{h_{i^{\prime}_{1}}l^{\prime}}$. Redefine $\mathsf{R}$ by $\mathsf{R}\to\mathsf{R}^{\prime\prime}=\mathsf{R}^{\prime}\setminus\{h_{i^{ \prime}_{1}},\dots,h_{i^{\prime}_{j^{\prime}}},h_{\rho^{\prime}(s^{\prime})}\}$. * Repeat the above steps until the $\mathsf{R}$ set becomes empty. Then putting all chains together, we get a graph. The sum of all possible graphs gives the partial momentum kernel (80). Obviously, if we define a unique reference order $\mathsf{R}$ for permutations $\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\boldsymbol{\sigma}_{\mathsf{H}}$ with all possible $\boldsymbol{\sigma}_{\mathsf{H}}$, the above graphic expansions of partial momentum kernels $\widetilde{S}_{\mathsf{H}}[\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle \boldsymbol{\sigma}_{\mathsf{H}}\|2,\dots,r-1,\boldsymbol{\sigma}_{\mathsf{H}}]$ are related with the graphic expansion of $\mathcal{C}(1,\boldsymbol{\sigma},r)$ (see (25)) in section (2.2) via replacing the factor $\epsilon_{h_{a}}\cdot F_{h_{i_{j}}}\cdot\dots\cdot F_{h_{i_{1}}}\cdot k_{b}$ for every chain by $s_{h_{a}h_{i_{j}}}s_{h_{i_{j}}h_{i_{j-1}}}\dots s_{h_{i_{1}}b}$. ### Proof of the relations (5) and (74) We have already shown that the equivalence condition (71) is a special case of the relation (5) with even $s$. In addition, we also have the relation (74) with odd $s$. Now let us prove both relations (5) and (74) by expanding the partial momentum kernels into graphs. #### 5.2.1 The proof of (5) To prove the relation (5) for even $s$, we first investigate two examples. Example-1: $\mathsf{H}=\{h_{1},h_{2}\}$ The simplest example for even $s$ is the case $\mathsf{H}=\{h_{1},h_{2}\}$ (hence $s=2$). If we choose reference order as $\mathsf{R}=\{h_{1},h_{2}\}$, the graphs corresponding to $\sigma_{\mathsf{H}}=\{h_{1},h_{2}\}$ ($\sigma_{\mathsf{H}}=\{h_{2},h_{1}\}$) are explicitly given by $(a)$ and $(b)$ ($(c)$ and $(d)$)in figure 1. The LHS of (5) for this case reads \[\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{1} ,h_{2}\}}\widetilde{S}_{\{h_{1},h_{2}\}}\bigl{[}\boldsymbol{\alpha}\big{\|}2, \dots,r-1,h_{1},h_{2}\bigr{]}A(1,\boldsymbol{\alpha},r)\] (81) \[+ \sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{2} ,h_{1}\}}\widetilde{S}_{\{h_{1},h_{2}\}}\bigl{[}\boldsymbol{\alpha}\big{\|}2, \dots,r-1,h_{2},h_{1}\bigr{]}A(1,\boldsymbol{\alpha},r).\] Expanding the partial momentum kernels into graphs (see figure 1), we rewrite the above expression as \[\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{1} ,h_{2}\}}\Bigl{[}\mathcal{D}^{[(a)]}(1,\boldsymbol{\alpha},r)+\mathcal{D}^{[(b )]}(1,\boldsymbol{\alpha},r)\Bigr{]}A(1,\boldsymbol{\alpha},r)\] (82) \[+ \sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{2} ,h_{1}\}}\Bigl{[}\mathcal{D}^{[(c)]}(1,\boldsymbol{\alpha},r)+\mathcal{D}^{[(d )]}(1,\boldsymbol{\alpha},r)\Bigr{]}A(1,\boldsymbol{\alpha},r),\] where $\mathcal{D}^{[(a)]}(1,\boldsymbol{\alpha},r)$, $\mathcal{D}^{[(b)]}(1,\boldsymbol{\alpha},r)$, $\mathcal{D}^{[(c)]}(1,\boldsymbol{\alpha},r)$ and $\mathcal{D}^{[(d)]}(1,\boldsymbol{\alpha},r)$ are coefficients associating to the graphs $(a)$, $(b)$, $(c)$ and $(d)$ in figure 1. The graphs $(a)$, $(c)$ and $(d)$ in the above equation contain two length-1 chains. They together contribute \[\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{1} ,h_{2}\}}\mathcal{D}^{[(c)]}(1,\boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r )+\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{2},h_{1}\}} \Bigl{[}\mathcal{D}^{[(c)]}(1,\boldsymbol{\alpha},r)+\mathcal{D}^{[(d)]}(1, \boldsymbol{\alpha},r)\Bigr{]}A(1,\boldsymbol{\alpha},r)\] (83) \[= \sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,\{h_{1},h_{2}\}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime \boldsymbol{\alpha}}_{\{h_{1},h_{2}\}}[\{h_{1},h_{2}\}]}}\,\,\mathcal{D}^{[ \mathcal{F}]}(1,\boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r),\] which is nothing but the RHS of the example (43), thus have to vanish. The only term that survives is the graph $(b)$ which contains no odd length chain \[\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\{h_{1} ,h_{2}\}}\mathcal{D}^{[(b)]}(1,\boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r )=\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text{perms}\,\{ h_{1},h_{2}\}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{ \alpha}}_{\{h_{1},h_{2}\}}[\emptyset]}}\,\,\mathcal{D}^{[\mathcal{F}]}(1, \boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r),\] agrees with the RHS of (5) with $\mathsf{H}=\{h_{1},h_{2}\}$. Example-2: $\mathsf{H}=\{h_{1},h_{2},h_{3},h_{4}\}$ Inspired by the previous example with $s=2$, one can expand all partial momentum kernels on the LHS of (5) in terms of graphs for a given reference order $\mathsf{R}$. For the case $\mathsf{H}=\{h_{1},h_{2},h_{3},h_{4}\}$, the total length of all chains of each expansion graph should equal to $4$. On the other hand, the total length $L^{\text{total}}$ of all chains is given by \[L^{\text{total}}=L^{\text{odd}}+L^{\text{even}},\] (85) where $L^{\text{odd}}$ and $L^{\text{even}}$ denote the total lengths of all odd- and even-length chains, respectively. If a graph contains odd number of odd-length chains, the total length must be odd according to the above equation. This conflicts with the fact $L^{\text{total}}=4$. Therefore, the number of odd-length chains must be even. In this example, each graph can contain 0, 2 or 4 odd-length chains. Thus for $\mathsf{H}=\{h_{1},h_{2},h_{3},h_{4}\}$, the LHS of (5) is expanded as \[\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,\mathsf{H}}\,\,\,\,\,\,\,\,\,\sum\limits_{\mathcal{F}\in{\mathcal{G}^ {\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\emptyset]}}\mathcal{D}^{[\mathcal{F} ]}(1,\boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r)\] (86) \[+ \sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,\mathsf{H}}\biggl{[}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime \boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{1},h_{2}\}]}}+\sum\limits_{\mathcal{F} \in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{1},h_{3}\}]}}+ \sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H }}[\{h_{1},h_{4}\}]}}\] \[+ \sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,\mathsf{H}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol {\alpha}}_{\mathsf{H}}[\{h_{1},h_{2},h_{3},h_{4}\}]}}\mathcal{D}^{[\mathcal{F} ]}(1,\boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r).\] The last three lines vanishes due to the gauge invariance induced relation (42) with $\mathsf{A}=\{h_{i},h_{j}\}$ ($h_{i},h_{j}\in\mathsf{H}$) and $\mathsf{A}=\{h_{1},h_{2},h_{3},h_{4}\}$ (the case with $\mathsf{A}=\{h_{3},h_{4}\}$ and $\mathsf{R}=\{h_{1},h_{2},h_{3},h_{4}\}$ is explicitly given by the example (48)), while the first line is the RHS of (5) for $s=4$. General proof of (5) If $\mathsf{H}$ contains an arbitrary even number of elements (_i.e._, $s$ is even), the number of odd-length chains in any graph has to be even, as analyzed in the $s=4$ example. Thus the partial momentum kernel can be written as \[S_{\mathsf{H}}\Bigl{[}\boldsymbol{\alpha}\in\{2,\dots,r-1\} \shuffle\boldsymbol{\sigma}_{\mathsf{H}}\Big{\|}2,\dots,r-1,\boldsymbol{\sigma} _{\mathsf{H}}\Bigr{]}\] \[=\] Then the combination of amplitudes on the LHS of (5) turns to \[\sum\limits_{\begin{subarray}{c}\boldsymbol{\alpha}\,\in\,\{2, \dots,r-1\}\\ \,\,\shuffle\,\text{perms\,}{\mathsf{H}}\end{subarray}}\Bigl{[}\sum\limits_{ \mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\emptyset] }}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\alpha},r)+\sum\limits_{\{h_{i_{1} },h_{i_{2}}\}\subset\mathsf{H}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime \boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{i_{1}},h_{i_{2}}\}]}}\mathcal{D}^{[ \mathcal{F}]}(1,\boldsymbol{\alpha},r)\] \[~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dots+\sum\limits_{ \mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\mathsf{H} ]}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\alpha},r)\Bigr{]}A(1,\boldsymbol {\alpha},r)\] (88) Every term in the above expression have the general form \[\sum\limits_{\begin{subarray}{c}\boldsymbol{\alpha}\,\in\,\{2, \dots,r-1\}\\ \,\,\shuffle\,\text{perms}\,\mathsf{H}\end{subarray}}\sum\limits_{\mathcal{F} \in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{i_{1}}h_{i_{2}}, \dots,h_{i_{j}}\}]}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\alpha},r)A(1, \boldsymbol{\alpha},r).~{}~{}\text{(for even $s$ and $j$)}\] (89) If $j\neq 0$, the set $\{h_{i_{1}}h_{i_{2}},\dots,h_{i_{j}}\}\subseteq\mathsf{H}$ is nonempty. Such a term has to vanish due to the gauge invariance induced relation (42) for the nonempty subset $\mathsf{A}$ with even number of elements. The first term in (88) (the case $j=0$) is given by summing over all graphs consisting of only even length chains, which is the RHS of (5). #### 5.2.2 The proof of (74) The first nontrivial example of (74) for odd $s$ is given by $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$. Let us study this case before the general proof of (74). Example: $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$ We expand the partial momentum kernels on the LHS of (74) in terms of graphs for a fixed reference order $\mathsf{R}=\{h_{\rho(1)},h_{\rho(2)},h_{\rho(3)}\}$. For a given graph, the total length of all chains must be $3$. As a consequence, the number of odd length chains in each graph must be odd (in this example it can be $1$ or $3$). Thus the LHS of (74) for $s=3$ is decomposed into \[\sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,\mathsf{H}}\biggl{[}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime \boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{1}\}]}}+\sum\limits_{\mathcal{F}\in{ \mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{2}\}]}}+\sum\limits_ {\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{3}\} ]}}\biggr{]}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\alpha},r)A(1, \boldsymbol{\alpha},r)\] (90) \[+ \sum\limits_{\boldsymbol{\alpha}\in\{2,\dots,r-1\}\shuffle\,\text {perms}\,\mathsf{H}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol {\alpha}}_{\mathsf{H}}[\{h_{1},h_{2},h_{3}\}]}}\mathcal{D}^{[\mathcal{F}]}(1, \boldsymbol{\alpha},r)A(1,\boldsymbol{\alpha},r),\] where each term on the first line vanishes due to the gauge invariance induced relation (42) with $\mathsf{A}=\{h_{i}\}$ ($i=1,2,3$) (see the examples (44), (45) and (46) for $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$), while the last line vanishes because of the relation (42) with $\mathsf{A}=\{h_{1},h_{2},h_{3}\}$ (see the example (47) for $\mathsf{R}=\{h_{1},h_{2},h_{3}\}$). Hence all terms of the LHS of (74) for $s=3$ vanish and the equation (74) for $s=3$ is proven. General proof of (74) If $\mathsf{H}$ contains an arbitrary odd number of elements (_i.e._, $s$ is odd), the number of odd length chains in any graph must be odd as shown in the $s=3$ example. The graphic expansions of partial momentum kernels then read \[S_{\mathsf{H}}\Bigl{[}\boldsymbol{\alpha}\in\{2,\dots,r-1\} \shuffle\boldsymbol{\sigma}_{\mathsf{H}}\Big{\|}2,\dots,r-1,\boldsymbol{\sigma} _{\mathsf{H}}\Bigr{]}\] (91) \[=\] \[~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{(for odd $s$)}.\] The combination of amplitudes in the LHS of (74) leads to \[\sum\limits_{\begin{subarray}{c}\boldsymbol{\alpha}\,\in\,\{2, \dots,r-1\}\\ \,\,\shuffle\,\text{perms\,}\mathsf{H}\end{subarray}}\Bigl{[}\sum\limits_{\{h_ {i_{1}}\}\subset\mathsf{H}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime \boldsymbol{\alpha}}_{\mathsf{H}}[\{h_{i_{1}}\}]}}\mathcal{D}^{[\mathcal{F}]}( 1,\boldsymbol{\alpha},r)+\sum\limits_{\{h_{i_{1}},h_{i_{2}},h_{i_{3}}\}\subset \mathsf{H}}\sum\limits_{\mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\alpha}} _{\mathsf{H}}[\{h_{i_{1}},h_{i_{2}},h_{i_{3}}\}]}}\mathcal{D}^{[\mathcal{F}]}( 1,\boldsymbol{\alpha},r)\] \[~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dots+\sum\limits_{ \mathcal{F}\in{\mathcal{G}^{\prime\boldsymbol{\alpha}}_{\mathsf{H}}[\mathsf{H} ]}}\mathcal{D}^{[\mathcal{F}]}(1,\boldsymbol{\alpha},r)\Bigr{]}A(1,\boldsymbol {\alpha},n),\] (92) in which, all terms must vanish due to the gauge invariance induced relation (42) for $\mathsf{A}$ with odd number of elements. Thus the relation (74) is proven. ## 6 The equivalence between DF and CMS constructions of NLSM amplitudes The equivalence between DF and CMS constructions of NLSM amplitudes, _i.e._, the first equality of (61) can be explicitly expressed by the following amplitude relation (93) where $\mathsf{\Gamma}$ is defined in section 4. In this section, we will prove the relation (93). The identity (76) (as a consequence (74)) with odd $s$ is crucial for the proof. To show the pattern, let us first discuss the four- and six-point examples as a warmup. ### Warm-up examples Now we take the cases with $n=4$ and $n=6$ as examples. #### Four-point example The simplest example is the four-point case, which have already been discussed in [23] and [24]. The LHS of the relation (93) for $n=4$ is explicitly written as \[S[23\|32]A(1,2,3,4)+S[32\|23]A(1,3,2,4).\] (94) Applying the relation (76) with $\mathsf{H}=\{2\}$ and $\mathsf{H}=\{3\}$ on the first and the second terms respectively, we immediately get \[-S[32\|32]A(1,3,2,4)-S[23\|23]A(1,2,3,4),\] (95) which is the RHS of (93) for four-point case. #### Six-point example The relation (93) for six-point amplitudes is much more nontrivial. By substituting the six-point numerators of DF type (64) into the LHS of (93), we get \[\sum\limits_{\boldsymbol{\sigma}\in S_{4}}\Bigl{(}S\left[ \boldsymbol{\sigma}\|\boldsymbol{\rho}=\{\sigma(5),\sigma(2),\sigma(4),\sigma(3 )\}\right]+S\left[\boldsymbol{\sigma}\|\boldsymbol{\rho}=\{\sigma(5),\sigma(3), \sigma(4),\sigma(2)\}\right]\] \[~{}~{}~{}\,\,+S\left[\boldsymbol{\sigma}\|\boldsymbol{\rho}=\{ \sigma(4),\sigma(3),\sigma(5),\sigma(2)\}\right]+S\left[\boldsymbol{\sigma}\| \boldsymbol{\rho}=\{\sigma(4),\sigma(2),\sigma(5),\sigma(3)\}\right]\] \[~{}~{}~{}\,\,+S\left[\boldsymbol{\sigma}\|\boldsymbol{\rho}=\{ \sigma(3),\sigma(2),\sigma(5),\sigma(4)\}\right]\Bigr{)}A(1,\boldsymbol{\sigma },n).\] (96) To prove this expression equals to the RHS of (93) with $n=6$, we perform our discussions by the following steps. _Step-1_ Collect those terms with a same $\boldsymbol{\rho}$. For example, if $\boldsymbol{\rho}=\{2,3,4,5\}$, one finds that the corresponding $\boldsymbol{\sigma}$ in (96) can be \[\{5,3,4,2\},~{}~{}~{}~{}\{5,3,2,4\},~{}~{}~{}~{}\{3,5,4,2\},~{}~{ }~{}~{}\{3,5,2,4\},~{}~{}~{}~{}\{3,2,5,4\}.\] (97) An interesting observation is that above permutations are those satisfying the ‘zigzag pattern’: $\sigma^{-1}(5)<\sigma^{-1}(4)$, $\sigma^{-1}(4)>\sigma^{-1}(3)$ and $\sigma^{-1}(3)<\sigma^{-1}(2)$. For convenience, we define the collection of such permutations by $\mathsf{Z}\{2\|3\|4\|5\}$: \[\mathsf{Z}\{2\|3\|4\|5\}\equiv\left\{\boldsymbol{\sigma}\,\|\,\sigma \in S_{4}\text{~{}s.t~{}}\sigma^{-1}(5)<\sigma^{-1}(4),\,\sigma^{-1}(4)>\sigma ^{-1}(3),\,\sigma^{-1}(3)<\sigma^{-1}(2)\right\}.\] (98) Under this definition, terms with $\boldsymbol{\rho}=\{2,3,4,5\}$ in (96) then give rise \[T(2\|3\|4\|5)\equiv\sum\limits_{\boldsymbol{\sigma}\in\mathsf{Z}\{2 \|3\|4\|5\}}S[\boldsymbol{\sigma}\|2,3,4,5]A(1,\boldsymbol{\sigma},6).\] (99) Terms corresponding to arbitrary $\boldsymbol{\rho}$ can be obtained by relabeling the above expression \[T(\rho(2)\|\rho(3)\|\rho(4)\|\rho(5))\equiv\sum\limits_{\boldsymbol {\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3)\|\rho(4)\|\rho(5)\}}S[\boldsymbol{\sigma} \|\boldsymbol{\rho}]A(1,\boldsymbol{\sigma},6).\] (100) All together, (96) becomes \[\sum\limits_{\boldsymbol{\rho}\in S_{4}}T(\rho(2)\|\rho(3)\|\rho(4) \|\rho(5)).\] (101) _Step-2_ For a given $\boldsymbol{\rho}$, we collect terms corresponding to those permutations $\boldsymbol{\sigma}$ ($\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3)\|\rho(4)\|\rho(5)\}$) in which $\rho(2)$, $\rho(3)$ and $\rho(4)$ have a same relative order. For instance, in the case $\boldsymbol{\rho}=\{2,3,4,5\}$, $T(2\|3\|4\|5)$ then becomes \[T(2\|3\|4\|5) = \Bigl{[}S[5,3,4,2\|2,3,4,5]A(1,5,3,4,2,6)+S[3,5,4,2\|2,3,4,5]A(1,3, 5,4,2,6)\Bigr{]}\] (102) \[+\Bigl{[}S[5,3,2,4\|2,3,4,5]A(1,5,3,2,4,6)+S[3,5,2,4\|2,3,4,5]A(1,3 ,5,2,4,6)\] \[~{}~{}+S[3,2,5,4\|2,3,4,5]A(1,3,2,5,4,6)\Bigr{]},\] where the first line gets contribution from permutations $\boldsymbol{\sigma}\in\mathsf{Z}\{2\|3\|4\|5\}$ with the relative order $\{3,4,2\}$; the second and the third lines get contributions from $\boldsymbol{\sigma}\in\mathsf{Z}\{2\|3\|4\|5\}$ with the relative order $\{3,2,4\}$. By means of the property (76) with ($\mathsf{H}=\{5\}$), we write the first line in the above expression as \[-S[3,4,5,2\|2,3,4,5]A(1,3,4,5,2,6)-S[3,4,2,5\|2,3,4,5]A(1,3,4,2,5,6).\] (103) Similarly, the second and the third lines sum to \[-S[3,2,4,5\|2,3,4,5]A(1,3,2,4,5,6).\] (104) If we define \[\mathsf{Z}\{2\|3,4,5\}\equiv\left\{\boldsymbol{\sigma}\,\|\,\sigma \in S_{4}\text{~{}s.t~{}}\sigma^{-1}(3)<\sigma^{-1}(4)<\sigma^{-1}(5),\,\sigma ^{-1}(3)<\sigma^{-1}(2)\right\},\] (105) the sum of (103) and (104) are further expressed by \[T(2\|3\|4\|5)=(-1)T(2\|3,4,5)\equiv(-1)\sum\limits_{\boldsymbol{ \sigma}\in\mathsf{Z}\{2\|3,4,5\}}S[\boldsymbol{\sigma}\|2,3,4,5]A(1,\boldsymbol{ \sigma},6).\] (106) For the same reason, $T(\rho(2)\|\rho(3)\|\rho(4)\|\rho(5))$ for arbitrary $\boldsymbol{\rho}$ is written as \[T(\rho(2)\|\rho(3)\|\rho(4)\|\rho(5)) = (-1)T(\rho(2)\|\rho(3),\rho(4),\rho(5))\] (107) \[\equiv (-1)\sum\limits_{\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3 ),\rho(4),\rho(5)\}}S[\boldsymbol{\sigma}\|\rho(2),\rho(3),\rho(4),\rho(5)]A(1, \boldsymbol{\sigma},6).\] Therefore (101) turns to \[(-1)\sum\limits_{\boldsymbol{\rho}\in S_{4}}T(\rho(2)\|\rho(3), \rho(4),\rho(5)).\] (108) _Step_-3 Now we collect terms in the combination of amplitudes (108) for a given element $\rho(2)\in\{2,3,4,5\}$. In the case of $\rho(2)=2$, we have \[(-1)\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}}T (2\|\boldsymbol{\sigma})=(-1)\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}} \{3,4,5\}}\sum\limits_{\boldsymbol{\alpha}\in\mathsf{Z}\{2\|\sigma(3),\sigma(4) ,\sigma(5)\}}S[\boldsymbol{\alpha}\|2,\sigma(3),\sigma(4),\sigma(5)]A(1, \boldsymbol{\alpha},6).\] (109) For each relative order $\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}$, the sum over $\boldsymbol{\alpha}\in\mathsf{Z}\{2\|\sigma(3),\sigma(4),\sigma(5)\}$ means summing over all possible permutations $\boldsymbol{\alpha}\in\{2\}\shuffle\{\sigma(3),\sigma(4),\sigma(5)\}$ with $\alpha^{-1}(2)>\alpha^{-1}(\sigma(3))$. When all possible $\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}$ are taken into account, according to the relation (76) with $\mathsf{H}=\{3,4,5\}$, the above equation converts to the sum of all terms with $\boldsymbol{\alpha}\in\{2\}\shuffle\{\sigma(3),\sigma(4),\sigma(5)\}$ s.t. $\alpha^{-1}(2)<\alpha^{-1}(\sigma(3))$ for all $\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}$, accompanied by a total minus. Hence, we arrive \[(-1)\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}}T (2\|\boldsymbol{\sigma})\] \[= \sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}}T(2, \boldsymbol{\sigma})\equiv\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}}\{ 3,4,5\}}S[2,\sigma(3),\sigma(4),\sigma(5)\|2,\sigma(3),\sigma(4),\sigma(5)]A(1, 2,\sigma(3),\sigma(4),\sigma(5),6).\] The cases $\rho(2)=3,4,5$ are obtained similarly. Finally, (108) becomes \[\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}}\{3,4,5\}}T(2, \boldsymbol{\sigma})+\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}}\{2,4,5 \}}T(3,\boldsymbol{\sigma})+\sum\limits_{\boldsymbol{\sigma}\in\text{perms~{}} \{2,3,5\}}T(4,\boldsymbol{\sigma})+\sum\limits_{\boldsymbol{\sigma}\in\text{ perms~{}}\{2,3,4\}}T(5,\boldsymbol{\sigma})\] (111) \[= \sum\limits_{\boldsymbol{\sigma}\in S_{4}}S[\boldsymbol{\sigma}\| \boldsymbol{\sigma}]A(1,\boldsymbol{\sigma},6),\] which is the RHS of the equivalence condition (93) for $n=6$. To summarize the above steps, the six-point example for (93) is proved by \[\Bigl{[}\text{LHS of \eqref{Eq:EquivDFCMS} (for $n=6$)}\Bigr{]} = \sum\limits_{\boldsymbol{\rho}\in S_{4}}\,T(\rho(2)\|\rho(3)\|\rho( 4)\|\rho(5))\,\,\,=(-1)\sum\limits_{\boldsymbol{\rho}\in S_{4}}\,T(\rho(2)\|\rho (3),\rho(4),\rho(5))\] (112) \[= \sum\limits_{\boldsymbol{\rho}\in S_{4}}\,T(\rho(2),\rho(3),\rho( 4),\rho(5))=\Bigl{[}\text{RHS of \eqref{Eq:EquivDFCMS} (for $n=6$)}\Bigr{]}.\] ### General proof of the relation (93) Now let us extend the six-point example to a general proof of (93). As in six-point example, we introduce zigzag permutations for any given $\boldsymbol{\rho}\in S_{n-2}$ by \[~{}~{}\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(2j)\|\rho(2j+1),\dots ,\rho(n-1)\}\] \[\equiv \{\boldsymbol{\sigma}\|\boldsymbol{\sigma}\in S_{n-2},\text{s.t.~{ }}\sigma^{-1}({\rho(n-1)})>\sigma^{-1}({\rho(n-2)})>\dots\sigma^{-1}(\rho(2j+2 ))>\sigma^{-1}(\rho(2j+1)),\] \[~{}~{}\sigma^{-1}(\rho(2j+1))<\sigma^{-1}(\rho(2j)),\sigma^{-1}( \rho(2j))>\sigma^{-1}(\rho(2j-1)),\dots,\sigma^{-1}(\rho(3))<\sigma^{-1}(\rho( 2))\}\] \[~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{(for $j\geq 0$ and even $n$)},\] where the $j=0$ case is understood as $\mathsf{Z}\{\rho(2),\rho(3),\dots,\rho(n-1)\}\equiv\boldsymbol{\rho}$. We further define a linear combination of amplitudes \[T(\rho(2)\|\dots\|\rho(2j)\|\rho(2j+1),\dots,\rho(n-1))\equiv\sum \limits_{\small\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\dots\|\rho(2j)\|\rho(2 j+1),\dots,\rho(n-1)\}}S[\boldsymbol{\sigma}\|\boldsymbol{\rho}]A(1,\boldsymbol {\sigma},n),\] (114) in which, the coefficients are momentum kernels. The six-point example (see (112)) implies the following recursive relation between $T(\rho(2)\|\dots\|\rho(2j)\|\rho(2j+1),\dots,\rho(n-1))$: \[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{\boldsymbol{\rho}\in S_{ n-2}}T(\rho(2)\|\dots\|\rho(2j)\|\rho(2j+1),\dots,\rho(n-1))\] (115) \[= (-1)\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}\,T(\rho(2)\|\dots\| \rho(2j-2)\|\rho(2j-1),\dots,\rho(n-1))~{}~{}~{}~{}(0\leq j\leq{n-2\over 2}).\] The proof of (115) is provided in appendix B. We consider two boundaries of this relation: * The upper boundary is $j={n-2\over 2}$, for which the LHS of (115) is ${\sum_{\boldsymbol{\rho}\in S_{n-2}}}T(\rho(2)\|\rho(3)\|\dots\|\rho(n-1))$. In appendix C, we show that the collection of all $\boldsymbol{\sigma}$ corresponding to a same $\boldsymbol{\rho}$ on the LHS of (93) is $\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(n-1)\}$ (_i.e._, $j={n-2\over 2}$). Thus the LHS of (115) for $j={n-2\over 2}$ is \[{\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}}T(\rho(2)\|\rho(3)\| \dots\|\rho(n-1))=\sum\limits_{\boldsymbol{\sigma}\in S_{n-2}}\sum\limits_{ \boldsymbol{\rho}\in\mathsf{\Gamma}}S[\boldsymbol{\sigma}\|\boldsymbol{\rho}]A( 1,\boldsymbol{\sigma},n),\] (116) which is the LHS of the equivalence condition (93). * The lower boundary is $j=0$. In this case, the sum on the RHS of (115) is given by \[\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}T(\rho(2),\rho(3),\dots ,\rho(n-1))=\sum\limits_{\boldsymbol{\sigma}\in S_{n-2}}S[\boldsymbol{\sigma}\| \boldsymbol{\sigma}]A(1,\boldsymbol{\sigma},n)\] (117) which is nothing but (upto a factor ${(-1)^{n-2\over 2}}$) the RHS of (93). When we start from the upper boundary and apply the relation (115) by ${n-2\over 2}$ times, we arrive the lower boundary with the correct factor $(-1)^{n-2\over 2}$. Thus the equivalence condition (93) is proven. ## 7 Conclusions In this paper, we derived highly-nontrivial generalized BCJ relation (42) by imposing gauge invariance and dimensional reduction on the graphic expansion of EYM amplitudes. Two additional relations (5) and (74) expressed by partial momentum kernels are consequent results of the gauge invariance induced relation (42). As an application, we proved the equivalence between amplitudes constructed by three different types of BCJ numerators. Thus the three approaches (Feynman rules, Abelian Z theory and CHY formula) to NLSM amplitudes are equivalent to each other. This way we prove the CHY formula of NLSM directly instead of relying on incomplete evidence, like the enhanced soft behavior [46]. There are several further directions. (i) First, generalized BCJ relations induced from the gauge invariance of multi-trace amplitudes deserves further consideration. (ii) Second, it seems that the CHY-inspired dimensional reduction is not the unique way to reduce the Lorentz invariants to pure Mandelstam variables. Along the line of unifying relation [47], one can also turn the polarizations to momenta. In addition, other formulations of gauge invariance identities were depicted in [43; 26; 44; 30; 45]. Thus it will be interesting to give a more comprehensive understanding of the gauge invariance induced relations by considering [47] and [43; 26; 44; 30; 45]⁸. (iii) As we have seen, the gauge invariance induced relations bridge the DF type BCJ numerators of NLSM amplitudes and the compact CMS type ones . Maybe they will help us to find compact polynomial BCJ numerators of YM amplitudes which are independent of any reference ordering from that of DF type. We know the sum of BCJ numerators of all possible reference orderings satisfy this requirement, but how about more compact ones? (iv) Last but not least, the gauge invariance induced relations should also exists in string theory. How about their applications in string amplitudes? [FOOTNOTE:8][ENDFOOTNOTE] ## Acknowledgments YD would like to acknowledge Jiangsu Ministry of Science and Technology under contract BK20170410, NSFC under Grant Nos.11105118, 111547310 as well as the "Fundamental Research Funds for the Central Universities". ## Appendix A All graphs for $\mathsf{H}=\{h_{1},h_{2},h_{3}\}$ <figure><img src="content_image/1803.01701/5ptGraphs.jpg"><figcaption>Figure 2: All possible graphs with H={h1,h2,h3}. Graphs in each row contributeto permutations {2,…,r−1}\shuffleσH for a given relative order σH.</figcaption></figure> When we choose the relative order $R=\{h_{1},h_{2},h_{3}\}$, all possible graphs are given by figure 2. The correspondence of graphs and the relative permutations $\boldsymbol{\sigma}_{\mathsf{H}}$ is given by \[\{h_{1},h_{2},h_{3}\}:~{}~{}~{}~{}(a1)\sim(a6);~{}~{}~{}~{}\{h_{1 },h_{3},h_{2}\}:~{}~{}~{}~{}(b1)\sim(b6);~{}~{}~{}~{}\{h_{2},h_{1},h_{3}\}:~{} ~{}~{}~{}(c1)\sim(c6);\] \[\{h_{2},h_{3},h_{1}\}:~{}~{}~{}~{}(d1)\sim(d6);~{}~{}~{}~{}\{h_{3 },h_{1},h_{2}\}:~{}~{}~{}~{}(e1)\sim(e6);~{}~{}~{}~{}\{h_{3},h_{2},h_{1}\}:~{} ~{}~{}~{}(f1)\sim(f6).\] (118) ## Appendix B Proof of (115) To prove the relation (115), we consider the LHS for a given $j$: \[\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}T(\rho(2)\|\dots\|\rho(2j )\|\rho(2j+1),\dots,\rho(n-1)).\] (119) Assuming $\mathsf{I}_{2j-1}\equiv\{i_{2},i_{3},\dots,i_{2j}\}$ with $2j-1$ elements is a subset of $\{2,\dots,n-1\}$, we can divide the set $\{2,\dots,n-1\}$ into two parts $\{i_{2},i_{3},\dots,i_{2j}\}$ and its complement $\overline{\mathsf{I}_{2j-1}}=\{2,\dots,n-1\}\setminus\{i_{2},i_{3},\dots,i_{2j}\}$. Then (119) can be arranged as \[\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}T(\rho(2)\|\dots\|\rho(2j )\|\rho(2j+1),\dots,\rho(n-1))\] (120) \[= \sum\limits_{\mathsf{I}_{2j-1}\subseteq\{2,\dots,n-1\}}\Bigl{[} \sum\limits_{\boldsymbol{\rho}_{A}\in\text{perms~{}$\mathsf{I}_{2j-1}$}}\sum \limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline{\mathsf{I}_{2j-1}}$} }T(\rho_{A}(2)\|\dots\|\rho_{A}(2j)\|\rho_{B}(2j+1),\dots,\rho_{B}(n-1))\Bigr{]}.\] in which, the first summation is over all possible choices of the subset $\mathsf{I}_{2j-1}$ for fixed $j$, the second and the third summations are given by summing over all possible permutations of elements in $\mathsf{I}_{2j-1}$ and $\overline{\mathsf{I}_{2j-1}}$. For given $\mathsf{I}_{2j-1}$ and given $\boldsymbol{\rho}_{A}\in\text{perms~{}$\mathsf{I}_{2j-1}$}$, we write the sum over $\boldsymbol{\rho}_{B}$ explicitly \[\sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline{ \mathsf{I}_{2j-1}}$}}T(\rho_{A}(2)\|\dots\|\rho_{A}(2j-2)\|\rho_{A}(2j-1)\|\rho_{A }(2j)\|\rho_{B}(2j+1),\dots,\rho_{B}(n-1))\] (121) \[= \sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline{ \mathsf{I}_{2j-1}}$}}\sum\limits_{\boldsymbol{\sigma}\in\mathsf{Z}\{\rho_{A}(2 )\|\dots\|\rho_{A}(2j)\|\boldsymbol{\rho}_{B}\}}S[\boldsymbol{\sigma}\|\boldsymbol {\rho}_{A},\boldsymbol{\rho}_{B}]A(1,\boldsymbol{\sigma},n).\] According to the definition of zigzag pattern (6.2), the sum over $\boldsymbol{\sigma}$ in the above equation can be realized by the following two steps: (i) first fix a relative order $\boldsymbol{\sigma}_{A}$ of $\rho_{A}(2),\rho_{A}(3),\dots,\rho_{A}(2j)$, s.t., \[\boldsymbol{\sigma}_{A}\in\Bigl{\{}\text{perms~{}}\boldsymbol{ \rho}_{A}\,\text{s.t.}\,\sigma_{A}^{-1}(\rho_{A}(2j-1))<\sigma_{A}^{-1}(\rho_{ A}(2j)),\sigma_{A}^{-1}(\rho_{A}(2j-2))>\sigma_{A}^{-1}(\rho_{A}(2j-1)),\] \[\ldots,\sigma_{A}^{-1}(\rho_{A}(3))<\sigma_{A}^{-1}(\rho_{A}(2)) \Bigr{\}},\] (122) and sum over all possible permutations $\boldsymbol{\sigma}\in\boldsymbol{\sigma}_{A}\shuffle\boldsymbol{\rho}_{B}$ s.t. $\sigma^{-1}(\rho_{B}(2j+1))<\sigma^{-1}(\rho_{A}(2j))$, (ii) sum over all possible $\boldsymbol{\sigma}_{A}$ satisfying (122). Since $\boldsymbol{\rho}_{B}$ and $\boldsymbol{\sigma}_{A}$ are permutations of elements from two disjointed sets, the sums over them commute with each other. Therefore, (121) becomes \[\sum\limits_{\boldsymbol{\sigma}_{A}}\biggl{[}\sum\limits_{ \boldsymbol{\rho}_{B}\in\text{perms~{}$\overline{\mathsf{I}_{2j-1}}$}}\sum \limits_{\begin{subarray}{c}\boldsymbol{\sigma}\in\boldsymbol{\sigma}_{A} \shuffle\,\boldsymbol{\rho}_{B},~{}\text{s.t.}\\ \sigma^{-1}(\rho_{B}(2j+1))<\sigma^{-1}(\rho_{A}(2j))\end{subarray}}S[ \boldsymbol{\sigma}\|\boldsymbol{\rho}_{A},\boldsymbol{\rho}_{B}]A(1, \boldsymbol{\sigma},n)\biggr{]},\] (123) where the first summation is taken over all $\boldsymbol{\sigma}_{A}$ satisfying (122). For a given $\boldsymbol{\sigma}_{A}$ satisfying (122), one can apply the relation (76) to the expression in the square brackets. Thus the above expression evaluates to \[(-1)\sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline {\mathsf{I}_{2j-1}}$}}\biggl{[}\sum\limits_{\boldsymbol{\sigma}_{A}}\sum \limits_{\begin{subarray}{c}\boldsymbol{\sigma}\in\boldsymbol{\sigma}_{A} \shuffle\,\boldsymbol{\rho}_{B},~{}\text{s.t.}\\ \sigma^{-1}(\rho_{A}(2j))<\sigma^{-1}(\rho_{B}(2j+1))\end{subarray}}S[ \boldsymbol{\sigma}\|\boldsymbol{\rho}_{A},\boldsymbol{\rho}_{B}]A(1, \boldsymbol{\sigma},n)\biggr{]}\] (124) \[= (-1)\sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline {\mathsf{I}_{2j-1}}$}}\biggl{[}\sum\limits_{\boldsymbol{\sigma}\in Z\{\rho_{A} (2)\|\dots\|\rho_{A}(2j-1),\rho_{A}(2j),\,\boldsymbol{\rho}_{B}\}}S[\boldsymbol{ \sigma}\|\boldsymbol{\rho}_{A},\boldsymbol{\rho}_{B}]A(1,\boldsymbol{\sigma},n) \biggr{]}\] \[= (-1)\sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline {\mathsf{I}_{2j-1}}$}}T(\rho_{A}(2)\|\dots\|\rho_{A}(2j-2)\|\rho_{A}(2j-1),\,\rho _{A}(2j),\,\boldsymbol{\rho}_{B}),\] in which the second equality is obtained by considering the definition of zigzag permutations (6.2): \[Z\{\rho_{A}(2)\|\dots\|\rho_{A}(2j-1),\rho_{A}(2j),\,\boldsymbol{ \rho}_{B}\}\] \[= \{\boldsymbol{\sigma}\|\boldsymbol{\sigma}\in S_{n-2},\text{s.t.~{ }}\sigma^{-1}({\rho_{B}(n-1)})>\dots>\sigma^{-1}(\rho_{B}(2j+1))>\sigma^{-1}( \rho_{A}(2j))>\sigma^{-1}(\rho_{A}(2j-1)),\] \[\sigma^{-1}(\rho_{A}(2j-2))>\sigma^{-1}(\rho_{A}(2j-1)),\sigma^{- 1}(\rho_{A}(2j-3))<\sigma^{-1}(\rho_{A}(2j-2)),\dots,\sigma^{-1}(\rho_{A}(3))< \sigma^{-1}(\rho_{A}(2))\}\] \[= \{\boldsymbol{\sigma}\|\boldsymbol{\sigma}\in\boldsymbol{\sigma}_{ A}\shuffle\,\boldsymbol{\rho}_{B},\text{s.t\,}\sigma_{A}^{-1}(\rho_{A}(2j-1))< \sigma_{A}^{-1}(\rho_{A}(2j)),\sigma_{A}^{-1}(\rho_{A}(2j-2))>\sigma_{A}^{-1}( \rho_{A}(2j-1)),\] \[\ldots,\sigma_{A}^{-1}(\rho_{A}(3))<\sigma_{A}^{-1}(\rho_{A}(2)) \text{\,and\,}\sigma^{-1}(\rho_{A}(2j))<\sigma^{-1}(\rho_{B}(2j+1))\}.\] Consequently, (120) becomes \[\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}T(\rho(2)\|\dots\|\rho(2j )\|\rho(2j+1),\dots,\rho(n-1))\] \[=\] Now let us understand the summations on the RHS of (B). Given $\mathsf{I}_{2j-1}$, we collect terms with $\rho_{A}(2j-1)=a$, $\rho_{A}(2j)=b$, (for given $a,b,\in\mathsf{I}_{2j-1}$) then obtain a term \[\sum\limits_{\boldsymbol{\rho}_{A^{\prime}}\in\text{perms~{}$ \mathsf{I}_{2j-3}$}}\sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$ \overline{\mathsf{I}_{2j-1}}$}}T(\rho_{A^{\prime}}(2)\|\dots\|\rho_{A^{\prime}}( 2j-2)\|a,\,b,\,\boldsymbol{\rho}_{B}),\] (127) where we define $\mathsf{I}_{2j-3}\equiv\mathsf{I}_{2j-1}\setminus\{a,b\}$. Correspondingly, we also have other terms in $\eqref{Eq:DFCMSGenProof2}$ with distinct $\mathsf{I}_{2j-1}$ (identical $\mathsf{I}_{2j-1}$ for the special case with $\rho_{A}(2j-1)=b$, $\rho_{A}(2j)=a$) but a same $\mathsf{I}_{2j-3}\equiv\mathsf{I}_{2j-1}\setminus\{x,y\}$, where $\rho_{A}(2j-1)=x$, $\rho_{A}(2j)=y$ for an ordered pair $(x,y)$ satisfying $x,y\in\{a,b\}\cup\overline{\mathsf{I}_{2j-1}}=\overline{\mathsf{I}_{2j-3}}$. The sum of all such terms gives rise \[\sum\limits_{\boldsymbol{\rho}_{A^{\prime}}\in\text{perms~{}$ \mathsf{I}_{2j-3}$}}\,\,\biggl{[}\sum\limits_{x,y\in\overline{\mathsf{I}_{2j-3 }}}\,\,\sum\limits_{\{\rho_{A}(2j-1),\rho_{A}(2j)\}\in\text{perms }\{x,y\}}\, \,\sum\limits_{\boldsymbol{\rho}_{B}\in\text{perms~{}$\overline{\mathsf{I}_{2j -3}}\setminus\{x,y\}$}}\] \[~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T(\rho_{A^{\prime}}(2)\|\dots\|\rho_{A^{\prime}}( 2j-2)\|\rho_{A}(2j-1),\rho_{A}(2j),\boldsymbol{\rho}_{B})\biggr{]}.\] (128) Defining $\rho_{A}(2j-1)\equiv\rho_{B^{\prime}}(2j-1)$, $\rho_{A}(2j)\equiv\rho_{B^{\prime}}(2j)$, $\rho_{B}(2j+1)\equiv\rho_{B^{\prime}}(2j+1)$, …, $\rho_{B}(n-1)\equiv\rho_{B^{\prime}}(n-1)$ and noting that for given $\boldsymbol{\rho}_{A^{\prime}}\in\text{perms~{}$\mathsf{I}_{2j-3}$}$ the other three summations becomes $\sum_{\boldsymbol{\rho}_{B^{\prime}}\in\text{perms~{}}\overline{\mathsf{I}_{2j -3}}}$, we reformulate the above expression as \[\sum\limits_{\boldsymbol{\rho}_{A^{\prime}}\in\text{perms~{}$ \mathsf{I}_{2j-3}$}}\,\sum\limits_{\boldsymbol{\rho}_{B^{\prime}}\in\text{ perms~{}}\overline{\mathsf{I}_{2j-3}}}\,T(\rho_{A^{\prime}}(2)\|\dots\|\rho_{A^{ \prime}}(2j-2)\|\boldsymbol{\rho}_{B^{\prime}}).\] (129) Summing over all possible choices of $\mathsf{I}_{2j-3}\subseteq\{2,\dots,n-1\}$, we finally express the RHS of (B) by \[(-1)\sum\limits_{\mathsf{I}_{2j-3}\subseteq\{2,\dots,n-1\}}\,\sum \limits_{\boldsymbol{\rho}_{A^{\prime}}\in\text{perms~{}$\mathsf{I}_{2j-3}$}} \,\sum\limits_{\boldsymbol{\rho}_{B^{\prime}}\in\text{perms~{}}\overline{ \mathsf{I}_{2j-3}}}\,T(\rho_{A^{\prime}}(2)\|\dots\|\rho_{A^{\prime}}(2j-2)\| \boldsymbol{\rho}_{B^{\prime}})\] (130) \[= (-1)\sum\limits_{\boldsymbol{\rho}\in S_{n-2}}\,T(\rho(2)\|\dots\| \rho(2j-2)\|\rho(2j-1),\rho(2j),\dots,\rho(n)).\] Hence the relation (119) is proven. ## Appendix C Understanding the zigzag pattern of $\boldsymbol{\sigma}$ for given $\boldsymbol{\rho}$ in (93) We first show that, if a given $\boldsymbol{\rho}=\{\rho(2),\rho(3),\dots,\rho(n-1)\}$ on the LHS of (93) can be considered as a permutation in $\mathsf{\Gamma}(\boldsymbol{\sigma})$ for some permutation $\boldsymbol{\sigma}$, the $\boldsymbol{\sigma}$ must satisfy the zigzag pattern, _i.e._, $\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(n-1)\}$. This can be understood as follows: * As defined in section 4.1, the element $n$ is always considered as the last element in both $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$, thus we have $\sigma^{-1}(\rho(n-1))<\sigma^{-1}(n)$. Since there is no element between $\rho(n-1)$ and $n$ in the permutation $\boldsymbol{\rho}$, according to the rule given in section 4.1 (see the point (ii) below (62)), we deduce $\sigma^{-1}(\rho(n-2))>\sigma^{-1}(\rho(n-1))$; * We now consider $\rho(n-2)$. In the permutation $\boldsymbol{\rho}$, there is one element $\rho(n-1)$, which satisfies $\sigma^{-1}(\rho(n-1))<\sigma^{-1}(\rho(n-2))$, between $\rho(n-2)$ and $n$ (note that $\sigma^{-1}(\rho(n-2))<\sigma^{-1}(n)$). According to the rule given in section 4.1 (see the point (i) below (62)), we deduce that $\sigma^{-1}(\rho(n-3))<\sigma^{-1}(\rho(n-2))$. * We further consider $\rho(n-3)$. Since $\sigma^{-1}(\rho(n-3))<\sigma^{-1}(\rho(n-2))$ and there is no element between $\rho(n-3)$ and $\rho(n-2)$ in the permutation $\boldsymbol{\rho}$, we must have $\sigma^{-1}(\rho(n-4))>\sigma^{-1}(\rho(n-3))$, in accordance to the point (ii) below (62). * We turn to $\rho(n-4)$. Since $\sigma^{-1}(\rho(n-4))>\sigma^{-1}(\rho(n-3))$, we should have $\sigma^{-1}(\rho(n-5))<\sigma^{-1}(\rho(n-4))$ due to the point (i) below (62)). * Repeat the above discussions, we find the general condition \[\sigma^{-1}(\rho(2j+2))>\sigma^{-1}(\rho(2j+1)),~{}~{}~{}~{} \sigma^{-1}(\rho(2j+1))<\sigma^{-1}(\rho(2j)),~{}~{}~{}~{}(\text{for $j\geq 1$ }).\] (131) Thus the permutation $\boldsymbol{\sigma}$ must be in $\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(n-1)\}$ for given $\boldsymbol{\rho}$. Conversely, we show that $\boldsymbol{\rho}$ must be in $\mathsf{{\Gamma}}(\boldsymbol{\sigma})$ for any permutation $\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(n-1)\}$. This is because: * For any $\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(n-1)\}$, if $\sigma(a)=\rho(2j+1)$ ($\sigma(a)\in\boldsymbol{\sigma}$), we must have some $b>a$ and $c>a$ s.t. $\sigma(b)=\rho(2j+2)$ and $\sigma(c)=\rho(2j)$. In the permutation $\boldsymbol{\rho}$, $\rho(2j+2)$ and $\rho(2j)$ are the nearest elements on the RHS and LHS satisfying $a=\sigma^{-1}(2j+1)<b=\sigma^{-1}(2j)$ and $a=\sigma^{-1}(\rho(2j+1))<c=\sigma^{-1}(\rho(2j+2))$. In addition, there is no element between $\rho(2j+2)$, $\rho(2j+1)$ and $\rho(2j)$, $\rho(2j+1)$ in the permutation $\boldsymbol{\rho}$. Thus the condition (ii) below (62) in section 4.1 is satisfied. * For any $\boldsymbol{\sigma}\in\mathsf{Z}\{\rho(2)\|\rho(3)\|\dots\|\rho(n-1)\}$, if $\sigma(a)=\rho(2j)$ ($\sigma(a)\in\boldsymbol{\sigma}$), we have two possibilities. * If the nearest element $\sigma(b)$ (and $\sigma(c)$) on the LSH (and RHS) to $\rho(2j)$ in permutation $\boldsymbol{\rho}$ s.t. $b>a$ (and $c>a$) has the form $\sigma(b)=\rho(2k)$ (and $\sigma(c)=\rho(2k^{\prime})$), we must have odd number of elements between $\sigma(b)=\rho(2k)$ (and $\sigma(c)=\rho(2k^{\prime})$) and $\sigma(a)=\rho(2j)$ in $\boldsymbol{\rho}$ (because there must be odd numbers between two even numbers). Thus the condition (i) in section 4.1 is satisfied; * Assuming the nearest element $\sigma(b)$ (or $\sigma(c)$) on the LSH (or RHS) to $\rho(2j)$ in $\boldsymbol{\rho}$, which satisfies $b>a$ (or $c>a$) has the form $\sigma(b)=\rho(2k+1)$ (or $\sigma(c)=\rho(2k^{\prime}+1)$), we always have $\rho(2k+2)$ (or $\rho(2k^{\prime})$), which is more nearer to $\sigma(a)=\rho(2j)$ in $\boldsymbol{\rho}$ and satisfies $\sigma^{-1}(\rho(2k+2))>\sigma^{-1}(\rho(2k+1))=\sigma(b)>\sigma(a)=\rho(2j)$ (or $\sigma^{-1}(\rho(2k^{\prime}))>\sigma^{-1}(\rho(2k^{\prime}+1))=\sigma(c)> \sigma(a)=\rho(2j)$). Thus we return to the previous case. ## References * (1) Z. Bern, J. J. M. Carrasco, and H. Johansson, _New Relations for Gauge-Theory Amplitudes_, _Phys. Rev._D78 (2008) 085011, [arXiv:0805.3993]. * (2) Z. Bern, J. J. M. Carrasco, and H. Johansson, _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_, _Phys. Rev. Lett._105 (2010) 061602, [arXiv:1004.0476]. * (3) R. Kleiss and H. Kuijf, _Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders_, _Nucl. Phys._B312 (1989) 616–644. * (4) B. Feng, R. Huang, and Y. Jia, _Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program_, _Phys. Lett._B695 (2011) 350–353, [arXiv:1004.3417]. * (5) Y.-X. Chen, Y.-J. Du, and B. Feng, _A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory_, _JHEP_02 (2011) 112, [arXiv:1101.0009]. * (6) N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove, _Minimal Basis for Gauge Theory Amplitudes_, _Phys. Rev. Lett._103 (2009) 161602, [arXiv:0907.1425]. * (7) S. Stieberger, _Open & Closed vs. Pure Open String Disk Amplitudes_, arXiv:0907.2211. * (8) G. Chen and Y.-J. Du, _Amplitude Relations in Non-linear Sigma Model_, _JHEP_01 (2014) 061, [arXiv:1311.1133]. * (9) F. Cachazo, S. He, and E. Y. Yuan, _Scattering equations and Kawai-Lewellen-Tye orthogonality_, _Phys. Rev._D90 (2014), no. 6 065001, [arXiv:1306.6575]. * (10) F. Cachazo, S. He, and E. Y. Yuan, _Scattering of Massless Particles in Arbitrary Dimensions_, _Phys. Rev. Lett._113 (2014), no. 17 171601, [arXiv:1307.2199]. * (11) F. Cachazo, S. He, and E. Y. Yuan, _Scattering of Massless Particles: Scalars, Gluons and Gravitons_, _JHEP_07 (2014) 033, [arXiv:1309.0885]. * (12) F. Cachazo, S. He, and E. Y. Yuan, _Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM_, _JHEP_07 (2015) 149, [arXiv:1412.3479]. * (13) Q. Ma, Y.-J. Du, and Y.-X. Chen, _On Primary Relations at Tree-level in String Theory and Field Theory_, _JHEP_02 (2012) 061, [arXiv:1109.0685]. * (14) R. Britto, F. Cachazo, and B. Feng, _New recursion relations for tree amplitudes of gluons_, _Nucl. Phys._B715 (2005) 499–522, [hep-th/0412308]. * (15) R. Britto, F. Cachazo, B. Feng, and E. Witten, _Direct proof of tree-level recursion relation in Yang-Mills theory_, _Phys. Rev. Lett._94 (2005) 181602, [hep-th/0501052]. * (16) Y.-J. Du and C.-H. Fu, _Explicit BCJ numerators of nonlinear simga model_, _JHEP_09 (2016) 174, [arXiv:1606.05846]. * (17) H. Kawai, D. C. Lewellen, and S. H. H. Tye, _A Relation Between Tree Amplitudes of Closed and Open Strings_, _Nucl. Phys._B269 (1986) 1. * (18) Z. Bern, L. J. Dixon, D. C. Dunbar, M. Perelstein, and J. S. Rozowsky, _On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences_, _Nucl. Phys._B530 (1998) 401–456, [hep-th/9802162]. * (19) N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng, and T. Sondergaard, _Gravity and Yang-Mills Amplitude Relations_, _Phys. Rev._D82 (2010) 107702, [arXiv:1005.4367]. * (20) N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng, and T. Sondergaard, _New Identities among Gauge Theory Amplitudes_, _Phys. Lett._B691 (2010) 268–273, [arXiv:1006.3214]. * (21) N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng, and T. Sondergaard, _Proof of Gravity and Yang-Mills Amplitude Relations_, _JHEP_09 (2010) 067, [arXiv:1007.3111]. * (22) N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard, and P. Vanhove, _The Momentum Kernel of Gauge and Gravity Theories_, _JHEP_01 (2011) 001, [arXiv:1010.3933]. * (23) J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, _Abelian Z-theory: NLSM amplitudes and $\alpha$’-corrections from the open string_, _JHEP_06 (2017) 093, [arXiv:1608.02569]. * (24) Y.-J. Du and F. Teng, _BCJ numerators from reduced Pfaffian_, _JHEP_04 (2017) 033, [arXiv:1703.05717]. * (25) S. Stieberger and T. R. Taylor, _New relations for Einstein-Yang-Mills amplitudes_, _Nucl. Phys._B913 (2016) 151–162, [arXiv:1606.09616]. * (26) D. Nandan, J. Plefka, O. Schlotterer, and C. Wen, _Einstein-Yang-Mills from pure Yang-Mills amplitudes_, _JHEP_10 (2016) 070, [arXiv:1607.05701]. * (27) L. de la Cruz, A. Kniss, and S. Weinzierl, _Relations for Einstein-Yang-Mills amplitudes from the CHY representation_, _Phys. Lett._B767 (2017) 86–90, [arXiv:1607.06036]. * (28) O. Schlotterer, _Amplitude relations in heterotic string theory and Einstein-Yang-Mills_, _JHEP_11 (2016) 074, [arXiv:1608.00130]. * (29) C.-H. Fu, Y.-J. Du, R. Huang, and B. Feng, _Expansion of Einstein-Yang-Mills Amplitude_, _JHEP_09 (2017) 021, [arXiv:1702.08158]. * (30) M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, _Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy_, _JHEP_07 (2017) 002, [arXiv:1703.00421]. * (31) F. Teng and B. Feng, _Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame_, _JHEP_05 (2017) 075, [arXiv:1703.01269]. * (32) Y.-J. Du, B. Feng, and F. Teng, _Expansion of All Multitrace Tree Level EYM Amplitudes_, arXiv:1708.04514. * (33) V. Del Duca, L. J. Dixon, and F. Maltoni, _New color decompositions for gauge amplitudes at tree and loop level_, _Nucl. Phys. B_571 (2000) 51–70, [hep-ph/9910563]. * (34) M. Kiermaier, _talk at amplitudes 2010, may 2010 at qmul, london, uk. http://www.strings.ph.qmul.ac.uk/~theory/amplitudes2010/talks/mk2010.pdf_, . * (35) Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, _Gravity as the Square of Gauge Theory_, _Phys. Rev._D82 (2010) 065003, [arXiv:1004.0693]. * (36) C. R. Mafra, O. Schlotterer, and S. Stieberger, _Explicit BCJ Numerators from Pure Spinors_, _JHEP_07 (2011) 092, [arXiv:1104.5224]. * (37) Y.-J. Du, B. Feng, and C.-H. Fu, _BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory_, _JHEP_08 (2011) 129, [arXiv:1105.3503]. * (38) C.-H. Fu, Y.-J. Du, and B. Feng, _An algebraic approach to BCJ numerators_, _JHEP_03 (2013) 050, [arXiv:1212.6168]. * (39) C.-H. Fu, Y.-J. Du, and B. Feng, _Note on Construction of Dual-trace Factor in Yang-Mills Theory_, _JHEP_10 (2013) 069, [arXiv:1305.2996]. * (40) Y.-J. Du, B. Feng, and C.-H. Fu, _The Construction of Dual-trace Factor in Yang-Mills Theory_, _JHEP_07 (2013) 057, [arXiv:1304.2978]. * (41) C.-H. Fu, Y.-J. Du, and B. Feng, _Note on symmetric BCJ numerator_, _JHEP_08 (2014) 098, [arXiv:1403.6262]. * (42) F. Cachazo, S. He, and E. Y. Yuan, _Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations_, _JHEP_01 (2015) 121, [arXiv:1409.8256]. * (43) L. A. Barreiro and R. Medina, _RNS derivation of N-point disk amplitudes from the revisited S-matrix approach_, _Nucl. Phys._B886 (2014) 870–951, [arXiv:1310.5942]. * (44) R. H. Boels and R. Medina, _Graviton and gluon scattering from first principles_, _Phys. Rev. Lett._118 (2017) 061602, [arXiv:1607.08246]. * (45) R. H. Boels and H. Luo, _A minimal approach to the scattering of physical massless bosons_, arXiv:1710.10208. * (46) C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, _On-Shell Recursion Relations for Effective Field Theories_, _Phys. Rev. Lett._116 (2016), no. 4 041601, [arXiv:1509.03309]. * (47) C. Cheung, C.-H. Shen, and C. Wen, _Unifying Relations for Scattering Amplitudes_, _JHEP_02 (2018) 095, [arXiv:1705.03025].
1403.3978	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31534, "num_imgs": 3, "llama3_tokens_count": 9861 }	[ "content_image/1403.3978/x1.png", "content_image/1403.3978/x2.png", "content_image/1403.3978/x3.png" ]	# A Novel Scheme for Downlink Opportunistic Interference Alignment Haijing Liu, Hui Gao, Wei Long, Tiejun Lv Key Laboratory of Trustworthy Distributed Computing and Service, Ministry of Education School of Information and Communication Engineering Beijing University of Posts and Telecommunications, Beijing, China 100876 Email: {Haijing_LIU, huigao, longwei, lvtiejun}@bupt.edu.cn This work is financially supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61271188. ###### Abstract In this paper we propose a downlink codebook-based opportunistic interference alignment (OIA) in a three-cell MIMO system. A codebook composed of multiple transmit vector sets is utilized to improve the multiuser selection diversity. The sum rate increases as the size of the codebook grows. In addition, during the user selection, effective channel gain and alignment metric are combined to generate a novel criterion, which improves the system performance, especially at low SNR. Furthermore, a threshold-based feedback approach is introduced to reduce the feedback load in the proposed scheme. Both the analytical results and simulations show that the proposed scheme provides significant improvement in terms of sum rates with no feedback load growth and slight increase of complexity. ## I Introduction With the exponential growth in mobile data traffic, interference has been one of the major challenges in wireless communication. Interference alignment (IA) [1] is a technique recently introduced to improve the performance of interference networks. Unfortunately, extensive channel station information and a large amount of computation is required to achieve the optimal DoFs [2], which makes IA too complicated to be implemented in practice. Motivated by opportunistic beamforming (OBF) [3], OIA schemes are developed in [4, 5, 6, 7, 8, 9], which only require limited feedback and modest computational complexity. Though OIA takes advantage of multiuser diversity via opportunistic user equipment (UE) scheduling, [5] proves that the number of required UEs grows with an exponential scale in order to achieve an optimal DoF. In practical systems, the number of UEs is usually limited, so the improvement of sum rate performance via OIA is not obvious. On the other hand, UEs are selected from the perspective of interference reduction in OIA, while the selection is done from the point of view of channel gains in Maximum SNR (MAX-SNR) scheduling. OIA outperforms MAX-SNR in an interference limited environment while MAX-SNR provides better performance in a noise limited environment. Neither of them have a wide SNR range of application. In this paper, we propose a downlink codebook-based opportunistic interference alignment (COIA) scheme. Compared with the conventional OIA schemes,three improvements are made: ($i$) A codebook composed of multiple transmit beamforming vector sets for three base stations (BSs) is utilized to bring more selection diversity. More specifically, besides the UE scheduling, BSs select a transmit beamforming vector set from the codebook to enhance system sum rate. Consequently, fewer UEs are needed in COIA to achieve the same sum rate compared with the conventional OIA. Particularly, with the theoretical analysis of the expectation of the alignment metric value, we can pre-calculate the required numbers of candidate UEs with various codebook sizes for the same sum rate as that of the conventional OIA. Note that code-book based uplink OIA schemes have recently been proposed in [10, 11]. Our downlink COIA is completely different from them because the codebook in our scheme is utilized to exploit the selection diversity, while the codebook in [10, 11] is used to reduce the feedforward load. We propose our downlink COIA scheme to improve the sum rate performance, while they focus on the UE and feedback bit scaling law with their uplink COIA schemes. ($ii$) An effective UE selection metric adaptively balancing the noise and interference power is introduced to overcome the shortcoming of OIA schemes at low SNR. With the above two improvements, COIA achieves better sum rate performance than MAX-SNR scheduling and the conventional OIA the same number of candidate UEs. ($iii$) When the codebook size is very large, the feedback load becomes unacceptable of our previous COIA scheme in [12]. In this paper, a threshold-based feedback scheme, which has never been discussed in OIA to the best of our knowledge, is explored to reduce the system feedback load in our COIA. We address the relationship between the threshold value and feedback load by an explicit expression, so that the feedback load of the proposed scheme can be adjusted to the same as that of the conventional OIA by setting an appropriate threshold. Throughout the paper, we describe matrices and vectors by bold upper and lower case letters. $\mathbf{A}^{H}$, $\lambda_{\textnormal{max}}(\mathbf{A})$, $\mathbf{v}_{\textnormal{max}}(\mathbf{A})$, $\lVert{\mathbf{A}}\rVert$ and $\mathbf{A}^{-1}$ denote the conjugate transpose, the largest eigenvalue, the eigenvector corresponding to the largest eigenvalue, $L_{2}$-norm and the inverse of matrix $\mathbf{A}$, respectively. ## II System Model We consider a 3-cell MIMO downlink system with a single BS and $K$ UEs in each cell. Both the BS and the UEs are each equipped with two antennas. In the $i$-th cell, $i=1,2,3,$ the BS sends a data stream to a scheduled UE with a normalized transmit beamforming vector $\mathbf{w}_{i}$, where $\lVert{\mathbf{w}_{i}}\rVert=1$. For convenience we denote the $k$-th UE in the $i$-th cell as UE $[k,i]$, where $1\leq k\leq K,k\in\mathbb{N}$ and $i=1,2,3$. Quasi-static channels between BSs and UEs are assumed. The received signal at UE $[k,i]$ is \[\small\mathbf{y}_{k_{i}}=\sqrt{P_{S}}\mathbf{H}_{k_{i},i}\mathbf{w}_{i}x_{i}+ \sqrt{P_{I}}\sum_{j=1,j\not=i}^{3}\mathbf{H}_{k_{i},j}\mathbf{w}_{j}x_{j}+ \mathbf{n}_{k_{i}},\] (1) where $\mathbf{H}_{k_{i},j}\in\mathbb{C}^{2\times 2}$ is the channel matrix from the BS in the $j$-th cell to UE $[k,i]$. Elements of $\mathbf{H}_{k_{i},j}$ are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance. $x_{j}$ is the signal transmitted by the $j$-th BS with a transmit power constraint $\mathbb{E}[\|x_{j}\|^{2}]=1$. $\mathbf{n}_{k_{i}}\in\mathbb{C}^{2\times 1}\sim\mathcal{CN}(\mathbf{0},\sigma^ {2}_{n}\mathbf{I})$ is the additive complex Gaussian noise at UE $[k,i]$. $P_{S}$ stands for the received data power and $P_{I}$ is the received average interference power from each interfering BS. Denoting the receive beamforming vector of UE $[k,i]$ by $\mathbf{v}_{k_{i}}\in\mathbb{C}^{2\times 1}$, the received signal after receive beamforming is \[\small\mathbf{v}^{H}_{k_{i}}\mathbf{y}_{k_{i}}\!=\!\sqrt{P_{S}}\mathbf{v}^{H}_ {k_{i}}\mathbf{H}_{k_{i},i}\mathbf{w}_{i}x_{i}+\sqrt{P_{I}}\!\sum_{j=1,j\not=i }^{3}\!\mathbf{v}^{H}_{k_{i}}\mathbf{H}_{k_{i},j}\mathbf{w}_{j}x_{j}+\mathbf{v }^{H}_{k_{i}}\mathbf{n}_{k_{i}}.\] (2) We also assume there exist low-rate but reliable and delay-free backhaul links between each UE with its relevant BS as well as among the BSs. Based on (2), the signal-to-interference-plus-noise ratio (SINR) of the data stream of UE $[k,i]$ is given by \[\small\textnormal{SINR}_{k_{i}}=\frac{P_{S}\|\mathbf{v}^{H}_{k_{i}}\mathbf{H}_{ k_{i},i}\mathbf{w}_{i}\|^{2}}{\sigma^{2}_{n}+P_{I}\sum_{j=1,j\not=i}^{3}\| \mathbf{v}^{H}_{k_{i}}\mathbf{H}_{k_{i},j}\mathbf{w}_{j}\|^{2}}.\] (3) ## III Conventional Opportunistic User Selection Schemes In both OBF and conventional OIA, the transmit beamforming vectors $\mathbf{w}$ are all generated randomly, while the UE selection criteria are significantly different. We discuss three opportunistic UE selection schemes (i.e., MAX-SINR, MAX-SNR and the conventional OIA) in this section. ### _Max-Sinr_ MAX-SINR has been shown to be an optimal opportunistic UE selection scheme in the sense of sum rate so far[5]. The receive beamforming vector of UE $[k,i]$ is $\mathbf{v}^{\textnormal{MAX-SINR}}_{k_{i}}=\mathbf{v}_{\textnormal{max}}( \mathbf{A}_{k_{i}}^{-1}\mathbf{B}_{k_{i}})$ to maximize $\textnormal{SINR}_{k_{i}}$, where and $\mathbf{B}_{k_{i}}=P_{S}\mathbf{H}_{k_{i},i}\mathbf{w}_{i}\mathbf{w}^{H}_{i} \mathbf{H}^{H}_{k_{i},i}$. The corresponding SINR is \[\small\textnormal{SINR}_{k_{i}}=\lambda_{\textnormal{max}}(\mathbf{A}_{k_{i}}^ {-1}\mathbf{B}_{k_{i}}).\] The UE with the largest SINR is selected, i.e., \[\small k^{\textnormal{MAX-SINR}}_{i}=\mathop{\arg\max}_{1{\leq}k_{i}{\leq}K} \textnormal{SINR}_{k_{i}}.\] ### _Max-Snr_ In MAX-SNR, the receive beamforming vector of UE $[k,i]$ is designed as $\mathbf{v}^{\textnormal{MAX-SNR}}_{k_{i}}=\frac{\mathbf{H}_{k_{i},i}\mathbf{w} _{i}}{\lVert\mathbf{H}_{k_{i},i}\mathbf{w}_{i}\rVert}$ to maximize the SNR. The corresponding SNR is $\textnormal{SNR}_{k_{i}}=\frac{P_{S}\lVert\mathbf{H}_{k_{i},i}\mathbf{w}_{i} \rVert^{2}}{\sigma^{2}_{n}}.$ In homogeneous network, each UE calculates its effective channel gain \[\small\beta_{k_{i}}=\lVert\mathbf{H}_{k_{i},i}\mathbf{w}_{i}\rVert^{2}\] (4) and informs the corresponding BS. The BS selects the UE with the largest SNR, i.e., \[\small k^{\textnormal{MAX-SNR}}_{i}=\mathop{\arg\max}_{1{\leq}k_{i}{\leq}K} \beta_{k_{i}}.\] (5) ### _Conventional OIA_ In OIA[5], the UE whose interference signals are most aligned with each other is selected. The alignment of interfering signals is measured by their chordal distance. The metric value of UE $[k,i]$ is \[\small\gamma_{k_{i}}=\frac{\lVert\mathbf{w}^{H}_{i^{\prime}}\mathbf{H}^{H}_{k_ {i},i^{\prime}}\mathbf{H}_{k_{i},i^{\prime\prime}}\mathbf{w}_{i^{\prime\prime} }\rVert^{2}}{\lVert\mathbf{H}_{k_{i},i^{\prime}}\mathbf{w}_{i^{\prime}}\rVert^ {2}\cdot\lVert\mathbf{H}_{k_{i},i^{\prime\prime}}\mathbf{w}_{i^{\prime\prime}} \rVert^{2}},\] (6) where $i^{\prime}$ is the $i$-th element of vector $[2,3,1]$, and $i^{\prime\prime}$ is the $i$-th element of vector $[3,1,2]$. Each UE sends the value back to the relevant BS. The preferred UE in the $i$th cell is \[\small k^{\textnormal{OIA}}_{i}=\mathop{\arg\max}_{1{\leq}k_{i}{\leq}K}\gamma_ {k_{i}}.\] (7) ## IV Novel Codebook-Based OIA Scheme In this section, we propose an OIA scheme with a codebook of transmit beamforming vector sets. A novel selection criterion adaptive to noise and interference power as well as a threshold-based feedback are further developed to enhance the sum rate performance and control the feedback load of the proposed scheme. ### _Codebook-Based OIA_ In codebook-based downlink OIA, BSs choose transmit beamforming vectors from multiple vectors in a codebook every time slot. The codebook composed of transmit beamforming vector sets is denoted by $\mathcal{C}=\{\mathbf{c}_{1},\dots,\mathbf{c}_{S}\},$ where $\mathbf{c}_{s}$ is the concatenation of the $s$-th set of random unit-norm transmit beamforming vectors, i.e., $\mathbf{c}_{s}=[\mathbf{w}^{H}_{1,s},\mathbf{w}^{H}_{2,s},\mathbf{w}^{H}_{3,s} ]^{H}\in\mathbb{C}^{6\times 1}$, and $S$ is the size of the codebook. All the UEs and BSs know the codebook $\mathcal{C}$. The UE selection and data transmission in COIA is shown as follows: _Step 1_: Each BS broadcasts pilots for channel estimation. Every UE obtains channel estimations $\hat{\mathbf{H}}_{k_{i},i}$ and $\hat{\mathbf{H}}_{k_{i},j}$. _Step 2_: Using the estimated channel information, each UE calculates $S$ alignment metric values for $S$ transmit beamforming vector sets in $\mathcal{C}$. The alignment metric value of UE $[k,i]$ for the $s$-th transmit beamforming vector set is \[\small\gamma_{k_{i},s}=\frac{\lVert\mathbf{w}^{H}_{i^{\prime},s}\hat{\mathbf{H }}^{H}_{k_{i},i^{\prime}}\hat{\mathbf{H}}_{k_{i},i^{\prime\prime}}\mathbf{w}_{ i^{\prime\prime},s}\rVert^{2}}{\lVert\hat{\mathbf{H}}_{k_{i},i^{\prime}} \mathbf{w}_{i^{\prime},s}\rVert^{2}\cdot\lVert\hat{\mathbf{H}}_{k_{i},i^{ \prime\prime}}\mathbf{w}_{i^{\prime\prime},s}\rVert^{2}}.\] (8) Each UE feeds the analog metric values back to the BS in its own cell. _Step 3_: BSs exchange the analog feedback, then select the preferred transmit beamforming vector set as well as the corresponding served UEs. Regarding a specific transmit beamforming vector set $\mathbf{c}_{s}$, we first find the UE with the largest alignment metric value in the $i$-th cell and the corresponding metric value, denoted by \[\small\bar{k}_{i,s}=\mathop{\arg\max}_{1{\leq}k_{i}{\leq}K}\gamma_{k_{i},s}, \quad\bar{\gamma}_{i,s}=\max_{1{\leq}k_{i}{\leq}K}{\gamma_{k_{i},s}}.\] (9) After that, we calculate the average of the largest alignment metric values of three cells for $\mathbf{c}_{s}$, which is given by \[\small\bar{\gamma}_{s}=\frac{1}{3}\sum_{i=1}^{3}\bar{\gamma}_{i,s}.\] (10) The preferred transmit beamforming vector set is then selected among all sets in the codebook as \[\small s^{}=\mathop{\arg\max}_{1{\leq}s{\leq}S}\bar{\gamma}_{s},\] (11) which means we choose the codeword to maximize the average of the largest alignment metric values of three cells. Once $s^{}$ is determined, the selected transmit beamforming vector for the $i$-th transmitter is $\mathbf{w}_{i,s^{}}$ and the preferred UE being served in the $i$-th cell is $\bar{k}_{i,s^{}}$. _Step 4_: Each BS serves the selected UE with the preferred transmit beamforming vector. ### _Analysis of Codebook-Based OIA_ We provide a theoretical analysis of the codebook-based OIA. The expectation of the alignment metric value of the selected UE increases as $S$ grows. In other words, compared with the conventional OIA ($S=1$), the interfering signals of the selected UE are aligned more and more closely when the codebook size increases in COIA. The expectation of $\bar{\gamma}_{s^{}}$ (i.e., the average of alignment metric value of the selected UE $\bar{k}_{i,s^{}}$), is approximately given by \[\small\begin{split}&\mathbb{E}[\bar{\gamma}_{s^{}}]\\ &=\frac{S}{(\mathbb{B}(a,b))^{S}}{\!}\cdot{\!}P(a,b,S){\!}\cdot{ \!}\mathbb{B}(aS+k_{1}+\cdots+k_{S-1}+1,b),\end{split}\] (12) where $\mathbb{B}(\cdot)$ is the beta function, $a=\frac{3K(K+2)}{K+1}$, $b=\frac{3(K+2)}{K+1}$, and \[\small\begin{split}& P(a,b,S)\\ &={\!}\sum_{k_{1}=0}^{\infty}{\!}\cdots{\!}\sum_{k_{S-1}=0}^{ \infty}{\!}\frac{(1-b)_{k_{1}}{\!}\cdots{\!}(1-b)_{k_{S-1}}}{(a{\!}+{\!}k_{1}) {\!}\cdots{\!}(a{\!}+{\!}k_{S-1})k_{1}!{\!}\cdots{\!}k_{S-1}!}.\end{split}\] (13) See Appendix for the derivation of (12). \| S=1 \| S=2 \| S=3 \| S=4 ---\|---\|---\|---\|--- \| K=10 --- 0.9091 \| 0.9351 \| 0.9461 \| 0.9529 \| K=15 --- 0.9375 \| 0.9559 \| 0.9635 \| 0.9680 \| K=20 --- 0.9524 \| 0.9666 \| 0.9725 \| 0.9759 TABLE I: E[¯γs∗] of various K and S. Given number of UEs $K$ and codebook size $S$, we can get the expectation efficiently because $\mathbb{B}(\cdot)$ can be calculated directly in MATLAB. TABLE I shows $\mathbb{E}[\bar{\gamma}_{s^{}}]$ for various $K$ and $S$. It can be seen that the expectation increases with the growth of $S$ for the same $K$. Since the average rate of the selected UE increases as the expectation of its alignment metric value grows [5], with the help of (12), we can get the number of UEs $K$ with variable codebook sizes $S$ for the same expectation, i.e., the same sum rate performance. For example, when $K=20,S=1$, $\mathbb{E}[\bar{\gamma}_{s^{}}]=\textnormal{0.9524}$. Then letting $S=4$ and setting the left hand side value of $\eqref{eq15}$ as 0.9524, we can get the required number of UEs $K=10$ for the same performance. Only half of UEs are needed in COIA with $S=4$ codebook compared with the conventional OIA. ### _Hybrid Criterion in COIA_ The effective channel gain in MAX-SNR of UE $[k,i]$ with the $s$-th transmit beamforming vector is defined as \[\small\beta_{k_{i},s}=\lVert\hat{\mathbf{H}}_{k_{i},i}\mathbf{w}_{i,s}\rVert^{ 2}.\] (14) We introduce a hybrid criterion with (8) and (14), which is given by \[\small\alpha_{k_{i},s}=[0,(1-\theta)]^{+}\cdot\gamma_{k_{i},s}+\theta\cdot \beta_{k_{i},s},\] (15) where $\theta=\frac{P_{S}/P_{I}}{P_{S}/\sigma_{n}^{2}}=\frac{\sigma_{n}^{2}}{P_{I}}$ and $[x,0]^{+}=\max(x,0).$ The BSs select the transmit beamforming vector set and UEs in the same way as that mentioned in Part. IV-A, except replacing $\gamma_{k_{i},s}$ with $\alpha_{k_{i},s}$ in (9). We can see that when the power of interference is smaller than that of noise, i.e., the system is at low SNR, the hybrid metric value only depends on the effective channel gain. With the increase of interference power, the proportion of the effective channel gain decreases and the effect of OIA UE selection is enhanced. At very high SNR, the hybrid metric value is almost equal to the OIA metric value. With the proposed hybrid criterion adaptive to noise and interference power, the COIA scheme achieves better sum rate performance in both low and high SNR regions. ### _Threshold-Based Feedback in COIA_ In the OIA scheme proposed in [5], every UE feeds back an alignment metric value to the corresponding BS, which we refer to as full feedback. $K$ values are needed to complete a UE selection in each cell. In COIA, if full feedback is adopted, the amount of feedback will be $K\cdot S$ due to the utilization of the $S$ size transmit beamforming vector codebook. The feedback load becomes unacceptable when $S$ is large. Here we propose a threshold-based feedback technique to reduce the feedback needs (by more than 75%) while preserving the sum rate performance in COIA. Similar techniques are introduced in [13], [14]. However, they take only signal and noise into consideration and ignore interference, which degrades their performance in multi-cell systems. In the proposed threshold-based feedback scheme, each UE compares its selection metric value to a predefined threshold $T$ and decides locally whether it sends feedback to the BS, only those who fall above $T$ are allowed to be fed back. BSs make selections with the feedback. If no feedback is received by all three BSs, transmit beamforming vector set and UE in each cell is selected randomly. Choosing a proper threshold is critical. We first characterize the statistics of the alignment metric $\gamma$ and the effective channel gain $\beta$ in terms of cumulative distributive function (CDF) and probability density function (PDF). As (6) shows, the alignment metric $\gamma$ is related to the chordal distance between two vectors. Using the results of [15],the CDF of $\gamma$, denoted by $F_{\gamma}(x)$ is given by \[F_{\gamma}(x)=P(\gamma\leq x)=x,\quad 0\leq x\leq 1.\] (16) The PDF of $\gamma$ is \[f_{\gamma}(x)=1,\quad 0\leq x\leq 1.\] (17) The effective channel gain $\beta$ is defined as (4). With a certain unit-norm vector $\mathbf{w}$, $\beta$ has a central chi-square distribution. The CDF of $\beta$ is given by \[F_{\beta}(y)=P(\beta\leq y)=1-e^{-y}(1+y),\quad y\geq 0.\] (18) The PDF of $\beta$ is \[f_{\beta}(y)=ye^{-y},\quad y\geq 0.\] (19) The normalized average feedback load $\bar{F}$ is defined as the ratio of the average load per selection to the total amount of full feedback ($KS$) in each cell. Apparently, with a threshold $T$, we have \[\bar{F}^{\textnormal{OIA}}(T)=1-F_{\gamma}(T)\] (20) and \[\bar{F}^{\textnormal{MAX-SNR}}(T)=1-F_{\beta}(T).\] (21) For a given feedback load requirement $\bar{F}$ (e.g., 1/4), we can get the threshold $T^{\textnormal{OIA}}$ and $T^{\textnormal{MAX-SNR}}$ with (20) and (21), respectively. In COIA, the metric value $\alpha$ is given by (15). With (17) and (19), when $0<\theta<1$, the CDF of $\alpha$ is given by (we omit the derivation due to space limitations) \[\small F_{\alpha}(z)=\left\{\begin{aligned} &\frac{1}{1-\theta}- \frac{e^{-\frac{z}{\theta}}(\theta+z)}{(1-\theta)\theta},\quad 0\leq z<1- \theta\\ &-\frac{e^{-\frac{z}{\theta}}\left(\theta+z-e^{\frac{1-\theta}{ \theta}}(2\theta-1+z)\right)}{(1-\theta)\theta},z\geq 1-\theta\\ \end{aligned}\right..\] (22) When $\theta\geq 1$, $\alpha=\theta\cdot\beta$, the CDF of $\alpha$ is given by \[\small F_{\alpha}(z)=1-e^{-\frac{z}{\theta}}(1+\frac{z}{\theta}),\quad z\geq 0.\] (23) The normalized average feedback load $\bar{F}$ is of COIA is \[\small\bar{F}^{\textnormal{COIA}}(T)=1-F_{\alpha}(T).\] (24) We can choose a proper threshold $T^{\textnormal{COIA}}$ with (24). It should be remarked that $T^{\textnormal{COIA}}$ is related to $\theta$, i.e., $T^{\textnormal{COIA}}$ is adaptive to noise and interference power, because we consider both signal channel quality and interference condition in COIA. It is different form $T^{\textnormal{OIA}}$ and $T^{\textnormal{MAX-SNR}}$ as (20) and (21) are only functions of $T$. ### _Complexity Analysis_ We analysis the computational complexity in the UE selection step of each UE in COIA briefly. Only complex multiplication is considered for simplicity. Assume the number of receive antennas is $N$. For every vector set in the codebook, the effective channel gain consumes $\mathcal{O}(N)$ computation, and operations of $\mathcal{O}(N)$ are needed to get the alignment metric. So $2S\cdot\mathcal{O}(N)$ computation is required for each UE in COIA. Just like MAX-SNR and the conventional OIA, the computational complexity of COIA is $\mathcal{O}(N)$. Note that we do not take the computation of channel estimation into account here because it is necessary for receive beamforming regardless of UE selection schemes. ## V Numerical Simulations In this section, we simulate the performance of the proposed COIA scheme. Preferred UErs are selected with different schemes, then the selected UE $k^{}_{i}$ executes MAX-SINR receive beamforming. It should be mentioned that we focus on the UE selection scheme while the receive beamforming vectors design after UE selection is not studied in depth. The sum rate is obtained according to the equation \[\small R=\sum^{3}_{i=1}\log_{2}(1+\textnormal{SINR}_{k^{}_{i}}),\] (25) where $\textnormal{SINR}_{k^{}_{i}}$ can be obtained by (3). Perfect channel estimation is assumed at all the UEs. <figure><img src="content_image/1403.3978/x1.png"><figcaption>Fig. 1: Expectations and sum rates of full feedback codebook-based OIA ofvarious S and K. PS=PI.</figcaption></figure> Fig. 1. a shows the expectation of the alignment metric value of the selected UEs in the codebook-based OIA. It is clear that the expectation increases with the increase of codebook size $S$ for the same number of UEs $K$. Further, the configuration $K=10,S=4$ has almost the same expectation as $K=20,s=1$ has. Fig. 1. b shows the sum rates of full feedback codebook-based OIA. The sum rates increase with the growth of $K$ and $S$, especially at high SNR. Only $K=10$ UEs are needed in $S=4$ codebook-based OIA to achieve almost the same sum rate performance as $K=20$ UEs in the conventional OIA (i.e., $S=1$), which is consistent with the analytical result in Part. IV-B. <figure><img src="content_image/1403.3978/x2.png"><figcaption>Fig. 2: Sum rates of threshold-based feedback OIA and MAX-SNR. K=10,PS=PI.</figcaption></figure> Fig. 2 shows the sum rates of threshold-based feedback OIA and MAX-SNR with various feedback load requirment $\bar{F}$. In OIA, the threshold value $T^{\textnormal{OIA}}$ is 0.5, 0.75 and 0.875 when $\bar{F}(T^{\textnormal{OIA}})$ is 1/2, 1/4 and 1/8, respectively. In MAX-SNR, the threshold value $T^{\textnormal{MAX-SNR}}$ is 1.6785, 2.6925 and 3.6070 when $\bar{F}(T^{\textnormal{MAX-SNR}})$ is 1/2, 1/4 and 1/8, respectively. The sum rate loss is negligible with 1/2 and 1/4 feedback load in the threshold-based feedback scheme. It means that a large reduction of the feedback is possible while preserving most of the sum rate performance. <figure><img src="content_image/1403.3978/x3.png"><figcaption>Fig. 3: Sum rates of various opportunistic UE selection schemes. K=10,PS=PI.</figcaption></figure> In Fig. 3, the sum rates of various schemes are shown with $K=10$ UEs in each cell and $P_{S}=P_{I}$. The word “C4” in legends means the codebook size $S=4$. The threshold-based feedback scheme is marked as “TFB”. For a fair comparison, we choose $\bar{F}^{\textnormal{COIA}}=1/S=1/4$ and calculate $T^{\textnormal{COIA}}$ according to (22), (23) and (24). MAX-SNR scheme outperforms the conventional OIA [5] and even codebook-based OIA in the low SNR region but gets significant performance degradation at high SNR. The proposed COIA with $S=4$ codebook approaches better sum rate performance than MAX-SNR and the conventional OIA in all range of SNR. In COIA, the sum rate performance of threshold-based feedback is almost the same as that of full feedback, which means the proposed COIA with threshold-based feedback outperforms MAX-SNR and the conventional OIA with the same feedback load. ## VI Conclusions In this paper, we have proposed a codebook-based opportunistic interference alignment with a hybrid selection criterion and threshold-based feedback in a three-cell MIMO downlink system. A codebook composed of multiple transmit vector sets is utilized to improve the multiuser selection diversity. Effective channel gain and alignment metric are combined to generate a novel metric for a wide SNR range of application. A threshold is employed to reduce the feedback load in COIA. Both the analytical results and simulations indicate that the proposed COIA scheme provides higher sum rates in wide SNR region than the conventional OIA scheme with the same feedback load. In the future, we will focus on the COIA scheme with multiple data streams for each UE. Defined in (9), it can be proved easily that $\bar{\gamma}_{i,s}\sim\textnormal{Beta}(K,1)$ in a similar way to [5], where $\textnormal{Beta}(\cdot)$ is the beta distribution. We omit the proof due to space limitations. Lemma 1.: _([16]): Let $S=\sum_{i=i}^{k}X_{i}$ where $X_{i}$ are i.i.d. random variables of $Beta(\alpha,\beta)$. The distribution of $S$ can be approximated by:_ \[\small Beta(e,f);e=Ff,f=\frac{F}{\sigma^{2}(1+F)^{3}}\] (26) _where $E=\sum\mathbb{E}[X_{i}]$, $F=\frac{E}{1-E}$, and $\sigma^{2}=\sum Var(X_{i}).$_ As $\bar{\gamma}_{i,s},i=1,2,3$ are i.i.d. beta-distributed random variables, using Lemma 1, we can consider $\bar{\gamma}_{s}$ defined by (10) as a new beta-distributed random variable, i.e., $\bar{\gamma}_{s}\sim\textnormal{Beta}(a,b),$ where $a=\frac{3K(K+2)}{K+1}$, $b=\frac{3(K+2)}{K+1}.$ Let $x=\bar{\gamma}_{s^{}}$ for convenience, the explicit expression of the expectation of the maximum of i.i.d. beta-distributed random variables $\bar{\gamma}_{1},\dots,\bar{\gamma}_{S}$ is \[\small\begin{split}\mathbb{E}[\bar{\gamma}_{s^{}}]& =\mathbb{E}[\max_{1\leq s\leq S}\bar{\gamma}_{s}]=\mathbb{E}[x]\\ &=\int_{0}^{1}x\cdot f_{X}(x)dx=\int_{0}^{1}x\cdot Sf(x)(F(x))^{S -1}dx\\ &=\int_{0}^{1}x\cdot\frac{Sx^{a-1}(1-b)^{b-1}}{\mathbb{B}(a,b)}( \mathbb{I}_{x}(a,b))^{S-1}dx,\end{split}\] (27) where $\mathbb{B}(\cdot)$ is the beta function and $\mathbb{I}_{x}(\cdot)$ is the regularized incomplete beta function. Using the series expansion \[\small\mathbb{I}_{x}(a,b)=\frac{x^{a}}{\mathbb{B}(a,b)}\sum_{k=0}^{\infty} \frac{(1-b)_{k}x^{k}}{(a+k)k!},\] (27) can be expressed as \[\small\begin{split}&\mathbb{E}[\bar{\gamma}_{s^{}}]\\ &=\frac{S}{(\mathbb{B}(a,b))^{S}}\int_{0}^{1}x^{a}(1-x)^{b-1} \left(x^{a}\sum_{k=0}^{\infty}\frac{(1-b)_{k}x^{k}}{(a+k)k!}\right)^{S-1}\\ &=\frac{S}{(\mathbb{B}(a,b))^{S}}\\ &\quad\times\sum_{k_{1}=0}^{\infty}\cdots\sum_{k_{S-1}=0}^{\infty }\frac{(1-b)_{k_{1}}\cdots(1-b)_{k_{S-1}}}{(a+k_{1})\cdots(a+k_{S-1})k_{1}! \cdots k_{S-1}!}\\ &\quad\times\int_{0}^{1}x^{aS+k_{1}+\cdots+k_{S-1}}(1-x)^{b-1}dx \\ &=\frac{S}{(\mathbb{B}(a,b))^{S}}\\ &\quad\times\sum_{k_{1}=0}^{\infty}\cdots\sum_{k_{S-1}=0}^{\infty }\frac{(1-b)_{k_{1}}\cdots(1-b)_{k_{S-1}}}{(a+k_{1})\cdots(a+k_{S-1})k_{1}! \cdots k_{S-1}!}\\ &\quad\times\mathbb{B}(aS+k_{1}+\cdots+k_{S-1}+1,b),\end{split}\] where $(\cdot)_{k}$ is the Pochhammer symbol defined as $(x)_{k}=x(x+1)\cdots(x+k-1).$ ## References [1] V. Cadambe and S. Jafar, “Interference alignment and degrees of freedom of the $k$-user interference channel,” _IEEE Trans. Inf. Theory_, vol. 54, pp. 3425–3441, Aug. 2008. * [2] S. Peters and R. Heath, “Cooperative algorithms for MIMO interference channels,” _IEEE Trans. Veh. Technol._, vol. 60, pp. 206–218, Jan. 2011. * [3] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” _IEEE Trans. Inf. Theory_, vol. 48, pp. 1277–1294, Jun. 2002. * [4] J. H. Lee and W. Choi, “Opportunistic interference aligned user selection in multiuser MIMO interference channels,” in _IEEE Global Telecommun. Conf._, Miami, Florida, 2010, pp. 1–5. * [5] ——, “On the achievable DoF and user scaling law of opportunistic interference alignment in 3-transmitter MIMO interference channels,” _IEEE Trans. Wireless Commun._, vol. 12, pp. 2743–2753, Jun. 2013. * [6] J. Leithon, C. Yuen, H. Suraweera, and H. Gao, “A new opportunistic interference alignment scheme and performance comparison of MIMO interference alignment with limited feedback,” in _IEEE Globecom Workshops_, Anaheim, CA, 2012, pp. 1123–1127. * [7] H. Gao, J. Leithon, C. Yuen, and H. A. Suraweera, “New uplink opportunistic interference alignment: An active alignment approach,” in _IEEE Wireless Commun. and Networking Conf._, Shanghai, China, 2013, pp. 3099–3104. * [8] H. Gao, T. Lv, D. Fang, S. Yang, and C. Yuen, “Limited feedback-based interference alignment for interfering multi-access channels,” _arXiv:1402.0391 [cs, math]_, Feb. 2014. [Online]. Available: http://arxiv.org/abs/1402.0391 * [9] X. Chen and C. Yuen, “Performance analysis and optimization for interference alignment over MIMO interference channels with limited feedback,” _arXiv:1402.0295 [cs, math]_, Feb. 2014. [Online]. Available: http://arxiv.org/abs/1402.0295 * [10] H. J. Yang, B. C. Jung, W.-Y. Shin, and A. Paulraj, “Codebook-based opportunistic interference alignment,” _arXiv:1310.2028 [cs, math]_, Oct. 2013. [Online]. Available: http://arxiv.org/abs/1310.2028 * [11] H.-H. Lee, K.-H. Park, Y.-C. Ko, and M.-S. Alouini, “Codebook-based interference alignment for uplink MIMO interference channels,” _Journal of Commun. and Networks_, vol. 16, no. 1, pp. 18–25, Feb. 2014. * [12] H. Liu, T. Lv, and Y. Lu, “Codebook-based hybrid opportunistic interference alignment,” in _Int. Conf. Cyberspace Tech._, Beijing, China, Nov. 2013, pp. 518–523. * [13] D. Gesbert and M.-S. Alouini, “How much feedback is multi-user diversity really worth?” in _IEEE Int. Conf. Commun._, Paris, France, 2004, pp. 234–238. * [14] V. Hassel, M.-S. Alouini, D. Gesbert, and G. Oien, “Exploiting multiuser diversity using multiple feedback thresholds,” in _IEEE Veh. Technology Conf._, Stockholm, Sweden, 2005, pp. 1302–1306. * [15] C. K. Au-Yeung and D. Love, “On the performance of random vector quantization limited feedback beamforming in a MISO system,” _IEEE Trans. Wireless Commun._, vol. 6, pp. 458–462, Feb. 2007. * [16] A. K. Gupta and S. Nadarajah, _Handbook of Beta Distribution and Its Applications_, 1st ed. Boca Raton: CRC Press LLC, 2004.
1204.4868	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 43098, "num_imgs": 6, "llama3_tokens_count": 11749 }	[ "content_image/1204.4868/x1.png", "content_image/1204.4868/x2.png", "content_image/1204.4868/x3.png", "content_image/1204.4868/x4.png", "content_image/1204.4868/x5.png", "content_image/1204.4868/x6.png" ]	# A Transfer Matrix Approach to Electron Transport in Graphene through Arbitrary Electric and Magnetic Potential Barriers Sameer Grover${}^{1}$, Sankalpa Ghosh${}^{1}$ and Manish Sharma${}^{2}$ ${}^{1}$ Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India ${}^{2}$ Centre for Applied Research in Electronics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India ###### Abstract A transfer matrix method is presented for solving the scattering problem for the quasi one-dimensional massless Dirac equation applied to graphene in the presence of an arbitrary inhomogeneous electric and perpendicular magnetic field. It is shown that parabolic cylindrical functions, which have previously been used in literature, become inaccurate at high incident energies and low magnetic fields. A series expansion technique is presented to circumvent this problem. An alternate method using asymptotic expressions is also discussed and the relative merits of the two methods are compared. pacs: 03.65.Ge, 72.80.Vp, 73.22.Pr _Keywords_: graphene,transfer matrix method, Frobenius method, parabolic cylindrical functions, wave equation ## 1 Introduction Graphene’s [1, 2] near perfect two-dimensional configuration and its unique electronic properties[3, 4] have made it one of the widely studied materials in recent times. Its electrons have been found to obey a linear dispersion relation near the Fermi energy which makes them behave like massless relativistic particles in two dimensions. As a result, they obey the massless Dirac equation instead of the Schödinger equation. One of the important consequences of the relativistic behaviour of transport electrons is their inability to be confined by an electrostatic barrier, a phenomenon known as Klein tunnelling [5]. The alternate strategy of confining these Dirac fermions using magnetic fields has been proposed[6, 7]. Consequently, there has been a lot of interest in electron transport through magnetic barriers in graphene. Moreover, building functional electronic devices using graphene relies on being able to control the electronic transport by the application of electromagnetic fields. In this context, electron transmission through varying regularly and irregularly shaped barriers of both scalar and vector potentials becomes an important problem. Proper analysis of such barriers calls for the development of efficient numerical techniques. In this work, we are interested in developing a general algorithm for the calculation of electron transmission in graphene through inhomogeneous electric and magnetic fields. We only consider magnetic fields perpendicular to the plane of the graphene sheet. We also restrict ourselves to the quasi one-dimensional problem which implies that the fields are invariant in the y-direction and the electronic plane wave is incident on it at an arbitrary angle. From a mathematical viewpoint, this involves solving the massless Dirac equation which consists of two first-order coupled ordinary differential equations with arbitrary values of electric and magnetic fields. We use the well-known transfer matrix method to solve this problem. This method has previously been applied to problems in optics [8] and quantum mechanics [9]. It is computationally easy to implement, involving only the multiplication of $2\times 2$ matrices. It has been used to study the scattering problem for the Schrödinger equation.[10, 11]. The method has also been extended to solving any homogeneous ordinary linear differential equation[12]. Transfer matrix methods to solve the electron transport problem in graphene have been studied extensively: [5, 6] have applied it to single magnetic barriers; it has been used in [13] to study the transmission through multiple magnetic barriers in graphene; in [14], it is used for electrostatic barriers in bilayer graphene; in [15, 16] for graphene superlattices; in [17] for fractally arranged magnetic barriers; and in [18] for tunnelling through electric barriers in the presence of a magnetic field. Although we proceed along similar lines, we show that the parabolic cylindrical functions (Weber functions) that have been used in literature can cause significant numerical difficulties at low magnetic fields or at high incident energies and we use a series expansion to solve the differential equation in order to avoid this problem. This forms the main result of this paper. Thus, we provide a uniform framework though this series expansion method and widen the applicability of the transfer matrix method for a large range of incident energies and magnetic fields. Since the ballistic transport regime of Graphene based device is now being studied extensively both experimentally and theoretically [19], our scheme will be quite useful to understand some of such future experiments. We also discuss an alternative method based on approximating Weber functions by their asymptotic form. This method is applicable only within the asymptotic regime whereas the series method is applicable to the entire range of magnetic fields and energies. We show that our method provides accurate results in this range also. In the special case when the average length across which the vector potential varies is smaller that the typical magnetic length $l_{B}=\sqrt{({\hbar c})/({eB})}$, the magnetic barrier can be approximated by a delta function. Analytic solutions for magnetic barriers modelled as a series of delta functions are well known[20], [21]. The transfer matrix method is more general and can be used even when this condition doesn’t hold. Solving the Schrödinger or the Dirac equation includes two different kinds of problems: the eigenvalue problem and the scattering problem. The eigenvalue problem involves finding the energy eigenvalues of the Hamiltonian and is used to find the allowed energy levels of bound states. The scattering problem, which is the one we tackle in this paper, involves the calculation of the transmission and reflection coefficients, formally defined as the ratio of the flux of particles transmitted or reflected from a potential configuration to the flux that is incident on it. It leads to a second order homogeneous differential equation, the ubiquitous wave equation, which in one dimension is given by: \[\psi^{\prime\prime}(x)+k^{2}(x)\psi(x)=0\quad(k\in\bf{C})\] (1) The transfer matrix method involves division of the one-dimensional domain into slices and taking an appropriate approximation of $k^{2}(x)$ in each slice. The equation for each slice is then solved and the continuity conditions are used at the interfaces of two such slices. The exact solution of the equation in each slice depends on the form of $k^{2}(x)$ chosen. For example, for the Schrödinger equation, a piecewise constant approximation of $k^{2}(x)$ leads to complex exponential solutions in each slice and a piecewise linear approximation leads to a solution basis consisting of the Airy functions[22]. In the case of graphene, we consider both scalar potentials (electrostatic fields) and vector potentials (magnetic fields), which lead to a piecewise linear vector potential and a piecewise constant scalar potential. The form of the resulting equation is: \[\psi(x)^{\prime\prime}+\left[\alpha^{2}-p-(\beta+px)^{2}\right]\psi(x)=0\] (2) where $\alpha,\beta,p\in\bf{R}$ and are explained later in detail. This equation admits parabolic cylindrical functions as the solution basis. We show that using these becomes computationally infeasible as $p\to 0$ which corresponds to low magnetic fields and therefore an alternate solution basis is called for. We obtain this using the Frobenius method and find basis functions that tend to complex exponentials as $p\to 0$. It is also necessary to restrict the transfer matrix method to cases where the magnetic field is non-zero only over some closed bounded (compact) interval. This divides space into three regions and the solution in the first and last regions are complex exponentials representing incoming and outgoing waves. From a physical point of view, this condition is necessary because if $\forall x,B\neq 0$, such as with a uniform magnetic field, the wavefunction gets localized along the spatial direction $x$. In Section 2 are outlined the equations to be solved and the notation used. In Section 3, the transfer matrix method is discussed. In Section 4, methods are outlined to solve Equation 2: Section 4.1 details the method previously found in literature along with its limitations. Section 4.2 is a method based on asymptotic expansions, and Section 4.3 is the proposed alternative series method. Finally, in Section 5, we apply this method to a number of cases and present the results obtained. <figure><img src="content_image/1204.4868/x1.png"><figcaption>Figure 1: Piecewise constant scalar potential and piecewise linear vectorpotential showing the notation for x coordinates and slices used in the paper</figcaption></figure> ## 2 Governing Equations The governing massless Dirac equation is given by $H\psi=E\psi$ where the Hamiltonian is given by \[H=v_{f}\ \vec{\sigma}.(\vec{p}+e\vec{A}(x))+V(x)\] (3) and $\psi=[\psi_{1},\psi_{2}]^{T}$ is the two component wavefunction, $\vec{\sigma}=\sigma_{x}\hat{i}+\sigma_{y}\hat{j}$ with $\sigma_{x,y}$ denoting the Pauli spin matrices. Both the magnetic field $B$ and scalar potentials $V$ are discretized and the magnetic field $B$ is converted to vector potential $A$ in the Landau gauge. The discretisation scheme that we have used is shown in Figure 1. Slices are numbered from $0$ to $N+1$, with the leftmost and rightmost slices unbounded. The boundaries between slices are denoted by $x_{i}$ with $1\leq i\leq N$. Therefore the ith slice is bounded by $x=x_{i-1}$ and $x=x_{i}$. We denote the magnetic field, scalar potential, and y component of the vector potential in slice $i$ by the notation $B_{i}$, $V_{i}$, $A_{i}$ respectively. For well defined transmission and reflection coefficients, it is necessary to have zero magnetic field in the first and last slice, $B_{0}=B_{N+1}=0$, so that the solution can be expressed as complex exponentials which represent incoming and outgoing plane waves. The only non-zero component of $\vec{A}$ is $A_{y}(x)$ and is denoted by $A$. The functional form of the vector potential in the ith slice is $A_{i}=C_{i}+B_{i}(x-x_{i-1})$ where $x_{i-1}$ represents the left edge of the ith slice, with $x_{-1}$ any conveniently chosen value (because $B_{0}=0$), and $C_{i}=\sum_{j=0}^{i-1}B_{j}(x_{j}-x_{j-1})$, $C_{0}=0$. The equations given above are converted to dimensionless form by defining two new variables. We substitute $x^{\prime}=x/x_{s}$ and $A^{\prime}=A/A_{s}$. These can be also be thought of as scaling factors and as we shall see later, their exact values are important in computations. In terms of these scaled units, $A^{\prime}_{i}=c_{i}+b_{i}(x^{\prime}-\delta_{i-1})$. It can immediately be seen that $b_{i}=B_{i}x_{s}/A_{s}$, $c_{i}=C_{i}/A_{s}$ and $\delta_{i-1}=x_{i-1}/x_{s}$. In terms of individual components and scaled units, Equation 3 is: \[\frac{-i}{x_{s}}\frac{\partial\psi_{2}}{\partial x^{\prime}}-ik_{ y}\psi_{2}+i\frac{e}{\hbar}(-A_{s}A^{\prime}\psi_{2})=(\epsilon-\tilde{v})\psi _{1}\] \[\frac{-i}{x_{s}}\frac{\partial\psi_{1}}{\partial x^{\prime}}+ik_{ y}\psi_{1}+i\frac{e}{\hbar}(A_{s}A^{\prime}\psi_{1})=(\epsilon-\tilde{v})\psi_ {2}\] where $\epsilon=E/\hbar v_{f}$ and $\tilde{v}=V/\hbar v_{f}$ (these have the units of $[L]^{-1}$). The y-invariance of the problem leads to $\frac{\partial}{\partial y}=ik_{y}$ where $k_{y}=\epsilon\sin(\phi)$ with $\phi$ being the angle of incidence. We seek the transmission as a function of $\phi$. The equations are decoupled bearing in mind that $\tilde{v}$ is constant in each slice and $A_{i}^{\prime}=c_{i}+b_{i}(x-\delta_{i-1})$ is a function of $x$. $\psi_{1}$ and $\psi_{2}$ are then governed by the following relations: (4) \[\psi_{2}=\frac{1}{(\epsilon-\tilde{v})}\left[\frac{-i}{x_{s}}\frac{\partial \psi_{1}}{\partial x^{\prime}}+ik_{y}\psi_{1}+i\frac{e}{\hbar}(A_{s}A^{\prime} \psi_{1})\right]\] (5) We use the standard technique of calculating $\psi_{1}$ from Equation 4 and calculating $\psi_{2}$ by backsubstituting $\psi_{1}$ in Equation 5. In Section 4, these equations are solved for a particular slice and in Section 3, these solutions are used to construct the transfer matrix and completely solve the transmission problem. ## 3 The transfer matrix method The transfer matrix method relies on the availability of two linearly independent analytic solutions of Equation 4 and Equation 5. If the two linearly independent solutions of $\psi_{1}$ are denoted by $\psi_{1}^{A}$ and $\psi_{1}^{B}$, and the corresponding solutions for $\psi_{2}$ are $\psi_{2}^{A}$ and $\psi_{2}^{B}$, the transfer matrix denoted by $M_{i}$ is such that the solution of the ith slice is given by: \[\left[\begin{array}[]{c}\psi_{1}\\ \psi_{2}\\ \end{array}\right]=M_{i}\left[\begin{array}[]{c}A_{i}\\ B_{i}\\ \end{array}\right],M(x)=\left[\begin{array}[]{cc}\psi_{1}^{A}(x)&\psi_{1}^{B}( x)\\ \psi_{2}^{A}(x)&\psi_{2}^{B}(x)\end{array}\right]\] (6) From the continuity of $\psi_{1}$ and $\psi_{2}$ across the boundaries, we have \[M_{i}(x_{i})\left[\begin{array}[]{c}A_{i}\\ B_{i}\\ \end{array}\right]=M_{i+1}(x_{i})\left[\begin{array}[]{c}A_{i+1}\\ B_{i+1}\\ \end{array}\right]\quad\forall\quad 0\leq i\leq N\] (7) This allows us to formulate a recurrence relation between the coefficients $A_{i},B_{i}$ and $A_{i+1},B_{i+1}$. Continuing in a similar manner, we relate $A_{0},B_{0}$ with $A_{N+1},B_{N+1}$ which then gives us the reflection and transmission coefficients. \[\left[\begin{array}[]{c}A_{0}\\ B_{0}\\ \end{array}\right]=P\left[\begin{array}[]{c}A_{N+1}\\ B_{N+1}\\ \end{array}\right],P=\prod_{j=0}^{N}M_{j}(x_{j})^{-1}M_{j+1}(x_{j})\] (8) We refer to the expression $M_{j}(x_{j-1})M_{j}^{-1}(x_{j})$ occurring in the expression for $P$ as the transfer matrix for the jth slice. It can be easily proven that this term is independent of the basis functions chosen in that slice. The expression for the transfer matrix given in Equation 8 can usually be simplified if the solution in each slice can be solved in a local coordinate system with its origin on the left edge of that slice. This can always be done by shifting the origin in the wave equation, Equation 1. Then the matrix $M(x)$ depends only on $x-x_{i-1}$. Thus, if $M(x)=N(x-x_{i-1})$, substitution in Equation 8 gives this formula: \[P=\prod_{j=0}^{N}N_{j}(x_{j}-x_{j-1})^{-1}N_{j+1}(0)\] (9) where $x_{-1}$ is a suitably chosen constant as explained earlier. We have used this expression in our computations. ### Form of incident, transmitted and reflected waves In the first and last region, the magnetic field is chosen to be zero so that the solution reduces to complex exponentials of the form $\exp(\pm ikx)$ that represent the incident, reflected and transmitted waves. In contrast to the Schrödinger equation in which $\exp(+ikx)$ represents right propagating waves, and $\exp(-ikx)$ represents left propating waves, in the case of the Dirac equation $\psi_{1}=\exp(+ikx)$ may represent either right or left propagating waves. If, in a slice, $E>V$, the probability flux corresponding to $\psi_{1}=\exp(+ikx)$ is positive implying that the wave is right propagating. On the other hand, if $E<V$, the flux corrresponding to the same wavefunction is negative and so it represents a left propagating wave. The incident wave, reflected wave and transmitted wave are given by $\exp({s_{0}ik_{0}x})$, $r\exp({-s_{0}ik_{0}x})$ and $t\exp({s_{N+1}ik_{N+1}x})$ respectively where $s_{i}=\textrm{sign}(E-V_{i})$ and $k_{j}=x_{s}\sqrt{(\epsilon-\tilde{v_{j}})^{2}-(k_{y}+\frac{e}{\hbar}A_{s}c_{j} )^{2}}$. The corresponding probability currents in the x-direction, within a constant, given by $\psi^{\dagger}\sigma_{x}\psi$ are $J_{i}=2k_{0}/\|\epsilon-\tilde{v}_{0}\|$, $J_{r}=-2k_{0}\|r\|^{2}/\|\epsilon-\tilde{v}_{0}\|$ and $J_{t}=2k_{N+1}\|t\|^{2}/\|\epsilon-\tilde{v}_{N+1}\|$. The transmission and reflection coefficients are given by: \[R =-J_{r}/J_{i}=\|r\|^{2}\] \[T =J_{t}/J_{i}=\|t\|^{2}\frac{k_{N+1}/\|\epsilon-\tilde{v}_{N+1}\|}{k_{ 0}/\|\epsilon-\tilde{v}_{0}\|}\] If, however $k_{N+1}$ is imaginary, $J_{t}=0$ and hence $T=0$. The elements of the transfer matrix $P$ (Equation 9) relates the coefficients of the complex exponentials with a positive or negative sign in the first layer to those in the last layer (denoted by $e_{first}\pm$ and $e_{last}\pm$): \[\left[\begin{array}[]{c}e_{first}+\\ e_{first}-\end{array}\right]=P\left[\begin{array}[]{c}e_{last}+\\ e_{last}-\end{array}\right] ,\quad P=\left[\begin{array}[]{cc}a&b\\ c&d\end{array}\right]\] The values of $r$ and $t$ to be used in Equation 3.1 depend on the form that the incident, reflected and transmitted waves have; i.e., whether they are represented by complex exponentials with positive or negative signs. The results are summarised in the following table: \[\begin{array}[]{llcll}E>V_{0},&E>V_{N+1}&:&t=1/a&r=c/a\\ E>V_{0},&E<V_{N+1}&:&t=1/b&r=d/b\\ E<V_{0},&E>V_{N+1}&:&t=1/c&r=a/c\\ E<V_{0},&E<V_{N+1}&:&t=1/d&r=b/d\\ \end{array}\] (10) ## 4 Solving the Governing Equations We now solve equations 4 and 5 and find solution bases $\psi_{1}^{A,B},\psi_{2}^{A,B}$ to construct the transfer matrix used in Equation 9. To this end, we introduce another change of variable with $x^{\prime\prime}=x^{\prime}-\delta$ representing a translation of the origin to the left boundary of each slice. For notational convenience, subscripts indicating the slice number are omitted in this section. Defining dimensionless constants $\alpha=x_{s}(\epsilon-\tilde{v})$, $p=\frac{e}{\hbar}x_{s}A_{s}b$ and $\beta=x_{s}(k_{y}+\frac{e}{\hbar}A_{s}c)$, equations 4 and 5 can be written in the dimensionless form \[\frac{d^{2}\psi_{1}}{dx^{\prime\prime 2}}+\left[\alpha^{2}-p-(\beta+px^{\prime \prime})^{2}\right]\psi_{1}=0\] (11) \[\psi_{2}=\frac{i}{\alpha}\left[\frac{\partial\psi_{1}}{\partial x^{\prime \prime}}+\psi_{1}(\beta+px^{\prime\prime})\right]\] (12) ### Parabolic Cylindrical function solution We first discuss the well-known technique of using parabolic cylindrical functions [17],[6],[23] to solve Equation 11 and 12. The parabolic cylindrical equation in standard form is \[\frac{\partial^{2}\psi}{\partial x^{2}}-\psi\left[\frac{x^{2}}{4}+a\right]=0\] (13) Following the notation used by [24], the two linearly independent solutions to the equation are given by $U(a,x)=D_{\nu}(x)$ and $V(a,x)=V_{\nu}(x)$ with $\nu=-(1/2+a)$. Alternatively, $D_{\nu}(x)$ and $D_{\nu}(-x)$ can also be used as linearly independent solutions. For solving Equation 11, three cases of $p>0$, $p<0$ and $p=0$ (corresponding to positive, negative and zero magnetic field) need to be dealt with separately. When $p=0$, the solutions are complex exponentials: \[\psi_{1}=e^{\pm i\sqrt{\alpha^{2}-\beta^{2}}x^{\prime\prime}}\\ \psi_{2}=\frac{1}{\alpha}\left[\pm\sqrt{\alpha^{2}-\beta^{2}}+i\beta\right]e^{ \pm i\sqrt{\alpha^{2}-\beta^{2}}x^{\prime\prime}}\] (14) When $p\neq 0$, the equation can be converted to standard form by substituting $z=\sqrt{{2}/{\|p\|}}(\beta+px^{\prime\prime})$. The solutions are $\psi_{1}=D_{\nu}(z),V_{\nu}(z),D_{\nu}(-z)$, where either $D_{\nu}(z)$, $V_{\nu}(z)$ or $D_{\nu}(z)$, $D_{\nu}(-z)$ can be used; $\nu$ is given by: \[\nu=\left\{\begin{array}[]{ll}\frac{\alpha^{2}}{2p}-1&p>0\\ \frac{\alpha^{2}}{2\|p\|}&p<0\end{array}\right.\] (15) The corresponding expression for $\psi_{2}$ given by Equation 12 is: \[\psi_{2}=\frac{i}{\alpha}\left[-\sqrt{2\|p\|}{sign}(p)\frac{\partial\psi_{1}}{ \partial z}+\psi_{1}z\sqrt{\frac{\|p\|}{2}}\right]\] (16) This can be further simplified by using standard recurrence relations relating $D_{\nu}(z)$ and $V_{\nu}(z)$ to their derivatives and the simplified expressions are: \[\begin{array}[]{rll}&\quad\psi_{1}=&\quad\psi_{2}=\\ \\ p>0:&D_{\nu}(z)&\frac{i}{\alpha}\sqrt{2p}D_{\nu+1}(z)\\ &D_{\nu}(-z)&\frac{-i}{\alpha}\sqrt{2p}D_{\nu+1}(-z)\\ &V_{\nu}(z)&\frac{i}{\alpha}\sqrt{2p}(\nu+1)V_{\nu+1}(z)\\ \\ p<0:&D_{\nu}(z)&\frac{i}{\alpha}\sqrt{2\|p\|}(\nu)D_{\nu-1}(z)\\ &D_{\nu}(-z)&\frac{-i}{\alpha}\sqrt{2\|p\|}(\nu)D_{\nu-1}(-z)\\ &V_{\nu}(z)&\frac{i}{\alpha}\sqrt{2\|p\|}V_{\nu-1}(z)\end{array}\] (17) We now discuss the limitations of this method. The function $D_{\nu}(z)$ has a power-law dependence with $\nu$ and increases at a near-exponential rate with an increase in $\nu$ and reaches $10^{308}$ at around $\nu=300$ which is the maximum representable double precision value on a computer. It can be seen from Equation 15 that the parameter $\nu$ contains the term: \[\frac{\alpha^{2}}{2\|p\|}=\frac{x_{s}(\epsilon-\tilde{v})^{2}}{2\frac{e}{\hbar}A _{s}b}=\frac{(\epsilon-\tilde{v})^{2}}{2\frac{e}{\hbar}B}\] (18) where the relation $b=B\frac{x_{s}}{A_{s}}$ has been used. From this, it can immediately be seen that $\nu$ increases with an increase in the incident energy $\epsilon$ and increases with a decrease in the magnetic field $B$. Furthermore, the expression for $\nu$ is independent of any normalization or scaling factors. The first problem with the parabolic cylindrical function method is obvious: as the magnetic field decreases, $\nu$ gets larger and $D_{\nu}(z)$ becomes too large to be calculated in double precision. For an incident energy of 82 meV (corresponding to a Fermi wavelength $k_{f}=2\pi/\epsilon$ of 50 nm), the minimum allowable magnetic field before this occurs is 0.017 T (corresponding to $\nu=300$). This makes it impossible to observe a transition between zero magnetic field and a finite magnetic barrier. Secondly, we are limited by the accuracy to which parabolic cylindrical functions themselves are computed. Using the Fortran codes given in [24], the lowest magnetic field at which errors start showing up can be as high as 0.6 T. These errors manifest themselves as unphysical results like abrupt discontinuities in the transmission plots. The transfer matrix can also become near singular making it difficult to invert. In this case, we calculate the pseudoinverse using singular value decomposition. We have verified that round-off errors are the source of the problem by calculating the parabolic cylindrical functions in several ways, including one in which calculations are performed in arbitrary precision before the result is rounded off to double-precision. Using this, we could go closer to the theoretical limit at $\nu=300$ mentioned above. Examining calculations in literature using this method, we find that in most of the cases authors have limited their calculations to incident energies and magnetic field values that result in small values of $\nu$. In [17], $\nu=12.5$ has been used and [6] have used $\nu=6.8$. This gives a rough indication of the range in which parabolic cylindrical functions work. ### Asymptotic solution When the magnetic field is small, the parameter $\nu$ in $D_{\nu}(z)$ becomes large and using an asymptotic expansion instead of the parabolic cylindrical function is a possibility. This can be achieved using an asymptotic form for $D_{\nu}(z)$ for large $\|\nu\|$ which can be expressed as a product of a $\nu$-dependent term, $h(\nu)$, that causes exponential growth of the function and some other factor. This large $\nu$-dependent term need not be explicitly computed because upon substitution in the expressions for the transfer matrix for the jth slice given by $M_{j}(x_{j-1})M_{j}^{-1}(x_{j})$ in Equation 8, it gets cancelled out. Asymptotic expansions that satisfy these criteria are available in [25, 26]. Different expressions are applicable in different regions of the $\nu$-$z$ plane. To elaborate, suppose there is a positive magnetic field in the jth slice and transfer matrix for that slice is (see Equation 17) \[M_{j}(x)=\left[\begin{array}[]{cc}D_{\nu}(z(x))&D_{\nu}(-z(x))\\ \frac{i}{\alpha}\sqrt{2p}D_{\nu+1}(z(x))&\frac{-i}{\alpha}\sqrt{2p}D_{\nu+1}(- z(x))\\ \end{array}\right]\] (19) We now use the recurrence relation $D_{\nu+1}(z)=\frac{1}{2}zD\nu(z)-D^{\prime}_{\nu}(z)$ and then substitute the asymptotic forms from [25, 26]. The factor of $h(\nu)$ can be factored out and gets cancelled. Similarly, when the magnetic field is negative, the recurrence relation to be used is $D_{\nu-1}(z)=(\frac{1}{2}zD\nu(z)+D^{\prime}_{\nu}(z))/\nu$ One of the limitations of this method is that it doesn’t work at the turning points of the parabolic cylindrical differential equation $z=\pm 2\sqrt{\nu+1/2}$ when $\nu>-1/2$ and gives inaccurate answers close to those points. The other is that these expressions work only in the asymptotic regime and not for all values of magnetic field and incident energy. This scheme is similar to that used in [27, 28] where an asymptotic form proposed in[29, 30] has been used. However, the expression they have used is valid for $\nu\rightarrow\infty$ and $z\to 0$ with $z\sqrt{\nu}$ finite. This, however, will not work for a general case where we need an asymptotic form that works for large $\nu$ and all $z$. ### Series solution Equation 11 predicts that the solutions with magnetic field ($p\neq 0$) should smoothly tend to the solutions without magnetic field ($p=0$). All the problems with the parabolic cylindrical functions method stem from our choice of basis functions $D_{\nu},V_{\nu}$ that do not tend to complex exponentials as $p\to 0$. This leads us to choose solutions of Equation 11 that do tend to complex exponentials as $p\to 0$. These are discussed in this section. This method to solve the equation relies on the Frobenius method which yields two linearly independent solutions of the form $\sum_{0}^{\infty}q_{n}(x^{\prime\prime})^{n}$. With $\theta=-(\alpha^{2}-\beta^{2})+p$ and $\phi=2\beta p$. The coefficients $q_{i}$ for the two solutions $\phi_{1}$ and $\phi_{2}$ are given by \[\begin{array}[]{lcccc}\phi_{1}:&q_{0}=1&q_{1}=0&q_{2}=\frac{\theta}{2!}&q_{3}= \frac{\phi}{3!}\end{array}\] (20) \[\begin{array}[]{lcccc}\phi_{2}:&q_{0}=0&q_{1}=1&q_{2}=0&q_{3}=\frac{\theta}{3! }\end{array}\] (21) and for $n\geq 2$ given by the recurrence relation: \[n\geq 2:q_{n+2}=\frac{\theta}{(n+1)(n+2)}q_{n}+\frac{\phi}{(n+1)(n+2)}q_{n-1}+ \frac{p^{2}}{(n+1)(n+2)}q_{n-2}\] (22) It can be shown that when $p=0$, the solutions tend to sine and cosine series. When $p=0$ and $\beta=0$, the first solution is $\cos(\alpha x^{\prime\prime})$ and the second one is $\sin(\alpha x^{\prime\prime})/\alpha$. Similar results holds for when $p=0$ and $\beta\neq 0$. Then the first solution is $\cos(\sqrt{\alpha^{2}-\beta^{2}}x^{\prime\prime})$ and the second one is $\sin(\sqrt{\alpha^{2}-\beta^{2}}x^{\prime\prime})/\sqrt{\alpha^{2}-\beta^{2}}$. We choose the two linearly independent solutions to be used in the transfer matrix equation, Equation 9 as $\psi_{1}^{A}=\phi_{1}+ik\phi_{2}$ and $\psi_{1}^{B}=\phi_{1}-ik\phi_{2}$ where $k=\sqrt{\alpha^{2}-\beta^{2}+p}$ because they reduce to complex exponentials in the limit $p\to 0$. In this way, the three cases of $p=0$, $p<0$ and $p>0$ do not need to be treated separately. Furthermore, Fuchs’s Theorem[31] guarantees convergence of the series solution. Some care needs to be taken while summing up these series term by term. Under usual circumstances, sine and cosine series are not directly summed up because the terms increase before they start decreasing[32]. However, for small arguments, the convergence is quick and manual summation becomes feasible. In the series that we have used, summation is possible only if $\alpha$, $\beta$, $p$ and $x^{\prime\prime}$ are small. We choose the scaling factors $x_{s}$ and $A_{s}$ judiciously to make this possible. This is critical to the process of manual summation. By making $x_{s}$ small, the variables $\alpha$, $\beta$ and $p$ can be made as small as desired. However, $x^{\prime\prime}=(x_{i}-x_{i-1})/x_{s}$ and decreasing $x_{s}$ increases $x^{\prime\prime}$. To avoid this, slices that are very wide will sometimes need to be subdivided into narrower slices. In case any of these parameters are chosen incorrectly, the coefficients $q_{i}$ overflow or underflow which can be detected quite easily. Also $x_{-1}$ should be taken to be equal to $x_{0}$. It should be noted that the series needs to be summed up only once for each slice. Equation 9 requires the evaluation of the series at $x=0$ which doesn’t require a series summation. Calculation of $\psi_{2}$ is done using Equation 12. This requires evaluation of the derivative which can be easily done during the series summation. ## 5 Numerical Examples ### Implementation In the subsequent sections, we demonstrate the results of the numerical technique we have developed by applying it to a few specific cases. They include both cases with scalar potential only, vector potential only and with both scalar and vector potentials. The series summation algorithm was implemented in Fortran 95 and the rest of the program in python. Series summation was performed till the error between partial sums was $10^{-20}$. The scale factor $x_{s}$ has been taken to be $10^{-8}$ nm in the results presented. ### Results for a single barrier We consider an electrostatic potential barrier of width 100nm and height 180meV and apply a varying magnetic field across this 100nm region. The incident energy chosen is around 82.66 meV corresponds to a Fermi wavelength of 50 nm. The transmission plots using the series method are shown in Figure 2. The polar plots depict the transmission as a function of the angle of incidence. The magnetic field values chosen for illustration are $0$ and between $0.1$($\nu=72.08$) to $1.25$($\nu=5.76$) A smooth transition from zero magnetic field to higher fields can be observed. The solutions calculated using the series method and parabolic cylindrical functions were found to match at higher fields but the latter algorithm fails at low magnetic fields. So the low to high magnetic field transition cannot be observed by using parabolic cylindrical functions. The asymptotic method solution matches with the results shown for low magnetic fields. For other combinations of incident energy and magnetic fields, the asymptotic forumation can give incorrect results if the parabolic cylindrical functions are calculated near the turning points. For comparison, similar plots using the parabolic cylindrical functions and asymptotic methods are shown in Figure 2. <figure><img src="content_image/1204.4868/x2.png"><figcaption>Figure 2: Transmision through a single electrostatic barrier of width 100nmand height 180meV with a varying magnetic field across the 100nm barrierregion. Incident energy is 82.66 meV</figcaption></figure> <figure><img src="content_image/1204.4868/x3.png"><figcaption>Figure 3: Transmission through a single electrostatic barrier of width 100nmand height 180meV with a varying magnetic field across the 100nm barrierregion. Incident energy is 82.66 meV. Left: Results calculated usingasymptotic form of parabolic cylindrical functions valid at low magneticfields. Right: Results calculated using parabolic cylindirical functions validat high magnetic fields. Comparison with Fig. 2 shows that both in the high aswell as low magnetic field limit results can be reproduced by the currentmethod very accurately.</figcaption></figure> ### Gaussian Barrier We consider a single gaussian-shaped magnetic field barrier with no electrostatic field and compute the transmission by using a coarse and a fine piecewise approximation. In the coarse approximation, it is reduced to a a single square barrier and the fine approximation consists of it being approximated by as a series of barriers of varying height. The two approximations are shown in Figure 4. The $1/e$ width of the gaussian curve is 140nm and the peak magnetic field is 1T. The coarse approximation consists of a single barrier of width 140nm and magnetic field $\pi/2$ T. The fine approximation consists of the division of the gaussian barrier into 21 slices with a maximum field variation of not more than 0.1T taken to be constant. The incident energy is 82.66 meV. The two transmission plots are also shown. This example clearly demonstrates the utility of being able to perform computations for low magnetic fields even if the peak field value is high. <figure><img src="content_image/1204.4868/x4.png"><figcaption>Figure 4: A gaussian shaped magnetic barrier with two differentapproximations. (a) The gaussian curve with the coarse and fine approximations(b) Transmission plots for both approximations</figcaption></figure> ### Experimental Data We calculate the transmission in graphene based on the experimental data given in [33]. They have shown that the presence of disorder in graphene gives rise to localised charge distributions on the surface, or as they are called, electron and hole puddles. They have also measured this charge distribution. It is known that a scalar potential applied to graphene sheet shifts the Dirac point and leads to charge accumulation. We therefore model the charge as arising from a scalar potential distribution proportional to it. We have calculated the transmission corresponding to a one-dimensional potential extracted from their experimental data scaled by an arbitrary factor of $10^{-29}$. The scalar potential and the transmission data are shown in Figure 5. <figure><img src="content_image/1204.4868/x5.png"><figcaption>Figure 5: Scalar potential distribution from experimental data given in [33]and the corresponding transmission.</figcaption></figure> ### Random magnetic fields It has been shown that any elastic deformation in the graphene sheet manifests itself as effective gauge fields acting on charge carriers[34]. This can be caused by a corrugated substrate or by the intrinsic thermodynamic ripples in graphene. It has also been shown that several electronic devices can be built by controlling the strain.[35]. The relation between a strain field and gauge fields is given in [34]. For a strain field with tensor components $u_{xx}$ and $u_{yy}$ denoting the normal strain and $u_{xy}$ the shear strain, the relation to scalar and vector potentials is as follows ($\beta,t,a,c,g$ are constants) \[A_{x} =c\frac{\beta t}{a}(u_{xx}-u{yy})\] \[A_{y} =-c\frac{\beta t}{a}u_{xy}\] \[V =g(u_{xx}+u{yy})\] Therefore, a strain field can be modelled as a gauge field. In the special case that only x-dependent shear strain is present, the only component of the equivalent magnetic field is $A_{y}(x)$ which is a Landau gauge representable vector potential. We carry out a transmission calculation in the presence of disorder with the magnetic field and scalar potential chosen randomly. Fifty slices, each 10 nm wide are taken with the magnetic field in each slice uniformly distributed between -1 T and 1 T. The scalar potential in each slice is uniformly distributed between 0 and 200meV. A typical result is given in Figure 6, where the magnetic field, scalar and vector potentials are shown alongwith the resultant transmission. <figure><img src="content_image/1204.4868/x6.png"><figcaption>Figure 6: Computation with random fields. (a) Transmission plot (b) MagneticField and vector potential (c) Scalar potential</figcaption></figure> ### Application to bilayer graphene This series technique can also be extended to bilayer graphene in the presence of electrostatic and magnetic fields to obtain the transmission at high energies and low magnetic fields. This method has been used in a recent communication [36]. ## 6 Conclusion We have applied the transfer matrix method to solve transmission problems in graphene in the presence of inhomogeneous electric and magnetic fields. We have brought out some of the difficulties associated with parabolic cylindrical functions and proposed a method to get around its limitations by changing the basis functions to a series solution which tends to complex exponentials. Despite the overhead of numerically computing a series sum, our method is robust and easy to implement with different cases not needing separate treatment compared to the use of parabolic cylindrical functions with or without asymptotic expansions. We also believe that the method is quite general and can be profitably employed whenever the wave equation is being solved with the transfer matrix method. ## 7 Acknowledgments This work is supported by grant SR/S2/CMP-0024/2009 by Science and Engineering Research Council, DST, Govt. of India. ## References * [1] A K Geim and K S Novoselov. The rise of graphene. _Nature materials_, 6(3):183–191, March 2007. * [2] A K Geim. Graphene: status and prospects. _Science (New York, N.Y.)_, 324(5934):1530–1534, June 2009. * [3] a. Castro Neto, F. Guinea, N. Peres, K. Novoselov, and a. Geim. The electronic properties of graphene. _Reviews of Modern Physics_, 81(1):109–162, January 2009. * [4] N. Peres. Colloquium: The transport properties of graphene: An introduction. _Reviews of Modern Physics_, 82(3):2673–2700, September 2010. * [5] M. I. Katsnelson, K. S. Novoselov, and a. K. Geim. Chiral tunnelling and the Klein paradox in graphene. _Nature Physics_, 2(9):620–625, August 2006. * [6] A. De Martino, L. Dell‚ÄôAnna, and R. Egger. Magnetic Confinement of Massless Dirac Fermions in Graphene. _Physical Review Letters_, 98(6):066802, February 2007. * [7] A. De Martino, L Dellanna, and R Egger. Magnetic barriers and confinement of Dirac‚ÄìWeyl quasiparticles in graphene. _Solid State Communications_, 144(12):547–550, December 2007. * [8] A. Ghatak, K. Thyagarajan, and M. Shenoy. Numerical analysis of planar optical waveguides using matrix approach. _Journal of Lightwave Technology_, 5(5):660–667, 1987. * [9] B. Jonsson and S.T. Eng. Solving the Schrodinger equation in arbitrary quantum-well potential profiles using the transfer matrix method. _IEEE Journal of Quantum Electronics_, 26(11):2025–2035, 1990. * [10] Petarpa Boonserm. _Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients_. PhD thesis, Victoria University of Wellington, June 2009. * [11] P Boonserm and M Visser. One Dimensional Scattering Problems : A Pedagogical Presentation of the Relationship between Reflection and Transmission Amplitudes. _Thai Journal of Mathematics_, 8:83–97, 2010. * [12] Sina Khorasani and A L I Adibi. Analytical solution of linear ordinary differential equations by differential transfer matrix method. _Electronic Journal of Differential Equations_, 79:1–18, 2003. * [13] M. Ramezani Masir, P. Vasilopoulos, a. Matulis, and F. Peeters. Direction-dependent tunneling through nanostructured magnetic barriers in graphene. _Physical Review B_, 77(23):235443, June 2008. * [14] Michaël Barbier, P. Vasilopoulos, F. Peeters, and J. Pereira. Bilayer graphene with single and multiple electrostatic barriers: Band structure and transmission. _Physical Review B_, 79(15):155402, April 2009. * [15] M Ramezani Masir, P Vasilopoulos, and F M Peeters. Kronig-Penney model of scalar and vector potentials in graphene. _Journal of physics. Condensed matter_, 22(46):465302, November 2010. * [16] Yu-Xian Li. Transport in a magnetic field modulated graphene superlattice. _Journal of Physics: Condensed Matter_, 22(1):015302, January 2010. * [17] Lifeng Sun, Chao Fang, Yu Song, and Yong Guo. Transport properties through graphene-based fractal and periodic magnetic barriers. _Journal of Physics: Condensed Matter_, 22:445303, 2010. * [18] M. Ramezani Masir, P. Vasilopoulos, and F. Peeters. Fabry-Pérot resonances in graphene microstructures: Influence of a magnetic field. _Physical Review B_, 82(11):115417, September 2010. * [19] A. F. Young and F. Kim. Electronic Transport in Graphene Heterostructure. _Annual Reviews of Condensed Matter Physics_, 2:101, November 2011. * [20] Godfrey Gumbs. Relativistic scattering states for a finite chain with $\delta$-function potentials of arbitrary position and strength. _Physical Review A_, 32(2):1208–1210, August 1985. * [21] Sankalpa Ghosh and Manish Sharma. Electron optics with magnetic vector potential barriers in graphene. _Journal of Physics: Condensed Matter_, 21(29):292204, July 2009. * [22] Wayne W Lui and Masao Fukuma. Exact solution of the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential barrier. _Journal of Applied Physics_, 60(September):1555–1559, 1986. * [23] L. Oroszlány, P. Rakyta, a. Kormányos, C. Lambert, and J. Cserti. Theory of snake states in graphene. _Physical Review B_, 77(8):081403(R), February 2008. * [24] Shanjie Zhang and Jianming Jin. _Computation of Special Functions_. John Wiley and Sons, NY, 1996. * [25] N.M. Temme. Numerical and asymptotic aspects of parabolic cylinder functions. _Journal of computational and applied mathematics_, 121(1-2):221–246, 2000. * [26] Nico M Temme. Numerical and Asymptotic Aspects of Parabolic Cylinder Functions. _arXiv:math/0109188v1_, September 2001. * [27] L. Dell‚ÄôAnna and a. De Martino. Wave-vector-dependent spin filtering and spin transport through magnetic barriers in graphene. _Physical Review B_, 80(15):155416, October 2009. * [28] Luca Dell’Anna and Alessandro De Martino. Magnetic superlattice and finite-energy dirac points in graphene. _Phys. Rev. B_, 83(15):155449, Apr 2011. * [29] G.N. Watson. The Harmonic Functions Associated with the Parabolic Cylinder. _Proc. London Math. Soc_, s2-8:393–421, 1910. * [30] Nathan Schwid. The Asymptotic Forms of the Hermite and Weber Functions. _Transactions of the American Mathematical Society_, 37(2):339–362, March 1935. * [31] George B Arfken and Hans J Weber. _Mathematical Methods for Physicists_. Academic Press, 2001. * [32] William H. Press and Brian P. Flannery. _Numerical Recipes in C: The art of scientific computing_. Cambridge University Press, 1992. * [33] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and a. Yacoby. Observation of electron‚Äìhole puddles in graphene using a scanning single-electron transistor. _Nature Physics_, 4(2):144–148, November 2008. * [34] F. Guinea, Baruch Horovitz, and P. Le Doussal. Gauge fields, ripples and wrinkles in graphene layers. _Solid State Communications_, 149(27-28):1140–1143, July 2009. * [35] Vitor Pereira and a. Castro Neto. Strain Engineering of Graphene‚Äôs Electronic Structure. _Physical Review Letters_, 103(4):046801, July 2009. * [36] Neetu Agrawal, Sameer Grover, Sankalpa Ghosh, and Manish Sharma. Reversal of Klein reflection in bilayer graphene. _Journal of Physics: Condensed Matter_, 24(17):175003, April 2012.
1809.06647	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 40990, "num_imgs": 7, "llama3_tokens_count": 9691 }	[ "content_image/1809.06647/x1.png", "content_image/1809.06647/x2.png", "content_image/1809.06647/x3.png", "content_image/1809.06647/x4.png", "content_image/1809.06647/x5.png", "content_image/1809.06647/x6.png", "content_image/1809.06647/x7.png" ]	# Attribute-aware Face Aging with Wavelet-based Generative Adversarial Networks Yunfan Liu${}^{1}$ Qi Li${}^{1,2,3}$¹ Zhenan Sun${}^{1,2,4}$ ${}^{1}$ Center for Research on Intelligent Perception and Computing, CASIA ${}^{2}$ National Laboratory of Pattern Recognition, CASIA ${}^{3}$ Artificial Intelligence Research, CAS, Jiaozhou, Qingdao, China ${}^{4}$ Center for Excellence in Brain Science and Intelligence Technology, CAS yunfan.liu@cripac.ia.ac.cn, {qli, znsun}@nlpr.ia.ac.cn Authors contributed equally. [FOOTNOTE:1][ENDFOOTNOTE] ###### Abstract Since it is difficult to collect face images of the same subject over a long range of age span, most existing face aging methods resort to unpaired datasets to learn age mappings. However, the matching ambiguity between young and aged face images inherent to unpaired training data may lead to unnatural changes of facial attributes during the aging process, which could not be solved by only enforcing identity consistency like most existing studies do. In this paper, we propose an attribute-aware face aging model with wavelet-based Generative Adversarial Networks (GANs) to address the above issues. To be specific, we embed facial attribute vectors into both the generator and discriminator of the model to encourage each synthesized elderly face image to be faithful to the attribute of its corresponding input. In addition, a wavelet packet transform (WPT) module is incorporated to improve the visual fidelity of generated images by capturing age-related texture details at multiple scales in the frequency space. Qualitative results demonstrate the ability of our model in synthesizing visually plausible face images, and extensive quantitative evaluation results show that the proposed method achieves state-of-the-art performance on existing datasets. ## 1 Introduction Face aging, also known as age progression [16], aims at rendering a given face image with aging effects while still preserving personalized features. Applications of face aging techniques range from social security to digital entertainment, such as predicting contemporary appearance of missing children and cross-age identity verification. Due to the practical value of face aging, many approaches have been proposed to address this problem in the last two decades [8, 20, 19, 21, 7]. With the rapid development of deep learning, deep generative models are widely adopted to synthesize aged face images [23, 3, 4]. However, the most critical problem of these methods is that multiple face images of the same person at different ages are required at training stage, which is extremely expensive to collect in practice and thus their applications are largely limited. <figure><img src="content_image/1809.06647/x1.png"><figcaption>Figure 1: Examples of face aging with mismatched facial attributes generatedby face aging model without facial attribute embedding. Four attributes (Race,Gender, Glasses, and Bald) are considered and three sample results arepresented for each. Labels of ‘Race’ and ‘Gender’ are all obtained viaadvanced publicly available APIs of Face++ [13] and placed underneath eachimage.</figcaption></figure> To deal with this problem, many recent studies resort to unpaired face aging data to train the model [23, 28, 25, 9]. However, these approaches mainly focus on face aging itself while neglecting other critical conditional information of the input (_e.g_., facial attributes), thus fail to regulate the training process. Consequently, training face image pairs with mismatched attributes would mislead the model into learning translations other than aging, causing serious ghosting artifacts and even incorrect facial attributes in generation results. Fig. 1 shows some face aging results with mismatched attributes. In the rightmost face aging result under ‘gender’, beard is mistakenly attached to the input female face image. This is because the model learns that growing a beard is a typical sign of aging but has no way to know that this does not happen to a woman, as face image pairs of young woman and old man could be treated as positive training samples if no conditional information is considered. To suppress such undesired changes of semantic information during the aging process, many recent face aging studies attempt to supervise the output by enforcing identity consistency [28, 1, 25, 9]. However, as shown in Fig. 1, personalized features are well preserved in the output for all sample results, nevertheless, obvious unnatural changes of facial attributes are still observed. In other words, well maintained identity-related features do NOT imply reasonable aging results when training with unpaired data. Therefore, merely enforcing identity consistency is insufficient to eliminate matching ambiguities in unpaired training data, thus fails to achieve satisfactory face aging performance. To solve the above-mentioned issues, in this paper, we propose a framework based on generative adversarial networks (GANs). Different from existing methods in the literature, we involve semantic conditional information of the input by embedding facial attribute vectors in both the generator and discriminator, so that the model could be guided to output elderly face images with attributes faithful to each corresponding input. Furthermore, to enhance aging details, based on the observation that signs of aging are mainly represented by wrinkles, laugh lines, and eye bags, which could be treated as local textures, we employ wavelet packet transform to extract features at multiple scales in the frequency space efficiently. To summarized, the main contributions are as follows: * Facial attributes are incorporated as conditional information into both the generator and discriminator for face aging, since identity preservation is insufficient for generating reasonable results. * Wavelet packet transform is adopted to extract features of texture details at multiple scales in the frequency domain for generating fine-grained details of aging effects. * Extensive experiments have been conducted to demonstrate the ability of the proposed method in rendering accurate aging effects and preserving information of both identity and facial attributes. Quantitative results indicate that our method achieves state-of-the-art performance. ## 2 Related Work In the last few decades, face aging has been a very popular research topic and a great amount of algorithms have been proposed to tackle this issue. In general, these methods could be divided into three categories: physical model-based methods, prototype-based methods, and deep learning-based methods. Physical model-based methods mechanically simulate the changes of facial appearance w.r.t. time by modeling the anatomical structure of human faces. Todd et al. [22] modeled the translation of facial appearance by revised cardioidal strain transformation. Subsequent works investigated the problem from various biological aspects including muscles and overall facial structures [8, 20]. However, physical model-based algorithms are computational expensive and large amount of image sequences of the same subject are required to model aging effects. Data-driven prototyping approaches [19, 21, 7] come into view the next, where faces are divided into age groups and each group is represented by an average face (prototype) computed from the training data. After that, translation patterns between prototypes are regarded as effects of aging. The main problem of prototyping methods is that personalized features are eliminated when calculating average faces, thus the identity information is not well preserved. In recent years, deep generative models with temporal architectures are adopted to synthesize images of elderly faces [23, 3, 4]. However, in most of these works, face image sequence over a long age span for each subject is required thus their potential in practical use is limited. With the success of GANs [5] in generating visually appealing images, many efforts have been made to tackle the problem of face aging using GAN-based framework [28, 25, 9, 17, 24, 10]. Zhang _et al_. [28] proposed a conditional adversarial autoencoder (CAAE) to achieve age progression and regression by traversing in low-dimensional manifold. The work most similar to ours is [25], in which a GAN-based model with pyramid architecture is proposed, and identity loss is adopted to achieve permanence. Besides preserving identity information, we focus on alleviating the influence of matching ambiguity of unpaired training samples and ensuring attribute consistency by embedding facial attribute vectors in the model. <figure><img src="content_image/1809.06647/x2.png"><figcaption>Figure 2: An overview of the proposed face aging framework. An hourglass-shaped generator G learns the age mapping and outputs lifelike elderly faceimages. A discriminator D is employed to distinguish synthesized face imagesfrom generic ones, based on multi-scale wavelet coefficients computed by thewavelet packet transform module. The p-dimensional attribute vector describingthe input face image is embedded to both the generator and discriminator toreduce matching ambiguity inherent to unpaired training data.</figcaption></figure> ## 3 Approach In a unpaired face aging dataset, each young face image might map to many elderly face candidates during the training process, and image pairs with mismatched semantic information may mislead the model into learning translations other than aging. To solve this problem, we present a GAN-based face aging model that takes young face images and their semantic information (_i.e_. facial attributes) as input and outputs visually plausible aged faces accordingly. The network consists of two parts: a facial attribute embedded generator $G$ and a wavelet-based discriminator $D$. The generator network embeds facial attributes into young face images and synthesizes aged faces. The discriminator network is used to encourage the generation results to be indistinguishable from generic ones and to possess attributes same as the corresponding input. An overview of the proposed framework is presented in Fig. 2. ### Facial Attribute Embedded Generator Existing face aging studies [9, 25, 28] only take young face images as inputs and then directly learn age mappings using GAN-based networks. Although constraints on identity information and pixel values are usually imposed to restrict modifications made to input images, facial attributes may still undergo unnatural translations (as shown in Fig. 1). Unlike previous works, we propose to incorporate both low-level image information (pixel values) and high-level semantic information (facial attributes) into the face aging model to regularize image translation patterns and reduce the ambiguity of mappings between unpaired young and aged faces. To be specific, the model takes young face images and their corresponding attribute vectors as input, and generates elderly face images with attributes in agreement with the input ones. Rather than supervising the attributes of generation results by simply adopting an additional loss term, we embed the attribute vector in the generator so that semantic facial information is well considered in the generation process and encourages the model to produce face images with consistent attributes more effectively. To be specific, we employ an hourglass-shaped fully convolutional network as the generator, which has achieved success in previous image translation studies [6, 29]. It consists of an encoder network, a decoder network, and four residual blocks in between as the bottleneck. The input facial attribute vector is replicated and concatenated to the output blob of the last residual block as they both contain high-level semantic features. After the combination, the decoder network transforms the concatenated feature blob back to the image space. Since face aging could be considered as rendering aging effects conditioned on the input young face image, we add the input image to the output of the decoder to form a residual connection. Compared to synthesize the whole face image, this structure automatically makes the generator focus more on modeling the difference between input and output face images, namely the representative signs of aging, and be less likely to be distracted by visual content irrelevant to aging, such as background. Finally, the numeric scale of the resultant tensor is normalized by a hyperbolic tangent (tanh) mapping and thus the generated elderly face image is obtained. <figure><img src="content_image/1809.06647/x3.png"><figcaption>Figure 3: Demonstration of wavelet packet transform. (a) Low-pass and high-pass decomposition filters (hlow and hhigh) are applied iteratively to theinput on k-th level to compute wavelet coefficients on the next level; (b) asample face image with its wavelet coefficients at different decomposinglevels.</figcaption></figure> ### Wavelet-based Discriminator To force the generator to absorb the semantic information of the input face image, a conditional discriminator is employed. The discriminator has two main functions: 1) distinguish synthesized face images from generic ones; 2) check whether the attribute of each generation result is faithful to that of the corresponding input. To be specific, considering the fact that typical signs of aging, such as wrinkles, laugh lines, and eye bags, could be regarded as local image textures, we adopt wavelet packet transform (WPT, see Fig. 3) to capture age-related textural features. Specifically, multi-level WPT is performed to provide a more comprehensive analysis of textures in the given image, and wavelet coefficients at each decomposing level are fed into a convolutional pathway of the discriminator. Note that this is different from [9], since wavelet coefficients are only used for discrimination in our work and no prediction or reconstruction is involved. To make the discriminator gain the ability of telling whether attributes are preserved in generated images, the input attribute vector is also replicated and concatenated to the output of an intermediate convolutional block of each pathway. At the end of the discriminator, same-sized outputs of all pathways are fused to form a single tensor, and adversarial loss is then estimated against the label tensor. Compared to extracting multi-scale features by a sequence of convolutional layers as in [25], the advantage of using WPT is that the computational cost is significantly reduced since calculating wavelet coefficients could be regarded as forwarding through a single convolutional layer. Therefore, WPT greatly reduces the number of convolutions performed in each forwarding process. Although this part of the model has been simplified, it still takes the advantage of multi-scale image texture analysis, which is helpful in improving the visual fidelity of generated images. ### Overall Objective Functions Training of GAN model simulates the process of optimizing a minimax-max two-player game between the generator $G$ and the discriminator $D$. Unlike regular GANs [5], we adopt least square loss instead of negative log likelihood loss for that margins between generated samples and the decision boundary in the feature space are also minimized, which further improves the quality of synthesized images [12]. Practically, we pair up young face images $x_{i}$ and their corresponding attribute vectors $\alpha_{i}$ of dimension $p$, denoted as $(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)$, and take them as input to the model. Only generic aged faces with attributes same as the input, _i.e_. $(x_{i},\alpha_{i})\sim P_{old}(x,\alpha_{i})$, are considered as positive samples, and real young faces, _i.e_. $(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)$, are regarded as negative samples to help $D$ gain discriminating ability on aging effects. Mathematically, the objective function for $G$ and $D$ could be written as follows, \[L_{GAN}(G)=\mathbb{E}_{(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)}[(D(G(x_{i}, \alpha_{i}),\alpha_{i})-1)^{2}]\] (1) \[\begin{split} L_{GAN}(D)=&\,\mathbb{E}_{(x_{i}, \alpha_{i})\sim P_{old}(x,\alpha_{i})}[(D(x_{i},\alpha_{i})-1)^{2}]+\\ &\,\mathbb{E}_{(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)}D(G(x_{ i},\alpha_{i}),\alpha_{i})^{2}+\\ &\,\mathbb{E}_{(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)}D(x_{i} ,\alpha_{i})^{2}\end{split}\] (2) where $P_{young}$ and $P_{old}$ denote the distribution of generic face images of young and old subjects, respectively. In addition, pixel loss and identity loss are adopted to maintain consistency in both image-level and personalized feature-level. To be specific, we utilize the VGG-Face descriptor [14], denoted by $\phi$, to extract the identity related semantic representation of a face image. These two loss terms could be formulated as, \[L_{pix}=\mathbb{E}_{(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)}\|\|G(x_{i}, \alpha_{i})-x_{i}\|\|_{F}^{2}\] (3) \[L_{id}=\mathbb{E}_{(x_{i},\alpha_{i})\sim P_{young}(x,\alpha)}\|\|\phi(G(x_{i}, \alpha_{i}))-\phi(x_{i})\|\|_{F}^{2}\] (4) In conclusion, overall objective functions of the proposed model could be written as follows, \[L_{G}=L_{GAN}(G)+\lambda_{pix}L_{pix}+\lambda_{id}L_{id}\\\] (5) \[L_{D}=L_{GAN}(D)\] (6) where $\lambda_{id}$ and $\lambda_{pix}$ are coefficients balancing the importance of critics on identity and pixels, respectively. We optimize the model by minimizing $L_{G}$ and $L_{D}$ alternatively until the optimality is reached. <figure><img src="content_image/1809.06647/x4.png"><figcaption>Figure 4: Sample results on Morph (first row) and CACD (second row). The firstimage in each result is the input test face image and subsequent 3 images aresynthesized elderly face images of the same subject in age group 31-40, 41-50and 51+, respectively.</figcaption></figure> <figure><img src="content_image/1809.06647/x5.png"><figcaption>Figure 5: Performance comparison with prior work on Morph (zoom in for abetter view of the aging details). The second row shows the results of priorwork, where four methods are considered and two sample results are presentedfor each. These four methods are (from left to right): CONGRE [18], HFA [26],GLCA-GAN [9], and PAG-GAN [25]. The last row shows the results of our method.</figcaption></figure> ## 4 Experiments ### Dataset MORPH [15] is a large aging dataset containing 55,000 face images of more than 13,000 subjects. Data samples in MORPH are color images of near-frontal faces exhibiting neutral expressions under uniform and moderate illumination with simple background. CACD [2] contains 163,446 face images of 2,000 celebrities captured in much less controlled conditions. Besides large variations in pose, illumination, and expression (PIE variations), images in CACD are collected via Google Image Search, making it a very challenging dataset due to the mismatching between actual face presented in each image and associated labels provided (name and age). As for facial attributes, MORPH provides researchers with labels including age, gender, and race for each image. We choose ‘gender’ and ‘race’ to be the attributes that are required to be preserved, since these two attributes are guaranteed to remain unchanged during natural aging process, and are relatively objective compared to attributes such as ‘attractive’ or ‘chubby’ used in popular facial attribute dataset CelebA [11]. For CACD, since face images with race other than ‘white’ only takes a small portion of the entire dataset, we only select ‘gender’ as the attribute to preserve. To be specific, we go through the name list of the celebrities and label the corresponding images accordingly. This introduces noise in gender labels due to the mismatching between the annotated name and the actual face presented in each image, which further increases the difficulty for our method to achieve good performance on this dataset. It is worthwhile to note that the proposed model is highly expandable, as researchers may choose whatever attributes to preserve simply by incorporating them in the conditional facial attribute vector and arrange training images pairs accordingly. ### Implementation Details All face images are cropped and aligned according to the five facial landmarks detected by MTCNN [27]. Following the convention in [25, 9], we divide the face images into four age groups, _i.e_., 30-, 31-40, 41-50, 51+, and only consider translations from 30- to the other three age groups. To evaluate the performance of the proposed method objectively, all metric measurements are conducted via stable public APIs of Face++ [13]. Thresholds adopted in our face verification experiments (threshold=76.5, FAR=1e-5) are the same as those used in [25]. Therefore, quantitative results of our experiments are comparable to those reported in [25]. We choose Adam to be the optimizer of both $G$ and $D$ with learning rate and batch-size set to $1e^{-4}$ and 16, respectively. Pixel-level critic is applied every 5 iterations, and $D$ is updated at every iteration. As for trade-off parameters, $\lambda_{pix}$ and $\lambda_{id}$ are firstly set to make $L_{pix}$ and $L_{id}$ to be of the same order of magnitude as $L_{GAN}(G)$, and then divided by 10 to emphasize the importance of the adversarial loss. All experiments are conducted under 5-fold cross validation on a Nvidia Titan Xp GPU. Morph \| CACD ---\|--- Age group \| 31 - 40 \| 41 - 50 \| 51 + \| Age group \| 31 - 40 \| 41 - 50 \| 51 + Estimated Age Distributions \| Estimated Age Distributions Generic \| 38.60 \| 47.74 \| 57.25 \| Generic \| 38.51 \| 46.54 \| 53.39 Synthetic \| 38.47 \| 47.55 \| 56.57 \| Synthetic \| 38.88 \| 47.42 \| 54.05 Difference of mean ages \| Difference of mean ages CAAE \| 10.08 \| 15.49 \| 21.42 \| CAAE \| 5.76 \| 11.53 \| 17.93 GLCA-GAN \| 0.23 \| 3.61 \| 8.61 \| GLCA-GAN \| 1.72 \| 2.07 \| 2.85 PAG-GAN \| 0.38 \| 0.52 \| 1.48 \| PAG-GAN \| 0.70 \| 0.22 \| 0.57 Ours \| 0.13 \| 0.19 \| 0.68 \| Ours \| 0.37 \| 0.58 \| 0.66 Table 1: Age estimation results on Morph and CACD (differences of mean ages are measured in absolute value). <figure><img src="content_image/1809.06647/x6.png"><figcaption>Figure 6: Distributions of the estimated ages. (a) synthetic faces on Morph;(b) synthetic faces on CACD; (c) generic faces on Morph; (d) generic faces onCACD.</figcaption></figure> ### Qualitative Results of Face Aging Sample results on Morph and CACD are shown in Fig. 4. It is clear that our method is able to simulate translations between age groups and synthesize elderly face images with high visual fidelity. In addition, our method is robust to variations in terms of race, gender, expression, and occlusion. Performance comparison with prior work on Morph is shown in Fig. 5. Traditional face aging methods, CONGRE [18] and HFA [26], only render subtle aging effects within tight facial area, which fails to accurately simulate the aging process. In contrast, GAN-based methods, GLCA-GAN [9] and GAN with pyramid architecture proposed in [25], referred to as PAG-GAN, have achieved significant improvement on the quality of generation results. However, our method further generates face images of higher resolution ($2\times$) with enhanced details compared to GLCA-GAN, and reduces ghosting artifacts in the results compared to PAG-GAN (_e.g_. finer details of hair and beard). ### Aging Accuracy and Identity Preservation In this subsection, we report evaluation results on aging accuracy and identity preservation. The performance of the proposed model is compared with previously state-of-the-art methods CAAE [28], GLCA-GAN [9] and PAG-GAN [25] to demonstrate the effectiveness. Aging Accuracy: Age distributions of both generic and synthetic faces in each age group are estimated, where less discrepancy between real and fake images indicates more accurate simulation of aging effects. On Morph and CACD, face images of age under or equal to 30 are considered as testing samples, and their corresponding aged faces in the other three age groups are synthesized. We estimated the apparent age of both generation results and natural face images in the dataset using Face ++ APIs for fair comparison. Age estimation results on Morph and CACD are shown in Table 1 and Fig. 6. We compare our method with previous works in terms of differences between mean ages. On Morph, it could be seen that estimated age distributions of synthetic elderly face images well match that of natural images for all age groups. Our method consistently outperforms other approached in all three aging processes, demonstrating the effectiveness of our method. Signs of aging in results of CAAE are not obvious enough, leading to large age estimation errors. On CACD, due to the existence of mismatching between face images and associated labels, slight performance drop could be observed. Still, the proposed method achieves results comparable to previous state-of-the-art. This shows that our method is relatively robust to noise in attribute labels and thus lower the requirement on the accuracy of the prior attribute detection process. Morph \| CACD ---\|--- Age group \| 31 - 40 \| 41 - 50 \| 51 + \| Age group \| 31 - 40 \| 41 - 50 \| 51 + Verification Confidence \| Verification Confidence 30 - \| 95.77 \| 94.64 \| 87.53 \| 30 - \| 93.67 \| 91.54 \| 90.32 31 - 40 \| - \| 95.47 \| 89.53 \| 31 - 40 \| - \| 91.74 \| 90.54 41 - 50 \| - \| - \| 90.50 \| 41 - 50 \| - \| - \| 91.12 Verification Rate (%) \| Verification Rate (%) CAAE \| 15.07 \| 12.02 \| 8.22 \| CAAE \| 4.66 \| 3.41 \| 2.40 GLCA-GAN \| 97.66 \| 96.67 \| 91.85 \| GLCA-GAN \| 97.72 \| 94.18 \| 92.29 PAG-GAN \| 100.00 \| 98.91 \| 93.09 \| PAG-GAN \| 99.99 \| 99.81 \| 98.28 Ours \| 100.00 \| 100.00 \| 98.26 \| Ours \| 99.76 \| 98.74 \| 98.44 Table 2: Face verification results on Morph and CACD. \| Preservation Rate (%) of ‘Gender’ \| Preservation Rate (%) of ‘Race’ ---\|---\|--- \| Morph \| CACD \| Morph Age group \| 31 - 40 \| 41 - 50 \| 51 + \| 31 - 40 \| 41 - 50 \| 51 + \| 31 - 40 \| 41 - 50 \| 51 + GLCA-GAN \| 96.30 \| 95.43 \| 95.77 \| 87.27 \| 86.79 \| 85.89 \| 91.79 \| 89.52 \| 89.34 PAG-GAN \| 95.96 \| 93.77 \| 92.47 \| 83.97 \| 81.28 \| 70.05 \| 95.83 \| 88.51 \| 87.98 Ours \| 97.37 \| 97.21 \| 96.07 \| 90.71 \| 87.63 \| 87.19 \| 95.86 \| 94.10 \| 93.22 Table 3: Facial attributes preservation rates for ‘Gender’ and ‘Race’ on Morph and CACD. Identity Preservation: Face verification experiments are conducted to check whether the identity information has been preserved during the face aging process. Similar to previous literature, comparisons between synthetic elderly face images from different age groups of the same subject are also conducted to inspect if the identity information is consistent among three separately trained age mappings. Results of face verification experiments are shown in Table 2. On Morph, our method achieves the highest verification rate on all three translations and outperforms other approaches by a clear margin, especially in the hardest case (from 30- to 51+). This demonstrates that the proposed method successfully achieves identity permanence during face aging. On the more challenging dataset CACD containing mismatched labels, the performance of our method is comparable to PAG-GAN with minor difference. Notably, as the time interval between two face images of a single subject increases, both verification confidence and accuracy decrease, which is reasonable as greater changes in facial appearance may occur as more time elapsed. ### Facial Attribute Consistency We evaluate the performance of facial attribute preservation by comparing facial attributes estimated before and after age progression, and results are listed in Table 3. On Morph, facial attributes of the majority of testing samples (up to 97.37% for ‘gender’ and 95.86% for ‘race’) are well preserved in the aging process. In addition, our method outperforms both GLCA-GAN and PAG-GAN by clear margins on translations to all age groups. On CACD, due to the influence of mistakenly labeled data samples, clear performance drop could be observed compared to the results on Morph. However, our method still gives better performance on facial attributes preservation than other methods. The advantage of our method in preserving the ‘gender’ attribute becomes greater as the age gap increases, and finally reaches 17.14% (87.19% over 70.05%) when translating to the oldest age group 51+. From Table 3, we could conclude that undesired changes of facial attributes are more likely to happen as the age gap increases, and incorporating conditional information is beneficial for maintaining consistency of target facial attributes in the aging process. <figure><img src="content_image/1809.06647/x7.png"><figcaption>Figure 7: Sample visual results of the ablation study. For each face, theestimated age (first row) and detected attributes (second row) are listedunderneath. Values in the last row are face verification confidence betweengeneration results and the test face.</figcaption></figure> ### Ablation Study In this part, experiments are conducted to fully explore the contribution of facial attribute embedding (FAE) and wavelet packet transform (WPT) in simulating accurate age translations. We investigate the impact of including/excluding attribute embedding (w/wo FAE) and wavelet packet transform (w/wo WPT) on age distribution, face verification rate, and attribute preservation rate. All experiments in this subsection are conducted only on Morph as labels are noisy on CACD dataset. \| Gender Preservation Rate (%) \| Race Preservation Rate (%) \| Deviation of Estimated Ages ---\|---\|---\|--- Age group \| 31-40 \| 41-50 \| 51+ \| 31-40 \| 41-50 \| 51+ \| 31-40 \| 41-50 \| 51+ woFAE / woWPT \| 95.72 \| 94.21 \| 93.60 \| 95.04 \| 93.55 \| 90.83 \| 0.44 \| 1.72 \| 3.03 woFAE / wWPT \| 96.15 \| 94.90 \| 93.61 \| 93.89 \| 88.63 \| 90.21 \| 0.68 \| 0.41 \| 2.31 wFAE / woWPT \| 97.21 \| 96.91 \| 95.85 \| 95.22 \| 94.35 \| 91.43 \| 0.82 \| 0.52 \| 4.82 Ours \| 97.37 \| 97.21 \| 96.07 \| 95.86 \| 94.10 \| 93.22 \| 0.13 \| 0.19 \| 0.68 Table 4: Comparison of results on facial attribute preservation and aging accuracy between variants of the proposed model (differences of mean ages are measured in absolute value). Age group \| 31-40 \| 41-50 \| 51+ ---\|---\|---\|--- woFAE / woWPT \| 100.00 \| 100.00 \| 99.92 woFAE / wWPT \| 100.00 \| 99.88 \| 98.06 wFAE / woWPT \| 100.00 \| 100.00 \| 98.86 Ours \| 100.00 \| 100.00 \| 98.26 Table 5: Face verification rates (%) of variants of the proposed model on Morph Visual illustrations of face images generated by variants of the proposed model are shown in Fig. 7. It is clear that when both FAE and WPT are not involved (woFAE_woWPT), generation results suffer from severe ghosting artifacts. Due to the intrinsic matching ambiguity of unpaired training data, the model without FAE mistakenly attaches moustache to the input female face image to show the aging effect. Notably, growing a moustache does not decrease the face verification confidence, as the generated face image still shares similar identity-related features with the input. This again confirms our observation that enforcing identity consistency is insufficient to obtain satisfactory face aging results. On the contrary, incorporating FAE suppresses the undesired facial attribute drift by reducing the matching ambiguity. To be specific, in Fig. 7, there is no more moustache in generation results after adopting FAE thus facial attribute consistency is achieved. Unfortunately, removing moustache also wipes out aging-related textural details (wrinkles, laugh lines, and eye bags), leading to relatively inaccurate aging results (much younger than expected). To solve this issue and generate more visually plausible face images with vivid signs of aging, WPT is employed as the initial layer of the discriminator. The contribution of WPT could be easily seen by comparing the results obtained under setting ‘woFAE / woWPT’ and ‘woFAE / wWPT’, as well as ‘wFAE / woWPT’ and ‘Ours’. Although results obtained under setting ‘woFAE / wWPT’ still suffer from wrong facial attributes, ghosting artifacts are significantly alleviated and lifelike aging effects are clearly observed. Quantitative results for ablation study are shown in Table 4 and 5. According to results in Table 4, introducing facial attribute embedding (wFAE) increases preservation rates for both ‘gender’ and ‘race’ under all three age mappings, especially in the case of translating to 51+. This proves the effectiveness of attribute embedding as it aligns unpaired age data in terms of facial attributes and thus reduces the intrinsic ambiguity in data mapping. In addition, it is clear that adopting WPT reduces the discrepancies between age distributions of generic and synthetic images in all cases. However, WPT provides little help in maintaining facial attribute consistency. This is because WPT only captures feature based on low-level visual data and could not bridge the semantic gap, so that the framework still suffers from mismatched data samples. Combining results in Table 4 and 5, it could be seen that while attribute preservation rates still have room for improvement, verification rates are about to reach perfection. This observation validates our statement that identity preservation does not guarantee that facial attributes remain stable during the aging process. Therefore, besides constraints on identity, supervision on facial attributes are also helpful to reduce the intrinsic matching ambiguity of unpaired data and achieve satisfactory face aging results. ## 5 Conclusion In this paper, we propose a GAN-based framework to synthesize aged face images. Due to the ineffectiveness of identity constraints in reducing the matching ambiguity of unpaired aging data, we propose to employ facial attributes to tackle this issue. Specifically, we embed facial attribute vectors to both the generator and discriminator to encourage generated images to be faithful to facial attributes of the corresponding input image. To further improve the visual fidelity of generated face images, wavelet packet transform is introduced to extract textual features at multiple scales efficiently. Extensive experiments are conducted on Morph and CACD, and qualitative results demonstrate that our method could synthesize lifelike face images robust to both PIE variations and noisy labels. Furthermore, quantitative results obtained via public APIs validate the effectiveness of the proposed method in aging accuracy as well as identity and attribute preservation. Acknowledgements. This work is supported by the National Natural Science Foundation of China (Grant No. 61702513, U1836217, 61427811). ## References * [1] Grigory Antipov, Moez Baccouche, and Jean-Luc Dugelay. Face aging with conditional generative adversarial networks. _IEEE International Conference on Image Processing (ICIP)_, pages 2089–2093, 2017. * [2] Bor-Chun Chen, Chu-Song Chen, and Winston H Hsu. Face recognition and retrieval using cross-age reference coding with cross-age celebrity dataset. _IEEE Transactions on Multimedia (TMM)_, 17(6):804–815, 2015. * [3] Chi Nhan Duong, Khoa Luu, Kha Gia Quach, and Tien D Bui. Longitudinal face modeling via temporal deep restricted boltzmann machines. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_, pages 5772–5780, 2016. * [4] Chi Nhan Duong, Kha Gia Quach, Khoa Luu, T Hoang Ngan Le, and Marios Savvides. Temporal non-volume preserving approach to facial age-progression and age-invariant face recognition. In _Proceedings of the IEEE International Conference on Computer Vision (ICCV)_, pages 3755–3763, 2017. * [5] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In _Advances in neural information processing systems (NIPS)_, pages 2672–2680, 2014. * [6] Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In _European Conference on Computer Vision (ECCV)_, pages 694–711, 2016. * [7] Ira Kemelmacher-Shlizerman, Supasorn Suwajanakorn, and Steven M Seitz. Illumination-aware age progression. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_, pages 3334–3341, 2014. * [8] Andreas Lanitis, Christopher J. Taylor, and Timothy F Cootes. Toward automatic simulation of aging effects on face images. _IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI)_, 24(4):442–455, 2002. * [9] Peipei Li, Yibo Hu, Qi Li, Ran He, and Zhenan Sun. Global and local consistent age generative adversarial networks. In _International Conference on Pattern Recognition (ICPR)_, pages 1073–1078, 2018. * [10] Qi Li, Yunfan Liu, and Zhenan Sun. Age progression and regression with spatial attention modules. In _arXiv preprint arXiv:1903.02133_, 2019. * [11] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In _Proceedings of the IEEE International Conference on Computer Vision (ICCV)_, 2015. * [12] Xudong Mao, Qing Li, Haoran Xie, Raymond YK Lau, Zhen Wang, and Stephen Paul Smolley. Least squares generative adversarial networks. In _Proceedings of the IEEE International Conference on Computer Vision (ICCV)_, pages 2813–2821, 2017. * [13] Megvii Inc. Face++ research toolkit. http://www.faceplusplus.com. * [14] Omkar M Parkhi, Andrea Vedaldi, Andrew Zisserman, et al. Deep face recognition. In _British Machine Vision Conference (BMVC)_, pages 41.1–41.12, 2015. * [15] Karl Ricanek and Tamirat Tesafaye. Morph: A longitudinal image database of normal adult age-progression. In _the International Conference on Automatic Face and Gesture Recognition (FG)_, pages 341–345, 2006. * [16] Xiangbo Shu, Jinhui Tang, Hanjiang Lai, Luoqi Liu, and Shuicheng Yan. Personalized age progression with aging dictionary. In _Proceedings of the IEEE International Conference on Computer Vision (ICCV)_, pages 3970–3978, 2015. * [17] Jingkuan Song, Jingqiu Zhang, Lianli Gao, Xianglong Liu, and Heng Tao Shen. Dual conditional gans for face aging and rejuvenation. In _Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI)_, pages 899–905, 2018. * [18] Jinli Suo, Xilin Chen, Shiguang Shan, Wen Gao, and Qionghai Dai. A concatenational graph evolution aging model. _IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI)_, 34(11):2083–2096, 2012. * [19] Jinli Suo, Song-Chun Zhu, Shiguang Shan, and Xilin Chen. A compositional and dynamic model for face aging. _IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI)_, 32(3):385–401, 2010. * [20] Yusuke Tazoe, Hiroaki Gohara, Akinobu Maejima, and Shigeo Morishima. Facial aging simulator considering geometry and patch-tiled texture. In _ACM SIGGRAPH_, 2012. * [21]Bernard Tiddeman, Michael Burt, and David Perrett. Prototyping and transforming facial textures for perception research. _IEEE Computer graphics and applications_, 21(5):42–50, 2001. * [22] James T Todd, Leonard S Mark, Robert E Shaw, and John B Pittenger. The perception of human growth. _Scientific American_, 242(2):132–145, 1980. * [23] Wei Wang, Zhen Cui, Yan Yan, Jiashi Feng, Shuicheng Yan, Xiangbo Shu, and Nicu Sebe. Recurrent face aging. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_, pages 2378–2386, 2016. * [24] Z. Wang, W. Luo X. Tang, and S. Gao. Face aging with identity-preserved conditional generative adversarial networks. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_, 2018. * [25] Hongyu Yang, Di Huang, Yunhong Wang, and Anil K. Jain. Learning face age progression: A pyramid architecture of gans. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_, pages 31–39, 2018. * [26] Hongyu Yang, Di Huang, Yunhong Wang, Heng Wang, and Yuanyan Tang. Face aging effect simulation using hidden factor analysis joint sparse representation. _IEEE Transactions on Image Processing (TIP)_, 25(6):2493–2507, 2016. * [27] Kaipeng Zhang, Zhanpeng Zhang, Zhifeng Li, and Yu Qiao. Joint face detection and alignment using multitask cascaded convolutional networks. _IEEE Signal Processing Letters_, 23(10):1499–1503, 2016. * [28] Zhifei Zhang, Yang Song, and Hairong Qi. Age progression/regression by conditional adversarial autoencoder. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_, pages 4352–4360, 2017. * [29] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In _Proceedings of the IEEE International Conference on Computer Vision (ICCV)_, pages 2242–2251, 2017.
1103.6282	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 26976, "num_imgs": 5, "llama3_tokens_count": 7807 }	[ "content_image/1103.6282/x1.png", "content_image/1103.6282/x2.png", "content_image/1103.6282/x3.png", "content_image/1103.6282/x4.png", "content_image/1103.6282/x5.png" ]	# Light trapping within the grooves of 1D diffraction gratings under monochromatic and sunlight illumination. Mario M. Jakas${}^{\dagger}$ and Francisco Llopis${}^{\ddagger}$ Departamento de Física Fundamental y Experimental, Electrónica y Sistemas. Universidad de La Laguna, 38205 La Laguna, Tenerife, Spain $^†$mmateo@ull.es $^‡$fllopis@ull.es ###### Abstract The Rayleigh-Modal method is used to calculate the electromagnetic field within the grooves of a perfectly conducting, rectangular-shaped 1D diffraction grating. An _enhancement coefficient_ ($\eta$) is introduced in order to quantify such an energy concentration. Accordingly, $\eta>$1 means that the amount of electromagnetic energy present within the grooves is larger than that one will have, over the same volume, if the diffraction grating is replaced by a perfectly reflecting mirror. The results in this paper show that $\eta$ can be as large as several decades at certain, often narrow, ranges of wavelengths. However, it reduces to approximately 20% under sunlight illumination. In this latter case, such values are achieved when the _optical spacing_ between the grooves $dn$ is greater than 500 nm, where $d$ is the groove spacing and $n$ is the refractive index of the substance within the grooves. For $dn$ smaller than 500 nm the enhancement coefficient turns negligibly small. pacs: 2.79.Dj, 88.40.H-, 88.40.fc, 42.79.Pw, 42.79.Qx ## 1 Introduction It is well documented that electromagnetic fields within the grooves of diffraction grating can be largely increased when illuminated by light [1, 2, 3, 4]. This phenomenon was recently proposed as way to enhance the absorption of light in PV cells and optoelectronic devices [5, 6, 7, 8]. In fact, for the purpose of increasing the efficiency of photo-cells where sunlight may not be absorbed so easily, the _field enhancement_ appears to be a useful approach. In this realm, however, rather than differential one needs integrated figures. This is so because large field enhancements, as those previously reported, are not enough if not accompanied by a net increase of the light energy integrated over a representative volume of the cell, along the pertinent range of wavelengths, incidence angles and states of polarization. In a previous paper of the Authors [9], the field enhancement was analysed for the more general case of beams of monochromatic light arriving to the grating along several directions and different states of polarization. There, an _enhancement coefficient_, namely $\eta$, is introduced, so that $\eta>$1 means that the amount of electromagnetic energy present within the grooves is larger than that one will have, over the same volume, if the diffraction grating is replaced by a perfectly reflecting mirror. In this paper, however, the previous study is extended to the case of illuminating the grating with solar light. The results of these calculations show that, when using monochromatic, polarized and well-collimated beams of light, $\eta$ exhibits a series of peaks at certain wavelengths, where the enhancement coefficient can be as large as several decades, or even grater. However, after taking an average over incidence angles and polarization states, $\eta$ is significantly reduced as it may reach then values which can hardly be larger than approximately 3. If the previous results are also averaged over the solar spectrum, the enhancement coefficient is further reduced to approximately 1.20. Although these results are to some extent discouraging, it does not mean that the diffraction grating structure cannot be advantageously used in designing photo-electronic devices. This is so because, depending on the case, the aforementioned 20% gain may suffice. In this regard, the present study could be interesting to those who may need increase the absorption of light within a solar cell and might possibly be planning to incorporate a diffraction grating for such a purpose. This paper is organized as follows: the basics equations necessary to obtain the field within the grooves of a diffraction grating are derived in Section 2. The results of numerically calculating the enhancement coefficient as a function of the various parameters in the model are shown and discussed in Section 3 and, finally, Section 4 contains a summary and concluding remarks. ## 2 Basic equations The calculation model is sketched in figure 1. There, one can observe the incident light, a linearly polarized plane-wave, arriving to the grating from above at an angle $\theta_{i}$ with respect to the $y$-axis and azimuthal angle $\varphi_{i}$ with respect to the x-axis. The light has a wavevector $\mathbf{k}_{i}=(k_{ix},k_{iy},k_{iz})$, which is related to the wavelength in the upper medium ($\lambda$) as $\left\|\mathbf{k}_{i}\right\|=k_{i}=2\pi/\lambda$. The electric field amplitude $\mathbf{E}_{i}$ is assumed to be perpendicular to $\mathbf{k}_{i}$ and $\phi$ is the angle formed by $\mathbf{E}_{i}$ and the ($y=0$)-plane. The diffraction grating is assumed to be an infinite set of equally spaced, rectangular-shaped grooves made on the surface of a perfect conductor. The spatial period of the grooves is $d$, whereas the width and depth are $b$ and $h$, respectively. Finally, the upper medium is assumed to be a non-lossy dielectric material that fills the upper part of the grating, up to a front surface which will not be taken into account in this calculations. <figure><img src="content_image/1103.6282/x1.png"><figcaption>Figure 1: A linearly polarized electromagnetic wave, with wave vector ki,electric field Ei, incidence angle θi and azimuthal angle φi, arrives from theupper half-space upon a rectangular-shaped diffraction grating. Grooves have aspatial period d, width b and depth h, and ϕ denotes the polarization anglewith respect to the (x-z)-plane.</figcaption></figure> The way the electromagnetic fields within the grooves are calculated was already shown in Ref.[7, 9], however, for the purpose of the present analysis, it is important to re-do it once more. In the first place, the _Rayleigh expansion_ is assumed for upper space, \[\mathbf{E}^{(y>0)}(\mathbf{r})=\mathbf{E}_{i}e^{i\mathbf{k}_{i}\cdot\mathbf{r} }+\sum_{n}\mathbf{R}_{n}\,e^{i\mathbf{q}_{n}\cdot\mathbf{r}}\hskip 43.362pt \mbox{for $y>0$}\,,\] (1) where $\mathbf{q}_{n}$ is the wave vector of the $n$-th order reflected beam and $\mathbf{R}_{n}=(R_{nx},R_{ny},R_{nz})$ is the corresponding electric field vector. Similarly, $\mathbf{q}_{n}=(q_{nx},q_{ny},k_{iz})$, where $q_{xn}=k_{ix}+2\pi n/d$, $n=0,\pm 1,\pm 2,\ldots$, and $q_{ny}=+\sqrt{k_{i}^{2}-q_{nx}^{2}-k_{iz}^{2}}$. Notice that the plus sign in front of the square root means that the positive, either real or imaginary solution to the square root has to be used. Within the grooves one has the so-called _modal waves_, namely, \[E_{x}^{(y<0)}(\mathbf{r})= \sum_{m,N}{{\cal E}_{x,m}\,e^{i\left(k_{iz}z+Nk_{ix} x\right)}\,X_{m,N}^{{}^{\prime}}(x)\,Y_{m}(y)}\] \[E_{y}^{(y<0)}(\mathbf{r})= \sum_{m,N}{{\cal E}_{y,m}\,e^{i\left(k_{iz}z+Nk_{ix} x\right)}\,X_{m,N}(x)\,Y_{m}^{{}^{\prime}}(y)}\] (2) \[E_{z}^{(y<0)}(\mathbf{r})= i\,\sum_{m,N}{{\cal E}_{z,m}\,e^{i\left(k_{iz}z+Nk_ {ix}x\right)}\,X_{m,N}(x)\,Y_{m}(y)}\] where \[X_{m,N}(x)=\left\{\begin{array}[]{cl}\sin{\left[K_{mx}(x-x_{N}) \right]}&\mbox{for $x_{N}<x<x_{n}+b$}\\ 0&\mbox{otherwise}\end{array}\right.\] \[X_{m,N}^{{}^{\prime}}(x)=\left\{\begin{array}[]{cl}\cos{\left[K_ {mx}(x-x_{N})\right]}&\mbox{for $x_{N}<x<x_{N}+b$}\\ 0&\mbox{otherwise}\end{array}\right.\] (3) \[Y_{m}(y)=\sin{\left[K_{my}(y+h)\right]}\mbox{~{}~{}~{}~{}and~{}~ {}~{}~{}}Y_{m}^{{}^{\prime}}(y)=\cos{\left[K_{my}(y+h)\right]}\,,\] where $K_{mx}=m\pi/b$ for $m=0,1,2,\ldots$ and $K_{my}=+\sqrt{k_{i}^{2}-k_{iz}^{2}-K_{mx}^{2}}$, where, again, the positive solution to the square root must be taken. Similarly, $x_{N}$ denotes the $x$-coordinate of the right wall in the $N$-th groove, i.e. $x_{N}=Nd$ for $N=0,\pm 1,\pm 2,\ldots$. Both, the continuity of the $x$ and $z$ component of the electric field along the groove aperture, and the boundary conditions of electromagnetic fields on the surface of a perfect conductor (see Refs.[10, 11]), allow one to write \[E_{ix}\,e^{ik_{ix}x}+\sum_{n=-\infty}^{+\infty}{R_{ nx}\,e^{iq_{nx}x}}=\left\{\begin{array}[]{ll}\sum_{m=0}^{+\infty} {{\cal E}_{x,m}\,\,\cos{\left(K_{mx}x\right)}\,\sin{\left(K_{my}h\right)}}& \mbox{$0<x<b$}\\ 0&\mbox{$b\leq x<d$}\end{array}\right.\] and, \[E_{iz}\,e^{ik_{ix}x}+\sum_{n=-\infty}^{+\infty}{R_{ nz}\,e^{iq_{nx}x}}=\left\{\begin{array}[]{ll}i\sum_{m=0}^{+\infty }{{\cal E}_{z,m}\,\,\sin{\left(K_{mx}x\right)}\,\sin{\left(K_{my}h\right)}}& \mbox{$0<x<b$}\\ 0&\mbox{$b\leq x<d$}\end{array}\right.\] By taking the Fourier transform of these equations in the $x$ variable and after rearranging terms, one obtains \[R_{nx}=-E_{ix}\,\delta_{n,0}+\frac{1}{d}\sum_{m=0}^ {+\infty}{{\cal E}_{x,m}\,\,\widetilde{C}_{m,n}\,\sin{\left(K_{my}h\right)}}\] (6) \[R_{nz}=-E_{iz}\,\delta_{n,0}+\frac{i}{d}\sum_{m=0}^ {+\infty}{{\cal E}_{z,m}\,\,\widetilde{S}_{m,n}\,\sin{\left(K_{my}h\right)}}\] (7) where, \[\widetilde{S}_{m,n}=\int_{0}^{b}{dx\,e^{-iq_{nx}x}\,\sin{\left(K_{mx}x\right)} }\mbox{~{}~{}~{}and~{}~{}~{}}\widetilde{C}_{m,n}=\int_{0}^{b}{dx\,e^{-iq_{nx}x }\,\cos{\left(K_{mx}x\right)}}\,.\,\] (8) Similarly, the continuity of the $x$ and $y$ components of the magnetic field along the grooves aperture yields \[2\,\frac{\left(k_{iy}^{2}+k_{iz}^{2}\right)\,E_{iz}+k_{ix}k_{iz} E_{ix}}{k_{iy}}\,\exp{\left(ik_{ix}x\right)}=\] \[\sum_{m=0}^{+\infty}\left\{\left[\frac{K_{mx}k_{iz}} {K_{my}}\sin{\left(K_{mx}x\right)}\cos{\left(K_{my}h\right)}-\sin{\left(K_{my} h\right)}\,U^{(xx)}_{m}(x)\right]{\cal E}_{x,m}\right.\] \[+\left.\left[\frac{K_{my}^{2}+k_{iz}^{2}}{K_{my}}\sin{\left(K_{mx }x\right)}\cos{\left(K_{my}h\right)}-\sin{\left(K_{my}h\right)}\,U^{(xz)}_{m}( x)\right]{\cal E}_{z,m}\right\}\,,\] and, \[2\,\frac{\left(k_{iy}^{2}+k_{ix}^{2}\right)\,E_{ix}+k_{ix}k_{iz} E_{iz}}{k_{iy}}\,\exp{\left(ik_{ix}x\right)}=\] \[\sum_{m=0}^{+\infty}\left\{\left[\frac{K_{mx}^{2}+K_ {my}^{2}}{K_{my}}\cos{\left(K_{mx}x\right)}\cos{\left(K_{my}h\right)}-\sin{ \left(K_{my}h\right)}\,U^{(zx)}_{m}(x)\right]{\cal E}_{x,m}\right.\] \[+\left.\left[\frac{K_{mx}k_{iz}}{K_{my}}\cos{\left(K_{mx}x\right) }\cos{\left(K_{my}h\right)}+\sin{\left(K_{my}h\right)}\,U^{(zz)}_{m}(x)\right] {\cal E}_{z,m}\right\}\,,\] where \[U_{m}^{(xx)}(x)=\frac{1}{d}\sum_{n=-\infty}^{+ \infty}{\frac{q_{nx}k_{iz}}{q_{ny}}\,\widetilde{C}_{m,n}\,e^{iq_{nx}x}}\,, U_{m}^{(xz)}(x)=\frac{i}{d}\sum_{n=-\infty}^{+ \infty}{\frac{q_{ny}^{2}+k_{iz}^{2}}{q_{ny}}\,\widetilde{S}_{m,n}\,e^{iq_{nx}x}}\] (11) \[U_{m}^{(zx)}(x)=\frac{i}{d}\sum_{n=-\infty}^{+ \infty}{\frac{q_{nx}^{2}+q_{ny}^{2}}{q_{ny}}\,\widetilde{C}_{m,n}\,e^{iq_{nx}x }}\,, ~{}~{}U_{m}^{(zz)}(x)=\frac{1}{d}\sum_{n=-\infty}^{+ \infty}{\frac{q_{nx}k_{iz}}{q_{ny}}\,\widetilde{S}_{m,n}\,e^{iq_{nx}x}}\] It must be noticed that no equations for the $y$-components of the field are necessary since the null divergence condition, i.e. $\nabla\cdot\mathbf{E}$ = 0, implies that only two, out of the three components of both $\mathbf{E}^{(y>0)}$ and $\mathbf{E}^{(y<0)}$ are independent. Finally, the values of ${\cal E}_{x,m}$ and ${\cal E}_{z,m}$ can be found by evaluating Eqs. (2-2) over a finite set of equally spaced $x$’s along the groove aperture, and the resulting system of linear algebraic equations is then solved by resorting to the Gauss elimination algorithm in Ref.[12]. Once ${\cal E}_{x,m}$ and ${\cal E}_{z,m}$ are obtained, the reflection amplitudes, i.e. $R_{nx}$ and $R_{nz}$, can readily calculated from Eqs.(6-7). Having arrived at this point, the _enhancement coefficient_ ($\eta$) is introduced, namely, \[\eta=\frac{1}{2\,A\,\|\mathbf{E}_{i}\|^{2}}\int_{0}^{b}dx\int_{-h}^{0}dy\,\| \mathbf{E}(x,y)\|^{2}\,,\] (12) where $\|\mathbf{E}_{i}\|$ is the electric field amplitude of the incoming light, $\mathbf{E}(x,y)$ is the electric field within the groove and $A$ is the cross sectional area of the groove, i.e. $A=hb$. Notice that no integration over the $z$-coordinate is necessary since, for 1D gratings $\|\mathbf{E}(x,y)\|^{2}$ does not depend on $z$. As was previously mentioned [9], such a coefficient allows one not only to quantify the field enhancement but also, and most important, it tells when there is a net gain of electromagnetic energy within the grooves. Actually, one can readily see that $\eta$ = 1 denotes the case for which, on an average, the electrical energy density within the grooves is the same as that present above, far away from the diffraction grating. <figure><img src="content_image/1103.6282/x2.png"><figcaption>Figure 2: Enhancement coefficient for a h/b=1 (◯), 2 (△), 4 (▽) and 8 (◊)diffraction grating, illuminated by non-polarized light, uniformly distributedover a 30-degree-wide, vertical cone.</figcaption></figure> It must stressed that the main goal in this paper is the calculation of $\eta$ as well as analysing its sensitivity upon the various parameters of the model. Before going into the results, however, it must be noticed that for the purpose of exploiting the trapping of light within the grooves, one can reasonably assume that $d=b$. By doing so, the number of variables is reduced by one and $b$ can be used as the unit of length. Accordingly, apart from $\theta_{i}$, $\varphi_{i}$, $\left\|\mathbf{E}_{i}\right\|$ and $\phi$, the solutions to Eqs.(2-2) must be functions of $bk_{i}$ (or $\lambda/b$), and $h/b$. The results of numerically calculating these equations under the aforementioned assumptions are produced in the following section. It must be mentioned however, that all along the following sections, when referring to wavelength it is assumed to be that in vacuum. Only within the numerical code, the wavelength is translated into that of Silicon using the refractive index from Ref. [13]. ## 3 Results and discussions The enhancement coefficient is calculated for a 30-degree conical beam of non-polarized light. The results are plotted in figure 2, where $\eta$ appears as a function of the wavelength and for grooves of depths $h/b$=1, 2, 4 and 8. It must be mentioned that these results are obtained by calculating the mean-value of $\eta$ in Eq.(12) over the pertinent range of polarization states, incidence angles and wavelengths. This is performed by means of a Monte-Carlo integration scheme, where relative uncertainties of the order of ten percent, or less, is normally achieved. As one can readily see in figure 2, the enhancement coefficient exhibits a single large peak around $\lambda/b\approx 1.8$, becoming as large as approximately 3 for the four groove depths analysed in this paper. For $\lambda/b\leq$1.8, $\eta$ decreases and becomes slightly greater than unity for $\lambda/b\leq$ 1.2. However, one can see that $\eta$ falls below unity as soon as $\lambda/b>$2. Curiously enough, the enhancement coefficient does not exhibit a strong dependence with the groove depth $h$ and, as a matter of fact, differences between the results corresponding to the different groove depths are masked by the noisy aspect of the curves. This noise stems in part from the statistical fluctuations in the Monte-Carlo calculations, and also from the narrow resonances occurring within the groove which, as will be seen below, have been considerably reduced as a result of taking averaged values. <figure><img src="content_image/1103.6282/x3.png"><figcaption>Figure 3: Enhancement coefficient for h/b=1 (full line) and 8 (short-dashedline) diffraction gratings, illuminated by a normally incident light withelectric field directed along the (a) x\- and (b) z-axis, respectively.</figcaption></figure> In order to analyse the results in figure 2, $\eta$ is calculated for light arriving to the grating along the normal direction and for two limiting cases of polarization states, namely, those with the electric field perpendicular and parallel to the grooves, respectively. The results are plotted in figure 3, where, in order to avoid a busy plot, only the results corresponding to $h/b$=1 and 8 appear. The results in figure 3(a) clearly show that when the electric field is perpendicular to the grooves, the field enhancements are oscillating functions of $\lambda/b$, becoming scarcely larger than unity over nearly the entire range of wavelengths calculated in the present paper. On the contrary, when the electric field is directed along the $z$-axis, as shown in figure 3(b), $\eta$ exhibits a series of large peaks, regularly spaced, with an amplitude which appears to increase with increasing the wavelength. This is so, however, for $\lambda/b\leq$2, because for $\lambda/b>$2 the field enhancement falls below unity, and it does at a rate that seems to be an increasing function of the groove depth. This however is not at all unexpected, since $E_{z}$ must be zero all over the surface of the grating, therefore, a $E_{z}$-polarized light cannot penetrate within the groove as soon as its wavelength becomes comparable to, or larger than the groove width. <figure><img src="content_image/1103.6282/x4.png"><figcaption>Figure 4: Enhancement coefficient for h/b=1 (full line) and 8 (short-dashedline) diffraction gratings. Light arrives at 30 degree incidence angle (θi)upon the grating, whereas the azimuthal angle (φi) and the electric fielddirection angle (ϕ), both in degrees, are: (a) 0, 90; (b) 0, 0; (c) 90, 90;and (d) 90, 0. (see figure 1 for an explanation of these angles)</figcaption></figure> Figure 4 shows four limiting cases of beams arriving to the grating with 30 degree incidence angle ($\theta_{i}$). These comprise the TE-and TM-polarization state and, 0 and 90 degree azimuthal angles ($\varphi_{i}$). The first conclusion one may extract from these curves is that if the electric field of the light lies on a plane that is parallel to the grooves, such as the (b) and (c) cases, then, there appears a cut-off wavelength above which the field enhancement falls down below unity quite rapidly. However, if the electric field is perpendicular to the grooves, $\eta$ remains greater the unity over the entire range of wavelengths analysed in this paper. This is obvious, since no large electric fields can be developed within the grooves if they are parallel to the lateral and bottom surfaces. As was already observed elsewhere [1, 2, 7, 9], the results in figures 4(a-d) show a number of peaks, which can reach values as large as several decades. These peaks are observed to occur at wavelengths which depend on the groove depth $h$, the incidence and azimuth angles, and the state of polarization. Curiously though, such peaks are often so narrow that many of them nearly disappear after taking an average, as seen in figure 1. Finally, one may find the enhancement coefficient of the grating exposed to sunlight. In this case, one must take an average of $\eta$ over all polarisation states and wavelengths in the solar spectrum, \[\eta_{sun}=\frac{\int_{0}^{\lambda_{0m}}d\lambda_{0}\,\Phi_{E}( \lambda_{0})\,\lambda_{0}\,\eta}{\int_{0}^{\lambda_{0m}}d\lambda_ {0}\,\Phi_{E}(\lambda_{0})\,\lambda_{0}\,}\,,\] (13) where $\lambda_{0}$ is the wavelength of the light in vacuum, $\Phi_{E}(\lambda_{0})$ is the spectral flux density of the sun light, $\eta$ is the enhancement coefficient for a given wavelength and after averaging over all polarisation states, and $\lambda_{0m}$ is the largest wavelength of the light which can promote electrons from the valence to the conduction band. Notice that the factor $\lambda_{0}$ appearing in both integrals is required in order to obtain the photon density spectrum from $\Phi_{E}(\lambda_{0})$, which is obtained from the so-called Reference AM 1.5 spectra in [14]. It must be also noticed that, since the wavelength in vacuum is used, the dimensions of the grating must be scaled using the refractive index $n$ of the medium that fills the groove. For the sake of simplicity, such an index is assumed to be constant along the solar spectrum and absorption is ignored, as a consequence $\eta_{sun}$ will be a function of both the optical spacing of the grooves, i.e. $dn$, and the aspect ratio $h/b$. Equation (13) is calculated using the Monte Carlo method, and the results are plotted in figure 5. There, one can see that the enhancement coefficient becomes larger than unity for $dn$ approximately grater than 500 nm, whereas for $nd$ smaller than this value, $\eta_{sun}$ drops down fairly fast. For optical spacing between the grooves greater than 500 nm the enhancement coefficient becomes as large as 1.2 and stays around this value up to $dn$=3600 nm, which is the largest $dn$-value calculated in this paper. Remarkable though, in a similar fashion as was previously observed in figure 2, $\eta_{sun}$ appears to nearly not to depend on the aspect ratio $h/d$. ## 4 Summary and concluding remarks <figure><img src="content_image/1103.6282/x5.png"><figcaption>Figure 5: Enhancement factor of the diffraction grating under AM 1.5 solarirrandiance assuming normal incidence. Calculations are performed for severalvalues of the optical spacing of the grooves, i.e. dn, and two aspect ratiosh/d=1 (◯) , and 8 (◊).)</figcaption></figure> The performance of a diffraction grating as a light trapping structure for PV cells applications is analysed. To this end, the electric fields produced within the grooves of a perfectly conducting, diffraction grating by a well collimated, monochromatic linearly-polarized beam of light are calculated. The grating has a $d$-period and is assumed to be made of infinitely long rectangular-shaped, $h$-deep and $b$-wide grooves. Although, for the purpose in this paper it is assumed that $d\approx b$. Furthermore, in order to properly assess the grating, an _enhancement coefficient_ ($\eta$) is introduced. $\eta$ is defined as the average electromagnetic-energy within the volume of the grooves relative to that one will have if grating is replaced by a flat perfect reflector. In this regard, a $\eta$ greater than unity implies that light is being trapped within the grooves of the diffraction grating. Results in this paper agree with previous calculations [1, 2, 3, 4, 7] that, for certain polarization state, wavelength and incidence angle, $\eta$ can be substantially larger than unity. When using conical, non-polarized beams, such enhancements however are reduced and $\eta$ can hardly be larger than three. Moreover, these values are observed for wavelength in the propagating medium within the range $b<\lambda<2b$. For $\lambda<b$, $\eta$ appears to be slightly greater than unity irrespective of the polarization state and incidence angle, provided this later is not greater than 30 deg. For $\lambda>2b$ however, the enhancement coefficient seems to be always smaller then unity. This is particularly so, when the electric field of the incoming light is parallel to the groove direction. Finally, the results of calculating the enhancement coefficient of the diffraction grating under the so-called Reference AM1.5 solar spectrum, namely $\eta_{sun}$, show that $\eta_{sun}$ can be, in the best of the cases, of the order of 1.2 . This maximum occurs for optical spacing of the groove $dn$ approximately equal to 500 nm, whereas for $dn\leq$ 500 nm $\eta_{sun}$ becomes negligible small. In all the cases, $\eta_{sun}$ does not seem to depend on the aspect ratio $h/b$. It is worth mentioning that, although the structure of the ideal cell sketched in this paper is different to that in Ref.[8], the figures reported by these Authors are similar to those in the present calculations. In short, diffraction grating may conceivably act as a light trapping structure and, consequently, increase the efficiency of a PV cell, figures appear to be around 20% for solar spectrum and groove spacing is expected to be in the sub-micron scale. ### Acknowledgements We are indebted to I. Tobias, A. Martí and A. Luque for encouraging the Authors to work on this problem. This work has been supported in part by the European Union Program IBPOWER (211640) and the NUMANCIA II project funded by the Comunidad Autónoma de Madrid. ## References * [1] N.E. Glass and A. A. Maradudin, “Diffraction of light by a bigrating: Surface polarization resonances and electric field enhancements”. Phys. Rev., B27, 5150–5153 (1983). * [2] A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light”. Phys. Rev., B31, 5573–5576 (1985). * [3] H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region”. Applied Optics, 32, 3459–3465 (1993). * [4] R. A. Depine and D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves”. J. Opt. Soc. Am., A11, 2844–2850 (1993). * [5] D. Crouse, “Numerical Modeling and Electromagnetic Resonance Modes in Complex Grating Structures and Optoelectronic Device Applications”. IEEE Trans. Electronic Devices, 52, 2365–2373 (2005). * [6] Y.-C. Lee, C.-F. Huang, J.-Y. Chang, and M. L. Wu, “Enhanced light trapping based on guided mode resonance effect for thin-film silicon solar cells with two filling factor gratings”. Optics Express, 16, 7969 – 7975 (2008). * [7] F. Llopis, I. Tobías and M. M. Jakas, “Light intensity enhancement inside the grooves of metallic gratings”. J. Opt. Soc. Am. , B27, 1198–2006 (2010). * [8] W. Wang, S. Wu, K. Reinhardt, Y. Lu and S: Chen, “Broadband Light Absorption Enhancement in Thin-Film Silicon Solar Cells”. Nano Lett. , 10, 2012–2018 (2010). * [9] M. M. Jakas and F. Llopis, “Light trapping within the grooves of diffraction gratings”. Paper presented at _29th Progress in Electromagnetics Research Symposium_ Marrakesh, Morocco, March 20-23, 2011, available on the web site: (after April 30, 2011) http://piers.org/piersproceedings/ * [10] J. A. Kong, _Electromagnetic Wave Theory_, (John Wiley & Sons, New York 1986). * [11] J. D. Jackson, _Classical electrodynamics_, (John Wiley & Sons, New York 1962). * [12] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, _Numerical Recipes in Fortran 77: The Art of Scientific Computation_, ( Cambridge University Press, Cambridge 1992). * [13] H. R. Philipp, and E. A. Taft, “Optical Constants of Silicon in the Region 1 to 10 ev” Phys. Rev. 120, 37–38 (1960). See also: http://www.ioffe.ru/SVA/NSM/Semicond/Si/optic.html * [14] The solar spectral irradiance are obtained by fitting an analytical expression to data for Air Mass 1.5, available from: http://rredc.nrel.gov/solar/spectra/am1.5/
1003.3036	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 33888, "num_imgs": 7, "llama3_tokens_count": 7834 }	[ "content_image/1003.3036/x1.png", "content_image/1003.3036/x2.png", "content_image/1003.3036/x3.png", "content_image/1003.3036/x5.png", "content_image/1003.3036/x7.png", "content_image/1003.3036/x8.png", "content_image/1003.3036/x9.png" ]	# Investigating the Morphological Categories in the NeuroMorpho Database by Using Superparamagnetic Clustering Krissia Zawadzki${}^{1}$, Mauro Miazaki${}^{1}$ and Luciano da F. Costa${}^{1}$${}^{2}$ ${}^{1}$ Institute of Physics at Sao Carlos - University of Sao Paulo, Avenue Trabalhador Sao Carlense 400, Caixa Postal 369, CEP 13560-970, Sao Carlos, Sao Paulo, Brazil ${}^{2}$ National Institute of Science and Technology for Complex Systems, Brazil krissia.zawadzki@usp.br mauro@ursa.ifsc.usp.br luciano@ifsc.usp.br ###### Abstract The continuing neuroscience advances, catalysed by multidisciplinary collaborations between the biological, computational, physical and chemical areas, have implied in increasingly more complex approaches to understand and model the mammals nervous systems. One particularly important related issue regards the investigation of the relationship between morphology and function of neuronal cells, which requires the application of effective means for their classification, for instance by using multivariated, pattern recognition and clustering methods. The current work aims at such a study while considering a large number of neuronal cells obtained from the NeuroMorpho database, which is currently the most comprehensive such a repository. Our approach applies an unsupervised clustering technique, known as Superparamagnetic Clustering, over a set of morphological measurements regarding four major neuronal categories. In particular, we target two important problems: (i) we investigate the coherence between the obtained clusters and the original categories; and (ii) we verify for eventual subclusters inside each of these categories. We report a good agreement between the obtained clusters and the original categories, as well as the identification of a relatively complex structure of subclusters in the case of the pyramidal neuronal cells. , , ## 1 Introduction The progress in neuroscience [1] aimed at understanding the complexity of the nervous system has continuously stimulated interdisciplinary integration between several scientific fields, such as biology, computer science [2], chemistry, and engineering. Despite the advances in these fields, there are still many remaining challenges. In this context, studies focused on morphological characterization and classification of neuronal cells have contributed substantially to enhance the neuroscientific knowledge about the relations between neuronal shape and physiology [3, 4, 5]. These discoveries are invaluable for several related areas, such as compared anatomy, neurobiology and diagnosis. The relationship between neuronal shape and function was first suggested and investigated by Cajal [6]. Subsequently, the attention from the neuroscientists shifted to electrophysiological analysis, which focused on neuronal electrical responses to stimulus. However, evidences about morphophysiological relationships have accumulated, such as the correspondence between morphological and physiological classifications of retinal ganglion cells [7, 8]. At the same time, advances in scientific visualization and shape analysis [9] paved the way to more comprehensive and sophisticated morphological approaches, giving rise to the area of computational neuromorphometry, whose aim is to quantify and study geometrical features of neurons [10]. The onset of the free-content initiatives, such as in software, artistic works, and research papers, have motivated the creation and expansion of public databases. Indeed, a free server of data allows researchers of several fields to have immediate access to the raw materials they need in their studies. The use of these databases also facilitate the replication of experiments and results, as anyone can access the same data [11]. Another important point is the prevention of data loss, as the servers cater for backup and data maintenance. Currently, the largest repository of neuronal cells is NeuroMorpho [12]. It contains not only the complete geometrical representation of the cells, but also several morphological measurements and data information, such as cell type, species, region, staining method, etc. The cell categories appearing in such databases frequently take into account both morpho or eletrophysiological properties. However, there is no consensus in the classification of neurons, to the extent that reclassifications of cells have been reported periodically (e.g. [13]). The difficulties in obtaining a more definitive taxonomy are, ultimately, a consequence of the incipient situation of neuromophological research. To begin with, there is no established set of geometrical measurements which should be used [10]. In addition, the lack of a representative number of cells allied to the choice of specific stochastic and classification methods also tends to yield varying taxonomies. Another important problem regarding the classification of neuronal cells is the identification of meaningful subcategories. In this context, the availability of significant amounts of neurons with their morphological properties in the NeuroMorpho repository motivated an underlying and more systematic study of classification based on morphometric features. Though several categories of neuronal cells have been traditionally adopted in the literature [3, 14, 15, 5, 10], as a consequence of the largely subjective and incomplete methods used for their definition, it is not clear how homogeneous these classes are. Therefore, the investigation of the cell distribution within each of the main classes provides a particularly important issue. Especially for the more heterogeneous cases, it is possible that the existing categories are composed of subcategories which have been overlooked as a consequence of the subjectiveness and coarseness of the previous applied measurements and classification methods. The current work sets out to investigate this important issue, namely by investigating the uniformity and possible presence of subcategories inside well-established morphological categories. In this work, we explore the morphological measurements in NeuroMorpho by using a established classification procedure of statistical physics, known as Superparamagnetic Clustering (SPC) [16]. This method is inspired in a natural magnetic phenomenon presented by materials due to temperature variations. Described by non-homogeneous Potts model, where spins at the same state are grouped together, this physical phenomenon can be applied to data clustering applications. We used a software available in VCCLAB [17]. We chose four large categories of neurons according to the cell type-region classification and clustered them. The results were then compared to the original classification. The main objective in this work is to compare the obtained clusters with the original classification in the repository, checking the agreement between the original categories and our clustering. Since SPC is an unsupervised method, it will explore data and find clusters analysing their features without any subjective judgment. This approach can either confirm the strength of the established classification or reveal unnoticed subcategories and problems in classifications. Our analysis presents interesting results regarding the classification of the selected categories as well as their homogeneity, leading to the necessity to investigate the possible factors which are responsible to the observed subcategories and suggesting a new classification based in the inner relations in classes that exhibits this behaviour. This article starts by presenting the adopted database, followed by the morphological concepts relative to the measurements used in the neuronal characterization. Afterwards, the theoretical method of SPC and the software used to analyse the NeuroMorpho data are explained. Next, the results are presented and discussed. ## 2 Materials and Methods ### NeuroMorpho Database NeuroMorpho [18] is a contributory database containing information about digitally reconstructed neuronal cells, collected by laboratories worldwide. Publicly available, this data is intended for researches working on issues such as studies of neuronal system complexity, visualization and neuronal modeling. The maintenance of the database is provided by the Computational Neuroanatomy Group of the Krasnow Institute for Advanced Study, from George Mason University. This initiative is part of the Neuroscience Information Framework project [19], endorsed by the Society for Neuroscience, and includes institutes such as Cornell Univ., Yale Univ., Stanford Univ., and Univ. of California. The database has been periodically updated. Currently, it contains data relative to 5673 neurons (version 4.0, released in 02/16/2010). We can access the original and standardized cell morphological reconstructions, as well as several data and properties such as cell type, species, brain region, animal width and weight, development age, gender, used reconstruction methods, magnification, date of upload, images of spatial structures of neurons, and references to the related literature. In this work, we separated the data according to the cell type and brain region and selected the most numerous classes from the database. This yelded the following categories: * Pyramidal cells from Hippocampus (Pyr-Hip); * Medium Spiny cells from Basal Forebrain (Spi-Bas); * Ganglion cells from Retina (Gan-Ret); * Uniglomerular Projection neurons from Olfactory Bulb (Uni-Olf). Table 1 presents the distribution of species within each type-region category. Since this is a public database, all data used in this article is available and easily accessible through the NeuroMorpho website. The original category names are the same as found in the database. Species \| Spi-Bas \| Pyr-Hip \| Uni-Olf \| Gan-Ret ---\|---\|---\|---\|--- Rat \| 232 \| 209 \| 0 \| 0 Mouse \| 1 \| 0 \| 0 \| 181 Salamander \| 0 \| 0 \| 0 \| 64 Drosophilla \| 0 \| 0 \| 233 \| 0 Table 1: Number of species in each type-region category. ### Measurements <figure><img src="content_image/1003.3036/x1.png"><figcaption>Figure 1: NeuroMorpho measurements.</figcaption></figure> In order to study the morphology of neurons, it is necessary to represent and characterize them in some way suitable for processing and analysis. NeuroMorpho provides L-Measure [20], a tool to extract several measurements from the neurons in the database. The measurements used in this work are illustrated in Figure 1, numbered from 1 to 19 and named as in the software documentation. The concepts of compartment, branch and bifurcation are illustrated in Figure 1. Compartments are segments represented as cylinders with diameter and extremity points coordinates. Branches are formed with one or more compartments between the soma, the bifurcations and the tips. Bifurcations are points where a branch splits into two other branches. Measurements 1, 2 and 3 are the height, width and depth of a neuron, calculated after its alignment along the principal axis using PCA. The number of bifurcations and branches in a neuron correspont to the measurements 4 and 5. The features related to the compartment are from 6 to 9, respectively: diameter, length, surface area and volume. The branches have their associated measurements numbered from 10 to 14. Measure 10 is the Euclidean distance between a compartment and the soma, while the path distance (11) is the sum of the lengths of the compartments between two endpoints. Contraction (12) is the ratio between the Euclidean distance and its path distance. Measure 13 is the order of the branch regarding the soma, which has order 0. The branches attached to the soma have order 1. The branches connected to these branches have order 2, and so on. Fragmentation (14) is the number of compartments in a branch. Only compartments between bifurcations or between a bifurcation and a tip are considered. Measurement 15 is the soma surface area. The soma can be of two types: a sphere or a set of compartments. In the latter case, the area is calculated as the sum of the area surfaces of the soma compartments. The other measurements are related to bifurcations. Pk_classic (16) is the ratio $\frac{d_{1}^{r}+d_{2}^{r}}{b^{r}}$, where $r$ is the Rall’s power law value, set in this measure as $1.5$, and $b$, $d_{1}$ and $d_{2}$ are the diameters of the bifurcation compartments (the parent and the two daughters, respectively). The partition asymmetry (17) considers the number of tips on the left and on the right daughter subtrees of a bifurcation as $n1$ and $n2$ in the expression $\frac{\|n1-n2\|}{n1+n2-2}$. In Figure 1, the analysed bifurcation has vertical stripes, while the left daughter subtree has horizontal stripes and the right one has a pattern of squares. Then, in this example, $n1=3$ and $n2=2$ gives $\frac{\|3-2\|}{3+2-2}=0.33$. Measure 18 is the calculation of the angle between two daughter compartments in a bifurcation, while measure 19 is the angle regarding the endpoints of two daughter branches. Table 2 shows the mean values of the measurements in each category. Measurement \| Spi-Bas \| Pyr-Hip \| Uni-Olf \| Gan-Ret ---\|---\|---\|---\|--- Height \| 167.43 \| 614.88 \| 51.78 \| 249.11 Width \| 182.23 \| 576.74 \| 117.13 \| 247.25 Depth \| 66.94 \| 353.68 \| 56.54 \| 21.01 N. Bifurcations \| 8.59 \| 99.29 \| 25.42 \| 60.34 N. Branches \| 24.79 \| 209.06 \| 52.17 \| 131.47 Diameter \| 0.78 \| 0.71 \| 1.07 \| 0.75 Length \| 1266.38 \| 23654.53 \| 499.74 \| 4339.40 Surface \| 2958.64 \| 36941.94 \| 1764.18 \| 9305.34 Volume \| 999.01 \| 9826.63 \| 655.88 \| 2455.30 Euclid. Dist. \| 179.08 \| 974.86 \| 112.58 \| 218.86 Path Dist. \| 224.35 \| 2727.95 \| 158.07 \| 291.82 Contraction \| 0.86 \| 0.85 \| 0.92 \| 0.86 Branch Order \| 3.28 \| 18.96 \| 12.35 \| 8.97 Fragmentation \| 1003.34 \| 3311.95 \| 446.99 \| 2629.73 Soma Surface \| 715.14 \| 1005.75 \| 0.00 \| 780.84 Pk Classic \| 1.30 \| 1.60 \| 1.77 \| 1.62 Part. Asymmetry \| 0.45 \| 0.54 \| 0.60 \| 0.49 Local Bif. Ampl. \| 63.55 \| 69.76 \| 93.24 \| 82.75 Remote Bif. Ampl. \| 56.30 \| 55.24 \| 91.62 \| 68.52 Table 2: Mean of the measurements in each type-region category. ### Superparamagnetic Clustering Method The superparamagnetic method is a clustering procedure based on a physical model of a real material exhibiting magnetic response to some external parameter. Different from classical approaches, which are restricted to statistical and mathematical analysis of the system, SPC allows to evaluate how efficient is the grouping in terms of its intrinsical properties and has as additional advantages insensitivity to the initial conditions and robustness to noise. Therefore, it is necessary to understand the theoretical foundation of the physical concepts underlying the approach, and how it can be applied to data clustering. The technique by Blatt, Wiseman and Domany [16] suggests an analogy with the generalized Ising model, also known as non-homogeneous Potts model. In this method, the temperature acts as a parameter controlling the spin configurations of a two-dimesional atoms network. It is possible to identify three reactions of the material in response to temperature variations: low, medium and high temperature intensities. They characterize the respective ferromagnetic, paramagnetic and superparamagnetic behaviour. The obtained arrangements of spin orientation is understood as defining the clustering structure of the data. The Potts model gives a reference to the energy $E$ of the configuration, so that stability requires low energy: \[\mathcal{H}(s)=-J\sum_{<i,j>}^{N}x_{i}x_{j}\] (1) The transition between the magnetic phases due to variation of temperature can be characterized by the susceptibility $\chi$, a physical parameter that reflects the respective overall magnetization of the system relative to the number of spins with a value between 1 and $q$, calculated as: (2) where the variance of magnetization is defined as: \[m=\frac{(N_{max}/N)q-1}{q-1}\] (3) The lowest energy probabilistic distribution of the system states requires predominance of configurations with similar spins for low temperatures and strong interactions. This distribution is expressed as: \[P(s)=\frac{1}{Z}\exp\left(\frac{\mathcal{H}(s)}{T}\right)\] (4) where $Z$ is a normalization constant: \[Z=\sum_{S}\exp\left(\frac{-\mathcal{H}(s)}{T}\right)\] (5) In order to apply these concepts to data sets and optimize the execution of the algorithm, the Swendsen-Wang method was adopted. The motivation of this approach is to analyse different configurations of the system based on spin neighborhood interactions. The method proceeds as described in the following. First, we assume a data set containing $N$ variables $x_{i}$ whose $d$ components are measurements of the system features. In analogy to atoms in a two-dimensional network, we assign a random state $s_{i}$ among the $q$ possibilities to each respective point $x_{i}$. The ranges and steps of temperature variation are predefined in order to choose the number of interactions in which we want to locate the superparamagnetic behaviour of system. Then, we analyse the probability (4) of connection between the sites with the same spin based on the mutual interaction and neighbourhood criterion. The latter suggests a maximum number $K$ such that the interaction $J_{ij}$ between $x_{i}$ and $x_{j}$ is computed only if they are K-nearest neighbors of each other. This interaction is inversely proportional to the average nearest-neighbor distance $a$ and is mathematically defined as: \[J_{ij}=\frac{1}{K}\exp\left(-\frac{d_{ij}}{2a}\right)\] (6) By changing the configuration, we estimate the physical parameters of magnetization (3) and the susceptibility (2). This is repeated until the number $M$ of interactions is reached. The threshold temperatures $T_{fs}$ and $T_{ps}$, for which the system is in the superparamagnetic phase, can be located by identifying the points of maximal susceptibility and sharp decrease, respectively. In the transition to paramagnetic regime a guess to $T_{ps}$ can be $T=\exp^{1/2}/4\ln(1+\sqrt{q})$. Aimed at quantifying the ordering properties of the new system configuration, we need to estimate the spin-spin correlation $G_{ij}=\frac{(q-1)\hat{C_{ij}}+1}{q}$, given in Equation 7, which is typically performed by a Monte Carlo procedure. \[\hat{C}_{ij}={\sum_{\ell=1}^{M}\frac{I_{ij}(\ell)}{M}}\] (7) where $I_{ij}=1$ if the points are at the same group or $0$ otherwise. So, it is necessary to identify the Swendsen-Wang groups with connected sites, construct the data clusters, compute the magnetization average $\left\langle\bar{m}\right\rangle=\frac{N_{max}}{N}$ and then repeat the above procedure until the maximal range of temperature is reached. We can verify the different regimes of the superparamagnetic phase through the susceptibility measure. After locating $T_{fs}$ and $T_{ps}$, we can analyse the superparamagnetic sub-phases assuming their mean temperature $T_{clus}=(T_{fs}+T_{ps})/2$ as the point where the clusters are formed. The SPC algorithm follows all these steps and concepts, and has many applications. Currently, there are effective and optimized implementations available on the web, such as Tetko’s program [21]. We used this software in the current article, described in the next section. ### SPC software For our data analysis, we used the SPC software implemented by Tetko [21]. It is developed in Java, runs on-line as an applet and is freely available through the VCCLAB (Virtual Computational Chemistry Laboratory) web site at $<$http://www.vcclab.org/lab/spc$>$. VCCLAB aims to provide free on-line tools to analyse chemical data [17]. In order to satisfy the required input format, we calculated the Euclidean distance between all data points and saved them in a text file to upload into the SPC program. We used all parameters in the default configuration, since these values are recomended by the author. The output is analysed and discussed in the following section. ## 3 Results and Discussions In order to compare the results of SPC and other approaches of morphological analysis, we applied PCA and LDA to the set of neuronal cells, which was selected based in the cell type and brain region. We also verified the agreement between each of the obtained clusters and the original categories. <figure><img src="content_image/1003.3036/x2.png"><figcaption>Figure 2: Evolution of the susceptibility parameter to each type-region.Observe the points of maximal and decreasing with which we can identify thesuperparamagnetic regime.</figcaption></figure> In both PCA and LDA, the Pyr-Hip cells are scattered, as we can see in Figures 3 and 4, respectively. This suggests that these neurons exhibit morphological features overlapping the other categories, instead of presenting more homogeneous characteristics which would otherwise imply in their separation from the other groups. The most homogeneous category is the Spi-Bas, which appears as a little, compact region in both methods. The Uni-Olf seems to have a more central position, intermediate to the other clusters. This behaviour is also observed for the Gan-Ret, whose location in LDA shows it to be distinct from the latter category. <figure><img src="content_image/1003.3036/x3.png"><figcaption></figcaption></figure> <figure><img src="content_image/1003.3036/x5.png"><figcaption></figcaption></figure> In order to analyse the internal composition of the categories, we applied the SPC method in each one of them and verified the formation of inner clusters by using PCA (Figure 5) and LDA (Figure 6) approaches. The Pyr-Hip category revealed more heterogeneity, splitting in more clusters than the others during the SPC process (five clusters at the mean temperature $T_{clus}=0.07$). This category also presented a susceptibility curve with a longer superparamagnetic phase, characterizing a behaviour different from those observed for the other classes. On the other hand, the Spi-Bas category presented three clusters not too far from each other and a big and sparse set of non-clustered cases. In the LDA, the homogeneity is indicated by the fact that the respective individuals appeared compacted in a specific region. Regarding the susceptibility curve, its main difference from the other classes is that it has only one peak at the superparamagnetic phase. In the Gan-Ret category, we observed a single cluster in the superparamagnetic process, and two peaks of susceptibility with approximate values, relatively lower than the peaks on the other classes. The resulting clusters agree with the multivariated methods, especially in a region which overlaps with other categories. We can also see many non-clustered individuals. The Uni-Olf category presented the most characteristic susceptibility curve, giving rise to two clusters in the obtained distribution. The points are relatively sparse, and a more numerous subcategory can be seen, which is surrounded by many unclassified cases and another small cluster. Another interesting approach regarding the distribution of the neuronal cells and their categories concerns the investigation about the relationship between type-region and species, as well as the obtained clusters. This study was done for both PCA and LDA (Figures 3 and 4) and compared with the direct references of the data set. We identified the species of the selected cells, finding them to correspond to rats, mice, salamanders and drosophillas. So, we count the number of these animals in each category (see Table 1) in order to verify the possible agreement with the results yielded by LDA and PCA. In most part of the measurements, it is possible to verify that the Pyr-Hip category has a significative higher mean value (see Table 2). <figure><img src="content_image/1003.3036/x7.png"><figcaption>Figure 5: SPC clusters of each category, visualized with PCA: (a) Spi-Bas, (b)Uni-Olf, (c) Gan-Ret, and (d) Pyr-Hip. In each pair of graphics, the left-handgraph shows the components 1 and 2, and the right-hand one exhibits thecomponents 1 and 3.</figcaption></figure> <figure><img src="content_image/1003.3036/x8.png"><figcaption>Figure 6: SPC clusters of each category, visualized with LDA: (a) Spi-Bas, (b)Uni-Olf, (c) Gan-Ret, and (d) Pyr-Hip. In each pair of graphics, the left-handgraph shows the components 1 and 2, and the right-hand one exhibits thecomponents 1 and 3.</figcaption></figure> In order to check about possible influences of the original data properties (e.g. researcher, staining method, etc) on the clusters obtained in the case of the Pyr-Hip cells, we visualized the PCA and LDA multivariated projections marked accordingly to these properties. This category presented five well defined and distinguished clusters that could indicate influence of the original data properties or new distinct subcategories. The available information about the data is: researcher who provided the data, animal strain, minimum age, maximum age, age scale, gender, minimum weight, maximum weight, development, secondary brain region, tertiary brain region, original format, experimenting protocol, staining method, slicing direction, slice thickness, objective type, magnification, reconstruction method, date of deposition and date of upload. In Figure 7, we show only the properties for which we found some relationship with the clusters. The unclustered elements were eliminated and the data was reprojected with LDA, in order to obtain a better visualization of the clusters for this analysis (Figure 7(a)). <figure><img src="content_image/1003.3036/x9.png"><figcaption>Figure 7: LDA visualization of the clusters found by SPC in the Pyr-Hipcategory (a), and the original data properties: researcher (b), strain (c),development (d), staining method (e), and reconstruction method (f). In eachpair of graphics, the left-hand graph shows the components 1 and 2, and theright-hand one exhibits the components 1 and 3.</figcaption></figure> Cluster \| Number of elements ---\|--- Blue square \| 111 Red diamond \| 29 Green square \| 25 Magenta x \| 18 Cyan star \| 6 Table 3: Pyr-Hip subclusters size. Table 3 shows the number of elements in the clusters of the Pyr-Hip category. The blue square cluster concentrates more than half of the elements, being underlain by a variety of property classes. On the other hand, the small cyan star cluster always presented its elements stable in the same property class through different information properties, but did not distinguish from the others since its classifications were never exclusive. The green square cluster is formed by the data from researchers (Figure 7(b)): Larkman (black cross), Barrionuevo (green squares) and part from Turner (red triangles). This cluster did not lead to a unique and concise classification. The neuronal data produced by the researcher Gulyas constitute almost all elements in the red diamond cluster and has unique classification in the original data properties: staining method (classified as Biotinylated dextran amine - Figure 7(e)) and reconstruction method (classified as Arbor - Figure 7(f)). In the strain category (Figure 7(c)), the Gulyas data shares its classification with the Spruston data. Both are classified as using the strain ’Wistar Rat’. All but one of the four Spruston elements are in the red diamond cluster. The magenta x cluster comprises all old rats, as we can see in the development data graphic in Figure 7(d). But, it also includes three exceptions (young rats). Examining the data, it was possible to find out that these old rats were 24 months old, while the young exceptions were 1 day old. Thus, although it gets all old rats, there are exceptions at the other side of the age cluster. Generally, we found no clear correspondence with any of those a priori properties. The fact that the obtained clustering structure could not be clearly explained by the original data properties suggests that the obtained subclusters are a consequence of some intrinsic morphological variation among the considered cells, possibly implying the definition of new categories. ## 4 Conclusion Among several challenges in neuroscience, the morphophysiological relationship of the neuronal cells has figured as an enhanced approach in order to establish the characterization and classification of these structures. Currently, the development of these studies have taken advantage of information avaliable in public databases, allowing the application of many pattern recognition and clustering methods. The problem has been continuosly investigated, leading to new techniques to classify a data set. The theoretical bases of these procedures can be established in physical real models, as Superparamagnetic Clustering. Aimed at investigating the clustering structures of the neuronal cells, we extracted many morphological measurements from NeuroMorpho, which is the largest repository currently, and compared the results of multivariated and clustering methods in the most numerous categories, whose analysis was performed considering all category elements and individual categories. In the first case, our purpose was to verify for agreement between the original classification and the categories obtained by the PCA and LDA. Afterwards, we isolated each selected category in order to locate internal clusters and the respective information that could explain their organization. The Pyr-Hip cells seemed to form the most heterogeneous category in both PCA and LDA results, in which remained sparse, as well as in the SPC results, where it had the higher number of clusters. The Gan-Ret category was located as intermediate among other categories and presented two species, despite the result of the SPC that revelead a single cluster surrounded by many unclassified cases. Given that the subclusters obtained for the Pyr-Hip category could not be explained by specific features of the original cells, we understand that they potentially imply in a revision of the current classification in order to account for possible new types of neuronal cells. New approaches can be used to complete these results, expanding the analysis to more properties, performing correlations and eliminating redundancy between the measurements, as well as the application of other clustering methods, such as the hierarchical Ward method, motivating further studies. ## 5 Acknowledgments Mauro Miazaki thanks FAPESP (07/50988-1) for financial support. Luciano da F. Costa is grateful to FAPESP (05/00587-5) and CNPq (301303/06-1) for sponsorship. ## References * [1] E. R. Kandel, J. H. Schwartz, and T. M. Jessel. _Principles of neural science_. McGraw-Hill, New York, 2000. * [2] T. Trappenberg. _Fundamentals of Computational Neuroscience_. Oxford University Press, USA, 2002. * [3] R. H. Masland. Neuron cell types. _Curr. Biology_, 14:497–500, 2004. * [4] Q. Wen and D. B. Chklovskii. A cost-benefit analysis of neuronal morphology. _J. Neurophysiology_, 99:497–500, 2008. * [5] M. Botaa and L. W. Swanson. The neuron classification problem. _Brain Research Reviews_, 56:79–88, 2007. * [6] S. R. Cajal. _Recollections of my life_. MIT Press, Massachussets, 1989. * [7] B. B. Waessle, H. Boycott and R. B. Illing. Morphology and mosaic of on- and off-beta cells in the cat retina and some functional considerations. _The Royal Society_, 212:177–195, 1981. * [8] H. Waessle and L. Peichl. The structural correlate of the receptive field centre of alpha ganglion cells in the cat retina. _J. Physiology_, 341:309–324, 1983. * [9] C. R. Hosking and J. L. Schwartz. The future’s bright: Imaging cell biology in the 21st century. _Trends in Cell Biology_, 9(11):553–554, 2009. * [10] L. da F. Costa. Computer vision based morphometric characterization of neural cells. _Review of Scientific Instruments_, 66:3770–3773, 1995. * [11] A. E. Dashti, S. Ghandeharizadeh, J. Stone, L. W. Swanson, and R. H. Thompson. Database challenges and solutions in neuroscienfic applications. _NeuroImage_, 97:97–115, 1997. * [12] NeuroMorpho.Org. * [13] L. da F. Costa and T. J. Velte. Automatic characterization and classification of ganglion cells from salamander retina. _J. Comp. Neurol._, 404:33–51, 1999. * [14] J. E. Cook. Getting to grips with neuronal diversity. what is a neuronal type? _Plenun Press_, pages 91–120, 1998. * [15] J. Stone. _Parallel processing in the visual system: the classification of retinal ganglion cells and its impact on the neurobiology of vision_. Plenum Press, New York, 1983. * [16] M. Blatt, S. Wiseman, and E. Domany. Superparamagnetic clustering of data. _Physical Review Letters_, 76(18):3251–3254, 1996. * [17] I. V. Tetko, J. Gasteiger, R. Todeschini, A. Mauri, D. Livingstone, P. Ertl, V. A. Palyulin, E. V. Radchenko, N. S. Zefirov, A. S. Makarenko, V. Y. Tanchuk, and V. V. Prokopenko. Virtual computational chemistry laboratory – design and description. _J. Comput. Aid. Mol. Des._, 19:453–63, 2005. * [18] G. A. Ascoli, D. E. Donohue, and M. Halavi. NeuroMorpho.Org: a central resource for neuronal morphologies. _Journal of Neuroscience_, 27(35):9247–9251, 2007. * [19] M. Halavi, S. Polavaram, D. E. Donohue, G. Hamilton, J. Hoyt, K. P. Smith, and G. A. Ascoli. NeuroMorpho.Org implementation of digital neuroscience: Dense coverage and integration with the NIF. _Journal of Neuroinformatics_, 6(3):241–252, 2008. * [20] R. Scorcioni, S. Polavaram, and G. Ascoli. L-Measure: a web-accessible tool for the analysis, comparison and search of digital reconstructions of neuronal morphologies. _Nature Protocols_, 3:866–76, 2008. * [21] I. V. Tetko, A. Facius, A. Ruepp, and H.-W. Mewes. Super paramagnetic clustering of protein sequences. _BMC Bioinformatics_, 6:82, 2005.
1501.01640	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 57306, "num_imgs": 9, "llama3_tokens_count": 14839 }	[ "content_image/1501.01640/x1.png", "content_image/1501.01640/x2.png", "content_image/1501.01640/x3.png", "content_image/1501.01640/x4.png", "content_image/1501.01640/x5.png", "content_image/1501.01640/x6.png", "content_image/1501.01640/x7.png", "content_image/1501.01640/x8.png", "content_image/1501.01640/x9.png" ]	# Characterizing the Brown Dwarf Formation Channels From the Initial Mass Function and Binary-Star Dynamics Ingo Thies¹ , Jan Pflamm-Altenburg¹ , Pavel Kroupa¹ , Michael Marks¹ [FOOTNOTE:1][ENDFOOTNOTE] ###### Abstract The stellar initial mass function (IMF) is a key property of stellar populations. There is growing evidence that the classical star-formation mechanism by the direct cloud fragmentation process has difficulties reproducing the observed abundance and binary properties of brown dwarfs and very-low-mass stars. In particular, recent analytical derivations of the stellar IMF exhibit a deficit of brown dwarfs compared to observational data. Here we derive the residual mass function of brown dwarfs as an empirical measure of the brown dwarf deficiency in recent star-formation models with respect to observations and show that it is compatible with the substellar part of the Thies-Kroupa IMF and the mass function obtained by numerical simulations. We conclude that the existing models may be further improved by including a substellar correction term that accounts for additional formation channels like disk or filament fragmentation. The term “peripheral fragmentation” is introduced here for such additional formation channels. In addition, we present an updated analytical model of stellar and substellar binarity. The resulting binary fraction and the dynamically evolved companion mass-ratio distribution are in good agreement with observational data on stellar and very-low-mass binaries in the Galactic field, in clusters, and in dynamically unprocessed groups of stars if all stars form as binaries with stellar companions. Cautionary notes are given on the proper analysis of mass functions and the companion mass-ratio distribution and the interpretation of the results. The existence of accretion disks around young brown dwarfs does not imply that these form just like stars in direct fragmentation. Subject headings:binaries: general — brown dwarfs — methods: numerical — methods: statistical — stars: low-mass — stars: luminosity function, mass function — ## 1. Introduction The stellar initial mass function (IMF) is a key tool for star-formation research because it mirrors the processes of the formation of stellar populations (Bastian et al. 2010, Kroupa et al. 2013). Consequently, the IMF has been subject to extensive research both observationally and theoretically. In recent years the majority of the star-formation community has favored the assumption of continuous star formation from the lowest-mass brown dwarfs (BDs) to the most massive stars (Padoan & Nordlund 2002, 2004, Hennebelle & Chabrier 2008). However, there is evidence for at least two separate formation channels for most BDs on the one hand and most stars on the other. For instance, careful analysis of the observational data reveals a disagreement between the theoretical predictions of the binary separation distributions of BDs and stars and the observed ones (Bouy et al. 2003, Burgasser et al. 2003, Martín et al. 2003 and Close et al. 2003). In this light the key assumption of a uniform or continuous formation mechanism assumed in most star-formation models needs to be questioned. Based on observational data and N-body computations, Parker & Goodwin (2011) conclude that the birth population of very-low-mass binaries with system masses below 0.2 $M_{\sun}$ must be very different from that of M-dwarfs. Another important issue is the “BD desert” (McCarthy, Zuckerman, & Becklin 2003), an observed dearth of BD companions to stars. Unlike earlier studies like e.g. by Grether & Lineweaver (2006) who interpret this as being related to the companion mass ratio rather than the absolute mass, the more recent survey by Dieterich et al. (2012) deduce a lower absolute mass limit of companions to stars close to 0.1 $M_{\sun}$. Analytical approaches by Padoan & Nordlund (2002, PN02, from here) and Hennebelle & Chabrier (2008, 2009, HC08, and HC09, hereafter) deduces the IMF from an analytical description of the distribution of prestellar cloud cores. While the stellar IMF is reproduced by these approaches, a significant deficit of BDs and very-low-mass stars (VLMSs, between about 0.08 and 0.2 $M_{\sun}$) with respect to observationally constrained IMF descriptions (Kroupa 2001, Chabrier 2005, Thies & Kroupa 2007, 2008, the latter three from here on referred to as C05, TK07, and TK08, respectively, and Kroupa et al. 2013) appears if realistic properties of the prestellar clouds are assumed. The fundamental reason why direct fragmentation of a turbulent molecular cloud rarely produces BDs is that the formation of a BD requires a high-density gravitationally self-bound but very low mass fluctuation that cannot draw significant amounts of additional mass from an accretion reservoir (see also Adams & Fatuzzo 1996). However, HC08 and HC09 speculate that this deficit might be solved by a refinement of their models by including turbulent fragmentation descriptions. A different interpretation is that their model is overall correct but reveals that an additional formation channel is required to match the observations. The IMF by C05 is an empirical and almost equivalent update of the IMF by Chabrier (2003). Fully hydrodynamical computations of whole star-forming clouds by e.g. Bate et al. (2003) and Bate (5) reproduce the formation of BDs largely from fragmenting circumstellar disks, although these simulations tend to overproduce BDs unless radiative heating is included (Bate 6, 2012). In this paper we introduce the residual mass function (RMF) as a correction term for this BD deficiency in the analytical approaches by PN02 and HC08 with respect to the observation-based IMFs in C05 and TK07. In Section 2 the model is described and the RMF is defined. The results are presented in Section 3, followed by an analysis of the companion mass-ratio distribution in the overlap region of these two populations and a discussion of the _BD desert_ and its claimed consistency with empirically determined IMF models (Reggiani & Meyer 2011), in Section 4. In addition, the term _peripheral fragmentation_ is introduced as the main formation channel of BDs. ## 2. Method The basic idea is to quantify the deficit of analytically derived mass functions with respect to observationally constrained IMFs. The resulting deficit defines the RMF. ### Observational IMF models The stellar IMF is based on an extensive analysis of observational data from young stellar clusters and in the Galactic field (Kroupa et al. 1993, Kroupa 2001). Because the majority of multiple systems remain unresolved such IMFs need to be interpreted as system mass functions unless a careful numerical binary correction is used. The empirical IMF by C05 used here is a system IMF based on the earlier IMF by Chabrier (2003), assuming BDs and stars forming the same way, thereby simply occupying different regions in the same continuous mass function, an assumption also made in the past by Kroupa (2001, 2002) TK07, on the other hand, provide an individual-body IMF that has been transformed into the corresponding system IMF by Monte Carlo random pairing among the _star-like_ and the separate _BD-like_ population introduced in TK07. This accounts for the observational evidence for two separate (albeit related) formation channels. In particular, there is a lack of observed BD companions to stars, especially for small separations (McCarthy et al. 2003, Grether & Lineweaver 2006), whereas the statistical properties of binary BDs differ largely from stellar ones (Bouy et al. 2003, Burgasser et al. 2003, Martín et al. 2003, Close et al. 2003). As argued in Thies & Kroupa (2007), the different binary properties of BDs and VLMSs on the one hand, and stars on the other, consequently suggest the existence of two separate formation channels. This thus leads to the requirement for two separate IMFs, each corrected for unresolved binaries, to describe the real individual-body mass function of a star-forming event. These IMF descriptions, however, are purely observationally motivated and therefore do not explain the underlying physical processes leading to the actual mass spectrum. ### Analytical IMF models In recent years, several attempts have been made to understand the physics behind star formation and to reproduce its outcome. This section deals with the analytical models by PN02 and HC08. In both PN02 and HC08 the prestellar clump mass function is described by an analytical function with the Jeans mass, the length scale of the initial inhomogeneities and the Mach number, ${\cal M}$, as the main parameters (see Equation (24) in PN02 and Equation (44) in HC08). Both studies derive the IMF from the mass distribution of gravitationally unstable clumps based on empirical power spectra of turbulent flows within the molecular clouds. While the particular formalism differs in PN02 and HC08, fragmentation due to the supersonic interaction of gas sheets is the engine for forming Jeans-unstable clumps in both models. The resulting prestellar clump mass function is then transformed into a stellar IMF assuming a star-formation efficiency of 30 %–50 %. The models have been tested in the range $3\leq{\cal M}\leq 20$. In this paper, we restrict the model to the Mach number ${\cal M}=12$ in the case of HC08 (Figure 5 top therein) and ${\cal M}=10$ in PN02 because these Mach numbers are suggested as the most likely values in these papers (see p. 873 in PN02 and p. 406 in HC08). With these values both models match each other well in the stellar regime despite their different formulations. Higher Mach numbers would mainly shift the mass spectrum toward lower masses and decrease the MF toward the high-mass end. As shown in Figure 1 the PN02 ${\cal M}=10$ mass function (long-dashed curve) and the ${\cal M}=12$ HC08 mass function (solid curve) closely match in the stellar regime but are slightly different at the low-mass end. The resulting mass function deviates significantly from the observed stellar+substellar mass function in the substellar mass regime, as shown in Figures 2 and 3 for both theoretical models compared to the observationally constrained IMFs by C05 and TK07. To quantify this difference the model IMF, either HC08 or PN02, is subtracted from an observational reference mass function that is taken from C05 and TK07 in order to determine two estimates of the RMF: $\xi_{\mathrm{res}}$, \[\xi_{\mathrm{res}}(m)=\xi_{\mathrm{obs}}(m)-\xi_{\mathrm{theo}}(m)\,,\] (1) where $\xi_{\mathrm{theo}}$ is any of the theoretical IMFs and $\xi_{\mathrm{obs}}$ refers to any of the observationally constrained IMFs. In general, a mass function $\xi$ is defined as the differential number $N$ over the differential object mass $m$: \[\xi(m)=\frac{\mathrm{d}N}{\mathrm{d}m}\,,\] (2) and, in the logarithmic scale \[\xi_{\mathrm{L}}(m)=\frac{\mathrm{d}N}{\mathrm{d}\log_{10}m/M_{\sun}}=(\ln 10) \,m/M_{\sun}\,\xi(m)\,.\] (3) The TK07 IMF is the canonical IMF (Kroupa et al. 2013) that takes into account that BDs and some VLMSs need to be added as an additional population called _BD-like_, whereas most stars belong to the _star-like_ population: (4) where $\mathcal{R}_{\mathrm{pop}}$ is the BD-like to star-like population ratio ($\mathcal{R}_{\mathrm{pop}}\approx 0.2$, TK07) and $k_{m}=\left(\frac{0.5}{0.07}\right)^{-1.3}$ ensures continuity of the stellar IMF at 0.5 $M_{\sun}$. Here, $m_{\mathrm{max}}$ follows from the $m_{\mathrm{max}}$-$M_{\mathrm{ecl}}$ relation (Weidner et al. 2010, 2013, Equation (10) in Pflamm-Altenburg et al. 2007) and approaches 150 $M_{\sun}$ for the most massive clusters. _Note the overlapping mass ranges of the populations indicating that bodies between about 0.07 and 0.15 $M_{\sun}$ may belong either to the star-like or the BD-like population._ At the high-mass end of the BD-like population, the sharp truncation used in TK07 has been replaced in this study by a steep power-law function to reduce numerical artifacts in the sum IMF. We chose a power-law exponent of 10 to keep the effect on the BD-like to star-like ratio negligibly small. There is no such declining function applied to the star-like population that is intrinsically smoother due to the dynamical population synthesis (DPS) method described in Section 2.3. In addition to this, star-like objects below 0.06 $M_{\sun}$ and BD-like ones above 0.3 $M_{\sun}$ are not considered. It should be noted here that the theoretical mass function obtained by Thies et al. (2010) from smoothed particle hydrodynamics (SPH) simulations also shows a steep decline above 0.1 $M_{\sun}$ rather than an exact truncation. Since the C05 stellar IMF used by HC08 is a system mass function rather than an individual-body-mass function, the TK07 canonical IMF has also been transformed into the corresponding TK07 system mass function using the Monte-Carlo model described in Section 2.3. The stellar component of the canonical TK07 system IMF is normalized to the C05 system IMF in the whole mass regime (0.01–150 $M_{\sun}$), i.e. \[\int\limits_{0.01~{}M_{\sun}}^{150~{}M_{\sun}}\xi_{\mathrm{TK07,sys}}(m)\, \mathrm{d}m=\int\limits_{0.01~{}M_{\sun}}^{150~{}M_{\sun}}\xi_{\mathrm{C05,sys }}(m)\,\mathrm{d}m\,.\] (5) Here it is assumed that in the C05 IMF stars and BDs share the same IMF without a discontinuity or overlap, so the mass regime above 0.08 $M_{\sun}$ corresponds to the stellar component in the TK07 model. The BD multiplicity fraction is only about 10 %-20 % (Bouy et al. 2003, Close et al. 2003, Martín et al. 2003; Kraus, White, & Hillenbrand 2006 and Law et al. 2008) and is assumed to be equal for both C05 and TK07 IMFs. Further, it has to be noted here that the RMF is based on the mass range between 0.01 and about 0.3 $M_{\sun}$ ($\log_{10}m/M_{\sun}$ between $-2$ and $-0.5$) only, because the residuals in the stellar regime are due to slightly different functional descriptions rather than being physically constrained and are therefore neglected here. The residuals between the observational stellar IMF fits of C05 and TK07 and the HC08 and PN02 models are of potential interest to later work, but they are not further considered in this paper. <figure><img src="content_image/1501.01640/x1.png"><figcaption>Figure 1.— Comparison of the analytical system IMF by PN02 for M=10 (theirpreferred value, dashed curve) with that by HC08 (M=12, solid curve). Bothfunctions are scaled in this plot for equal peak height for comparison only.Although they nearly match in the stellar regime there are slight deviationsin the substellar regime as well as in the positions (i.e. masses) of thepeak.</figcaption></figure> ### Monte-Carlo model of the binarity of stars and brown dwarfs Besides the mass function itself, the binarity as a function of the primary-star mass and the CMRD are also important characteristics of stellar populations. They are studied here using a Monte Carlo approach with star-like and BD-like objects drawn from separate IMFs. The binarity or binary fraction, $f$, is defined as the ratio of the number of binary or higher-order systems, $N_{\mathrm{bin}}$, to the total number of systems, $N_{\mathrm{sys}}$. Here, the term _system_ includes multiple systems and singles (their number being noted as $N_{\mathrm{sng}}$) as well. Then (6) For the star-like population we apply the binary DPS method developed by Marks, Kroupa, & Oh (2011) and Marks & Kroupa (2011), hereafter referred to as dynamical or DPS pairing. In DPS the binary stars are formed in a population of embedded clusters within which they are dynamically processed to yield the Galactic disk stellar single-plus-binary population. An attractive feature of this theory is its underlying assumption of the universality of binary properties of late-type stars being consistent with observational data (Marks & Kroupa 2012, Leigh et al. 2014). For initial binaries with intermediate to large separations the DPS pairing method applies random pairing¹ below a primary mass of 5 $M_{\sun}$ and ordered pairing (such that $q\geq 0.9$) above. Here, $q=m_{\mathrm{comp}}/m_{\mathrm{prim}}\leq 1$, where $m_{\mathrm{comp}}$ is the companion mass and $m_{\mathrm{prim}}$ is the mass of the primary star. This initial binary population is then altered by dynamical evolution. Close binaries with orbital periods below about 10 days undergo _eigenevolution_(Kroupa 1995) and tend to equalize the companion masses. Note that this _eigenevolution_ term alters the very-low-mass end of the starlike IMF. For the purpose of this work, however these effects only play a negligible role. Here, the initial or primordial binary fraction is 100 %, i.e. it is assumed that all stars form in binaries. The final (after dynamical processing in the embedded cluster) overall binary fraction is about 40 % (i.e. $f=0.4$) but varies as a function of the primary-star mass. For M-dwarfs, in particular, it is as low as 25 %while G-dwarfs show about 56 % binarity. The binary fraction approaches 90 % for O stars. For the BD-like population we chose an overall binary fraction of 20 % (i.e. $f=0.20$), in accordance with TK07, TK08. About half of the members of observed average stellar populations are binaries, most of them remaining unresolved in typical star-cluster surveys. However, very young and likely dynamically unevolved populations like the Taurus-Auriga association exhibit almost 100 % binarity (Kroupa et al. 2013, Duchêne & Kraus 2013, Reipurth et al. 2014). The number of systems must not be confused with the number of individual bodies, $N_{\mathrm{bod}}=N_{\mathrm{sng}}+2N_{\mathrm{bin}}$. Since higher-order multiples are relatively rare (Goodwin & Kroupa 2005) they are summarized within the binary population in this work, so the total number of bodies is [FOOTNOTE:1][ENDFOOTNOTE] \[N_{\mathrm{bod}}=2N_{\mathrm{bin}}+N_{\mathrm{sng}}\,.\] (7) The CMRD describes the relative number of binaries as a function of the companion-to-primary mass ratio. Observations reveal a continuous decline of $f$ as a function of the primary-object mass which has been interpreted as a continuous transition from the stellar to the substellar regime (Joergens 2008, Kraus & Hillenbrand 2012, but see Thies & Kroupa 2008). There is also a shift towards more equal-mass binaries ($q=1$) for VLMS and BDs (Dieterich et al. 2012). These properties of the stellar population are well reproduced by DPS such that the origin and properties of binary populations are well understood. In this study we used a Monte Carlo approach with the TK07 IMF (Equation (4)), i.e., with two separate populations, _BD-like_ and _star-like_, ranging from 0.01 to 0.15 $M_{\sun}$ and from 0.06 to the maximum stellar mass of 150 $M_{\sun}$, respectively. The slightly larger mass ranges compared to Equation (4) are due to the steep power-law decline added to the mass borders to reduce numerical artifacts. The masses are drawn randomly from each IMF where the relative normalization between both populations is simply obtained by the number of objects drawn from each IMF. BD-like objects are assumed to form as single substellar cores some of which are subsequently paired to binaries within a dense dynamically preprocessed environment like a massive extended accretion disk (Stamatellos et al. 58, Thies et al. 2010, Basu & Vorobyov 2012). Similarly, stellar binaries are assumed to be assembled from individually formed and subsequently paired stars. After the individual-body populations have been drawn from the IMF, two different methods are used for assembling the binaries. For stars we apply the DPS pairing method mentioned above. BD binaries, on the other hand, are created by first drawing two objects from their separate population characterized by $\xi_{\mathrm{BD}}$ (Equation 4) with the pairing probability $p_{\mathrm{pair}}$ being determined by a power law, \[p_{\mathrm{pair}}=\mbox{const}\,q^{\gamma}\,,\] (8) following the power-law bias scheme used by Goodwin (2013) for the mass ratio distribution of binaries from second (i.e. binary-forming) fragmentation. Whereas the case $\gamma=0$ corresponds to random pairing from the IMF, $\gamma>0$ describes a biased pairing rule with an increasing pairing probability toward equal-mass components. The extreme case of perfect equal-mass pairing corresponds to $\gamma=\infty$ but is in disagreement with data from the Very-Low-Mass Binary Archive (VLMBA; Burgasser et al. 2007) which also contains a few unequal substellar binaries. In practice, first an array of BD-likes is generated by drawing randomly from $\xi_{\mathrm{BD}}$ (Equation 4). All subsequent pairings to make binaries are performed on the array only. Random pairing is performed by randomly drawing a companion from the array for each object of the same population. If the companion is more massive than the considered object it becomes the primary, and otherwise it is the secondary component. In the case of biased pairing an additional decision is made whether a companion randomly drawn from the population is accepted to be a partner or rejected, depending on the probability $p_{\mathrm{pair}}$ in Equation (8). A rejected partner may later be chosen as a partner to another object. Objects that, on the other hand, are already bound in a binary are skipped in the pairing procedure henceforth. The biased pairing procedure is iterated until the required binary fraction $f$ is achieved. Because the objects are chosen in random order, this does not introduce any additional bias besides $p_{\mathrm{pair}}$ to the binary populations. The method ensures that the slope of the canonical IMF is retained, in contrast to methods that select components according to a mass-ratio distribution only. ## 3. Results ### Residual Mass function for Semianalytical Models In this section we present the RMFs obtained from the BD/VLMS deficit of the analytical IMF models by PN02 and HC08. The RMF is derived for a particular and, according to PN02 and HC08, typical parameter set, in particular the turbulent Mach number. Other parameters yield different RMFs, so only the general functional shape for a likely parameter set is presented here. It is further shown that such an RMF provides a correction term to the stellar IMFs in order to fit a reasonable overall IMF without the BD deficit observed in the purely stellar ones. Figure 2 shows the results of our calculations for the analytical model from HC08 with the C05 and TK07 system IMF as the observational reference (upper and lower panels, respectively). In both cases the RMF is limited to the mass range below 0.3 $M_{\sun}$ ($\log_{10}m/M_{\sun}<-0.5$) because the slightly different shapes of the IMFs in the stellar region are beyond the scope of this paper. The local minima near $\log_{10}m/M_{\sun}=-0.8$ (upper panel) and $-1.4$ (lower panel) occur because the difference between the HC08 mass function and the empirical mass function almost vanishes locally. This behavior is highly sensitive to the normalization of the IMFs and is most prominent in the case of the discontinuous TK07 IMF. Peaking near the hydrogen-burning mass limit the RMF declines steeply and effectively vanishes above $\log_{10}m/M_{\sun}=-0.5$. The peak mass, however, is better represented in the residual to TK07 in Figure 2 whereas the RMF from C05 appears to be shifted toward lower masses by about a factor of two (i.e. an offset of about -0.3 on the logarithmic scale). In the case of the PN02 analytical model, as shown in Figure 3, the result with respect to the C05 IMF covers a mass range similar to HC08 but is also continuous. This is expected because both analytical models have a functional form similar to the C05 IMF, namely an extended log-normal-type shape. In the case of the TK07 system IMF (Figure 3) there is a prominent “dip” in the RMF around $\log_{10}m/M_{\sun}=-1.3$ (i.e. $m=0.05\,M_{\sun}$). The reason is the near-equality of TK07 and PN02 at this point, which varies sensitively with the normalization of the TK07 system IMF and its substellar component. As with the HC08 model, the RMF of PN02 versus C05 peaks at lower masses by a factor of about two. <figure><img src="content_image/1501.01640/x2.png"><figcaption>Figure 2.— Upper panel: the analytical IMF model for M=12 by HC08 (solid line)compared to the empirical IMF by Chabrier (2005) (dashed line). Thesefunctions, originally defined as system mass functions, have been normalizedto match in the stellar regime. The dotted line represents the residual massfunction, i.e. the difference between both mass functions. The gaps in theRMFs are caused by the intersection of the HC08 model IMF with the empiricalIMFs. They are highly sensitive to the normalization of the IMFs. Lower panel:same as in the upper panel but with the combined bimodal IMF according toThies & Kroupa (2007) (dashed line). Note that the RMF is truncated forlog10m/M\sun≥−0.5 because the different functional shapes of the stellar partsare not considered here.</figcaption></figure> <figure><img src="content_image/1501.01640/x3.png"><figcaption>Figure 3.— Upper panel: the same as in Figure 2 but with the clump massfunction model by PN02 instead of HC08. Lower panel: same as Figure 2, lowerpanel, but with the clump mass function model by PN02 instead of HC08. Here,the RMF does not contain any gaps.</figcaption></figure> ### Comparison with Simulation Data <figure><img src="content_image/1501.01640/x4.png"><figcaption>Figure 4.— Residual mass functions of HC08 (upper panel, from Figure 2) andPN02 (lower panel) with respect to the system mass functions by C05 (dashedline, from Figure 3) and TK07 (dotted line) as well as the average of both(solid line) in comparison to the SPH mass function from Thies et al. (2010,including post-population SPH data), shown as a histogram (normalized inlog10ξL to the RMF).</figcaption></figure> The RMF obtained in the previous section is compared here with the mass spectrum of substellar clumps formed in the SPH models of Thies et al. (2010). Therein the induced fragmentation of massive extended circumstellar disks due to perturbation by passing stars in an embedded cluster has been studied. Using an SPH code with a radiative cooling approximation (Stamatellos et al. 60) gravitational instabilities have been demonstrated to form through tidal perturbations and to form compact clumps with typical masses between 0.01 and 0.15 $M_{\sun}$. The mass function of the thus-formed clumps, the “SPH mass function” (SPH MF) hereafter, has been shown in Thies et al. (2010) to be in agreement with the substellar component of the observed system IMF from TK07. In the current study, the SPH mass function is based on 80 objects formed in 29 computations, some of them performed in continuation of TK07. In Figure 4 the (scaled) SPH mass function is compared to the corresponding RMF. The overall shape of the RMF essentially matches the SPH mass function which features a general increase with increasing mass, a peak around 0.1 $M_{\sun}$ and a rapid decay for higher masses, thus characterizing a population that is restricted to the substellar and very-low-mass stellar regime. As mentioned by Thies et al. (2010), these results are quite similar to those in Stamatellos & Whitworth (2009) who computed BD formation in self-fragmenting disks. Therefore, the results from triggered fragmentation can be taken as representative for the general disk fragmentation outcome. Both scenarios assume that the substellar clumps formed in a fragmenting disk are ejected by either mutual dynamical interactions of clumps in the disk or in subsequent stellar encounters. According to Basu & Vorobyov (2012) some of these clumps are disrupted by the tidal forces during the ejection or are destroyed because of migration onto the central star. The impact of this effect on the mass distribution will be subject to future research. Given the uncertainties of the PN02 and HC08 MFs the RMF based on them can be considered to be in agreement with the substellar IMF in both TK07 and Thies et al. (2010). Interestingly, the average of both RMFs (each weighted as 50 %) provides an even better fit to the SPH data (see the solid curves in Figure 4). Use of such an average may be motivated by the expectation that the true IMF below $\approx 0.2\,M_{\sun}$ is likely between the TK07 and C05 quantifications. ### Monte Carlo Study Applied to Observations <figure><img src="content_image/1501.01640/x5.png"><figcaption>Figure 5.— Upper panel: the combined IMF of BD-like and star-like objects fromour Monte Carlo calculations for individual-body masses (solid curve),primary-body masses (dashed curve), and system masses (dash-dotted curve). TheBD-like binarity is 20 % here, and the star-like one is here about 80 % onaverage, to match the case of dynamically unevolved populations (see text).For comparison, the histogram for grouped stars in Kirk & Myers (2012,“KM12”), which also shows a discontinuity near the stellar–substellar border,is included as the dotted histogram. Lower panel: the IMFs for the BD-like andstar-like populations plotted separately. The line patterns are the same as inthe upper panel for each separate IMF. For comparison, the individual-body sumIMF is superimposed (thin dotted curve, the same as the solid curve in the toppanel). Note that the very-low-mass end of the star-like IMF is slightlydepleted due to _eigenevolution_ (Marks et al. 2011).</figcaption></figure> The IMF from the Monte Carlo study described in Section 2.3 is shown in Figure 5 in comparison to the system mass function (dash-dotted curve), the primary-body mass function (dashed curve) and the individual-body mass function (solid curve). The upper panel displays the sum IMF with both populations combined, as it would be seen by an observer. In the lower panel, both populations are plotted separately. We note that Figure 12 in Kirk & Myers (2012), which shows the observed average mass functions of young stellar clusters, groups and isolated objects across the stellar–substellar boundary, reveals a prominent step near the stellar–substellar border that is very similar to the one that emerges from our analysis as plotted in the upper panel of Figure 5. The system MF (dash-dotted curve) is to be compared with the mass function by Kirk & Myers (2012). Because Kirk & Myers (2012) only studied low-density, dynamically unevolved populations, the stellar system IMF is modelled here with a binary fraction of 80 % (see also fig. 10 in Marks et al. 2011). For the Galactic field, for which the rest of our study has been performed, the binary fraction is only about 40–50 % for low-mass stars. Figure 6 depicts as the solid line the binarity as a function of the primary-body mass (i.e. the system mass in the case of single objects) from the DPS pairing model for the Galactic field population for embedded clusters with a half-mass radius of 0.1 pc for the underlying cluster population. The long-dashed line indicates the case of a half-mass radius of 0.2 pc which yields an up to 20 % higher multiplicity. These model Galactic field populations, computed with DPS (Section 2.3), assume that all stars form in binary systems in a population of embedded star clusters that evolve and disperse their stars into the field. These two cases are not to be taken as statistically strict confidence limits but rather as two likely cases that have been studied in Marks & Kroupa (2011), and the initial stellar binary population is indicated by the shaded region at the top. The stellar binary fraction resulting from DPS is about $f=0.4$ on average in the low-mass region below 1 $M_{\sun}$ for the case of 0.1 pc. The single-hatched region indicates the large uncertainty of the observed binarity in the substellar region. For comparison, Galactic-field observational data are overplotted for A stars (VAST survey, De Rosa et al. 2014, asterisks), G and late-F stars (Duquennoy & Mayor 1991, filled squares), M to G stars (Kroupa et al. 2003; open squares), M dwarfs (Fischer & Marcy 1992; upright triangle), and L dwarfs (Reid et al. 2008; upside-down triangle). Despite consisting of two separate populations (BD-like and star-like) the binary fraction declines _continuously_ from the stellar toward the substellar regime. As discussed in Marks & Kroupa (2011) and Marks & Kroupa (2012), this is primarily due to the dynamical processing of the star-like population. In part, the mass overlap of the star-like and the BD-like population also contributes to this transition. It remains to be shown in future work that the apparent observed trend toward narrow binary separation distributions for late-type M dwarfs reported by Janson et al. (2014) can also be reproduced with DPS by this overlap. ### Binarity and Companion Mass-ratio Distribution The observable BD CMRD, which is the CMRD for all primaries (BD-like and star-like) below 0.08 $M_{\sun}$, is shown in Figure 7. As can be seen the case of strongly biased pairing with $\gamma=2$ (solid line) matches the measured data from the VLMBA better (and also see Dieterich et al. 2012) than the random-pairing $q$ ($\gamma=0$, dotted line) and weakly biased pairing models ($\gamma=1$, dashed line). In addition, the case $\gamma=4$ (Duchêne & Kraus 2013; dash-dotted line) is shown. The binarity of BDs near the hydrogen-burning mass limit of 0.08 $M_{\sun}$ is quite low in the model, only about 11 %–12 % which is largely due to the “dip” in the binarity in the VLMS region of the DPS pairing method (see Figure 6). This local minimum is a direct consequence of the two overlapping populations. Reid et al. (2008) indeed report a similar binary fraction of $12.5{}^{+5}_{-3}$ % for L dwarfs. If a local minimum is confirmed by future surveys with sufficient precision in mass this would further support an overlap of two separate populations in this mass region. However, in the current data this feature is within the uncertainties of the observational data. <figure><img src="content_image/1501.01640/x6.png"><figcaption>Figure 6.— Binary fraction for BDs and stars as a function of the primary-object mass. The solid line shows the result of our Monte Carlo computationfor stars via DPS pairing for the field population based on the integration ofthe star-formation outcome over all cluster masses. A half-mass radius rh=0.1pc is assumed for all embedded host clusters. The average field-star binaryfraction is f≈0.5 for low-mass to intermediate-mass stars, in accordance withTK07. The long-dashed line refers to the DPS model with rh=0.2 pc,corresponding to a higher binary fraction. The short-dashed curve representsthe BD-like population with biased pairing. The single-hatched region in thelower left corner indicates the uncertainty of the substellar binary fractionin the observational data. The smoothly shaded area at the top refers to theinitial stellar binary population with a binary fraction of near 100 % (Marks& Kroupa 2011). Observational field-star data are overplotted for A stars(VAST survey, De Rosa et al. 2014, asterisks), G and late-F stars (Duquennoy &Mayor 1991, filled squares), M to G stars (Kroupa et al. 2003; open squares),M dwarfs (Fischer & Marcy 1992; upright triangle), and L dwarfs (Reid et al.2008; upside-down triangle).</figcaption></figure> <figure><img src="content_image/1501.01640/x7.png"><figcaption>Figure 7.— Number distribution f(q) for binary BDs between 0.03 and 0.08M\sun as a function of the companion-to-primary mass ratio. The solid lineshows the result of our computation for γ=2, and the dashed and dotted linesrefer to the case γ=1 and random pairing, respectively. For completeness, alsothe more extreme case of γ=4 (Duchêne & Kraus 2013) is also shown as the dash-dotted line. The histogram shows the observational data from the Very-Low-MassBinary Archive (Burgasser et al. 2007). The peak at q=1 is primarily due tothe contribution of binaries from the low-mass end of the star-like populationbecause any binary with both components near the lower mass border of apopulation can only have nearly equal component masses. The area below eachcurve and below the histogram is normalized to 1.</figcaption></figure> <figure><img src="content_image/1501.01640/x8.png"><figcaption>Figure 8.— Comparison of the Thies & Kroupa (2007), Chabrier (2005), and theextrapolated Bochanski et al. (2010) IMF, the latter being used by Reggiani &Meyer (2011). The original Bochanski IMF has been derived from data between0.1 and 0.8 M\sun. Note the steeper decline in the Bochanski IMF below itsvalid mass range relative to the Chabrier IMF.</figcaption></figure> ### Contribution of Peripheral Fragmentation to BDs One important implication of the model discussed in this paper is the hybrid nature of BDs and VLMSs. Although there is a minor contribution by the star-like population from direct fragmentation, the majority of BDs are contributed by the BD-like population through dynamically preprocessed gas (e.g. fragmenting circumstellar disks (Stamatellos & Whitworth 2009, Thies et al. 2010). Because this preprocessed material often occurs in the peripheral regions of star formation (e.g. in the outer parts of disks) we propose the term _peripheral fragmentation_ for the BD-like formation channel. Of all BDs between 0.01 and 0.08 $M_{\sun}$ 64 % are contributed by the BD-like population, and 19 % of M dwarfs between 0.08 and 0.45 $M_{\sun}$ are BD-like. This result, however, is highly sensitive to the chosen lower mass border of the star-like regime. Here, we assumed it to be 0.06 $M_{\sun}$. If, on the other hand, a sharp truncation of the star-like regime at 0.08 $M_{\sun}$ is chosen, the BD-like fraction of BDs is 100 % but still 18 % of M dwarfs are BD-like. Because star-like BDs and VLMSs may be detectable by larger circumsubstellar disks or, if applicable, wide binary separations, future high-resolution observations will help to further constrain the BD-like and star-like mass borders. ## 4. Discussion <figure><img src="content_image/1501.01640/x9.png"><figcaption>Figure 9.— Snapshot from SPH model X002 by Thies et al. (2010) showingsubstellar clumps (with sink particles marked with white points) around afragmented accretion disk, all of them surrounded by their own accretionenvelopes. Note that the escaping object (0.013 M\sun, i.e. a low-mass BD) tothe right has retained its accretion disk (visible as diffuse structure aroundthe sink particle) even after dynamical ejection. The half-mass radius of theenvelope is about 10 AU, and its mass is 2⋅10−4M\sun, i.e. almost 2 % of themas of the escaping object.</figcaption></figure> In this paper we have introduced the RMF as a correction term for the analytical star-formation models by PN02 and HC08 to match the observationally constrained IMF by Chabrier (2005) and Thies & Kroupa (2007). The effective deficit of BDs and VLMSs in these theoretical models with respect to the observed BD statistics suggests the requirement for additional formation channels in these analytical IMF models. HC08 speculate that this deficit, being at least at the edge of significance (see uncertainties shown in Figure 4–23 in Kroupa et al. 2013), may be solvable by an improved algorithm that accounts for the effects of turbulence and other dynamical processing of the prestellar gas. The expected contribution of these additional effects to the IMF may be understood as a separate population or, at least, as an additional formation channel of BDs and VLMSs and can formally be described by an additional correction term. It is also applicable as a test for future star-formation models. This correction term has been identified and quantified here as the RMF which has a similar shape for both analyzed observational reference IMFs by C05 and TK07. The most striking outcome is a general agreement between the RMF (Equation (1)) as obtained from the theoretical IMF and the observational IMFs, and the mass function of fragments from SPH simulations (Thies et al. 2010). In other words, both PN02 and HC08 IMF models appear to describe essentially the population of stars without a significant fraction of BDs. They can be considered to be successful models of the direct fragmentation process in molecular clouds. These models cannot, by their nature, accommodate the peripheral fragmentation, e.g., in accretion disks around protostars, which yields the BD-like population (see Section 3.5). The most natural way to explain these results is, to the best of our knowledge, a composite population consisting of a star-like and a BD-like component such that a significant fraction of the BD part comes from an additional process acting during star formation (Reipurth & Clarke 2001, Stamatellos et al. 58, Thies et al. 2010, Basu & Vorobyov 2012). Essentially this result follows for BDs being less likely to form directly in a star-forming molecular cloud than from dynamically preprocessed material. This is because in order to form only a BD a cloud core needs to have a high pre-collapse density (for it to be gravitationally unstable) while having no further supply of gas in order to limit its further growth in mass. This has already been emphasized by Adams & Fatuzzo (1996). While there has been a recent discovery of a possible proto-BD in Oph B-11 with a mass of 0.03 $M_{\sun}$ (André et al. 2012) its further evolution and possible additional accretion from the surrounding molecular gas remains unclear. And, the existence of the BD-like population formed through peripheral fragmentation does not exclude the formation of BDs via the direct fragmentation channel, i.e. as star-like objects as is evident from the analytical PN02 and HC08 IMFs. ### Is There a Separate Substellar Population? There is an ongoing discussion whether a separate population is really needed to explain the observed properties. A similar discontinuity between planets and BDs is widely accepted. Chabrier et al. (2014) even report a possible mass overlap of the BD and the giant planet regime as well and suggest a distinction between BDs and planets by their formation history rather than by deuterium burning. This can be seen as an analog to the separate substellar population addressed by our work. Jumper & Fisher (2013) claim to be able to model the trend toward more tightly bound binaries in the BD regime. However, because this study already assumes the masses to be drawn from a continuous IMF it a priori excludes the option of a two-component IMF. The AstraLux survey reports a narrower binary separation distribution for M-dwarf binaries compared to binaries with FGK primaries (Janson et al. 2012 and Janson et al. 2014, respectively). Such a narrow peak in the separation distribution may be related to sensitivity artifacts as well as, at least for late-type M dwarfs, due to the mass overlap with the BD-like population that extends up to about 0.2 $M_{\sun}$. In contrast, in a study of dynamical binary evolution Parker & Goodwin (2011) found that the field populations of M dwarfs and very-low-mass binaries must reflect very different birth populations, since their dynamical processing is essentially the same. ### The Brown Dwarf Desert Similarly, the BD desert itself has been disputed by several groups who suggest that it is more related to the companion mass ratio rather than to the absolute mass. However, a recent survey by Dieterich et al. (2012) further supports the existence of a true BD desert, i.e. a dearth of substellar companions to primary stars ranging from M to G dwarfs. They found an effective lower mass limit of stellar companions near 0.1 $M_{\sun}$ which cannot be related to incompleteness but rather points toward a discontinuity in the pairing statistics close to the hydrogen-burning mass limit. They particularly point out that this discontinuity is essentially independent of the primary star mass. This clearly supports a separate substellar population contributing the majority of BDs and is also in agreement with SPH studies of whole star-forming clouds by, e.g., Bate et al. (2003) and Bate (5) who reproduce the formation of BDs largely from fragmenting circumstellar disks. They, however, point out that these simulations tend to overproduce BDs and attribute this to the lack of radiative heating in their model (Bate 6, 2012). It should also be noted that Bate (2012) could reproduce a mass function in good agreement with the C05 IMF and that the previous overproduction of BDs was avoided there because the radiative heating reduced the frequency of disk fragmentation. In a related Monte Carlo study performed here, we also found a good agreement of the two-population composite model with observational data on field very-low-mass stellar and BD binaries (Figure 6). The binarity is well reproduced down to the stellar–substellar border as a continuous function of the primary-star mass in agreement with the observations. More importantly, the mass-ratio distribution of very-low-mass binaries deduced from observational data is well reproduced in our model (Figure 7), once more supporting a bimodal star and BD formation scenario. Whereas star-like binaries are well represented by DPS (Marks et al. 2015), the BD-like binaries apparently do fit a biased pairing where the probability of pairing rises toward more equal component masses following Equation (8). Here, the cases $\gamma=0$ (simple random pairing) and $1\leq\gamma\leq 2$ (biased toward equal-mass binaries) were compared. The biased pairing case is in better agreement with the observational data. It remains to be studied whether postformation dynamical processing through stellar-dynamical encounters in their birth embedded clusters is responsible for this apparent preference of more equal-mass binaries over more unequal ones in the BD mass range. However, both simple and biased pairing are in reasonable agreement within the uncertainties of the data if a low binary fraction of 10 % within the BD-like population is assumed. A binary fraction of less than 10 % in the theoretical disk fragmentation outcome and the very narrow range of semimajor axes have also been found by Thies et al. (2010), as well as the survival of circumsubstellar accretion disks, as shown in Figure 9. The overall binary fraction for systems within 0.03 and 0.08 $M_{\sun}$, irrespective of the population to which they belong, is around 20 % for both simple and biased pairing (i.e. the value chosen for the BD-like population), if the local minimum in the binarity function near the stellar–substellar boundary (Figure 6) is assumed to be an artifact, whereas it is as low as 11 % otherwise. However, observations of VLMSs indicate a binary fraction as low as 12.5 % (Reid et al. 2008), so this “dip” may indeed be real and thus be a result of the overlap of two separate populations. ### Companion Mass Ratio Distribution There is an ongoing discussion whether random pairing is applicable for stars. Reggiani & Meyer (2011) claim to rule out this model in favor of a universal CMRD. However, their results only cover a narrow range of validity of the underlying model IMF (as specified by Bochanski et al. 2010, see Figure 8) that only covers M, K and late-G stars, but no BDs. Consequently, the CMRD derived from it is quite limited, especially for M dwarfs, for which no hypothesis test on the pairing rule can be made on this basis. Reggiani & Meyer (2011) perform pairing experiments that actually exceed the range of validity of the underlying mass function by Bochanski et al. (2010), and therefore their results are not applicable. Goodwin (2013) suggests that a random partition of protostellar cloud fragments is in better agreement with observations of stellar binarity than an initial binary population drawn from the IMF with subsequent dynamical and eigenevolution. Because that study was based on the IMF by Chabrier (2003) without a separate BD treatment there cannot by any satisfactory agreement with the observed substellar CMRD. Even more importantly, the initial binary population needs to be modeled and tested against dynamically barely evolved stellar populations like Taurus-Auriga rather than against the Galactic field. The field population then results from an integration of all dynamically evolved clustered populations following the embedded cluster-mass function (Marks & Kroupa 2011, Marks et al. 2015). As a final remark, it is being argued that the existence of disks around young BDs implies that these form like stars supposedly supporting the continuous IMF scenario. Figure 9 is one of many examples where a young BD with an accretion disk formed from peripheral fragmentation in a circumstellar disk is nudged away to become a free-floating BD with a disk. ## 5. Conclusion As the main conclusion from the first part of this study we emphasize the difficulties of theoretical star-formation models to describe both stars and BDs by a single mechanism, namely from direct cloud fragmentation. It remains necessary to treat BDs separately which implies a separate albeit related formation channel for the majority of BDs. This already follows from the work of Adams & Fatuzzo (1996). The theoretical evidence by Stamatellos & Whitworth (2009), Thies et al. (2010), and Basu & Vorobyov (2012) supports a separate population by fragmentation of extended young circumstellar disks. If O stars are close by, the accretion envelope of VLM protostars may be photoevaporated, leaving an unfinished substellar embryo (Kroupa & Bouvier 2003). In addition, the embryo-ejection model by Reipurth & Clarke (2001) gives an example of BD formation by ejection of unfinished stellar embryos out of multiple protostar systems. Because these mechanisms are not covered by the analytical cloud fragmentation models by PN02 and HC08 they do not contribute to the resulting theoretical clump mass function. The same is true for BD formation in dense gaseous filaments that form due to the gravitational pull of surrounding stars (Bonnell et al. 2008) rather than being described by random density fluctuations assumed by PN02 and C05. Therefore, an analytical model also covering BDs and VLMSs must include such separate formation channels via dynamically preprocessed material (disks, unstable multiple stellar embryo systems, photoevaporation, tidally shaped gas filaments and so on). The normalization may be adjusted empirically or deduced from physical relations based on typical disk-to-stellar mass relations and by the application of turbulence and the Jeans criterion within this preprocessed material. The development of such an analytical or semianalytical model should be the subject of future studies. The observational data thus very strongly suggest that the dominant fraction (Section 3.5) of BDs and VLMSs results from peripheral fragmentation that contributes a separate IMF from that of stars. This project has been funded by DFG project KR1635/27. ## References * Adams & Fatuzzo (1996) Adams, F. C., & Fatuzzo, M. 1996, ApJ, 464, 256 * André et al. (2012) André, P., Ward-Thompson, D., & Greaves, J. 2012, Science, 337, 69 * Bastian et al. (2010) Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48, 339 * Basu & Vorobyov (2012) Basu, S., & Vorobyov, E. I. 2012, ApJ, 750, 30 * Bate (2009a) Bate, M. R. 2009a, MNRAS, 392, 590 * Bate (2009b) —. 2009b, MNRAS, 392, 1363 * Bate (2012) —. 2012, MNRAS, 419, 3115 * Bate et al. (2003) Bate, M. R., Bonnell, I. A., & Bromm, V. 2003, MNRAS, 339, 577 * Bochanski et al. (2010) Bochanski, J. J., Hawley, S. L., Covey, K. R., West, A. A., Reid, I. N., Golimowski, D. A., & Ivezić, Ž. 2010, AJ, 139, 2679 * Bonnell et al. (2008) Bonnell, I. A., Clark, P., & Bate, M. R. 2008, MNRAS, 389, 1556 * Bouy et al. (2003) Bouy, H., Brandner, W., Martín, E. L., Delfosse, X., Allard, F., & Basri, G. 2003, AJ, 126, 1526 * Burgasser et al. (2003) Burgasser, A. J., Kirkpatrick, J. D., Reid, I. N., Brown, M. E., Miskey, C. L., & Gizis, J. E. 2003, ApJ, 586, 512 * Burgasser et al. (2007) Burgasser, A. J., Reid, I. N., Siegler, N., Close, L., Allen, P., Lowrance, P., & Gizis, J. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Univ. Arizona Press, Tucson), 427–441 * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763 * Chabrier (2005) Chabrier, G. 2005, in Astrophysics and Space Science Library, Vol. 327, The Initial Mass Function 50 Years Later, ed. E. Corbelli, F. Palla, & H. Zinnecker, 41 * Chabrier et al. (2014) Chabrier, G., Johansen, A., Janson, M., & Rafikov, R. 2014, in Protostars and Planets VI, ed. H. Beuther, C. Klessen, R. Dullemond, & T. Henning (Tucson, AZ: University of Arizona Press), TBA * Close et al. (2003) Close, L. M., Siegler, N., Freed, M., & Biller, B. 2003, ApJ, 587, 407 * De Rosa et al. (2014) De Rosa, R. J., et al. 2014, MNRAS, 437, 1216 * Dieterich et al. (2012) Dieterich, S. B., Henry, T. J., Golimowski, D. A., Krist, J. E., & Tanner, A. M. 2012, AJ, 144, 64 * Duchêne & Kraus (2013) Duchêne, G., & Kraus, A. 2013, ARA&A, 51, 269 * Duquennoy & Mayor (1991) Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485 * Fischer & Marcy (1992) Fischer, D. A., & Marcy, G. W. 1992, ApJ, 396, 178 * Goodwin (2013) Goodwin, S. P. 2013, MNRAS, 430, L6 * Goodwin & Kroupa (2005) Goodwin, S. P., & Kroupa, P. 2005, A&A, 439, 565 * Grether & Lineweaver (2006) Grether, D., & Lineweaver, C. H. 2006, ApJ, 640, 1051 * Hennebelle & Chabrier (2008) Hennebelle, P., & Chabrier, G. 2008, ApJ, 684, 395 * Hennebelle & Chabrier (2009) —. 2009, ApJ, 702, 1428 * Janson et al. (2014) Janson, M., Bergfors, C., Brandner, W., Kudryavtseva, N., Hormuth, F., Hippler, S., & Henning, T. 2014, ApJ, 789, 102 * Janson et al. (2012) Janson, M., et al. 2012, ApJ, 754, 44 * Joergens (2008) Joergens, V. 2008, A&A, 492, 545 * Jumper & Fisher (2013) Jumper, P. H., & Fisher, R. T. 2013, ApJ, 769, 9 * Kirk & Myers (2012) Kirk, H., & Myers, P. C. 2012, ApJ, 745, 131 * Kraus & Hillenbrand (2012) Kraus, A. L., & Hillenbrand, L. A. 2012, ApJ, 757, 141 * Kraus et al. (2006) Kraus, A. L., White, R. J., & Hillenbrand, L. A. 2006, ApJ, 649, 306 * Kroupa (1995) Kroupa, P. 1995, MNRAS, 277, 1507 * Kroupa (2001) —. 2001, MNRAS, 322, 231 * Kroupa (2002) —. 2002, Science, 295, 82 * Kroupa & Bouvier (2003) Kroupa, P., & Bouvier, J. 2003, MNRAS, 346, 369 * Kroupa et al. (2003) Kroupa, P., Bouvier, J., Duchêne, G., & Moraux, E. 2003, MNRAS, 346, 354 * Kroupa et al. (1993) Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545 * Kroupa et al. (2013) Kroupa, P., Weidner, C., Pflamm-Altenburg, J., Thies, I., Dabringhausen, J., Marks, M., & Maschberger, T. 2013, in Planets, Stars and Stellar Systems. Volume 5: Galactic Structure and Stellar Populations, ed. T. D. Oswalt & G. Gilmore (Springer), 115 * Law et al. (2008) Law, N. M., Hodgkin, S. T., & Mackay, C. D. 2008, MNRAS, 384, 150 * Leigh et al. (2014) Leigh, N. W. C., Mastrobuono-Battisti, A., Perets, H. B., & Böker, T. 2014, MNRAS, 441, 919 * Marks & Kroupa (2011) Marks, M., & Kroupa, P. 2011, MNRAS, 417, 1702 * Marks & Kroupa (2012) —. 2012, A&A, 543, A8 * Marks et al. (2011) Marks, M., Kroupa, P., & Oh, S. 2011, MNRAS, 417, 1684 * Marks et al. (2015) Marks, M., Pflamm-Altenburg, J., & Kroupa, P. 2015, in preparation * Martín et al. (2003) Martín, E. L., Barrado y Navascués, D., Baraffe, I., Bouy, H., & Dahm, S. 2003, ApJ, 594, 525 * McCarthy et al. (2003) McCarthy, C., Zuckerman, B., & Becklin, E. E. 2003, in Brown Dwarfs, IAU Symposium 211, ed. E. Martín (ASP, San Francisco), 279–+ * Padoan & Nordlund (2002) Padoan, P., & Nordlund, Å. 2002, ApJ, 576, 870 * Padoan & Nordlund (2004) —. 2004, ApJ, 617, 559 * Parker & Goodwin (2011) Parker, R. J., & Goodwin, S. P. 2011, MNRAS, 411, 891 * Pflamm-Altenburg et al. (2007) Pflamm-Altenburg, J., Weidner, C., & Kroupa, P. 2007, ApJ, 671, 1550 * Reggiani & Meyer (2011) Reggiani, M. M., & Meyer, M. R. 2011, ApJ, 738, 60 * Reid et al. (2008) Reid, I. N., Cruz, K. L., Burgasser, A. J., & Liu, M. C. 2008, AJ, 135, 580 * Reipurth & Clarke (2001) Reipurth, B., & Clarke, C. 2001, AJ, 122, 432 * Reipurth et al. (2014) Reipurth, B., Clarke, C. J., Boss, A. P., et al. 2014, in University of Arizona Space Science Series, Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Tucson, AZ: University of Arizona Press) * Stamatellos et al. (2007a) Stamatellos, D., Hubber, D. A., & Whitworth, A. P. 2007a, MNRAS, 382, L30 * Stamatellos & Whitworth (2009) Stamatellos, D., & Whitworth, A. P. 2009, MNRAS, 392, 413 * Stamatellos et al. (2007b) Stamatellos, D., Whitworth, A. P., Bisbas, T., & Goodwin, S. 2007b, A&A, 475, 37 * Thies & Kroupa (2007) Thies, I., & Kroupa, P. 2007, ApJ, 671, 767 * Thies & Kroupa (2008) —. 2008, MNRAS, 390, 1200 * Thies et al. (2010) Thies, I., Kroupa, P., Goodwin, S. P., Stamatellos, D., & Whitworth, A. P. 2010, ApJ, 717, 577 * Weidner et al. (2010) Weidner, C., Kroupa, P., & Bonnell, I. A. D. 2010, MNRAS, 401, 275 * Weidner et al. (2013) Weidner, C., Kroupa, P., & Pflamm-Altenburg, J. 2013, MNRAS, 434, 84
1203.5557	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 79380, "num_imgs": 9, "llama3_tokens_count": 21043 }	[ "content_image/1203.5557/x1.png", "content_image/1203.5557/x2.png", "content_image/1203.5557/x3.png", "content_image/1203.5557/x4.png", "content_image/1203.5557/x5.png", "content_image/1203.5557/x6.png", "content_image/1203.5557/x7.png", "content_image/1203.5557/x8.png", "content_image/1203.5557/x9.png" ]	# Quantum Theory without Planck’s Constant John P. Ralston _Department of Physics & Astronomy_ _The University of Kansas, Lawrence KS 66045_ ###### Abstract Planck’s constant was introduced as a fundamental scale in the early history of quantum mechanics. We find a modern approach where Planck’s constant is absent: it is unobservable except as a constant of human convention. Despite long reference to experiment, review shows that Planck’s constant cannot be obtained from the data of Ryberg, Davisson and Germer, Compton, or that used by Planck himself. In the new approach Planck’s constant is tied to macroscopic conventions of Newtonian origin, which are dispensable. The precision of other fundamental constants is substantially improved by eliminating Planck’s constant. The electron mass is determined about 67 times more precisely, and the unit of electric charge determined 139 times more precisely. Improvement in the experimental value of the fine structure constant allows new types of experiment to be compared towards finding “new physics.” The long-standing goal of eliminating reliance on the artifact known as the International Prototype Kilogram can be accomplished to assist progress in fundamental physics. ## 1 The Evolution of Physical Constants More than a century after Planck’s work, why are we still using Planck’s constant? The answer seems self-evident. Planck’s discovery led to quantum mechanics, which explains experimental data. Yet there exists a picture of quantum theory where Planck’s constant is spurious. It cannot by found by fitting quantum mechanics to data. I propose that Planck’s constant originates in human conventions. The conventions come from an era assuming Newtonian theory was fundamental. The relevance of the macroscopic theory has faded over time, and it no longer constitutes first principles. The quantum theory that replaced it turns out to have an unrecognized degree of symmetry. Assuming quantum mechanics is primary and fundamental, the symmetries reveal certain Newtonian conventions that fundamental physics can do without. It is often thought that fundamental constants have an absolute experimental character, which must override theory. The electron mass $m_{e}$ is an example. By a certain trick it has been planted as pre-existing and well-defined, with one invariant value in several different theories. In our approach no constant is defined without the theory that commits to defining its meaning. The Newtonian inertial electron mass is a problematic element of Newtonian theory. Avoiding its problems makes a theory not needing Planck’s constant. We base our approach on quantum mechanics defined without a scale for amplitudes. Observables are given by $<\hat{A}>=<\psi\|\hat{A}\|\psi>/<\psi\|\psi>$ in which the scale of the wave function cancels out¹. It is conventional to normalize the wave function, and reduce the ambiguity to an unobservable overall phase, but mathematicians recognize the wave function as a projective or ray representation: producing a symmetry we call “quantum homogeneity,” which is admittedly not that new. Given a formula for the action $S$, the principle is [FOOTNOTE:1][ENDFOOTNOTE] \[\delta S=0,\] (1) whereby $S$ is dimensionless. The number $0$ on the right hand side of Eq. 1 is special. The attempt to give $S$ and $0$ dimensions, a scaling $0\rightarrow\lambda 0$, is unobservable. If $S$ could be given a meaningful scale, it would violate quantum homogeneity. Eq. 24 in Section 2.2.3 expresses the repercussions of this symmetry for $\hbar$ and $m_{e}$. At this point the question of “what is meant by quantum mechanics” enters, because some will insist on giving action dimensions in their theory. Almost a century after discovery, there remain too many possibilities to resolve what quantum mechanics might be. In order not to offend anyone’s definition, we set up our own system. It is based largely on dispensing with pre-history dominated by Newtonian work, energy, and the $MKS$ unit system that so sorely needed Planck’s constant. Due to that history there are different default meanings of “action.” In Eq. 1 the symbol $S$_does not_ refer to a Newtonian expression. It refers to the Action of the quantum mechanical wave function, and quantum fields, as explained below. For emphasis, we completely abandon the idea that the _classical action_$S_{cl}$_in classical Newtonian units_ has any predictive power for 21st century formulation of fundamental theory. (Once again this should not be controversial, but to avoid backlash we will call it “our theory.”) In Eq. 1 the symbol $S$ refers to the complete action of the Universe, which we believe has no absolute scale. When Eq. 1 is taken seriously the Hamiltonian $H=-\partial S/\partial t$ has dimensions of $time^{-1}$. It happens that measurements of time and frequency are by far the most accurate in current experimental physics. We believe this is not a technological accident. It seems to be something fundamental, and inherent in quantum theory; it certainly emerged when technology became mature and self-consistent with quantum theory. In recognition of dimensionless action, our primary dynamical equation computes the time evolution of the wave function without introducing extraneous units: \[\mathrm{i}\dot{\psi}=\hat{\Omega}\psi\] (2) Here $\hat{\Omega}$ is the _frequency operator_. The equation computes frequencies in just the same units where frequencies are measured. For example, the excitations of the Hydrogen atom were first observed in frequency (wave number) units. There is no need to convert the observed frequency to Newtonian “energy” and back to predict frequencies. The intermediate step known as “Planck’s constant” is avoided. If not obvious at this stage, any physics which can avoid a fundamental constant is new and worth examining. Dropping redundant constants at the first stage cleans up an unlimited number of relations downstream. For example: The Planck spectrum of a black body is naturally a function of frequency with a single frequency parameter. The frequency-temperature does not need to be converted to temperature in degrees Kelvin, nor do degrees Kelvin need conversion to Newtonian “energy” and back to completely describe the spectrum. Yet multiple conversions of units are a fact of the standard Planck spectrum, conventionally expressed with a term going like $(exp(\hbar\omega/k_{B}T)-1)^{-1}$. Quantum theory came out of thermodynamics and its definition of energy. When the general proportionality of energy and frequency was discovered, it was associated with whole-numbers of “quanta” also found to exist, leading to the term “quantum mechanics.” Yet now “quantum mechanics” is a misnomer. Whole number units of physical quantities are not matters of principle. They are dynamical facts predicted by dynamical equations, when and if they occur. That was supposed to replace and eliminate the “unit of action” postulated to mock up quanta using pre-quantum Newtonian physics. Yet the cult of the quantum of action remains strong. It seems to be an inherent feature of what is commonly called quantum theory. In comparison, we find faults in the Newtonian precepts that led to a quantum of action in the first place. Newtonian _inertial mass_, set up by convention in certain units, is the culprit and something we can do without in a fundamental theory. #### 1.0.1 Avoidable Constants Early physics was full of arbitrary units for seemingly unrelated quantities that were later united by a theory. Energy, temperature, and time evolution at first were unrelated. The number known as Boltzmann constant $k_{B}$ had a different status in 1900 than today. Planck’s writings show he believed it was fundamental. He treated both $h$ and $k_{B}$ as _universal constants_ with great symmetry, including both in setting up his system of“fundamental” units. At the time “temperature” as the average energy of atoms was a _hypothesis_, and energy itself was not absolutely established as more fundamental. The interpretation of Boltzmann’s constant has evolved. It is uncontroversial to call Boltzmann’s constant an _avoidable_ constant of unit conversion. Avoidable constants represent human convention. Avoidable constants can be measured: one can measure the number of centimeters in an inch by comparing two rulers. However avoidable constants cannot be measured until a convention is imposed to define them: they are “absolutely unobservable,” but conditionally observable by introducing outside conventions for definition. There is nothing incorrect in setting up an arbitrary scale for action based on a reproducible subsystem, so long as it is recognized as arbitrary. The arbitrariness of units of energy and time are recognized by everyone, while action got a special dispensation. It appears this comes from circularly defining canonical transformations to exclude a phase-space scale transformation. In much the same way that general relativity allows scale transformations on space-time, because they can’t be prohibited, our approach to quantum mechanics is unchanged when the scale of action is globally revised. It is a minor change to permit a global transformation, but there are consequences. One consequence of avoiding redundant physical constants is that other constants can be determined better. For example: the fine structure constant is conventionally defined by $\alpha=e^{2}/hc$, where (in history) the electric charge $e$, the speed of light $c$ and Planck’s constant $h$ had previously been identified. When $\alpha$ is given that definition, the experimental errors in its determination are inherited from $e$, $h$ and $c$. Are any of these constants avoidable? That depends on the theory. In the theory where $c$ is an absolute constant of free space we agree that $c$ is observable. Yet the number for $c$ depends on the units of length and time. By sensible agreement decades ago, the distance scale of the $meter$ was eliminated in terms of a reference value of $c$ and a frequency standard. It is important that theory played a role. If the theoretical symmetries of special relativity had not been accepted, the errors of the Internatlonal Prototype Meter (a platinum bar) would now dominate physics. With $c$ eliminated the errors in $h$ currently dominate errors in $\alpha$. In our approach $\alpha$ comes directly from data, without intermediate definitions of $h$ or $e$. Then from data we can determine $\alpha$ more precisely. It also turns out that the value of $e$ is an avoidable fiction based on concepts of point charges, Coulomb’s Law, and Newtonian physics that offer nothing fundamental. We avoid $e$ entirely, since it does not really occur anywhere: when used in calculations, it is shorthand for $\sqrt{\alpha}$. When the numerical value such as $c$ is reduced to a definition, something experimentally testable is given up. Such decisions, if well-motivated by theory, stand to be reviewed from time to time, and the topic of Lorentz symmetry violation happens to be very active. Whether or not one is interested in Lorentz violation, it is still the right decision to eliminate $c$. That is because fundamental constants must be used self-consistently. This is not always appreciated. The electron mass we mentioned might mean a Newtonian inertial parameter to one physicist, a number in Schroedinger’s or Dirac’s theory to others, and a feature of the Higgs coupling to someone else. None of these definitions are the same. Relating one usage to another needs a theory, which will involve the error bars and systematics of the theory. I will show how to significantly improve the electron mass defined as a parameter in the fundamental Lagrangian. Our definition should not be controversial. Those familiar with perturbative renormalization know this is surprisingly intricate, and ultimately dependent on conventions. More surprisingly, the best current determinations of the “electron mass” continue to refer to the inertial mass of a _Newtonian electron_ ! Planck’s constant is again responsible for the fault. If one insists on measuring Newtonian electron masses in kilograms, the fundamental arbitrariness and inaccuracy of the kilogram enters both in the measurement and in Planck’s constant. Giving up Planck’s constant is a not mathematically challenging, but just as replacing $c$ by a constant represents a theory, it may seem momentous. We find it sharpens a vision of quantum mechanics itself, where many historical holdovers stand out more clearly and can be eliminated. Describing a theory without Planck’s constant is quite a bit simpler than explaining every apparent paradox of the theory built to perpetuate Planck’s constant. Because the historical elements dominate current thinking, much of the paper is concerned with reconsidering elementary issues from a new point of view. The paper is written so readers not concerned with the puzzles of conventional theory can turn straight to Section 4. #### 1.0.2 Organization of the Paper Section 2 presents a “derivation” of Schroedinger’s dynamics along the lines of effective field theory. It is elementary, yet novel, avoiding the illusory “golden road” of predicting quantum mechanics from Newtonian physics. The new approach shows that $\hbar$ does not come from comparing quantum theory to data. Section 3 reviews the concept of mass and energy. Snippets of history known as “modern physics” are interesting to consider from the new viewpoint. How $\hbar$ came to have its special numerical value is explained in Section 3.1. Measurement,“spin”, the uncertainty principle and several other issues are considered in Section 3.2. There is no perfect ordering of many possible topics, and a short discussion was felt preferable to an encyclopedic compendium answering every possible question. Section 3.3 considers whether the Planck scale of gravity involves Planck’s constant. The fact that eliminating Planck’s constant improves the determination of other constants is discussed in Section 4. ## 2 Dynamics The non-relativistic Schroedinger equation is often “derived from” Newtonian physics using substitutions rules involving Planck’s constant. With respect for the founders, it might be considered absurd _a priori_ in the 21st century to “derive” quantum physics, which is more fundamental, from classical physics, which is an approximation. A different derivation makes no reference to that starting point. #### 2.0.1 Derivation Quantum systems have wave attributes. A generic linear wave equation for amplitude $\phi$ is \[{\partial^{2}\phi\over\partial t^{2}}+c_{}^{2}\vec{\nabla}^{2} \phi=W(\phi).\] (3) Symbol $c_{}$ has dimensions of speed, and might depend on the system (label “$$”). The coupling $W(\phi)$ on the right hand side of Eq. 3 is often omitted in “wave equations”, but something must couple waves to the world. The simplest possibility is a linear dependence: \[W(\phi)=W(\vec{x})\,\phi(\vec{x}).\] The coupling $W(\vec{x})$ will be fit to experimental data, as we review momentarily. It is convenient to extract the constant part of $W$ from the spatially varying part $\tilde{W}(\vec{x})$: \[W(\vec{x})=\omega_{}^{2}+\tilde{W}(\vec{x}).\] (4) Substitution gives \[{\partial^{2}\phi\over\partial t^{2}}+c_{}^{2}\vec{\nabla}^{2} \phi=\omega_{}^{2}\phi+\tilde{W}(\vec{x})\phi.\] A simple transformation removes the $\omega_{}^{2}$ term. Define \[\phi=e^{-i\omega_{}t}\psi,\] (5) where $\psi$ is a symbol for $e^{i\omega_{}t}\phi$. If $\phi$ was real-valued then $\psi$ becomes complex by an act of notation. In discussing spin and charge we will assume $\phi$ is complex from the beginning. Putting Eq. 5 into the wave Eq. 3 gives \[{\partial\phi\over\partial t} = e^{-i\omega_{}t}(-i\omega_{}\psi+{\partial\psi\over\partial t});\] \[{\partial^{2}\phi\over\partial t^{2}} = e^{-i\omega_{}t}(-\omega_{}^{2}\psi-2i\omega_{}{\partial\psi \over\partial t}+{\partial^{2}\psi\over\partial t^{2}}).\] (6) The $\omega_{}^{2}$ term cancels. We drop $\partial^{2}\psi/\partial t^{2}<<\omega_{}{\partial\psi/\partial t}$ for the low-frequency limit. The steps reduce the basic wave equation to \[i{\partial\psi\over\partial t}=-{c_{}^{2}\over 2\omega_{}}\vec {\nabla}^{2}\psi+{\tilde{W}(\vec{x})\over 2\omega_{}}\psi.\] Simplifying the symbols gives \[i{\partial\psi\over\partial t}=\hat{\Omega}\psi.\] (7) Here $\hat{\Omega}$ is the _frequency operator_, in these approximations given by \[\hat{\Omega}=-{c_{}\lambda_{}\over 2}\vec{\nabla}^{2}+U(\vec{x}).\] where² [FOOTNOTE:2][ENDFOOTNOTE] \[\lambda_{}={c_{}\over\omega_{}};\:\:\:\:\:\:U(x)={\tilde{W}( \vec{x})\over 2\omega_{}}\] Symbol $U(x)$ will be called the “interaction function”. In the next Section we will find the interaction function for electrons from experiment. While much of that will be familiar, it is important to be conceptually complete and independent. Rather than repeat historical arguments in the historical order, we will keep track of each source of information and its consequences for the quantum system without imposing any Newtonian prejudices. Comment:Some will recognize our approach as a low-frequency effective field theory. It is possible to object that the quantum mechanical wave functions are not classical fields. This is true, but irrelevant. No matter how $\psi$ is interpreted, $\hbar$ is not needed to set up the dynamics. There is a related issue from old probabilistic arguments introducing $\hbar$ early, and reasoning on its basis to supposedly predict elements of the dynamics. It is evidently not needed, because we’ve not used it. Heisenberg’s matrix mechanics predated Schroedinger’s equation via a presentation that made Planck’s constant appear indispensable. For our purposes it is only necessary to define the dynamics once. _Given_ the frequency equation without reference to $\hbar$, the corresponding Heisenberg operator equations of motion follow without $\hbar$. More discussion appears in Section 3.3. ### Solutions and Data Fitting <figure><img src="content_image/1203.5557/x1.png"><figcaption>Figure 1: J. A. Crowther’s 1910 data on electron scattering from Rutherford’s1911 Philosophical Magazine article. The column ϕ/√t (in units ofradianscm−1/2) refers to the target thickness t for which half the total fluxwas scattered to angle exceeding ϕ. N is Rutherford’s computation of nuclearcharge Z, whose relation to the atomic weight was not established at the time.The table yields a parameter ^σ∼1±0.3×10−26cm2.</figcaption></figure> We briefly discuss fitting the constants of the frequency equation to experimental data. We follow the unusual but self-consistent path of not introducing Newtonian assumptions. All the data necessary was available before 1925, but we also consider the 1927 work of Davison and Germer. Rutherford interpreted the data of Geiger and Marsden’s data on alpha-particles, and Crowther for beta particles, electrons. Rutherford’s main contribution was a classical scattering model we see no reason to repeat. Rutherford fit an electron-atom cross section given by \[{d\sigma\over d\Omega}=({c\over v_{0}})^{4}Z^{2}{\hat{\sigma} \over sin^{4}(\theta/2)}.\] (8) From basic _wave mechanics_ of our scattering theory in Born approximation³ the $sin^{4}(\theta/2)$ angular distribution _predicts the interaction function_ going like $1/r$: [FOOTNOTE:3][ENDFOOTNOTE] \[U(\vec{x}) = {\kappa Zc_{e}\over r},\] (9) where the constant $\kappa$ is dimensionless. There is tremendous information in the shape of the interaction function. Using the modern values of $Z$, we fit the 1910 data (Fig. 1) to find $\hat{\sigma}=(1.03\pm 0.28)\times 10^{-26}\,cm^{2}.$ The error is statistical; Crowther’s later data indicates this has systematic errors of relative order 50-100%. The number implies \[\kappa \sim ({2\times 10^{-13}cm\over\lambda_{e}})\,({c\over c_{e}})^{2}.\] (10) One combination of constants has been determined. * Bound states and characteristic frequencies (normal modes) are found by solving \[\hat{\Omega}\psi_{n}=\omega_{n}\psi_{n}.\] Wavelengths and frequencies are the observables of spectroscopic experiments. The wave equation predicts zero current and no radiation from frequency eigenstates, resolving a pre-quantum puzzle. When there is a current, it predicts radiation frequencies that are differences of electron eigenfrequencies $\Delta\omega_{n_{1}n_{2}}=\omega_{n_{1}}-\omega_{n_{2}}$. <figure><img src="content_image/1203.5557/x2.png"><figcaption>Figure 2: Excerpt from the minutes of Society for Physics and Mathematics(Lund, Sweden) of November 5th 1888, showing Rydberg’s formula. From LundUniversity Physics[4].</figcaption></figure> Rydberg observed for the Hydrogen atom \[\Delta\omega_{n_{1}n_{2}}=2.07\times 10^{16}({1\over n_{1}^{2}}-{ 1\over n_{2}^{2}})\,s^{-1}.\] (11) Here $n_{1}\geq 1$, $n_{2}\geq 1$ are integers⁴. It is very significant that Rydberg’s frequency spectrum is consistent with the prediction from the scattering interaction, so that the interaction function has been determined twice. The data for $\omega_{H}$ fixes one new combination of parameters: [FOOTNOTE:4][ENDFOOTNOTE] \[{\kappa^{2}c_{e}\over 2\lambda_{e}}=2.07\times 10^{16}\,s^{-1}.\] (12) <figure><img src="content_image/1203.5557/x3.png"><figcaption>Figure 3: Davisson and Germer’s data for electron intensities scattered fromNickel as a function of azimuthal angle relative to the beam at fixed polarangle θ=50o.</figcaption></figure> * In 1927 Davisson and Germer[5] observed electrons scattering from Nickel crystals. They used the crystals as diffraction gratings and measured the scattered angular distribution of electrons. Our fit to Fig. 3 gives and observed wavelength $\lambda=1.65\times 10^{-8}$ cm. For this measurement the $DG$ beam had a frequency⁵$\omega=3.97\omega_{H}$. The experiment fixes one parameter in the free-space dispersion relation, [FOOTNOTE:5][ENDFOOTNOTE] \[c_{e}\lambda_{e}=1.14cm^{2}/s.\] * The wave model predicts a characteristic length scale $a_{0}=\lambda_{e}/\kappa$ for the ground state of Hydrogen. By 1899 Dewar had already produced solid Hydrogen. Its density of 0.07 $gm/cm^{3}$ at the boiling point is reported in 1904, “Physical Constants at Low Temperatures. (1)–The Densities of Solid Oxygen, Nitrogen, Hydrogen, etc”[3]. Estimating $r_{H}\sin 3a_{0}$ and atomic volume $4\pi r_{H}^{3}/3$ gives $r_{H}\sim 2.88\times 10^{-8}\,cm$. Then \[{\lambda_{e}\over\kappa}\sim{2.88\times 10^{-8}\over 3}\sim 10^{- 8}\,cm.\] (13) This is a relatively crude estimate, but no harm is done by including it. <figure><img src="content_image/1203.5557/x4.png"><figcaption>Figure 4: Compton’s data from the 1923 paper. The legend cites the observablescale 0.0485A0=4πλe.</figcaption></figure> * Compton’s 1923 paper[9] ”A Quantum Theory of the Scattering of X-Rays by Light Elements” treated electrons and photons as relativistic particles conserving energy and momentum. Compton’s early interpretation continues to be cited as a proof of both a particle interpretation and the necessity of Planck’s constant. Our approach uses neither because they add no information. Take a moment to review the calculation. From Lorentz covariant perturbation theory, the differential cross section $d\sigma$ is given in standard form by \[d\sigma={d^{3}k_{1}^{\prime}\over(2\pi)^{3}2k_{1}^{{}^{\prime}0} }...{d^{3}k_{n}^{\prime}\over(2\pi)^{3}2k_{n}^{{}^{\prime}0}}(2\pi)^{4}\delta^ {4}(k_{1}+k_{2}-\sum_{f}k_{f}^{\prime})\|M\|^{2},\] where $M$ is a Lorentz-invariant scattering amplitude. Symbols $k^{\mu}=(\omega/c,\,\vec{k})$ are the frequencies and wave numbers of participating waves. The delta function comes from translational invariance, and gives \[k_{e}+k_{\gamma} = k_{e}^{\prime}+k_{\gamma}^{\prime};\] \[k_{\gamma}\cdot k_{e} = k_{\gamma}^{\prime}\cdot k_{e}^{\prime}=k_{\gamma}^{\prime}\cdot (k_{e}+k_{\gamma});\] \[(k_{\gamma}-k_{\gamma}^{\prime})\cdot k_{e} = k_{\gamma}^{\prime}\cdot k_{\gamma}.\] (14) The electron wave dispersion relation is \[({\omega_{e}\over c})^{2}-\vec{k}_{e}^{2}-1/\lambda_{e}^{2}=0.\] In the rest frame the electron wave vibrates with $\vec{k}=0$ and frequency $c/\lambda_{e}$. Evaluating Eq. 14 gives \[({\omega_{\gamma}\over c}-{\omega_{\gamma}^{\prime}\over c}){c \over\lambda_{e}}={\omega_{\gamma}\omega_{\gamma}^{\prime}\over c^{2}}(1-cos \theta).\] Dividing both sides by $\omega_{\gamma}\omega_{\gamma}^{\prime}$ gives the change in the photon wavelength $\Delta\lambda_{\gamma}$: \[\Delta\lambda_{\gamma}=4\pi\lambda_{e}\,sin^{2}(\theta/2).\] While Compton need extraneous constants to make the calculation, none appear in this approach. Fig. 4 shows Compton’s data from the original paper, and the fit: \[\Delta\lambda_{\gamma}=0.485A^{0}sin^{2}(\theta/2).\] Compton’s data gives \[4\pi\lambda_{e}=4.85\times 10^{-11}\,cm;\] \[\lambda_{e}=3.86\times 10^{-11}\,cm.\] (15) ### Summary of Preliminary Fits Source \| Data \| Relation ---\|---\|--- Crowther/Rutherford \| Electron-Atom Scattering \| (ce/c)2κλe∼2×10−11cm Rydberg \| Hydrogen Frequency Spectrum \| κ2c2λe∼4.14×1016s−1 DavissonandGermer \| Electron-Nickel Scattering \| ceλe∼1.14cm2/s Dewar \| Solid Hydrogen Density \| λe/κ∼10−8cm Compton \| Electron-Light Scattering \| λe∼3.86×10−11cm. Table 1: Summary of experiments and parameter relations in a preliminary fit to the quantum theory, as discussed in Section 1. Table 1 summarizes the experiments and parameter relations each implies. Since early data seldom included error bars, no great effort has been made to assess errors in this preliminary exercise. Each of the independent relations given in Table 1 predicts a relation of parameters $f_{j}(\kappa,\,c_{e})=constant_{j}$ We define $\chi^{2}=\sum_{j}\,(f_{j}(\kappa,\,c_{e})/constant_{j}-1)^{2}$, in order to scale out the absolute size of numbers and units. While $\chi^{2}$ is not weighted by errors, which are unavailable, it can be rescaled by any uniform error estimate $\sigma^{2}$. Figure 5 shows two contour plots of $\chi^{2}(\kappa,\,\lambda_{e})$ with $c_{e}\to c$ (left panel) and $\chi^{2}(\kappa,\,c_{e})$ with $\lambda_{e}\to Compton^{\prime}s\,value$ (right panel). <figure><img src="content_image/1203.5557/x5.png"><figcaption>Figure 5: Contours of χ2, the summed-squared differences of data versus fitobtained from the independent relations given in Table 1. Left panel: As afunction of parameters κ and λe with ce→c. Right panel: As a function ofparameters κ and ce with λe given by Compton’s value. Dots shows the points ofminimum χ2∼0.24 in both cases. Contours are χ2=1, 2, 3… Lines show modernvalues c=3×1010cm/s,λe=3.87×10−11,κ=1/137 all lie inside the range of χ2≲1.</figcaption></figure> The best fit value $\chi^{2}\sim 0.24$ comes in both cases, with preliminary values of $\kappa$ and $c_{e}$: \[\kappa = 0.0061={1\over 164};\:\:\:\:\:\lambda_{e}=3.65\times\,10^{-11}\, cm\:\:\:\:\:(c_{e}\to 3\times 10^{10}\,cm/s)\] \[\kappa = 0.0066={1\over 151};\:\:\:\:\:c_{e}=2.78\times 10^{10}\,cm/s:\: \:\:\:(\lambda_{e}\to 3.86\times 10^{-11}\,cm).\] These results are very acceptable compared to modern values. Disregarding all the data in favor of Compton’s value of $\lambda_{e}$ and Rydberg’s determination of $\Omega_{H}$, which are probably more reliable yields a one-parameter relation \[\kappa^{2}c_{e}=1.59\times 10^{6}\,cm/s\] Assuming $c_{e}\to c$ as indicated by the rest of the data gives \[\kappa=\sqrt{{1.59\times 10^{6}\,cm/s\over 3\times 10^{10}cm/s}}= 0.0073={1\over 137.0}.\] Our interaction constant $\kappa$ came to be called the _fine structure constant,_ symbol $\alpha$, which we adopt: \[\alpha\equiv\kappa\sim{1\over 137}.\] It is important that $\alpha$ comes directly from the quantum data, without using extraneous definitions in terms of the classically-motivated concepts of $e$ and $h$. #### 2.2.1 Where is Planck’s Constant? The development shows quantum mechanics can be developed without Planck’s constant. It is absent in a theory developed directly from quantum mechanical data. Each quantum system, such as the electron, muon, proton… is characterized by a frequency scale $\omega_{}$ or equivalent length scale $\lambda_{}$. A few of these fundamental constants are given in Table 2. They can be found from data by ratios. For instance, a simple comparison of muonium to positronium level frequencies will give $\omega_{\mu}/\omega_{e}\sim 207$. System \| ω∗ (s−1) \| λ∗ (m) ---\|---\|--- e \| 7.79 ×1020 \| 3.862×10−13 μ \| 1.604 ×1023 \| 1.869×10−15 τ \| 2.698 ×1024 \| 1.111×10−16 p \| 1.43 ×1024 \| 2.104×10−16 n \| 1.427 ×1024 \| 2.101×10−16 γ \| 0 \| ∞ W± \| 1.221 ×1026 \| 2.455×10−18 Z0 \| 1.385 ×1026 \| 2.165×10−18 Table 2: Characteristic frequency (ω∗) and length (λ∗ ) scales of selected physical systems. The electromagnetic interaction is characterized by a pure number $\alpha\sim 1/137$. The strong and weak interactions have similar dimensionless parameters. It is impossible to compute Planck’s constant from these numbers. #### 2.2.2 Non-Relativistic Action As conventional, our quantum dynamics (Eq. 7) can be expressed using an action principle. It is the outcome of varying the action $S_{QM}$: \[S_{QM}=\int dt\,L_{QM}=\int dt\int d^{3}x\,i\psi^{}\dot{\psi}- \psi^{}\hat{\Omega}\psi.\] (16) Variation by $\psi^{}$ gives \[\mathrm{i}\dot{\psi}-\hat{\Omega}\psi=0,\] (17) from which the frequency operator $\hat{\Omega}$ is identified. After solving the equations the numerical value of the action can be computed: \[S_{QM}=0.\] Observe that the Lagrangian $L_{QM}$ has the form \[L_{QM}=\sum_{x}\,p_{x}\dot{q}_{x}-H(q_{x},\,p_{x}),\] where \[q_{x} = \psi(x);\] \[p_{x} = {\delta L_{QM}\over\delta\dot{q}_{x}}=i\psi(x)^{}.\] (18) Here $\sum_{x}$ is discrete notation for the volume integral. The fact that $i\,\psi^{}$ are the canonical momenta of the field provides a physical interpretation of observables $<\psi\|\hat{A}\|\psi>=\sum_{x}\,p_{x}\hat{A}_{xx^{\prime}}q_{x^{\prime}}$. This is a projective map _from_ many canonical variables _into_ certain canonical quantities with definite transformation properties. Neither probability interpretation nor semi-classical arguments involving Planck’s constant are needed to explore this. For example, a bulk translation of the coordinate system is defined by \[\psi(x)\rightarrow\psi_{\vec{a}}(\vec{x})=\psi(\vec{x}-\vec{a}).\] Noether’s theorem using $L_{QM}$ (Eq. 16) then predicts the conjugate total momentum: \[\vec{P}=\int d^{3}x\,\psi^{}(-i\vec{\nabla})\psi=<-i\vec{\nabla}>.\] (19) Since $<\psi\|\psi>$ drops out of observables we normalized it to one. By using Noether’s theorem Eq. 19 reproduces a familiar step of traditional theory involving heuristic (and actually redundant) quantum postulates. Since $\vec{P}$ are full-fledged canonical variables, there are conjugate variables $\vec{Q}$, which we represent with an projective map involving operator $\vec{\cal Q}$, and its transformation properties: \[\vec{Q}=d^{3}x\,\psi^{}\vec{\cal Q}\psi;\] \[\vec{Q}\rightarrow\vec{Q}+\vec{a}.\] The basic test of conjugacy lies in the Poisson bracket: \[\{Q_{i},\,P_{j}\}_{PB}=-i\sum_{x}\,\left({\delta Q_{i}\over\delta \psi_{x}}{\delta P_{j}\over\delta\psi_{x}^{}}-{\delta P_{j}\over\delta\psi_{x }}{\delta Q_{i}\over\delta\psi_{x}^{}}\right).\] Computing the derivatives gives \[\{Q_{i},\,P_{j}\}_{PB}=-i\sum_{x}\psi_{x}^{}[{\cal Q}_{i},\,{ \cal P}_{j}]\psi_{x}.\] With $<\psi\|\psi>=1$ we obtain the map between the operator algebra and Poisson bracket \[\{Q_{i},\,P_{j}\}_{PB}=\delta_{ij}\rightarrow[{\cal Q}_{i},\,{ \cal P}_{j}]=i\delta_{ij}\] (20) That is, Noether’s theorem and group representations predict the semi-classical operator substitution rules postulated early in quantum mechanics are kinematic consequences of the wave theory[12]. The reason our approach does not need Planck’s constant, while the historical one did, lies in accepting an infinite number of degrees of freedom for the “electron” in the first step. It is an experimental fact of quantum waves. In order to obtain a constant $\lambda$ with any desired dimensions on the right hand side of Eq. 20, it is sufficient to multiply the action (Eq. 16) by the same constant, which has no consequences whether $\lambda=\hbar$ or any other value. #### 2.2.3 Path Integrals and Special Relativity Path integrals are often cited as a starting point for quantum field theory. One might ask whether quantum field theory, which is a highly comprehensive approach, might more fundamental than basic quantum mechanics, even though developed as a generalization. It is possible to question whether field theory or relativity somehow puts Planck’s constant into the theory. Then consider the action _in engineering units_ for relativistic quantum electrodynamics, which is⁶ [FOOTNOTE:6][ENDFOOTNOTE] \[S_{QED}=\int d^{4}x\,\bar{\psi}(i\hbar\partial\hskip-7.499886pt \hbox to 7.499886pt{\hss\sl/\/\hss}-eA\hskip-7.499886pt\hbox to 7.499886pt{ \hss\sl/\/\hss}-m_{e})\psi-{1\over 4}F_{\mu\nu}F^{\mu\nu}.\] (21) Here $\psi$ is a Dirac field, $m_{e}$ is a constant called the Lagrangian mass parameter, and $F_{\mu\nu}$ is the electromagnetic field strength tensor. Now to wrongly “prove” that Planck’s constant is involved, each field configuration in the path integral is weighted by $exp(iS_{QED}/\hbar)$, in which we see $\hbar$ explicitly. The claim is false, because the action of Eq. 21 was multiplied by $\hbar$ when formulated by historical substitution rules. The constant naturally cancels out in expressions using $S/\hbar$. In more detail, the path integral is invariant under a change of measure. Let $A_{\mu}^{\prime}=eA_{\mu}/\hbar$. At each point in space-time, let \[d[A]d[\psi]d[\bar{\psi}] = d[\Phi];\] \[d[\hbar A^{\prime}/e]d[\psi]d[\bar{\psi}] = d[\Phi]^{\prime}.\] Observable correlations are represented by \[<O(A,\,\psi,\,\bar{\psi})> = {\int d[\Phi]\,e^{iS_{QED}/\hbar}O(A,\,\psi,\,\bar{\psi})\over \int d[\Phi]\,e^{iS_{QED}/\hbar}},\] \[= {\int d[\Phi]^{\prime}\,e^{iS_{QED}/\hbar}O(\hbar A^{\prime}/e,\, \psi,\,\bar{\psi})\over\int d[\Phi]^{\prime}\,d[\bar{\psi}]e^{iS_{QED}/\hbar}}\] The action exponent, including division by $\hbar$, transforms to \[S_{QED}/\hbar=\int d^{4}x\,\bar{\psi}(i\partial\hskip-7.499886pt \hbox to 7.499886pt{\hss\sl/\/\hss}-A\hskip-7.499886pt\hbox to 7.499886pt{\hss \sl/\/\hss}^{\prime}-\omega_{e})\psi-{1\over 4\alpha}F_{\mu\nu}^{\prime}F^{\mu \nu^{\prime}},\] (22) where \[\omega_{e}={m_{e}\over\hbar};\:\:\:\:\alpha={e^{2}\over\hbar}.\] (23) Notice the electric charge $e$ itself does not appear in Eq. 22. The two parameters $\omega_{e}$ (or $\lambda_{e}$) and $\alpha$ determine the path integral just as in our approach to quantum mechanics. Since $\hbar$ cancelled out, it cannot be measured in experiments using Eq. 22. However nothing stops one from keeping it in the formalism while respecting a parameter symmetry. When one uses three parameters to represent two constants, there is one symmetry: \[\hbar\rightarrow\xi\hbar;\:\:\:\:\:m_{e} \rightarrow \xi m_{e};\] \[{\hbar\over m_{e}} \rightarrow {\hbar\over m_{e}}={1\over\omega_{e}}\:\:\:invariant.\] (24) ## 3 What Do We Mean by Energy and Mass? #### 3.0.1 Newtonian Inertial Mass The “mass” of Newton’s era was a “measure of the quantity of matter.” It is strongly tied up with intuitive, post-medieval expressions of “force” as a primary concept. The tradition constantly cites simplistic ideas of _additivity_ of mass and force, and probably cannot proceed without them. We claim those concept of are no longer a starting point for fundamental physics. They are irrelevant in general, but relevant here to find out how $\hbar$ crept into physics early. The Hamiltonian describing Newton’s world is \[H_{N}=\sum_{i}\,{\vec{p}_{Ni}^{2}\over 2m_{Ni}}+\sum_{ij}\,V_{ij }(\|\vec{q}_{i}-\vec{q}_{i}\|)\] This reproduces Newton’s three “Laws” of motion in Newton’s coordinates. Since it is not fundamental, we do not find a high obligation to reproduce the model from fundamental physics. But already by making a Hamiltonian the starting point, the theory has been revised, and the interpretation of the constants of the theory revised. Given the Newtonian mass as a parameter in the action, its meaning and units are derived from the action. On the naive basis that $\delta S=0$, let us explore classical physics of dimensionless action. By inspection of this _macroscopic_ action we have units \[q_{i} \rightarrow meters;\] \[p_{Ni} \rightarrow meters^{-1};\] \[H_{N} \rightarrow seconds^{-1};\] \[mass_{N} \rightarrow seconds/meter^{2}.\] This defines the $MK\hskip-7.499886pt\hbox to 7.499886pt{\hss\sl/\/\hss}S$ system where the kilogram ($K$) units of mass is never introduced by external standards. Can one do classical physics this way? Force is the gradient of the Hamiltonian: \[\vec{F}_{N}={d\vec{p}_{N}\over dt}=-\vec{\nabla}H_{N}.\] Force has units of $(meter\cdot seconds)^{-1}$. Consider a force of (say) 3 inverse $meter-seconds$ in the $x$ direction, and apply it to a mass of (say) 5 _seconds per square-meter_. Compute the acceleration \[a_{x}={F_{Nx}\over m_{N}}=3(meter\cdot seconds)^{-1}\,{1\over 5} \,{meter^{2}\over seconds}={3\over 5}{meters\over seconds^{2}}.\] Glue together two masses of $m_{1}=5\,seconds/meter^{2}$ and $m_{2}~{}=10\,seconds/meter^{2}$. What makes us think their inertial masses add? Go to center of mass ($cm$) and relative coordinates. Neglecting deformation of the glue, the relative coordinate drops out. Hamilton’s equations tell us the $cm$ coordinate has effective mass $m_{1}+m_{2}=15\,seconds/meter^{2}$. Applying the same force as before (using “spring balances”, etc) it will accelerate at 3/15 $meters/seconds^{2}$. Proceeding this way we can re-build classical physics, including the theory of work, thermodynamics, steam-engines, entropy and the Planck distribution, without ever introducing the kilogram. There is a different theory where the mass parameter $m_{R}$ naturally has units of frequency. The “free particle” Hamiltonian in this theory is \[H_{R}=\sqrt{\vec{p}_{R}^{2}c^{2}+m_{R}^{2}}.\] (25) Since $m_{R}$ in Eq. 25 is a constant, it does not depend on the initial conditions of the theory. That produces one fact relating particles with energy and momentum $E_{R},\,p_{R}$ to those with a different energy and momentum $E_{R}^{\prime},\,p_{R}^{\prime},$ which is the Lorentz transformation. On dimensional grounds $m_{R}$ is simply a frequency scale for the energy, which has units of frequency. An explanation is needed why $m_{N}$ ever became popular. In the regime of small $p_{R}$, Eq. 25 becomes \[H_{R}\sim{\vec{p}_{R}^{2}c^{2}\over 2m_{R}}.\] If the two theories describe the same thing, then \[m_{R} = m_{N}c^{2};\] \[seconds^{-1} = {seconds\over meters^{2}}\times{meters^{2}\over seconds^{2}}.\] (26) So far we’ve shown units are consistent, without going to the step of unit standardization. In both the Newtonian and relativistic $MK\hskip-7.499886pt\hbox to 7.499886pt{\hss\sl/\/\hss}S$ systems it is sufficient to choose a certain lump of material (standard “mass artifact”), and declare that it has one (1) unit of $seconds/meter^{2}$ of Newtonian mass. Such an object traveling at 1 $meter/second$ has an energy of $mv^{2}/2=(1/2)\,seconds^{-1}$, defining the energy unit. The same object has $m_{R}=9\times 10^{16}\,seconds^{-1}$ of relativistic mass. Standardization is then tied to the mass artifact, which might have been a different lump of material, related by a simple scale factor. Different scale factors then translate to different units chosen for time. Our quantum mechanical fits to electron waves have given $\lambda_{e}=3.65\times\,10^{-11}\,cm$, or $\omega_{e}=c/\lambda_{e}=8.22\times 10^{20}\,seconds^{-1}$. This excellent time standard is the frequency of a free electron vibrating at rest. An agreement to make the number exact will define the time unit of the $second$ without referring to years, days, and minutes, or atomic clocks based on much more complicated theory. Indeed the resolution of an atomic clock is based on an approximate theory for the lifetime of certain spectral lines, while the lifetime of an electron appears to be infinite. Perhaps in the future the electron mass - or an easier to control atomic mass - will directly define the standard of time without redundant intermediates. We have reviewed how arbitrary standardization not coming from Nature affects Newtonian unit conventions. Only one more step of specifying the particular mass convention is needed to _derive_ Planck’s constant. ### Enter Kilogram By choosing a particular convention the conversion factor from our units of $mass$ and $energy$ to engineering units becomes specified. In brief: * Directly from fitting quantum data $\lambda_{e}\sim 3.86\times 10^{-11}\,cm$. The corresponding frequency scale in the quantum Lagrangian is $\omega_{e}=c/\lambda_{e}\sim 7.8\times 10^{20}s^{-1}$. * The Newtonian electron mass parameter $m_{Ne}=\omega_{e}/c^{2}=0.865\,s/cm^{2}$. This is a convenient macroscopic value. * By ratios, the proton and Hydrogen mass is $m_{H}\sim 1836\,m_{e}\sim 1588\,s/cm^{2}$. * To a good approximation, one mole ($6.02\times 10^{23}$) of protons defines a gram of mass, to relative errors of a few parts per thousand. The arbitrary human conventions of the “kilogram” enters here. It is also possible to define the kilogram by a reference standard object (“artifact”), and deduce Avogadro’s number from the mass of a mole. * In $cgs$ convention 1 $erg=1\,gm\,cm^{2}/s^{2}$, or \[1\,erg = 1600{s\over cm^{2}\,proton}6.02\times 10^{23}{protons\,cm^{2} \over s^{2}}\] \[\sim 9.63\times 10^{26}\,s^{-1}.\] The inverse relation is \[1\,{rad\over s} = 1.05\times 10^{-27}\,erg.\] The last line defines the conversion constant $h$ going from frequency to $MKS$ units. We have not bothered with high precision in the calculation. Unit conventions should be expressed with exact numerical values standardized in their definitions. ### Paradoxes, Measurements, Group Generators, Gravity Quantum theory is a large subject, so that challenging any one point can lead to distinct types of disagreement about what quantum theory actually implies. We granted early that anyone wanting to keep $\hbar$ in their conventions can do so without contradiction, although we find it redundant. What are more troublesome are false paradoxes and sometimes obstacles thrown up to maintain a pedagogical tradition. We are not particularly concerned with paradoxes affecting the “old” theory’s intricate need to maintain Planck’s constant, since as we mentioned we have a new theory. We plan to give a more comprehensive review of those fascinating side issues elsewhere[14], restricting our discussion here. The basic algorithm to explore questions goes as follows: Every traditional quantum mechanical formula involving $\hbar$ is either an independent fact of mathematics, or our theory, which has been multiplied by some power of $\hbar$ on both sides. Most physicists agree that equations are unchanged in content by multiplying both sides by the same constant. Yet a curious degree of resistance is sometimes found to applying the algorithm after its effects are realized, almost as if $\hbar$ should get a special variance from the rules of algebra. We will highlight a few topics briefly to contrast the absence of Planck’s constant in _our particular_ quantum theory: * The historical path to quantum mechanics invariably begins with Planck. Reviewing his original paper, it is interesting that Planck himself could not find his constant from his own analysis of black body spectral data. What Planck’s paper literally found[10] was a ratio $h/k_{B}=4.866\,.\,10^{-11}sec\,degree$. At that moment Planck was in a position to eliminate both constants in terms of a frequency parameter $\omega_{T}$ for the temperature: a spectrum involving $(exp(\omega/\omega_{T})-1)^{-1}$. However Planck’s archaic definitions of energy from thermodynamics made it impossible. His deep commitment to (and invention of) Boltzmann’s constant⁷ made it impossible for him to see either constant as avoidable. [FOOTNOTE:7][ENDFOOTNOTE] * The historical path often cites Einstein’s approach to the photoelectric effect. The“production of cathode rays by illumination of solids” was but one topic in a phenomenological paper citing several reasons for a new picture of particle-like quanta. Einstein found that Planck’s conversion factor of energy and frequency was consistent with a 1902 experimental paper of Lenard. Einstein wrote[20] “To see now whether the relation derived here agrees, as to order of magnitude, with experiments, we put $P^{\prime}=0$, $\nu=1.03\times 10^{15}$, (corresponding to the ultraviolet limit of the solar spectrum) and $\beta=4\cdot 866\times 10^{-11}$. We obtain $\Pi\times 10^{7}=4.3\,Volt$, a result which agrees, as to order of magnitude, with Mr. Lenard’s results.” The paper uses Planck’s constant $R\beta/N$ as Planck did, with $R$ the gas constant and $N$ the number of “real molecules” in gram-equivalent units (per mole.) Symbol $P^{\prime}=0$ sets the zero of the work function, and $\Pi$ is the voltage to reduce the photocurrent to zero, i.e. the photo-electron’s energy. Just as with Compton’s experiment, Einstein’s use of conservation of frequency expressed the kinematic fact of time-translational invariance in photoelectric scattering[16]. However at this early point the conventional unit of the $Volt$ had not been converted to frequency. We convert without the intermediary of $R$, $N$, $\beta$ or $h$ as follows: From Eq.11 the Hydrogen ionization frequency $\omega_{E}=2.07\times 10^{16}\to 13.6eV$ makes a fiducial _definition_ of the volt. It allows re-scaling frequency to frequency, from which Einstein’s 4.3 $Volt=(4.3/13.6)\times 2.07\times 10^{16}/s=6.55\times 10^{15}/s$. Compare $2\pi\nu=6.47\times 10^{15}/s$; the numbers agree within about 1%. The upshot is the photoelectric effect does not need Planck’s constant. * Reference to Planck’s constant involving “measurement theory” are common. A finite value of $\hbar$ is cited as responsible for lower bounds on disturbances caused in measurement. The example known as Heisenberg’s microscope it typical. The key steps of post-quantum measurement are different, and involve projecting onto a wave function $\|\psi_{2}>$ given a wave function $\|\psi_{1}>$. There is a convenient identity \[\|\psi_{1}>=\|\psi_{2}><\psi_{2}\|\psi_{1}>+\|\psi_{\perp}>,\] where $\|\psi_{\perp}>$ is in the orthogonal complement to $\|\psi_{2}>$. Since $\|\psi_{2}><\psi_{2}\|$ is a normalized projector, the equation says that $<\psi_{2}\|\psi_{1}>$ is the pre-existing amount of $\|\psi_{2}>$ already presenting in $\|\psi_{1}>$. The squared overlap $\|<\psi_{2}\|\psi_{1}>\|^{2}$ is identified as the probability of $\|\psi_{2}>$ given $\|\psi_{1}>$. When this describes an experiment, the measurement had to be sufficiently gentle that the pre-existing projection is found _without_ mixing in $\|\psi_{\perp}>$ or any other disturbing effects. In other words, any “uncontrollable disturbance” of measurement will _not_ be so simple that mere projection describes it. Whether one uses wave functions or density matrices for observables, Planck’s constant does not appear in the Born rule. * In certain cases the fixed normalization of wave functions $<\psi\|\psi>=1$, coupled to an absolute value of Planck’s constant, is thought to be responsible for “quantization of energy levels”. It is true that boundary conditions enter quantization but not the overall scale. If the quantum wave of an infinite square well, say, were interpreted classically, the overall amplitude would enter the total classical energy. Once the wave function is normalized the energy is fixed. Here again we must recognize _quantum homogeneity symmetry_, which tells us the square well frequencies are the energies no matter how the normalization is set. This is admittedly a radical revision of the concept of energy, but not explained (nor improved) by insisting it be expressed in any particular system of units. * Questions of the type, “Without $\hbar$ how are you going to quantize the harmonic oscillator” refer to schoolbook exercises. They have little to do with Nature. When setting up new models the parameters tend to come from previous models (values of $\omega_{e}$, etc.) and whatever fudge-factors are needed in the lab. The question “could we find a primed-system where $i\hbar^{\prime}\dot{\psi}=\hat{H}^{\prime}\psi$ would give us a new value of $\hbar^{\prime}$” has been answered repeatedly. Physicists fit Hamiltonians to data using the existing unit conversion factors, by convention. Exact agreement of one universal value of Planck’s constant has been made trivial by universal practice: if not universal discussion. * Textbooks[15] tend to cite the Stern-Gerlach experiment as yielding an inexplicable quantization with directly observable units of $\hbar$. The experiment is truly inexplicable in a straw-man context of Newtonian point particles there is no reason to consider. In the context of the Schroedinger-Pauli equation one computes solutions to scattering off a non-uniform magnetic field. Solutions predict that beams with orthogonal spins separate. Since $\hbar$ is absent in the equation it is not in the solutions. The fact a particular polarization is strictly correlated with each beam is mathematically true and interesting. We cannot find new information in predicting it by an external principle that the eigenstate of the Pauli spin operator $\hbar\sigma_{z}/s$ is“measured.” It is circular, and the same statement that a eigenstate of $\sigma_{z}/2$ appears in the correlation. * Quantization of angular momentum is cited in elementary treatments as a measure of $\hbar$. From Noether’s theorem in the Schroedinger model, the operator yielding orbital angular momentum is $\vec{L}=-i\vec{x}\times\vec{\nabla}$. It is a fact of mathematics that eigenstates of $L_{z}$ have whole number labels and a whole number of nodes. There is no logic in calling this quantization “also” a prediction of a Principle or Axiom of physics: once a math fact is a math fact, making it a principle would be redundant. By the previous analysis the conversion constant from intrinsically dimensionless form to $MKS$ units originates in macroscopic physics. This may need reiteration so we will explain. In Newton’s world with Newton’s units it is a great mystery why quantum angular momentum is a whole number of $1.05\times 10^{-27}erg\,seconds$. We are challenged to explain why it is not $7.05\times 10^{-27}erg\,seconds$, or some other number, and is not the absolute number meaningful? Our answer is that the existence of whole numbers was explained by the wave theory and counting the nodes of spherical harmonics. It is fine and wonderful that $L_{z}Y_{\ell m}=mY_{\ell m}$, and $\vec{L}^{2}Y_{\ell m}=\ell(\ell+1)Y_{\ell m}$. By an intricate process the number “1” for each unit was converted to $1.05\times 10^{-27}erg\,seconds$ when humans introduced the gram, kilogram, and Avogadro’s number, following Eq. 3.1. Then we agree the number is important and necessary for commerce and engineering, which surely need the gram, kilogram, and Avogadro’s number. We don’t need them in fundamental physics, and prefer to designate 1 unit as 1 unit. * _Spin_ is often misidentified as “coming from” analogies with Planck’s constant. A quantum system with spin is described by a wave function $\psi_{a}(\vec{x},\,t)$, where $a$ is the polarization index of the spin representation. Although representation theory came into physics after Planck’s constant, it is absolutely independent and stands on its own. The mathematics of the rotation group predicts $2\times 2$ generators acting on a spinor space with the algebra \[[{\sigma_{i}\over 2},\,{\sigma_{j}\over 2}]=i\epsilon_{ijk}{ \sigma_{k}\over 2}.\] It is an empty act of notation to define $\vec{S}=\hbar\vec{\sigma}$. The notation predicts \[[\,S_{i},\,S_{j}]=i\hbar\epsilon_{ijk}S_{k}.\] (27) A finite rotation is then expressed by $U(\vec{\theta})=exp(i\vec{\theta}\cdot\vec{S}/\hbar)$, in which $\hbar$ cancels out. By the same steps, any quantum commutation relation $[\hat{A},\,\hat{B}]=i\hbar\hat{C}$ is an ordinary algebraic relation $[\tilde{A},\,\tilde{B}]=i\tilde{C},$ where $\hat{A}=\hbar\tilde{A}$, etc. The commutation relations of $\tilde{A}$ describe a geometry that involves no physical scale. The elementary cancellation of $\hbar$ in commutation relations is seldom noticed, for reasons that can be explained. In early times the facts of Lie groups and commutation relations were new to physics, and misidentified as “quantum effects”. The illegal step of taking $\hbar\to 0$ on the right hand side of Eq. 27 with other symbols fixed was argued to produce the “classical limit” including language such as “in the classical limit operators commute.” We suggest the language and concepts fail to pass modern quality-control standards, and should be discouraged. There is no consistent sense in which $[{\sigma_{i}},\,{\sigma_{j}}]=0$. The classical limit is much more subtle than replacing operators by numbers, _except for_ a brief moment in history when selling quantum mechanics needed it. * The uncertainty principle is a powerful math fact relating the spread of wave numbers $\Delta k$ and the spread of size $\Delta x$ of a wave packet: \[\Delta k\Delta x\gtrsim 1/2.\] Multiplying both sides by $\hbar$ gives \[\hbar\Delta k\Delta x\gtrsim\hbar/2.\] (28) Introducing $\hbar$ made the first time in history where multiplying a math identity by the same constant on both sides was reported to make a new physical principle. It comes from $[\,x,-i\partial/\partial x]=i$, which is the trivial identity it appears to be. Acknowledging some sarcasm, it seems deceptive in the current millennium to talk about a particle (six canonical degrees of freedom) and simultaneously introduce Eq. 28 as a descriptive feature, concealing an infinite number of degrees of freedom that produced the identity. The sensible reason to write $[\,\hat{x},\,\hat{p}]=i$ as a commutator is to set up a coordinate-free algebra between symbols $\hat{x}$, $\hat{p}$. The underlying bracket algebra of a Hamiltonian system is an invariant concept no matter how it is expressed. Expressing Lie group relations with commutators represents high level discoveries of notation, not discoveries about Nature _per se_. * After the identity $[\,x,-i\partial/\partial x]=i$ was discovered useful, it was implemented again by “field quantization”. For every field $\phi(x)$ and its conjugate momentum $\pi(x)$, the equal-time commutation relations are \[[\,\pi(x),\,\phi(x^{\prime})\,]=-i\delta(x-x^{\prime}).\] This is a sophisticated invariant way of defining symbol $\pi(x)=-i\delta/\delta\phi(x)$, to wit, an identity in which canonical momenta are generators of canonical coordinates, which is what we mean by momenta. What is new is the promotion of the dynamics to another infinite dimensional space. The proposal is physics, but $\hbar$ is not involved in it. On the huge space of quantum fields there are natural time evolution generators $\hat{\Omega}=\hat{\Omega}(\pi,\,\phi)=\hat{\Omega}(-i\delta/\delta\phi(x),\, \phi(x))$. That is so general it excludes very little. Meanwhile historical models put great faith in local quadratic functions of $\pi$ and $\phi$ which paid off. Whether or not local quantum field theories are a good model of the Universe, “quantization” does not need Planck’s constant in our approach. ### Gauge Theories Relativity and gauge invariance explain a small puzzle in our development. In Section 2 we found the so-called Coulomb interaction function $U=\tilde{W}/(2\omega_{})\rightarrow\alpha/r$. To maintain the same number $\alpha$ for systems of different $\omega$, as observed, requires our initial symbol $\tilde{W}$ to be proportional to $\omega$. There is no obvious motivation for this in a generic non-relativistic effective field theory. The relativistic gauge-covariant derivative explains the puzzle. In simplest form replace $(i\partial/\partial t)^{2}\rightarrow(i\partial/\partial t-eA^{0})^{2}$. With $i\partial/\partial t\rightarrow\omega_{}$ as first approximation, the expansion to the non-relativistic domain produces $\tilde{W}\sim 2eA^{0}\omega_{}$ just as needed for a universal constant $\alpha$. In any event, the relativistic field theory rather than basic quantum mechanics becomes the arena to determine what constants are universal. Every known force is due to a gauge invariance of one kind or other. General relativity predicts that gravity couples the curvature tensor $R_{\mu\nu}$ to the energy momentum tensor $T_{\mu\nu}$: \[R_{\mu\nu}-{1\over 2}g_{\mu\nu}R=-{G_{N}^{\prime}\over 8\pi}\,T_ {\mu\nu}.\] (29) The dimensions of $R$ are inverse length-squared, and the dimensions of $T_{\mu\nu}$ determine those of $G_{N}^{\prime}$. Specifying the action $S$ fixes the rest: \[S = {1\over 16\pi G_{N}^{\prime}}\int d^{4}x\,\sqrt{g}R\] \[+ \sum_{j}\,m_{j}\,\int d\tau_{j}\,{\partial x^{\mu}\over\partial \tau_{j}}g_{\mu\nu}(x_{j}){\partial x^{\nu}\over\partial\tau_{j}}.\] Thus $G_{N}^{\prime}$ has units of square-meters. Since this scale is non-trivial, there is a good motivation for seeking new physics in it. The units and size of Newton’s constant in $M\hskip-7.499886pt\hbox to 7.499886pt{\hss\sl/\/\hss}K\hskip-7.499886pt\hbox t o 7.499886pt{\hss\sl/\/\hss}S$ units ($c=1$) is \[G_{N}^{\prime}=2.5\times 10^{-64}cm^{2}\] The scale stands on its own, and does not need Planck’s constant; It is refreshing to find no reason quantum mechanics needs to be relevant. It is interesting that the root-inverse of Newton’s constant in $M\hskip-7.499886pt\hbox to 7.499886pt{\hss\sl/\/\hss}K\hskip-7.499886pt\hbox t o 7.499886pt{\hss\sl/\/\hss}S$ units is a macroscopically large number not far from the 14 billion parsec size of the observable Universe: up to a relatively small factor of about 1500 that might be possible to explain. The non-relativistic coupling of gravity to matter is well-known. Since we have dispensed with Newtonian mass, it is interesting we do not need the concept of gravitational mass to express the coupling. If there are $N_{1}$ ($N_{2}$) localized quantum systems (particles) of scale $\lambda_{1}$ ($\lambda_{2}$) separated by $r$, the equivalent interaction function is predicted to be $G_{N}^{\prime}c\,N_{1}N_{2}/(\lambda_{1}\lambda_{2}r)$. Referring to $\lambda_{}$ parameters makes this well-defined in a quantum context. The approximate proportionality to wave numbers $1/\lambda_{}$, which in Newtonian physics add linearly when weakly interacting systems are composed, finally explains the origin of additivity of Newtonian mass in the form of “weight.” ## 4 Precision Fundamental Constants <figure><img src="content_image/1203.5557/x6.png"><figcaption>Figure 6: Compilation of the values and errors on the fine structure constantα from Ref. [18]. Over time α has become determined better and better byexperiments tying its definition to frequency just as we suggest.</figcaption></figure> Great care is mandatory in the field of fundamental constants, but there has always been two separate purposes. One purpose is standardization for commerce and engineering. Another purpose is testing fundamental physics. The two purposes are not the same, because the engineering usage is _obliged_ to focus on the kilogram and Planck’s constant, while fundamental physics (in our opinion!) is obliged to base nothing on them. Before continuing we note the exceeding care for consistency that has become standard in perturbative calculations in quantum field theory. In that regime coupling constants are defined in terms of definite subtraction schemes, renormalization points, and order-by-order procedures that are a world of their own. Those issues tend to be buried in the theory-blind standardization of fundamental constants, but they are not trivial. As far as we can tell, our approach has no effect on renormalization conventions, due to an unstated agreement everywhere not to renormalize Planck’s constant. Consider the electron mass evaluated by the CODATA group[18]. The uncertainty of the 2010 determination is $4.0\times 10^{-38}$ kg. The relative uncertainty ($u_{r}$) is $4.4\times 10^{-8}$. This uncertainty has changed very slowly with time. What is used for direct measurement of the electron mass? The most accurate determination of the electron mass[13] by Farnham _et al_ and cited by CODATA-2006 is based on the cyclotron frequency of _classical electrons_ orbiting in Penning traps. We observe that a _classical_ model of Newtonian electrons is a theory subject to numerous assumptions. The dynamics of particles in magnetic fields are not Newtonian, and the characteristic lifetime and frequency spread of cyclotron orbits due to synchrotron radiation is but one of the issues clouding the interpretation of the “mass” deduced from data. Farnham _et al_ state that the Penning traps work best with 5-13 electrons. It is seldom discussed that extending microscopic quantum propagation into the classical regime is exquisitely sensitive to uncontrolled tiny effects. That is because the “Ehrenfest relations” so convincing for a beginning treatment of “free” Schroedinger particles do not lead to high-precision theory of interacting particles, as far as we know. How to precisely formulate the concept of 5-13 Newtonian electrons does not appear in the references. What about Planck’s constant? The _Overview_ of the unpublished CODATA2010 adjustments states that a new value of Avogadro’s number obtained from highly enriched silicon has a $u_{r}$ of $3\times 10^{-8}$, providing an inferred value of $h$ with essentially the same uncertainty. This (indirect) uncertainty is somewhat smaller than $u_{r}$ of $3.6\times 10^{?8}$ of the most accurate directly-measured watt-balance value of $h$. Yet the two values disagree. That has led to a recommended $u_{r}$ of $h$ of $4.4\times 10^{?8}$, which is almost no change. This value coincides exactly with the relative error on $m_{e}$ just discussed. Since it is a matter for specialists we are in no position to compete with the goals and methods of dedicated groups such as CODATA in standardizing fundamental constants. On the other hand it is valid to estimate the effects on fundamental constants by modestly revising theoretical assumptions. Then, and for fundamental purposes in this Section, we abandon reference to the kilogram, Planck’s constant, and the classical electron mass. We define the electron’s inertial mass $m_{Ne}\equiv hc/2\pi\lambda_{e}$ to be an identity. The identity allows fixing $h$ to an exact reference value. As reviewed with Eq. 23 the constant of electric charge $e$ also does not appear in quantum physics, and the definition $e=\sqrt{\hbar c\alpha}\rightarrow\sqrt{\alpha}$ is also taken as an identity. It is worth mentioning that our decision involves physics, and some future technology might find the classical electron mass with superb precision, adding information. Either that process would confirm our decision on theory, or contradict it. If contradicted then the meaning and values of fundamental constants will evolve once more. We are concerned here with what can be accomplished within the theory of this paper. #### 4.0.1 Preliminary Constant Values <figure><img src="content_image/1203.5557/x7.png"><figcaption>Figure 7: Improved determination of ωe=c/λe and α from data. The dashed lineshows Eq. 12 from the Rydberg constant, whose errors are thinner than the linewidth. Information from the hyperfine interval of positronium with errors isshown by the shaded band[7, 8]. The intersection (region inside ellipse)determines α=137.0360±.00025 and ωe=(7.763±0.00002)×1020s−1. A recentexperiment (Harvard 2008) fixes α with great accuracy (errors thinner than thevertical (red online) line), which then fixed ωe and the electron mass witherrors smaller than any previous determination.</figcaption></figure> Quantity \| Method \| ur \| uRCODATA2006 \| uRCODATA∗2010 ---\|---\|---\|---\|--- mNe \| R∞(α2006) \| 1.4×10−9 \| 5×10−8 \| 4.4×10−8 mNe \| R∞(α2008) \| 7.4×10−10 \| 5×10−8 \| 4.4×10−8 e \| √α2006 \| 3.4×10−10 \| 2.5×10−8 \| ... e \| √α2008 \| 1.8×10−10 \| … \| 2.2×10−8 ℏ \| exact \| 0 \| 5×10−8 \| 4.4×10−8. Table 3: Summary of relative uncertainties ur in the values of fundamental constants using our procedure. ∗ CODATA-2010 analysis is unpublished, but available on website. While determining fundamental constants generally uses global fits to many variables, two independent data points suffice to determine two constants of the theory. The Rydberg constant $R_{\infty}=m_{Ne}c\alpha^{2}/2h$ and the fine structure constant $\alpha$ will predict $m_{Ne}$. We choose $R_{\infty}$ because its relative uncertainty is of order $5\times 10^{-12}$, which is so small it can be neglected. There are a number of ways to get high precision data for $\alpha$. Consider the ground state hyperfine interval of positronium[7]. Ref.[8] reports the experimental frequency shift \[\Delta f=2.0338910(74)\times 10^{11}\,s^{-1}.\] (30) Ref.[7] cites a calculation in perturbation theory \[\Delta f = {\alpha^{4}\omega_{e}\over 2\pi}\] (31) \[\times (\,7/12-\alpha/\pi(1/2ln(2)+8/9)\] \[+ 5/24\alpha^{2}ln(1/\alpha)+0.6\alpha^{2}\,).\] Two constants are determined, as shown in Figure 7. The errors in $\Delta f$ dominate, yielding \[{1\over\alpha} = 137.0360\pm.00025;\] \[\omega_{e} = (7.763\pm 0.00002)\times 10^{20}s^{-1}.\] This illustrates how the constants can be determined to a relative accuracy of few parts of $10^{-6}$ without great complication. The electron magnetic moment parameter $g-2$ is arguably more reliable. A 2006 Harvard study[19] found \[1/\alpha_{2006}=137.035999711(96),\] a relative uncertainty of $7.0\times 10^{-10}$. It is worth noting that $g-2$ is measured directly in terms of a _frequency_, explaining why the existing uncertainties in $m_{e}$ do not degrade this determination of $\alpha$. When the 2006 errors of $\alpha$ are applied to the vertical line in Fig. 7 the errors are too small to be visible. It leads to a relative error in $m_{Ne}$ of about $1.4\times 10^{-9}$. Compared to the published CODATA-2006 (unpublished 2010) results (Table 3), the value of $m_{Ne}$ is determined about 36 (31) times more precisely by our method. In 2008 the Harvard group[21] announced improvement of combined theoretical and experimental uncertainties of $\alpha$ to 0.37 parts per billion: \[{1\over\alpha}=137.035999084(51).\] Adopting this figure produces the electron mass \[m_{e}=9.1093821500(70)\times 10^{-31}\,kg.\] The relative uncertainty of $7.4\times 10^{-10}$ is 67 times less than reported by the best previous value published by CODATA-2006. Our uncertainties of the electric charge $e$ are simply determined by $\Delta e/e\sim(1/2)\Delta\alpha/\alpha$. Using the 2008 value of $\alpha$, the standardized values of $c$, $\hbar$ and the electrical constant $\epsilon_{0}$ of $MKS$ units we find \[e=1.60217648684(26)\times 10^{-19}.\] The relative uncertainty $u_{r}=1.8\times 10^{-10}$. The numerical value is within the error bars of the CODATA-2010 determination that cites a relative uncertainty of $2.5\times 10^{-8}$. Compared to the published determination of $e$, our relative uncertainty is $2.5\times 10^{-8}/1.8\times 10^{-10}\sim 139$ times smaller: see Table 3. We have shown that the electron mass and electric charge are substantially improved when reliance on the kilogram and conversion factors of Planck’s constant are avoided. This is related to the generally known fact that the ratio of the electron mass to an atomic mass unit (Carbon-12) can be determined with greatly improved uncertainty by using a common method (Penning trap) and avoiding the kilogram. It is no accident that what is actually measured in these experiments are comparisons of _frequency_$\omega_{e}$ versus _frequency_$\omega_{{}^{12}C}$, just as we find is fundamental. #### 4.0.2 Correlated Parameters What about discrepancies between calculations in perturbative $QED$ and data, including the anomalous moment measurements we have just used for our preliminary constant determinations? They are welcome! One of the main interests in precision fundamental constants is in testing new physics. Clinging to outmoded theoretical procedures happens to introduce inconsistencies. If a disagreement between theory and experiment is sharpened by complete consistency of definitions it can only represent progress. One might ask whether our determination lost information compared to finding $\hbar,\,e,\,m_{e},\,\alpha$ separately. It might seem that separate determination and comparison would be “testing the theory”. However the process of testing theories means posing alternative physical hypotheses capable in principle of giving different answers. Groups such as CODATA are not charged with considering models of new physics. The default physics has become one uniform framework of quantum electrodynamics, with theoretical contributions from other sources, which is the same framework as ours. Rather than global fits testing a framework, global fits will produce global error bars. Yet sometimes the data fitting process can indicate redundant parameters. That is done by finding stalemates with unusually high parameter degeneracy. A careful reading of the 105 page CODATA2006 document[18] will find the experts are aware of a strong correlation between the quantities we have identified. Great effort has gone into studying the correlations between the _experimental_ inputs, which are seldom independent. We are concerned with the outputs. Table $LI$, page 102 of Ref. [18] shows the correlation of the evaluated $h$ and $m_{e}$ is $r=0.9996.$ (The correlation goes to $r=0.9999$ in the 2010 website material.) While it may be simplistic, this correlation is an outcome predicted by the symmetry of Eq. 24, varying $m_{e}$ and $h$ while keeping the observable Rydberg frequency $R_{\infty}=m_{Ne}c\alpha^{2}/2h$ fixed: Exactly as we did in Section 1. The other correlations in Table $LI$ with magnitudes exceeding $0.999$ come between $(h,e)$, $(h,N_{A})$, $(N_{A},m_{e})$, and $(N_{A},e)$, where $N_{A}$ is Avogadro’s number. These facts support our view that eliminating the correlations is actually a matter of principle, not improving technology. Trimming redundant constants such as $e$ and $\hbar$ clears the way for meaningful constants to be determined better, and move on to compare precision experiments better. The need for many independent experiments remains. For example, the theory of the Josephson effect has traditionally been formulated in terms of a constant $K_{J}=e/h$. This can be traced to theoretical decisions to separate the parameter $\alpha$ found in quantum electrodynamics into terms involving $e$, and an external magnetic field also proportional to $e$, representing a net dependence on $e^{2}/h$. Consistent definitions should improve tests of the theory of Josephson junctions and quantum Hall physics. The unsettled discrepancy between the electron and muon magnetic moments is another example where high precision constants are important. ### Exit Kilogram <figure><img src="content_image/1203.5557/x8.png"><figcaption>Figure 8: Observed variation of mass with time for the InternationalPrototype Kilogram (IPK) relative to 6 official copies. Scale “0” is relativeto the IPK, which has not so far changed relative to itself. Planck’s constantis changing with time by an amount linear in the change of the standardmasses. From Ref. [17].</figcaption></figure> <figure><img src="content_image/1203.5557/x9.png"><figcaption>Figure 9: Time variation observed in Planck’s constant. Vertical scale showsvalues of 105(h/10−34−6.620) from Ref. [18]. The fit with the smaller (butnon-zero) slope is close to the slope predicted by time variation of theInternational Prototype Kilogram, Fig. 8. The fit with the larger slope omitsthe most recent point, whose tiny errors make the global fit worse.</figcaption></figure> Finally we return to the kilogram. The kilogram is the only $SI$ unit not defined by independently reproducible experiments. In comparison with 6 official copies, the single International Prototype Kilogram has consistently lost mass over time[17]. An unknown mechanism is causing a loss rate $\|\Delta m/m\|\sim 2.5\times 10^{-7}/century$ (Fig. 8). The relative change of the mass standard itself every ten years is comparable to the relative errors on the electron mass found in the last ten years. Time-dependence of fundamental constants has been an important topic since Dirac[22] suggested the fine structure constant might be time dependent. Fig. 9 shows a fit to the time dependence of $h$. The slope of the minimum $\chi^{2}$ fit is $\Delta h/h=3.3\times 10^{-7}/century$. It is remarkably close to the slope of the $IPK$. This is perhaps fortuitous. The most recent point with the smallest errors dominates the fit. If this point is removed, the value of $\chi^{2}$ decreases by 4 units, indicating a much better fit from removing an outlier. The slope of Planck’s constant _increases_ to $\Delta h/h=3.0\times 10^{-6}/century$. Planck’s is the first fundamental constant to develop an observed time-dependence. While interesting we find it foolish to take this seriously. But it is supposed to be foolish to maintain forever that $\hbar$ must be fundamental now because it was once fundamental in the past. We find it simple and rational to fix Planck’s constant to a definition. It fulfills the long-standing goal of a reproducible standard of mass[23] not intrinsically depending on comparison with artifacts. ## 5 Concluding Remarks Planck’s constant entered physics by a particular historical path. Newtonian concepts of energy and inertial mass were assumed. To this day the historical path is used in teaching the subject. In the meantime fundamental physics has evolved. Research practice has every reason to drop holdovers from history. The introduction suggested that the agreement to drop Planck’s constant from fundamental units should not be difficult. Some physicists accustomed to ignoring it might not find the conclusion trivial, yet that appreciation is hardly universal. We have shown that unless Newtonian physics has primacy and new information not available from quantum theory, reference to Planck’s constant is redundant. When historical prejudices are dropped Planck’s constant disappears. There is still a place for standardizing Planck’s constant, just as standardizing other units is important to engineering and commerce. Standardization of $\hbar$ and $c$ share the common element of removing barriers to precision measurements of other constants. Unlike $c$, which in principle might disagree with theory, Planck’s constant in our quantum theory is unobservable, and we can’t even suggest an experiment to find it. The challenge of testing fundamental physics should not be saddled with constructs set up by human conventions. It would be delightful if the tradition of retaining Planck’s constant might not forever propagate into tests of fundamental physics. Then the current generation of precision quantum measurements might find discrepancies in the fundamental quantum parameters requiring new physics. _Acknowledgements:_ Research supported in part under DOE Grant Number DE-FG02-04ER14308. We thank Carl Bender, Don Colloday, Jacob Hermann, Danny Marfatia, Phil Mannheim, Doug McKay, Dan Neusenschwander, and Peter Rolnick for comments. ## References * [1] E. Rutherford, The Scattering of $\alpha$ and $\beta$ Particles by Matter and the Structure of the Atom, Philosophical Magazine. Series 6, vol. 21. May 1911. * [2] J. A. Crowther, Proc. Roy. Soc. 84, 570, 226 (1910); J. A. Crowther and B. F. A. Shonland, Proc. Roy. Soc. 100, 706, 526 (1922). * [3] J. Dewar, Proc. Roy. Soc. 73, 251 (1904). * [4] For a review of Rydberg’s physics, see “Janne Rydberg his life and work”, by I. Martinson and L.J. Curtis, NIM _B 235_, 17 (2005). For the numerical value reported by Rydberg see Lund University Physics, http://www.lth.se/?id=17657. * [5] C. Davisson and L.H. Germer, Nature 119, 558 (1927). * [6] R. B. Laughlin and D. Pines, PNAS 97, 28 (1999). * [7] G. S. Adkins, Y. M Aksi, and M. H. T. Bui, Phys. Rev. A47, 2640 (1993). * [8] M .W. Ritter, P.O. Egan, V. W. Hughes, and K. A. Woodle, Phys. Rev. A30, 1331 (1984). * [9] A. H. Compton, Phys. Rev. 21, 483 (1923). * [10] Max Planck , Annalen der Physik 4, 553 (1901); translated at http://axion.physics.ubc.ca/200-06/Planck-1901.html * [11] Max Planck, _The theory of Heat Radiation_, translated by Morton Mosius, P. Blackiston’s Sons, (1914). * [12] J. P. Ralston, J. Phys. A: Math. Theor. 40, 9883 (2007). * [13] D. L. Farnham, R. S. Van Dyck, Jr., and P. B. Schwinberg, Phys. Rev. Lett. 75, 3598 (1995). * [14] J. P. Ralston, in preparation. * [15]_Modern Quantum Mechanics_, by J. J. Sakurai, San Fu Tuan, editor (Addison Wesley, 1998). * [16] Willis E. Lamb, Jr. and Marlan O. Scully, The Photoelectric Effect without Photons , pp363-369, in _Polarization, matter and radiation; Jubilee volume in honor of Alfred Kasler_, Presses Universitaires de France, Paris (1969). * [17] Richard Davis, Metrologia 40, 299 (2003). * [18] P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. 80, 633 (2008) [arXiv:0801.0028 [physics.atom-ph]]. The unpublished 2010 values exist on the website http://physics.nist.gov/cuu/Constants/index.html * [19] G. G. Gabrielse, T. Hanneke, K. Kinoshita, M. Nio, and B. Odom, Phys. Rev. Lett. 97, 030802 (2006); Erratum ibid.99, 039902, ( 2007). * [20] A. Einstein, ”On a Heuristic Point of View about the Creation and Conversion of Light”, translated in _The Old Quantum Theory_, Pergamon 1967, by D. Ter Haar. * [21] D. Hanneke, S. Fogwell and G. Gabrielse, Phys. Rev. Lett. 100, 120801 (2008) [arXiv:0801.1134 [physics.atom-ph]]. * [22] P. A. M. Dirac, Nature 139, 323 (1937). For reviews of time variation see J.-P. Uzan, Rev. Mod. Phys. 75, 403 (2003); K. A. Olive, M. Pospelov, Y. Z. Qian, A. Coc, M. Casse and E. Vangioni-Flam, Phys. Rev. D 66, 045022 (2002). * [23] A Eichenberger, B Jeckelmann and P Richard, Metrologia 40, 356 (2003).
1203.4424	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23713, "num_imgs": 4, "llama3_tokens_count": 6525 }	[ "content_image/1203.4424/x1.png", "content_image/1203.4424/x2.png", "content_image/1203.4424/x3.png", "content_image/1203.4424/x4.png" ]	# Inhomogeneities on all scales at a phase transition altered by disorder I. Balog balog@ifs.hr K. Uzelac katarina@ifs.hr Institute of Physics, P.O.Box 304, Bijenička cesta 46, HR-10001 Zagreb, Croatia ###### Abstract We have done a finite-size scaling study of a continuous phase transition altered by the quenched bond disorder, investigating systems at quasicritical temperatures of each disorder realization by using the equilibriumlike invaded cluster algorithm. Our results indicate that in order to access the thermal critical exponent $y_{\tau}$, it is necessary to average the free energy at quasicritical temperatures of each disorder configuration. Despite the thermal fluctuations on the scale of the system at the transition point, we find that spatial inhomogeneities form in the system and become more pronounced as the size of the system increases. This leads to different exponents describing rescaling of the fluctuations of observables in disorder and thermodynamic ensembles. pacs: 05.50.+q, 64.60.F-, 75.10.Hk, 2.70.-c The question of how disorder modifies a phase transition is still not fully resolved, notwithstanding decades of research. Many issues remain open, even in the case of the ferromagnetic transition in the presence of quenched disorder, from the qualitative nature of the ordered phase near criticality [1; 2], the presence of Griffiths singularities [3], the role of multiple length scales [4], to violation [5] of the proposed [6] bound on the thermal critical exponent. An important aspect present in the majority of the cited problems is the non-self-averaging effect between different disorder realizations [7], leading to the question of proper averaging over disorder. The standard approach is to reduce the problem to an effective translationally invariant one, by averaging the free energy over disorder configurations at a given temperature [8]. Such averaging permits one to apply the perturbative renormalization group (RG), and crucially simplifies the finite-size scaling (FSS) studies based on numerical simulations as well. When applied to a classical ferromagnetic model with bond randomness, displaying a continuous phase transition in the pure case such as the one we consider in this paper, the conclusion seems to be [9] that the disorder, if relevant in the sense of the Harris criterion [10], merely changes the values of the critical exponents, while the nature of the transition remains similar to that of the pure case. We challenge this notion with a FSS study in which the average is taken at quasicritical (i.e. finite size critical) temperatures of each disorder sample. Except for some partial attempts, a systematic study using this type of averaging is still missing, since it requires an exceptional numerical effort. The recently proposed equilibriumlike invaded cluster (EIC) algorithm [11; 12] enables such a procedure to be technically feasible. There exist prior works revealing conceptual problems in the transition with bond disorder. It has been pointed out that the self-averaging ratios [7] of observables at criticality depend on averaging [13] although they should be universal [14]. Correlation length has also been shown to be non-self-averaging near criticality [15]. Pázmándi _et. al._[5] have shown that standard averaging obscures the intrinsic thermal exponent $y_{\tau}=1/\nu$, when it is superior to the exponent $\tilde{y}_{\tau}$ describing disorder fluctuations of quasicritical temperatures, \[\tilde{y}_{\tau}<y_{\tau}.\] (1) In such a case, in standard averaging, $\tilde{y}_{\tau}$ always takes over $y_{\tau}$ in FSS relations, and the information about $y_{\tau}$ is lost. It has therefore been suggested [13; 16] that in order to find the true critical exponents one has to perform a disorder average of observables, taken at the quasicritical temperature of each disorder sample. We use the EIC approach on the two-dimensional (2D), three-state Potts model with bond disorder [17], where disorder is relevant by the Harris criterion, since the specific-heat critical exponent of the pure model is positive ($\alpha=1/3$). Our main result shows inequality (1) by an explicit numerical calculation, justifying the assumption by Pázmándi _et al._[5] in a classical system. We suggest an alternative interpretation of the lack of self-averaging as an emergence of frozen inhomogeneities in observables at all scales, which may be analyzed further from the statistics of the largest cluster at criticality. Since the Chayes bound [6], i.e. $\tilde{y}_{\tau}\leq d/2$, is exclusively a result of averaging at a unique temperature, it is irrelevant for the intrinsic thermal exponent $y_{\tau}$. For this reason our findings might also be of interest to systems in which a violation of the Chayes criterion was observed [18; 19; 20]. Experimentally, the intrinsic exponent $y_{\tau}$ would be observed only in direct calorimetric measurements of the heat capacity, while the indirect measurements (e.g. birefringence [21]) would yield the exponent $\tilde{y}_{\tau}$ obeying the Chayes criterion. We introduce disorder in the Potts model [22] as a random bond dilution \[H=\sum_{<i,j>}-J_{i,j}\big{(}\delta_{\sigma_{i},\sigma_{j}}-1\big{)},\] (2) where random couplings $J_{i,j}$ are restricted to neighboring lattice sites $i$, $j$ and take zero value with concentration $\boldsymbol{c}$, or $J_{i,j}=J$ otherwise. $\sigma$ denotes the Potts variable with $q=3$ discrete states. For the purpose of numerical simulations, the partition function of the Potts model is written in terms of the random cluster model, by using the Fortuin and Kasteleyn (FK) graph expansion [23] \[Z=\sum_{\gamma\in\Gamma_{\boldsymbol{\alpha}}}p^{b(\gamma)}\cdot(1-p)^{B-b( \gamma)}\cdot q^{c(\gamma)},\] (3) where the bond probability \[p=1-e^{-\frac{J}{k_{B}T}}\] (4) takes the role of temperature, while $b(\gamma)$ and $c(\gamma)$ denote the number of bonds and connected components “FK clusters” in the graph $\gamma$, respectively. The summation runs over the set of all possible FK graphs $\Gamma_{\boldsymbol{\alpha}}$, compatible with the given disorder configuration $\boldsymbol{\alpha}$. When applied to a system with disorder, the EIC algorithm [11] simulates it at the quasicritical (i.e., finite-size critical) bond probabilities $p^{c}_{\boldsymbol{\alpha}}(L)$ (related to temperature by Eq. (4)) belonging to each disorder configuration $\boldsymbol{\alpha}$, defined by the onset of the percolation of the largest FK cluster. This property allows us to consider separately the scaling of thermal fluctuations of an observable around its thermodynamic mean and the disorder ensemble fluctuations of the thermodynamic means. Our algorithm is an extension of the invaded cluster (IC) algorithm [24] and has a similar mechanism of self-regulating to the quasicritical point, but with the crucial difference of generating the equilibrium thermodynamic ensemble [12]. The duration of a Monte Carlo (MC) step of the EIC algorithm is approximately the same as that of the IC algorithm, but because of the canonical constraint, the EIC algorithm requires a certain number of thermalization steps (never exceeding $5000$ in the present paper). The EIC algorithm generalizes to the problem with bond dilution in a straightforward way since disorder merely excludes some configurations from the set of all possible FK graphs of the pure case. Throughout this paper we denote by $[\cdot]$ the disorder average taken at $p^{c}_{\boldsymbol{\alpha}}(L)$ of each disorder configuration and by $\overline{{}\cdot{}}$ thermodynamic average for a given $\boldsymbol{\alpha}$. Numerical results presented are based on simulations on square lattices of linear size $L$ ranging from $64$ up to $896$, with two disorder concentrations, $c=0.125$ and $0.25$, using the statistics of $400$ and $600$ disorder configurations respectively, with $20000$ Monte Carlo steps (MCS) per disorder configuration after thermalization. Calculation of a running average reveals that such a disorder statistics is sufficient to determine each $[p^{c}_{\boldsymbol{\alpha}}(L)]$ to seven significant digits. The disorder is introduced microcanonically, i.e. the exact number of bond vacancies corresponding to a concentration is randomly distributed on the lattice. The thermal critical exponent $y_{\tau}$ is calculated from the magnetization-energy cumulant \[U^{me}_{\boldsymbol{\alpha}}=\frac{\overline{m_{\boldsymbol{\alpha}}e_{ \boldsymbol{\alpha}}}-\overline{m_{\boldsymbol{\alpha}}}\cdot\overline{e_{ \boldsymbol{\alpha}}}}{\overline{m_{\boldsymbol{\alpha}}}},\] (5) where $m_{\boldsymbol{\alpha}}$ and $e_{\boldsymbol{\alpha}}$ denote the order parameter and energy density of a disorder configuration $\boldsymbol{\alpha}$, respectively. The values of $y_{\tau}$ in Table 1 are obtained by fitting (Fig. 1 a) the averaged data to the power law form $[U^{me}_{\boldsymbol{\alpha}}]\propto L^{y_{\tau}-\boldsymbol{d}}$. The simple power law describes the scaling of $[U^{me}_{\boldsymbol{\alpha}}]$ for $c=0.25$ in the entire range of lattice sizes and for $c=0.125$ only the data from the smallest size deviate [25]. Since the scaling corrections are always found to be important in approaches using the averaging at a unique temperature (see e.g. [26]), their negligibility can only be attributed to the averaging procedure we used in this work. Exponent $y_{\tau}$ also shows a negligible concentration dependence, contrary to the previous studies where the standard averaging was used [27]. <figure><img src="content_image/1203.4424/x1.png"><figcaption>Figure 1: (color online) Slopes in (a)-(c) correspond to yτ−d2, ~yτ−d2 andd2−~x respectively, pointing out the difference between the three exponents.The disorder statistics determines the precision of [Umeα] for each L to thefourth significant digit and the precision of δpc and δ¯¯¯ec to the thirdsignificant digit. The error bars for Ld/2[Umeα] are smaller than the symbols.</figcaption></figure> c \| yτ \| ~yτ \| d−~x ---\|---\|---\|--- 0.125 \| 1.03(1) \| 0.88(4) \| 1.35(5) 0.25 \| 1.01(1) \| 0.95(2) \| 1.14(4) Table 1: Rescaling exponents of [Umeα], δpcα and δ¯¯¯ecα. Each set of data has been fitted in the entire range of sizes to a single power-law except for the [Umeα] for c=0.125, where the smallest lattice has been excluded. Error bars have been estimated by jackknife binning. To understand how the exponent $y_{\tau}$ becomes obscured when the averaging at a unique temperature is used, we examine the fluctuations of quasicritical bond probabilities $p^{c}_{\boldsymbol{\alpha}}$ (Fig. 1 b) in disorder ensemble (ensemble of random samples) \[\delta p^{c}=\sqrt{[p^{c2}_{\boldsymbol{\alpha}}]-[p^{c}_{\boldsymbol{\alpha}} ]^{2}}\propto L^{-\tilde{y}_{\tau}}.\] (6) By virtue of Eq. (4) $p^{c}$ obeys the same scaling law as the fluctuations of quasicritical temperatures $T^{c}_{\boldsymbol{\alpha}}$. The fact that beyond statistical errors $\tilde{y}_{\tau}$ is dominant over $y_{\tau}$ (Tab. 1), leads to the conclusion that, by averaging at a unique temperature one effectively measures the exponent $\tilde{y}_{\tau}$[5]. The range of values for different disorder concentrations that we have obtained for $\tilde{y}_{\tau}$ corresponds to the findings for the thermal exponent in previous studies, which used averaging at a unique temperature [28; 29; 30] for the same system. For example Jacobsen and Cardy [28] have estimated the value to $0.96(4)$. Unlike $y_{\tau}$, the exponent $\tilde{y}_{\tau}$ displays a strong dependence on disorder, similar to that found in previous studies. From $U^{me}_{\boldsymbol{\alpha}}$ we have determined the singular part of the energy density which scales $\propto L^{y_{\tau}-d}$. Considering disorder ensemble fluctuations of thermodynamic means of energy density $\overline{e}^{c}_{\boldsymbol{\alpha}}$ taken at $T^{c}_{\boldsymbol{\alpha}}$ (Fig. 1 c), \[\delta\overline{e}^{c}=\sqrt{[\overline{e}^{c2}_{\boldsymbol{\alpha}}]-[ \overline{e}^{c}_{\boldsymbol{\alpha}}]^{2}}\propto L^{-\tilde{x}},\] (7) we obtain that $L^{d}\delta\overline{e}^{c}$ increases, with the exponent $d-\tilde{x}$ (Tab. 1) significantly larger than ${y}_{\tau}$ or $\tilde{y}_{\tau}$ for both $c$. Since $\delta\overline{e}^{c}$ is the sum of the analytic and the singular part, the exponent $\tilde{x}$ has to be attributed to the analytic part, which is dominant over the singular one. In Ref. [28] the analysis of the energy fluctuations by the standard disorder averaging produces the exponent $d-0.91(1)=1.09(1)$, significantly larger than their value for the thermal exponent $0.96(4)$. If the disorder had merely changed the values of the critical exponents found for the pure case, $\tilde{y}_{\tau}$ and $d-\tilde{x}$ would have to be equal to $y_{\tau}$, as determined by the singular part of the free energy. However, we show the exponents to be different (Fig. 1). The evidence presented leads to the conclusion that changing a disorder configuration $\boldsymbol{\alpha}$ while keeping the concentration of disorder constant, drives the system out of the critical area. A consequence of this scenario is the formation of spatial inhomogeneities at all scales. Consider the free energy density for a single disorder configuration $\boldsymbol{\alpha}$ \[f_{\boldsymbol{\alpha}}=L^{-d}\hat{f}_{\boldsymbol{\alpha}}(L^{y_{\tau}}(T-T^{ c}_{\boldsymbol{\alpha}}(L)),L^{y_{h}}h)+T\cdot\tilde{e}_{\boldsymbol{\alpha}} (T,h;L),\] (8) where $h$ is the magnetic field and $T\tilde{e}_{\boldsymbol{\alpha}}$ is the analytical part. Since we have demonstrated that inequality (1) applies, changing $\boldsymbol{\alpha}$ produces a shift in the argument of expression (8) $\propto L^{y_{\tau}-\tilde{y}_{\tau}}$, which diverges for $L\to\infty$. However, when a system of size $L$ can be driven out of criticality by changing $\boldsymbol{\alpha}$, this shift is even larger for any of its parts of size $L^{\prime}<L$ since $L^{\prime-\tilde{y}_{\tau}}>L^{-\tilde{y}_{\tau}}$. The shift may be positive or negative in any of the parts since the disorder configurations in them are uncorrelated, so the system locally prefers either the low- or the high- temperature phase, resulting in spatial inhomogeneities. The contributions from local parts accumulate in the analytic part of the energy, causing the disorder ensemble fluctuations (i.e., the exponent $\tilde{x}$). If the opposite to inequality (1) were true, the inhomogeneities would have disappeared in the limit $L\to\infty$, since $L^{y_{\tau}-\tilde{y}_{\tau}}$ would be at most of order $1$. We demonstrate the existence of the spatial inhomogeneities on the example of the local order parameter $\overline{m}_{\boldsymbol{\alpha}}(\vec{r})$, defined as the thermodynamic probability that the spin on the site $\vec{r}$ belongs to the largest FK cluster \[\overline{m}_{\boldsymbol{\alpha}}(\vec{r})=\frac{\sum_{\{\sigma\},\{b\}}\pi_{ \boldsymbol{\alpha}}(\vec{r})\cdot w_{\boldsymbol{\alpha}}(\{\sigma\},\{b\})}{ \sum_{\{\sigma\},\{b\}}w_{\boldsymbol{\alpha}}(\{\sigma\},\{b\})},\] (9) where $\pi_{\boldsymbol{\alpha}}$ is $1$ if $\sigma(\vec{r})$ belongs to the largest FK cluster and $0$ otherwise. By $w_{\boldsymbol{\alpha}}$ we denote the statistical weight of a configuration in the joint space of spin and bond variables. The spatial average of $\overline{m}_{\boldsymbol{\alpha}}(\vec{r})$ is equivalent to the standard Potts order parameter [31]. In Fig. 2$\overline{m}_{\boldsymbol{\alpha}}(\vec{r})$ in a single disorder sample with $25\%$ of disorder at $p^{c}_{\boldsymbol{\alpha}}(L)$ is presented. The behavior of $\overline{m}_{\boldsymbol{\alpha}}(\vec{r})$ contrasts with the situation in the pure model, where all the lattice sites are equivalent and $\overline{m}_{\boldsymbol{\alpha}}(\vec{r})$ is constant. It illustrates how a fractal volume, pinned by disorder, emerges as the preferred ordering volume on which the local order parameter is close to $1$. <figure><img src="content_image/1203.4424/x2.png"><figcaption>Figure 2: (Color online) Spatial dependence of ¯¯¯¯¯mα(→r) for a singledisorder configuration at pcα obtained with 106 MCS for L=2048. The spatialaverage of ¯¯¯¯¯mα(→r) in this disorder configuration is 0.29164 and islabeled by the tick on the scale.</figcaption></figure> <figure><img src="content_image/1203.4424/x3.png"><figcaption>Figure 3: (color online) Distributions of spatial variations of the orderparameter for two lattice sizes averaged over several disorder configurations(in parentheses) for c=0.25.</figcaption></figure> <figure><img src="content_image/1203.4424/x4.png"><figcaption>Figure 4: (Color online) Size dependence of: a) order parameter [¯¯¯¯¯mα], b)width of the spatial variations of the local order parameter [\|δ¯¯¯¯¯mα(→r)\|],and c) width of the thermal fluctuations of the order parameter [δmα], forc=0.25. The dashed lines denote the best fits which include scalingcorrections.</figcaption></figure> We define $\mu_{\boldsymbol{\alpha}}(\delta\overline{m}_{\boldsymbol{\alpha}})$ as the probability distribution of the deviations of the local order parameter from its spatial mean $\delta\overline{m}_{\boldsymbol{\alpha}}(\vec{r})=\overline{m}_{\boldsymbol{ \alpha}}(\vec{r})-(1/V)\int\overline{m}_{\boldsymbol{\alpha}}(\vec{r})d\vec{r}$ at $p^{c}_{\boldsymbol{\alpha}}(L)$, for each $\boldsymbol{\alpha}$. A comparison of the averaged distributions $[\mu_{\boldsymbol{\alpha}}]$ for the two different lattice sizes (Fig. 3) reveals that the tail of the distribution, related to the significant deviations from the average ($\delta\overline{m}_{\boldsymbol{\alpha}}\approx 1$), becomes more pronounced with increasing system size, although the order parameter vanishes in the limit $L\to\infty$. As shown in Fig. 4, the width of the spatial inhomogeneities of order parameter $[\|\delta\overline{m}_{\boldsymbol{\alpha}}(\vec{r})\|]$ is larger than the width of thermal fluctuations of the order parameter $[\delta m_{\boldsymbol{\alpha}}]=\sqrt{[\overline{m^{2}}_{\boldsymbol{\alpha}} -\overline{m}^{2}_{\boldsymbol{\alpha}}]}$. We conclude that spins which contribute to ordering are increasingly frozen as $L$ increases. The magnetic exponent ($\frac{\beta}{\nu}$) calculated for $c=0.125$ and $0.25$ from the first moment of the order parameter ($0.129(3)$) which includes the contribution from inhomogeneities does not differ significantly from the value obtained from the second moment ($0.132(3)$), which includes only thermal fluctuations [25]. Both of these values are within the range of previous results for the exponent $\beta/\nu$[28; 29; 30]. We conclude that the difference in $\overline{m}_{\boldsymbol{\alpha}}$ between different disorder configurations $\boldsymbol{\alpha}$, when determined at $p^{c}_{\alpha}$, is of the same order as the width of the thermal fluctuations of the order parameter for each $\boldsymbol{\alpha}$. In conclusion, we have used the recently proposed EIC algorithm to study the critical behavior in the $2D$$q=3$ Potts model with quenched disorder by averaging at quasicritical temperatures of individual disorder configurations. This procedure has allowed us to separate the thermal from disorder-sample fluctuations and obtain the critical exponents $y_{\tau}$ and $y_{h}=d-\beta/\nu$ characterizing the transition in a single disorder configuration. The difference between the critical and disorder fluctuation exponents is interpreted by the emergence of spatial inhomogeneities on all scales at the transition point. By examining the local order parameter, we explicitly show that the inhomogeneities are present and that they become increasingly frozen (localized) as the size of the system increases. The above studies are currently being extended to higher dimensions, by using the same EIC algorithm. The approach may be easily applied to more specific problems, such as the extensively discussed 2D Ising model [32] as a marginal case of quenched disorder according to the Harris criterion, or to the change from a first- to second-order phase transition induced by disorder [17]. _Note added_. Another interesting extension of the application of the EIC algorithm would include models with random field, for which an alternative approach to eliminate the lack of self-averaging has been recently proposed [33]. ###### Acknowledgements. I.B. wishes to thank O. S. Barišić for useful discussions and comments. This work was supported by the Croatian Ministry of Science, Education and Sports through Grant No. 035-0000000-3187. We thank the referee for driving our attention to Ref. [33]. ## References * (1) G. Tarjus and Vik. Dotsenko, J. Phys. A 35, 1627 (2002) * (2) F.Krzakala, F.Ricci-Tersenghi, D.Sherrington and L.Zdeborová, J. Phys.A 44, 042003 (2011) * (3) R.B.Griffiths: Phys. Rev. Lett. 23, 17 (1969) * (4) A.L.Korzhenevskii, H-O. Heuer and K.Herrmanns: J. Phys. A 31, 927 (1998) * (5) F.Pázmándi, R.T.Scalettar and G.T.Zimányi: Phys. Rev. Lett. 79, 5130 (1997) * (6) J.T.Chayes, L.Chayes, D.S.Fisher and T.Spencer: Phys. Rev. Lett. 57, 2999 (1986) * (7) S.Wiseman and E. Domany: Phys. Rev. E 52, 3469 (1995) * (8) G.Grinstein and A. Luther: Phys. Rev. B 13, 1329 (1976) * (9) T. Vojta: J. Phys. A 39, R143 (2006) * (10) A.B.Harris: J. Phys. C 7, 1671 (1974) * (11) I.Balog and K.Uzelac: Phys. Rev. E 77, 050101(R) (2008) * (12) I.Balog and K.Uzelac: Phys. Rev. E 81, 041111 (2010) * (13) S. Wiseman and E. Domany: Phys. Rev. Lett. 81, 22 (1998) * (14) A.Aharony, A.B.Harris and S.Wiseman: Phys. Rev. Lett. 81, 252 (1998) * (15) G. Parisi, M.Picco and N. Sourlas: Europhys. Lett. 66, 465 (2004) * (16) K.Bernardet, F.Pázmandi and G.G.Batrouni: Phys. Rev. Lett. 84, 4477 (2000) * (17) For a recent review of the random bond Potts model see e.g. B.Berche and C.Chatelain, in _Order, Disorder, and Criticality_, edited by Yu. Holovatch (World Scientific, Singapore, 2004) p. 146 and references therein. * (18) O.Narayan and D.S.Fisher: Phys. Rev. B 46, 11520 (1992) * (19) R.R.P.Singh and M.E.Fisher: Phys. Rev. Lett. 60, 548 (1988) * (20) M.A.Paalanen, T.F.Rosenbaum, G.A.Thomas and R.N.Bhatt: Phys. Rev. Lett. 48, 1284 (1982) * (21) R.J.Birgeneau, R.A.Cowley, G.Shirane, H.Yoshizawa, D.P.Belanger, A.R.King and V.Jaccarino: Phys. Rev. B, 27, 6747 (1983) * (22) R.B.Potts: Proc. Cambridge Philos. Soc. 48, 106 (1952) * (23) P. W. Kasteleyn, C. M. Fortuin, Jour. Phys. Soc. Japan 26, 11, (1969) * (24) J. Machta,Y. S. Choi,A. Lucke,T. Schweizer and L. V. Chayes, Phys. Rev. Lett. 75, 2792 (1995) * (25) I.Balog and K.Uzelac: in preparation * (26) M.Hasenbuch, F.P.Toldin, A.Pelissetto and E.Vicari: J. Stat. Mech. P02016, (2007) * (27) H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi and J. J. Ruiz-Lorenzo: Phys. Rev. B 58, 274 (1998) * (28) J.L.Jacobsen and J. Cardy: Nuc. Phys. B 515, 701 (1998) * (29) A.W.W.Ludwig: Nuc. Phys. B 285, 97 (1987) * (30) Vl.Dotsenko, M.Picco and P.Pujol: Nuc. Phys. B 455, 701 (1995) * (31) J.P.Straley and M.E.Fisher: J Phys. A 6, 1310 (1973) * (32) For a recent reference see A. Gordillo-Guerrero, R. Kenna, J.J. Ruiz-Lorenzo, arXiv:0909.3774v2 and references therein * (33) L. A. Fernandez, V. Martin-Mayor, and D. Yllanes : Phys. Rev. B 84, 100408 (R) (2011)
1704.03416	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 28208, "num_imgs": 0, "llama3_tokens_count": 13281 }	[]	See pages 1,,2-75 of lecturenotes.pdf ## References * Adams [1971] Adams, T. F. 1971, Astrophys. J. , 168, 575 * Adams [1972] Adams, T. F. 1972, Astrophys. J. , 174, 439 * Agertz et al. [2009] Agertz, O., Teyssier, R., & Moore, B. 2009, MNRAS, 397, L64 * Ahn et al. [2000] Ahn, S.-H., Lee, H.-W., & Lee, H. M. 2000, Journal of Korean Astronomical Society, 33, 29 * Ahn et al. [2001] Ahn, S.-H., Lee, H.-W., & Lee, H. M. 2001, Astrophys. J. , 554, 604 * Ahn et al. [2002] Ahn, S.-H., Lee, H.-W., & Lee, H. M. 2002, Astrophys. J. , 567, 922 * Ahn et al. [2003] Ahn, S.-H., Lee, H.-W., & Lee, H. M. 2003, MNRAS, 340, 863 * Ahn & Lee [2015] Ahn, S.-H., & Lee, H.-W. 2015, Journal of Korean Astronomical Society, 48, 195 * Aggarwal [1983] Aggarwal, K. M. 1983, MNRAS, 202, 15P * Aggarwal et al. [1991] Aggarwal, K. M., Berrington, K. A., Burke, P. G., Kingston, A. E., & Pathak, A. 1991, Journal of Physics B Atomic Molecular Physics, 24, 1385 * Alipour et al. [2015] Alipour, E., Sigurdson, K., & Hirata, C. M. 2015, Phys. Rev. D, 91, 083520 * Angel [1969] Angel, J. R. P. 1969, Astrophys. J. , 158, 219 * Atek et al. [2008] Atek, H., Kunth, D., Hayes, M., Östlin, G., & Mas-Hesse, J. M. 2008, A&A, 488, 491 * Atek et al. [2009] Atek, H., Kunth, D., Schaerer, D., et al. 2009, A&A, 506, L1 * Aver et al. [2015] Aver, E., Olive, K. A., & Skillman, E. D. 2015, JCAP, 7, 011 * Bach & Lee [2014] Bach, K., & Lee, H.-W. 2014, Journal of Korean Astronomical Society, 47, 187 * Bach & Lee [2015] Bach, K., & Lee, H.-W. 2015, MNRAS, 446, 264 * Bacon et al. [2010] Bacon, R., Accardo, M., Adjali, L., et al. 2010, PROCSPIE, 7735, 773508 * Barkana & Loeb [2001] Barkana, R., & Loeb, A. 2001, Physics Reports, 349, 125. * Barkana [2004] Barkana, R. 2004, MNRAS, 347, 59 * Barkana & Loeb [2004] Barkana, R., & Loeb, A. 2004, Astrophys. J. , 609, 474 * Barnes & Haehnelt [2010] Barnes, L. A., & Haehnelt, M. G. 2010, MNRAS, 403, 870 * Barnes et al. [2011] Barnes, L. A., Haehnelt, M. G., Tescari, E., & Viel, M. 2011, MNRAS, 416, 1723 * Basko [1981] Basko, M. M. 1981, Astrophysics, 17, 69 * Beck et al. [2016] Beck, M., Scarlata, C., Hayes, M., Dijkstra, M., & Jones, T. J. 2016, Astrophys. J. , 818, 138 * Behrens & Niemeyer [2013] Behrens, C., & Niemeyer, J. 2013, A&A, 556, A5 * Behrens et al. [2014] Behrens, C., Dijkstra, M., & Niemeyer, J. C. 2014, A&A, 563, A77 * Bely & van Regemorter [1970] Bely, O., & van Regemorter, H. 1970, ARA&A, 8, 329 * Birnboim & Dekel [2003] Birnboim, Y., & Dekel, A. 2003, MNRAS, 345, 349 * Black [1981] Black, J. H. 1981, MNRAS, 197, 553 * Blanc et al. [2011] Blanc, G. A., Adams, J. J., Gebhardt, K., et al. 2011, Astrophys. J. , 736, 31 * Bolton et al. [2011] Bolton, J. S., Haehnelt, M. G., Warren, S. J., et al. 2011, MNRAS, 416, L70 * Bolton & Haehnelt [2013] Bolton, J. S., & Haehnelt, M. G. 2013, MNRAS, 429, 1695 * Bonilha et al. [1979] Bonilha, J. R. M., Ferch, R., Salpeter, E. E., Slater, G., & Noerdlinger, P. D. 1979, Astrophys. J. , 233, 649 * Bosma [1978] Bosma, A. 1978, Ph.D. Thesis, * Borisova et al. [2016] Borisova, E., Cantalupo, S., Lilly, S. J., et al. 2016, Astrophys. J. , 831, 39 * Bouwens et al. [2012] Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2012, Astrophys. J. , 754, 83 * Bower [2011] Bower, R. 2011, Nature (London), 476, 288 * Brandt & Chamberlain [1959] Brandt, J. C., & Chamberlain, J. W. 1959, Astrophys. J. , 130, 670 * Brasken & Kyrola [1998] Brasken, M., & Kyrola, E. 1998, A&A, 332, 732 * Breit & Teller [1940] Breit, G., & Teller, E. 1940, Astrophys. J. , 91, 215 * Burgess [1965] Burgess, A., 1965, MmRAS, 69, 1 * Cai et al. [2017] Cai, Z., Fan, X., Yang, Y., et al. 2017, Astrophys. J. , 837, 71 * Cantalupo et al. [2005] Cantalupo, S., Porciani, C., Lilly, S. J., & Miniati, F. 2005, Astrophys. J. , 628, 61 * Cantalupo et al. [2008] Cantalupo, S., Porciani, C., & Lilly, S. J. 2008, Astrophys. J. , 672, 48 * Cantalupo et al. [2012] Cantalupo, S., Lilly, S. J., & Haehnelt, M. G. 2012, MNRAS, 425, 1992 * Cantalupo et al. [2014] Cantalupo, S., Arrigoni-Battaia, F., Prochaska, J. X., Hennawi, J. F., & Madau, P. 2014, Nature (London), 506, 63 * Caruana et al. [2012] Caruana, J., Bunker, A. J., Wilkins, S. M., et al. 2012, MNRAS, 427, 3055 * Caruana et al. [2014] Caruana, J., Bunker, A. J., Wilkins, S. M., et al. 2014, MNRAS, 443, 2831 * Cen [1992] Cen, R. 1992, ApJS, 78, 341 * Chandrasekhar [1960] Chandrasekhar, S. 1960, New York: Dover, 1960 * Chang et al. [2017] Chang, S.-J., Lee, H.-W., & Yang, Y. 2017, MNRAS, 464, 5018 * Chapman et al. [2005] Chapman, S. C., Blain, A. W., Smail, I., & Ivison, R. J. 2005, Astrophys. J. , 622, 772 * Chluba & Sunyaev [2009] Chluba, J., & Sunyaev, R. A. 2009, A&A, 496, 619 * Chonis et al. [2013] Chonis, T. S., Blanc, G. A., Hill, G. J., et al. 2013, Astrophys. J. , 775, 99 * Choudhury et al. [2015] Choudhury, T. R., Puchwein, E., Haehnelt, M. G., & Bolton, J. S. 2015, MNRAS, 452, 261 * Chung et al. [2009] Chung, A., van Gorkom, J. H., Kenney, J. D. P., Crowl, H., & Vollmer, B. 2009, AJ, 138, 1741 * Chuzhoy & Zheng [2007] Chuzhoy, L., & Zheng, Z. 2007, Astrophys. J. , 670, 912 * Ciardi et al. [2003] Ciardi, B., Stoehr, F., & White, S. D. M. 2003, MNRAS, 343, 1101 * Cooper et al. [2008] Cooper, J. L., Bicknell, G. V., Sutherland, R. S., & Bland-Hawthorn, J. 2008, Astrophys. J. , 674, 157 * Cox [2000] Cox, A. N. 2000, Allen’s Astrophysical Quantities, 1 * Dayal et al. [2010] Dayal, P., Ferrara, A., & Saro, A. 2010, MNRAS, 402, 1449 * Dayal et al. [2011] Dayal, P., Maselli, A., & Ferrara, A. 2011, MNRAS, 410, 830 * Deharveng et al. [2008] Deharveng, J.-M., Small, T., Barlow, T. A., et al. 2008, Astrophys. J. , 680, 1072 * Dennison et al. [2005] Dennison, B., Turner, B. E., & Minter, A. H. 2005, Astrophys. J. , 633, 309 * Dessauges-Zavadsky et al. [2010] Dessauges-Zavadsky, M., D’Odorico, S., Schaerer, D., Modigliani, A., Tapken, C., & Vernet, J. 2010, A&A, 510, A26 * Dey et al. [2005] Dey, A., Bian, C., Soifer, B. T., et al. 2005, Astrophys. J. , 629, 654 * Dijkstra et al. [2006] Dijkstra, M., Haiman, Z., & Spaans, M. 2006, Astrophys. J. , 649, 14 * Dijkstra et al. [2007] Dijkstra, M., Lidz, A., & Wyithe, J. S. B. 2007, MNRAS, 377, 1175 * Dijkstra & Loeb [2008c] Dijkstra, M., & Loeb, A. 2008c, NA, 13, 395 * Dijkstra & Loeb [2008a] Dijkstra, M., & Loeb, A. 2008a, MNRAS, 391, 457 * Dijkstra & Loeb [2008b] Dijkstra, M., & Loeb, A. 2008b, MNRAS, 386, 492 * Dijkstra [2009] Dijkstra, M. 2009, Astrophys. J. , 690, 82 * Dijkstra & Loeb [2009] Dijkstra, M., & Loeb, A. 2009, MNRAS, 400, 1109 * Dijkstra & Wyithe [2010] Dijkstra, M., & Wyithe, J. S. B. 2010, MNRAS, 408, 352 * Dijkstra & Westra [2010] Dijkstra, M., & Westra, E. 2010, MNRAS, 401, 2343 * Dijkstra et al. [2011] Dijkstra, M., Mesinger, A., & Wyithe, J. S. B. 2011, MNRAS, 414, 2139 * Dijkstra & Wyithe [2012] Dijkstra, M., & Wyithe, J. S. B. 2012, MNRAS, 419, 3181 * Dijkstra & Jeeson-Daniel [2013] Dijkstra, M., & Jeeson-Daniel, A. 2013, MNRAS, 435, 3333 * Dijkstra [2014] Dijkstra, M. 2014, PASA, 31, e040 * Dijkstra [2016] Dijkstra, M. 2016, Understanding the Epoch of Cosmic Reionization: Challenges and Progress, 423, 145 * Dijkstra et al. [2016] Dijkstra, M., Sethi, S., & Loeb, A. 2016, Astrophys. J. , 820, 10 * Dijkstra et al. [2016b] Dijkstra, M., Gronke, M., & Venkatesan, A. 2016b, Astrophys. J. , 828, 71 * Draine et al. [2007] Draine, B. T., Dale, D. A., Bendo, G., et al. 2007, Astrophys. J. , 663, 866 * Duval et al. [2014] Duval, F., Schaerer, D., Östlin, G., & Laursen, P. 2014, A&A, 562, A52 * Erb et al. [2014] Erb, D. K., Steidel, C. C., Trainor, R. F., et al. 2014, Astrophys. J. , 795, 33 * Ewen & Purcell [1951] Ewen, H. I., & Purcell, E. M. 1951, Nature (London), 168, 356 * Faisst et al. [2014] Faisst, A. L., Capak, P., Carollo, C. M., Scarlata, C., & Scoville, N. 2014, Astrophys. J. , 788, 87 * Fardal et al. [2001] Fardal, M. A., Katz, N., Gardner, J. P., et al. 2001, Astrophys. J. , 562, 605 * Faucher-Giguère et al. [2008] Faucher-Giguère, C.-A., Prochaska, J. X., Lidz, A., Hernquist, L., & Zaldarriaga, M. 2008, Astrophys. J. , 681, 831 * Faucher-Giguère et al. [2010] Faucher-Giguère, C.-A., Kereš, D., Dijkstra, M., Hernquist, L., & Zaldarriaga, M. 2010, Astrophys. J. , 725, 633 * Field [1959] Field, G. B. 1959, Astrophys. J. , 129, 551 * Finkelstein et al. [2008] Finkelstein, S. L., Rhoads, J. E., Malhotra, S., Grogin, N., & Wang, J. 2008, Astrophys. J. , 678, 655 * Finkelstein et al. [2012] Finkelstein, S. L., Papovich, C., Salmon, B., et al. 2012, Astrophys. J. , 756, 164 * Finlator et al. [2009] Finlator, K., Özel, F., & Davé, R. 2009, MNRAS, 393, 1090 * Foreman-Mackey et al. [2013] Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 * Forero-Romero et al. [2011] Forero-Romero, J. E., Yepes, G., Gottlöber, S., et al. 2011, MNRAS, 415, 3666 * Francis et al. [1996] Francis, P. J., Woodgate, B. E., Warren, S. J., et al. 1996, Astrophys. J. , 457, 490 * Fujita et al. [2009] Fujita, A., Martin, C. L., Mac Low, M.-M., New, K. C. B., & Weaver, R. 2009, Astrophys. J. , 698, 693 * Furlanetto et al. [2004] Furlanetto, S. R., Zaldarriaga, M., & Hernquist, L. 2004, Astrophys. J. , 613, 1 * Furlanetto et al. [2005] Furlanetto, S. R., Schaye, J., Springel, V., & Hernquist, L. 2005, Astrophys. J. , 622, 7 * Furlanetto et al. [2006] Furlanetto, S. R., Oh, S. P., & Briggs, F. H. 2006, Physics Reports, 433, 181 * Fynbo et al. [1999] Fynbo, J. U., Møller, P., & Warren, S. J. 1999, MNRAS, 305, 849 * Garel et al. [2012] Garel, T., Blaizot, J., Guiderdoni, B., et al. 2012, MNRAS, 422, 310 * Gnedin [2000] Gnedin, N. Y. 2000, Astrophys. J. , 535, 530 * Gnedin [2016] Gnedin, N. Y. 2016, Astrophys. J. , 821, 50 * Goerdt et al. [2010] Goerdt, T., Dekel, A., Sternberg, A., et al. 2010, MNRAS, 407, 613 * Gould & Weinberg [1996] Gould, A., & Weinberg, D. H. 1996, Astrophys. J. , 468, 462 * Gordon [1929] Gordon, W. 1929, Annalen der Physik, 394, 1031 * Greig & Mesinger [2017] Greig, B., & Mesinger, A. 2017, MNRAS, 465, 4838 * Gronke & Dijkstra [2014] Gronke, M., & Dijkstra, M. 2014, MNRAS, 444, 1095 * Gronke et al. [2015] Gronke, M., Bull, P., & Dijkstra, M. 2015, Astrophys. J. , 812, 123 * Gronke & Dijkstra [2016] Gronke, M., & Dijkstra, M. 2016, Astrophys. J. , 826, 14 * Gronke et al. [2016] Gronke, M., Dijkstra, M., McCourt, M., & Oh, S. P. 2016, ApJL, 833, L26 * Guaita et al. [2015] Guaita, L., Melinder, J., Hayes, M., et al. 2015, A&A, 576, A51 * Gunn & Peterson [1965] Gunn, J. E., & Peterson, B. A. 1965, Astrophys. J. , 142, 1633 * Haiman et al. [2000] Haiman, Z., Spaans, M., & Quataert, E. 2000, ApJL, 537, L5 * Haiman & Rees [2001] Haiman, Z., & Rees, M. J. 2001, Astrophys. J. , 556, 87 * Haiman [2002] Haiman, Z. 2002, ApJL, 576, L1 * Hamilton [1947] Hamilton, D. R. 1947, Astrophys. J. , 106, 457 * Hansen & Oh [2006] Hansen, M., & Oh, S. P. 2006, MNRAS, 367, 979 * Harrington [1973] Harrington, J. P. 1973, MNRAS, 162, 43 * Hashimoto et al. [2013] Hashimoto, T., Ouchi, M., Shimasaku, K., et al. 2013, Astrophys. J. , 765, 70 * Hashimoto et al. [2015] Hashimoto, T., Verhamme, A., Ouchi, M., et al. 2015, Astrophys. J. , 812, 157 * Hayashino et al. [2004] Hayashino, T., Matsuda, Y., Tamura, H., et al. 2004, AJ, 128, 2073 * Hayes et al. [2005] Hayes, M., Östlin, G., Mas-Hesse, J. M., et al. 2005, A&A, 438, 71 * Hayes et al. [2007] Hayes, M., Östlin, G., Atek, H., et al. 2007, MNRAS, 382, 1465 * Hayes et al. [2010] Hayes, M., Östlin, G., Schaerer, D., et al. 2010, Nature (London), 464, 562 * Hayes et al. [2011] Hayes, M., Schaerer, D., Östlin, G., et al. 2011, Astrophys. J. , 730, 8 * Hayes et al. [2011b] Hayes, M., Scarlata, C., & Siana, B. 2011b, Nature (London), 476, 304 * Hayes et al. [2013] Hayes, M., Östlin, G., Schaerer, D., et al. 2013, ApJL, 765, L27 * Heckman et al. [2011] Heckman, T. M., Borthakur, S., Overzier, R., et al. 2011, Astrophys. J. , 730, 5 * Hennawi et al. [2015] Hennawi, J. F., Prochaska, J. X., Cantalupo, S., & Arrigoni-Battaia, F. 2015, Science, 348, 779 * Henry et al. [2015] Henry, A., Scarlata, C., Martin, C. L., & Erb, D. 2015, Astrophys. J. , 809, 19 * Henyey [1940] Henyey, L. G. 1940, Proceedings of the National Academy of Science, 26, 50 * Hirata [2006] Hirata, C. M. 2006, MNRAS, 367, 259 * Hoang-Binh [1990] Hoang-Binh, D. 1990, A&A, 238, 449 * Hogan & Weymann [1987] Hogan, C. J., & Weymann, R. J. 1987, MNRAS, 225, 1P * Hu et al. [1998] Hu, E. M., Cowie, L. L., & McMahon, R. G. 1998, ApJL, 502, L99 * Hummer [1962] RT, D. G. 1962, MNRAS, 125, 21 * Humphrey et al. [2013] Humphrey, A., Vernet, J., Villar-Martín, M., et al. 2013, ApJL, 768, L3 * Iliev et al. [2006] Iliev, I. T., Mellema, G., Pen, U.-L., et al. 2006, MNRAS, 369, 1625 * Iliev et al. [2008] Iliev, I. T., Shapiro, P. R., McDonald, P., Mellema, G., & Pen, U.-L. 2008, MNRAS, 391, 63 * Izotov et al. [2014] Izotov, Y. I., Thuan, T. X., & Guseva, N. G. 2014, MNRAS, 445, 778 * Jeeson-Daniel et al. [2012] Jeeson-Daniel, A., Ciardi, B., Maio, U., et al. 2012, MNRAS, 424, 2193 * Jensen et al. [2013] Jensen, H., Laursen, P., Mellema, G., et al. 2013, MNRAS, 428, 1366 * Jensen et al. [2014] Jensen, H., Hayes, M., Iliev, I. T., et al. 2014, MNRAS, 444, 2114 * Jones et al. [2012] Jones, T., Stark, D. P., & Ellis, R. S. 2012, Astrophys. J. , 751, 51 * Jones et al. [2013] Jones, T. A., Ellis, R. S., Schenker, M. A., & Stark, D. P. 2013, Astrophys. J. , 779, 52 * Kalberla et al. [2005] Kalberla, P. M. W., Burton, W. B., Hartmann, D., et al. 2005, A&A, 440, 775 * Kakiichi et al. [2016] Kakiichi, K., Dijkstra, M., Ciardi, B., & Graziani, L. 2016, MNRAS, 463, 4019 * Kashikawa et al. [2006] Kashikawa, N., Shimasaku, K., Malkan, M. A., et al. 2006, Astrophys. J. , 648, 7 * Kashikawa et al. [2011] Kashikawa, N., Shimasaku, K., Matsuda, Y., et al. 2011, Astrophys. J. , 734, 119 * Kennicutt [1998] Kennicutt, R. C., Jr. 1998, ARA&A, 36, 189 * Kobayashi et al. [2006] Kobayashi, M. A. R., Kamaya, H., & Yonehara, A. 2006, Astrophys. J. , 636, 1 * Kollmeier et al. [2010] Kollmeier, J. A., Zheng, Z., Davé, R., et al. 2010, Astrophys. J. , 708, 1048 * Kornei et al. [2010] Kornei, K. A., Shapley, A. E., Erb, D. K., et al. 2010, Astrophys. J. , 711, 693 * Krolik [1989] Krolik, J. H. 1989, Astrophys. J. , 338, 594 * Kunth et al. [1998] Kunth, D., Mas-Hesse, J. M., Terlevich, E., Terlevich, R., Lequeux, J., & Fall, S. M. 1998, A&A, 334, 11 * Kulas et al. [2012] Kulas, K. R., Shapley, A. E., Kollmeier, J. A., et al. 2012, Astrophys. J. , 745, 33 * Lake et al. [2015] Lake, E., Zheng, Z., Cen, R., et al. 2015, Astrophys. J. , 806, 46 * Laursen & Sommer-Larsen [2007] Laursen, P., & Sommer-Larsen, J. 2007, ApJL, 657, L69 * Laursen et al. [2009a] Laursen, P., Razoumov, A. O., & Sommer-Larsen, J. 2009a, Astrophys. J. , 696, 853 * Laursen et al. [2009b] Laursen, P., Sommer-Larsen, J., & Andersen, A. C. 2009b, Astrophys. J. , 704, 1640 * Laursen [2010] Laursen, P. 2010, Ph.D. Thesis, Niels Bohr Institute, University of Copenhagen, arXiv:1012.3175 * Laursen et al. [2011] Laursen, P., Sommer-Larsen, J., & Razoumov, A. O. 2011, Astrophys. J. , 728, 52 * Laursen et al. [2013] Laursen, P., Duval, F., Östlin, G. 2013, Astrophys. J. , 766, p124-143 * Lee [1974] Lee, J.-S. 1974, Astrophys. J. , 192, 465 * Lee et al. [1994] Lee, H.-W., Blandford, R. D., & Western, L. 1994, MNRAS, 267, 303 * Lee & Ahn [1998] Lee, H.-W., & Ahn, S.-H. 1998, ApJL, 504, L61 * Lee [2003] Lee, H.-W. 2003, Astrophys. J. , 594, 637 * Loeb & Rybicki [1999] Loeb, A., & Rybicki, G. B. 1999, Astrophys. J. , 524, 527 * Madau et al. [1997] Madau, P., Meiksin, A., & Rees, M. J. 1997, Astrophys. J. , 475, 429 * Majumdar et al. [2014] Majumdar, S., Mellema, G., Datta, K. K., et al. 2014, MNRAS, 443, 2843 * Martin et al. [2014] Martin, D. C., Chang, D., Matuszewski, M., et al. 2014, Astrophys. J. , 786, 106 * Martin et al. [2015] Martin, C. L., Dijkstra, M., Henry, A., et al. 2015, Astrophys. J. , 803, 6 * Martin et al. [2015] Martin, D. C., Matuszewski, M., Morrissey, P., et al. 2015, Nature (London), 524, 192 * Mas-Ribas & Dijkstra [2016] Mas-Ribas, L., & Dijkstra, M. 2016, Astrophys. J. , 822, 84 * Mas-Ribas et al. [2017] Mas-Ribas, L., Dijkstra, M., Hennawi, J. F., et al. 2017, arXiv:1703.02593 * Matsuda et al. [2004] Matsuda, Y., et al. 2004, AJ, 128, 569 * Matsuda et al. [2011] Matsuda, Y., Yamada, T., Hayashino, T., et al. 2011, MNRAS, 410, L13 * Matsuda et al. [2012] Matsuda Y., Yamada T., Hayashino T., Yamauchi R., Nakamura Y., Morimoto N., Ouchi M., Ono Y., Umemura M., Mori M., 2012, MNRAS, 425, 878 * Matthee et al. [2015] Matthee, J., Sobral, D., Santos, S., et al. 2015, MNRAS, 451, 400 * McCarthy [1993] McCarthy, P. J. 1993, ARA&A, 31, 639 * McCourt et al. [2016] McCourt, M., Oh, S. P., O’Leary, R. M., & Madigan, A.-M. 2016, arXiv:1610.01164 * McKee & Ostriker [1977] McKee, C. F., & Ostriker, J. P. 1977, Astrophys. J. , 218, 148 * McLinden et al. [2011] McLinden, E. M., Finkelstein, S. L., Rhoads, J. E., et al. 2011, Astrophys. J. , 730, 136 * McLinden et al. [2014] McLinden, E. M., Rhoads, J. E., Malhotra, S., et al. 2014, MNRAS, 439, 446 * McQuinn et al. [2007] McQuinn, M., Hernquist, L., Zaldarriaga, M., & Dutta, S. 2007, MNRAS, 381, 75 * Meiksin [2006] Meiksin, A. 2006, MNRAS, 370, 2025 * Meiksin [2009] Meiksin, A. A. 2009, Reviews of Modern Physics, 81, 1405 * Mellema et al. [2006] Mellema, G., Iliev, I. T., Alvarez, M. A., & Shapiro, P. R. 2006, NA, 11, 374 * Mesinger & Furlanetto [2007] Mesinger, A., & Furlanetto, S. 2007, Astrophys. J. , 669, 663 * Mesinger & Furlanetto [2008] Mesinger, A., & Furlanetto, S. R. 2008, MNRAS, 386, 1990 * Mesinger et al. [2015] Mesinger, A., Aykutalp, A., Vanzella, E., et al. 2015, MNRAS, 446, 566 * Mesinger et al. [2011] Mesinger, A., Furlanetto, S., & Cen, R. 2011, MNRAS, 411, 955 * Mesinger [2016] Mesinger, A. 2016, Understanding the Epoch of Cosmic Reionization: Challenges and Progress, 423, * [198] Miralda-Escudé, J. 1998, Astrophys. J. , 501, 15-22 * Momose et al. [2014] Momose, R., Ouchi, M., Nakajima, K., et al. 2014, MNRAS, 442, 110 * Morales & Wyithe [2010] Morales, M. F., & Wyithe, J. S. B. 2010, ARA&A, 48, 127 * Mortlock [2016] Mortlock, D. 2016, Understanding the Epoch of Cosmic Reionization: Challenges and Progress, 423, 187 * Neufeld [1990] Neufeld, D. A. 1990, Astrophys. J. , 350, 216 * Neufeld [1991] Neufeld, D. A. 1991, ApJL, 370, L85 * Nestor et al. [2011] Nestor, D. B., Shapley, A. E., Steidel, C. C., & Siana, B. 2011, Astrophys. J. , 736, 18 * Nestor et al. [2013] Nestor, D. B., Shapley, A. E., Kornei, K. A., Steidel, C. C., & Siana, B. 2013, Astrophys. J. , 765, 47 * Nilsson & Møller [2009] Nilsson, K. K., & Møller, P. 2009, A&A, 508, L21 * North et al. [2012] North, P. L., Courbin, F., Eigenbrod, A., & Chelouche, D. 2012, A&A, 542, A91 * Ono et al. [2012] Ono, Y., Ouchi, M., Mobasher, B., et al. 2012, Astrophys. J. , 744, 83 * Orsi et al. [2012] Orsi, A., Lacey, C. G., & Baugh, C. M. 2012, MNRAS, 425, 87O * Osterbrock [1962] Osterbrock, D. E. 1962, Astrophys. J. , 135, 195 * Osterbrock & Ferland [2006] Osterbrock, D. E., & Ferland, G. J. 2006, Astrophysics of gaseous nebulae and active galactic nuclei, 2nd. ed. by D.E. Osterbrock and G.J. Ferland. Sausalito, CA: University Science Books, 2006, * Ostlin et al. [2009] Östlin, G., Hayes, M., Kunth, D., et al. 2009, AJ, 138, 923 * Östlin et al. [2014] Östlin, G., Hayes, M., Duval, F., et al. 2014, Astrophys. J. , 797, 11 * Ota et al. [2010] Ota, K., Iye, M., Kashikawa, N., et al. 2010, Astrophys. J. , 722, 803 * Ota et al. [2017] Ota, K., Iye, M., Kashikawa, N., et al. 2017, arXiv:1703.02501 * Ouchi et al. [2008] Ouchi, M., Shimasaku, K., Akiyama, M., et al. 2008, ApJS, 176, 301-330 * Ouchi et al. [2010] Ouchi, M., Shimasaku, K., Furusawa, H., et al. 2010, Astrophys. J. , 723, 869 * Pallottini et al. [2015] Pallottini, A., Ferrara, A., Pacucci, F., et al. 2015, MNRAS, 453, 2465 * Partridge & Peebles [1967] Partridge, R. B., & Peebles, P. J. E. 1967, Astrophys. J. , 147, 868 * Pawlik & Schaye [2008] Pawlik, A. H., & Schaye, J. 2008, MNRAS, 389, 651 * Peebles [1993] Peebles, P. J. E. 1993, Princeton Series in Physics, Princeton, NJ: Princeton University Press, —c1993, * Pentericci et al. [2011] Pentericci, L., Fontana, A., Vanzella, E., et al. 2011, Astrophys. J. , 743, 132 * Pentericci et al. [2014] Pentericci, L., Vanzella, E., Fontana, A., et al. 2014, Astrophys. J. , 793, 113 * Pierleoni et al. [2009] Pierleoni, M., Maselli, A., & Ciardi, B. 2009, MNRAS, 393, 872 * Planck Collaboration et al. [2016] Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13 * Prescott et al. [2013] Prescott, M. K. M., Dey, A., & Jannuzi, B. T. 2013, Astrophys. J. , 762, 38 * Prescott et al. [2015] Prescott, M. K. M., Martin, C. L., & Dey, A. 2015, Astrophys. J. , 799, 62 * Pritchard & Loeb [2010] Pritchard, J. R., & Loeb, A. 2010, Phys. Rev. D, 82, 023006 * Pritchard & Loeb [2012] Pritchard, J. R., & Loeb, A. 2012, Reports on Progress in Physics, 75, 086901 * Prochaska [2006] Prochaska, J. X. 2006, Astrophys. J. , 650, 272 * Rahmati et al. [2013] Rahmati, A., Pawlik, A. H., Raicevic, M., & Schaye, J. 2013, MNRAS, 430, 2427 * Raiter et al. [2010] Raiter, A., Schaerer, D., & Fosbury, R. A. E. 2010, A&A, 523, 64 * Rauch et al. [2008] Rauch, M., Haehnelt, M., Bunker, A., et al. 2008, Astrophys. J. , 681, 856 * Rémy-Ruyer et al. [2014] Rémy-Ruyer, A., Madden, S. C., Galliano, F., et al. 2014, A&A, 563, A31 * Reuland et al. [2003] Reuland, M., van Breugel, W., Röttgering, H., et al. 2003, Astrophys. J. , 592, 755 * Rivera-Thorsen et al. [2015] Rivera-Thorsen, T. E., Hayes, M., Östlin, G., et al. 2015, Astrophys. J. , 805, 14 * Rosdahl & Blaizot [2012] Rosdahl, J., & Blaizot, J. 2012, MNRAS, 423, 344 * Rubiño-Martín et al. [2006] Rubiño-Martín, J. A., Chluba, J., & Sunyaev, R. A. 2006, MNRAS, 371, 1939 * Rybicki & Lightman [1979] Rybicki, G. B., & Lightman, A. P. 1979, New York, Wiley-Interscience, 1979. 393 p., * Rybicki & dell’Antonio [1994] Rybicki, G. B., & dell’Antonio, I. P. 1994, Astrophys. J. , 427, 603 * Rybicki & Loeb [1999] Rybicki, G. B., & Loeb, A. 1999, ApJL, 520, L79 * Rybicki [2006] Rybicki, G. B. 2006, Astrophys. J. , 647, 709 * Salpeter [1955] Salpeter, E. E. 1955, Astrophys. J. , 121, 161 * Santos [2004] Santos, M. R. 2004, MNRAS, 349, 1137 * Santos et al. [2016] Santos, S., Sobral, D., & Matthee, J. 2016, MNRAS, 463, 1678 * Scarlata et al. [2009] Scarlata, C., Colbert, J., Teplitz, H. I., et al. 2009, ApJL, 704, L98 * Schaerer [2002] Schaerer, D. 2002, A&A, 382, 28 * Schaerer [2003] Schaerer, D. 2003, A&A, 397, 527 * Schaerer & Verhamme [2008] Schaerer, D., & Verhamme, A. 2008, A&A, 480, 369 * Schaye [2001] Schaye, J. 2001, Astrophys. J. , 559, 507 * Schenker et al. [2012] Schenker, M. A., Stark, D. P., Ellis, R. S., et al. 2012, Astrophys. J. , 744, 179S * Schneider et al. [2016] Schneider, R., Hunt, L., & Valiante, R. 2016, MNRAS, 457, 1842 * Scholz et al. [1990] Scholz, T. T., Walters, H. R. J., Burke, P. J., & Scott, M. P. 1990, MNRAS, 242, 692 * Scholz & Walters [1991] Scholz, T. T., & Walters, H. R. J. 1991, Astrophys. J. , 380, 302 * Schuster [1879] Schuster, A. 1879, MNRAS, 40, 35 * Semelin et al. [2007] Semelin, B., Combes, F., & Baek, S. 2007, A&A, 474, 365 * Shapiro & Kang [1987] Shapiro, P. R., & Kang, H. 1987, Astrophys. J. , 318, 32 * Shibuya et al. [2014] Shibuya, T., Ouchi, M., Nakajima, K., et al. 2014, Astrophys. J. , 788, 74 * Shull & McKee [1979] Shull, J. M., & McKee, C. F. 1979, Astrophys. J. , 227, 131 * Shull & van Steenberg [1985] Shull, J. M., & van Steenberg, M. E. 1985, Astrophys. J. , 298, 268 * Smith et al. [2015] Smith, A., Safranek-Shrader, C., Bromm, V., & Milosavljević, M. 2015, MNRAS, 449, 4336 * So et al. [2014] So, G. C., Norman, M. L., Reynolds, D. R., & Wise, J. H. 2014, Astrophys. J. , 789, 149 * Sobacchi & Mesinger [2014] Sobacchi, E., & Mesinger, A. 2014, MNRAS, 440, 1662 * Sobacchi & Mesinger [2015] Sobacchi, E., & Mesinger, A. 2015, MNRAS, 453, 1843 * Sobral et al. [2015] Sobral, D., Matthee, J., Darvish, B., et al. 2015, Astrophys. J. , 808, 139 * Sokasian et al. [2001] Sokasian, A., Abel, T., & Hernquist, L. E. 2001, NA, 6, 359 * Song et al. [2014] Song, M., Finkelstein, S. L., Gebhardt, K., et al. 2014, Astrophys. J. , 791, 3 * Spitzer & Greenstein [1951] Spitzer, L., Jr., & Greenstein, J. L. 1951, Astrophys. J. , 114, 407 * Stark et al. [2010] Stark, D. P., Ellis, R. S., Chiu, K., Ouchi, M., & Bunker, A. 2010, MNRAS, 408, 1628 * Steidel et al. [2000] Steidel, C. C., Adelberger, K. L., Shapley, A. E., Pettini, M., Dickinson, M., & Giavalisco, M. 2000, Astrophys. J. , 532, 170 * Steidel et al. [2010] Steidel, C. C., Erb, D. K., Shapley, A. E., et al. 2010, Astrophys. J. , 717, 289 * Stenflo [1980] Stenflo, J. O. 1980, A&A, 84, 68 * Steidel et al. [2011] Steidel C. C., Bogosavljević M., Shapley A. E., Kollmeier J. A., Reddy N. A., Erb D. K., Pettini M., 2011, Astrophys. J. , 736, 160 * Tasitsiomi [2006] Tasitsiomi, A. 2006, Astrophys. J. , 645, 792 * Tasitsiomi [2006b] Tasitsiomi, A. 2006b, Astrophys. J. , 648, 762 * Thoul & Weinberg [1995] Thoul, A. A., & Weinberg, D. H. 1995, Astrophys. J. , 442, 480 * Tozzi et al. [2000] Tozzi, P., Madau, P., Meiksin, A., & Rees, M. J. 2000, Astrophys. J. , 528, 597 * Trac & Cen [2007] Trac, H., & Cen, R. 2007, Astrophys. J. , 671, 1 * Trac & Gnedin [2011] Trac, H. Y., & Gnedin, N. Y. 2011, Advanced Science Letters, 4, 228 * Trainor et al. [2015] Trainor, R. F., Steidel, C. C., Strom, A. L., & Rudie, G. C. 2015, Astrophys. J. , 809, 89 * Trebitsch et al. [2016] Trebitsch, M., Verhamme, A., Blaizot, J., & Rosdahl, J. 2016, A&A, 593, A122 * Treu et al. [2013] Treu, T., Schmidt, K. B., Trenti, M., Bradley, L. D., & Stiavelli, M. 2013, ApJL, 775, L29 * Tumlinson & Shull [2000] Tumlinson, J., & Shull, J. M. 2000, ApJL, 528, L65 * Unno [1952] Unno, W. 1952, PASJ, 4, 100 * van Breugel et al. [2006] van Breugel, W., de Vries, W., Croft, S., et al. 2006, Astronomische Nachrichten, 327, 175 * Vanzella et al. [2010] Vanzella, E., Grazian, A., Hayes, M., et al. 2010, A&A, 513, A20 * Verhamme et al. [2006] Verhamme, A., Schaerer, D., & Maselli, A. 2006, A&A, 460, 397 * Verhamme et al. [2008] Verhamme, A., Schaerer, D., Atek, H., & Tapken, C. 2008, A&A, 491, 89 * Verhamme et al. [2012] Verhamme, A., Dubois, Y., Blaizot, J., et al. 2012, A&A, 546, A111 * Verhamme et al. [2015] Verhamme, A., Orlitová, I., Schaerer, D., & Hayes, M. 2015, A&A, 578, A7 * Visbal et al. [2016] Visbal, E., Haiman, Z., & Bryan, G. L. 2016, MNRAS, 460, L59 * White [1934] White, H. E. 1934, IEEE Transactions on Applied Superconductivity, * Wisotzki et al. [2016] Wisotzki, L., Bacon, R., Blaizot, J., et al. 2016, A&A, 587, A98 * Wofford et al. [2013] Wofford, A., Leitherer, C., & Salzer, J. 2013, Astrophys. J. , 765, 118 * Wold et al. [2014] Wold, I. G. B., Barger, A. J., & Cowie, L. L. 2014, Astrophys. J. , 783, 119 * Wouthuysen [1952] Wouthuysen, S. A. 1952, AJ, 57, 31 * Wyithe & Loeb [2007] Wyithe, J. S. B., & Loeb, A. 2007, MNRAS, 375, 1034 * Xue et al. [2017] Xue, R., Lee, K.-S., Dey, A., et al. 2017, Astrophys. J. , 837, 172 * Yajima et al. [2012] Yajima, H., Li, Y., Zhu, Q., & Abel, T. 2012, MNRAS, 424, 884 * Yajima et al. [2012] Yajima, H., Li, Y., Zhu, Q., et al. 2012, Astrophys. J. , 754, 118 * Yang et al. [2016] Yang, H., Malhotra, S., Gronke, M., et al. 2016, Astrophys. J. , 820, 130 * Yusef-Zadeh et al. [1984] Yusef-Zadeh, F., Morris, M., & White, R. L. 1984, Astrophys. J. , 278, 186 * Zahn et al. [2011] Zahn, O., Mesinger, A., McQuinn, M., et al. 2011, MNRAS, 414, p727-738 * Zanstra [1949] Zanstra, H. 1949, BAN, 11, 1 * Zheng & Miralda-Escudé [2002] Zheng, Z., & Miralda-Escudé, J. 2002, Astrophys. J. , 578, 33 * Zheng et al. [2010] Zheng, Z., Cen, R., Trac, H., & Miralda-Escudé, J. 2010, Astrophys. J. , 716, 574 * Zheng et al. [2011] Zheng, Z., Cen, R., Weinberg, D., Trac, H., & Miralda-Escudé, J. 2011, Astrophys. J. , 739, 62 * Zheng & Wallace [2014] Zheng, Z., & Wallace, J. 2014, Astrophys. J. , 794, 116 * Zheng et al. [2017] Zheng, Z.-Y., Wang, J., Rhoads, J., et al. 2017, arXiv:1703.02985 * Zitrin et al. [2015] Zitrin, A., Labbé, I., Belli, S., et al. 2015, ApJL, 810, L12
1812.09136	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 59056, "num_imgs": 11, "llama3_tokens_count": 21055 }	[ "content_image/1812.09136/x1.png", "content_image/1812.09136/x2.png", "content_image/1812.09136/x3.png", "content_image/1812.09136/x4.png", "content_image/1812.09136/x5.png", "content_image/1812.09136/x6.png", "content_image/1812.09136/x7.png", "content_image/1812.09136/x8.png", "content_image/1812.09136/x9.png", "content_image/1812.09136/x10.png", "content_image/1812.09136/x11.png" ]	# Nonuniform superconductivity and Josephson effect in conical ferromagnet Hao Meng University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France School of Physics and Telecommunication Engineering, Shaanxi University of Technology, Hanzhong 723001, China Shanghai Key Laboratory of High Temperature Superconductors, Shanghai University, Shanghai 200444, China A. V. Samokhvalov Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia A. I. Buzdin alexandre.bouzdine@u-bordeaux.fr University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France Department of Materials Science and Metallurgy, University of Cambridge, CB3 0FS, Cambridge, United Kingdom Sechenov First Moscow State Medical University, Moscow, 119991, Russia ###### Abstract Using the Gorkov equations, we provide an exact solution for a one-dimensional model of superconductivity in the presence of a conical helicoidal exchange field. Due to the special type of symmetry of the system, the superconducting transition always occurs into a nonuniform superconducting phase (in contrast with the Fulde-Ferrell-Larkin-Ovchinnikov state, which appears only at low temperatures). We directly demonstrate that the uniform superconducting state in our model carries a current and thus does not correspond to the ground state. We study in the framework of the Bogoliubov-de Gennes approach the properties of the Josephson junction with a conical ferromagnet as a weak link. In our numerical calculations, we do not use any approximations (such as, e.g., a quasiclassical approach), and we show a realization of an anomalous $\phi_{0}$ junction (with a spontaneous phase difference $\phi_{0}$ in the ground state). The spontaneous phase difference $\phi_{0}$ strongly increases at high values of the exchange field near the borderline with a half-metal, and it exists also in the half-metal regime. pacs: 74.50.+r, 73.45.+c, 76.50.+g ## I Introduction The interest in superconductor-ferromagnet (SF) structures has been stimulated by the unusual SF proximity effect, leading to the fabrication of the Josephson junctions with unique properties (see, e.g., [1; 2; 3; 4; 5]), which paved the way for superconducting spintronics. Moreover, the combination of spin-orbit coupling and a Zeeman field may lead to the anomalous Josephson effect—the so-called $\phi_{0}$ junction with a spontaneous phase difference at the ground state [6; 7; 8; 9]. This is related to an emergence of topological nonuniform superconducting phases [10]. In [11] it has been noted that a superconductor with a conical helical magnet structure is described by the same Hamiltonian as a topological superconducting phase appearing in systems with spin-orbit and Zeeman interactions. The problem of a superconducting uniform phase in the presence of the helicoidal exchange field has a complete analytical solution in the framework of the formalism of Gorkov’s Green functions [12]. In [13] the peculiar properties of the Josephson junction between two helicoidal superconductors were considered, while in [14; 15; 16; 17; 18] the Josephson junction with a magnetic helix weak link was studied in the framework of the quasiclassical approximation. In Sec. II of this paper, we use Gorkov’s formalism to get the analytical expressions for Green’s functions in the conical helical superconducting magnet, taking into account the possibility of the topological nonuniform superconducting phase realization. Further, we perform a detailed analysis of the one-dimensional (1D) system and demonstrate the emergence of the nonuniform superconducting phase with a modulation wave vector $q$ when the helix becomes conical. The modulation vector is proportional to the canting of the helix and inversely proportional to the helix period. Our conclusion is based on the analysis of the critical temperature dependence on the superconductivity modulation vector $q$, which is obtained from the linear equation for the superconducting order parameter. The modulated superconducting state corresponds to the minimum energy of the system and does not carry current. Complimentarily, we calculate the current at $T=0$ in the uniform superconducting phase and show that it is not equal to zero, which proves that the uniform phase cannot be a ground state and thus the modulated phase is the most stable at all temperatures. The emergence of the modulated superconducting state may be illustrated by simple arguments in the framework of Ginzburg-Landau theory. In the standard situation, the lowest over the gradients of the order parameter $\Psi$ term gives the following well known quadratic contribution to the free energy, $\delta{F_{in\hom}}=\gamma\left\|\mathbf{\nabla}\Psi\right\|^{2}$, while the higher derivative terms may be neglected. The term that is linear over the gradient is absent because it is not invariant under the inversion symmetry operation. In the absence of inversion symmetry, Rashba spin-orbit interaction (SO) leads to the following additional contribution to the electron’s energy: $\sim[\vec{\sigma}\times\vec{p}]\cdot\vec{n}$, where $\vec{p}$ is the momentum, $\vec{n}$ is the unit vector along the axis with broken inversion symmetry, and $\vec{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ is the vector of Pauli matrices [19]. In the presence of the exchange field $\vec{h}$ this results in a term that is linear over the gradient of the superconducting order parameter $\Psi$ in the Ginzburg-Landau (GL) free energy $\sim[\vec{n}\times\vec{h}]\cdot(\nabla\Psi)\,\Psi^{}$ (see, for example, [19; 20]). In the case of the conical helicoid, the role of the $\vec{n}$ vector is played by the vector $[\,\vec{h}\times\mathrm{rot}\vec{h}\,]$, and the linear-over-gradient term becomes $\sim\left[\,\vec{h}\times[\,\vec{h}\times\mathrm{rot}\vec{h}]\,\right]\,\cdot( \nabla\Psi)\,\Psi^{}$. This is a manifestation of the equivalence of a model of a conical superconductor to a model of a topological superconductor [11]. In the considered case of the conical helicoid with the exchange field $\vec{h}=(h\cos\mathbf{Qr},h\sin\mathbf{Qr},h_{z})$ (the wave vector $\mathbf{Q}=Q\mathbf{z}_{0}$ is along the $z$ axis), the normal state is lacking inversion symmetry and the following additional invariant that is linear over the gradient is possible: \[\delta F_{add}=\left[i\lambda\Psi\left(\mathbf{h}\cdot rot\mathbf{h}\right) \cdot(h_{z}\cdot\left(\mathbf{\nabla}\Psi\right)_{z})+c.c.\right]\,,\] (1) where the parameter $\lambda$ depends on the strength of the SO coupling. In the result, the energy contribution due to the modulation of the order parameter $\Psi=\Psi_{0}e^{iqz}$ becomes $\delta{F_{in\hom}}=\gamma{q^{2}}\Psi_{0}^{2}-2\lambda{q}h^{2}h_{z}Q\Psi_{0}^{2}$, and the minimum energy (and the maximum of the critical temperature) corresponds to the nonuniform superconducting state with a modulation vector $q\sim{h^{2}}h_{z}Q$. Note that there is no threshold on the value of counting field $h_{z}$ to generate the modulation, which is in sharp contrast with a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [21; 22]. The FFLO modulated state appears when the usual gradient term in the Ginzburg-Landau functional changes its sign, i.e., the coefficient $\gamma$ becomes negative when the exchange field overcomes some threshold [23]. At low temperature for a standard superconductor we may use the London theory, and the gauge invariance imposes the following form of the term in the energy, depending on the vector-potential $\mathbf{A}$: \[\triangle{F}=a\left(\nabla\varphi+\frac{2e}{\hbar}\mathbf{A}\right)^{2}\,,\] where $\varphi$ is the phase of the superconducting order parameter $\Psi=\|\Psi\|\,\exp{(i\varphi})$. As a consequence, the current density $j=-c\,\delta F/\delta\mathbf{A}$, and in the absence of the magnetic field, choosing $A=0$, we see that the minimum energy corresponds to $\nabla\varphi=0$ and therefore $j=0$. In the considered case of the conical helicoid, the contribution $\Delta{F}$ to the energy should have a linear over $\left(\nabla\varphi+2e\mathbf{A}/\hbar\right)$ term: (2) As a result, the current density reads \[j\sim 2a\left(\nabla\varphi+\frac{2e}{\hbar}\mathbf{A}\right)+b\,,\] and in the absence of the field ($A=0$) and phase modulation ($\nabla\varphi=0$) the current is nonzero, $j\sim{b}$. This reflects the fact that the uniform state is not a ground-state of our system. Indeed, for $A=0$ the minimum of the energy (2) corresponds to $\nabla\varphi=-b/2a$, and for this phase modulation the current vanishes. In Sec. III we calculate the Josephson current for the 1D model of the weak link made of the conical helix. Our numerical calculations use the exact solutions of the Bogoliubov-de Gennes (BdG) equations and we demonstrate the realization of the anomalous $\phi_{0}$ junction. The spontaneous phase shift $\phi_{0}$ strongly increases when we approach the half-metal regime or when we are completely in the half-metal state. In this case, the current-phase relation for the supercurrent is $I(\phi)=I_{c}\sin(\phi-\phi_{0})$ and the additional phase shift $\phi_{0}$ is proportional to the ferromagnetic component of exchange field $h_{z}$. We provide a detailed study of the properties of the $\phi_{0}$ junction as a function of conical helix parameters. The conical helicoidal phase exists, for example, in antiferromagnetic Ho, and the Ho/Nb structure has attracted a lot of attention [24; 25; 26; 27; 28]. In these systems, the electron mean free path is of the same order as the period of the helix, and we believe that qualitatively the results of our work may be applicable to these structures. The possibility to use the conical helix as a building block of the $\phi_{0}$ junction may be important for the design of the superconducting spintronics devices. <figure><img src="content_image/1812.09136/x1.png"><figcaption>Figure 1: The sketch of a superconductor with a conical magnetic texture. Thegreen thick arrow indicates the direction of the exchange field.</figcaption></figure> ## II Superconducting conical helicoidal phase—Gorkov’s Green Functions We study a clean _s_-wave magnetic superconductor with conical magnetic order. The conical magnetism and the spatially modulated order parameter can be characterized by $\vec{h}=(h\cos\mathbf{Q\cdot{r}},h\sin\mathbf{Q\cdot{r}},h_{z})$ and $\Delta(\mathbf{r})=\Delta{e^{i\mathbf{q\cdot{r}}}}$, respectively (see Fig. 1). Using the mean-field approximation, we may write the Hamiltonian of the system as [29] \[\hat{H} =\sum_{\alpha\beta}\int{d^{3}}\mathbf{r}\left\{\hat{\psi}_{\alpha }^{\dagger}(\mathbf{r})\xi_{p}\hat{\psi}_{\alpha}(\mathbf{r})+\hat{\psi}_{ \alpha}^{\dagger}(\mathbf{r})(\vec{h}\cdot\vec{\sigma})_{\alpha\beta}\hat{\psi }_{\beta}(\mathbf{r})\right.\] (3) where $\xi_{p}=\frac{p^{2}}{2m}-E_{F}$, and $\hat{\psi}_{\alpha}^{\dagger}(\mathbf{r})$ and $\hat{\psi}_{\alpha}(\mathbf{r})$ represent creation and annihilation operators with spin $\alpha$. The spatially modulated superconducting order parameter is described by $\left\langle\hat{\psi}_{\alpha}^{\dagger}(\mathbf{r})\hat{\psi}_{\beta}^{ \dagger}(\mathbf{r})\right\rangle=\left(i\sigma_{y}\right)_{\alpha\beta}\Delta {e^{i\mathbf{q\cdot{r}}}}$. The Gorkov equations of the system of the Green’s functions $G_{\alpha,\beta}(\mathbf{r},\mathbf{r}^{\prime})=-\left\langle{T}\hat{\psi}_{ \alpha}(\mathbf{r})\hat{\psi}_{\beta}^{\dagger}(\mathbf{r}^{\prime})\right\rangle$ and $F_{\alpha,\beta}^{\dagger}(\mathbf{r},\mathbf{r}^{\prime})=\left\langle{T}\hat {\psi}_{\alpha}^{\dagger}(\mathbf{r})\hat{\psi}_{\beta}^{\dagger}(\mathbf{r}^{ \prime})\right\rangle$ have the form \[\left(i\omega_{n}-\xi_{p}-\hat{V}\right)\hat{G}(\mathbf{r},\mathbf{r}^{\prime} )+\Delta{e^{i\mathbf{q\cdot{r}}}}\cdot{\hat{I}}\hat{F}^{{\dagger}}(\mathbf{r}, \mathbf{r}^{\prime})=\delta(\mathbf{r}-\mathbf{r}^{\prime}),\] (4) \[\left(i\omega_{n}+\xi_{p}+\tilde{V}\right)\hat{F}^{{\dagger}}(\mathbf{r}, \mathbf{r}^{\prime})-\Delta^{\ast}{e^{-i\mathbf{q\cdot{r}}}}\cdot{\hat{I}}\hat {G}(\mathbf{r},\mathbf{r}^{\prime})=0,\] (5) where the matrix $\hat{I}$ is written as \[\hat{I}=i\sigma_{y}=\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right).\] (6) The wave vectors $\mathbf{Q}$ and $\mathbf{q}$ are along the $z$-axis, and the potential of the conical magnetic order is given by \[\hat{V}(\mathbf{r})=\vec{h}\cdot\vec{\sigma}=\left(\begin{array}[]{cc}h_{z}&he ^{-iQz}\\ he^{iQz}&-h_{z}\end{array}\right)\] (7) while \[\tilde{V}(\mathbf{r})=\left(\begin{array}[]{cc}h_{z}&he^{iQz}\\ he^{-iQz}&-h_{z}\end{array}\right).\] (8) Using the Fourier transform, we obtain the exact solution of (4)-(5) described in Appendix A and get the Green functions (below only $\hat{F}_{21}^{{\dagger}}$ and $\hat{G}_{11}$ are presented) \[\hat{F}_{21}^{{\dagger}}\left(p-\frac{Q}{2}-\frac{q}{2},p^{\prime }\right)=-\delta\left(p-\frac{Q}{2}+\frac{q}{2}-p^{\prime}\right)\] (9) \[\times\frac{\left[\left(i\omega_{n}-\xi_{2}+h_{z}\right)\left(i \omega_{n}+\xi_{3}+h_{z}\right)+h^{2}-\left\|\Delta\right\|^{2}\right]\Delta^{{} ^{\ast}}}{D(\omega_{n})},\] \[\hat{G}_{11}\left(p-\frac{Q}{2}+\frac{q}{2},p^{\prime}\right)= \delta\left(p-\frac{Q}{2}+\frac{q}{2}-p^{\prime}\right)\] (10) \[\times\left[\frac{\left(i\omega_{n}-\xi_{2}+h_{z}\right)\left(i \omega_{n}+\xi_{3}+h_{z}\right)\left(i\omega_{n}+\xi_{4}-h_{z}\right)}{D( \omega_{n})}\right.\] \[\left.-\frac{\left(i\omega_{n}-\xi_{2}+h_{z}\right)h^{2}+\left(i \omega_{n}+\xi_{4}-h_{z}\right)\left\|\Delta\right\|^{2}}{D(\omega_{n})}\right],\] where $D(\omega_{n})=$ \[\left[\left(i\omega_{n}-\xi_{1}-h_{z}\right)\left(i\omega_{n}+\xi _{4}-h_{z}\right)+h^{2}-\left\|\Delta\right\|^{2}\right]\] (11) \[\times\left[\left(i\omega_{n}-\xi_{2}+h_{z}\right)\left(i\omega_{ n}+\xi_{3}+h_{z}\right)+h^{2}-\left\|\Delta\right\|^{2}\right]\] \[-\left(2i\omega_{n}-\xi_{1}+\xi_{3}\right)\left(2i\omega_{n}-\xi_ {2}+\xi_{4}\right)h^{2}.\] and \[\xi_{1}=\xi_{p-\frac{Q}{2}+\frac{q}{2}},\xi_{2}=\xi_{p+\frac{Q}{2}+\frac{q}{2} },\\\] (12) \[\xi_{3}=\xi_{p+\frac{Q}{2}-\frac{q}{2}},\xi_{4}=\xi_{p-\frac{Q}{2}-\frac{q}{2}}.\] (13) Note that we have obtained the exact solution of the 1D model, which is readily generalized to the 3D case: indeed we start from the Hamiltonian (3) describing the 3D system, and the corresponding Gorkov equations (4) and (5) are readily applied to the 3D case provided that we consider all vectors as 3D vectors $\vec{p}$, $\vec{Q}$, and $\vec{q}$. The superconducting conical ferromagnet is one of the rare examples when it is possible to get explicitly the complete solution in the framework of the microscopical Gorkov equations. ### The energy spectrum of the conical ferromagnet Let us first consider the normal conical ferromagnet without superconducting coupling ($\Delta=0$ and $q=0$). The Green’s function $\hat{G}_{11}$ in such a case reads \[\hat{G}_{11}\left(p-\frac{Q}{2},p^{\prime}\right)=\delta\left(p- \frac{Q}{2}-p^{\prime}\right)\] (14) \[\times\frac{i\omega_{n}-\xi_{p+\frac{Q}{2}}+h_{z}}{\left(i\omega_ {n}-\xi_{p-\frac{Q}{2}}-h_{z}\right)\left(i\omega_{n}-\xi_{p+\frac{Q}{2}}+h_{z }\right)-h^{2}}.\] To find the electrons spectrum $\epsilon$, we should perform the analytical continuation $i\omega_{n}\rightarrow\epsilon$ in the denominator of the equation (14), and then its zeros give us the equation for the energy spectrum, \[\left(\epsilon-\xi_{p-\frac{Q}{2}}-h_{z}\right)\left(\epsilon-\xi_{p+\frac{Q}{ 2}}+h_{z}\right)-h^{2}=0.\] (15) In the result, we obtain two branches of the energy spectrum, \[\epsilon_{1(2)} = \frac{1}{2}\left[\xi_{p-\frac{Q}{2}}+\xi_{p+\frac{Q}{2}}\right.\] \[\mp\left.\sqrt{\left(\xi_{p+\frac{Q}{2}}-\xi_{p-\frac{Q}{2}}-2h_{ z}\right)^{2}+4h^{2}}\right].\] As illustrated in Fig. 2(a), two branches ($\epsilon_{1}$ and $\epsilon_{2}$) of the energy spectrum are not symmetric with respect to $p=0$, and they also do not contain the gaps. It is a peculiar property of the periodical helicoidal exchange field—it does not create the gap band structure in contrast to the usual case of the periodical potential field. <figure><img src="content_image/1812.09136/x2.png"><figcaption>Figure 2: (a) Energy spectrums (ϵ1 and ϵ2) for the conical ferromagnet(h/EF=0.25, hz/EF=0.2, and Q/kF=π/2) and the normal metal (h/EF=0, hz/EF=0,and Q/kF=0); (b) the velocity of a quasiparticle as a function of the wavevector p. Here the quantities of the velocity are normalized to the value ofthe Fermi velocity vF0 of the quasiparticle in the normal metal.</figcaption></figure> According to the formula $v=\frac{1}{\hbar}\frac{dE}{dk}$, we can compute the velocity of quasiparticles [see Fig. 2(b)]. It is known that in the normal metal, the Fermi velocities of two quasiparticles (at $\pm{k_{F0}})$ have the same absolute values $v_{F0}$ of the Fermi velocities. However, in the conical ferromagnet, the absolute values of Fermi velocities of the quasiparticles are different in the same branches, for instance in the $\epsilon_{1}$ branch ($v_{F1}^{a}=1.102v_{F0}$ and $v_{F2}^{a}=0.9057v_{F0}$) and in the $\epsilon_{2}$ branch ($v_{F1}^{b}=0.6467v_{F0}$ and $v_{F2}^{b}=0.7397v_{F0}$) for chosen parameters of the conical ferromagnet. Namely, this property is characteristic of the systems with a spin-orbit interaction and leads to the appearance of the modulated superconducting states. ### Superconducting transition temperature in the modulated phase The critical temperature of the system is determined by the linearized self-consistency equation (taking in the limit $\Delta\rightarrow{0}$): \[\Delta^{\ast}=\|g\|T\sum_{\omega_{n}}\int_{-\infty}^{+\infty}\hat{F}_{21}^{{ \dagger}}\frac{dp}{2\pi},\] (17) where $g$ is the electron-phonon coupling constant. It is more convenient to write it in the following form: \[\ln\left(\frac{T_{c}}{T_{c0}}\right)=2T_{c}\sum_{\omega_{n}\geq 0}\left[{\rm Re }\int_{-\infty}^{+\infty}\frac{\hat{F}_{21}^{{\dagger}}}{\Delta^{\ast}}d\xi- \frac{\pi}{\omega_{n}}\right],\] (18) where $T_{c}$ is the critical temperature and $T_{c0}$ is the critical temperature in the absence of exchange field $\vec{h}$. Introducing $\Delta{T_{c}}=T_{c}-T_{c0}$, and performing the expansion over the modulation vector $q$ of the superconducting phase in the limit $h_{z}$, $h\ll{T_{c}}$ we finally obtain (see Appendix B for details) \[\frac{\Delta{T_{c}}}{T_{c0}} = 2\pi{T_{c}}\sum_{\omega_{n}\geq 0}\left[-\frac{h_{z}^{2}}{\omega _{n}^{3}}-\frac{4h^{2}}{\left(4\omega_{n}^{2}+v^{2}Q^{2}\right)\omega_{n}}\right.\] \[\left.-\frac{4Qh^{2}h_{z}q}{m\left(4\omega_{n}^{2}+v^{2}Q^{2} \right)\omega_{n}^{3}}-\frac{v^{2}q^{2}}{4\omega_{n}^{3}}\right].\] The very important point is the presence of linear-over-$q$ term, which means that the maximum of the critical temperature always occurs at finite $q$. The linear dependence of the critical temperature $T_{c}$ over $q$ (which describes the modulation of the superconducting order parameter) is the direct consequence of the linear-over-gradient term $\nabla\Psi$ in the GL free energy (1). In accordance with the form of the GL term, the coefficient on $q$ dependence is proportional to the product $h^{2}h_{z}Q$. At the same time, the presence of a linear-over-$q$ term guarantees that the modulated state corresponds to the absence of the current, while the uniform one ($q=0$) does not. For $vQ\ll{T_{c0}}$, the above equation can be simplified as \[\frac{\Delta{T_{c}}}{T_{c0}}=2\pi{T_{c}}\sum_{\omega_{n}\geq 0}\left[-\frac{h_ {z}^{2}+h^{2}}{\omega_{n}^{3}}-\frac{Qh^{2}h_{z}}{m\omega_{n}^{5}}q-\frac{v^{2 }}{4\omega_{n}^{3}}q^{2}\right].\] (20) The maximum of the transition temperature is reached at the modulation wave vector \[q_{0}=-\frac{31Qh^{2}h_{z}}{28\pi^{2}T_{c0}^{2}E_{F}}\frac{\zeta(5)}{\zeta(3)}.\] (21) Here $\zeta(s)$ is the Euler–Riemann zeta function and $E_{F}=\frac{mv^{2}}{2}$. In the opposite limit, $vQ\gg T_{c0}$, the above equation will change to \[\frac{\Delta{T_{c}}}{T_{c0}} = 2\pi{T_{c}}\sum_{\omega_{n}\geq 0}\left[-\left(\frac{h_{z}^{2}}{ \omega_{n}^{2}}+\frac{4h^{2}}{v^{2}Q^{2}}\right)\frac{1}{\omega_{n}}\right.\] \[\left.-\frac{4h^{2}h_{z}}{mv^{2}Q\omega_{n}^{3}}q-\frac{v^{2}}{4 \omega_{n}^{3}}q^{2}\right]\] and the modulation vector of the superconducting phase will be $q_{0}=-\frac{4h^{2}h_{z}}{v^{2}QE_{F}}$. Note that in the both cases, the expression for the modulation vector $q_{0}$ contains a small factor $\frac{h_{z}}{E_{F}}$, and this circumstance explains why the emergence of the modulated superconducting phase cannot be described in the framework of Eilenberger or Usadel quasiclassical equations, where such effects are simply neglected. ### Current in a uniform superconducting phase with the conical magnetic order <figure><img src="content_image/1812.09136/x3.png"><figcaption>Figure 3: The supercurrent J versus (a) the magnetic order h and (b) thespiral wave vector Q. We choose EF=100Δ and hz/EF=0.05. Here the supercurrentunit is J0=2em.</figcaption></figure> We now derive the expression for supercurrent in uniform ($q=0$) superconductors with the conical spiral magnetic order. The spiral magnetic order is characterized by the wave vector $\mathbf{Q}$ along the $z$ axis, $\mathbf{Q}=\chi Q\mathbf{e}_{z}$, and by the helicity $\chi=\pm 1$. In the limit $h_{z}\ll\left\|\Delta\right\|$, the Green function $\hat{G}_{11}\left(p-\frac{Q}{2},p^{\prime}\right)$ reads \[\hat{G}_{11}\left(p-\frac{Q}{2},p^{\prime}\right)=\hat{G}_{11}^{(0)}\left(p- \frac{Q}{2},p^{\prime}\right)+h_{z}\hat{G}_{11}^{(1)}\left(p-\frac{Q}{2},p^{ \prime}\right),\] (23) where \[\hat{G}_{11}^{(0)}\left(p-\frac{Q}{2},p^{\prime}\right)=\delta \left(p-\frac{Q}{2}-p^{\prime}\right)\] (24) \[\times\frac{-\xi_{p-\frac{Q}{2}}\left(\omega_{n}^{2}+\xi_{p+\frac {Q}{2}}^{2}+\left\|\Delta\right\|^{2}\right)-\xi_{p+\frac{Q}{2}}h^{2}}{\left[ \omega_{n}^{2}+E_{1}^{2}\right]\left[\omega_{n}^{2}+E_{2}^{2}\right]},\] \[\hat{G}_{11}^{(1)}\left(p-\frac{Q}{2},p^{\prime}\right)=\delta \left(p-\frac{Q}{2}-p^{\prime}\right)\] (25) \[\times\left\{\frac{\xi_{p+\frac{Q}{2}}^{2}-\omega_{n}^{2}-h^{2}+ \left\|\Delta\right\|^{2}}{\left[\omega_{n}^{2}+E_{1}^{2}\right]\left[\omega_{n} ^{2}+E_{2}^{2}\right]}\right.\] \[E_{1,2}^{2} = \tilde{\zeta}^{2}+\tilde{\eta}^{2}+\left\|\Delta\right\|^{2}+h^{2}\] \[\pm 2\sqrt{\tilde{\zeta}^{2}\left(\tilde{\eta}^{2}+h^{2}\right)+ \left\|\Delta\right\|^{2}h^{2}},\] $\tilde{\zeta}=\left(\xi_{p-\frac{Q}{2}}+\xi_{p+\frac{Q}{2}}\right)/2$ and $\tilde{\eta}=\left(\xi_{p-\frac{Q}{2}}-\xi_{p+\frac{Q}{2}}\right)/2$. <figure><img src="content_image/1812.09136/x4.png"><figcaption>Figure 4: The SFS Josephson junction consists of two _s_ -wave superconductorsand a conical ferromagnet. The green thick arrow indicates the direction ofthe exchange field in the conical ferromagnet.</figcaption></figure> We may write for the current [30] \[J = \frac{ie}{m}\left(\nabla_{r^{\prime}}-\nabla_{r}\right)\left. \left[\hat{G}_{11}(r,r^{\prime})+\hat{G}_{22}(r,r^{\prime})\right]\right\|_{r^{ \prime}\rightarrow{r}}\] \[= \frac{2e}{m}\iint{dp}d\omega\left[p\hat{G}_{11}(p,h_{z})+p\hat{G} _{22}(p,h_{z})\right]\] \[= \frac{4eh_{z}}{m}\left\{\frac{Q\pi}{m}\int{dp}\frac{p^{2}\xi_{Q}( p)}{E_{1}^{2}E_{2}+E_{1}E_{2}^{2}}\right.\] \[+\frac{Q\pi}{2}\int{dp}\frac{2h^{2}-\left(E_{1}-E_{2}\right)^{2}/ 2}{E_{1}^{2}E_{2}+E_{1}E_{2}^{2}}\] \[\left.+\frac{Q\pi}{m}\int{dp}\frac{p^{2}\xi_{Q}(p)\left[\frac{Q^{ 2}}{m}\xi_{Q}(p)-\left(E_{1}+E_{2}\right)^{2}\right]}{E_{1}E_{2}(E_{1}+E_{2})^ {3}}\right\},\] where $\xi_{Q}(p)=\xi(p)+\frac{Q^{2}}{8m}$. The details of these calculations are presented in Appendix C. From the above formula (II.3), we can obtain the dependence of supercurrent $J$ on the strength of the helical field $h/E_{F}$ and the helix wave vector $Q/k_{F}$ (see Fig. 3). We see that the current in the uniform state is proportional to $h_{z}h^{2}$ and the spiral wave vector $Q$ in accordance with the results of Sec. IIB. Therefore, the uniform superconducting phase is not a ground state, which should be a nonuniform superconducting phase at any temperatures. ## III The Bogoliubov–de Gennes approach for conical Josephson junction It is known that the effects related to the spin-orbit interaction often cannot be adequately described by the usual quasiclassical approach [31; 32]. As mentioned beforehand, the superconductor with a conical helical magnetic structure is similar to the topological superconducting phase appearing in the systems with spin-orbit and Zeeman interactions. So the anomalous supercurrent in the Josephson junction with conical magnetization should be calculated using exact solutions of the BdG approach but not the quasiclassical one. We consider the SFS Josephson junction made of two BCS superconductors (S) and a normal-state metal barrier (F) with conical magnetic spiral ordering, see Fig. 4. The $z$ axis is chosen to be perpendicular to the layer interfaces with the origin located at the center of the ferromagnetic layer. The superconducting gap is supposed to be constant in the leads $\left(\left\|{z}\right\|>L/2\right)$ and absent inside the conical ferromagnet $\left(\left\|{z}\right\|<L/2\right)$: \[\Delta(\mathbf{r})=\left\{\begin{array}[]{cc}\Delta\,e^{i\phi/2}\,,&z<-L/2\,, \\ 0\,,&\|z\|\leq{L/2}\,,\\ \Delta\,e^{-i\phi/2}\,,&z>L/2,\end{array}\right.\] (28) where $\Delta$ is the magnitude of the gap and $\phi$ is the phase difference between the two leads. As before, the spiral is characterized by the wave vector $\mathbf{Q}$ along the $z$ axis, $\mathbf{Q}=\chi{Q}\mathbf{e}_{z}$, and by the helicity $\chi=Q_{z}/Q=\pm{1}$. The BCS mean-field effective Hamiltonian of the considered system is described by the expression (3) [2; 29] with a step-like $\Delta(\mathbf{z})$ (28). To diagonalize the effective Hamiltonian, we use the Bogoliubov transformation $\hat{\psi}_{\alpha}(\mathbf{r})=\sum_{n}[u_{n\alpha}(\mathbf{r})\hat{\gamma}_{ n}+v_{n\alpha}^{\ast}(\mathbf{r})\hat{\gamma}_{n}^{{\dagger}}]$ and take into account the anticommutation relations of the quasiparticle annihilation operator $\hat{\gamma}_{n}$ and creation operator $\hat{\gamma}_{n}^{{\dagger}}$. Using the presentation $u_{n\alpha}(\mathbf{r})=u_{p}^{\alpha}e^{ipz}$, $v_{n\alpha}(\mathbf{r})=v_{p}^{\alpha}e^{ipz}$, the resulting Bogoliubov-de Gennes (BdG) equations can be expressed as [29] \[\begin{pmatrix}\hat{H}_{1}&i\hat{\sigma}_{y}\Delta(z)\\ -i\hat{\sigma}_{y}\Delta^{\ast}(z)&-\hat{H}_{2}\end{pmatrix}\begin{pmatrix} \hat{u}(z)\\ \hat{v}(z)\end{pmatrix}=\epsilon\begin{pmatrix}\hat{u}(z)\\ \hat{v}(z)\end{pmatrix},\] (29) where \[\hat{H}_{1(2)}=\begin{pmatrix}\xi_{p\mp{Q/2}}+{h_{z}}&{h}\\ {h}&\xi_{p\pm{Q/2}}-h_{z}\end{pmatrix}.\] Moreover, $\hat{u}(z)=[u_{p-Q/2}^{\uparrow}(z),u_{p+Q/2}^{\downarrow}(z)]^{T}$ and $\hat{v}(z)=[v_{p+Q/2}^{\uparrow}(z),v_{p-Q/2}^{\downarrow}(z)]^{T}$ are quasiparticle and quasihole wave functions, respectively. The solutions of the BdG equation (29) can be found in each layer separately and then matched with the boundary conditions. For a given energy $\epsilon$ inside the superconducting gap, we find the following plane-wave solutions in the left superconducting electrode: \[\psi_{L}^{S}(z) = C_{1}\hat{\rho}_{1}e^{-ik_{S}^{+}z}+C_{2}\hat{\rho}_{2}e^{ik_{S} ^{-}z}\] \[+C_{3}\hat{\rho}_{3}e^{-ik_{S}^{+}z}+C_{2}\hat{\rho}_{4}e^{ik_{S} ^{-}z},\] where $k_{S}^{\pm}=k_{F}\sqrt{1\pm{i}\sqrt{\Delta^{2}-\epsilon^{2}}/E_{F}}$ are the wave vectors for quasiparticles. $\hat{\rho}_{1}=[1,0,0,R_{1}e^{-i\phi/2}]^{T}$, $\hat{\rho}_{2}=[1,0,0,R_{2}e^{-i\phi/2}]^{T}$, $\hat{\rho}_{3}=[0,1,-R_{1}e^{-i\phi/2},0]^{T}$, and $\hat{\rho}_{4}=[0,1,-R_{2}e^{-i\phi/2},0]^{T}$ are the four basis wave functions of the left superconductor, in which $R_{1(2)}=(\epsilon\mp{i}\sqrt{\Delta^{2}-\epsilon^{2}})/\Delta$. The corresponding wave function in the right superconducting electrode is \[\psi_{R}^{S}(z) = D_{1}\hat{\eta}_{1}e^{ik_{S}^{+}z}+D_{2}\hat{\eta}_{2}e^{-ik_{S} ^{-}z}\] \[+D_{3}\hat{\eta}_{3}e^{ik_{S}^{+}z}+D_{4}\hat{\eta}_{4}e^{-ik_{S} ^{-}z},\] where $\hat{\eta}_{1}=[1,0,0,R_{1}e^{i\phi/2}]^{T}$, $\hat{\eta}_{2}=[1,0,0,R_{2}e^{i\phi/2}]^{T}$, $\hat{\eta}_{3}=[0,1,-R_{1}e^{i\phi/2},0]^{T}$, and $\hat{\eta}_{4}=[0,1,-R_{2}e^{i\phi/2},0]^{T}$. ### The eigenenergy spectrum and eigenfunction of the conical ferromagnet From the equation (29) we obtain four eigenvalues and four eigenfunctions for our system. The first eigenfunction is determined by the expression \[\hat{u}_{1}(z)=M_{1}\left(\begin{array}[]{c}e^{i(p_{1}-\frac{Q}{2})z}\\ T_{1}e^{i(p_{1}+\frac{Q}{2})z}\end{array}\right)+M_{2}\left(\begin{array}[]{c} e^{i(p_{2}-\frac{Q}{2})z}\\ T_{2}e^{i(p_{2}+\frac{Q}{2})z}\end{array}\right),\] (32) where $T_{1(2)}=-h/(\xi_{p_{1(2)}+\frac{Q}{2}}-h_{z}-\epsilon)$. The wave vectors $p_{1}$ and $p_{2}$ can be found numerically from equation $\epsilon_{1}(p_{1(2)})=\epsilon$, there the branches of the energy spectrum $\epsilon_{1(2)}(p)$ are determined by the relation (II.1). The second eigenfunction reads \[\hat{u}_{2}(z)=M_{3}\left(\begin{array}[]{c}T_{3}e^{i(p_{3}-\frac{Q}{2})z}\\ e^{i(p_{3}+\frac{Q}{2})z}\end{array}\right)+M_{4}\left(\begin{array}[]{c}T_{4} e^{i(p_{4}-\frac{Q}{2})z}\\ e^{i(p_{4}+\frac{Q}{2})z}\end{array}\right),\] (33) where $T_{3(4)}=-h/(\xi_{p_{3(4)}-\frac{Q}{2}}+h_{z}-\epsilon)$ and the wave vectors $p_{3}$ and $p_{4}$ are the solutions of the equation $\epsilon_{2}(p_{3(4)})=\epsilon$. The third eigenfunction may be written as \[\hat{v}_{1}(z)=M_{5}\left(\begin{array}[]{c}e^{i(p_{5}+\frac{Q}{2})z}\\ T_{5}e^{i(p_{5}-\frac{Q}{2})z}\end{array}\right)+M_{6}\left(\begin{array}[]{c} e^{i(p_{6}+\frac{Q}{2})z}\\ T_{6}e^{i(p_{6}-\frac{Q}{2})z}\end{array}\right),\] (34) where $T_{5(6)}=-h/(\xi_{p_{5(6)}-\frac{Q}{2}}-h_{z}+\epsilon)$ and the wave vectors $p_{5}$ and $p_{6}$ arise from the equation $\epsilon_{1}(p_{5(6)})=-\epsilon$. The fourth eigenfunction can be described as \[\hat{v}_{2}(z)=M_{7}\left(\begin{array}[]{c}T_{7}e^{i(p_{7}+\frac{Q}{2})z}\\ e^{i(p_{7}-\frac{Q}{2})z}\end{array}\right)+M_{8}\left(\begin{array}[]{c}T_{8} e^{i(p_{8}+\frac{Q}{2})z}\\ e^{i(p_{8}-\frac{Q}{2})z}\end{array}\right),\] (35) where $T_{7(8)}=-h/(\xi_{p_{7(8)}+\frac{Q}{2}}+h_{z}+\epsilon)$. The corresponding wave vectors $p_{7}$ and $p_{8}$ satisfy the equation $\epsilon_{2}(p_{7(8)})=-\epsilon$. As a result, the total wave function in the ferromagnetic region can be described as \[\psi_{F}(z)=I_{1}\otimes\hat{u}_{1}(z)+I_{1}\otimes\hat{u}_{2}(z)+I_{2}\otimes \hat{v}_{1}(z)+I_{2}\otimes\hat{v}_{2}(z),\] (36) where $I_{1}=[1,0]^{T}$ and $I_{2}=[0,1]^{T}$. ### Josephson current of the system The wave functions [$\psi_{L}^{S}(z)$, $\psi_{F}(z)$ and $\psi_{R}^{S}(z)$] and their first derivatives should satisfy the continuity conditions at the S/F and F/S interfaces, \[\psi_{L}^{S}(-\frac{L}{2})=\psi_{F}(-\frac{L}{2}),\frac{\partial\psi_{L}^{S}}{ \partial{z}}\left\|{}_{z=-\frac{L}{2}}\right.=\frac{\partial\psi_{F}}{\partial{ z}}\left\|{}_{z=-\frac{L}{2}}\right.,\] (37) \[\psi_{F}(\frac{L}{2})=\psi_{R}^{S}(\frac{L}{2}),\frac{\partial\psi_{F}}{ \partial{z}}\left\|{}_{z=\frac{L}{2}}\right.=\frac{\partial\psi_{R}^{S}}{ \partial{z}}\left\|{}_{z=\frac{L}{2}}\right..\] (38) From these boundary conditions, we can set up 16 linear equations in the following form: \[\hat{A}X=\hat{B},\] (39) where $X$ contains 16 scattering coefficients and $\hat{A}$ is a $16\times 16$ matrix. The solution of the characteristic equation \[\det\hat{A}=0\] (40) allows one to identify two Andreev bound-state solutions for energies $E_{A\sigma}$ ($\sigma$=1, 2). The Josephson current can be calculated as \[I(\phi)=\frac{2e}{\hbar}\frac{\partial\Omega}{\partial\phi},\] (41) where $\Omega$ is the phase-dependent thermodynamic potential. This potential can be obtained from the excitation spectrum by using the formula [33; 34] \[\Omega=-2T\sum_{\sigma}\ln\left[2\cosh\frac{E_{A\sigma}(\phi)}{2T}\right].\] (42) where $\Delta$, $h$, $h_{z}$, and $\mathbf{Q}$ are assumed to be the equilibrium values, which minimize the free energy of the SFS structure and depend on microscopic parameters [35]. The summation in (42) is taken over all positive Andreev energies [$0<E_{A\sigma}(\phi)<\Delta$]. For each value of $\phi$, we solve Eq. (40) numerically to obtain the two spin-polarized Andreev levels. Since the Andreev energy spectra are doubled as they include the Bogoliubov redundancy, and only half part of the energy states should be taken into account, we can acquire the Josephson current via Eqs. (41) and (42). ### Results and discussions <figure><img src="content_image/1812.09136/x5.png"><figcaption>Figure 5: (a) Energy spectrum of the helical ferromagnet, (b) Andreev bound-state energies vs the superconducting phase difference ϕ, and (c) current-phase relation for the helical ferromagnetic junction when h/EF takes threedifferent values. The results plotted are for EF=1000Δ, hz/EF=0, Q/kF=π/2, andkFL=60. The horizontal dash-dotted line in (a) denotes the Fermi level.</figcaption></figure> In this section, we present our results for the energy spectrum, Andreev bound-state spectrum, and the current-phase relation. Unless otherwise stated, we use the superconducting gap $\Delta$ as the unit of energy. All lengths and the exchange field strengths are measured in units of the inverse Fermi wave vector $k_{F}$ and the Fermi energy $E_{F}$, respectively. The current-phase relations are calculated at $T=0$ and the current is presented in units of $I_{0}=2e\Delta/\hbar$ as a function of the parameters of the ferromagnetic barrier $L$, $h$, $h_{z}$, and $\mathbf{Q}$, which are supposed to be equilibrium values. Note that the different components of the exchange field produce different effects on the current-phase relations, and should be analyzed separately. <figure><img src="content_image/1812.09136/x6.png"><figcaption>Figure 6: (a) Andreev bound-state energies vs the superconducting phasedifference ϕ and (b) current-phase relation for a conical ferromagneticjunction when hz/EF takes three different values. The right inset shows thedependence of I(ϕ=0) on the exchange field hz/EF. The results plotted are forEF=1000Δ, h/EF=0.15, kFL=60, and Q/kF=π/2.</figcaption></figure> We start our numerical solutions of the BdG equation (29) from the case of the helical exchange fields $h$ without canting, i.e., for $h_{z}=0$. In Fig. 5, we present the results of calculations of electrons energy spectra, Andreev bound-state spectra, and the current-phase relations for the three different values of the exchange field $h\gg\Delta$ to demonstrate the transition from the polarized metal ferromagnet to the half-metal. For chosen parameters of the F layer, the junction under consideration satisfies the short Josephson junction condition $L\ll\xi_{0}=\hbar{v_{F}}/\Delta$. For a metal interlayer, the current-phase relation is strongly nonsinusoidal and looks like the current-phase relation of short clean SNS [1] and SFS [34] junctions. In the case of the half-metal ($h/E_{F}=0.55$), the current-phase relation approaches a sinusoidal one, and as expected the critical current is strongly decreased. Note that contrary to [34], we do not see the complete vanishing of the Josephson current in the half-metal state. As we can see in Fig. 5, the Josephson current always goes to zero for $\phi=0$ and we have the standard Josephson junction behaviors in this regime. <figure><img src="content_image/1812.09136/x7.png"><figcaption>Figure 7: (a) Andreev bound-state energies vs the superconducting phasedifference ϕ and (b) current-phase relation for a conical ferromagneticjunction when kFL=10. The inset in (a) shows the dependence of I(ϕ=0) on thethickness kFL. The top and bottom insets in (b) illustrate the sum of theAndreev bound-state energies and the zoom of the current-phase relation nearϕ=0, respectively. The results plotted are for EF=100Δ, h/EF=0.15, andQ/kF=0.3.</figcaption></figure> <figure><img src="content_image/1812.09136/x8.png"><figcaption>Figure 8: (a) Andreev bound-state energies vs the superconducting phasedifference ϕ and (b) current-phase relation for a conical half-metallicjunction when hz/EF takes several different values. The right inset shows thedependence of I(ϕ=0) on the exchange field hz/EF. The results plotted are forEF=1000Δ, h/EF=0.55, kFL=60, and Q/kF=π/2.</figcaption></figure> The situation changes drastically if the ferromagnetic component of the exchange field along the $z$ axis exists ($h_{z}\neq 0$). Figure 6 shows the Andreev spectrum and the current-phase relation of a short Josephson junction with polarized ferromagnetic metal as a barrier. Small deformation of energy spectrum due to the exchange fields canting results in the qualitative modification of the Andreev spectrum and the current-phase relation: a small non zero Josephson current $I(\phi=0)$ appears in the absence of the phase difference $\phi=0$. Hence, the $\phi_{0}$ Josephson junction [6; 7; 8] is obtained with a finite phase difference $\|\phi_{0}\|\ll\pi$ in the ground state. For the exchange field $h/E_{F}<0.1$ the spontaneous current seems to be very small and the precision of our numerical analysis is not enough to study this regime. Starting at $h/E_{F}>0.1$, we clearly observe the emergence of the spontaneous current and its amplitude increase when we approach the half-metal case. The current $I(\phi=0)$ oscillates and changes sign as the canting field $h_{z}/E_{F}$ increases. For $\Delta\ll{h_{z}}\ll{h}$, the value $I(\phi=0)$ remains small in comparison with the critical current. So, the particularities of the electrons spectra in the conical ferromagnet as a weak link lead to the appearance of the spontaneous Josephson current in the absence of the phase difference. Such behavior can be understood as a phase accumulation due to the superconducting order-parameter modulation described in Sec. II.B. This modulation is proportional to $h_{z}$ in formula (21) and vanishes at $h_{z}=0$. <figure><img src="content_image/1812.09136/x9.png"><figcaption>Figure 9: (a) Andreev bound-state energies vs the superconducting phasedifference ϕ and (b) current-phase relation for a conical half-metallicjunction when kFL varies from 40 to 80 with steps of 10. The inset shows thedependence of I(ϕ=0) on the thickness kFL. Here we set the parametersEF=1000Δ, h/EF=0.55, hz/EF=0.084, and Q/kF=π/2.</figcaption></figure> In Fig. 7 we present the evolution of the Andreev spectra and the spontaneous current when the parameter $\Delta/E_{F}$ increases. The short Josephson junction condition is valid for shorter barrier $k_{F}L=10$$(L/\xi_{0}\simeq 0.05)$. A comparison of Figs. 6 and 7 shows that the Andreev spectra and the current-phase relation look similar for close values of $h_{z}/\Delta$ and $L/\xi_{0}$. The current $I(\phi=0)$ oscillates with the variation of the thickness of the ferromagnet ${L}$ and changes its sign for negative $h_{z}$: $I(\phi=0,-h_{z})=-I(\phi=0,h_{z})$ (see the inset in Fig. 7). The amplitude of the spontaneous Josephson current grows as the factor $k_{F}L$ increases. Figures 8 and 9 show how the Andreev spectrum and current-phase relation of the Josephson junction depend on the canting field $h_{z}/E_{F}$ and the barrier thickness $k_{F}L$ for the rather large ratio $h/E_{F}=0.55$, which corresponds to the half-metal state of the ferromagnet. We see that the current-phase relation for a conical half-metallic junction is close to the sinusoidal one and differs qualitatively from the previous case of the polarized ferromagnetic metal. The spontaneous current $I(\phi=0)$ and the spontaneous phase difference $\phi_{0}$ change continuously with the exchange field canting and the thickness. Hence, we can obtain a finite current at zero superconducting phase and a continuous change of the phase difference $\phi_{0}$ from $0$ to $\pi$ by tuning the exchange field canting. As expected, nonzero $h_{z}$ generates the $\phi_{0}$ junction in this case too, and the ground phase difference is very sensitive to the length of the weak link. ## IV Conclusion On the basis of the exact solution in terms of Gorkov’s Green functions of the 1D model of a superconductor with a conical exchange field, we demonstrate that the ground states corresponds to the modulated superconducting phase at all temperatures. The instability of the uniform state is related to the special symmetry of the system generating the triplet superconducting correlations. We calculate the wave vector of the superconducting state modulation near the superconducting transition temperature, and we show that it is proportional to the ferromagnetic component of the conical field. These results of the exact solution are in sharp contrast to the results of the solution in the framework of the quasiclassical Eilenberger or Usadel approach, which always predict the uniform superconducting state in the case of the weak exchange field. In the second part of the article, we study the properties of the S/F/S junction with the F-conical ferromagnet. Our numerical solutions of full Bogoliubov-de Gennes equations (without the usual quasiclassical approximation) reveal the emergence of the $\phi_{0}$ junction with the finite phase difference at the ground state and nonzero current for $\phi=0$. We study how the anomalous current depends on the characteristics of the conical magnet. The revealed direct coupling between the exchange field and the Josephson phase difference paves the way for interesting implementations of the $\phi_{0}$ junctions in superconducting spintronics. ## Acknowledgments The authors thank A. Melnikov and S. Mironov for useful discussions and suggestions. A.B. wishes to thank the Leverhulme Trust for supporting his stay at Cambridge University. This work was supported by French ANR project SUPERTRONICS and OPTOFLUXONICS (A.I.B.) and EU Network COST CA16218 (NANOCOHYBRI). A.V.S. acknowledges the funding from by Russian Foundation for Basic Research (Grants No. 17-52-12044 NNIO and No. 18-02-00390) and Russian Science Foundation under Grant No. 17-12-01383 (Sec. II C). H.M. acknowledges the National Natural Science Foundation of China (Grant No. 11604195) and the Youth Hundred Talents Programme of Shaanxi Province. ## References * (1) A. A. Golubov, M. Yu. Kupriyanov, and E. Ilichev, Rev. Mod. Phys. 76, 411 (2004). * (2) A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). * (3) F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). * (4) J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307 (2015). * (5) M. Eschrig, Rep. Prog. Phys. 78, 104501 (2015). * (6) I. V. Krive, L. Y. Gorelik, R. I. Shekhter, and M. Jonson, Fiz. Nizk. Temp. 30, 535 (2004) [Low Temp. Phys. 30, 398 (2004)]. * (7) A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, and M. Avignon, Phys. Rev. Lett. 101, 107001 (2008). * (8) A. I. Buzdin, Phys. Rev. Lett. 101, 107005 (2008); F. Konschelle and A. Buzdin, ibid. 102, 017001 (2009). * (9) S. Mironov and A. Buzdin, Phys. Rev. B 92, 184506 (2015). * (10) K. N. Nesterov, M. Houzet, and J. S. Meyer, Phys. Rev. B 93, 174502 (2016). * (11) I. Martin and A. F. Morpurgo, Phys. Rev. B 85, 144505 (2012). * (12) L. N. Bulaevskii, A. I. Rusinov, and Kulić, J. Low Temp. Phys. 39, 255 (1980). * (13) M. L. Kulić and I. M. Kulić, Phys. Rev. B 63, 104503 (2001). * (14) L. Bulaevskii, R. Eneias, and A. Ferraz, Phys. Rev. B 95, 104513 (2017). * (15) A. F. Volkov, A. Anishchanka, and K. B. Efetov, Phys. Rev. B 73, 104412 (2006). * (16) I. V. Bobkova, A. M. Bobkov, and M. A. Silaev, Phys. Rev. B 96, 094506 (2017). * (17) D. S. Rabinovich, I. V. Bobkova, A. M. Bobkov, and M. A. Silaev, Phys. Rev. B 98, 184511 (2018). * (18) D. S. Rabinovich, I. V. Bobkova, A. M. Bobkov, and M. A. Silaev, arXiv:1811.09304. * (19) V. P. Mineev and M. Sigrist, Lect. notes Phys. 847, 129 (2012). * (20) V. M. Edelstein, J. Phys. Condens. Matter 8, 339 (1996). * (21) P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). * (22) A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)]. * (23) A. I. Buzdin and H. Kachkachi, Phys. Lett. A 225, 341 (1997). * (24) G. B. Halász, J. W. A. Robinson, J. F. Annett, and M. G. Blamire, Phys. Rev. B 79, 224505 (2009). * (25) I. Sosnin, H. Cho, V. T. Petrashov, and A. F. Volkov, Phys. Rev. Lett. 96, 157002 (2006). * (26) J. D. S. Witt, J. W. A. Robinson, and M. G. Blamire, Phys. Rev. B 85, 184526 (2012). * (27) F. Chiodi, J. D. S. Witt, R. G. J. Smits, L. Qu, G. B. Halász, C.-T. Wu, O. T. Valls, K. Halterman, J. W. A. Robinson, and M. G. Blamire, Europhys Lett. 101, 37002 (2013). * (28) A. Di Bernardo, S. Diesch, Y. Gu, J. Linder, G. Divitini, C. Ducati, E. Scheer, M. G. Blamire, and J. W. A. Robinson, Nat. Commun. 6, 8053 (2015). * (29) P. G. de Gennes, _Superconductivity of Metals and Alloys_ (Benjamin, New York, 1966), Chap. 5. * (30) A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, _Methods of Quantum Field Theory in Statistical Physics_ (Prentice-Hall, Englewood Cliffs, NJ, 1964). * (31) C. R. Reeg and D. L. Maslov, Phys. Rev. B 92, 134512 (2015). * (32) M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Phys. Rev. B 95, 184508 (2017). * (33) J. Bardeen, R. Kümel, A. E. Jacobs, and L. Tewordt, Phys. Rev. 187, 556 (1969). * (34) J. Cayssol and G. Montambaux, Phys. Rev. B 70, 224520 (2004). * (35) L. N. Bulaevskii, A. I. Buzdin, M. L. Kulić, and S. V. Panyukov, Adv. Phys. 34, 176 (1985). ## Appendix A The Gor’kov equations (4) and (5) can be expressed in matrix form: \[\left(\begin{array}[]{cc}i\omega_{n}-\xi_{p}-h_{z}&-he^{-iQz}\\ -he^{iQz}&i\omega_{n}-\xi_{p}+h_{z}\end{array}\right)\left(\begin{array}[]{cc} \hat{G}_{11}&\hat{G}_{12}\\ \hat{G}_{21}&\hat{G}_{22}\end{array}\right)\] (43) \[+\Delta{e^{iqz}}\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right)\left(\begin{array}[]{cc}\hat{F}_{11}^{{\dagger}}&\hat{F }_{12}^{{\dagger}}\\ \hat{F}_{21}^{{\dagger}}&\hat{F}_{22}^{{\dagger}}\end{array}\right)=\delta( \mathbf{r}-\mathbf{r}^{\prime})\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right),\] \[\left(\begin{array}[]{cc}i\omega_{n}+\xi_{p}+h_{z}&he^{iQz}\\ he^{-iQz}&i\omega_{n}+\xi_{p}-h_{z}\end{array}\right)\left(\begin{array}[]{cc} \hat{F}_{11}^{{\dagger}}&\hat{F}_{12}^{{\dagger}}\\ \hat{F}_{21}^{{\dagger}}&\hat{F}_{22}^{{\dagger}}\end{array}\right)\] (44) \[-\Delta^{\ast}{e^{-iqz}}\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right)\left(\begin{array}[]{cc}\hat{G}_{11}&\hat{G}_{12}\\ \hat{G}_{21}&\hat{G}_{22}\end{array}\right)=0.\] Applying the Fourier transform to (43) and (44), we get a set of equations \[\left(i\omega_{n}-\xi_{p-\frac{Q}{2}+\frac{q}{2}}-h_{z}\right) \hat{G}_{11}(p-\frac{Q}{2}+\frac{q}{2},p^{\prime})\] (45) \[-h\hat{G}_{21}(p+\frac{Q}{2}+\frac{q}{2},p^{\prime})+\Delta\hat{F }_{21}^{{\dagger}}(p-\frac{Q}{2}-\frac{q}{2},p^{\prime})\] \[=\delta(p-\frac{Q}{2}+\frac{q}{2}-p^{\prime}),\] \[\left(i\omega_{n}-\xi_{p+\frac{Q}{2}+\frac{q}{2}}+h_{z}\right) \hat{G}_{21}(p+\frac{Q}{2}+\frac{q}{2},p^{\prime})\] (46) \[-h\hat{G}_{11}(p-\frac{Q}{2}+\frac{q}{2},p^{\prime})-\Delta\hat{F }_{11}^{{\dagger}}(p+\frac{Q}{2}-\frac{q}{2},p^{\prime})=0,\] \[\left(i\omega_{n}+\xi_{p+\frac{Q}{2}-\frac{q}{2}}+h_{z}\right) \hat{F}_{11}^{{\dagger}}(p+\frac{Q}{2}-\frac{q}{2},p^{\prime})\] (47) \[+h\hat{F}_{21}^{{\dagger}}(p-\frac{Q}{2}-\frac{q}{2},p^{\prime})- \Delta^{\ast}\hat{G}_{21}(p+\frac{Q}{2}+\frac{q}{2},p^{\prime})=0,\] \[\left(i\omega_{n}+\xi_{p-\frac{Q}{2}-\frac{q}{2}}-h_{z}\right) \hat{F}_{21}^{{\dagger}}(p-\frac{Q}{2}-\frac{q}{2},p^{\prime})\] (48) \[+h\hat{F}_{11}^{{\dagger}}(p+\frac{Q}{2}-\frac{q}{2},p^{\prime})+ \Delta^{\ast}\hat{G}_{11}(p-\frac{Q}{2}+\frac{q}{2},p^{\prime})=0.\] The solutions of (45)-(48) provide the expression for $\hat{F}_{11}^{{\dagger}}(p+\frac{Q}{2}-\frac{q}{2},p^{\prime})$, $\hat{F}_{21}^{{\dagger}}(p-\frac{Q}{2}-\frac{q}{2},p^{\prime})$, $\hat{G}_{11}(p-\frac{Q}{2}+\frac{q}{2},p^{\prime})$ and $\hat{G}_{21}(p+\frac{Q}{2}+\frac{q}{2},p^{\prime})$. Following the same derivation procedure, we can get another set of equations from (43) and (44) for Green functions $\hat{F}_{22}^{{\dagger}}(p-\frac{Q}{2}-\frac{q}{2},p^{\prime})$, $\hat{F}_{12}^{{\dagger}}(p+\frac{Q}{2}-\frac{q}{2},p^{\prime})$, $\hat{G}_{22}(p+\frac{Q}{2}+\frac{q}{2},p^{\prime})$ and $\hat{G}_{12}(p-\frac{Q}{2}+\frac{q}{2},p^{\prime})$. These equations coincide with (45)-(48) provided ($\hat{F}_{11}$, $\hat{F}_{21}$, $\hat{G}_{11}$, $\hat{G}_{21}$) are replaced by ($-\hat{F}_{22}$, $-\hat{F}_{12}$, $\hat{G}_{22}$, $\hat{G}_{12}$) and ($\omega_{n}$, $Q$, $q$, $h$, $h_{z}$) are replaced by ($\omega_{n}$, $-Q$, $q$, $h$, $-h_{z}$). ## Appendix B To obtain $\hat{F}_{21}^{{\dagger}}$ in a linear-over-$\Delta$ approximation, it is sufficient to neglect the quadratic term $\left\|\Delta\right\|^{2}$ in Eqs. (9) and (11). Performing the expansion over $h^{2}$ and also making the substitutions $Q/2\rightarrow\tilde{Q}$ and $q/2\rightarrow\tilde{q}$, the expressions of $\hat{F}_{21}^{{\dagger}}$ can be simplified into the following form: \[\hat{F}_{21}^{{\dagger}}\left(p-\tilde{Q}-\tilde{q},p^{\prime} \right)=-\delta\left(p-\tilde{Q}+\tilde{q}-p^{\prime}\right)\tilde{F}_{21}^{{ \dagger}}\,,\] (49) \[\tilde{F}_{21}^{{\dagger}}=-\frac{(A_{3}A_{4}+h^{2})\Delta^{\ast} }{A_{1}A_{2}A_{3}A_{4}-(A_{1}A_{3}+A_{2}A_{4})h^{2}}\] (50) \[\qquad\simeq-\frac{\Delta^{\ast}}{A_{1}A_{2}}{(1+\frac{h^{2}}{A_{ 3}A_{4}}+\frac{h^{2}}{A_{2}A_{4}}+\frac{h^{2}}{A_{1}A_{3}})}\,,\] where $A_{1}\sim{A_{4}}$ are determined by the expressions \[A_{1} = i\omega_{n}-\xi\left(p-\tilde{Q}+\tilde{q}\right)-h_{z}\simeq{i} \omega_{n}-\xi+X_{1},\] \[A_{2} = i\omega_{n}-\xi\left(p+\tilde{Q}+\tilde{q}\right)+h_{z}\simeq{i} \omega_{n}+\xi+X_{2},\] \[A_{3} = i\omega_{n}+\xi\left(p+\tilde{Q}-\tilde{q}\right)+h_{z}\simeq{i} \omega_{n}-\xi+X_{3},\] \[A_{4} = i\omega_{n}+\xi\left(p-\tilde{Q}-\tilde{q}\right)-h_{z}\simeq{i} \omega_{n}+\xi+X_{4}.\] Here $\xi=p^{2}/2m-E_{F}$ and \[X_{1}=v(\tilde{Q}-\tilde{q})+\frac{\tilde{Q}\tilde{q}}{m}-h_{z} \,,X_{2}=-v(\tilde{Q}+\tilde{q})+\frac{\tilde{Q}\tilde{q}}{m}-h_{z}\,,\] \[X_{3}=-v(\tilde{Q}+\tilde{q})-\frac{\tilde{Q}\tilde{q}}{m}+h_{z} \,,X_{4}=v(\tilde{Q}-\tilde{q})-\frac{\tilde{Q}\tilde{q}}{m}+h_{z}\,.\] As a result, the function $\tilde{F}_{21}^{{\dagger}}$ can be expressed as \[\tilde{F}_{21}^{{\dagger}} = -\Delta^{\ast}\left[\frac{1}{(i\omega_{n}-\xi+X_{1})(i\omega_{n}+ \xi+X_{2})}\right.\] \[+\frac{1}{(i\omega_{n}-\xi+X_{1})(i\omega_{n}+\xi+X_{2})}\] \[\times\frac{1}{(i\omega_{n}-\xi+X_{3})(i\omega_{n}+\xi+X_{4})}\] \[+\frac{1}{(i\omega_{n}-\xi+X_{1})(i\omega_{n}+\xi+X_{2})^{2}(i \omega_{n}+\xi+X_{4})}\] \[\left.+\frac{1}{(i\omega_{n}-\xi+X_{1})^{2}(i\omega_{n}+\xi+X_{2} )(i\omega_{n}-\xi+X_{3})}\right]\,.\] Performing the integration over $\xi$ in (B), we find \[\int\frac{\tilde{F}_{21}^{{\dagger}}}{\Delta^{\ast}}d\xi\simeq 2 \pi{i}\left\{\frac{1}{2\left(i\omega_{n}-v\tilde{q}+\frac{\tilde{Q}\tilde{q}}{ m}-h_{z}\right)}\right.\] (52) \[+\frac{h^{2}}{8\left(i\omega_{n}-v\tilde{q}+\frac{\tilde{Q}\tilde {q}}{m}-h_{z}\right)^{2}\left(i\omega_{n}+v\tilde{Q}-v\tilde{q}\right)}\] \[+\left.\frac{h^{2}}{8\left(i\omega_{n}-v\tilde{q}+\frac{\tilde{Q} \tilde{q}}{m}-h_{z}\right)^{2}\left(i\omega_{n}-v\tilde{Q}-v\tilde{q}\right)} \right\}.\] If one performs the Taylor expansion of (52) to the second power of $\tilde{q}$ in the limit $h\ll{T_{c0}}$, the equation for the critical temperature becomes \[\ln\left(\frac{T_{c}}{T_{c0}}\right)=2\pi{T_{c}}\sum_{\omega_{n} \geq 0}\left\{\frac{\omega_{n}}{\omega_{n}^{2}+h_{z}^{2}}-\frac{1}{\omega_{n}}\right.\] (53) \[-\frac{\omega_{n}^{3}h^{2}}{\left(\omega_{n}^{2}+h_{z}^{2}\right) ^{2}\left(\omega_{n}^{2}+v^{2}\tilde{Q}^{2}\right)}-\frac{4\omega_{n}^{3} \tilde{Q}h^{2}h_{z}\tilde{q}}{m\left(\omega_{n}^{2}+h_{z}^{2}\right)^{3}\left( \omega_{n}^{2}+v^{2}\tilde{Q}^{2}\right)}\] Using the definition $\Delta{T_{c}}=T_{c}-T_{c0}$ and the relation $\ln\left(\frac{T_{c}}{T_{c0}}\right)\approx\frac{\Delta{T_{c}}}{T_{c0}}$, in the limit $h_{z}\ll{T_{c0}}$ we have \[\frac{\Delta{T_{c}}}{T_{c0}} = 2\pi{T_{c}}\sum_{\omega_{n}\geq 0}\left[-\frac{h_{z}^{2}}{\omega _{n}^{3}}-\frac{h^{2}}{\left(\omega_{n}^{2}+v^{2}\tilde{Q}^{2}\right)\omega_{n }}\right.\] \[\left.-\frac{4\tilde{Q}h^{2}h_{z}\tilde{q}}{m\left(\omega_{n}^{2} +v^{2}\tilde{Q}^{2}\right)\omega_{n}^{3}}-\frac{v^{2}\tilde{q}^{2}}{\omega_{n} ^{3}}\right].\] Finally, by the opposite substitutions ${\tilde{Q}}\rightarrow\frac{Q}{2}$ and ${\tilde{q}}\rightarrow\frac{q}{2}$ we obtain \[\frac{\Delta{T_{c}}}{T_{c0}} = 2\pi{T_{c}}\sum_{\omega_{n}\geq 0}\left[-\frac{h_{z}^{2}}{\omega _{n}^{3}}-\frac{4h^{2}}{\left(4\omega_{n}^{2}+v^{2}Q^{2}\right)\omega_{n}}\right.\] \[\left.-\frac{4Qh^{2}h_{z}q}{m\left(4\omega_{n}^{2}+v^{2}Q^{2} \right)\omega_{n}^{3}}-\frac{v^{2}q^{2}}{4\omega_{n}^{3}}\right].\] ## Appendix C From (45)–(48) we get the Green’s function $\hat{G}_{11}(p,p^{\prime})$ for the uniform superconductor ($q$=0) with a helical magnetic order \[\hat{G}_{11}(p,p^{\prime})=\delta(p-p^{\prime})\] (56) \[\times\left[\frac{(i\omega-\xi_{p+Q}+h_{z})(i\omega+\xi_{p+Q}+h_{ z})(i\omega+\xi_{p}-h_{z})}{D_{1}(\omega)}\right.\] \[\left.-\frac{(i\omega-\xi_{p+Q}+h_{z})h^{2}+(i\omega+\xi_{p}-h_{z })\left\|\Delta\right\|^{2}}{D_{1}(\omega)}\right],\] where $D_{1}(\omega)=$ \[\left[(i\omega-\xi_{p}-h_{z})(i\omega+\xi_{p}-h_{z})+h^{2}-\left\| \Delta\right\|^{2}\right]\] (57) \[\times\left[(i\omega-\xi_{p+Q}+h_{z})(i\omega+\xi_{p+Q}+h_{z})+h^ {2}-\left\|\Delta\right\|^{2}\right]\] \[-(2i\omega-\xi_{p}+\xi_{p+Q})(2i\omega-\xi_{p+Q}+\xi_{p})h^{2},\] $\xi_{p}=\xi(p)=p^{2}/2m-E_{F}$, and we use $\omega$ instead of $\omega_{n}$ for short. The solutions for the Green function $\hat{G}_{22}$ are described by the same expressions (56) and (57) by replacing $Q\rightarrow-Q$ and $h_{z}\rightarrow-h_{z}$. Taking into account the symmetry relation between the Green functions \[\hat{G}_{11}(-p,-h_{z})=\hat{G}_{22}(p,h_{z})\,,\] the supercurrent in a magnetic superconductor with spiral magnetic order, \[J=\left.\frac{ie}{m}(\nabla_{r^{\prime}}-\nabla_{r})\left[\hat{G}_{11}(r,r^{ \prime})+\hat{G}_{22}(r,r^{\prime})\right]\right\|_{r^{\prime}\rightarrow{r}},\] (58) can be written via the Green function $\hat{G}_{11}$ (56) as follows: \[J=\frac{2e}{m}\iint\,dp\,d\omega\left[p\,\hat{G}_{11}(p,h_{z})-p\,\hat{G}_{11} (p,-h_{z})\right].\] (59) Although it is possible to carry out these calculations for arbitrary $h_{z}$, we restrict our consideration to only terms linear on $h_{z}$ in $\hat{G}_{11}$. In this case the expression (56) can be expanded into the following form: \[\hat{G}_{11}(p,p^{\prime})=\hat{G}_{11}^{(0)}(p,p^{\prime})+h_{z}\hat{G}_{11}^ {(1)}(p,p^{\prime})+\Delta\hat{G}_{11}(p,p^{\prime})\] (60) where \[\hat{G}_{11}^{(0)}(p,p^{\prime})=\delta(p-p^{\prime})\frac{-\xi_{p}(\omega^{2} +\xi_{p+Q}^{2}+\left\|\Delta\right\|^{2})-\xi_{p+Q}h^{2}}{\left[\,\omega^{2}+E_{ 1}^{2}\,\right]\left[\,\omega^{2}+E_{2}^{2}\,\right]}\] (61) and \[\hat{G}_{11}^{(1)}(p,p^{\prime})=\delta(p-p^{\prime})\left[\frac{ \xi_{p+Q}^{2}-\omega^{2}-h^{2}+\left\|\Delta\right\|^{2}}{\left[\,\omega^{2}+E_{ 1}^{2}\,\right]\left[\,\omega^{2}+E_{2}^{2}\,\right]}\right.\] (62) \[\left.-2\omega^{2}\frac{(\xi_{p+Q}^{2}-\xi_{p}^{2})(\omega^{2}+ \xi_{p+Q}^{2}+\left\|\Delta\right\|^{2}+h^{2})}{\left[\,\omega^{2}+E_{1}^{2}\, \right]^{2}\left[\,\omega^{2}+E_{2}^{2}\right]^{2}}\,\right]\,.\] The last item $\Delta\hat{G}_{11}(p,p^{\prime})$ in (60) includes terms that are odd in frequency $\omega$, which does not contribute to the integral $\int\,d\omega...$, and/or terms containing a higher power of $h_{z}$. The significant components $\hat{G}_{11}^{(0)}(p,p^{\prime})$ and $\hat{G}_{11}^{(1)}(p,p^{\prime})$ are described by the energy spectra \[E_{1,2}^{2} = \zeta^{2}+\eta^{2}+\left\|\Delta\right\|^{2}+h^{2}\] \[\pm 2\sqrt{\zeta^{2}\left(\eta^{2}+h^{2}\right)+\left\|\Delta \right\|^{2}h^{2}}\,,\] where $\zeta=\left(\xi_{p}+\xi_{p+Q}\right)/2$ and $\eta=\left(\xi_{p}-\xi_{p+Q}\right)/2$. <figure><img src="content_image/1812.09136/x10.png"><figcaption>Figure A1: (a) Andreev bound-state energies vs the superconducting phasedifference ϕ and (b) current-phase relation for the conical ferromagneticjunction when h/EF takes three different values. The results plotted are forkFL=60, EF=1000Δ, hz/EF=0.084, and Q/kF=π/2.</figcaption></figure> <figure><img src="content_image/1812.09136/x11.png"><figcaption>Figure A2: Current-phase relation for the conical ferromagnetic junction whenQ takes three different values. We set the parameters EF=1000Δ, h/EF=0.55,hz/EF=0.084, and kFL=60.</figcaption></figure> Substituting expansion (60) into Eq. (59), we get \[J=\frac{4eh_{z}}{m}\iint\,dp\,d\omega\left[\left(p-Q/2\right)\hat{G}_{11}^{(1) }\left(p-Q/2,p^{\prime}\right)\right].\] (64) Performing long but straightforward calculations, we find the following analytical expression for supercurrent (64), \[J = \frac{8e\pi\tilde{Q}\tilde{h}_{z}}{m}\left\{\int{d}\tilde{p}\frac {\tilde{p}^{2}\tilde{\xi}_{Q}(\tilde{p})}{e_{1}^{2}e_{2}+e_{1}e_{2}^{2}}\right.\] \[+ \frac{1}{4}\int\,d\tilde{p}\,\frac{2\tilde{h}^{2}-\left(e_{1}-e_{ 2}\right)^{2}/2}{e_{1}^{2}e_{2}+e_{1}e_{2}^{2}}\] \[+\] where $e_{1,2}=E_{1,2}/E_{F}$ and $\tilde{\xi}_{Q}(\tilde{p})=\tilde{p}^{2}+\tilde{Q}^{2}/4-1$. Here we use the dimensionless variables $\tilde{h}$, $\tilde{h}_{z}$, and $\tilde{\Delta}$ in the units of Fermi energy $E_{F}=p_{F}^{2}/2m$ as well as $\tilde{Q}$, $\tilde{p}$ in the units of Fermi momentum $p_{F}$. ## Appendix D In Fig. A1 we plot the Andreev spectrum and the current-phase relation for increasing exchange fields $h/E_{F}$ when the energy band structure changes from ferromagnet to half-metal. We note that with an increase of $h/E_{F}$, the asymmetry of the Andreev spectrum structure is enhanced and the phase shift $\phi_{0}$ increases accordingly. In Fig. A2 it is shown how the transition from ferromagnet to half-metal with the increase of the helical modulation vector $Q$ changes the spontaneous current.
1508.02514	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31825, "num_imgs": 1, "llama3_tokens_count": 11813 }	[ "content_image/1508.02514/x1.png" ]	# Mixed type surfaces with bounded mean curvature in $3$-dimensional space-times A. Honda M. Koiso M. Kokubu M. Umehara K. Yamada National Institute of Technology, Miyakonojo College, 473-1, Yoshio-cho, Miyakonojo, Miyazaki 885-8567, Japan atsufumi@cc.miyakonojo-nct.ac.jp Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan koiso@math.kyushu-u.ac.jp Department of Mathematics, School of Engineering, Tokyo Denki University, Tokyo 120-8551, Japan kokubu@cck.dendai.ac.jp Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-34, O-okayama Meguro-ku, Tokyo 152-8552, Japan umehara@is.titech.ac.jp Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8551, Japan kotaro@math.titech.ac.jp February 27, 2024 ###### Abstract. In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvature and investigate their properties. Key words and phrases: causality, type change, mean curvature, Lorentzian manifolds 2010 Mathematics Subject Classification: Primary:53A35; Secondary 57R42, 35M10. ## 1. Introduction We say that a connected surface immersed in a Lorentzian 3-manifold $(M^{3},g)$ is of _mixed type_ if both the space-like and time-like point sets are non-empty. In general, the mean curvature of such surfaces diverges: for example, the graph of a smooth function $t=f(x,y)$ in the Lorentz-Minkowski space-time $({\boldsymbol{R}}^{3}_{1};t,x,y)$ gives a space-like (resp. time-like) surface if $B>0$ (resp. $B<0$), where (1.1) \[B:=1-f_{x}^{2}-f_{y}^{2}.\] In this situation, the unit normal vector is given by (1.2) \[\nu=\frac{1}{\sqrt{\|B\|}}(1,f_{x},f_{y}),\] and the mean curvature function is computed as (1.3) \[H=\frac{\left(f_{x}^{2}-1\right)f_{yy}-2f_{x}f_{y}f_{xy}+\left(f_{y}^{2}-1 \right)f_{xx}}{2\|B(x,y)\|^{3/2}},\] which is unbounded around the set $\{B(x,y)=0\}$, in general. On the other hand, several zero mean curvature surfaces of mixed type in ${\boldsymbol{R}}^{3}_{1}$ were found in [11], [7], [10], [12], [9], [5], [2] and [3]. Moreover, such examples can be found in other space-times: in fact, a zero mean curvature surface of mixed type in the de Sitter 3-space (resp. in the anti-de Sitter 3-space) is given in this paper (cf. Example 2.6 and Example 2.7). It is known that zero mean curvature surfaces in ${\boldsymbol{R}}^{3}_{1}$ change types across their fold singularities, except for the special case as in [2]. On the other hand, in [8], it was shown that space-like non-zero constant mean curvature surfaces do not admit fold singularities, which suggests that space-like non-zero constant mean curvature surfaces never change types. More precisely, the following questions naturally arise: * _Is there a mixed type surface with non-zero constant mean curvature?_ * _Is there a mixed type surface whose mean curvature vector field is smooth and does not vanish along the curve of type change?_ In this paper, we show that the answer to Question (a) is negative. This is a consequence of the following assertion: Theorem 1.1.: _Let $U$ be a connected domain in ${\boldsymbol{R}}^{2}$, and $f:U\to(M^{3},g)$ a real analytic immersion into an oriented real analytic Lorentzian manifold $(M^{3},g)$. We denote by $U_{+}$$($resp. $U_{-})$ the set of points where $f$ is space-like $($resp. time-like$)$. Suppose that $U_{+},U_{-}$ are both non-empty, and the mean curvature function $H$ on $U_{+}\cup\,U_{-}$ is bounded. Then for each $p\in\overline{U_{+}}\cap\overline{U_{-}}$, there exists a sequence $\{p_{n}\}_{n=1,2,3,..}$ in $U_{+}$$($resp. $U_{-})$ converging to $p$ so that $\lim_{n\to\infty}H(p_{n})=0$, where $\overline{U_{+}},\overline{U_{-}}$ are the closures of $U_{+},U_{-}$ in $U$._ There exist space-like and time-like constant mean curvature immersions in ${\boldsymbol{R}}^{3}_{1}$ which are not of mixed type although their induced metrics degenerate along certain smooth curves (cf. Examples 2.3 and 2.4 in Section 2). Also, there are similar such examples of space-like constant mean curvature one surfaces in the de Sitter 3-space $S^{3}_{1}$ with singularities which are not of mixed type ([1]). The existence of such examples implies that we cannot drop the assumption that both $U_{+},U_{-}$ are non-empty. The proof of Theorem 1.1 is given in Section 2. On the other hand, we show that the answer to Question (b) is affirmative. In fact, we show in Section 3 that the mean curvature vector fields of real analytic surfaces of mixed type with bounded mean curvature functions can be analytically extended across the sets of type change under a suitable genericity assumption (cf. Proposition 3.6). Moreover, we show the following: Theorem 1.2.: _There exists a real analytic function $g(x,y)$ on ${\boldsymbol{R}}^{2}$ whose graph realized in ${\boldsymbol{R}}^{3}_{1}$ satisfies the following properties:_ 1. _the set_ $\Sigma_{g}$ _of non-degenerate points of type change of the graph of_ $g$ _is non-empty,_ _and the induced metric of the graph of_ $g$ _is non-degenerate on_ ${\boldsymbol{R}}^{2}\setminus\Sigma_{g}$_,_ 2. _the mean curvature function of the graph of_ $g$ _is bounded on_ ${\boldsymbol{R}}^{2}\setminus\Sigma_{g}$_._ 3. _the mean curvature vector field can be extended to_ $\Sigma_{g}$ _real analytically, and does not vanish at each point of_ $\Sigma_{g}$_._ This suggests that surfaces with smooth mean curvature vector fields form an important sub-class of the set of mixed type surfaces. ## 2. Behavior of mean curvature along curves of type change Let $(M^{3},g)$ be an oriented real analytic Lorentzian $3$-manifold. Then, the vector product $\boldsymbol{v}\times_{g}\boldsymbol{w}$ is defined for linearly independent tangent vectors $\boldsymbol{v},\boldsymbol{w}$ at $p\in M^{3}$, satisfying the following three properties: 1. $\boldsymbol{v}\times_{g}\boldsymbol{w}$ is orthogonal to $\boldsymbol{v}$ and $\boldsymbol{w}$, 2. $\{\boldsymbol{v},\boldsymbol{w},\boldsymbol{v}\times_{g}\boldsymbol{w}\}$ is a basis of the tangent space $T_{p}M$ which is compatible with the orientation of $M^{3}$, 3. it holds that \[g_{p}(\boldsymbol{v}\times_{g}\boldsymbol{w},\boldsymbol{v}\times_{g} \boldsymbol{w})=-g_{p}(\boldsymbol{v},\boldsymbol{v})g_{p}(\boldsymbol{w}, \boldsymbol{w})+g_{p}(\boldsymbol{v},\boldsymbol{w})^{2}.\] For each tangent vector $\boldsymbol{v}\in T_{p}M^{3}$ ($p\in M^{3}$), we set \[\|\boldsymbol{v}\|:=\sqrt{\|g_{p}(\boldsymbol{v},\boldsymbol{v})\|}.\] We fix a domain $U$ in ${\boldsymbol{R}}^{2}$. Let $f:U\to M^{3}$ be a real analytic immersion. Set $f_{u}:=df(\partial_{u})$, $f_{v}:=df(\partial_{v})$, where $\partial_{u}:=\partial/\partial u$, $\partial_{v}:=\partial/\partial v$. Using three real analytic functions \[g_{11}:=g(f_{u},f_{u}),\quad g_{12}=g_{21}=g(f_{u},f_{v}),\quad g_{22}:=g(f_{v },f_{v})\] on $U$, we define a function $\beta:U\to{\boldsymbol{R}}$ by (2.1) \[\beta:=g_{11}g_{22}-g_{12}^{2}.\] Then \[U_{+}:=\{p\in U\,;\,\beta(p)>0\},\qquad U_{-}:=\{p\in U\,;\,\beta(p)<0\}\] give the set of space-like points and the set of time-like points, respectively. The unit normal vector field (2.2) \[\nu=\frac{f_{u}\times_{g}f_{v}}{\|f_{u}\times_{g}f_{v}\|}\] of $f$ is well-defined on $U_{+}\cup U_{-}$. Using this, we set \[h_{11}:=g(f_{uu},\nu),\quad h_{12}=h_{21}=g(f_{uv},\nu),\quad h_{22}:=g(f_{vv} ,\nu),\] where \[f_{uu}=\nabla_{\partial_{u}}f_{u},\quad f_{uv}=\nabla_{\partial_{v}}f_{u}= \nabla_{\partial_{u}}f_{v},\quad f_{vv}=\nabla_{\partial_{v}}f_{v},\] and $\nabla$ is the Levi-Civita connection of the Lorentzian manifold $(M^{3},g)$. Each $h_{ij}$ ($i,j=1,2$) is a function defined on $U_{+}\cup U_{-}$. The mean curvature function $H$ is also defined on $U_{+}\cup U_{-}$, and is given by (2.3) \[H:=\frac{g_{11}h_{22}-2g_{12}h_{12}+g_{22}h_{11}}{2\|\beta\|}=\frac{\alpha}{2\| \beta\|^{3/2}},\] where (2.4) \[\alpha:=\sqrt{\|\beta\|}(g_{11}h_{22}-2g_{12}h_{12}+g_{22}h_{11}).\] Then the following assertion holds: Lemma 2.1.: _The function $\alpha:U_{+}\cup U_{-}\to{\boldsymbol{R}}$ can be analytically extended to $U$._ Proof.: We set $\tilde{\nu}:=f_{u}\times_{g}f_{v}$. Then \[\beta=-g(f_{u}\times_{g}f_{v},f_{u}\times_{g}f_{v})\] and $\nu=\tilde{\nu}/\sqrt{\|\beta\|}$ holds (cf. (2.2)). Therefore, we have that \[\alpha =\sqrt{\|\beta\|}\biggl{(}g(f_{vv},\nu)g_{11}-2g(f_{uv},\nu)g_{12}+ g(f_{uu},\nu)g_{22}\biggr{)}\] \[=g(f_{vv},\tilde{\nu})g_{11}-2g(f_{uv},\tilde{\nu})g_{12}+g(f_{uu },\tilde{\nu})g_{22},\] proving the assertion. ∎ Using the lemma, we now give the proof of Theorem 1.1: Proof of Theorem 1.1.: We may assume that the mean curvature function $H$ is not identically zero. Let $(x^{1},x^{2})$ be the coordinates of $U$. We fix a point $p\in\overline{U_{+}}\cap\overline{U_{-}}$. Let $\varepsilon>0$ be an arbitrary positive number and $V$ a neighborhood of $p$. It is sufficient to show that there exist points $q_{+}\in V_{+}$ and $q_{-}\in V_{-}$ such that $\|H(q_{+})\|$ and $\|H(q_{-})\|$ are both less than $\varepsilon$. We may assume that $V$ is connected. If $\beta\geq 0$ or $\beta\leq 0$ on $V$, this contradicts the fact that $p\in\overline{U_{+}}\cap\overline{U_{-}}$. So, we can take two points $q_{0},q_{1}\in V$ such that $\beta(q_{0})>0$ and $\beta(q_{1})<0$. We then take a smooth curve $\gamma(s)$ ($s\in[0,2\pi]$) on $V$ such that $\gamma(0)=q_{0}$ and $\gamma(2\pi)=q_{1}$. Since the image of $\gamma$ lies in $V$, we can write $\gamma=(\gamma^{1},\gamma^{2})$ and each $\gamma^{i}$ ($i=1,2$) has the following Fourier series expansion: \[\gamma^{i}(s)=u^{i}_{0}+\sum_{k=1}^{\infty}\left(u_{k}^{i}\cos ks+v_{k}^{i} \sin ks\right)\qquad(i=1,2).\] We then set \[\gamma^{i}_{N}(s)=u^{i}_{0}+\sum_{k=1}^{N}\left(u_{k}^{i}\cos ks+v_{k}^{i}\sin ks \right)\qquad(i=1,2),\] where $N$ is a sufficiently large positive integer. Then the real analytic curve defined by $\gamma_{N}(s):=(\gamma^{1}_{N}(s),\gamma^{2}_{N}(s))$ satisfies (2.5) \[\beta(\gamma_{N}(0))>0,\qquad\beta(\gamma_{N}(2\pi))<0.\] Since \[\hat{\beta}(s):=\beta(\gamma_{N}(s))\qquad(0\leq s\leq 2\pi)\] is a real analytic function defined on $[0,2\pi]$, the set of zeros of the function $\hat{\beta}(s)$ consists of a finite set of points \[0<s_{1}<\cdots<s_{n}<2\pi.\] By (2.5), we can choose the number $j$ such that the sign of $\hat{\beta}(s)$ changes from positive to negative at $s=s_{j}$. Then there exists a positive integer $m$ such that \[\lim_{s\to s_{j}}\frac{\hat{\beta}(s)}{(s-s_{j})^{m}}=b\,(\neq 0),\] where $b$ is a non-zero real number. Since $\hat{\beta}(s)$ changes sign at $s=s_{j}$, the integer $m$ is odd. By Lemma 2.1, we may regard $\alpha$ as a real analytic function on $U$. So we set \[\hat{\alpha}(s):=\alpha(\gamma_{N}(s)).\] By (2.3), we have that \[H(\gamma_{N}(s)):=\frac{\hat{\alpha}(s)}{2\|\hat{\beta}(s)\|^{3/2}}\] for $s\neq s_{1},...,s_{n}$. Since $H$ is bounded, we have $\hat{\alpha}(s_{j})=0.$ Since $\hat{\alpha}(s)$ is a real analytic function, there exists a positive integer $\ell$ such that \[\lim_{s\to s_{j}}\frac{\hat{\alpha}(s)}{(s-s_{j})^{\ell}}=a\,(\neq 0),\] where $a$ is a non-zero real number. Then it holds that \[\lim_{s\to s_{j}}\|s-s_{j}\|^{(3m/2)-\ell}\|H(\gamma_{N}(s))\|=\frac{\|a\|}{\|b\|^{3/2 }}\,(\neq 0).\] Since $H$ is bounded, we have $2\ell\geq 3m$. Moreover, since $m$ is odd, we have $2\ell>3m.$ Then we have $\lim_{s\to s_{j}}\|H(\gamma_{N}(s))\|=0.$ In particular, if we set \[q_{+}:=\gamma_{N}(s_{j}-\delta),\qquad q_{-}:=\gamma_{N}(s_{j}+\delta),\] then $\|H(q_{+})\|$ and $\|H(q_{-})\|$ are less than $\varepsilon$ for sufficiently small $\delta>0$. So we get the assertion. ∎ As a consequence, we get the following corollary: Corollary 2.2.: _Under the assumption of Theorem 1.1, the function $\alpha:U_{+}\cup U_{-}\to{\boldsymbol{R}}$ can be analytically extended to $U$ and vanishes on $\overline{U_{+}}\cap\overline{U_{-}}$._ Proof.: By Lemma 2.1, the function $\alpha$ can be analytically extended to $U$. Suppose that $\alpha(p)\neq 0$ for $p\in\overline{U_{+}}\cap\overline{U_{-}}$. Then the mean curvature function cannot be bounded, since $\beta(p)=0$. ∎ We give here several examples: Example 2.3 (A space-like CMC surface with parabolic symmetry).: Consider the map $f_{P}:{\boldsymbol{R}}^{2}\to{\boldsymbol{R}}^{3}_{1}$ such that \[f_{P}(u,v):=\left(-\eta(v)+u^{2}v+v,-\eta(v)+u^{2}v-v,2uv\right),\] where \[\eta(v):=\frac{1}{2}\left(\arctan(v)-\frac{v}{v^{2}+1}\right),\qquad\left\| \arctan(v)\right\|<\frac{\pi}{2}.\] This surface has singularities on the $u$-axis. Moreover, the inverse image $f^{-1}_{P}(\{\boldsymbol{0}\})$ coincides with the $u$-axis, where $\boldsymbol{0}:=(0,0,0)$. One can easily check that $f_{P}$ gives a space-like immersion of constant mean curvature $1/2$ on ${\boldsymbol{R}}^{2}\setminus\{v=0\}$. Moreover, the image of $f_{P}$ is contained in the set (cf. Figure 1, left) \[\mathcal{P}:=\left\{(t,x,y)\in{\boldsymbol{R}}^{3}_{1}\,;\,-t^{2}+x^{2}+y^{2}= 2(t-x)\eta\left(\frac{t-x}{2}\right)\right\}.\] The light-like line \[L:=\{(c,c,0)\,;\,c\in{\boldsymbol{R}}\}=\left\{\lim_{u\to\infty}f_{P}(u,\frac{ c}{u^{2}})\,;\,c\in{\boldsymbol{R}}\right\}\] is contained in $\mathcal{P}$, and the image of $f$ coincides with $\mathcal{P}\setminus L$. The set $\mathcal{P}$ itself is a surface in ${\boldsymbol{R}}^{3}_{1}$ without self-intersections which has a cone-like singular point at the origin $\boldsymbol{0}$, and has bounded mean curvature function on $\mathcal{P}\setminus\{\boldsymbol{0}\}$. Moreover, the induced metric on $\mathcal{P}$ degenerates only on the line $L$. This implies that we cannot drop the assumption that $U_{+},U_{-}$ are non-empty in the statement of Theorem 1.1. This example is an analogue of the maximal surface called _the Enneper surface of the 2nd kind_ or _parabolic catenoid_ (cf. [11], [2]). <figure><img src="content_image/1508.02514/x1.png"><figcaption>Figure 1. The figure of P (left) and H (right).</figcaption></figure> Example 2.4 (A space-like CMC surface with hyperbolic symmetry).: We next consider the map defined by \[f_{H}(u,v):=(v\cosh u,v\sinh u,\varphi(v))\qquad((u,v)\in{\boldsymbol{R}} \times(-1,1)),\] where \[\varphi(v):=\log\left(\frac{1+v}{1-v}\right)-v.\] Like the case of $f_{P}$, this surface has singularities on the $u$-axis and $f^{-1}_{H}(\{\boldsymbol{0}\})$ coincides with the $u$-axis. One can easily check that $f_{H}$ gives a space-like immersion of constant mean curvature $1/2$ on ${\boldsymbol{R}}^{2}\setminus\{v=0\}$. Moreover, the image of $f_{H}$ is contained in the set (cf. Figure 1, right) \[\mathcal{H}:=\left\{(t,x,y)\in{\boldsymbol{R}}^{3}_{1}\,;\,y=\varphi(\pm\sqrt{ t^{2}-x^{2}})\right\}=\left\{(t,x,y)\in{\boldsymbol{R}}^{3}_{1}\,;\,t^{2}=x^{2 }+\psi(y)^{2}\right\},\] where $\psi:{\boldsymbol{R}}\to(-1,1)$ is the inverse function of $\varphi:(-1,1)\to{\boldsymbol{R}}$. Two light-like lines \[L_{\pm}:=\{(c,\pm c,0)\,;\,c\in{\boldsymbol{R}}\}\] are contained in $\mathcal{H}$ and \[\mathcal{H}=L_{+}\cup L_{-}\cup(\mbox{Image of }f_{H})\cup(\mbox{Image of }f^{ \prime}_{H}),\] where \[f^{\prime}_{H}(u,v):=(-v\cosh u,v\sinh u,\varphi(v))\qquad((u,v)\in{ \boldsymbol{R}}\times(-1,1)).\] Like as in the case of $\mathcal{P}$, the set $\mathcal{H}$ has no self-intersections, and has bounded mean curvature function on $\mathcal{H}\setminus\{\boldsymbol{0}\}$. The origin $\boldsymbol{0}$ is a cone-like singular point. Moreover, its induced metric degenerates along the lines $L_{\pm}$. This example is an analogue of the maximal surface called _the catenoid of the 2nd kind_ or _hyperbolic catenoid_ (cf. [11], [2]). Similar examples, that is, a family of space-like surfaces with constant mean curvature one containing light-like lines in the de Sitter 3-space $S^{3}_{1}$ have recently been found in [1]. The following is one typical mixed type surface whose mean curvature vanishes identically. Example 2.5.: Consider the function \[f_{K}(x,y):=x\tanh y.\] Then the graph of $f_{K}$ in ${\boldsymbol{R}}^{3}_{1}$ gives a zero mean curvature surface, which is space-like on the set $U_{+}:=\{(x,y)\in{\boldsymbol{R}}^{2}\,;\,x^{2}>\cosh^{2}y\}$ and time-like on the set $U_{-}:=\{(x,y)\in{\boldsymbol{R}}^{2}\,;\,x^{2}<\cosh^{2}y\}$. This example is called the _helicoid of the 2nd kind_, which was found by Kobayashi [11]. On the other hand, we can find a similar example in another space form: Example 2.6.: Consider the map $f_{Z}:{\boldsymbol{R}}\times S^{1}\to S^{3}_{1}$ given by \[f_{Z}(u,v):=\left(\sinh u\sin v,\,\cos u\cos v,\,\sin u\cos v,\,\cosh u\sin v \right),\] where \[S^{3}_{1}:=\{(t,x,y,z)\in{\boldsymbol{R}}^{4}_{1}\,;\,-t^{2}+x^{2}+y^{2}+z^{2} =1\}\] is the de Sitter 3-space, which is the space-time of constant sectional curvature $1$. Then the first fundamental form of $f_{Z}$ is given by $ds^{2}=\cos 2v\,du^{2}+dv^{2}$. In particular, $f_{Z}$ is space-like (resp. time-like) if $\cos 2v>0$ (resp. $\cos 2v<0$). Moreover, the mean curvature function of $f_{Z}$ vanishes identically. Example 2.7.: We define an immersion $f_{\rm ads}:{\boldsymbol{R}}^{2}\to H^{3}_{1}$ by \[f_{\rm ads}(u,v)=\left(\cosh u\cosh v,\,\sinh au\sinh v,\,\cosh au\sinh v,\, \sinh u\cosh v\right),\] where $a=1/\tanh\alpha$$(\alpha\neq 0)$ is a constant, and \[H^{3}_{1} =\left\{(t,x,y,z)\in{\boldsymbol{R}}^{4}_{2}\,;\,-t^{2}-x^{2}+y^{ 2}+z^{2}=-1\right\}\] is the anti-de Sitter 3-space, which is the space-time of constant sectional curvature $-1$. Then the first fundamental form of $f_{\rm ads}$ is given by \[\dfrac{\cosh 2\alpha-\cosh 2v}{2\sinh^{2}\alpha}du^{2}+dv^{2}.\] In particular, $f_{\rm ads}$ is space-like (resp. time-like) if $\cosh 2\alpha>\cosh 2v$ (resp. $\cosh 2\alpha<\cosh 2v$). ## 3. Properties of points where surfaces change type In this section, we shall investigate the properties of functions $t=f(x,y)$ whose graphs induce mixed type surfaces in ${\boldsymbol{R}}^{3}_{1}$ with bounded mean curvature. Definition 3.1 (cf. [3, Definition 2.3]).: Let $U$ be a domain in the $xy$-plane ${\boldsymbol{R}}^{2}$, and $f:U\to{\boldsymbol{R}}$ a $C^{\infty}$-function. We set \[B:=1-f_{x}^{2}-f_{y}^{2}.\] A point $p\in U$ is called a _non-degenerate point of type change_ if (3.1) \[B(p)=0,\qquad\nabla B(p)\neq 0\] hold, where $\nabla B:=(B_{x},B_{y})$. By definition, the first fundamental form of the graph of $f$ is degenerate at a non-degenerate point of type change. We set \[A:=(f_{x}^{2}-1)f_{yy}-2f_{x}f_{y}f_{xy}+(f_{y}^{2}-1)f_{xx}.\] Then the functions $A,B$ can be considered as a special case of the functions (cf. (2.1) and (2.4)) $\alpha,\beta$ by setting $(u,v)=(x,y)$. By (1.3), we have (3.2) \[H=\frac{A}{2\|B\|^{3/2}}.\] Proposition 3.2 (cf. Proposition 2.4 in [3]).: _Suppose that the mean curvature function of the graph of $f$ is bounded. Let $p\in U$ be a point satisfying $B(p)=0$. Then the following two assertions are equivalent:_ 1. _the point_ $p$ _is a non-degenerate point of type change._ 2. $p$ _is a dually regular point in the sense of_ _[_7_]__, that is,_ $p$ _is a point where_ $f_{xx}(p)f_{yy}(p)-f_{xy}(p)^{2}\neq 0$ _._ Proof.: The proof is almost parallel to that of Proposition 2.4 in [3]. It holds that (3.3) \[\nabla B={\operatorname{Hess}}(f){\begin{pmatrix}f_{x}\\ f_{y}\end{pmatrix}},\qquad{\operatorname{Hess}}(f):={\begin{pmatrix}f_{xx}&f_{ xy}\\ f_{xy}&f_{yy}\end{pmatrix}}.\] Now suppose that (2) holds. Then ${\operatorname{Hess}}(f)$ is a regular matrix at $p$. Since $B(p)=0$, $(f_{x},f_{y})\neq 0$ at $p$. Thus, (3.3) implies that $\nabla B\neq 0$ at $p$, that is, (1) holds. We next suppose on the contrary that (2) does not hold. By a suitable linear coordinate change of $(x,y)$, we may assume without loss of generality that $f_{xy}(p)=0$. Then either $f_{xx}(p)=0$ or $f_{yy}(p)=0$. By (3.2), and Theorem 1.1, we have $A(p)=0$. This with $B(p)=0$ and $f_{xy}(p)=0$ implies that \[f_{x}(p)^{2}f_{xx}(p)+f_{y}(p)^{2}f_{yy}(p)=0.\] This with $f_{xx}(p)=0$ or $f_{yy}(p)=0$ implies that \[{\operatorname{Hess}}(f){\begin{pmatrix}f_{x}\\ f_{y}\end{pmatrix}}={\begin{pmatrix}f_{x}(p)f_{xx}(p)\\ f_{y}(p)f_{yy}(p)\end{pmatrix}}={\begin{pmatrix}0\\ 0\end{pmatrix}}.\] So (1) does not hold. ∎ A regular curve $\Gamma:(a,b)\to{\boldsymbol{R}}^{3}_{1}$ is called _null_ or _isotropic_ if $\dot{\Gamma}(t):=d\Gamma(t)/dt$ is a light-like vector for each $t\in(a,b)$. Definition 3.3.: A null curve $\Gamma:(a,b)\to{\boldsymbol{R}}^{3}_{1}$ is called non-degenerate at $t=c$ if $\dot{\Gamma}(c)$ and $\ddot{\Gamma}(c)$ are linearly independent. If $\Gamma(t)$ is non-degenerate for all $t\in(a,b)$, the curve $\Gamma$ is called a _non-degenerate_ null curve. Let $p\in U$ be a non-degenerate point of type change. Then, by the implicit function theorem, there exists a regular curve $\gamma:(-\varepsilon,\varepsilon)\to U$ such that $B\circ\gamma(t)=0$ and $\gamma(0)=p$, where $\varepsilon$ is a positive number. We call this curve $\gamma$_the characteristic curve_ of type change. The following assertion is a generalization of [3, Proposition 2.5] for zero-mean curvature surfaces. Proposition 3.4.: _Suppose that the graph $t=f(x,y)$ over a domain $U$ has bounded mean curvature function. If the graph changes type along a regular curve $\gamma(t)$$(\|t\|<\varepsilon)$ such that $f\circ\gamma(t)$ is a non-degenerate null curve in ${\boldsymbol{R}}^{3}_{1}$, then $\gamma(t)$ consists of non-degenerate points of type change._ Proof.: The proof is completely parallel to that of [3, Proposition 2.5]. ∎ The converse assertion is given as follows, which is a generalization of [3, Proposition 2.6] for zero-mean curvature surfaces. Proposition 3.5.: _Suppose that the graph $t=f(x,y)$ over a domain $U$ has bounded mean curvature function. Let $p\in U$ be a non-degenerate point of type change and $\gamma(t)$$(\|t\|<\varepsilon)$ the characteristic curve of type change such that $\gamma(0)=p$. Then $f\circ\gamma(t)$ is a non-degenerate null curve._ Proof.: Using the fact that $A(\gamma(t))=0$ holds, the proof of this assertion is completely parallel to that of [3, Proposition 2.6]. ∎ Moreover, the following assertion holds: Proposition 3.6.: _Let $t=f(x,y)$ be a real analytic function over the domain $U$ which gives a graph with bounded mean curvature function. Suppose that the zeros of $B(x,y)$ are all non-degenerate points of type change. Then, the mean curvature vector $H\nu$ can be analytically extended to all of $U$._ Proof.: Let $p\in U$ be a non-degenerate point of type change. Then we can take a real analytic local coordinate system $(u,v)$ centered at $p$ such that the $u$-axis is the characteristic curve of type change. By the condition $\nabla B(u,0)\neq(0,0)$ (cf. (3.1)), there exists a real analytic function $b(u,v)$ defined near the $u$-axis such that $B(u,v)=vb(u,v)$ and $b(u,0)\neq 0$. On the other hand, Theorem 1.1 yields that there exists a real analytic function $a(u,v)$ defined near the $u$-axis such that (3.4) \[A(u,v)=v^{2}a(u,v).\] By (3.2), we have \[H(u,v)=\frac{\sqrt{\|v\|}a(u,v)}{2\|b(u,v)\|^{3/2}}.\] By (1.2), we have that (3.5) \[H\nu=\frac{\sqrt{\|v\|}a(u,v)}{2\|b(u,v)\|^{3/2}}\frac{1}{\sqrt{\|v\|\|b(u,v)\|}}(1,f_ {x},f_{y})=\frac{a(u,v)}{2b(u,v)^{2}}(1,f_{x},f_{y}),\] proving the assertion. ∎ Finally, we prove Theorem 1.2 in the introduction: Proof of Theorem 1.2.: Let $f:{\boldsymbol{R}}^{2}\to{\boldsymbol{R}}$ be a real analytic function whose graph gives a zero-mean curvature surface, with function $B:=1-f_{x}^{2}-f_{y}^{2}$ satisfying $\nabla B\neq(0,0)$ if $B=0$. Take a real analytic function $\psi:{\boldsymbol{R}}\to{\boldsymbol{R}}$ such that (3.6) \[\psi(0)=\psi^{\prime}(0)=\psi^{\prime\prime}(0)=0.\] We then set \[g(x,y):=f(x,y)+\psi(B(x,y)),\] and \[\tilde{B}:=1-g_{x}^{2}-g_{y}^{2}.\] Since \[g_{x}=f_{x}+\psi^{\prime}(B)B_{x},\quad g_{y}=f_{y}+\psi^{\prime}(B)B_{y},\] we have that (3.7) \[\tilde{B}=B-2\psi^{\prime}(B)(f_{x}B_{x}+f_{y}B_{y})-\psi^{\prime}(B)^{2}(B_{x }^{2}+B_{y}^{2}).\] Here, the relation $C_{1}\equiv C_{2}\mod B$ for two real analytic functions $C_{i}(x,y)$ ($i=1,2$) means that $(C_{1}-C_{2})/B$ is a real analytic function on ${\boldsymbol{R}}^{2}$. Since $\psi^{\prime}(B)\equiv 0\mod B$, $\tilde{B}$ can be divided by $B$. Thus, to show the mean curvature vector field can be smoothly extended across the set $B=0$, it is sufficient to show that \[\tilde{A}:=(g_{y}^{2}-1)g_{xx}-2g_{x}g_{y}g_{xy}+(g_{x}^{2}-1)g_{yy}\] can be divided by $B^{2}$. Since \[g_{xx} =f_{xx}+\psi^{\prime\prime}(B)B_{x}^{2}+\psi^{\prime}(B)B_{xx},\] \[g_{xy} =f_{xy}+\psi^{\prime\prime}(B)B_{x}B_{y}+\psi^{\prime}(B)B_{xy},\] \[g_{yy} =f_{yy}+\psi^{\prime\prime}(B)B_{y}^{2}+\psi^{\prime}(B)B_{yy},\] the fact that $A=0$ yields (3.8) \[\tilde{A}\equiv\psi^{\prime\prime}(B)\Gamma+\psi^{\prime}(B)\Delta\mod B^{3},\] where \[\Gamma :=(f_{y}^{2}-1)B_{x}^{2}-2f_{x}f_{y}B_{x}B_{y}+(f_{x}^{2}-1)B_{y} ^{2},\] \[\Delta :=2(B_{x}f_{x}f_{yy}-B_{x}f_{xy}f_{y}-B_{y}f_{x}f_{xy}+B_{y}f_{xx }f_{y})\] \[\phantom{aaaaaaaaaaaaaaaaaa}+B_{xx}\left(f_{y}^{2}-1\right)-2B_{ xy}f_{x}f_{y}+B_{yy}\left(f_{x}^{2}-1\right).\] Since \[\Gamma =(-B-f_{x}^{2})B_{x}^{2}-2f_{x}f_{y}B_{x}B_{y}+(-B-f_{y}^{2})B_{y }^{2}\] \[=-B(B_{x}^{2}+B_{y}^{2})-(f_{x}B_{x}+f_{y}B_{y})^{2}\] and \[f_{x}B_{x}+f_{y}B_{y} =-2(f_{x}(f_{x}f_{xx}+f_{y}f_{xy})+f_{y}(f_{x}f_{xy}+f_{y}f_{yy}))\] (3.9) \[=2A+2B(f_{xx}+f_{yy})=2B(f_{xx}+f_{yy}),\] we have that (3.10) \[\Gamma\equiv-B(B_{x}^{2}+B_{y}^{2})\mod B^{2}.\] Since \[\psi^{\prime}(B)\equiv 0\mod B^{2},\qquad\psi^{\prime\prime}(B)\equiv 0\mod B,\] (3.8) and (3.10) yield that $\tilde{A}$ can be divided by $\tilde{B}^{2}$. To give an explicit example, we consider the function $f_{K}(x,y):=x\tanh y$ given in Example 2.5. Then, we have \[B(x,y)=(\cosh^{2}y-x^{2})\text{sech}^{4}y\] and $x=\pm\cosh y$ give the characteristic curves of type change. We consider the new function (3.11) \[g(x,y):=x\tanh y+c\tanh^{3}(B(x,y))\qquad(0<c\leq 1),\] where $c$ is a constant. Then the mean curvature vector field is real analytic along the set of type change $\Sigma_{f}:=\{(\pm\cosh y,y)\,;\,y\in{\boldsymbol{R}}\}$. By (3.10), \[\Gamma\equiv-4B\text{sech}^{4}y\mod B^{2}\] holds. By a straightforward calculation, \[\Delta=2(B+1){\operatorname{sech}}^{4}y\] holds. Since $\psi(B)=c\tanh^{3}(B)$, we have \[\psi^{\prime}(B)\equiv 3cB^{2},\quad\psi^{\prime\prime}(B)\equiv 6cB\mod B^{3}.\] Thus, (3.8) yields that (3.12) \[\left.\frac{\tilde{A}}{\tilde{B}^{2}}\right\|_{(x,y)=(\pm\cosh y,y)}=\left. \frac{\tilde{A}}{B^{2}}\right\|_{(x,y)=(\pm\cosh y,y)}=\frac{-18c}{\cosh^{4}y} \qquad(y\in{\boldsymbol{R}}),\] which never vanishes on the set $\Sigma_{f}$. To complete the proof, it is sufficient to show that $\tilde{B}/B$ has no zeros if $c$ is sufficiently small. We shall now compute $\tilde{B}/B$ using (3.7). We set \[\varphi(t):=\frac{\tanh t}{t}\] which is a real analytic bounded function. We set \[U:=x\,{\operatorname{sech}}^{2}y,\quad V:={\operatorname{sech}}\,y,\quad S:={ \operatorname{sech}}(V^{2}-U^{2}).\] Here $U$ is unbounded, but $V,S$ are bounded on ${\boldsymbol{R}}^{2}$. By a straight-forward calculation, one can get that \[\frac{\tilde{B}}{B}=1-12cB\varphi(B)^{2}S^{2}(C_{1}+C_{2}),\] where \[C_{1} :=2U(U^{2}-V^{2})\tanh y,\] \[C_{2} :=3cB^{2}\varphi(B)^{2}S^{2}\biggl{(}U^{2}V^{4}+(-2U^{2}+V^{2})^{ 2}\tanh^{2}y\biggr{)}.\] Since \[\frac{\cosh(V^{2}-U^{2})}{\cosh(U^{2})}=\cosh(V^{2})-\sinh(V^{2})\tanh(U^{2}),\] using the fact that $\|V\|\leq 1$, we have \[e^{-1}\leq\exp(-V^{2})=\cosh(V^{2})-\sinh(V^{2})<\frac{\cosh(V^{2}-U^{2})}{ \cosh(U^{2})}.\] In particular \[S\|U\|^{m}=\frac{\|U\|^{m}}{\cosh(U^{2})}\frac{\cosh(U^{2})}{\cosh(V^{2}-U^{2})}< \frac{e\|U\|^{m}}{\cosh(U^{2})}\] is a bounded function for $m\geq 0$. Then we can write \[\frac{\tilde{B}}{B}=1-12c\varphi(B)^{2}SB(SC_{1}+SC_{2}).\] Since $\tanh y$, $\varphi(B)$, and $SB=B{\operatorname{sech}}B$ are all bounded, there exists a positive constant $m$ which does not depend on the choice of $c\in(0,1]$ such that $\varphi(B)^{2}SB(SC_{1}+SC_{2})<m$ holds for all $(x,y)\in{\boldsymbol{R}}^{2}$, and so \[\left\|\frac{\tilde{B}}{B}-1\right\|<12mc.\] If $0<c<1/(12m)$, then the zero set of $\tilde{B}$ coincides with that of $B$, proving the assertion. ∎ _Acknowledgements_.: The first, the fourth and the fifth authors thank Udo Hertrich-Jeromin and Kosuke Naokawa for fruitful conversations at TU-Wien. The authors thank Wayne Rossman for valuable comments. ## References * [1] S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, and K. Yamada, _Analytic extension of exceptional constant mean curvature one elliptic catenoids in de Sitter 3-space_, preprint. * [2] S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada and S.-D. Yang, Zero mean curvature surfaces in Lorentz-Minkowski $3$-space which change type across a light-like line, Osaka J. Math. 52 (2015), 285–297. * [3] S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada and S.-D. Yang, _Zero mean curvature surfaces in Lorentz-Minkowski $3$-space and 2-dimensional fluid mechanics_, Math. J. Okayama Univ. 57 (2015), 173–200. * [4] S. Fujimori, W. Rossman, M. Umehara, K. Yamada and S.-D.Yang, _Spacelike mean curvature one surfaces in de Sitter 3-space_, Comm. in Anal. and Geom. 17 (2009), 383-427. * [5] S. Fujimori, W. Rossman, M. Umehara, K. Yamada and S.-D. Yang, _Embedded triply periodic zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-space_, Michigan Math. J. 63 (2014), 189–207. * [6] S. Fujimori, K. Saji, M. Umehara and K. Yamada, _Singularities of maximal surfaces_, Math. Z. 259 (2008), 827–848. * [7] C. Gu, _The extremal surfaces in the $3$-dimensional Minkowski space_, 　Acta Math. Sinica 1 (1985), 173–180. * [8] A. Honda, M. Koiso and K. Saji, _Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space_, preprint. * [9] Y. W. Kim, S.-E Koh, H. Shin and S.-D. Yang, _Spacelike maximal surfaces, timelike minimal surfaces, and Björling representation formulae_, J. Korean Math. Soc. 48 (2011), 1083–1100. * [10] V. A. Klyachin, _Zero mean curvature surfaces of mixed type in Minkowski space_, Izvestiya Math. 67 (2003), 209–224. * [11] O. Kobayashi, _Maximal surfaces in the $3$-dimensional Minkowski space $\mathbb{L}^{3}$_, Tokyo J. Math., 6 (1983), 297–309. * [12] V. Sergienko and V.G. Tkachev, _Doubly periodic maximal surfaces with singularities_, Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), 571–584, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000. * [13] M. Umehara and K. Yamada, _Maximal surfaces with singularities in Minkowski space_, Hokkaido Math. J. 35 (2006), 13–40.
1609.04273	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 1748, "num_imgs": 0, "llama3_tokens_count": 359 }	[]	AIM sec:valid:aimavu The ‘astrometric instrument model’ (AIM) is a scaled-down counterpart of IDT and First Look restricted to some astrometric elements of the daily processing; its focus is on the independent verification of selected AF monitoring and diagnostics, of the image parameter determination, and instrument modelling and calibration. This separate processing chain is described in 2014SPIE.9150E..0KB. In particular AIM provides image parameters through its own image parameter estimation code, allowing routine comparisons with – and thus external verification of – IDT image location values and corresponding formal errors. More details will be given in Busonero et al. (2016, in preparation). As stated in Sect. sec:lsfpsf, the reconstruction of the LSF and PSF are two of the Gaia key calibrations. For that reason, AIM implements its own independent PSF/LSF image profile models based in a one-dimensional case on a set of monochromatic basis functions, where the zero-order base is the sinc function squared. The complete model includes the contribution of finite pixel size, modulation transfer function and CCD operation in TDI mode. The higher-order functions are generated by suitable combinations of the parent function and its derivatives according to a construction rule ensuring orthonormality. The spatially variable LSF/PSF is reconstructed as the sum of spatially invariant functions, with coefficients varying over the fields of view. The polychromatic functions are built according to linear superposition of the monochromatic counterparts, weighted by the normalised detected source spectrum. The model is briefly described in 2013PASP..125..444G; details will be given in Busonero et al. (2016, in preparation).
1102.1193	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 45865, "num_imgs": 1, "llama3_tokens_count": 15945 }	[ "content_image/1102.1193/fuzzyqutrit.jpg" ]	# Extremal Black Holes as Qudits Michael Rios¹ _California State University, Los Angeles_ _Physics Graduate Program_ _5151 State University Drive_ _Los Angeles, CA 90032, USA_ [FOOTNOTE:1][ENDFOOTNOTE] February 21, 2024 ###### Abstract We extend the black hole/qudit correspondence by identifying five and six-dimensional 1/2-BPS black string and hole charge vectors in $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities with qubits and qutrits over composition algebras. In $D=6$, this is accomplished via Hopf fibrations, which map qubits over composition algebras to rank one elements of Jordan algebras of degree two. An analogous procedure maps qutrits over composition algebras to $D=5$ charge vectors, which are rank one elements of Jordan algebras of degree three. In both cases, the U-duality groups are interpreted as qudit SLOCC transformation groups. We provide explicit gates for such transformations and study their applications in toroidally compactified M-theory. $Keywords:$ Black Holes, Qudits, U-duality. ###### Contents Contents * 1 Introduction * 2 $\mathcal{N}=8$ and $\mathcal{N}=2$ Magic Supergravities * 2.1 Composition Algebras * 2.2 Jordan Algebras * 2.3 $D=6$ Black Strings * 2.4 $D=5$ Black Holes * 3 Qubits, Qutrits and Black Holes * 3.1 Qubits and $D=6$ Black Strings * 3.2 Qutrits and $D=5$ Black Holes * 4 Conclusion ## 1 Introduction In recent papers [1]-[9][55]-[57], Ferrara, Leváy and Duff et al. established a correspondence between extremal black holes and qubit and qutrit entanglement. Such a correspondence stems from Duff’s observation that the entropy of an extremal black hole in the four-dimensional $\mathcal{N}=2$ STU model and the measure of three-qubit entanglement are both given by Cayley’s hyperdeterminant [8]. Later work was motivated by studies of BPS black holes in a class of $\mathcal{N}=2$ supergravities whose scalar fields lie on a symmetric space [10]-[14], called magic supergravities [15]-[18]. It was shown that extremal black holes in $D=3,4,5,6$ magic supergravities can be described by Jordan algebras and their corresponding Freudenthal triple systems, with the Bekenstein-Hawking entropy of the black holes given by algebraic invariants [4, 10, 13, 14, 19, 20]. In $D=4$, the quartic invariant was shown to be proportional to Cayley’s hyperdeterminant [2, 3, 4] and used to describe three-qubit entanglement, yielding a novel derivation of the three-qubit entanglement classes [5]. Subsequent work led to the description of STU black holes in terms of four qubits [57] and a rigorous classification of four qubit entanglement based on $D=3$ black hole U-duality orbits [55, 56]. In [4], Borsten et al. proposed the identification of elements of Jordan algebras of degree two and three with $2\times 2$ and $3\times 3$ reduced density matrices for two qubits and qutrits, in quantum mechanics defined over arbitrary division algebras. Here, we also make use of the Jordan formulation of quantum mechanics, but give a different interpretation for elements of Jordan algebras over composition algebras, which include division algebras as well as their split forms. Given that 1/2-BPS black hole charge vectors in $D=5,6$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities are rank one elements of Jordan algebras of degree two and three [13, 52], it is possible to identify 1/2-BPS black hole charge vectors with pure qudits (qubits and qutrits) over composition algebras. The unitary and SLOCC groups of such qudits correspond to the automorphism and reduced structure groups of the corresponding Jordan algebras, respectively. As higher rank Jordan elements can always be expressed as orthogonal sums of rank one elements via spectral decomposition [26, 29], we identify 1/4 and 1/8-BPS black hole charge vectors with observables on orthogonal qudit states. Geometrically, 1/2-BPS black hole charge vectors are mapped to points in projective space, while 1/4 and 1/8-BPS charge vectors are mapped to projective lines and planes. The corresponding U-duality groups are isomorphic to the collineation groups of the projective spaces [24]. In the quantum information context, U-duality transformations are realized by acting on Jordan algebra matrices with elements of the reduced structure group [45], where such elements are interpreted as qudit SLOCC gates. ## 2 $\mathcal{N}=8$ and $\mathcal{N}=2$ Magic Supergravities We review $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities in $D=5,6$ and give their corresponding Jordan algebras, symmetry groups and entropy expressions for black string and hole solutions. The relevant Jordan algebras are of degree two and three and taken over finite dimensional composition algebras. We will first review composition algebras and Jordan algebras, particularly those relevant to black hole solutions in supergravity. ### Composition Algebras Let $V$ be a finite dimensional vector space over a field $\mathbb{F}=\mathbb{R},\mathbb{C}$. An $algebra$$structure$ on $V$ is a bilinear map \[V\times V\to V\] (1) \[(x,y)\mapsto x\bullet y.\] A $composition$$algebra$ is an algebra $\mathbb{A}=(V,\bullet)$, admitting an identity element, with a non-degenerate quadratic form $\eta$ satisfying \[\forall x,y\in\mathbb{A}\quad\eta(x\bullet y)=\eta(x)\eta(y).\] (2) If $\exists x\in\mathbb{A}$ such that $x\neq 0$ and $\eta(x)=0$, $\eta$ is said to be $isotropic$ and gives rise to a $split$$composition$$algebra$. When $\forall x\in\mathbb{A}$, $x\neq 0$, $\eta(x)\neq 0$, $\eta$ is $anisotropic$ and yields a $composition$$division$$algebra$. Theorem 2.1.: _A finite dimensional vector space $V$ over $\mathbb{F}=\mathbb{R},\mathbb{C}$ can be endowed with a composition algebra structure if and only if $\textrm{dim}_{\mathbb{F}}(V)=1,2,4,8$. If $\mathbb{F}=\mathbb{C}$, then for a given dimension all composition algebras are isomorphic. For $\mathbb{F}=\mathbb{R}$ and $\textrm{dim}_{\mathbb{F}}(V)=8$ there are only two non-isomorphic composition algebras: the octonions $\mathbb{O}$ for which $\eta$ is anisotropic and the split-octonions $\mathbb{O}_{s}$ for which $\eta$ is isotropic and of signature $(4,4)$. Moreover for all composition algebras, the quadratic form $\eta$ is uniquely defined by the algebra structure._ The composition algebras immediately applicable to $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities include those over $\mathbb{R}$ with anisotropic quadratic form, the $\textrm{dim}_{\mathbb{R}}(V)=1,2,4,8$ division algebras $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$, and the eight-dimensional split octonions $\mathbb{O}_{s}$ with isotropic quadratic form. It is worth noting that even though $\mathbb{O}$ and $\mathbb{O}_{s}$ are non-isomorphic, they are both real subalgebras of the unique $\textrm{dim}_{\mathbb{C}}(V)=8$ dimensional bioctonion composition algebra $\mathbb{O}_{\mathbb{C}}$[19, 54]. ### Jordan Algebras A Jordan algebra over a field $\mathbb{F}=\mathbb{R},\mathbb{C}$ is a vector space over $\mathbb{F}$ equipped with a bilinear form (Jordan product) $(X,Y)\to X\circ Y$ satisfying $\forall X,Y\in\mathcal{A}$: \[X\circ Y=Y\circ X\] \[X\circ(Y\circ X^{2})=(X\circ Y)\circ X^{2}.\] Given an associative algebra $\mathcal{A}$, we can define the Jordan product $\circ$ on $\mathcal{A}$ using the associative product in $\mathcal{A}$, given by $X\circ Y\equiv\frac{1}{2}(XY+YX)$. Under the Jordan product, ($\mathcal{A},\circ$) becomes a _special_ Jordan algebra $\mathcal{A}^{+}$. Any Jordan algebra that is simple and not special is called an _exceptional_ Jordan algebra or Albert Algebra. The exceptional Jordan algebras are 27-dimensional and over $\mathbb{R}$ are isomorphic to either $J^{\mathbb{O}}_{3}$ or $J^{\mathbb{O}_{s}}_{3}$[26]. An idempotent (or projector) in a Jordan algebra is a non-zero element such that $P^{2}=P$. Given two idempotents $P_{1},P_{2}$, they are orthogonal if $P_{1}P_{2}=0$. An idempotent is called _primitive_ if it is not the sum of two orthogonal idempotents. In finite dimensions, any idempotent can be decomposed into an orthogonal sum of primitive idempotents. The number of primitive idempotents the identity decomposes into is called the _capacity_, and determines the _degree_ of the Jordan algebra [25]. The Jordan algebras relevant to $D=5,6$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities are those of degree two and three, consisting of $2\times 2$ and $3\times 3$ Hermitian matrices with entries in a composition algebra. The automorphism and determinant preserving groups for these algebras will be denoted by $\textrm{Aut}(J^{\mathbb{A}}_{n})$ and $\textrm{Str}_{0}(J^{\mathbb{A}}_{n})$, respectively. ### $D=6$ Black Strings The $D=6$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities arise as uplifts of $D=5$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities with $n_{V}=27$, $n_{V}=15$, $n_{V}=9$ and $n_{V}=6$ vector fields [21]. They exhibit $\textrm{Spin}(5,5)$, Spin$(9,1)$, Spin$(5,1)$, Spin$(3,1)$ and Spin$(2,1)$ U-duality symmetry since the $D=6$ vector multiplets in the Coulomb phase (after Higgsing) transform as spinors of dimension 16, 8, 4 and 2, respectively [21]. One can associate a black string solution with charges $q_{I}$ ($I=1,...,n_{V}$ for $n_{V}=10,6,4,3$) an element \[J=\sum_{I=1}^{n}q^{I}e_{I}=\left(\begin{array}[]{cc}r_{1}&A\\ \overline{A}&r_{2}\end{array}\right)\quad r_{i}\in\mathbb{R},A\in\mathbb{A}\] (3) of a Jordan algebra $J^{\mathbb{A}}_{2}$ of degree two, where the $e_{I}$ form the $n_{V}$-dimensional basis of the Jordan algebra over a composition algebra $\mathbb{A}$. Elements of $J^{\mathbb{A}}_{2}$ transform as the (dim $\mathbb{A}+2$) representation of $SL(2,\mathbb{A})$, the $\mathbf{10}$, $\mathbf{10}$, $\mathbf{6}$, $\mathbf{4}$, $\mathbf{3}$ of $SO(5,5)$, $SO(9,1)$, $SO(5,1)$, $SO(3,1)$ and $SO(2,1)$, respectively [4]. This can be seen by noting the $(p,q)$ spacetime inner product can be expressed as \[J\cdot K=\frac{1}{2}(\textrm{tr}(J)\textrm{tr}(K)-\textrm{tr}(J\circ K))\] (4) and by setting $J=K$, $r_{1}=x+y$ and $r_{2}=x-y$ we recover the $(p,q)$ spacetime norm squared \[J\cdot J=\textrm{det}(J)=x^{2}-y^{2}+A\overline{A}.\] (5) As $SL(2,\mathbb{A})=\textrm{Str}_{0}(J_{2}^{\mathbb{A}})$ consists of determinant preserving transformations over $J^{\mathbb{A}}_{2}$, such transformations necessarily preserve the $(p,q)$ spacetime norm. Moreover, since $SL(2,\mathbb{O}_{s})=\textrm{Spin}(5,5)$, $SL(2,\mathbb{O})=\textrm{Spin}(9,1)$, $SL(2,\mathbb{H})=\textrm{Spin}(5,1)$, $SL(2,\mathbb{C})=\textrm{Spin}(3,1)$ and $SL(2,\mathbb{R})=\textrm{Spin}(2,1)$, the original claim for the $SO(p,q)$ groups follows. Over any composition algebra, the black string entropy is given by [4] \[S=\pi\sqrt{\|I_{2}(J)\|}\] (6) where \[I_{2}(J)=\textrm{det}(J)=r_{1}r_{2}-A\overline{A}.\] (7) The U-duality orbits are distinguished by rank [52] via \[\begin{array}[]{rcl}\textrm{Rank}\, J=2\quad\textrm{iff}\quad I_{2}(J) \neq 0\quad\quad\quad\qquad\hskip 1.0ptS\neq 0,\,\textrm{1/4-BPS}\\ \textrm{Rank}\, J=1\quad\textrm{iff}\quad I_{2}(J)=0,\, J\neq 0 \qquad S=0,\,\textrm{1/2-BPS}\hfill.\end{array}\] (8) Being that 1/2-BPS black hole charge vectors satisfy $I_{2}(J)=J\cdot J=0$, they transform as lightlike vectors in $(p,q)$ spacetime. ### $D=5$ Black Holes In $D=5$, the $\mathcal{N}=8$ and $\mathcal{N}=2$ supergravities are coupled to 6, 9, 15 and 27 vector fields with U-duality symmetry groups $SL(3,\mathbb{R})$, $SL(3,\mathbb{C})$, $SU^{\ast}(6)$, $E_{6(-26)}$ and $E_{6(6)}$, respectively [4, 13, 14]. The orbits of BPS black hole solutions were classified [13, 22] by studying the underlying Jordan algebras of degree three under the actions of their reduced structure groups, $\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$, which correspond to the U-duality groups of the $\mathcal{N}=8$ and $\mathcal{N}=2$ supergravities. This is seen by associating a given black hole solution with charges $q_{I}$ ($I=1,...,n_{V})$ an element \[J=\sum_{I=1}^{n}q^{I}e_{I}=\left(\begin{array}[]{ccc}r_{1}&A_{1}&\overline{A}_ {2}\\ \overline{A}_{1}&r_{2}&A_{3}\\ A_{2}&\overline{A}_{3}&r_{3}\end{array}\right)\quad r_{i}\in\mathbb{R},A_{i} \in\mathbb{A}\] (9) of a Jordan algebra of degree three $J_{3}^{\mathbb{A}}$ over a composition algebra, where $e_{I}$ form a basis for the $n_{V}$-dimensional Jordan algebra. This establishes a correspondence between Jordan algebras of degree three and the charge spaces of the extremal black hole solutions [13]. In all cases, the entropy of a black hole solution can be written [4, 13, 23] in the form \[S=\pi\sqrt{\|I_{3}(J)\|}\] (10) where $I_{3}$ is the cubic invariant given by \[I_{3}=\textrm{det}(J)\] (11) and \[\textrm{det}(J)=r_{1}r_{2}r_{3}-r_{1}\|\|A_{1}\|\|^{2}-r_{2}\|\|A_{2}\|\|^{2}-r_{3}\|\|A _{3}\|\|^{3}+2\textrm{Re}(A_{1}A_{2}A_{3}).\] (12) The U-duality orbits are distinguished by rank via \[\begin{array}[]{rcl}\textrm{Rank}\, J=3\quad\textrm{iff}\quad I_{3}(J) \neq 0\hskip 3.0pt\qquad\quad\qquad S\neq 0,\,\textrm{1/8-BPS}\\ \textrm{Rank}\, J=2\quad\textrm{iff}\quad I_{3}(J)=0,J^{\natural}\neq 0 \qquad S=0,\,\textrm{1/4-BPS}\\ \textrm{Rank}\, J=1\quad\textrm{iff}\quad J^{\natural}=0,\, \, J\neq 0\,\,\quad\qquad S=0,\,\textrm{1/2- BPS}\end{array}\] (13) where the quadratic adjoint map is given by \[J^{\natural}=J\times J=J^{2}-\textrm{tr}(J)J+\frac{1}{2}(\textrm{tr}(J)^{2}- \textrm{tr}(J^{2}))I.\] (14) Note the quadratic adjoint map is a generalization of the $I_{2}$ invariant, whose vanishing defined 1/2-BPS black hole solutions in $D=6$. It will be seen that the vanishing of the quadratic adjoint map or the $I_{2}$ invariant are defining conditions for 1/2-BPS black hole charge vectors to be interpreted as points in projective space. ## 3 Qubits, Qutrits and Black Holes In this section, qubits and qutrits over composition algebras will be defined and identified with 1/2-BPS black string and hole charge vectors in $D=5,6$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities. ### Qubits and $D=6$ Black Strings A quantum bit or $qubit$ is a state of a quantum system with a two-dimensional representation space $\mathcal{H}^{2}$. The basis states of this space are denoted as $\|0\rangle$ and $\|1\rangle$ and a general state can be written in this basis as: \[\|\Psi\rangle=a_{0}\|0\rangle+a_{1}\|1\rangle\quad\quad a_{i}\in\mathcal{H}^{2}.\] (15) $\|\Psi\rangle$ is usually taken to be an element of the Hilbert space $\mathbb{C}^{2}$, where \[\|0\rangle=\left(\begin{array}[]{c}1\\ 0\end{array}\right)\quad\|1\rangle=\left(\begin{array}[]{c}0\\ 1\end{array}\right)\] (16) represent the qubit computational basis. A $pure$ qubit satisfies the normalization condition $\|\|a_{0}\|\|^{2}+\|\|a_{1}\|\|^{2}=1$, where $\|\|a_{i}\|\|^{2}$ is the probability of measuring the qubit in state $\|i\rangle$. Two pure states $\|\psi\rangle$ and $\|\phi\rangle$ can be obtained with certainty from each other by means of local operations assisted with classical communication (LOCC) if and only if they are related by local unitaries LU [32, 33]. If we merely require that two states $\|\psi\rangle$ and $\|\phi\rangle$ be obtained from each other with a non-vanishing probability of success, the conversion of the states is performed through stochastic local operations and classical communication (SLOCC) [32, 33]. For a single qubit, the LOCC and SLOCC equivalence groups are equivalent to $SU(2)$ and $SL(2,\mathbb{C})$[5, 35]. Given a pure qubit, one can assign a rank one density matrix \[P=\|\Psi\rangle\langle\Psi\|=\left(\begin{array}[]{cc}\|\|a_{0}\|\|^{2}&a_{0} \overline{a}_{1}\\ a_{1}\overline{a}_{0}&\|\|a_{1}\|\|^{2}\end{array}\right).\] (17) satisfying \[\begin{array}[]{ccl}P^{2}=P\\ \textrm{tr}(P)=1.\end{array}\] (18) As $P$ is a rank one $2\times 2$ Hermitian idempotent, we have a map from the Hilbert space $\mathbb{C}^{2}$ to the space of rank one elements of the Jordan algebra $J^{\mathbb{C}}_{2}$. Geometrically, $P$ is mapped to a point of the projective line $\mathbb{CP}^{1}$[24]. In fact, the mapping of a pure qubit to $\mathbb{CP}^{1}$ gives rise to the Hopf fibration $S^{1}\hookrightarrow S^{3}\to S^{2}$. In general, given a pure qubit over a composition divison algebra $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$, the mapping $\mathbb{A}^{2}\rightarrow\mathbb{AP}^{1}$ gives rise to all four Hopf fibrations [24] \[\mathbb{R}^{2}\rightarrow\mathbb{RP}^{1}:\qquad\, \,\,\,\, S^{0}\hookrightarrow S^{1}\to S ^{1}\] \[\mathbb{C}^{2}\rightarrow\mathbb{CP}^{1}:\qquad\, \,\,\,\, S^{1}\hookrightarrow S^{3}\to S ^{2}\] \[\mathbb{H}^{2}\rightarrow\mathbb{HP}^{1}:\qquad\, \,\,\, S^{3}\hookrightarrow S^{7}\to S^{4}\] \[\mathbb{O}^{2}\rightarrow\mathbb{OP}^{1}:\qquad S^{7} \hookrightarrow S^{15}\to S^{8}.\] (19) For split composition algebras $\mathbb{A}=\mathbb{C}_{s},\mathbb{H}_{s},\mathbb{O}_{s}$, there exist analagous mappings [58], providing non-compact Hopf fibrations which represent maps between hyperboloids in different dimensions with hyperboloid fibers \[\mathbb{C}_{s}^{2}\rightarrow\mathbb{CP}_{s}^{1}:\qquad\, \,\, H^{1,0}\hookrightarrow H^{2,1}\to H^{1,1}\] \[\mathbb{H}_{s}^{2}\rightarrow\mathbb{HP}_{s}^{1}:\qquad\, \, H^{2,1}\hookrightarrow H^{4,3}\to H^{2,2}\] \[\mathbb{O}_{s}^{2}\rightarrow\mathbb{OP}_{s}^{1}:\qquad H^{4,3} \hookrightarrow H^{8,7}\to H^{4,4}.\] (20) The non-compact Hopf fibrations thus describe the geometry of qubit state spaces over split composition algebras. As $SU(2)=\textrm{Aut}(J^{\mathbb{C}}_{2})$, it is natural to identify LOCC groups for pure qubits over composition algebras with the automorphism group of the corresponding Jordan algebra of degree two $\textrm{Aut}(J^{\mathbb{A}}_{2})$, yielding $SO(2)$, $SU(2)$, $Usp(4)$ and $SO(9)$ for composition division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$[24], and $SO(2,1)$, $SO(3,2)$ and $SO(5,4)$ for split composition algebras $\mathbb{C}_{s}$, $\mathbb{H}_{s}$ and $\mathbb{O}_{s}$, respectively [58]. The relevant SLOCC groups are then given by the reduced structure groups $SL(2,\mathbb{A})=\textrm{Str}_{0}(J_{2}^{\mathbb{A}})$, giving $SL(2,\mathbb{R})=\textrm{Spin}(2,1)$, $SL(2,\mathbb{C})=\textrm{Spin}(3,1)$, $SL(2,\mathbb{H})=\textrm{Spin}(5,1)$ and $SL(2,\mathbb{O})=\textrm{Spin}(9,1)$ for composition division algebras [24], and $SL(2,\mathbb{C}_{s})=\textrm{Spin}(2,2)$, $SL(2,\mathbb{H}_{s})=\textrm{Spin}(3,3)$, $SL(2,\mathbb{O}_{s})=\textrm{Spin}(5,5)$ for split composition algebras. With a direct correspondence between qubit state spaces and spaces of rank one elements of Jordan algebras of degree two, it is seen that six-dimensional 1/2-BPS black hole charge vectors transform as qubits over composition algebras. This allows one to identify 1/2-BPS black holes with qubits and also gives a quantum computational interpretation for $D=6$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravitiy U-duality groups. Specifically, U-duality groups consist of those qubit transformations which allow the conversion between states with a non-vanishing probability of success. This is sensible since, in general, special linear transformations are not isometries of qubit state spaces and not expected to preserve the normalization of pure qubits. However, such transformations do preserve the determinant, hence rank, acting as single qubit SLOCC gates which map between not necessarily normalized states in projective space. Single qubit gates are typically given by unitary transformations [36, 37], but for reversible computation any $2\times 2$ invertible linear transformation will suffice [38]. Important unitary single qubit gates include the Hadamard gate \[H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\ 1&-1\end{array}\right),\] (21) the Pauli-X,Y,Z gates (where Pauli-X acts as a NOT gate) \[X=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\quad Y=\left(\begin{array}[]{cc}0&-i\\ i&0\end{array}\right)\quad Z=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),\] (22) and the phase shift gates \[R_{\theta}=\left(\begin{array}[]{cc}1&0\\ 0&e^{i\theta}\end{array}\right).\] (23) Some example non-unitary gates [38] include \[N_{1}=\left(\begin{array}[]{cc}1&0\\ 0&r_{1}\end{array}\right)\quad N_{2}=\left(\begin{array}[]{cc}r_{2}&0\\ 0&1\end{array}\right)\quad r_{i}\in\mathbb{R},r_{i}\neq 0.\] (24) Over any composition algebra $\mathbb{A}$, the Pauli gates are generalized to the set \[X=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\quad Y_{i}=\left(\begin{array}[]{cc}0&\overline{l}_{i}\\ l_{i}&0\end{array}\right)\quad Z=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)\quad i=1,...,\textrm{dim}(\mathbb{A})-1\] where $l_{i}^{2}=(\textrm{e}_{i})^{2}=-1$ or $l_{i,s}^{2}=(\textrm{ie}_{i})^{2}=1$ and the phase shift gates become \[R_{i,\theta}=\left(\begin{array}[]{cc}1&0\\ 0&e^{l_{i}\theta}\end{array}\right).\] (25) It can be verified that the generalized Pauli gates $X,Y_{i}$ and $Z$ are generators for $\textrm{Aut}(J_{2}^{\mathbb{A}})$, while $H,R_{i,\theta}\in\textrm{Aut}(J_{2}^{\mathbb{A}})$ and $N_{k}\in\textrm{Str}(J_{2}^{\mathbb{A}})$. Using elements of $\textrm{Aut}(J_{2}^{\mathbb{A}})$ and $\textrm{Str}(J_{2}^{\mathbb{A}})$ with entries in a complex subalgebra [45], single qubit gates act on general elements $J\in J_{2}^{\mathbb{A}}$ via conjugation \[J^{\prime}=SJS^{\dagger}.\] (26) Gates in $\textrm{Aut}(J_{2}^{\mathbb{A}})$ preserve the Frobenius norm $\lVert J\rVert^{2}=\textrm{tr}(J^{2})$ while the $\textrm{Str}(J_{2}^{\mathbb{A}})$ gates preserve the determinant $\textrm{det}(J)$ up to a real constant. When the determinant is preserved exactly, one recovers SLOCC gates in $\textrm{Str}_{0}(J_{2}^{\mathbb{A}})$. Given a black string in $D=6$, $\mathcal{N}=8$ or $\mathcal{N}=2$ magic supergravity, its charge vector is an element $J\in J_{2}^{\mathbb{A}}$[4] with spectral decomposition \[J=\lambda_{1}P_{1}+\lambda_{2}P_{2}\quad\quad\lambda_{i}\in\mathbb{R},\] (27) where $P_{i}$ are orthonormal rank one idempotents of $J_{2}^{\mathbb{A}}$. Identifying $P_{1},P_{2}$ with pure qubit density matrices allows one to interpret a 1/4-BPS black string charge vector as a Boolean observable on two orthogonal qubit states. The eigenvalues $\lambda_{i}$ are those values the observable takes on two coordinate charts of the projective line $\mathbb{AP}^{1}$[51]. The group of qubit SLOCC transformations $\textrm{Str}_{0}(J_{2}^{\mathbb{A}})$ transforms the observable while preserving the black string entropy $S=\pi\sqrt{\|\textrm{det}(J)\|}=\pi\sqrt{\|\lambda_{1}\lambda_{2}\|}$. This is consistent with the interpretation of $D=6$ black strings as bound states or intersections of basic constituent $p$-branes of charge $Q_{1},Q_{2}$, for which the area law for the entropy yields $S=\pi\sqrt{\|Q_{1}Q_{2}\|}$[50]. These configurations preserve 1/4 of the supersymmetry and in $\mathcal{N}=8$ supergravity can arise from two orthogonally intersecting M-branes in $D=11$[43, 44, 46, 47, 48]. ### Qutrits and $D=5$ Black Holes A quantum trit or $qutrit$ is a state of a quantum system with a three-dimensional representation space. The basis states of this space are denoted as $\|0\rangle$, $\|1\rangle$ and $\|2\rangle$ and a general state can be written in this basis as: \[\|\Psi\rangle=a_{0}\|0\rangle+a_{1}\|1\rangle+a_{2}\|2\rangle\quad\quad a_{i}\in \mathcal{H}^{3}.\] (28) $\|\Psi\rangle$ is usually taken to be an element of $\mathbb{C}^{3}$, where \[\|0\rangle=\left(\begin{array}[]{c}1\\ 0\\ 0\end{array}\right)\quad\|1\rangle=\left(\begin{array}[]{c}0\\ 1\\ 0\end{array}\right)\quad\|2\rangle=\left(\begin{array}[]{c}0\\ 0\\ 1\end{array}\right)\] (29) represent the qutrit computational basis. A $pure$ qutrit satisfies the normalization condition $\|\|a_{0}\|\|^{2}+\|\|a_{1}\|\|^{2}+\|\|a_{2}\|\|^{2}=1$, where $\|\|a_{i}\|\|^{2}$ is the probability of measuring the qutrit in state $\|i\rangle$. For a single qutrit, the LOCC and SLOCC equivalence groups are $SU(3)$ and $SL(3,\mathbb{C})$, respectively [3, 4, 34, 39]. Given a pure qutrit, we can construct its rank one density matrix \[P=\|\Psi\rangle\langle\Psi\|=\left(\begin{array}[]{ccc}\|\|a_{0}\|\|^{2}&a_{0} \overline{a}_{1}&a_{0}\overline{a}_{2}\\ a_{1}\overline{a}_{0}&\|\|a_{1}\|\|^{2}&a_{1}\overline{a}_{2}\\ a_{2}\overline{a}_{0}&a_{2}\overline{a}_{1}&\|\|a_{2}\|\|^{2}\end{array}\right).\] (30) satisfying \[\begin{array}[]{ccl}\textrm{tr}(P)=1\\ P^{2}=P.\end{array}\] (31) This mapping can be generalized for any pure qutrit in $\mathbb{A}^{3}$, where $\mathbb{A}$ is a composition algebra. As the resulting density matrix is rank one, satisfying $P^{\#}=0$, one can map any pure qutrit in $\mathbb{A}^{3}$ to a rank one matrix of $J_{3}^{\mathbb{A}}$, hence, a point of $\mathbb{AP}^{2}$. In the case of the octonions, this is leads to a successful construction of the projective plane $\mathbb{OP}^{2}$ (Cayley-Moufang plane) [24], for which the standard construction using only equivalence classes of elements of $\mathbb{A}^{3}$ fails. The failure of the standard construction is due to the identification $v\sim\lambda v,v\in\mathbb{A}^{3}$ requiring associativity to be an equivalence relation [24, 53]. Therefore, for non-associative composition algebras, the rank one operator representation for qutrits is essential in order to recover a sensible theory of quantum mechanics. Hence, quantum mechanics over non-associative composition algebras is necessarily expressed in the Jordan formulation of quantum mechanics [4, 25, 40, 59]. In such theories of quantum mechanics a qutrit projective space is maximal, in the sense that there exist topological obstructions preventing the construction of higher projective $n$-spaces $\mathbb{AP}^{n},n>2$, over non-associative composition algebras [53]. As $SU(3)$ and $SL(3,\mathbb{C})$ are the automorphism and reduced structure groups for $J_{3}^{\mathbb{C}}$, respectively, we will associate the LOCC and SLOCC groups for any qutrit over composition algebra $\mathbb{A}$ with $\textrm{Aut}(J_{3}^{\mathbb{A}})$ and $\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$. For composition algebras $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O},\mathbb{O}_{s}$, this gives $SO(3)$, $SU(3)$, $USp(6)$, $F_{4}$ and $F_{4(4)}$ as LOCC groups and $SL(3,\mathbb{R})$, $SL(3,\mathbb{C})$, $SL(3,\mathbb{H})\cong SU^{}(6)$, $E_{6(-26)}$, $E_{6(6)}$ as SLOCC groups, respectively. For $D=5$ supergravities, $\textrm{Aut}(J_{3}^{\mathbb{A}})$ and $\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$ act as rotation and U-duality groups, respectively [4, 13, 14, 20, 21]. This establishes a correspondence between qutrit SLOCC groups over composition algebras and U-duality groups for $D=5$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities. Qudit SLOCC Groups --- Composition Algebra A \| Qutrit SLOCC \| Qubit SLOCC R \| SL(3,R) \| SL(2,R)=Spin(2,1) C \| SL(3,C) \| SL(2,C)=Spin(3,1) H \| SL(3,H) \| SL(2,H)=Spin(5,1) O \| E6(−26) \| SL(2,O)=Spin(9,1) Os \| E6(6) \| SL(2,Os)=Spin(5,5) OC \| E6(C) \| SL(2,OC)=Spin(10,C) Table 1: SLOCC groups for qutrits and qubits over composition algebras. Single qutrit gates are typically given by unitary transformations [4, 34, 39, 41]. However, just as for qubits, any invertible linear transformation can serve as a gate for reversible computations [38]. We can construct qutrit gates by embedding qubit gates and nesting transformations appropriately. For example, the Hadamard and Pauli gates can be embedded in $3\times 3$ Hermitian gates $M_{1},M_{2},M_{3}$ of the form \[\left(\begin{array}[]{ccc}1&0&0\\ 0&r_{1}&A_{1}\\ 0&\overline{A}_{1}&r_{2}\end{array}\right),\left(\begin{array}[]{ccc}r_{3}&0& \overline{A}_{2}\\ 0&1&0\\ A_{2}&0&r_{4}\end{array}\right),\left(\begin{array}[]{ccc}r_{5}&A_{3}&0\\ \overline{A}_{3}&r_{6}&0\\ 0&0&1\end{array}\right),\] where $r_{i}\in\mathbb{R}$ and $A_{i}\in\mathbb{A}$ and gate $M_{i}$ leaves the standard idempotent $P_{i}$ invariant. Requiring that $r_{i}=-r_{i+1}$ and $r_{i}^{2}+A\overline{A}=1$, yields unitary gates $M\in\textrm{Aut}(J_{2}^{\mathbb{A}})\subset\textrm{Aut}(J_{3}^{\mathbb{A}})$. SLOCC gates are recovered by requiring $\textrm{det}(M)=1$, giving $M\in\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$. In general, such $M$ gates are in $\textrm{Str}(J_{3}^{\mathbb{A}})$ and general transformations of elements $J\in J_{3}^{\mathbb{A}}$ are recovered via nesting \[J^{\prime}=M_{k}(M_{j}(M_{i}JM_{i}^{\dagger})M_{j}^{\dagger})M_{k}^{\dagger}.\] (32) Other unitary qutrit transformations include $\textrm{Aut}(J_{2}^{\mathbb{A}})\subset\textrm{Aut}(J_{3}^{\mathbb{A}})$ rotations \[\left(\begin{array}[]{ccc}e^{\overline{l}\theta}&0&0\\ 0&e^{l\theta}&0\\ 0&0&1\end{array}\right)\] (33) \[\left(\begin{array}[]{ccc}\textrm{cos}\theta&l\,\textrm{ sin}\theta&0\\ \overline{l}\,\textrm{sin}\theta&\textrm{cos}\theta&0\\ 0&0&1\end{array}\right),\] (34) and $\textrm{Str}_{0}(J_{2}^{\mathbb{A}})\subset\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$ boosts \[\left(\begin{array}[]{ccc}e^{\beta}&0&0\\ 0&e^{-\beta}&0\\ 0&0&1\end{array}\right),\left(\begin{array}[]{ccc}\textrm{cosh}\beta&\overline {l}\,\textrm{sinh}\beta&0\\ l\,\textrm{sinh}\beta&\textrm{cosh}\beta&0\\ 0&0&1\end{array}\right).\] (35) In [45], it was shown that for $J_{3}^{\mathbb{O}}$ one can recover all of $\textrm{Str}_{0}(J_{3}^{\mathbb{O}})=E_{6(-26)}$ via suitable nesting of such rotations and boosts. Since $J_{3}^{\mathbb{A}}\subseteq J_{3}^{\mathbb{O}}$ for any composition division algebra $\mathbb{A}$, this applies to other $\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$ as well. It then only remains to show $E_{6(6)}$ can be recovered by nesting of rotations of boosts over the split-octonions, using arguments similar to those given Manogue and Dray in the case of $E_{6(-26)}$[45]. Such nesting permits the construction of general SLOCC gates for qutrits over any composition algebra $\mathbb{A}$, using embedded qubit SLOCC gates. In the corresponding $D=5$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities, the nested rotations and boosts yield general U-duality transformations. <figure><img src="content_image/1102.1193/fuzzyqutrit.jpg"><figcaption>Figure 1: An projective basis for AP2 as a ternary computational basis.</figcaption></figure> Given a black hole in $D=5$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravity, we can associate an element $J\in J_{3}^{\mathbb{A}}$ with spectral decomposition \[J=\lambda_{1}P_{1}+\lambda_{2}P_{2}+\lambda_{3}P_{3}\quad\quad\lambda_{i}\in \mathbb{R},\] (36) where $P_{i}$ are orthonormal rank one idempotents of $J_{3}^{\mathbb{A}}$. Identifying $P_{1},P_{2},P_{3}$ with pure qutrit density matrices allows one to interpret 1/4-BPS and 1/8-BPS black hole charge vectors as observables on three orthogonal pure qutrit states. The eigenvalues are the values the observable takes on three coordinate charts of $\mathbb{AP}^{2}$. The automorphism group $\textrm{Aut}(J_{3}^{\mathbb{A}})\cong\textrm{Isom}(\mathbb{AP}^{2})$ preserves pure qutrit normalization while $\textrm{Str}_{0}(J_{3}^{\mathbb{A}})$ acts as a SLOCC group preserving the black hole entropy $S=\pi\sqrt{\|\textrm{det}(J)\|}=\pi\sqrt{\|\lambda_{1}\lambda_{2}\lambda_{3}\|}$. This agress with the interpretation of $D=5$ black holes preserving 1/8 supersymmetry as bound states or intersections of basic constituent $p$-branes with charges $Q_{1},Q_{2},Q_{3}$, for which the entropy law yields $S=\pi\sqrt{\|Q_{1}Q_{2}Q_{3}\|}$[50]. As an illustrative example, consider the case of $D=5$, $\mathcal{N}=8$ supergravity, where black hole charge vectors are in $J_{3}^{\mathbb{O}_{s}}$, the 27* of $E_{6(6)}$. Given a qubit SLOCC gate $S\in SL(2,\mathbb{O}_{s})=\textrm{Spin}(5,5)$, we can construct a $3\times 3$ qutrit SLOCC gate \[M=\left(\begin{array}[]{cc}S&0\\ 0&1\end{array}\right)\in\textrm{Str}_{0}(J_{3}^{\mathbb{O}_{s}})=E_{6(6)}.\] (37) Acting on a black hole charge vector $J\in J_{3}^{\mathbb{O}_{s}}$ imposes the block structure \[J=\left(\begin{array}[]{cc}V&\psi\\ \psi^{\dagger}&c\end{array}\right)\] (38) where $V\in J_{2}^{\mathbb{O}_{s}}$ transforms as a ten-dimensional vector, $\psi\in\mathbb{O}^{2}_{s}$ as a sixteen-dimensional spinor, and $c\in\mathbb{R}$ a scalar. As $\textrm{Spin}(5,5)$ is the double cover of $SO(5,5)$, choosing one of the three embeddings of a qubit SLOCC gate into a $3\times 3$ qutrit SLOCC gate leads to the decomposition under the T-duality group $SO(5,5)$: $\textbf{27}\rightarrow\textbf{10}+\textbf{16}+\textbf{1}$. At the group level, this corresponds to the three embeddings of $\textrm{Spin}(5,5)$ in $E_{6(6)}$, giving three such decompositions of the $\mathbf{27}$. In the projective plane $\mathbb{OP}^{2}_{s}$, each copy of $\textrm{Spin}(5,5)$ transforms two orthogonal rank one projectors (hence a projective line), while leaving the third invariant. This is seen by noting lines in the plane $\mathbb{OP}^{2}_{s}$ can be represented by rank two elements [25] \[P_{1}+P_{2}\quad\rightarrow\quad\textrm{line in }\mathbb{OP}^{2}_{s}.\] (39) Acting on a diagonalized element of $J_{2}^{\mathbb{O}_{s}}$ with $M\in\textrm{Spin}(5,5)\subset E_{6(6)}$ gives \[MDM^{\dagger}=\lambda_{1}MP_{1}M^{\dagger}+\lambda_{2}MP_{2}M^{\dagger}+ \lambda_{3}P_{3}\] (40) leaving a rank one idempotent invariant, while transforming the projective line $l_{12}=P_{1}+P_{2}$. The other two embeddings of the qubit gate $S\in\textrm{Spin}(5,5)$ accordingly transform the remaining two lines $l_{13}=P_{1}+P_{3}$ and $l_{23}=P_{2}+P_{3}$. Quantum computationally, this means an embedded $\textrm{Spin}(5,5)$ acts on a corresponding qubit subspace of the qutrit state space $\mathbb{OP}^{2}_{s}$, and by nesting transformations from different $\textrm{Spin}(5,5)$ subgroups a general qutrit transformation is recovered. Similar results apply to black holes in $D=5$, exceptional $\mathcal{N}=2$ magic supergravity, where the relevant groups are $SO(9,1)$, $\textrm{Spin}(9,1)$ and $E_{6(-26)}$. ## 4 Conclusion It was shown that 1/2-BPS black string and hole charge vectors in $D=5,6$, $\mathcal{N}=8$ and $\mathcal{N}=2$ magic supergravities can be identified with qubits and qutrits over composition algebras, using the Jordan formulation of quantum mechanics. This allows one to interpret U-duality groups as SLOCC groups for generalized qubits and qutrits. Moreover, 1/4 and 1/8-BPS black hole charge vectors were identified with observables on orthogonal qudit states and such an interpretation found agreement with results of multi-charge BPS solutions in toroidally compactified M-theory. In the case of M-theory on $T^{6}$, corresponding to $D=5$, $\mathcal{N}=8$ supergravity, the qutrit interpretation provided a novel view of the decomposition of the 27 of $E_{6(6)}$, under its T-duality subgroup $SO(5,5)$, found to arise from choosing an embedding of qubit SLOCC gates into qutrit SLOCC gates. Further research into the black hole/qudit correspondence might find applications for BPS black strings and holes in generalized spin chains, as spin-1/2 and spin-1 chains are currently used for both quantum teleporation and transfer of qudit states [60, 61]. Spin chains over non-associative composition algebras, with $SO(5,5)$ and $SO(9,1)$ SLOCC symmetry, might even prove helpful in building novel string bit models [62] for superstrings. In such a case, superstrings would arise from infinitely long polymers of BPS black string bits. Other possible applications of the black hole/qudit correspondence may lie in the study of scattering amplitudes in twistor space [63, 64], where the compact and non-compact complex Hopf fibrations are already in use, mapping spinors to lightlike vectors in (3,1) and (2,2) signature spacetimes. Perhaps the use of more general Hopf fibrations, such as those used for qubits over non-associative composition algebras, will provide deeper insights into the structure of these amplitudes and their interpretation within M-theory. Surely, there are many other unforeseen applications, and it appears certain that the black hole/qudit correspondence will continue to illuminate facets of both quantum gravity and quantum information theory and their fascinating interrelationship. ## References * [1]M. J. Duff, S. Ferrara, $E_{7}$ _and the tripartite entanglement of seven qubits_, Phys. Rev. D76:025018 (2007), arXiv:quant-ph/0609227. * [2]M. J. Duff, S. Ferrara, _Black hole entropy and quantum information_, Lect. Notes. Phys.755:93-114 (2008), arXiv:hep-th/0612036. * [3]M. J. Duff, S. Ferrara, $E_{6}$ _and the bipartite entanglement of three qutrits_, Phys. Rev. D76:124023 (2007), arXiv:0704.0507 [hep-th]. * [4]L. Borsten, D. Dahanayake, M. J. Duff, H. Ebrahim, W. Rubens, _Black Holes, Qubits and Octonions_, Phys. Rep. 471:113-219 (2009), arXiv:0809.4685 [hep-th]. * [5]L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens, H. Ebrahim, _Freudenthal triple classification of three-qubit entanglement_, arXiv:0812.3322 [quant-ph]. * [6]L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens, _Black holes admitting a Freudenthal dual_, Phys. Rev. D80:026003 (2009), arXiv:0903.5517 [hep-th]. * [7]L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens, _Superqubits_, arXiv:0908.0706 [quant-ph]. * [8]M. J. Duff, _String triality, black hole entropy and Cayley’s hyperdeterminant_, Phys. Rev. D76:025017 (2007), arXiv:hep-th/0601134. * [9]P. Lévay, _Stringy Black Holes and the Geometry of Entanglement_, Phys. Rev. D74:024030 (2006), arXiv:hep-th/0603136. * [10]M. Günaydin, A. Neitzke, B. Pioline, A. Waldron, _BPS black holes, quantum attractor flows and automorphic forms_, arXiv:hep-th/0512296. * [11]M. Günaydin, A. Neitzke, B. Pioline, _Topological Wave Functions and Heat Equations_, arXiv:hep-th/0607200 . * [12]B. Pioline, _Lectures on Black Holes, Topological Strings and Quantum Attractors_, arXiv:hep-th/0607227 . * [13]M. Günaydin, _Unitary Realizations of U-duality Groups as Conformal and Quasiconformal Groups and Extremal Black Holes of Supergravity Theories_, arXiv:hep-th/0502235 . * [14]S. Bellucci, S. Ferrara, M. Günaydin, A. Marrani _Charge Orbits of Symmetric Special Geometries and Attractors_, arXiv:hep-th/0606209. * [15]M. Günaydin, G. Sierra, and P. K. Townsend, _The geometry of $N=2$ Maxwell-Einstein supergravity and Jordan algebras_, Nucl. Phys. B242:244 (1984). * [16]M. Günaydin, G. Sierra, and P. K. Townsend, _Exceptional supergravity theories and the MAGIC square_, Phys. Lett. B133:72 (1983). * [17]M. Günaydin, G. Sierra, and P. K. Townsend, _Gauging the $d=5$ Maxwell-Einstein supergravity theories: More on Jordan algebras_, Nucl. Phys. B253:573 (1985). * [18]S. Ferrara and A. Marrani, $N=8$ _non-BPS attractors, fixed scalars and magic supergravities_, Nucl. Phys. B788:63 (2008), arXiv:0705.3866 [hep-th]. * [19]M. Rios, _Jordan $C^{\ast}$-Algebras and Supergravity_, arXiv:1005.3514 [hep-th]. * [20]S. Ferrara, M. Günaydin,_Orbits and Attractors for $N=2$ Maxwell-Einstein Supergravity Theories in Five Dimensions_, Nucl. Phys. B759:1-19 (2006), arXiv:hep-th/0606108. * [21]M. Bianchi, S. Ferrara, _Enriques and Octonionic Magic Supergravity Models_, JHEP 0802:054 (2008), arXiv:0712.2976 [hep-th]. * [22]S. Ferrara and M. Günaydin, _Orbits of exceptional groups, duality and BPS states in string theory_, Int. J. Mod. Phys.A13:2075 (1998), arXiv:hep-th/9708025. * [23]S. Ferrara and R. Kallosh, _Universality of Sypersymmetric Attractors_, Phys. Rev. D54:5344 (1996), arXiv:hep-th/9603090. * [24]J. Baez, _The Octonions_, Bull. Amer. Math. Soc. 39:145-205 (2002), arXiv:math/0105155 [math.RA]. * [25]H. Ruegg, _Octonionic Quark Confinement_, Acta. Phys. Pol. B9:1037 (1978). * [26]N. Jacobson, _Structure and Representations of Jordan Algebras_, American Mathematical Society Colloquium Publications v. 39, American Mathematical Society (1968). * [27]S. Krutelevich, _Jordan algebras, exceptional groups, and higher composition laws_, arXiv:math/0411104 [math.NT]. * [28]M. Günaydin, K. Koepsell, H. Nicolai, _Conformal and Quasiconformal Realizations of Exceptional Lie Groups_, Commun. Math. Phys. 221:57-76 (2001), arXiv:hep-th/0008063. * [29]T. Dray, C. Manogue, _The Exceptional Jordan Eigenvalue Problem_, IJTP 38:2901-2916 (1999), arXiv:math-ph/9910004. * [30]F. Gürsey, C. Tze, _On the role of division, Jordan and related algebras in particle physics_, World Scientific, Singapore (1996). * [31]M. Günaydin, O. Pavlyk, _Quasiconformal Realizations of $E_{6(6)}$, $E_{7(7)}$, $E_{8(8)}$ and $SO(n+3,m+3)$, $N=4$ and $N>4$ Supergravity and Spherical Vectors_, arXiv:0904.0784 [hep-th]. * [32]C. H. Bennett, S. Popescu, D. Rohrlich, J. Smolin, A. Thapliyal, _Exact and Asymptotic Measures of Multipartite Pure State Entanglement_, arXiv:quant-ph/9908073. * [33]W. Dür, G. Vidal, J. Cirac, _Three qubits can be entangled in two inequivalent ways_, Phys. Rev. A 62, 062314 (2000), arXiv:quant-ph/0005115. * [34]E. Briand, J. Luque, J. Thibon, F. Verstraete, _The moduli space of three qutrit states_, arXiv:quant-ph/0306122. * [35]P. Kurzynski, _Multi-Bloch Vector Representation of the Qutrit_, arXiv:0912.3155 [quant-ph]. * [36]A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, H. Weinfurter, _Elementary gates for quantum computation_, Phys. Rev. A52: 3457 (1995), arXiv:quant-ph/9503016. * [37]M. A. Nielsen, I. L. Chuang, _Quantum Computation and Quantum Information_, Cambridge University Press (2000). * [38]H. Terashima, M. Ueda, _Nonunitary quantum circuit_, Int. J. Quantum Inform. 3: 633-647 (2005), arXiv:quant-ph/0304061. * [39]A. T. Bolukbasi, T. Dereli, _On the SU(3) parametrization of qutrits_, arXiv:quant-ph/0511111. * [40]S. Catto, _Exceptional Projective Geometries and Internal Symmetries_, arXiv:hep-th/0302079. * [41]F. S. Khan, M. Perkowski, _Title: Synthesis of Ternary Quantum Logic Circuits by Decomposition_, Proceedings of the 7th International Symposium on Representations and Methodology of Future Computing Technologies (2005), arXiv:quant-ph/0511041. * [42]S. Ferrara, _BPS black holes, supersymmetry and orbits of exceptional groups_, Fortsch. Phys. 47: 159-165 (1999), arXiv:hep-th/9801095. * [43]J. P. Gauntlett, _Intersecting Branes_, arXiv:hep-th/9705011. * [44]G. Papadopoulos, P. K. Townsend, _Intersecting M-branes_, Phys. Lett. B380: 273-279 (1996), arXiv:hep-th/9603087. * [45]C. Manogue, T. Dray, _Octonions, E6, and Particle Physics_, arXiv:0911.2253 [math.RA]. * [46]A. A. Tseytlin, _Harmonic superpositions of M-branes_, Nucl. Phys. B475:149-163 (1996), arXiv:hep-th/9604035. * [47]N. Khviengia, Z. Khviengia, H. Lu, C. N. Pope, _Intersecting M-branes and bound states_, Phys.Lett.B388:21-28 (1996), arXiv:hep-th/9605077. * [48]H. Lu, C. N. Pope, T. R. Tran, K. W. Xu , _Classification of p-branes, NUTs, Waves and Intersections_, Nucl. Phys. B511:98-154 (1998), arXiv:hep-th/9708055. * [49]M. J. Duff, S. Ferrara, R. R. Khuri, J. Rahmfeld, _Supersymmetry and Dual String Solitons_, Phys. Lett. B356: 479-486 (1995), arXiv:hep-th/9506057. * [50]R. R. Khuri, _Supersymmetry, Duality and Bound States_, Talk given at Strings 96, arXiv:hep-th/9609094. * [51]J. Madore, _Noncommutative Geometry for Pedestrians_, Lecture given at the International School of Gravitation, Erice (1999), arXiv:gr-qc/9906059. * [52]L. Borsten, D. Dahanayake, M. J. Duff, S. Ferrara, A. Marrani, W. Rubens, _Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity_, Class. Quantum Grav. 27:185003 (2010), arXiv:1002.4223v2 [hep-th]. * [53]A. Hatcher, _Algebraic Topology_, Cambridge University Press (2002), download. * [54]O. Shukuzawa, _Explicit Classification of Orbits in Cayley Algebras over the Groups of Type $G_{2}$_, Yokohama Math. J. 53 (2007). * [55]L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, W. Rubens, _Four-qubit entanglement from string theory_, Phys.Rev.Lett.105:100507 (2010), arXiv:1005.4915 [hep-th]. * [56]L. Borsten, M.J. Duff, A. Marrani, W. Rubens, _On the Black-Hole/Qubit Correspondence_, arXiv:1101.3559 [hep-th] * [57]P. Lévay, _STU Black Holes as Four Qubit Systems_, Phy.Rev.D82:026003 (2010), arXiv:1004.3639 [hep-th]. * [58]K. Hasebe, _The Split-Algebras and Non-compact Hopf Maps_, arXiv:0905.2792 [math-ph]. * [59]M. Günaydin, C. Piron, H. Ruegg, _Moufang plane and octonionic quantum mechanics_, Comm.Math.Phys. 61, 1 (1978), 69-85, download. * [60]A. S. Darmawan, S. D. Bartlett, _Optical spin-1 chain and its use as a quantum computational wire_, Phys.Rev.A82:012328 (2010), arXiv:1004.3626 [quant-ph]. * [61]L. C. Venuti, C. D. E. Boschi, M. Roncaglia, _Qubit Teleportation and Transfer across Antiferromagnetic Spin Chains_, Phys.Rev.Lett.99:060401 (2007), arXiv:quant-ph/0703202. * [62]O. Bergman, C. B. Thorn, _String Bit Models for Superstring_, Phys.Rev.D52:5980-5996 (1995) arXiv:hep-th/9506125. * [63]E. Witten, _Perturbative Gauge Theory As A String Theory In Twistor Space_, Commun.Math.Phys.252:189-258 (2004), arXiv:hep-th/0312171. * [64]N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, J. Trnka, _Local Integrals for Planar Scattering Amplitudes_, arXiv:1012.6032 [hep-th].
1712.09593	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 30794, "num_imgs": 4, "llama3_tokens_count": 8989 }	[ "content_image/1712.09593/x1.png", "content_image/1712.09593/x2.png", "content_image/1712.09593/x3.png", "content_image/1712.09593/x4.png" ]	# Spin Filtering via Resonant Reflection of Relativistic Surface States I. A. Nechaev Centro de Física de Materiales CFM – MPC and Centro Mixto CSIC-UPV/EHU, 20018 San Sebastián/Donostia, Spain Tomsk State University, 634050 Tomsk, Russia Saint Petersburg State University, 198504 Saint Petersburg, Russia E. E. Krasovskii Departamento de Física de Materiales, Facultad de Ciencias Quíimicas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Apdo. 1072, San Sebastián/Donostia, 20080 Basque Country, Spain Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, San Sebastián/Donostia, 20018 Basque Country, Spain IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain February 29, 2024 ###### Abstract A microscopic approach is developed to scattering of surface states from a non-magnetic linear defect at a surface with strong spin-orbit interaction. Spin-selective reflection resonances in scattering of Rashba-split surface states by an atomic stripe are theoretically discovered in a proof-of-principle calculation for a model crystal potential. Spin-filtering properties of such linear defects are analyzed within an envelope-function formalism for a perturbed surface based on the Rashba Hamiltonian. The continuous Rashba model is found to be in full accord with the microscopic theory, which reveals the essential physics behind the scattering resonance. The spin-dependent reflection suggests a novel mechanism to manipulate spins on the nanoscale. Scattering of spin-orbit coupled electrons by extended defects arises in many spintronics-related phenomena, such as spin transport, accumulation, and filtering, which underlie the manipulation of spin currents in spin-based devices Wolf _et al._ (2001); Bercioux and Lucignano (2015). Furthermore, a detailed understanding of reflection and transmission of relativistic electrons is important for the unambiguous interpretation of scanning tunneling spectroscopy of spin-orbit split surface states El-Kareh _et al._ (2013); Schirone _et al._ (2015). Similar problem arises in ballistic transport through interfaces where powerful _ab initio_ methods exist for scattering of bulk electrons from surfaces, such as multiple scattering Henk _et al._ (2006), embedded Green-function Wortmann _et al._ (2002), or Bloch-waves formalism Krasovskii (2004). These methods are, however, not directly applicable to scattering of surface states because of a complicated structure of the incident and reflected waves in the asymptotic (unperturbed) region. Therefore, scattering of surface states has been considered either within a tight-binding scheme Kobayashi (2011) or within a ${\mathbf{k}}\!\cdot\!{\mathbf{p}}$ theory combined with continuity conditions for the envelope function G. Usaj and C. A. Balseiro (2005); Reynoso _et al._ (2006); Sablikov and Tkach (2007); Xie _et al._ (2016) (see also the application to spin dependent transport in nanowires Zhang _et al._ (2005); Alomar _et al._ (2016)). However, in the ${\mathbf{k}}\!\cdot\!{\mathbf{p}}$ method the smoothness of the envelope spinor function generally conflicts with current conservation Molenkamp _et al._ (2001), which leads to a qualitatively incorrect separation of the probability current into the spin-orbit and classical-momentum contributions Krasovskii (2014). The tight-binding formalism, on the other hand, is not well suited for free-electron-like motion along the surface. This calls for a more universal approach to scattering of two-dimensional (2D) states, which could be formulated in an _ab initio_ framework. We present a method to microscopically calculate the scattering of spin-orbit split 2D states from a linear (1D-periodic) defect. “Microscopic” means that the system is defined by the crystal potential $V({\mathbf{r}})$, and the wave functions satisfy the Schrödinger equation in real space. Therefore, the method can be straightforwardly transferred to _ab initio_ calculations. Here, we report a proof-of-principle calculation of the transmission of Rashba-split states through atomically thin defects. We study spin-filtering properties of the defects and discover spin-dependent reflection resonances for certain scatterers. Previous studies of the effect of spin-orbit coupling on the scattering of 2D states included multibeam spin-polarized reflection from a lateral barrier Govorov _et al._ (2004); Chen _et al._ (2005); Teodorescu and Winkler (2009), spin accumulation at the edges of semi-infinite systems G. Usaj and C. A. Balseiro (2005); Reynoso _et al._ (2006); Sonin (2010); Khaetskii and Sukhorukov (2013), spin selective refraction at an interface of two 2D media Khodas _et al._ (2004), spin dependent transmission of electrons incident from a non-relativistic medium through a barrier with spin-orbit coupling Ramaglia _et al._ (2004); Sablikov and Tkach (2007), and a semi-classical reflection from a smooth barrier Silvestrov and Mishchenko (2006). The above studies relied on an envelope-function description of the surface states using effective Hamiltonians. By contrast, here, the perturbed surface is treated fully microscopically: the scattering problem is reduced to a supercell band structure problem, which naturally involves both the propagating and all the required evanescent 2D waves and yields a detailed description of scattering, beyond the envelope-function picture. Still, the resonant properties of the scatterer can be related to the parameters of a ${\mathbf{k}}\!\cdot\!{\mathbf{p}}$ Rashba model for the perturbed surface. This demonstrates the generality of the phenomenon and suggests a way to its experimental realization. <figure><img src="content_image/1712.09593/x1.png"><figcaption>Figure 1: (a) CECs of Rashba split surface states. k±r, k±t, and k±i are theBloch vectors of reflected, transmitted, and incident waves. (b) Finite-thickness slab with a linear defect in the topmost layer. (c) Supercellgeometry: topmost layer with a repeated row of impurity atoms. The two boxesindicate two asymptotic regions. (d) Upper panel: CEC at E−E¯Γ=1.5 eV for12-fold supercell. Lower panel: dots are the dispersion E(Ky) of the solutionsΦKn for Kx=0. Shaded area shows the ky-projected states of the ideal surface.The spin-orbit split states due to the defect are highlighted blue (true boundstate) and red (resonance when inside the gray area). In the 1.5 eV CEC,arrows indicate the bound state (B) and the resonance (R). (e) Bloch vectorskx extracted from the eigenvalues exp(iτk) of the host lattice translationoperator for the three supercells for the well (ϵ=1.04, upper quadrant) andthe barrier (ϵ=0.96, lower quadrant). (f) Density profiles of four scatteringstates at E−E¯Γ=0.7 eV for γ=44, 51, 61, and 65∘ for the well ϵ=1.07 in a12-fold supercell. Dashed lines are their asymptotic representations in 0thand 1st cells continued up to the defect.</figcaption></figure> Typical constant energy contours (CEC) of Rashba-split states comprise two circles centered at $\bar{\Gamma}$ with spin oriented along ${\mathbf{k}}\times{\mathbf{n}}$ for the inner circle (of radius $R^{+}$) and along $-{\mathbf{k}}\times{\mathbf{n}}$ for the outer circle ($R^{-}$), where ${\mathbf{n}}$ is the surface normal and ${\mathbf{k}}$ is the 2D Bloch vector. We denote the unperturbed states by $\|\,{\mathbf{k}}^{\chi}_{\xi}\,\rangle$, where $\chi=\pm$ indicates chirality and $\xi=\rm r/t$ is the propagation direction along $x$, see Fig 1(a). Consider a defect created by substituting a row of atoms (along $y$ axis) by a different atom, Fig. 1(b). For a surface state $\|\,{\mathbf{k}}_{\rm i}^{\pm}\,\rangle$ incident from the left half-plane the scattering solution $\|\,\Psi\,\rangle$ far from the defect contains two transmitted $\|\,{\mathbf{k}}^{\pm}_{\rm t}\,\rangle$ and two reflected $\|\,{\mathbf{k}}^{\pm}_{\rm r}\,\rangle$ waves, Fig. 1(a). The crystal momentum along $y$ is conserved, so $k^{+}_{{\rm t}y}=k^{-}_{{\rm t}y}=k^{+}_{{\rm r}y}=k^{-}_{{\rm r}y}=k^{\chi}_{ {\rm i}y}$. For $R^{-}>k^{-}_{{\rm i}y}>R^{+}$ there is only one transmitted and one reflected wave. In the unperturbed region, $\|\,\Psi\,\rangle$ contains also evanescent waves, and depending on how fast they decay away from the defect the scattering state $\|\,\Psi\,\rangle$ can be obtained from band structure solutions $\Phi_{{\mathbf{K}}^{n}}$ for a smaller or larger supercell, Fig. 1(c). The method works as follows: At a given $k_{{\rm i}y}^{\chi}$, the surface states perturbed by the periodic defect give rise to four (or two) supercell eigenfunctions $\Phi_{{\mathbf{K}}^{n}}$ with the supercell crystal momentum $K_{y}^{n}=k^{\chi}_{{\rm i}y}$, $n=1,2,3,4$ ($n=1,2$ if $k^{-}_{{\rm i}y}>R^{+}$), see Fig. 1(d). Far from the scatterers [in the asymptotic region denoted 0th cell in Fig. 1(c)] the functions $\Phi_{{\mathbf{K}}^{n}}({\mathbf{r}})$ can obviously be decomposed into a sum of the four unperturbed surface states $\|\,{\mathbf{k}}_{\xi}^{\chi}\,\rangle$. The latter are obtained as eigenfunctions of the translation operator of the ideal surface $\hat{T}\Theta({\mathbf{r}})=\Theta({\mathbf{r}}+\bm{\tau})=\exp(i\bm{\tau}{ \mathbf{k}})\Theta({\mathbf{r}})$ in terms of the supercell solutions: $\Theta=\sum_{n}c^{n}\Phi_{{\mathbf{K}}^{n}}$. The functions $\Theta$ are defined everywhere in the crystal, and in the asymptotic region they coincide with the unperturbed surface states: $\Theta_{\xi}^{\chi}({\mathbf{r}})=\langle\,{\mathbf{r}}\,\|\,{\mathbf{k}}_{\xi} ^{\chi}\,\rangle$. The full scattering solution is then a linear combination $\Psi=\sum_{\chi\xi}a^{\chi}_{\xi}\Theta_{\xi}^{\chi}$ defined by the condition that $\Psi$ contain only one right-traveling wave in the 0th supercell and no left-traveling waves in the 1st supercell (the next asymptotic region), Fig. 1(c). Note that $\Psi$ is valid everywhere, including the vicinity of the defect. <figure><img src="content_image/1712.09593/x2.png"><figcaption>Figure 2: Transmission probability as a function of the angle of incidencefor the ϵ=0.96 (a), 1.04 (b), and 1.07 (c) for E−E¯¯¯Γ=1.5, 0.7, and 0.4 eV.The shades of red (blue) show T+ (T−) for 8, 10, and 12-fold supercells. Solidlines are the continuous model fit of T+ (red) and T− (blue) obtained withU=0.27, −0.33, and −0.55 eV for ϵ=0.96, 1.04, and 1.07, respectively. Theparameters α and m∗ for the presented energies are listed in Table 1.</figcaption></figure> Let us consider a 7-layer slab with the geometry of a Au(111) surface with an overlayer. The atoms are represented by a 3D regular muffin-tin potential, which is expanded in a truncated 3D Fourier series and included into the microscopic Hamiltonian ${\hat{p}}^{2}+V(\mathbf{r})+\beta{\bm{\sigma}}\cdot\left[\,\bm{\nabla}V({ \mathbf{r}})\times\mathbf{\hat{p}}\,\right]$, with $\beta$ scaled such that the Rashba splitting of the surface states be close to that in Au(111). The supercell band structure is calculated on a rectangular ${\mathbf{k}}$-mesh with $\Delta K_{x}=\Delta K_{y}=0.0056$ a.u.${}^{-1}$, and the functions $\Phi_{{\mathbf{K}}^{n}}$ for a given energy and $K_{y}$ are obtained by triangular interpolation. A typical constant energy contour is shown in the upper panel of Fig. 1(d). The artificial periodicity of the defect gives rise to spectral gaps, so for certain $K_{y}$ there are no solutions $\Phi$. However, for a given $K_{y}$ one can always choose a supercell for which the solution exists. We will consider two types of defects: barrier and well. For a barrier, the potential at the impurity site $U_{\rm D}$ is shallower than the potential $U_{\rm S}$ at the host atom, and for a well it is deeper. Computationally, the muffin-tin potentials $U_{\rm D}$ are linearly scaled: $U_{\rm D}(r)=\epsilon U_{\rm S}(r)$. The Bloch vectors of the unperturbed surface states extracted from the eigenvalues $\exp(i\bm{\tau}{\mathbf{k}})$ are shown in Fig. 1(e). The good agreement between the three supercells both for a well and for a barrier demonstrates that the evanescent waves are negligible in the asymptotic region already for the 8-fold supercell. Figure 1(f) shows the density profiles of the outer-circle surface states at $E-E_{\bar{\Gamma}}=0.7$ eV scattered by a well $\epsilon=1.07$ for four angles of incidence $\gamma$. Although the unperturbed surface states are derived from 0th cell, the asymptotic representation is seen to be valid over a much wider region (see, especially, $\gamma=44^{\circ}$). The transmission probability $T^{\pm}$ as a function of $\gamma$ is shown in Fig. 2 for a barrier, $\epsilon=0.96$, and for two wells, $\epsilon=1.04$ and 1.07. Here $T^{+}$ and $T^{-}$ stand for the incident wave in the inner and in the outer circle, respectively. The colored symbols are the microscopic calculations, with shades of red used for $T^{+}$ and blue for $T^{-}$. The voids in the curves correspond to the gaps in the supercell band structure, and in approaching the gap the transmission sometimes shows a spurious growth [see vertical arrows in Figs. 2(a) and 2(b)]. This happens when two of the Bloch vectors $K_{y}^{n}$ are close to the edge of the Brillouin zone, and the numerical method finds the two solutions $\Phi_{{\mathbf{K}}^{n}}$ linearly dependent. Such artifacts are recognized by an accuracy criterion, and they are easily sorted out because they occur at different angles for different supercells. E−E¯¯¯Γ (eV) \| m∗ (a.u.) \| α (a.u.) \| R−−R+ (a.u.−1) ---\|---\|---\|--- 0.0 \| 1.59 \| 0.033 \| 0.105 0.4 \| 1.85 \| 0.026 \| 0.096 0.7 \| 2.00 \| 0.020 \| 0.080 1.5 \| 2.00 \| 0.012 \| 0.048 Table 1: Rashba Hamiltonian parameters m∗ and α used to model the transmission through the defect, Fig 2. They are derived by fitting the dispersion E(k) of the unperturbed surface state of the microscopic calculation. m∗ and α depend on energy because E(k) is not exactly parabolic. Atomic Hartree units are used: ℏ=m0=e=1. Most important is the strikingly different behavior of the transmission probability $T^{-}$ for the two types of defects: for a barrier, $T^{-}$ steadily decreases, Fig. 2(a), whereas for a well it shows a sharp minimum followed by a maximum, see Figs. 2(b) and 2(c). By contrast, $T^{+}$ steadily decreases in both cases. To understand this behavior, let us consider the contribution of evanescent waves to the scattering states $\Psi$. Their weight can be inferred from the deviation of the density profile $\|\Psi\|^{2}$ in Fig. 1(f) from the left and right asymptotics (dashed lines) continued up to the scatterer. For small angles the weight of the evanescent waves is negligible, and it starts growing when $k^{-}_{{\rm i}y}$ exceeds $R^{+}$ because the evanescent waves replace the missing propagating solutions of the inner circle. This point manifests itself by a cusp maximum in $T^{-}$, e.g., at 0.4 eV around $\gamma=45^{\circ}$ in Fig. 2(a). In approaching the minimum [see $\gamma=61^{\circ}$ in Fig. 1(f)] the density around the defect steeply grows and then rapidly decreases with increasing $\gamma$. This happens because the defect causes a sharp perturbation of the potential $V({\mathbf{r}})$ (comparable to the lattice period), which is known to give rise to a bound state localized at the defect and energetically split off from the band continuum Madelung (1978). For the Rashba states that are bounded only from below a barrier does not produce any bound states. By contrast, a well-like perturbation produces two structures [Fig. 1(d)]: bound state B and its spin-orbit counterpart resonance R [highlighted red in Fig. 1(d)]. The hybridization of the incident wave of the outer circle with the resonance – the inner branch of the Rashba split 1D impurity state – gives rise to the asymmetric $T^{-}(\gamma)$ feature. In order to relate the reflection resonance to phenomenologically relevant spin-orbit characteristics of the material let us consider a Rashba system with the Hamiltonian $\hat{H}_{\mathrm{R}}=k^{2}/2m^{\ast}+\alpha(k_{y}\sigma_{x}-k_{x}\sigma_{y})$. The relation $k^{2}=k_{x}^{2}+k_{y}^{2}$ determines whether a given branch is propagating or evanescent for a given $k_{y}$ and $E$ Sablikov and Tkach (2007). The defect is represented by a potential barrier (well) $V(x)=U$ for $-l<x<l$, with $V(x)=0$ elsewhere. The width of the defect equals the width of the unit cell: $2l=\tau_{x}$. The scattering solution is found by the condition of the continuity of the spinor wave function and flux across the defect Molenkamp _et al._ (2001). Thus, the envelope-function formalism solves the problem without resorting to an artificial supercell. A four-wave representation in the perturbed region $\sum_{\chi\xi}d^{\chi}_{\xi}\|\,\tilde{\mathbf{k}}^{\chi}_{\xi}\,\rangle$ is matched to the wave function in the left and right half-planes at the boundaries $x=\pm l$ indicated in Fig 1(f). Here $\tilde{\mathbf{k}}$ are the wave vectors of the eigenfunctions of the Hamiltonian $\hat{H}_{\mathrm{R}}+U$ (with the same $\alpha$ and $m^{\ast}$ as for the unperturbed surface). The scattering problem then reduces to an $8\times 8$ matrix equation $\hat{M}\mathbf{a}=\mathbf{f}$ for the vector $\mathbf{a}=(r^{\pm},d^{\pm}_{\mathrm{r}},d^{\pm}_{\mathrm{t}},t^{\pm})^{ \mathrm{T}}$. Here $\hat{M}$ is the matching matrix, and $r^{\pm}$ and $t^{\pm}$ are the coefficients of the two reflected and two transmitted waves in the two unperturbed half-planes, see the legends in Fig. 1(f). The right-hand side $\mathbf{f}$ represents the incident wave $\|\,{\mathbf{k}}_{\rm i}^{\chi}\,\rangle$, and it has four non-zero components: the value and the flux (for both spins) at $x=-l$. <figure><img src="content_image/1712.09593/x3.png"><figcaption>Figure 3: (a) Electronic structure of the Rashba system with linear defect.Solid lines show the bound states split off from the Rashba continuum. Thewidth of the blurred red line shows the ky-width of the resonance. The Rashbacontinuum of the ideal system E±(k) [cf. grey area in Fig. 1(d)] is shown bythe sign of the ky-projected x-spin spectral density Stotx. Light redindicates Stotx>0, and light blue Stotx<0. (b) Band structure and side viewgeometry of a BiTeI trilayer with a nanostripe. Shaded area covers the ky-projected states of the clean trilayer. The localization of the trilayerstates on the atoms beneath the stripe are shown by red (σ↑x) and blue (σ↓x)fat bands. Light red (light blue) fat bands correspond to σ↑x (σ↓x) danglingbond states localized on the stripe.</figcaption></figure> The transmitted current $T^{\chi}=\|t^{+}_{\chi}\|^{2}+\|t^{-}_{\chi}\|^{2}$ is shown in Fig. 2 by solid lines. With $U$ adjusted to fit the microscopic calculations the Rashba model perfectly reproduces the shape of the curves and the dependence of the position and the width of the resonance on the energy and on the scatterer. Surprisingly, the envelope-function method originally designed for slowly varying potentials shows excellent performance for the atomic stripe. To establish the analogy with the microscopic picture, let us consider the eigenspectrum of the perturbed system. It is obtained by dropping the incident wave and finding zeros of real and imaginary part of the determinant of the matrix $\hat{M}$. In Fig. 3(a), the ideal surface is presented by the energy-momentum distribution of the sign ($\uparrow$ or $\downarrow$) of the $k_{y}$-projected $\sigma_{x}$-spin spectral density $S_{x}^{\rm tot}=S^{+}_{x}+S^{-}_{x}$, where $S^{\pm}_{x}(E,k_{y})=\int dk_{x}\left\langle\,\mathbf{k}^{\pm}\,\right\|\sigma_ {x}\|\,\mathbf{k}^{\pm}\,\rangle\,\delta[E-E^{\pm}({\mathbf{k})}]/8\pi^{2}$. The incident wave comes from the $\sigma_{x}^{\downarrow}$ continuum (blue area), which overlaps with the spectral resonance having $\sigma_{x}^{\uparrow}$ spin. Just at the resonance, $T^{-}(\gamma)$ sharply drops to zero and then steeply rises to unity, exactly as in the microscopic model, see Figs. 2(b) and 2(c). Thus, the scattering by the 1D defect is transparently expressed through the relation between the CECs of the host and the defect region. Here, an important ingredient is the spin non-conservation, so the effect does not occur, e.g., for Zeeman splitting. <figure><img src="content_image/1712.09593/x4.png"><figcaption>Figure 4: Transmission through barriers of width 2l=1 nm and 5 nm with U=0.27eV. Dashed lines show the partial currents \|t+χ\|2 (red) and \|t−χ\|2 (blue). Thecurrent carried by the wave of the opposite chirality to the incident wave isshown by the shaded areas.</figcaption></figure> For the above monoatomic stripes, practically all the transmitted current is carried by the same wave as is incident, so the spin orientation of the incident electron is preserved on transmission. The picture becomes very different for thicker stripes, in which the evanescent waves (complex $\tilde{k}_{x}$) do not participate in the transmission through the defect. The continuous Rashba model predicts that for barriers thicker than 1 nm the spin-flip transmission, i.e., $\|\,{\mathbf{k}}^{\mp}_{\rm t}\,\rangle$ for $\|\,{\mathbf{k}}^{\pm}_{\rm i}\,\rangle$, becomes quite important, see Fig. 4. Another vivid feature of the nanosized barrier are the Fabry-Pérot oscillations of the transmission. Finally, as a possible platform for the experimental realization of the discovered resonant reflection, we suggest the layered semiconductors of the BiTe$X$ ($X=$I, Br, Cl) family, where giant Rashba-split surface states reside in an absolute gap Ishizaka _et al._ (2011); Crepaldi _et al._ (2012); Eremeev _et al._ (2012); Sakano _et al._ (2013); Eremeev _et al._ (2013). Already a trilayer Te-Bi-$X$ – the easily exfoliated structure element of these semiconductors – provides the desired 2D spin-orbit split valence and conduction states Chen _et al._ (2013); Ma _et al._ (2014). For holes and electrons of a stand-alone trilayer, a perturbation can be introduced by putting on it a nanostripe as shown in the inset of Fig. 3(b) ¹. This linear defect gives rise to 1D spin-orbit split bound states that split off from the valence band, as follows from our _ab initio_ calculation ², see vertical arrows in Fig. 3(b). These states form the 1D Rashba channel that guides the holes. As seen in Fig. 3(b), the inner branch (red arrow) becomes the above-mentioned spectral resonance when it enters the projected continuum. [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] To summarize, we have developed a microscopic approach to scattering of relativistic surface states by a linear defect and found strong spin selectivity of electron transmission for well-like perturbations. Thereby, the transmitted spin current can be enhanced, which suggests a potential technique for non-magnetic spin filtering and spin injection. ###### Acknowledgements. This work was supported by the Spanish Ministry of Economy and Competitiveness MINECO (Project No. FIS2016-76617-P). I.A.N. also acknowledges support from Saint Petersburg State University (Grant No. 15.61.202.2015). ## References * Wolf _et al._ (2001)S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: A spin-based electronics vision for the future,” Science 294, 1488–1495 (2001). * Bercioux and Lucignano (2015)Dario Bercioux and Procolo Lucignano, “Quantum transport in Rashba spin–orbit materials: a review,” Reports on Progress in Physics 78, 106001 (2015). * El-Kareh _et al._ (2013)L. El-Kareh, P. Sessi, T. Bathon, and M. Bode, “Quantum interference mapping of Rashba-split Bloch states in $\mathrm{Bi}/\mathrm{Ag}(111)$,” Phys. Rev. Lett. 110, 176803 (2013). * Schirone _et al._ (2015)S. Schirone, E. E. Krasovskii, G. Bihlmayer, R. Piquerel, P. Gambardella, and A. Mugarza, “Spin-flip and element-sensitive electron scattering in the BiAg${}_{2}$ surface alloy,” Phys. Rev. Lett. 114, 166801 (2015). * Henk _et al._ (2006)J. Henk, A. Ernst, K. K. Saha, and P. Bruno, “Computing conductances of tunnel junctions by the Korringa-Kohn-Rostoker method: formulation and test of a Green function approach,” J. Phys.: Condensed Matter 18, 2601 (2006). * Wortmann _et al._ (2002)D. Wortmann, H. Ishida, and S. Blügel, “Embedded Green-function approach to the ballistic electron transport through an interface,” Phys. Rev. B 66, 075113 (2002). * Krasovskii (2004)E. E. Krasovskii, ‘‘Augmented-plane-wave approach to scattering of Bloch electrons by an interface,” Phys. Rev. B 70, 245322 (2004). * Kobayashi (2011)Katsuyoshi Kobayashi, “Electron transmission through atomic steps of Bi${}_{2}$Se${}_{3}$ and Bi${}_{2}$Te${}_{3}$ surfaces,” Phys. Rev. B 84, 205424 (2011). * G. Usaj and C. A. Balseiro (2005)G. Usaj and C. A. Balseiro, “Spin accumulation and equilibrium currents at the edge of 2DEGs with spin-orbit coupling,” Europhys. Lett. 72, 631–637 (2005). * Reynoso _et al._ (2006)A. Reynoso, Gonzalo Usaj, and C. A. Balseiro, “Spin hall effect in clean two-dimensional electron gases with Rashba spin-orbit coupling,” Phys. Rev. B 73, 115342 (2006). * Sablikov and Tkach (2007)V. A. Sablikov and Yu. Ya. Tkach, “Evanescent states in two-dimensional electron systems with spin-orbit interaction and spin-dependent transmission through a barrier,” Phys. Rev. B 76, 245321 (2007). * Xie _et al._ (2016)Hang Xie, Feng Jiang, and Wei E.I. Sha, “Numerical methods for spin-dependent transport calculations and spin bound states analysis in Rashba waveguides,” Computer Physics Communications 198, 118 – 127 (2016). * Zhang _et al._ (2005)Lebo Zhang, P. Brusheim, and H. Q. Xu, “Multimode electron transport through quantum waveguides with spin-orbit interaction modulation: Applications of the scattering matrix formalism,” Phys. Rev. B 72, 045347 (2005). * Alomar _et al._ (2016)M. I. Alomar, Llorenç Serra, and David Sánchez, “Interplay between resonant tunneling and spin precession oscillations in all-electric all-semiconductor spin transistors,” Phys. Rev. B 94, 075402 (2016). * Molenkamp _et al._ (2001)L. W. Molenkamp, G. Schmidt, and G. E. W. Bauer, “Rashba Hamiltonian and electron transport,” Phys. Rev. B 64, 121202 (2001). * Krasovskii (2014)E. E. Krasovskii, “Microscopic origin of the relativistic splitting of surface states,” Phys. Rev. B 90, 115434 (2014). * Govorov _et al._ (2004)A. O. Govorov, A. V. Kalameitsev, and John P. Dulka, “Spin-dependent transport of electrons in the presence of a smooth lateral potential and spin-orbit interaction,” Phys. Rev. B 70, 245310 (2004). * Chen _et al._ (2005)Hong Chen, J. J. Heremans, J. A. Peters, A. O. Govorov, N. Goel, S. J. Chung, and M. B. Santos, “Spin-polarized reflection in a two-dimensional electron system,” Applied Physics Letters 86, 032113 (2005). * Teodorescu and Winkler (2009)V. Teodorescu and R. Winkler, “Spin angular impulse due to spin-dependent reflection off a barrier,” Phys. Rev. B 80, 041311 (2009). * Sonin (2010)E. B. Sonin, “Edge spin accumulation: Spin Hall effect without bulk spin current,” Phys. Rev. B 81, 113304 (2010). * Khaetskii and Sukhorukov (2013)A. Khaetskii and E. Sukhorukov, “Unitarity of scattering and edge spin accumulation,” Phys. Rev. B 87, 075303 (2013). * Khodas _et al._ (2004)M. Khodas, A. Shekhter, and A. M. Finkel’stein, “Spin polarization of electrons by nonmagnetic heterostructures: The basics of spin optics,” Phys. Rev. Lett. 92, 086602 (2004). * Ramaglia _et al._ (2004)V. Marigliano Ramaglia, D. Bercioux, V. Cataudella, G. De Filippis, and C. A. Perroni, “Spin polarization of electrons with Rashba double-refraction,” J. Phys.: Condensed Matter 16, 9143 (2004). * Silvestrov and Mishchenko (2006)P. G. Silvestrov and E. G. Mishchenko, “Polarized electric current in semiclassical transport with spin-orbit interaction,” Phys. Rev. B 74, 165301 (2006). * Madelung (1978)Otfried Madelung, _Introduction to Solid-State Theory_ (Springer-Verlag, Berlin Heidelberg, 1978). * Ishizaka _et al._ (2011)K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonobe, K. Koizumi, S. Shin, H. Miyahara, A. Kimura, K. Miyamoto, T. Okuda, H. Namatame, M. Taniguchi, R. Arita, N. Nagaosa, K. Kobayashi, Y. Murakami, R. Kumai, Y. Kaneko, Y. Onose, and Y. Tokura, “Giant Rashba-type spin splitting in bulk BiTeI,” Nat. Mater. 10, 521–526 (2011). * Crepaldi _et al._ (2012)A. Crepaldi, L. Moreschini, G. Autès, C. Tournier-Colletta, S. Moser, N. Virk, H. Berger, Ph. Bugnon, Y. J. Chang, K. Kern, A. Bostwick, E. Rotenberg, O. V. Yazyev, and M. Grioni, “Giant ambipolar Rashba effect in the semiconductor BiTeI,” Phys. Rev. Lett. 109, 096803 (2012). * Eremeev _et al._ (2012)S. V. Eremeev, I. A. Nechaev, Yu. M. Koroteev, P. M. Echenique, and E. V. Chulkov, “Ideal two-dimensional electron systems with a giant Rashba-type spin splitting in real materials: Surfaces of bismuth tellurohalides,” Phys. Rev. Lett. 108, 246802 (2012). * Sakano _et al._ (2013)M. Sakano, M. S. Bahramy, A. Katayama, T. Shimojima, H. Murakawa, Y. Kaneko, W. Malaeb, S. Shin, K. Ono, H. Kumigashira, R. Arita, N. Nagaosa, H. Y. Hwang, Y. Tokura, and K. Ishizaka, “Strongly spin-orbit coupled two-dimensional electron gas emerging near the surface of polar semiconductors,” Phys. Rev. Lett. 110, 107204 (2013). * Eremeev _et al._ (2013)S. V. Eremeev, I. P. Rusinov, I. A. Nechaev, and E. V. Chulkov, “Rashba split surface states in BiTeBr,” New J. Phys. 15, 075015 (2013). * Chen _et al._ (2013)Y. L. Chen, M. Kanou, Z. K. Liu, H. J. Zhang, J. A. Sobota, D. Leuenberger, S. K. Mo, B. Zhou, S-L. Yang, P. S. Kirchmann, D. H. Lu, R. G. Moore, Z. Hussain, Z. X. Shen, X. L. Qi, and T. Sasagawa, “Discovery of a single topological Dirac fermion in the strong inversion asymmetric compound BiTeCl,” Nat. Phys. 9, 704–708 (2013). * Ma _et al._ (2014)Yandong Ma, Ying Dai, Wei Wei, Xinru Li, and Baibiao Huang, “Emergence of electric polarity in BiTeX (X = Br and I) monolayers and the giant Rashba spin splitting,” Phys. Chem. Chem. Phys. 16, 17603–17609 (2014). * (33)We consider a stripe of one-nanometer width made of a single BiTeI trilayer and with stable stoichiometric edges as in Ref. Eremeev _et al._ (2017). * (34)Our DFT-GGA calculations employed the full-potential linearized augmented-plane-wave method implemented in the FLEUR code, http://www.flapw.de. * Eremeev _et al._ (2017)S. V. Eremeev, I. A. Nechaev, and E. V. Chulkov, “Two- and three-dimensional topological phases in BiTe$X$ compounds,” Phys. Rev. B 96, 155309 (2017).
1301.0236	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 14296, "num_imgs": 0, "llama3_tokens_count": 3645 }	[]	# Mode Locking At and Below the CW Threshold Shai Yefet and Avi Pe’er${}^{}$ ###### Abstract We explore experimentally a new regime of operation for mode locking in a Ti:Sapphire laser with enhanced Kerr nonlinearity, where the threshold for pulsed operation is lowered below the threshold for continuous-wave (CW) operation. Even though a CW solution cannot exist in this regime, pulsed oscillation can be realized directly from zero CW oscillation. In this regime, the point of maximum strength of the Kerr nonlinear process provides a ”sweet spot” for mode locking, which can be optimized to considerably lower the pump power threshold. The properties of the ”sweet spot” are explained with a qualitative model. Department of physics and BINA Center of nano-technology, Bar-Ilan university, Ramat-Gan 52900, Israel ${}^{}$e-mail: avi.peer@biu.ac.il 140.3538, 140.3580, 140.7090 The ultra-broad gain bandwidth of the Ti:Sapphire (TiS) laser renders it the ’work-horse’ of the last decades for generation of ultrashort pulses by mode locking (ML) [1]. The nonlinear mechanism responsible for ML is self-focusing of the beam due to the optical Kerr effect within the TiS crystal, introducing an intensity dependent loss mechanism that favors pulses over continuous-wave (CW) operation [2]. A known feature of ML is the abrupt transition between CW and ML operation in terms of pump power [3]. Only when the pump power crosses a certain threshold, ML can be initiated from a noise-seeded fluctuation (either by a knock on a cavity element or by external injection of long pulses). Another common feature is that the threshold pump power for ML is higher than the CW threshold. It seems as if a certain amount of CW oscillations is necessary, and only on top of an existing CW can an intensity fluctuation be amplified to create the pulse. The threshold-like behavior of ML was elegantly explained by the theory of statistical light-mode dynamics (SLD) [4], where the transition from CW to ML is described as a first order phase transition, in which the order parameter (analogous to temperature) is $T\sim 1/(\gamma_{s}P^{2})$, where $\gamma_{s}$ represents the strength of the relevant nonlinearity and $P$ is the total laser cavity power. It was demonstrated [5] that on top of an existing CW, ML can occur only when $T$ is lowered below a critical value of $T_{c}$. Yet, to our knowledge, the question whether the preliminary existence of a CW oscillation is a necessary condition for ML operation was not directly explored and is not trivial to answer a priori. Here we demonstrate experimentally that an initial CW power need not exist. The addition of a second intracavity Kerr medium was explored in the past using different combinations of gain and Kerr media [6, 7, 8, 9, 10, 11], and was shown to lower the ML threshold. Here, we further enhance the nonlinear Kerr mechanism, observing a new regime of mode locking, where: 1. the intracavity CW power needed to initiate ML can be reduced to zero, 2. pulses can be sustained even below the CW threshold and the pump power necessary for pulsed operation can be considerably improved. Our linear TiS cavity is illustrated in Fig.1. By adding a lens based $1$x$1$ telescope between the curved mirrors the focus inside the TiS crystal is imaged towards mirror $M1$, allowing us to enhance the nonlinearity of the cavity in a controlled manner by introducing an additional Kerr medium near the imaged focus while varying its position. We first introduced a $3mm$ long planar window of BK7 glass, which was AR coated and set near normal incidence. As opposed to Brewster windows, where the beam expands in one dimension due to refraction, thereby reducing the nonlinear response and generating an astigmatic Kerr lens, with normal incidence the intra-cavity beam retains its small size, which enhances the nonlinearity and provides an astigmatic-free Kerr lens. [FIGURE:S0.F1][ENDFIGURE] If we define $\delta$ as a measure for the distance between $M1$ and $M2$ with respect to an arbitrary reference point, two separate bands of $\delta$ values $[\delta_{1},\delta_{2}],\ [\delta_{3},\delta_{4}]$ allow stable CW operation of the cavity. These two stability zones are bounded by four stability limits ($\delta_{4}>\delta_{3}>\delta_{2}>\delta_{1}$) and the working point for ML in our experiment is near the second stability limit $\delta_{2}$[2]. Near this limit the additional nonlinear Kerr lens causes a decrease of the mode size at the OC for ML. In order to favor ML, one can exploit the Kerr lens in one of two ways: either by placing an aperture near the OC to selectively induce loss on the CW mode, or by increasing the distance $\delta$ between the curved mirrors a little bit beyond the stability limit of $\delta_{2}$, which passively induces diffraction losses to the CW mode. For pulsed operation, the additional Kerr lens re-stabilizes the cavity, eliminating the diffraction losses. In this manner increasing the distance $\delta$ is equivalent to closing a physical aperture on the beam, which increases the threshold for CW due to loss, and requires higher power in ML for the nonlinear lens to overcome the loss. [FIGURE:S0.F2][ENDFIGURE] The measured ML and CW operation parameters are plotted in Fig.2 as a function of $Z=\delta-\delta_{2}$. Measurements were taken for two positions of the BK7 window: 1. at the imaged focus, where considerable nonlinearity is added by the window (in-focus). 2. several centimeters away, much beyond the Rayleigh range of the intracavity mode (off-focus), where the added nonlinearity is negligible and the cavity acts as a standard TiS cavity (with some additional material dispersion). Fig.2(a) plots the CW threshold and the ML threshold as a function of $Z$ for off-focus position. The ML threshold is defined as the minimum pump power required to initiate pulsed operation. As expected, the CW threshold increases with $Z$,due to increased diffraction losses. The ML threshold also increases (since higher power is needed for ML to overcome the loss), yet with a varying slope as $Z$ increases. Fig.2(b) plots the CW and ML intracavity powers at the ML threshold as a function of $Z$ for off-focus position. As typical for ML lasers, the ML threshold is always larger than the CW threshold and CW oscillation must exist to initiate the ML process. In addition, although ML operation is favorable over the entire range of $Z$, it is most favorable at the ”sweet spot” ($Z_{ss}\approx 1.1mm$) where the CW oscillation required to start the ML process reaches a minimal value. The same CW and ML parameters are plotted in Fig.2(c) and (d) for in-focus position. The ML threshold (Fig.2(c)) is reduced by the added nonlinearity, and the ML threshold curve eventually crosses the CW threshold at $Z_{c}\approx 1.2mm$ where the intracavity CW power drops to zero, marking the transition point to a different regime. At $Z_{c}$, ML can be achieved from pure fluorescence with no CW oscillation. The corresponding CW and ML intracavity powers at the ML threshold are shown in Fig.2(d). Beyond the crossing point ($Z>Z_{c}$), stable ML can still be initiated, but only by first raising the pump power up to the CW threshold, locking, and then lowering the pump again. At the CW threshold the pump power is too high and mode locking generates a pulse with a CW spike attached to it, which can be eliminated by lowering the pump power below the CW threshold. The ML threshold in Fig.2(c) for $Z>Z_{c}$ is the minimal pump power needed to maintain a clean pulse. We can understand the need to first increase the pump power to the CW threshold and than lower it by noting that the CW threshold marks the crossover between decay and amplification in the cavity. For ML to occur, an intensity fluctuation must first be linearly amplified to a sufficient peak power to initiate the Kerr-lensing mechanism. For $Z>Z_{c}$, one must pump the laser sufficiently for a noise-induced fluctuation to be amplified (rather than decay) in order for it to reach the peak intensity required to mobilize the Kerr-lensing process. After reducing the pump power to the ML threshold, a clean pulse operation is obtained, but if ML is broken the cavity will not mode-lock again. To investigate the appearance of the new regime ($Z>Z_{c}$) for in-focus window position, we plot the ratio of CW to ML powers $\gamma_{e}\!\equiv\!P_{CW}\!/\!P_{ML}$ as a function of $Z$, for windows of variable thickness (Fig.3). $\gamma_{e}$ represents an experimental measure for the strength of the Kerr effect, which demonstrates a ”sweet spot” where $\gamma_{e}$ is minimal and the nonlinear mechanism is most efficient. The apparent tendency from Fig.3 is that for increased nonlinearity, the sweet spot is pushed to larger $Z$ and the $\gamma_{e}$ value at the sweet spot is reduced. We find that for a $2mm$ thick window the $\gamma_{e}$ curve touches on zero near the sweet spot, marking the onset of the new regime. For a $3mm$ thick window the curve crosses zero at $Z\!=\!Z_{c}$. Well above $Z\!>\!Z_{c}$ pulsed operation becomes unstable, and we could not observe the reappearance of the $\gamma_{e}$ curve for larger values of $Z$. At every experimental point the prisms were adjusted to provide the broadest pulse bandwidth. The maximum bandwidth was obtained near the sweet spot due to the maximized Kerr strength, reaching $\approx\!100nm$ for all of the window thicknesses. This indicates that the bandwidth was limited mainly by high order dispersion of the prisms-mirrors combination, and not by the added dispersion of the windows. Although the pulse temporal width was not measured, we expect the pulses to be nearly transform limited based on previous experience with similar TiS cavities. [FIGURE:S0.F3][ENDFIGURE] To provide a qualitative model for the dynamics of the ”sweet spot” with increasing Kerr nonlinearity we examine a commonly used theoretical measure for the Kerr Strength: \[\gamma_{s}\equiv\frac{P_{c}}{\omega}\frac{d\omega}{dP},\] (1) where $\omega$ is the mode radius at the OC and $P$ is the pulse peak power normalized to the critical power for self-focusing $P_{c}$[12]. This Kerr strength, which represents the change of the mode size due to a small increase in the ML power $P$, is a convenient measure for mode-locking with an aperture near the OC. Yet, since increasing $Z$ beyond $\delta_{2}$ is equivalent to closing an aperture, $\gamma_{s}$ is useful also for our configuration. Usually, $\gamma_{s}$ is calculated at zero power ($P=0$) [13] to estimate the tendency of small fluctuations to develop into pulses. We note however that the dependence of $\gamma_{s}(P)$ on power is most important. Specifically, a large (negative) value for $\gamma_{s}$ indicates that only a small increase in the ML power (or threshold) will be necessary to overcome a small reduction of the aperture size (or increase in $Z$). The power where $\gamma_{s}$ is most negative will represent a sweet spot for mode locking. [FIGURE:S0.F4][ENDFIGURE] Figure 4 plots $\gamma_{s}(P)$ for no added Kerr window (TiS only) and for a $2mm$ long added window, demonstrating a clear minimum (sweet spot) on both curves. Furthermore, as Kerr material is added (enhanced Kerr strength), the minimum point is deepened and pushed towards higher power $P$ (larger $Z$), similar to the observed $\gamma_{e}$. Although $\gamma_{s}$ and $\gamma_{e}$ are somewhat different measures for the Kerr strength, the calculation of $\gamma_{s}$ provides reasoning for the measured behavior of the sweet spot for the different window thicknesses. In conclusion we note that the performance of the ML laser reported here at the critical distance $Z_{c}$, where the thresholds for pulsed and CW operation meet in Fig.2(c), can be compared to recently published record-results [14] that reported a mode-locked TiS laser with low pump power of $2.4W$ using an OC of 99% and curved mirrors radii of $R=8.6cm$ with output power of $30mW$ and intracavity power of $3W$. Here, we have achieved ML from zero CW oscillation with similar repetition rate ($\approx 80MHz$) using pump power of $2.3W$ with far less stringent conditions. In our experiment, the OC had only 95% reflectivity (5 times more losses), coupling more power out ($\approx 85mW$) with lower intracavity power of $1.7W$ and using curved mirrors of radius $R=15cm$. With further optimization of the added window material and cavity parameters, this may allow the development of ultra-low threshold ML sources in the future. This research was supported by the Israeli science foundation (grant 807/09). ## References * [1] H. A. Haus, IEEE J. Sel. Topics Quantum Electron. 6, 1173–1185 (2000). * [2] T. Brabec, P. F. Curley, C. Spielmann, E. Wintner, and A. J. Schmidt, J. Opt. Soc. Am. B 10, 1029–1034 (1993). * [3] H. Haken and H. Ohno, Opt. Commun. 26, 117–118 (1978). * [4] A. Gordon and B. Fischer, Opt. Commun. 223, 151–156 (2003). * [5] A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, and B. Fischer, Phys. Rev. Lett. 105, 013905 (2010). * [6] G. Gabetta, D. Huang, J. Jacobson, M. Ramaswamy, E. P. Ippen, and J. G. Fujimoto, Opt. Lett. 16, 1756–1758 (1991). * [7] G. P. A. Malcolm and A. I. Ferguson, Opt. Lett. 16, 1967–1969 (1991). * [8] G. W. Pearson, C. Radzewicz, and J. S. Krasinski, Opt. Commun. 94, 221–226 (1992). * [9] C. Radzewicz, G. W. Pearson, and J. S. Krasinski, Opt. Commun. 102, 464–468 (1993). * [10] M. A. Larotonda, A. A. Hnilo, and F. P. Diodati, Opt. Commun. 183, 485–491 (2000). * [11] R. Ell, U. Morgner, F. X. Kartner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, A. Boiko, and B. Luther-Davies, Opt. Lett. 26, 373–375 (2001). * [12] G. Fibich and A. L. Gaeta, Opt. Lett. 25, 335–337 (2000). * [13] V. Magni, G. Cerullo, and S. D. Silvestri, Opt. Commun. 101, 365–370 (1993). * [14] C. G. Durfee, T. Storz, J. Garlick, S. Hill, J. A. Squier, M. Kirchner, G. Taft, K. Shea, H. Kapteyn, M. Murnane, and S. Backus, Opt. Express 20, 13677–13683 (2012).
1612.06122	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35395, "num_imgs": 0, "llama3_tokens_count": 12093 }	[]	# Metric versus observable operator representation, higher spin models Andreas Fring and Thomas Frith Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK E-mail: a.fring@city.ac.uk, thomas.frith@city.ac.uk ###### Abstract: We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the nonlinear Ermakov-Pinney equation. † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction Standard quantum mechanics allows for many equivalent variants to describe the same physical observables. The well-known reason for this is that expectation values are computed from ambiguous quantities in which the individual components can be modified while the overall expression for the expectation values are left unchanged. Gauge transformations are prominent examples for such possible alterations. For time-dependent situations the well known equivalence between the Schrödinger and the Heisenberg picture allows to change from time-dependent states and time-independent operators to time-independent states and time-dependent operators, respectively. Recently we [1] argued that in time-dependent $\mathcal{PT}$-symmetric/quasi-Hermitian systems [2, 3, 4] another variant is possible in which the time-dependence is transferred from observables to metric operators. We will refer to the former as the _observable operator representation_ and the latter as the _metric representation_ by indicating the time-dependent object in the name of the representation. These physically equivalent representations are made possible in this setting as it always involves non-trivial metric operators on the non-Hermitian side. In [1] we demonstrated that the time-dependent Schrödinger equation (TDSE) for a time-dependent Hermitian Hamiltonian, $h(t)=h^{\dagger}(t)$, and the easier TDSE for a time-independent Hermitian Hamiltonian, $H\neq H^{\dagger}$, \[h(t)\phi(t)=i\hbar\partial_{t}\phi(t),\qquad\text{and\qquad}H\Psi(t)=i\hbar \partial_{t}\Psi(t)\] (1) may be treated equivalently. In the proposed scenario the Hermitian system is governed by a time-dependent Hamiltonian $h(t)$ and a standard time-independent metric operator $\mathbb{I}$, i.e. the unit operator, whereas the non-Hermitian system is characterized by the time-independent Hamiltonian $H$ and a non-standard time-dependent metric operator $\rho(t)$. The associated inner products in both systems are equivalent in the sense that \[\left\langle\phi(t)\right.\left\|\mathbb{I}\phi(t)\right\rangle=\left\langle \Psi(t)\right.\left\|\rho(t)\Psi(t)\right\rangle,\] (2) where the two wave functions $\phi(t)$ and $\Psi(t)$, solving the respective equation in (1), are connected by the time-dependent invertible Dyson operator $\eta(t)$ as \[\phi(t)=\eta(t)\Psi(t).\] (3) The metric operator in (2) and the Dyson operator in (3) are simply related as $\rho(t):=\eta^{\dagger}(t)\eta(t)$. Thus in this picture the time-dependence has been moved from the Hamiltonian in the Hermitian system to the metric operator in the non-Hermitian system. There are two central equations that serve to determine the quantities involved in the equations above. The first one, the time-dependent quasi-Hermiticity relation \[H^{\dagger}\rho(t)-\rho(t)H=i\hbar\partial_{t}\rho(t),\] (4) results by demanding that the time-evolution is unitary, that is the expectation values in (2) are preserved in time. Setting the time derivative of (2) to zero and using the TDSE (1) leads to (4). The second equations, the time-dependent Dyson relation \[h(t)=\eta(t)H\eta^{-1}(t)+i\hbar\partial_{t}\eta(t)\eta^{-1}(t),\] (5) is obtained by substituting (3) into (1). It was noted some time ago [5, 6, 7, 8, 9] that as a consequence of the Dyson relation (5) the Hamiltonian satisfying the TDSE (1) is not observable¹, since observables $\mathcal{O}$ in the non-Hermitian system need to be quasi-Hermitian, meaning they have to be related to a corresponding observable $o$, i.e. a self-adjoint operator, in the Hermitian system as $o(t)=\eta(t)\mathcal{O}(t)\eta^{-1}(t)$. The non-observability is also a feature when the Hamiltonian is explicitly time-dependent, i.e. even for $H\to H(t)$. Furthermore, this implies that $H$ is not the operator that characterizes the energy but instead the operator [FOOTNOTE:1][ENDFOOTNOTE] \[\tilde{H}(t)=\eta^{-1}(t)h(t)\eta(t)=H+i\hbar\eta^{-1}(t)\partial_{t}\eta(t),\] (6) that does not satisfy the TDSE, and is therefore by definition not a Hamiltonian, is the energy operator in the non-Hermitian system. The relation between the expectation values in the different systems is easily verified to be \[\left\langle\phi(t)\right.\left\|h(t)\phi(t)\right\rangle=\left\langle\Psi(t) \left\|\rho(t)\tilde{H}(t)\Psi(t)\right\rangle\right.,\] (7) supporting the above statement. As demonstrated in [1] unitary time-evolution operators $u(t,t^{\prime})$ and $U(t,t^{\prime})$ that evolves a state as $\phi(t)=u(t,t^{\prime})\phi(t^{\prime})$ or $\Psi(t)=U(t,t^{\prime})\Psi(t^{\prime})$, respectively, from a time $t^{\prime}$ to $t$ may also be constructed when $\phi(t)$ and $\Psi(t)$ have been obtained. Since the equations (4) and (5) describe highly overdetermined systems it is a priori not evident whether they possess any solutions at all and if they do whether they are meaningful. Remarkably such solutions do exist and can be found as was demonstrated for time-dependent [9, 11] and time-independent Hamiltonians [1]. Here we provide further solutions, focussing on the limitations and in particular on the different solution procedures. In [1] we pursued the following process: Starting from a given a non-Hermitian Hamiltonian $H$ we solved the time-dependent quasi-Hermiticity relation (5) first, which seems most natural as it only involves one unknown quantity, namely $\rho(t)$. Assuming the Dyson operator $\eta(t)$ to be Hermitian, it can in principle be computed from $\rho(t)$ by taking its square root. Subsequently one may compute the Hermitian counterpart $h(t)$ by direct evaluation of the right hand side of the time-dependent Dyson relation (5). As we will demonstrate in more detail below, taking the square root in this case can be rather awkward and to avoid this step we pursue here a different approach by solving the time-dependent Dyson relation first. As we will see, this is more efficient, but evidently requires some initial guess about the structure of the Hermitian Hamiltonian. The models we consider here are slightly modified versions of the lattice Yang-Lee model [12, 13] \[H_{N}^{s}=-\frac{1}{2}\sum_{j=1}^{N}(c_{y}S_{j}^{y}+\omega c_{\omega}\vec{S}_{ j}\cdot\vec{S}_{j+1}+ic_{x}\gamma S_{j}^{x}),\quad\omega,\gamma,c_{x},c_{y},c_ {\omega}\in\mathbb{R},\] (8) where we allow for higher spin representations for the matrices $S_{j}^{x}$, $S_{j}^{y}$, $S_{j}^{z}$ at cite $j$ labelled by $s$. Our model parameters are $\omega,\gamma\in\mathbb{R}$ and the constants $c_{x},c_{y},c_{\omega}$ are conveniently adjusted for the particular representations. Here we will consider the one-site models and attempt, in analogy to the study in [1], to map the non-Hermitian Hamiltonians to Hermitian Hamiltonians of the form \[h(t)=-\frac{1}{2}\left[\omega\mathbb{I}+\chi(t)S_{z}\right],\] (9) where initially $\chi(t)$ is an arbitrary unknown function of $t$. It turns out that in all spin models considered the time-dependent function $\chi(t)$ is restricted to obey an equation that can be converted easily into the nonlinear Ermakov-Pinney equation. ## 2 A solvable equivalence pair of spin 1/2 models The simplest version of $H_{N}^{s}$ is the one-site spin 1/2 model. Taking the matrices $S_{j}^{x}$, $S_{j}^{y}$, $S_{j}^{z}$ simply to be the standard Pauli spin matrices $\sigma_{j}^{x}$, $\sigma_{j}^{y}$, $\sigma_{j}^{z}$ and adjusting the constants $c_{x}=c_{y}=1$, $c_{\omega}=1/3$, the Hamiltonian (8) acquires the form \[H_{1}^{1/2}=-\frac{1}{2}(\sigma_{y}+\frac{\omega}{3}~{}\vec{\sigma}\cdot\vec{ \sigma}+i\gamma\sigma_{x})=-\frac{1}{2}\left(\begin{array}[]{cc}\omega&i( \gamma-1)\\ i(\gamma+1)&\omega\end{array}\right).\] (10) The corresponding TDSE (1) is easily solved by \[\Psi_{\pm}(t)=\left(\begin{array}[]{c}\pm i(1-\gamma)\\ \phi\end{array}\right)e^{-itE_{\pm}},~{}~{}~{}\qquad E_{\pm}=-\frac{\omega}{2} \pm\frac{\phi}{2},\] (11) where $\phi:=\sqrt{1-\gamma^{2}}$. Thus, this model exhibits the typical feature for $\mathcal{PT}$-symmetric/quasi-Hermitian systems [3, 4] that despite being described by a non-Hermitian Hamiltonian there exists a range for the model parameters, in this case $\left\|\gamma\right\|\leq 1$, for which the eigenvalue spectrum is real. Next we will solve the time-dependent Dyson relation (5) and the time-dependent quasi-Hermiticity relation (4) in more detail and compare the advantages of one approach over the other. ### Solutions of the time-dependent quasi Hermiticity relation Assuming the time-dependent metric operator to be Hermitian we take it to be in the most generic form \[\rho(t)=\left(\begin{array}[]{cc}\rho_{1}(t)&\rho_{2}(t)-i\rho_{3}(t)\\ \rho_{2}(t)+i\rho_{3}(t)&\rho_{4}(t)\end{array}\right),\] (12) with unknown real functions $\rho_{i}$, $i=1,\ldots,4$. Substituting this Ansatz into (4) and reading off the real and imaginary parts in each matrix entry yields the four constraining first order differential equations \[\dot{\rho}_{1}=(1+\gamma)\rho_{2},\qquad\dot{\rho}_{2}=\rho_{1}\frac{\gamma-1} {2}+\rho_{4}\frac{\gamma+1}{2},\qquad\dot{\rho}_{3}=0,\qquad\dot{\rho}_{4}=( \gamma-1)\rho_{2}.\qquad\] (13) As common we adopt the convention to indicate derivatives with respect to time by an overdot. The general solution to these equations is easily obtained as \[\rho_{1}(t)=\frac{1+\gamma}{\phi}\Gamma_{b_{2}}^{-b_{1}}+b_{4},\quad\rho_{2}(t )=\Gamma_{b_{1}}^{b_{2}},\quad\rho_{3}(t)=b_{3},\quad\rho{}_{4}(t)=\frac{1- \gamma}{\phi}\Gamma_{-b_{2}}^{b_{1}}+\frac{1-\gamma}{1+\gamma}b_{4},\] (14) where we abbreviate $\Gamma_{x}^{y}:=x\sin(\phi t)+y\cos(\phi t)$ and introduced the real integration constants $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$. To find (14) we just need to solve a harmonic oscillator equation obtained from computing $\ddot{\rho}_{2}$ and the subsequent use the expressions for $\dot{\rho}_{1}$, $\dot{\rho}_{4}$. Once $\rho_{2}$ is known the remaining integrals are simply of first order. In principle, we can take now the square root by diagonalizing $\rho$ first as $\rho=UDU^{-1}$ and subsequently computing $\sqrt{\rho}=UD^{1/2}U^{-1}$. This is indeed feasible as shown in [1], but even for simple $2\times 2$-matrices it involves relatively lengthy expressions and requires specific choices for the constants in order to guarantee that the eigenvalues are all real. This is also the case for the model considered here as seen from the determinant of $\rho$ \[\det\rho=\frac{1-\gamma}{1+\gamma}b_{4}^{2}-b_{1}^{2}-b_{2}^{2}-b_{3}^{2}.\] (15) Evidently this expression might become negative, so that at least one of the two eigenvalues of $\rho$ would be negative and even for choices for which $\det\rho>0$ we may have two negative eigenvalues. We will not carry out this step here, but instead follow an easier way to find $\eta$ from (5) and compare thereafter with the solution (14). ### Solutions of the time-dependent Dyson relation In order to solve the time-dependent Dyson relation we need to make some pre-assumptions about the Hermitian Hamiltonian $h(t)$ and the map $\eta(t)$. We take $h(t)$ to be of the form as specified in (9) with $S_{z}=\sigma_{z}$ and assume $\eta(t)$ to be of the most generic Hermitian form \[\eta(t)=\left(\begin{array}[]{cc}\eta_{1}(t)&\eta_{2}(t)-i\eta_{3}(t)\\ \eta_{2}(t)+i\eta_{3}(t)&\eta_{4}(t)\end{array}\right),\] (16) Taking this Ansatz into (5) leads to seven different constraining equations (17) Even though this system of equations is overdetermined, it can be solved by \[\eta_{1}(t)=\frac{c_{1}(\gamma+1)}{\chi^{1/2}(\gamma-1)},\quad\eta_{2}(t)= \frac{c_{1}\dot{\chi}}{\chi^{3/2}(\gamma-1)},\quad\eta_{3}(t)=\frac{c_{1}\chi^ {1/2}}{\gamma-1},\quad\eta_{4}(t)=\frac{c_{1}}{\chi^{1/2}},\] (18) with one integration constant $c_{1}\in\mathbb{R}$ provided that the function $\chi$ satisfies the second order nonlinear differential equation \[\ddot{\chi}-\frac{3}{2}\frac{\dot{\chi}^{2}}{\chi}-\frac{1}{2}\phi^{2}\chi+ \frac{1}{2}\chi^{3}=0.\] (19) Using the variable transformation $\chi=2/\sigma^{2}$ this equation is converted into the Ermakov-Pinney equation [14, 15] \[\ddot{\sigma}+\frac{1}{4}\phi^{2}\sigma=\frac{1}{\sigma^{3}},\] (20) which is ubiquitous in the context of the TDSE, e.g. [16, 17, 18, 19, 20, 21], and also some quantization schemes [22, 23]. The general solution to this equation is known to be \[\sigma(t)=\left[A\sin^{2}(\phi t/2)+B\cos^{2}(\phi t/2)\pm 2C\sin(\phi t/2) \cos(\phi t/2)\right]^{1/2},\] (21) where the constants $A$, $B$ and $C$ are constraint as $AB-C^{2}=4/\phi^{2}$, see [24]. Transforming back to $\chi$ and introducing the new real constants $c_{2}$ and $c_{3}$ via the relations $A=2(-c_{3}\pm\sqrt{1+c_{2}^{2}+c_{3}^{2}})/\phi$ and $B=2(c_{3}\pm\sqrt{1+c_{2}^{2}+c_{3}^{2}})/\phi$, we obtain the general solution to (19) in the form \[\chi(t)=\frac{\phi}{c_{2}\sin(\phi t)+c_{3}\cos(\phi t)\pm\sqrt{1+c_{2}^{2}+c_ {3}^{2}}}.\] (22) Thus with (18) and (16) we have obtained a generic solution for $\eta$. Let us now compare this with the solution of the time-dependent quasi Hermiticity relation obtained in the previous subsection. Computing $\eta^{2}$ from the above expressions and identifying the result as $\rho$ we can compare with the solution (14) obtained previously. Matching the constants as \[b_{1}=-\frac{2c_{3}\gamma c_{1}^{2}}{(1-\gamma)^{2}},~{}~{}b_{2}=\frac{2c_{2} \gamma c_{1}^{2}}{(1-\gamma)^{2}},~{}~{}b_{3}=\frac{2\gamma c_{1}^{2}}{(1- \gamma)^{2}},~{}~{}b_{4}=\frac{2\phi c_{1}^{2}\sqrt{1+c_{2}^{2}+c_{3}^{2}}}{(1 -\gamma)^{3}},\] (23) the two solutions become identical. Evidently these constants could not have been guessed in the approach of the previous subsection. With these values the determinant becomes \[\det\rho=\frac{4(1+\gamma)c_{1}^{4}}{(1-\gamma)^{3}}(1+c_{2}^{2}+c_{3}^{2}),\] (24) which is positive for $\left\|\gamma\right\|\leq 1$. From the above it is clear that it is far easier to solve (5) directly, as it can essentially be reduced to some algebraic manipulations, a simple integration and the Ermakov-Pinney equation for which the general solution is known. We have now obtained all the ingredients to compute the solution to the TDSE for the Hermitian system from (3). Assembling our results we obtain from $\phi_{\pm}(t)=\sqrt{\mathcal{N}_{\pm}}\eta(t)\Psi_{\pm}(t)$ the normalized eigenvectors \[\phi_{\pm}(t)=c_{1}\sqrt{\mathcal{N}_{\pm}\chi}\left(\begin{array}[]{c}\frac{1 +\gamma}{\phi}i\left[e^{\pm it(E_{+}-E_{-})}(ic_{2}\mp c_{3})+1\mp\sqrt{1+c_{2 }^{2}+c_{3}^{2}}\right]\\ e^{\pm it(E_{+}-E_{-})}(\pm ic_{2}+c_{3})\pm 1-\sqrt{1+c_{2}^{2}+c_{3}^{2}} \end{array}\right)e^{-itE_{\pm}},\] (25) with normalization factors \[\mathcal{N}_{\pm}=\frac{1-\gamma}{\mp 4c_{1}^{2}\phi\left(\gamma\mp\sqrt{1+c_{ 2}^{2}+c_{3}^{2}}\right)}.\] (26) Form this we compute the expectation values \[\mathcal{N}_{\pm}\left\langle\Psi_{\pm}(t)\right.\left\|\rho(t) \Psi_{\pm}(t)\right\rangle = \left\langle\phi_{\pm}(t)\right.\left\|\phi_{\pm}(t)\right\rangle=1,\] (27) \[\mathcal{N}_{\pm}\left\langle\Psi_{\pm}(t)\right.\left\|\rho(t) \Psi_{\mp}(t)\right\rangle = \left\langle\phi_{\pm}(t)\right.\left\|\phi_{\mp}(t)\right\rangle= \frac{\gamma\left(\pm c_{3}+ic_{2}\right){}}{\sqrt{\phi^{2}+c_{2}^{2}+c_{3}^{2 }}},\] (28) which confirm that the time-evolution is indeed unitary. We also confirm the validity of the relation for the energy expectations (7) by computing (29) While we found some explicit solutions, this example also demonstrates that one can not map to any arbitrary given target Hamiltonian, as $\chi(t)$ is restricted by the nonlinear equation (19). ## 3 A solvable equivalence pair of spin 1 models Increasing the dimension of the spin representation poses a more difficult challenge, but as we will see many of the features we observed for the spin 1/2 model will survive. Let us next consider a generalization of the previous model to a spin 1 model where the matrices $S_{j}^{x}$, $S_{j}^{y}$, $S_{j}^{z}$ in (8) are taken to be the standard $3\times 3$-spin 1 matrices \[S^{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\ 1&0&1\\ 0&1&0\end{array}\right),\quad\quad S^{y}=\frac{1}{\sqrt{2}}\left(\begin{array} []{ccc}0&-i&0\\ i&0&-i\\ 0&i&0\end{array}\right),\quad\quad S^{z}=\left(\begin{array}[]{ccc}1&0&0\\ 0&0&0\\ 0&0&-1\end{array}\right).\] (30) Choosing the constants $c_{x},c_{y},c_{\omega}$ conveniently this Hamiltonian simplifies for $N=1$ to \[H_{1}^{1}=-\frac{1}{\sqrt{2}}(S^{y}+\frac{\omega}{\sqrt{2}}\mathbb{I}+i\gamma S ^{x})=-\frac{1}{2}\left(\begin{array}[]{ccc}\omega&i(\gamma-1)&0\\ i(\gamma+1)&\omega&i(\gamma-1)\\ 0&i(\gamma+1)&\omega\end{array}\right).\] (31) The corresponding TDSE (1) is solved by \[\Psi_{k}(t)=\left(\begin{array}[]{c}(-1)^{k}(1-\gamma)\\ 2ik\tilde{\phi}\\ 1-\gamma\end{array}\right)e^{-itE_{k}},~{}~{}~{}\qquad E_{k}=-\frac{\omega}{2} +k\tilde{\phi},\quad~{}~{}~{}k=0,\pm 1\] (32) where $\tilde{\phi}:=\sqrt{(1-\gamma^{2})/2}$. Once again in the parameter region $\left\|\gamma\right\|\leq 1$ the non-Hermitian Hamiltonian (31) possesses a real eigenvalue spectrum. Next we solve (4) and (5). ### Solutions of the time-dependent quasi Hermiticity relation Assuming the time-dependent metric operator to be Hermitian we substitute the most generic Ansatz \[\rho(t)=\left(\begin{array}[]{ccc}\rho_{1}(t)&\rho_{2}(t)-i\rho_{3}(t)&\rho_{4 }(t)-i\rho_{5}(t)\\ \rho_{2}(t)+i\rho_{3}(t)&\rho_{6}(t)&\rho_{7}(t)-i\rho_{8}(t)\\ \rho_{4}(t)+i\rho_{5}(t)&\rho_{7}(t)+i\rho_{8}(t)&\rho_{9}(t)\end{array}\right).\] (33) into the time-dependent quasi Hermiticity relation (4) obtaining in principle $18$ equation for the nine real functions $\rho_{i}(t),$$i=1,\ldots,9$. Excluding vanishing and related ones we are left with nine equations \[\begin{array}[]{l}\dot{\rho}_{1}=\rho_{2}(\gamma+1),\quad\dot{\rho}_{2}=\rho_{ 1}\frac{\gamma-1}{2}+(\rho_{4}+\rho_{6})\frac{\gamma+1}{2},\quad\dot{\rho}_{3} =\rho_{5}\frac{\gamma+1}{2},\\ \dot{\rho}_{4}=\rho_{2}\frac{\gamma-1}{2}+\rho_{7}\frac{\gamma+1}{2},\quad\dot {\rho}_{5}=\rho\frac{\gamma-1}{2}+\rho_{8}\frac{\gamma+1}{2},\quad\dot{\rho}_{ 6}=\rho_{2}(\gamma-1)+\rho_{7}(\gamma+1)\begin{array}[]{c}\\ \\ \end{array}\\ \dot{\rho}_{7}=(\rho_{4}+\rho_{6})\frac{\gamma-1}{2}+\rho_{9}\frac{\gamma+1}{2 },\quad\dot{\rho}_{8}=\rho_{5}\frac{\gamma-1}{2},\quad\dot{\rho}_{9}=\rho_{7}( \gamma-1).\end{array}\] (34) Once again as in the spin $1/2$ case we have as many equations as unknown functions and it is straightforward to solve these equations, as substitutions lead to simple integrals. We find the solutions \[\begin{array}[]{l}\rho_{1}(t)=\frac{\left(2b_{4}+3b_{5}\right)(\gamma+1)}{8(1- \gamma)}+\tilde{\Gamma}_{b_{6}}^{b_{7}}+\breve{\Gamma}_{b_{8}}^{b_{9}},\quad \quad\rho_{2}(t)=\frac{\phi}{1+\gamma}\left[\tilde{\Gamma}_{-b_{7}}^{b_{6}}+2 \breve{\Gamma}_{-b_{9}}^{b_{8}}\right],\\ \rho_{3}(t)=\frac{1+\gamma}{2\phi}\left[\tilde{\Gamma}_{b_{2}}^{-b_{1}}\right] +b_{3},\quad\quad\rho_{4}(t)=\frac{\gamma-1}{\gamma+1}\breve{\Gamma}_{b_{8}}^{ b_{9}}+\frac{1}{8}(6b_{4}+b_{5}),\quad\quad\rho_{5}(t)=\tilde{\Gamma}_{b_{1}}^ {b_{2}},\\ \rho_{6}(t)=2\frac{\gamma-1}{\gamma+1}\breve{\Gamma}_{b_{8}}^{b_{9}}-\frac{1}{ 4}(2b_{4}-b_{5}),\quad\quad\rho_{7}(t)=\frac{1}{\sqrt{2}}\left(\frac{1-\gamma} {1+\gamma}\right)^{3/2}\left[\tilde{\Gamma}_{-b_{7}}^{b_{6}}+2\breve{\Gamma}_{ b_{9}}^{-b_{8}}\right],\\ \rho_{8}(t)=\frac{\phi}{1+\gamma}\tilde{\Gamma}_{-b_{2}}^{b_{1}}+\frac{1- \gamma}{1+\gamma}b_{3},\quad\quad\rho_{9}(t)=\left(\frac{1-\gamma}{1+\gamma} \right)^{2}\left[-\tilde{\Gamma}_{b_{6}}^{b_{7}}+\breve{\Gamma}_{b_{8}}^{b_{9} }\right]+\frac{1-\gamma}{8(1+\gamma)}(2b_{4}+3b_{5}),\end{array}\] (35) with nine integration constants $b_{i},$$i=1,\ldots,9$. We abbreviated $\tilde{\Gamma}_{x}^{y}:=x\sin(\tilde{\phi}t)+y\cos(\tilde{\phi}t)$ and $\breve{\Gamma}_{x}^{y}:=x\sin(2\tilde{\phi}t)+y\cos(2\tilde{\phi}t)$. For this solution it is even less evident to chose suitable constants and simplifying choices by setting some of the $b_{i}$ to zero usually yield negative eigenvalues for $\rho$. Thus we will not compute the root, but return to this solution below for comparison. ### Solutions of the time-dependent Dyson relation Instead we solve the time-dependent Dyson equation (5). We assume a similar form for our Hermitian target Hamiltonian as in (9) and take $S^{z}$ to be a spin 1 matrix, denote $\chi=X$ and take $\eta(t)$ to be of the Hermitian form \[\eta(t)=\left(\begin{array}[]{ccc}\eta_{1}(t)&\eta_{2}(t)-i\eta_{3}(t)&\eta_{4 }(t)-i\eta_{5}(t)\\ \eta_{2}(t)+i\eta_{3}(t)&\eta_{6}(t)&\eta_{7}(t)-i\eta_{8}(t)\\ \eta_{4}(t)+i\eta_{5}(t)&\eta_{7}(t)+i\eta_{8}(t)&\eta_{9}(t)\end{array}\right).\] (36) Substituting these expressions into the time-dependent Dyson equation (5) yields in principle $18$ equation for the real functions $\eta_{i}(t),$$i=1,\ldots,9$. We obtain (37) and \[(1+\gamma)\eta_{3}-X\eta_{1}=(1-\gamma)\eta_{3}+(1+\gamma)\eta_{8}=(1-\gamma) \eta_{8}+X\eta_{9}=0.\] (38) Unlike the system of equations for the metric operator this set is overdetermined. Nonetheless, they may be solved by (39) where $X(t)$ is restricted to obey the second order non-linear differential equation \[\ddot{X}-\frac{3}{2}\frac{\dot{X}^{2}}{X}-\frac{1}{2}\tilde{\phi}^{2}X+\frac{X ^{3}}{8}=0.\] (40) This equations closely resembles (19) and we can once more transform it to the Ermakov-Pinney equation (20) with $\sigma\rightarrow\tilde{\sigma}$, $\phi\rightarrow\tilde{\phi}$ by using $X=4/\tilde{\sigma}^{2}$ in this case. Following the same steps of the previous subsection we obtain the general solution for (40) as \[X(t)=\frac{2\tilde{\phi}}{c_{2}\sin(\tilde{\phi}t)+c_{3}\cos(\tilde{\phi}t)\pm \sqrt{1+c_{2}^{2}+c_{3}^{2}}}.\] (41) Again we compute $\eta^{2}$ and compare the result with $\rho$ from the previous subsection. Identifying the constants as (42) the two solutions coincide. This demonstrates once more why simple choices for the constants $c_{i}$ did not yield meaningful solutions for $\eta$. Using the values (42) we compute the determinant \[\det\rho=8\frac{(1-\gamma)^{3}}{(1+\gamma)^{9}}c_{1}^{6}(1+c_{2}^{2}+c_{3}^{2} )^{3},\] (43) which is positive for the parameter range of interest. Having computed the Dyson map and the solution to the TDSE for $H$ we obtain the solution for the TDSE involving $h(t)$ from (3) \[\begin{array}[]{l}\phi_{\pm}(t)=\left(\begin{array}[]{c}-X(t)\left[1\mp\sqrt{1 +c_{2}^{2}+c_{3}^{2}}+e^{it(E_{\pm}-E_{0})}(ic_{2}\mp c_{3})\left[1\mp 2\phi/X (t)\right]\right]\\ 2(1-\gamma)(-c_{2}\mp ic_{3})e^{it(E_{\pm}-E_{0})}\begin{array}[]{c}\\ \\ \end{array}\\ (1-\gamma)X(t)\left[1\pm\sqrt{1+c_{2}^{2}+c_{3}^{2}}-e^{it(E_{\pm}-E_{0})}(ic_ {2}\mp c_{3})\left[1\pm 2\phi/X(t)\right]\right]\end{array}\right)\frac{c_{1}e ^{-itE_{\pm}}}{1+\gamma}\begin{array}[]{c}\\ \\ \\ \\ \\ \\ \end{array}\\ \phi_{0}(t)=\left(\begin{array}[]{c}X(t)\left[ic_{3}\sin(\phi t)-ic_{2}\cos( \phi t)-1\right]+2\phi\sqrt{1+c_{2}^{2}+c_{3}^{2}}\\ 2i(1-\gamma)\begin{array}[]{c}\\ \\ \end{array}\\ (1-\gamma)\left[X(t)\left[-ic_{3}\sin(\phi t)+ic_{2}\cos(\phi t)-1\right]+2 \phi\sqrt{1+c_{2}^{2}+c_{3}^{2}}\right]\end{array}\right)\frac{c_{1}e^{-itE_{0 }}}{1+\gamma}\end{array}\] (44) As in the previous section we can use these expressions to confirm that the time-evolution is unitary and also verify (7) . ## 4 A solvable equivalence pair of spin 3/2 models Finally we also consider a spin 3/2 model and take the matrices $S_{j}^{x}$, $S_{j}^{y}$, $S_{j}^{z}$ in (8) to be $4\times 4$-spin 3/2 matrices \[S^{x}=\frac{1}{2}\left(\begin{array}[]{cccc}0&\sqrt{3}&0&0\\ \sqrt{3}&0&2&0\\ 0&2&0&\sqrt{3}\\ 0&0&\sqrt{3}&0\end{array}\right),~{}S^{y}=\frac{i}{2}\left(\begin{array}[]{ cccc}0&-\sqrt{3}&0&0\\ \sqrt{3}&0&-2&0\\ 0&2&0&-\sqrt{3}\\ 0&0&\sqrt{3}&0\end{array}\right),~{}S^{z}=\frac{1}{2}\left(\begin{array}[]{ cccc}3&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-3\end{array}\right),\] (45) at cite $j$. Choosing the constants $c_{x},c_{y},c_{\omega}$ conveniently, for $N=1$ this Hamiltonian simplifies to \[H_{1}^{3/2}=-\frac{1}{6}(S^{y}+\frac{2\omega}{3}\mathbb{I}+i\gamma S^{x})=- \frac{1}{4}\left(\begin{array}[]{cccc}\omega&i\frac{\gamma-1}{\sqrt{3}}&0&0\\ i\frac{\gamma-1}{\sqrt{3}}&\omega&i\frac{\gamma-1}{6}&0\\ 0&i\frac{\gamma-1}{6}&\omega&\frac{\gamma-1}{\sqrt{3}}\\ 0&0&\frac{\gamma-1}{\sqrt{3}}&\omega\end{array}\right).\] (46) The corresponding TDSE (1) is solved to \[\Psi_{k}(t)=\left(\begin{array}[]{c}i(1-\gamma)^{3/2}\\ -2\sqrt{3}k\hat{\phi}(1-\gamma)^{1/2}\\ 2i\sqrt{3}(k^{2}-2\left\|k\right\|)\hat{\phi}(1+\gamma)^{1/2}\\ \mathop{\mathrm{s}ign}(k)(\left\|k\right\|-2)(1+\gamma)^{3/2}\end{array}\right)e ^{-itE_{k}},~{}~{}~{}\qquad E_{k}=-\frac{1}{2}k~{}\hat{\phi}-\frac{\omega}{4}, \quad~{}~{}~{}k=\pm 1,\pm 3\] (47) where $\hat{\phi}:=\sqrt{1-\gamma^{2}}/6$. The eigenvalue spectrum is real for the same parameter range as in the previous subsections. Here we will only solve the time-dependent Dyson equation (5) to see whether the features of the spin 1/2 and spin 1 models are also present in this model. We assume a similar form for our Hermitian target Hamiltonian as in (9), denote $\chi=\Xi$ and take $\eta(t)$ to be of the Hermitian form \[\eta(t)=\left(\begin{array}[]{cccc}\eta_{1}(t)&\eta_{2}(t)-i\eta_{3}(t)&\eta_{ 4}(t)-i\eta_{5}(t)&\eta_{6}(t)-i\eta_{7}(t)\\ \eta_{2}(t)+i\eta_{3}(t)&\eta_{8}(t)&\eta_{9}(t)-i\eta_{10}(t)&\eta_{11}(t)-i \eta_{12}(t)\\ \eta_{4}(t)+i\eta_{5}(t)&\eta_{7}(t)+i\eta_{8}(t)&\eta_{13}(t)&\eta_{14}(t)-i \eta_{15}(t)\\ \eta_{6}(t)+i\eta_{7}(t)&\eta_{11}(t)+i\eta_{12}(t)&\eta_{14}(t)+i\eta_{15}(t) &\eta_{16}(t)\end{array}\right).\] (48) Substituting these expressions into the time-dependent Dyson equation (5) yields in principle $32$ equation for the $\eta_{i}(t),$$i=1,\ldots,16$. Once again the system is highly overdetermined, but remarkably it can be solved similarly as in the previous sections. Here we only present the solutions to these equations. We find \[\begin{array}[]{l}\eta_{1}(t)=\frac{c_{1}}{\Xi^{3/2}},~{}\quad\eta_{2}(t)=- \frac{6\sqrt{3}c_{1}\dot{X}}{(1+\gamma)\Xi^{5/2}},\quad~{}\eta_{3}(t)=\frac{3 \sqrt{3}c_{1}}{(1+\gamma)\Xi^{1/2}},\quad~{}\eta_{4}(t)=\frac{9\sqrt{3}c_{1}(4 \dot{\Xi}^{2}-\Xi^{4})}{(1+\gamma)^{2}\Xi^{7/2}},\\ \eta_{5}(t)=-\frac{36\sqrt{3}c_{1}\dot{\Xi}}{(1+\gamma)^{2}\Xi^{3/2}},\quad~{} \eta_{6}(t)=\frac{54c_{1}(3\dot{\Xi}\Xi^{4}-4\dot{\Xi}^{3})}{(1+\gamma)^{3}\Xi ^{9/2}},\quad~{}\eta_{7}(t)=\frac{27c_{1}(12\dot{\Xi}^{2}-\Xi^{4})}{(1+\gamma) ^{3/2}\Xi^{5/2}},\quad\quad\begin{array}[]{c}\\ \\ \end{array}\\ \eta_{8}(t)=\frac{6c_{1}(12\dot{\Xi}^{2}+3\Xi^{4}-\hat{\phi}^{2}\Xi^{2})}{( \gamma+1)^{2}\Xi^{7/2}},\quad~{}\eta_{9}(t)=\frac{18c_{1}\dot{\Xi}(4\hat{\phi} ^{2}\Xi^{2}-12\dot{\Xi}^{2}-3\Xi^{4})}{(\gamma+1)^{3}\Xi^{9/2}},\\ \eta_{10}(t)=\frac{9c_{1}\dot{\Xi}(4\hat{\phi}^{2}\Xi^{2}-12\dot{\Xi}^{2}-3\Xi ^{4})}{(\gamma+1)^{3}\Xi^{5/2}},\quad\eta_{11}(t)=\frac{9\sqrt{3}c_{1}(1- \gamma)(\Xi^{4}-4\dot{X}^{2})}{(1+\gamma)^{3}\Xi^{7/2}},\quad~{}\begin{array}[ ]{c}\\ \\ \end{array}\\ \eta_{12}(t)=\frac{36\sqrt{3}c_{1}(1-\gamma)\dot{\Xi}}{(1+\gamma)^{3}\Xi^{3/2} },\quad\eta_{13}(t)=\frac{6c_{1}(\gamma-1)(12\dot{\Xi}^{2}+3\Xi^{4}-\hat{\phi} ^{2}\Xi^{2})}{(1+\gamma)^{3}\Xi^{7/2}},\\ \eta_{14}(t)=-\frac{6\sqrt{3}c_{1}(1-\gamma)^{2}\dot{\Xi}}{(1+\gamma)^{3}\Xi^{ 5/2}},\quad~{}\eta_{15}(t)=\frac{3\sqrt{3}c_{1}(1-\gamma)^{2}}{(1+\gamma)^{3} \Xi^{1/2}},\quad\eta_{16}(t)=\frac{c_{1}(\gamma-1)3}{(1+\gamma)^{3}\Xi^{3/2}}, \begin{array}[]{c}\\ \\ \end{array}\end{array}\] (49) where $\Xi(t)$ has to obey the second order non-linear differential equation \[\ddot{\Xi}-\frac{3}{2}\frac{\dot{\Xi}^{2}}{\Xi}-\frac{1}{2}\hat{\phi}^{2}\Xi+ \frac{\Xi^{3}}{8}=0.\] (50) As in the previous subsection we can transform it to the Ermakov-Pinney equation (20) with $\sigma\rightarrow\hat{\sigma}$, $\phi\rightarrow\hat{\phi}$ using $\Xi=4/\hat{\sigma}^{2}$ in this case and therefore we have \[\Xi(t)=\frac{2\hat{\phi}}{c_{2}\sin(\hat{\phi}t)+c_{3}\cos(\hat{\phi}t)\pm \sqrt{1+c_{2}^{2}+c_{3}^{2}}}.\] (51) Computing from this $\rho=\eta^{2}$, we evaluate the determinant to \[\det\rho=\frac{6^{6}(1-\gamma)^{6}c_{1}^{8}}{(1+\gamma)^{18}}(1+c_{2}^{2}+c_{3 }^{2})^{6},\] (52) which is always positive for the parameter range of interest. Naturally (3) yields once more the soltution to the TDSE for $h(t)$. ## 5 Conclusions We have demonstrated that metric representations lead to consistent descriptions equivalent to the operator representation by providing further solutions to the time-dependent quasi Hermiticity relation (4) and the time-dependent Dyson relation (5). For the spin models we considered here we observed that the determining relation for the metric operator (4) converts into as many equations as unknown functions. The equations are easily decoupled and integrated to determine the metric operator. However, the diagonalization needed in order to take the square root is usually and moreover requires specific choices for the constants involved to ensure the all eigenvalues are positive. As we have demonstrated simple choices are usually not evident or do not even exist. In order to bypass this step we pursued what turned out to be an easier approach and solved the time-dependent Dyson relation (5) instead. Assuming a general form for the Hermitian Hamiltonian in (5) converts it into an overdetermined set of equations for the components of the Dyson map. Remarkably these equations can be decoupled and solved by simple integrations for the components of $\eta$. The time-dependent equation occurring in the Hermitian Hamiltonian is restricted by a nonlinear equation that can be converted into the Ermakov-Pinney equation. This feature was observed in all three spin models considered here and based on this observation we conjecture that it might be universal and will hold for all higher spin representations. Evidently there are many interesting open problems left for future research, such as a more extensive treatment of systems with explicitly time-dependent non-Hermitian Hamiltonian and with regard to the spin models more sites pose a natural challenge. Acknowledgments: TF is supported by a City, University of London Research Fellowship. ## References * [1] A. Fring and T. Frith, Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians, arXiv:1610.07537 (2016). * [2] C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243–5246 (1998). * [3] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70, 947–1018 (2007). * [4] A. Mostafazadeh, Pseudo-Hermitian Representation of Quantum Mechanics, Int. J. Geom. Meth. Mod. Phys. 7, 1191–1306 (2010). * [5] C. Figueira de Morisson Faria and A. Fring, Time evolution of non-Hermitian Hamiltonian systems, J. Phys. A39, 9269–9289 (2006). * [6] C. Figueira de Morisson Faria and A. Fring, Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation, Laser Physics 17, 424–437 (2007). * [7] A. Mostafazadeh, Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator, Physics Letters B 650(2), 208–212 (2007). * [8] M. Znojil, Time-dependent version of crypto-Hermitian quantum theory, Physical Review D 78(8), 085003 (2008). * [9] A. Fring and M. H. Y. Moussa, Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians, Physical Review A 93(4), 042114 (2016). * [10] F. S. Luiz, M. A. Pontes, and M. H. Y. Moussa, Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps, arXiv:1611.08286 (2016). * [11] A. Fring and M. M. H. Y. Moussa, Non-Hermitian Swanson model with a time-dependent metric, Physical Review A 94(4), 042128 (2016). * [12] G. von Gehlen, Critical and off critical conformal analysis of the Ising quantum chain in an imaginary field, J. Phys. A24, 5371–5400 (1991). * [13] O. A. Castro-Alvaredo and A. Fring, A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field, J. Phys. A42, 465211 (2009). * [14] V. Ermakov, Transformation of differential equations,, Univ. Izv. Kiev. 20, 1–19 (1880). * [15] E. Pinney, The nonlinear differential equation $y^{\prime\prime 3}=0$, Proc. Amer. Math. Soc. 1, 681(1) (1950). * [16] A. Hone, Exact discretization of the Ermakov-Pinney equation, Phys. Lett. A 263, 347–354 (1999). * [17] R. M. Hawkins and J. E. Lidsey, Ermakov-Pinney equation in scalar field cosmologies, Phys. Rev. D 66, 023523 (2002). * [18] J. R. Choi and B. H. Kweon, Operator method for a nonconservative harmonic oscillator with and without singular perturbation, Int. J. Mod. Phys. B16, 4733–4742 (2002). * [19] J. R. Choi, Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and $(1/x)p+p(1/x)$ Terms, Int. J. Theor. Phys. 42, 853–861 (2003). * [20] N. Ferkous, A. Bounames, and M. Maamache, Time-dependent Schrödinger equation with non-central potentials, Physica Scripta 88, 35001–35004 (2013). * [21] S. Dey and A. Fring, Noncommutative quantum mechanics in a time-dependent background, Phys. Rev. D 90, 084005 (2014). * [22] M. V. Ioffe and H. Korsch, Nonlinear supersymmetric (Darboux) covariance of the Ermakov-Milne-Pinney equation, Phys.Lett. A311, 200–205 (2003). * [23] S. Dey, A. Fring, and L. Gouba, Milne quantization for non-Hermitian systems, Journal of Physics A: Mathematical and Theoretical 48(40), 40FT01 (2015). * [24] C. Eliezer and A. Gray, A note on the time-dependent harmonic oscillator, SIAM Journal on Applied Mathematics 30, 463–468 (1976).
1708.02421	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 43738, "num_imgs": 7, "llama3_tokens_count": 11522 }	[ "content_image/1708.02421/perspective_demo_v7.png", "content_image/1708.02421/pipeline_v16.png", "content_image/1708.02421/fcn_v4.png", "content_image/1708.02421/perspective_heatmap_v3.png", "content_image/1708.02421/pred_peripheral_central.png", "content_image/1708.02421/x1.png", "content_image/1708.02421/x2.png" ]	# FoveaNet: Perspective-aware Urban Scene Parsing Xin Li${}^{1,2}$ Zequn Jie${}^{3}$ Wei Wang${}^{4,2}$ Changsong Liu${}^{1}$ Jimei Yang${}^{5}$ Xiaohui Shen${}^{5}$ Zhe Lin${}^{5}$ Qiang Chen${}^{6}$ Shuicheng Yan${}^{2,6}$ Jiashi Feng${}^{2}$ ${}^{1}$ Department of EE Tsinghua University ${}^{2}$ Department of ECE National University of Singapore ${}^{3}$ Tencent AI Lab ${}^{4}$ University of Trento ${}^{5}$ Adobe Research ${}^{6}$ 360 AI institute {lixincn2015 zequn.nus}@gmail.com wei.wang@unitn.it lcs@ocrserv.ee.tsinghua.edu.cn {jimyang xshen zlin}@adobe.com {chenqiang-iri yanshuicheng}360.cn elefjia@nus.edu.cn ###### Abstract Parsing urban scene images benefits many applications, especially self-driving. Most of the current solutions employ generic image parsing models that treat all scales and locations in the images equally and do not consider the geometry property of car-captured urban scene images. Thus, they suffer from heterogeneous object scales caused by perspective projection of cameras on actual scenes and inevitably encounter parsing failures on distant objects as well as other boundary and recognition errors. In this work, we propose a new FoveaNet model to fully exploit the perspective geometry of scene images and address the common failures of generic parsing models. FoveaNet estimates the perspective geometry of a scene image through a convolutional network which integrates supportive evidence from contextual objects within the image. Based on the perspective geometry information, FoveaNet “undoes” the camera perspective projection — analyzing regions in the space of the actual scene, and thus provides much more reliable parsing results. Furthermore, to effectively address the recognition errors, FoveaNet introduces a new dense CRFs model that takes the perspective geometry as a prior potential. We evaluate FoveaNet on two urban scene parsing datasets, Cityspaces and CamVid, which demonstrates that FoveaNet can outperform all the well-established baselines and provide new state-of-the-art performance. ## 1 Introduction Urban scene parsing is a heated research topic that finds application in many fields, especially self-driving. It aims to predict the semantic category for each pixel within a scene image captured by car mounted cameras, which enables self-driving cars to perform reasoning about both the overall scene background and the individual objects moving in front of the cars. <figure><img src="content_image/1708.02421/perspective_demo_v7.png"><figcaption>Figure 1: Illustration of our motivation. Top two rows: a scene image withperspective geometry and its two zoomed-in regions. Bottom two rows: typicalfailures in urban scene parsing. Left: “broken-down” error on objects of largescales (the bus). Right: boundary errors on objects of small scales.</figcaption></figure> Recent progress in urban scene parsing is mostly driven by the advance of deep learning. Deep convolutional neural network (CNN) based parsing algorithms [25, 21] have demonstrated remarkable performance on several semantic parsing benchmarks [7, 5, 24]. However, directly applying the generic CNN based image parsing models usually leads to unsatisfactory results on urban scene images for self-driving cars, since they ignore the important perspective geometry of scene images. As captured by ego-centric cameras, perspective projection from actual scenes to the image plane changes the object scales: a nearby car seems much bigger than a car far away, even though they have the same scale in reality. The top row in Figure 1 illustrates such a perspective geometry structure within a scene image. Generic parsing models do not take such heterogeneous object scales into consideration. Consequently, they do not perform well on parsing distant objects (of small scales), and boundary and recognition errors are introduced. See the parsing result marked with the small box in Figure 1. In addition, objects that are near to the camera and usually distributed within the peripheral region have relatively large scales. Generic parsing models tend to “break down” a large-scale object into several pieces of similar classes, as shown in the parsing result marked with the big box in Figure 1. Both of the above problems are from ignoring the perspective geometry. Therefore, we propose a novel FoveaNet to handle heterogeneous scales in urban scene parsing by considering the perspective geometry. FoveaNet works like the fovea of human eyes: the center of the vision field (fovea region) is focused on and the visual acuity is the highest. Through localizing “the fovea region” during parsing, FoveaNet “undoes” the camera perspective projection by scale normalization and parses regions at suitable scales. Specifically, FoveaNet employs a perspective estimation network to infer the overall perspective geometry and output dense perspective scores for each individual pixel, indicating the nearness of a pixel to the vanishing point. Objects with large perspective scores are usually small in the projected scene image. To address the unsatisfactory performance on parsing distant objects, FoveaNet performs _scale normalization_ on the fovea region that consists of small-scale objects. Then the parsings of small distant objects and large near objects are untangled by a perspective-aware parsing scene network, and boundary errors induced by small scale objects are reduced. To address the “broken-down” issues with parsing large objects, FoveaNet employs a new perspective-aware dense CRFs model that takes as input the perspective information and outputs different potentials on the pixels of different perspective scores. The proposed CRFs smooths the pixels from distant objects with large perspective scores more slightly than on the large objects. Through this adaptive strategy, the proposed CRFs are able to handle the “broken-down” errors and meanwhile avoid over-smoothing on small objects. We evaluate the proposed FoveaNet on two challenging datasets, Cityspaces and CamVid, and prove that it can provide new state-of-the-art performance on urban scene parsing problems. We make following contributions to urban scene parsing: * We propose to consider perspective geometry in urban scene parsing and introduce a perspective estimation network for learning the global perspective geometry of urban scene images. * We develop a perspective-aware parsing network that addresses the scale heterogeneity issues well for urban scene images and gives accurate parsing on small objects crowding around the vanishing point. * We present a new perspective-aware CRFs model that is able to reduce the typical “broken-down” errors in parsing peripheral regions of a scene image. ## 2 Related Work Semantic ParsingRecently, deep learning has greatly stimulated the progress on parsing tasks. Among CNN based algorithms, the Fully Convolutional Network (FCN) [25] and the DeepLab model [21] have achieved most remarkable success. Afterwards, various approaches have been proposed to combine the strengths of FCN and CRFs [39, 23], or to refine predictions by exploiting feature maps output by more bottom layers [28, 11]. A common way to deal with scale issues in parsing is to zoom in the input images [9, 27, 6, 22, 4]. The input images are rescaled to multiple scales and processed by a shared deep network [6, 22, 4]. More recently, Xia _et al._ [34] addressed the scale issues in the scenario of object parsing by “zoom and refine”. However, it is not suitable for urban scene parsing. Our FoveaNet differs from end-to-end trained attention models which learn black-box localization functions [31, 35, 26, 17]. Instead, FoveaNet explicitly models the visible geometry structure for fovea region localization and better fits the urban scene parsing task. <figure><img src="content_image/1708.02421/pipeline_v16.png"><figcaption>Figure 2: Architecture overview of FoveaNet. FoveaNet consists of aperspective-aware parsing network and perspective-aware CRFs. With theperspective estimation network (PEN), FoveaNet infers the global perspectivegeometry by producing a heatmap. Based on the perspective heatmap, FoveaNetlocalizes a fovea region (cyan rectangle) where small distant objects crowd.FoveaNet performs scale normalization on the fovea region, on which itproduces a finer parsing via the Fovea branch. This result is then fused withthe parsing from a coarse branch into the final prediction. The perspective-aware CRFs take input the fused parsing result, the perspective heatmp as wellas object detection results, and output the final parsing result. Best viewedin color.</figcaption></figure> Perspective Geometry in Urban ScenesAs 3D perspective geometry is a key property of urban scene images, several works consider modeling 3D geometric information as an additional feature for scene understanding [33, 19, 14, 15, 38]. Sturgess _et al._ [33] made use of geometric features in road scene parsing, which are computed using 3D point clouds. Hoiem _et al._ [14] modeled geometric context through classifying pixels into different orientation labels. Some others infer proper object scales with perspective geometry [15, 38, 19]. For example, Hoiem _et al._ [15] established the relationship between camera viewpoint and object scales, and used it as a prior for an object proposal. Ladicky _et al._ [19] trained a classifier with hand-crafted features to jointly solve semantic parsing and depth estimation. Training samples are transformed into the canonical depth, due to the observation that performance is limited by the scale misalignment due to the perspective geometry. All of the methods above are based on hand-crafted features rather than deep learning. ## 3 The Proposed FoveaNet ### Overview The basic idea of FoveaNet is estimating the perspective geometry of an urban scene image and parsing regions at suitable scales, instead of processing the whole image at a single scale. The overall architecture of FoveaNet is illustrated in Figure 2. The FoveaNet consists of two components, _i.e._, the perspective-aware parsing net and the perspective-aware CRFs. The perspective-aware parsing net aims at better parsing small scale objects crowding around the vanishing point by exploiting the image inherent perspective geometry. We propose a perspective estimation network (PEN) to estimate the perspective geometry by predicting a dense perspective heatmap, where a pixel of an object nearer to the vanishing point would have a larger value. Thus PEN provides clues to locate a _fovea region_ within which most small scale objects crowd. The fovea region is then re-scaled and receives finer processing by the parsing net, _i.e._ a two-branch FCN. In this way, small distant objects are untangled from large near objects for parsing. The perspective-aware CRFs aim at addressing “broken-down” errors when parsing the peripheral region of a scene image. Within this new CRFs model, we introduce a spatial support compatibility function that incorporates the perspective information from PEN, and facilitates the parsing by imposing adaptive potentials at different locations with different perspective heatmap scores. Only the regions confidently from the same object are processed by the CRFs. Small distant objects will be smoothed in a lighter way than the large near objects. The “broken-down” errors in peripheral regions can be alleviated effectively. We now proceed to introduce each component of FoveaNet, respectively. ### FCN in FoveaNet FoveaNet is based on the fully convolutional network (FCN) [25] for parsing the images. As a deeper CNN model benefits more for the parsing performance, we here follow Chen _et al._ [3] and use the vanilla ResNet-101 [13] to initialize the FCN model in FoveaNet. We observe that preserving high spatial resolution of feature maps is very important for accurately segmenting small objects within scenes. Therefore, we disable the last down-sampling layer by setting its stride as $1$. This increases the size of the feature map output by res$5\_c$ to $1/16$ of the input image size (without this modification the size of the output feature map is only $1/32$ of the input image size). <figure><img src="content_image/1708.02421/fcn_v4.png"><figcaption>Figure 3: Architectural overview of the perspective estimation network (PEN).PEN has a similar network structure as the FCN. Given an input scene image,PEN produces a one channel heatmap indicating (roughly) the nearness to thevanishing point at pixel-level.</figcaption></figure> ### Perspective-aware Scene Parsing Network FoveaNet localizes the fovea region with proper scales and concentrates on the localized fovea region to normalize the various object scales. To this end, a perspective estimation network is used to estimate the overall perspective geometry of a scene image and localize the region (roughly) centered at the vanishing point where most of small scale objects crowd. PEN then works together with a two-branch FCN as a perspective-aware scene parsing network. Training PENPEN has a same structure as the baseline FCN model, as shown in Figure 3. Our ground truth takes the form of a heatmap: a larger value in the heatmap indicates a higher possibility of small objects to crowd. As it is not easy to estimate the vanishing point of a scene image correctly (sometimes the vanishing point may be invisible or not exist in the image), we use the object scale as a clue to roughly estimate the position of the vanishing point and the perspective geometry. For training PEN, we formulate the ground truth heatmap of an image as follows: \[\vspace{-2mm}H_{i}^{(n)} =\dfrac{AveSize(\ell(m))}{Size(m)},\text{where }i\in\text{ instance }m,\] \[G_{i} =\frac{1}{N}\sum_{n=1}^{N}H^{(n)}_{i},\quad V_{i}^{(n)}=H_{i}^{(n )}+\delta\times G_{i},\] (1) In the above equation, $m$ denotes an object instance in the $n$-th image, and $i$ indexes a pixel from this instance. $\ell{m}$ denotes the category label of instance $m$. ${AveSize}(\ell(m))$ denotes the category-level average instance size. Thus $H_{i}^{(n)}$, _i.e._ the value of pixel $i$ in the $n$-th heatmap $H^{(n)}$, depends on the ratio of the category-level average instance size over the current instance size $Size(m)$. Global perspective score prior $G_{i}$ for the $i$-th pixel is the average value over all the $N$ heatmaps. The ground truth $V_{i}^{(n)}$ for training PEN is formulated by weighted summing both the image specific characteristics $H_{i}^{(n)}$ and the global average $G_{i}$, being traded-off by a parameter $\delta$. PEN is trained by minimizing a smoothed $\ell_{1}$ loss [12] between the produced heatmap based on raw images and the ground truth heatmap. Figure 4 illustrates the result of PEN. Figure 4 (a) shows a training urban scene image with perspective geometry and Figure 4 (b) shows its ground truth parsing map. We follow Eqn. (1) to obtain the ground truth perspective confidence map shown in Figure 4 (c). From the perspective map estimated by PEN (Figure 4 (d)), one can observe that PEN successfully predicts the overall geometry of the input image — it outputs larger values for the pixels closer to the vanishing point. With this perspective heatmap, FoveaNet localizes the fovea region with maximal response (highlighted with cyan rectangle). In our experiments, we define the size of the fovea region as $1/2$ of the heatmap size. To locate the fovea region based on the heatmap, FoveaNet passes the heatmap from PEN through an average pooling layer. The receptive field of the maximal pooling result on the heatmap is selected as the fovea region, as illustrated by the cyan boxes in Figure 2 and Figure 4 (d). <figure><img src="content_image/1708.02421/perspective_heatmap_v3.png"><figcaption>Figure 4: Illustration of perspective heatmap estimation. (a) An urban sceneimage. (b) Parsing ground truth. (c) Ground truth perspective heatmapgenerated by Eqn. (1). (d) Estimated perspective heatmap and detected Fovearegion from PEN. Fovea region with maximal response is highlighted in cyan.Best viewed in color.</figcaption></figure> DiscussionAnother choice for estimating perspective information is to estimate depth information from a single image [10, 8]. However, the single image depth prediction results are not discriminative for localizing distant objects. In contrast, PEN can produce a heatmap with distinguishable per-pixel scores, leading to more precise fovea region localization. Therefore, we use the method introduced above to estimate the perspective geometry and train PEN. A qualitative comparison between predicted depth [10] and our estimated perspective heatmap on Cityscapes dataset is provided in supplementary material. Perspective-aware Scene ParsingFoveaNet performs _scale normalization_ to achieve better parsing performance on objects of heterogeneous scales. After localizing the fovea region, FoveaNet parses the fovea region and the raw image separately through a two-branch FCN, as shown in Figure 2. The raw input image passes through the _coarse branch_ to produce an overall parsing result. Meanwhile, the fovea region is re-scaled to the original input size and passes through the _fovea branch_ to produce finer parsing for the fovea region. The two branches have the same structure as the baseline FCN model and share parameters from conv$1$ to res$3\_3b3$. More architectural details are given in Section 3.2. The two-branch FCN is end-to-end trainable by minimizing per-pixel cross-entropy loss. ### Perspective-aware CRFs The perspective-aware scene parsing network can parse the distant objects better by estimating perspective information. However, another common issue in parsing scene images is that large objects in peripheral regions of a scene image usually suffer from “broken-down” errors, _i.e._, a large object tends to be broken into several small pieces which are misclassified into different yet similar classes. This problem is illustrated in the bottom left subfigure of Figure 1: some parts of the bus are misclassified into the train, harming the parsing performance on the peripheral region. Intuitively, it would be beneficial for the final performance to refine the prediction with the aid of appearance features. In object segmentation, dense CRFs are usually applied to the prediction scores produced by FCN, and have shown impressive effects on refining prediction. However, directly applying dense CRFs to urban scene images does not give satisfactory performance due to heterogeneous scales of objects from the fovea region and the peripheral region. A dense CRFs model performing well on the peripheral region tends to over-smooth the predictions on small objects from the fovea region, which harms the performance on small objects significantly. Based on the perspective information from PEN, we propose a new perspective-aware dense CRFs model to alleviate “broken-down” errors. The CRFs model is trained separately, following the DeepLab model [21]. Let $\ell$ denote the label vector for all the pixels, $f_{i}$ denote the learned representation of the pixel $i$, and $p_{i}$ denote the 2D-coordinate of the pixel $i$ in the image plane. The energy function of the perspective-aware dense CRFs is defined as \[\mathcal{E}(\ell)=\sum_{i}\psi_{u}(\ell_{i})+\sum_{i,j}\psi_{p,\text{persp}}( \ell_{i},\ell_{j}).\vspace{-0.3cm}\] Here $\psi_{u}$ is the standard unary potential. The pairwise potential in our proposed CRFs model has a new form: \[\psi_{p,\text{persp}}(\ell_{i},\ell_{j})=\mu(p_{i},p_{j})\nu(\ell_{i},\ell_{j} )\kappa(f_{i},f_{j}),\vspace{-0.1cm}\] where the kernel $\kappa(\cdot,\cdot)$ is the contrast-sensitive two-kernel potential proposed by Krahenbuhl _et al._ [18], and $\nu$ is the Potts label compatibility function. Here $\mu$ is a new spatial support compatibility function introduced for the perspective-aware CRFs that considers auxiliary object detection results and perspective information provided by PEN: \[\mu(p_{i},p_{j})=d_{k}\left(\frac{\sum_{m\epsilon\widehat{V}}v_{m}}{\|\widehat{ V}\|}/\frac{\sum_{n\epsilon B_{k}}v_{n}}{\|B_{k}\|}\right)\] (2) The object bounding boxes are detected by a Faster-RCNN [29] model. Among them, some bounding boxes $B_{k},k=1,2...K$ contain the pixels $p_{i}$,$p_{j}$. Then the box $B_{k}$ with the maximum detection score $d_{k}$ is selected as the target one. Here $\widehat{V}$ denotes the estimated heatmap, and $m,n$ index pixels from the estimated heatmap $\widehat{V}$ and bounding box $B_{k}$ respectively. This $\mu(p_{i},p_{j})$ incorporates perspective information as follows. It lowers the weights of the pairwise potential at a bounding box with higher heatmap values. Thus for a small object with a high perspective score, the pairwise potential becomes small (imposing lighter spatial smoothing), and the unary potential plays a major role. By focusing on each detection proposal with adaptive perspective weights, the proposed CRFs model effectively alleviates the “broken-down” problems and meanwhile avoids over-smoothing the details. ## 4 Experiments ### Experimental Settings We implement FoveaNet using the Caffe library [16] and evaluate its performance on two urban scene parsing datasets: Cityscapes [5] and Camvid [2]. For performing ablation studies on FoveaNet, we employ a vanilla FCN architecture with ResNet-101 being its front-end model as the baseline. It takes raw images as inputs and is trained with per-pixel cross-entropy loss. During testing, it produces parsing results at a single scale. We examine how its performance changes by incorporating different components from FoveaNet, in order to understand the contribution of each component. FoveaNet is initialized by a modified ResNet-101 network pre-trained on ImageNet (see Section 3.2 for more details). We fine-tune the initial model on an individual scene parsing dataset. The initial learning rate is $0.001$, and is decreased by a factor of $0.1$ after every $20$ epochs for twice. The momentum is $0.9$. ### Results on Cityscapes The Cityscapes dataset [5] is a recently released large-scale benchmark for urban scene parsing. Its images are taken by car-carried cameras and are collected in streets of $50$ different cities. It contains in total $5{,}000$ images with high quality pixel-level annotations. These images are split to $2{,}975$ for training, $500$ for validation and $1{,}525$ for testing. Cityscapes provides annotations at two semantic granularities _i.e._, classes and higher-level categories. Annotations can be divided into 30 classes and 8 higher-level categories. For instance, the classes of _car_, _truck_, _bus_ and other $3$ classes are grouped into the _vehicle_ category. Among them, 19 classes and 7 categories are used for evaluation. Our FoveaNet is trained on $2{,}975$ training images, and evaluated on the validation set. Then we add $500$ validation images to fine-tune our model and obtain the test performance. Following the provided evaluation protocol with the dataset [5], we report the performance of compared models in terms of four metrics _i.e._$\text{IoU}_{\text{class}}$, $\text{IoU}_{\text{category}}$, $\text{iIoU}_{\text{class}}$ and $\text{iIoU}_{\text{category}}$. Compared with the standard $\text{IoU}_{\text{class}}$ and $\text{IoU}_{\text{category}}$, the latter two IoU metrics put more emphasis on the performance on small scale instances. The resolution of images is $2048\times 1024$, which brings a challenge to training deep networks with limited GPU memory. Hence, we use a random image crop of $896\times 896$ in training. For building the perspective-aware CRFs model, we train a Faster-RCNN on Cityscapes with $8$ classes whose ground truth bounding boxes can be derived from instance annotations, including _truck_, _bus_, _motorcycle_. \| Metric \| road \| sidewalk \| building \| wall \| fence \| pole \| tr. light \| tr. sign \| veg. \| terrain \| sky \| person \| rider \| car \| truck \| bus \| train \| mcycle \| bicycle \| Class \| Category ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- FCN Baseline \| IoU \| 97.7 \| 81.9 \| 91.0 \| 48.5 \| 52.9 \| 58.2 \| 63.1 \| 73.5 \| 91.4 \| 61.6 \| 94.3 \| 78.1 \| 56.0 \| 93.4 \| 57.5 \| 81.2 \| 66.2 \| 60.4 \| 74.2 \| 72.7 \| 87.6 iIoU \| —————— \| 60.4 \| 37.8 \| 84.8 \| 36.2 \| 58.5 \| 43.1 \| 36.9 \| 56.4 \| 51.8 \| 72.8 \+ fixed fovea region \| IoU \| 97.8 \| 82.8 \| 91.3 \| 48.0 \| 51.0 \| 60.8 \| 66.7 \| 75.6 \| 91.7 \| 61.3 \| 94.5 \| 79.2 \| 57.0 \| 93.3 \| 55.3 \| 79.9 \| 65.1 \| 62.4 \| 75.1 \| 73.1 \| 88.3 iIoU \| —————— \| 62.8 \| 46.2 \| 86.5 \| 36.3 \| 59.6 \| 43.2 \| 42.2 \| 59.6 \| 54.6 \| 75.4 \+ PEN fovea region \| IoU \| 97.8 \| 82.9 \| 91.4 \| 48.6 \| 54.3 \| 62.5 \| 69.0 \| 77.3 \| 91.9 \| 60.7 \| 94.4 \| 80.6 \| 60.3 \| 93.6 \| 56.8 \| 80.2 \| 60.4 \| 65.8 \| 76.2 \| 73.9 \| 88.8 iIoU \| —————— \| 64.7 \| 48.6 \| 87.2 \| 42.8 \| 62.2 \| 45.3 \| 46.9 \| 61.4 \| 57.4 \| 76.6 \+ PEN fovea region & normal CRFs \| IoU \| 97.7 \| 82.7 \| 90.7 \| 48.7 \| 51.5 \| 54.1 \| 60.7 \| 75.3 \| 90.9 \| 62.9 \| 94.5 \| 78.9 \| 57.7 \| 93.3 \| 61.8 \| 83.4 \| 70.8 \| 65.2 \| 74.7 \| 73.5 \| 87.2 iIoU \| —————— \| 61.6 \| 47.0 \| 83.3 \| 36.2 \| 58.5 \| 43.7 \| 45.7 \| 59.0 \| 54.4 \| 73.6 \+ PEN fovea region & depth-aware CRFs \| IoU \| 97.7 \| 82.4 \| 91.2 \| 47.2 \| 53.9 \| 61.8 \| 67.9 \| 76.5 \| 91.7 \| 60.5 \| 94.2 \| 79.8 \| 58.6 \| 93.9 \| 60.6 \| 84.2 \| 69.7 \| 64.2 \| 75.5 \| 74.3 \| 88.4 iIoU \| —————— \| 63.2 \| 46.7 \| 87.0 \| 40.4 \| 60.6 \| 44.2 \| 42.8 \| 59.8 \| 55.6 \| 75.6 \+ PEN fovea region & persp-aware CRFs \| IoU \| 97.9 \| 83.0 \| 91.5 \| 47.7 \| 54.5 \| 62.8 \| 69.1 \| 77.5 \| 91.9 \| 60.9 \| 94.4 \| 80.7 \| 60.4 \| 94.4 \| 72.5 \| 86.2 \| 72.7 \| 66.8 \| 76.4 \| 75.9 \| 88.8 iIoU \| —————— \| 65.1 \| 48.9 \| 87.8 \| 43.0 \| 62.3 \| 49.2 \| 46.8 \| 61.6 \| 58.1 \| 76.8 Table 1: Performance comparison among several variants of FoveaNet on the Cityscapes _validation_ set. The metric of iIoU is not applicable for categories of _road_ to _bicycle_. Best viewed in color. Perspective DistortionWe now quantitatively analyze how much perspective distortion affects urban scene parsing and demonstrate perspective distortion is a severe issue for urban scene parsing. We evaluate the baseline FCN model (trained on the whole images) on two image sets: one contains only the central region and the other contains only the peripheral region, as illustrated in Figure 5. Table 2 shows a detailed comparison between these two image sets on _Object_ and _Vehicle_ category, which consist of $3$ and $4$ classes respectively. First, we find that the performance on the _Object_ category in the central region is much worser than in the peripheral region. More concretely, we find that the $\text{IoU}_{\text{category}}$ of _Object_ drops $10.6\%$ in the central region. This performance drop comes from the small object scales in the center region caused by perspective distortion. This problem can also be observed from parsing results in Figure 5. The parsing in the central region lacks enough details. Second, generic parsing models tend to “break down” a large-scale object into several pieces of similar classes, as illustrated in Figure 5. We can observe from Table 2 that the $\text{IoU}_{\text{Category}}$ of _Vehicle_ improves $2.7\%$, but corresponding $\text{IoU}_{\text{Class}}$ deteriorates largely in the peripheral region. This can be largely attributed to misclassification between fine-grained classes, which is reflected by the $\text{IoU}_{\text{Class}}$ metric. Objects in the peripheral region have an unbalanced larger scale due to perspective distortion. The performance drop on _Vehicle_ category is brought by the “broken-down” issue. <figure><img src="content_image/1708.02421/pred_peripheral_central.png"><figcaption>Figure 5: Typical parsing result of baseline FCN model. We evaluate FCN onperipheral and central region respectively, to analyze how much a perspectivedistortion affects urban scene parsing.</figcaption></figure> Region \| peripheral \| central ---\|---\|--- IoUcategory \| object \| 69.7 \| 59.1 IoUClass \| pole \| 62.8 \| 51.1 tr. light \| 66.3 \| 58.2 tr. sign \| 77.7 \| 67.5 IoUcategory \| vehicle \| 93.3 \| 90.6 IoUClass \| car \| 94.3 \| 91.8 truck \| 48.0 \| 66.0 bus \| 78.7 \| 83.0 train \| 61.0 \| 71.6 Table 2: Comparison on _Object_ and _Vehicle_ category between peripheral and central regions. Ablation AnalysisWe now analyze FoveaNet by investigating the effects of each component separately. Table 1 lists the performance of adding each component of FoveaNet to the baseline model (vanilla FCN) on the validation set. We also give a qualitative comparisons in Figure 6. From the results, we can make following observations. <figure><img src="content_image/1708.02421/x1.png"><figcaption>Figure 6: Example parsing results on Cityscapes. 1st-2nd row: urban sceneimages with two types of fovea regions derived from global prior (red) and PEN(yellow) based on its estimated heatmap (2nd row). 3rd row: parsing result onfovea regions with FCN baseline. 4th row: parsing result on fovea region withFoveaNet. FoveaNet produces more detailed parsing results on small scaleobjects _e.g._ , _pole_ , _traffic light_ , _traffic sign_. Best viewed incolor.</figcaption></figure> _Perspective-aware Parsing_: The 2nd row in Figure 6 shows that PEN successfully estimates the global perspective geometry. In the heatmap, small scale objects have larger response values (brighter). We compare the fovea region estimated by PEN (yellow rectangle) with a pre-fixed fovea region estimated from the global average (red rectangles; ref. Eqn. (1)). Comparing these two fovea regions shows PEN better localizes the regions covering small objects and is adaptive to different images. For example, the leftmost image presents a road turning left and thus small scale objects crowd in the left panel. PEN effectively locates this region but the globally fixed one fails. We also quantitatively compare the benefits of these two fovea region localization strategies in Table 1 (+ fixed fovea region vs. + PEN fovea region). One can observe that relying on the fovea regions provided by PEN significantly performs better by a margin of $2.8\%$ in terms of $\text{iIoU}_{\text{class}}$. Compared with the baseline FCN model, performing perspective-aware parsing with the help of PEN significantly improves the performance by $5.6\%$ and $3.8\%$ on the instance-level scores $\text{iIoU}_{\text{class}}$ and $\text{iIoU}_{\text{category}}$ respectively (highlighted in blue). This verifies perspective information is indeed beneficial for urban scene parsing. Figure 6 provides more qualitative results. We visualize the parsing results on the fovea region (from PEN) with FoveaNet or with FCN baseline model (the 4th row and the 3rd row respectively). One can observe that perspective-aware parsing gives results with richer details. Particularly, the pole, traffic light and traffic sign are parsed very well. This is also confirmed by their IoU improvement in Table 1 (highlighted in green), which is up to $6\%$. These qualitative and quantitative results clearly validate the effectiveness of the perspective-aware parsing network on objects of small scales, as it can better address the scale heterogeneity issue in urban scenes. _Perspective-aware CRFs:_ Based on the perspective-aware parsing on the fovea region, we further compare perspective-aware CRFs, normal dense CRFs and depth-aware CRFs in Table 1. The depth-aware CRFs model is similar to the perspective-aware CRFs model, except that the perspective heatmap in Eqn. (2) is replaced by single image depth prediction from the method in [10]. <figure><img src="content_image/1708.02421/x2.png"><figcaption>Figure 7: Parsing results of perspective-aware CRFs on Cityscapes validationset. Top: input images with object detection bounding boxes (yellow). 2nd row:parsing results from FCN. Large scale objects in peripheral region present“broken-down” errors. 3rd row: the perspective information by PEN which isintegrated into proposed perspective-aware CRFs. Bottom: FoveaNet appliesperspective-aware CRFs to remove the “broken-down” error. Best viewed incolor.</figcaption></figure> We observe that _truck_, _bus_ and _train_ are the three classes with most severe “broken-down” errors. Applying the normal dense CRFs improves the $\text{IoU}_{\text{class}}$ of these three classes by up to $10.4\%$ (highlighted in brown). This demonstrates that the normal dense CRFs model is effective in alleviating the “broken-down” error to some extent. However, the normal dense CRFs model harms the parsing results of small-scale objects. This can be observed from $\text{IoU}_{\text{class}}$ of _pole_, _traffic light_ and _traffic sign_ (highlighted in orange) which significantly drop _w.r.t._ results provided by its baseline (baseline FCN + PEN fovea region). This is due to over-smoothness artifacts of the normal dense CRFs as it is unaware of the scale variance within the image. In contrast, the perspective-aware CRFs model significantly boosts the $\text{IoU}_{\text{class}}$ of _truck_, _bus_, and _train_ by $15.7\%$, $6.0\%$, $12.3\%$ respectively (highlighted in red), without harming the results of small objects. Therefore, by incorporating perspective information, the perspective-aware CRFs model successfully reduces the “broken-down” errors without bringing over-smoothness, superior to the normal dense CRFs. The depth-aware CRFs model is superior to the normal dense CRFs one, but inferior to the perspective-aware CRFs one. This demonstrates that considering perspective geometry is useful but depth prediction is not so discriminative as perspective information predicted by our proposed model, as discussed in Section 3.3. Figure 7 gives additional parsing examples from the perspective-aware CRFs model. The trained Faster R-CNN model provides several object bounding boxes for the three urban scene images. PEN predicts perspective scores on these objects, where a brighter value indicates a higher probability of being near to the vanishing point (2nd row). We can observe that before applying perspective-aware CRFs, large scale objects suffers from “broken-down” errors (3rd row). Perspective-aware CRFs significantly reduces such errors in the peripheral region without over-smoothing small objects (_e.g._, _pole_) (4th row). Comparison with State-of-the-artWe fine-tune the FoveaNet using both training and validation images. Then on the test set we compare its performance with state-of-the-art published models which achieved best performance. Table 3 shows the results. Our FoveaNet outperforms all the published state-of-the-arts. FoveaNet performs especially well at instance-level (see iIoU results). Compared with the FCN model, FoveaNet brings significant improvement on $\text{iIoU}_{\text{class}}$ and $\text{iIoU}_{\text{category}}$, up to $5.2\%$. These two instance-level scores reflect the good parsing performance of FoveaNet on small scale objects. The improvement of $\text{IoU}_{\text{class}}$ and $\text{IoU}_{\text{category}}$ can be largely attributed to our perspective-aware CRFs, which can significantly reduce “broken-down” errors. Upon acceptance, we will release the code and model. Methods \| Class \| Category ---\|---\|--- IoU \| iIoU \| IoU \| iIoU Dilation10 [36] \| 67.1 \| 42.0 \| 86.5 \| 71.1 NVSegNet [1] \| 67.4 \| 41.4 \| 87.2 \| 68.1 DeepLabv2-(Resnet-101) [21] \| 70.4 \| 42.6 \| 86.4 \| 67.7 AdelaideContext [23] \| 71.6 \| 51.7 \| 87.3 \| 74.1 LRR-4x [11] \| 71.8 \| 47.9 \| 88.3 \| 74.1 Baseline FCN \| 71.3 \| 47.2 \| 87.8 \| 72.9 FoveaNet (ours) \| 74.1 \| 52.4 \| 89.3 \| 77.6 Table 3: Performance comparison with baseline models on Cityscapes _test_ set. ### Results on CamVid Cambridge-driving Labeled Video Database (CamVid) [2] consists of over $10$min of high quality videos. There are pixel-level annotations of 701 frames with resolution $960\times 720$. Each pixel is labeled with one of the 32 candidate classes. Perspective geometry can also be observed on these frames. Following previous works [1, 20], we use 11 classes for evaluation and report the per-pixel and average per-pixel accuracy. To implement FoveaNet, we reuse PEN and Faster-RCNN trained on Cityscapes urban scene images. The two-branch FCN model (coarse and fovea branch) are initialized from ResNet-101 and fine-tuned on CamVid training and validation sets. The performance of FoveaNet on the test set and the comparison with state-of-the-arts are shown in Table 4. FoveaNet outperforms the best baseline method on this dataset by $1.7\%$ and $3.7\%$ in global accuracy and average accuracy respectively. Due to limited space, we defer qualitative results on CamVid to Supplementary Material. \| Global Accuracy \| Average Accuracy ---\|---\|--- Zhang et al.[37] \| 82.1 \| 55.4 Bulo et al.[30] \| 82.1 \| 56.1 Shuai et al.[32] \| 91.6 \| 78.1 FoveaNet \| 93.3 \| 81.8 Table 4: Performance comparison with baseline models on CamVid test set. ## 5 Conclusion We proposed a new urban scene parsing model FoveaNet by considering the ubiquitous scale heterogeneity when parsing scene images, which can provide state-of-the-art performance as validated on the Cityscapes and CamVid datasets. FoveaNet exploits the perspective geometry information through two novel components, perspective-aware parsing net and perspective-aware CRFs model, which work jointly and successfully to solve the common scale issues, including parsing errors on small distant objects, “broken-down” errors on large objects and over-smoothing artifacts. ## 6 Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No. 61471214 and the National Basic Research Program of China (973 program) under Grant No.2013CB329403. The work of Jiashi Feng was partially supported by NUS startup R-263-000-C08-133, MOE Tier-I R-263-000-C21-112 and IDS R-263-000-C67-646. ## References * [1] V. Badrinarayanan, A. Handa, and R. Cipolla. Segnet: A deep convolutional encoder-decoder architecture for robust semantic pixel-wise labelling. _arXiv preprint arXiv:1505.07293_, 2015. * [2] G. J. Brostow, J. Fauqueur, and R. Cipolla. Semantic object classes in video: A high-definition ground truth database. _PRL_, 2008. * [3] L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille. Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. _arXiv:1606.00915_, 2016. * [4] L.-C. Chen, Y. Yang, J. Wang, W. Xu, and A. L. Yuille. Attention to scale: Scale-aware semantic image segmentation. In _CVPR_, 2016. * [5] M. Cordts, M. Omran, S. Ramos, T. Rehfeld, M. Enzweiler, R. Benenson, U. Franke, S. Roth, and B. Schiele. The cityscapes dataset for semantic urban scene understanding. In _CVPR_, 2016. * [6] J. Dai, K. He, and J. Sun. Boxsup: Exploiting bounding boxes to supervise convolutional networks for semantic segmentation. In _ICCV_, 2015. * [7] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In _CVPR_, 2009. * [8] D. Eigen, C. Puhrsch, and R. Fergus. Depth map prediction from a single image using a multi-scale deep network. In _NIPS_, 2014. * [9]C. Farabet, C. Couprie, L. Najman, and Y. LeCun. Learning hierarchical features for scene labeling. _TPAMI_, 2013. * [10] R. Garg, G. Carneiro, and I. Reid. Unsupervised cnn for single view depth estimation: Geometry to the rescue. In _ECCV_, 2016. * [11] G. Ghiasi and C. C. Fowlkes. Laplacian pyramid reconstruction and refinement for semantic segmentation. In _ECCV_, 2016. * [12] R. Girshick. Fast r-cnn. In _ICCV_, 2015. * [13] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In _CVPR_, 2016. * [14] D. Hoiem, A. A. Efros, and M. Hebert. Geometric context from a single image. In _ICCV_, 2005. * [15] D. Hoiem, A. A. Efros, and M. Hebert. Putting objects in perspective. _IJCV_, 2008. * [16] Y. Jia, E. Shelhamer, J. Donahue, S. Karayev, J. Long, R. Girshick, S. Guadarrama, and T. Darrell. Caffe: Convolutional architecture for fast feature embedding. In _ACM Multimedia_, 2014. * [17] F. Juefei-Xu, E. Verma, P. Goel, A. Cherodian, and M. Savvides. Deepgender: Occlusion and low resolution robust facial gender classification via progressively trained convolutional neural networks with attention. In _CVPR Workshops_, 2016. * [18]P. Krähenbühl and V. Koltun. Efficient inference in fully connected crfs with gaussian edge potentials. In _NIPS_, 2011. * [19] L. Ladicky, J. Shi, and M. Pollefeys. Pulling things out of perspective. In _CVPR_, 2014. * [20] L. Ladický, P. Sturgess, K. Alahari, C. Russell, and P. H. S. Torr. What, where and how many? combining object detectors and crfs. In _ECCV_, 2010. * [21] C. Liang-Chieh, G. Papandreou, I. Kokkinos, K. Murphy, and A. Yuille. Semantic image segmentation with deep convolutional nets and fully connected crfs. In _ICLR_, 2015. * [22] G. Lin, C. Shen, A. v. d. Hengel, and I. Reid. Exploring context with deep structured models for semantic segmentation. _arXiv preprint arXiv:1603.03183_, 2016. * [23] G. Lin, C. Shen, I. Reid, et al. Efficient piecewise training of deep structured models for semantic segmentation. _arXiv preprint arXiv:1504.01013_, 2015. * [24] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick. Microsoft coco: Common objects in context. In _ECCV_, 2014. * [25] J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In _CVPR_, 2015. * [26] V. Mnih, N. Heess, A. Graves, et al. Recurrent models of visual attention. In _NIPS_, 2014. * [27]M. Mostajabi, P. Yadollahpour, and G. Shakhnarovich. Feedforward semantic segmentation with zoom-out features. In _CVPR_, 2015. * [28] P. O. Pinheiro, T.-Y. Lin, R. Collobert, and P. Dollár. Learning to refine object segments. In _ECCV_, 2016. * [29] S. Ren, K. He, R. Girshick, and J. Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In _NIPS_, 2015. * [30] S. Rota Bulo and P. Kontschieder. Neural decision forests for semantic image labelling. In _CVPR_, 2014. * [31] S. Sharma, R. Kiros, and R. Salakhutdinov. Action recognition using visual attention. _arXiv preprint arXiv:1511.04119_, 2015. * [32] B. Shuai, Z. Zuo, B. Wang, and G. Wang. Dag-recurrent neural networks for scene labeling. In _CVPR_, 2016. * [33] P. Sturgess, K. Alahari, L. Ladicky, and P. H. Torr. Combining appearance and structure from motion features for road scene understanding. In _BMVC_, 2009. * [34] F. Xia, P. Wang, L.-C. Chen, and A. L. Yuille. Zoom better to see clearer: Human and object parsing with hierarchical auto-zoom net. In _ECCV_, 2016. * [35] K. Xu, J. Ba, R. Kiros, K. Cho, A. C. Courville, R. Salakhutdinov, R. S. Zemel, and Y. Bengio. Show, attend and tell: Neural image caption generation with visual attention. In _ICML_, 2015. * [36]F. Yu and V. Koltun. Multi-scale context aggregation by dilated convolutions. In _ICLR_, 2016. * [37] Y. Zhang and T. Chen. Efficient inference for fully-connected crfs with stationarity. In _CVPR_, 2012. * [38] Z. Zhang, A. G. Schwing, S. Fidler, and R. Urtasun. Monocular object instance segmentation and depth ordering with cnns. In _ICCV_, 2015. * [39] S. Zheng, S. Jayasumana, B. Romera-Paredes, V. Vineet, Z. Su, D. Du, C. Huang, and P. H. Torr. Conditional random fields as recurrent neural networks. In _ICCV_, 2015.
1808.04937	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 68895, "num_imgs": 10, "llama3_tokens_count": 22441 }	[ "content_image/1808.04937/x1.png", "content_image/1808.04937/x2.png", "content_image/1808.04937/x3.png", "content_image/1808.04937/x4.png", "content_image/1808.04937/x5.png", "content_image/1808.04937/x6.png", "content_image/1808.04937/x7.png", "content_image/1808.04937/x8.png", "content_image/1808.04937/x9.png", "content_image/1808.04937/x10.png" ]	# Photoneutron reaction cross section measurements on ${}^{94}$Mo and ${}^{90}$Zr relevant to the _p_-process nucleosynthesis A. Banu Corresponding author: banula@jmu.edu E. G. Meekins [ Department of Physics and Astronomy, James Madison University, Harrisonburg, Virginia 22807, USA J. A. Silano [ H. J. Karwowski Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27516, USA S. Goriely Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, Campus de la Plaine, CP-226, 1050 Brussels, Belgium February 29, 2024 ###### Abstract The photodisintegration cross sections for the ${}^{94}$Mo($\gamma$,n) and ${}^{90}$Zr($\gamma$,n) reactions have been experimentally investigated with quasi-monochromatic photon beams at the High Intensity $\gamma$-ray Source (HI$\gamma$S) facility of the Triangle Universities Nuclear Laboratory (TUNL). The energy dependence of the photoneutron reaction cross sections was measured with high precision from the respective neutron emission thresholds up to 13.5 MeV. These measurements contribute to a broader investigation of nuclear reactions relevant to the understanding of the $p$-process nucleosynthesis. The results are compared with the predictions of Hauser-Feshbach statistical model calculations using two different models for the dipole $\gamma$-ray strength function. The resulting ${}^{94}$Mo($\gamma$,n) and ${}^{90}$Zr($\gamma$,n) photoneutron stellar reaction rates as a function of temperature in the typical range of interest for the $p$-process nucleosynthesis show how sensitive the photoneutron stellar reaction rate can be to the experimental data in the vicinity of the neutron threshold. pacs: 25.20.Lj, 21.10.Pc, 25.40.Lw, 27.60.+j † [FOOTNOTE:†][ENDFOOTNOTE] Present address:] Geisinger Medical Center, Danville, Pennsylvania 17822, USA Present address:] Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA ## I Introduction How the nuclear reactions that occur in stars and in stellar explosions have been forging the elements out of hydrogen and helium leftover from the Big Bang is a longstanding Burbidge _et al._ (1957), still timely research topic in nuclear astrophysics. Although there is a fairly complete understanding of the production of elements up to iron by nuclear fusion reactions in stars, important details concerning the production of the elements beyond iron remain puzzling. Current understanding is that the nucleosynthesis beyond iron proceeds mainly via neutron capture reactions and subsequent $\beta^{-}$ decays in the $s$- and $r$-processes. However, some 35 proton-rich stable isotopes, between ${}^{74}$Se and ${}^{196}$Hg, cannot be synthesized by neutron-capture processes since they are located on the neutron-deficient side of the valley of $\beta$-stability. They are thus shielded from the $s$- or $r$-process. These proton-rich stable nuclides are generally referred to as $p$-nuclei Arnould (1976); Woosley and Howard (1978); Lambert (1992); Arnould and Goriely (2003). As a group they are the rarest of all stable isotopes. The mechanism responsible for their synthesis is termed the $p$-process. The gross similarities between the abundance curves of the $p$-nuclei and the $s$- and $r$-nuclei imply that the much more abundant $s$- and $r$-nuclei may serve as seeds for the $p$-process, but the astrophysical details of the $p$-process are still under discussion. So far it has been impossible to reproduce the solar abundances of all $p$-isotopes using a single nucleosynthesis process. Several different sites and (independently operating) processes seem to be required, with the largest fraction of the $p$-isotopes being synthesized by sequences of photodisintegrations and $\beta^{+}$ decays. Due to the dominance of photodisintegrations, this mechanism of the $p$-process is sometime referred to as the $\gamma$-process Woosley and Howard (1978). It is generally accepted that the $\gamma$-process occurs mainly in explosive O/Ne burning during supernova Type II explosions at temperatures in the range of T $\approx$ 2-3 GK, but supernovae Type Ia and Ib/c are also expected to contribute Arnould and Goriely (2003). Calculations based on the $\gamma$-process concept can reproduce the bulk of the $p$-nuclei within a factor of $\approx$ 3 Rayet _et al._ (1995); Rauscher and Thielemann (2000), but the most abundant $p$-isotopes, ${}^{92,94}$Mo and ${}^{96,98}$Ru (as well as potentially the A $<$ 124 mass region), are underproduced, making their nucleosynthesis one of the great outstanding mysteries in nuclear astrophysics. It is not yet clear whether specific environments need to be invoked for their production, such as He-accreting sub-Chandrasekhar white-dwarf Goriely _et al._ (2002) or $p$-rich neutrino driven winds of type-II supernovae Fröhlich _et al._ (2006) or $s$-process-enriched type Ia supernova Arnould and Goriely (2003); Travaglio _et al._ (2011), or if the calculated underproductions are due to deficiencies in the astrophysical models or in the underlying nuclear physics input, $i.e.$ the reaction rates used in the model. Contrary to the $s$- or $r$-process, the concepts of steady flows or reaction rate equilibria cannot be applied to the $p$-process, which operates far from equilibrium. As a result, an extended network of some 20000 reactions linking about 2000 nuclei in the A $\leq$ 210 mass region must be computed in detail Arnould and Goriely (2003). It is impossible to measure all these reaction rates in the laboratory. Hence, it becomes obvious that the vast majority of the reaction rates must be determined theoretically. Usually the unknown reaction rates are calculated within the framework of Hauser-Feshbach (HF) statistical model calculations with typical uncertainties of about 30% for stable nuclei Arnould and Goriely (2003); Rauscher and Thielemann (2000), but that can reach a factor of 10 for neutron-deficient nuclei Arnould and Goriely (2003). The model requires input based on nuclear structure, optical model potentials, and nuclear level densities to calculate transmission coefficients (average widths) which, in turn, determine the reaction cross sections, and thus, the reaction rates. The uncertainties involved in any HF cross section calculation are not related to the model of formation and de-excitation of the compound nucleus itself, but rather to the evaluation of the nuclear quantities necessary for the calculation of the transmission coefficients. The photon transmission coefficient is particularly relevant in the case of photonuclear reactions and is calculated assuming the dominance of dipole $E1$ transitions. The transmission coefficient for $\gamma$-ray emission with multipolarity $L$ is related to the (downward) $\gamma$-ray strength function ($\gamma$SF) $f$ as follows: \[T_{\gamma}^{L}=2\pi E_{\gamma}^{2L+1}f(E_{\gamma}).\] (1) Much effort has been and still is devoted to measuring and understanding the electric dipole strength function that exhibits a pronounced peak at the giant dipole resonance (GDR) energy. There are many approaches used to derive $f$, each leading to an energy dependence of the $E1$ transmission coefficients given near the GDR energy by a Lorentzian function. Experimental photoabsorption data confirm the simple semi-classical prediction of a Lorentzian shape at energies near the resonance energy but this description is less satisfactory at lower energies, and especially near the photodisintegration reaction threshold Arnould and Goriely (2003). Therefore, it is of substantial interest to develop microscopic models which are expected to provide reasonable reliability and predictive power for the $E1$-$\gamma$SF. Efforts in this direction, such as QRPA calculations Goriely and Khan (2002); Tsoneva _et al._ (2015); Martini _et al._ (2016); Goriely _et al._ (2016, 2018), have been applied successfully to several photoneutron cross section measurements carried out recently with quasi-monochromatic laser-Compton scattering $\gamma$-rays Utsunomiya _et al._ (2009); Tonchev _et al._ (2010); Utsunomiya _et al._ (2013); Raut _et al._ (2013); Filipescu _et al._ (2014); Sauerwein _et al._ (2014); Nyhus _et al._ (2015); Renstrøm _et al._ (2018); Utsunomiya _et al._ (2018). Despite the endemic problem of reproducing the solar abundances of ${}^{92,94}$Mo and ${}^{96,98}$Ru, as well as the part of the A $<$ 124 region, recent studies performed by Travaglio $et$$al.$Travaglio _et al._ (2011, 2015) in supernova Type Ia calculations using both deflagration and delayed detonation models demonstrated that both light and heavy $p$-nuclei, including the much debated isotopes ${}^{92}$Mo and ${}^{96,98}$Ru, are produced with similar enhancement factors relative to solar abundances, provided an $s$-process enrichment of the progenitor is assumed. The model, however, predicts the production of ${}^{94}$Mo with a much lower abundance in comparison to all the other light $p$-nuclei. Another remarkable finding of Ref. Travaglio _et al._ (2011) points out that the $\gamma$-process can make important contributions to the production of the neutron magic nucleus ${}^{90}$Zr, previously known as a genuine $s$-process nuclide. In light of the intriguing findings of Refs. Travaglio _et al._ (2011, 2015), we were motivated to investigate the photoneutron reactions on ${}^{94}$Mo and ${}^{90}$Zr. The measurements were focused on studying the energy dependence of the photoneutron reaction cross sections near the respective neutron emission thresholds and up to 13.5 MeV, taking into account the fact that the energy window of effective stellar burning for photoneutron reactions is located close to the reaction threshold at $E_{\gamma}^{eff}$= ($l$+1/2)$kT$+ $S_{n}$, where $l$ is the neutron orbital angular momentum, $k$ is the universal Boltzmann constant, $T$ is the stellar temperature, and $S_{n}$ is the neutron separation energy. The experimental photroneutron cross sections are compared to the predictions of HF statistical model calculations using different models for the $\gamma$SF, thus allowing the $\gamma$SF to be constrained and further to estimate the corresponding photoneutron stellar reaction rates which directly influence the $p$-process nucleosynthesis. ## II Experimental details In this experiment excitation functions of ($\gamma$,n) photodisintegration reactions on the nuclei ${}^{94}$Mo and ${}^{90}$Zr were measured close to and above the corresponding neutron emission thresholds - $S_{n}$ = 9.68 MeV for ${}^{94}$Mo($\gamma$,n) and $S_{n}$ = 11.97 MeV for ${}^{90}$Zr($\gamma$,n). The measurements reported in this paper extend to a $\gamma$-ray energy of 13.5 MeV and were performed using TUNL’s High Intensity $\gamma$-ray Source (HI$\gamma$S) facility. Quasi-monoenergetic, circularly-polarized and highly intense beams of real photons, of selectable energy, were produced via intracavity back-scattering of free-electron laser (FEL) photons from relativistic electrons Weller _et al._ (2009). These beams were collimated to a diameter of 1.5 cm by a 101.6-cm long Al collimator, which was located 53 m downstream from the collision point inside the optical cavity of the HI$\gamma$S FEL storage ring and 5 m upstream from the experimental setup in the Upstream Target Room (UTR) at HI$\gamma$S. An Al collimator was used, instead of the Pb collimator generally in use at HI$\gamma$S, to limit the beam-induced neutron background in the ${}^{3}$He proportional counters used for the neutron detection. That is because the ${}^{27}$Al($\gamma$,n) reaction has a neutron emission threshold of 13.1 MeV, higher than nearly all of the $\gamma$-ray beam energies of interest in this experiment and higher than the neutron emission threshold for Pb. A schematic drawing (not to scale) of the experimental setup as it was assembled for the present experiment is shown in Fig. 1. <figure><img src="content_image/1808.04937/x1.png"><figcaption>Figure 1: (Color online) Schematic drawing (not to scale) of the experimentalsetup in the HIγS Upstream Target Room (UTR). See the text for details of thedetectors sketched in the figure.</figcaption></figure> The quasi-monoenergetic $\gamma$-ray beam had an energy width in the range of 4% - 5% (FWHM). The $\gamma$-ray beam flux was continuously monitored and yielded values in the range of 10${}^{7}$- 10${}^{8}$$\gamma$/s on target. The very high $\gamma$-ray flux available at HI$\gamma$S makes this facility ideal for the investigation of photoneutron reaction cross sections with $p$-nuclei as targets. The two targets of interest consisted of 98.97% enriched ${}^{94}$Mo with an areal density of 598 mg/cm${}^{2}$ and 97.70% enriched ${}^{90}$Zr with an areal density of 1087 mg/cm${}^{2}$. As illustrated in Fig. 1, the targets were mounted in the longitudinal and axial center of the INVS detector, inside a polycarbonate vacuum pipe kept under rough vacuum to prevent background due to scattering in the air of the $\gamma$-ray beam. Since the ${}^{94}$Mo($\gamma$,n)${}^{93}$Mo reaction produces the unstable residual isotope of ${}^{93}$Mo, which has a long half-life of $T_{1/2}$ = 3500 yr and decays by electron capture without $\gamma$-ray emission, the only way to experimentally study at HI$\gamma$S the excitation function for the ${}^{94}$Mo($\gamma$,n)${}^{93}$Mo reaction was by direct neutron counting. The overall 98-hour beam time of the experiment was divided to accommodate beam energy measurements, ($\gamma$,n) reaction cross section measurements, as well as beam-induced background measurements of relevance for accurate neutron counting. In the following subsections the experimental details of these measurements are provided. ### $\gamma$-ray beam energy measurements The $\gamma$-ray beam energies were measured with a large high-purity germanium (HPGe) detector of 123% relative efficiency appropriately positioned downstream of the target and the neutron detector array. As indicated in Fig. 1, the HPGe detector was remotely positioned on the beam axis for the $\gamma$-ray beam energy measurements and out of the beam axis during the ($\gamma$,n) reaction measurements. Sets of copper attenuators of precisely known thicknesses, stationed behind the exit mirror of the FEL optical cavity Weller _et al._ (2009), were remotely inserted in the beam to reduce the $\gamma$-ray beam flux during the beam energy measurements. At $\gamma$-ray beam energies above 9 MeV, of interest for this work, the measured $\gamma$-ray beam energy spectrum becomes strongly convolved with the detector response function resulting in a broad energy peak with overlapping full-energy peak, first- and second-escape peaks, and their respective Compton edges. To unfold the photon beam full-energy peak, a 7.6-cm thick segmented NaI(Tl) annulus was mounted around the 123% HPGe detector that enabled the extraction of the $\gamma$-ray beam energy distribution. The detection of photons escaping the HPGe detector due to pair production or Compton scattering in any of the four NaI(Tl) segments of the annulus detector was recorded in anticoincidence with the HPGe detector. As shown in Fig. 2, at a $\gamma$-ray beam energy of 9 MeV, the $\gamma$-ray beam energy distribution was extracted by fitting the high-energy tail of the full-energy peak with a Gaussian function. The fit parameters which minimize the $\chi^{2}$-value of the fit give the full-energy peak of the $\gamma$-ray beam and its energy width. The 123% HPGe detector had an energy resolution of approximately 4 keV and was calibrated using the $\gamma$-ray lines from thermal neutron capture on a ${}^{58}$Ni target as well as from standard calibration sources and the activity of naturally occurring radioactive nuclei present in the UTR. <figure><img src="content_image/1808.04937/x2.png"><figcaption>Figure 2: (Color online) γ-ray spectrum recorded by the 123% HPGe detector inanticoincidence with the NaI(Tl) annulus at a γ-ray beam energy of 9 MeV. Thered curve represents a Gaussian fit from where the centroid of the full-energypeak of the γ-ray beam was determined. The peak at 8.5 MeV corresponds to thefirst-escape peak.</figcaption></figure> ### $\gamma$-ray beam flux measurements The $\gamma$-ray beam flux was measured with a single Bicron BC-400 thin plastic scintillation detector (dubbed “paddle detector” because of its geometrical shape) coupled to a photomultiplier tube Pywell _et al._ (2009). The paddle detector was located behind the Al collimator (see Fig. 1). By design, the scintillating paddle detects recoil electrons and positrons from the photoelectric effect, Compton and pair-production processes. Its efficiency has been shown Pywell _et al._ (2009) to be well described by GEANT4 GEANT4 (2003) simulations. The flux stability in the paddle detector was monitored throughout the experiment by cross-checking it against flux measurements of the d($\gamma$,n)p reaction, which has a very well-studied excitation function Birenbaum _et al._ (1985); Bernabei _et al._ (1986); Schiavilla (2005). The corresponding experimental setup for the detection of neutrons from the deuteron photodisintegration consisted of two Bicron BC-501A organic liquid scintillator detectors coupled to photomultiplier tubes that were placed 46 cm from a deuterated benzene cell at a scattering angle of 90 . As illustrated in Fig. 1, the setup was located downstream at the end of the UTR with the deuterated benzene located on the axis of the $\gamma$-ray beam. ### Neutron detection In this experiment the neutrons were detected using an assembly of 18 tubular proportional counters, filled with ${}^{3}$He gas at $\sim$6 atm Arnold _et al._ (2011). The neutron detector array, known as the model IV inventory sample counter (INVS), was originally developed at Los Alamos National Laboratory and it is presently available at TUNL. The tubes are arranged in two concentric rings of radii 7.24 cm and 10.60 cm, each containing nine equally spaced counters. The counters are embedded in a cylindrical polyethylene body 46.2 cm long and 30.5 cm in diameter which serves as a neutron moderator. A schematic drawing of the INVS detector is presented in Fig. 3. <figure><img src="content_image/1808.04937/x3.png"><figcaption>Figure 3: (Color online) Axial and side cut-away views of the INVS detectorshowing the 3He tubes arranged in two concentric rings around a centralcavity. See the text for information on dimensions.</figcaption></figure> The neutron detector array was readout by three TTL logic pulses corresponding to detections occurring in the inner ring (I), the outer ring of the array (O), and the logical OR of the I and O pulses. All three logic output signals were recorded in scalers that were integrated in the CODA (CEBAF Online Data Acquisition) data acquisition system. To limit the rate of detection of background neutrons generated outside of the INVS detector, layers of borated polyethylene and Cd sheets were placed around it, as illustrated in Fig. 1. The neutron detection efficiency depends on the neutron energy. Neutron energies of this work span a broad range from 20 keV to $\sim$4 MeV that correspond to detection efficiencies as high as $\sim$55% and as low as $\sim$25%, respectively. The efficiencies were simulated in GEANT4 assuming isotropically distributed neutrons, and were consistent with the experimentally measured efficiencies of Arnold $et$$al.$Arnold _et al._ (2011). However, it should be noted that for the energy range studied in this work, the ${}^{94}$Mo($\gamma$,n) and ${}^{90}$Zr($\gamma$,n) reactions only proceed directly to the ground state of their respective residual nuclei at low photon beam energies. At higher energies, population of the ground state proceeds predominantly via the population of excited states in the residual nuclei which then $\gamma$ decay to the ground state. Because the information on the neutron energy is lost by the thermalization of the neutrons in the moderator, determining the neutron detection efficiency for such a detector is a complex problem to tackle. Thus, instead of determining the efficiency for the ($\gamma,n_{0}$) channel only that corresponds to neutrons emitted when the ground state in populated directly, contributions from channels which populate excited states in the daughter nucleus must also be taken into account. An effective efficiency has been defined as \[\epsilon_{n}^{eff}=\sum_{i}b_{i}\epsilon_{n_{i}}(E_{n_{i}}),\] (2) where the $b_{i}$ are the neutron branchings of the ($\gamma,n_{i}$) channel at a given $E_{\gamma}$, and $\epsilon_{n_{i}}$ are the energy-dependent detection efficiencies for neutrons from the ($\gamma,n_{i}$) channel. We calculated the neutron branching ratios $b_{i}$ of the ($\gamma,n_{i}$) channels using the TALYS nuclear reaction code Koning and Rochman (2012) with the $\gamma$SF axially-symmetric-deformed Hartree-Fock-Bogoliubov (HFB) plus QRPA model based on the D1M Gogny interaction Martini _et al._ (2016); Goriely _et al._ (2016); Péru and Goutte (2008); Goriely _et al._ (2018), the HFB plus combinatorial nuclear level density model Goriely _et al._ (2008), and with the spherical neutron-nucleus optical-model potential of Koning and Delaroche Koning and Delaroche (2003). More details about this theoretical framework are given in Section IV. For the photon energy range reached in this experiment and under consideration of quantum mechanical selection rules we considered several contributions of the excited states in the residual ${}^{89}$Zr and ${}^{93}$Mo nuclei which can be populated by neutron $s$- and$/$or $p$-waves from $E1$ excitations of $1^{-}$ states in the ${}^{90}$Zr and ${}^{94}$Mo nuclei of interest, as presented in the following. Note: TALYS database uses the RIPL-3 library Capote _et al._ (2009) for the treatment of the discrete levels. #### ii.3.1 ${}^{90}$Zr($\gamma$,n) The highest photon beam energy reached in this experiment at $E_{\gamma}$ = 13.5 MeV will give access to an excitation window in ${}^{89}$Zr up to $\sim$1.5 MeV from the neutron emission threshold at $S_{n}$ = 11.97 MeV. At photon beam energies $E_{\gamma}$$<$$S_{n}$ + 588 keV, the ${}^{90}$Zr($\gamma$,n)${}^{89}$Zr reaction can proceed only to the ground state of ${}^{89}$Zr. This path would be strongly hampered due to the large angular momentum required for the emitted neutron ($f$ wave) from the compound nucleus ${}^{90}$Zr${}^{}$ with $J^{\pi}$ = 1${}^{-}$ to the ground state of ${}^{89}$Zr with $J^{\pi}$ = 9/2${}^{+}$. Starting at photon beam energies larger than 12.8 MeV the population of the ground state in ${}^{90}$Zr proceeds predominantly via emitted neutrons ($s$ wave) from the first two excited states of ${}^{89}$Zr at 588 keV and 1.095 MeV. Table I presents the energies of the emitted neutrons with their corresponding detection efficiencies and branching ratios as simulated in GEANT4 and calculated in TALYS, respectively. The value of the effective neutron efficiency calculated from Eq. (2) for each of the photon beam energy reached in this experiment is presented in the last column of Table I. Eγ [MeV] \| Ei [MeV] \| Jπii \| Eni [MeV] \| li \| ϵni [%] \| bi \| ϵeffn [%] ---\|---\|---\|---\|---\|---\|---\|--- 12 \| 0 \| 9/2+ \| 0.03 \| 3 (f wave) \| 52.89 \| 1 \| 52.89 12.1 \| 0 \| 9/2+ \| 0.13 \| 3 (f wave) \| 52.15 \| 1 \| 52.15 12.2 \| 0 \| 9/2+ \| 0.23 \| 3 (f wave) \| 51.53 \| 1 \| 51.53 12.4 \| 0 \| 9/2+ \| 0.43 \| 3 (f wave) \| 49.21 \| 1 \| 49.21 12.5 \| 0 \| 9/2+ \| 0.53 \| 3 (f wave) \| 47.69 \| 1 \| 47.69 12.8 \| 0 \| 9/2+ \| 0.82 \| 3 (f wave) \| 44.18 \| 0.17 \| 49.94 \| 0.5878 \| 1/2− \| 0.24 \| 0 (s wave) \| 51.12 \| 0.83 \| 13 \| 0 \| 9/2+ \| 1.02 \| 3 (f wave) \| 41.33 \| 0.23 \| 46.94 \| 0.5878 \| 1/2− \| 0.44 \| 0 (s wave) \| 48.61 \| 0.77 \| 13.5 \| 0 \| 9/2+ \| 1.51 \| 3 (f wave) \| 36.71 \| 0.26 \| 42.97 \| 0.5878 \| 1/2− \| 0.93 \| 0 (s wave) \| 42.68 \| 0.45 \| \| 1.0949 \| 3/2− \| 0.43 \| 0 (s wave) \| 49.02 \| 0.29 \| Table 1: 90Zr(γ,n)89Zr: Photon beam energies (Eγ), energy levels in 89Zr (Ei), energies of emitted neutrons calculated as Eni=(8990)(Eγ−Sn−Ei), orbital angular momentum of the emitted neutrons (li), percent neutron detection efficiency at each neutron energy (ϵni), neutron branching ratio for each neutron energy (bi), and the percent effective neutron detection efficiency, as calculated from Eq. (2), for each photon beam energy (ϵeffn). #### ii.3.2 ${}^{94}$Mo($\gamma$,n) The highest photon beam energy reached in this experiment at $E_{\gamma}$ = 13.5 MeV will give access to an excitation window in ${}^{93}$Mo up to $\sim$3.8 MeV from the neutron emission threshold at $S_{n}$ = 9.68 MeV. At photon beam energies $E_{\gamma}$$<$$S_{n}$ + 943 keV, the ${}^{94}$Mo($\gamma$,n)${}^{93}$Mo reaction can proceed only to the ground state of ${}^{93}$Mo. This path would be strongly hampered due to the large angular momentum required for the emitted neutron ($p$ wave) from the compound nucleus ${}^{94}$Mo${}^{}$ with $J^{\pi}$ = 1${}^{-}$ to the ground state of ${}^{93}$Mo with $J^{\pi}$ = 5/2${}^{+}$. Starting at photon beam energies larger than 10.8 MeV the population of the ground state in ${}^{93}$Mo proceeds predominantly via emitted neutrons ($s$ wave or $p$ wave) from 22 excited states of ${}^{93}$Mo. Tables II and III present the energies of the emitted neutrons with their corresponding detection efficiencies and branching ratios as simulated in GEANT4 and calculated in TALYS, respectively. The value of the effective neutron efficiency calculated from Eq. (2) for each of the photon beam energy reached in this experiment is presented in the last columns of the tables. Eγ [MeV] \| Ei [MeV] \| Jπii \| Eni [MeV] \| li \| ϵni [%] \| bi \| ϵeffn [%] ---\|---\|---\|---\|---\|---\|---\|--- 9.7 \| 0 \| 5/2+ \| 0.02 \| 1 (p wave) \| 53.79 \| 1 \| 53.79 9.75 \| 0 \| 5/2+ \| 0.07 \| 1 (p wave) \| 53.29 \| 1 \| 53.29 9.8 \| 0 \| 5/2+ \| 0.12 \| 1 (p wave) \| 53.03 \| 1 \| 53.03 9.85 \| 0 \| 5/2+ \| 0.17 \| 1 (p wave) \| 52.44 \| 1 \| 52.44 9.95 \| 0 \| 5/2+ \| 0.27 \| 1 (p wave) \| 51.45 \| 1 \| 51.45 10 \| 0 \| 5/2+ \| 0.32 \| 1 (p wave) \| 50.30 \| 1 \| 50.30 10.2 \| 0 \| 5/2+ \| 0.52 \| 1 (p wave) \| 47.74 \| 1 \| 47.74 10.5 \| 0 \| 5/2+ \| 0.81 \| 1 (p wave) \| 44.03 \| 1 \| 44.03 10.8 \| 0 \| 5/2+ \| 1.11 \| 1 (p wave) \| 40.51 \| 0.59 \| 45.35 \| 0.9433 \| 1/2+ \| 0.18 \| 1 (p wave) \| 52.32 \| 0.41 \| 11 \| 0 \| 5/2+ \| 1.31 \| 1 (p wave) \| 38.37 \| 0.46 \| 44.73 \| 0.9433 \| 1/2+ \| 0.37 \| 1 (p wave) \| 50.15 \| 0.54 \| 11.5 \| 0 \| 5/2+ \| 1.80 \| 1 (p wave) \| 34.35 \| 0.31 \| 42.83 \| 0.9433 \| 1/2+ \| 0.87 \| 1 (p wave) \| 43.23 \| 0.37 \| \| 1.4925 \| 3/2+ \| 0.33 \| 1 (p wave) \| 50.19 \| 0.26 \| \| 1.6950 \| 5/2+ \| 0.13 \| 1 (p wave) \| 52.18 \| 0.06 \| 11.65 \| 0 \| 5/2+ \| 1.95 \| 1 (p wave) \| 33.14 \| 0.29 \| 42.46 \| 0.9433 \| 1/2+ \| 1.02 \| 1 (p wave) \| 41.68 \| 0.33 \| \| 1.4925 \| 3/2+ \| 0.47 \| 1 (p wave) \| 48.39 \| 0.28 \| \| 1.6950 \| 5/2+ \| 0.27 \| 1 (p wave) \| 50.43 \| 0.10 \| 11.8 \| 0 \| 5/2+ \| 2.10 \| 1 (p wave) \| 32.25 \| 0.29 \| 40.71 \| 0.9433 \| 1/2+ \| 1.17 \| 1 (p wave) \| 39.94 \| 0.30 \| \| 1.4925 \| 3/2+ \| 0.62 \| 1 (p wave) \| 46.42 \| 0.28 \| \| 1.6950 \| 5/2+ \| 0.42 \| 1 (p wave) \| 49.04 \| 0.13 \| 11.95 \| 0 \| 5/2+ \| 2.25 \| 1 (p wave) \| 32.99 \| 0.26 \| 41.57 \| 0.9433 \| 1/2+ \| 1.31 \| 1 (p wave) \| 38.36 \| 0.25 \| \| 1.4925 \| 3/2+ \| 0.77 \| 1 (p wave) \| 44.66 \| 0.24 \| \| 1.6950 \| 5/2+ \| 0.57 \| 1 (p wave) \| 47.25 \| 0.11 \| \| 2.1420 \| 5/2+ \| 0.129 \| 1 (p wave) \| 52.30 \| 0.04 \| \| 2.1454 \| 3/2+, 5/2+ \| 0.125 \| 1 (p wave) \| 52.32 \| 0.07 \| \| 2.1811 \| 3/2+ \| 0.09 \| 1 (p wave) \| 52.83 \| 0.03 \| 12.25 \| 0 \| 5/2+ \| 2.25 \| 1 (p wave) \| 29.54 \| 0.23 \| 41.32 \| 0.9433 \| 1/2+ \| 1.61 \| 1 (p wave) \| 35.57 \| 0.16 \| \| 1.4925 \| 3/2+ \| 1.07 \| 1 (p wave) \| 40.99 \| 0.17 \| \| 1.6950 \| 5/2+ \| 0.87 \| 1 (p wave) \| 43.44 \| 0.08 \| \| 2.1420 \| 5/2+ \| 0.43 \| 1 (p wave) \| 48.93 \| 0.07 \| \| 2.1454 \| 3/2+, 5/2+ \| 0.42 \| 1 (p wave) \| 48.92 \| 0.12 \| \| 2.1811 \| 3/2+ \| 0.39 \| 1 (p wave) \| 49.72 \| 0.12 \| \| 2.4374 \| 1/2+ \| 0.13 \| 1 (p wave) \| 51.98 \| 0.04 \| \| 2.5297 \| 1/2−, 3/2− \| 0.04 \| 0 (s wave) \| 52.82 \| 0.01 \| Table 2: 94Mo(γ,n)93Mo: Photon beam energies (Eγ), energy levels in 89Zr (Ei), energies of emitted neutrons calculated as Eni=(9394)(Eγ−Sn−Ei), orbital angular momentum of the emitted neutrons (li), percent neutron detection efficiency at each neutron energy (ϵni), neutron branching ratio for each neutron energy (bi), and the percent effective neutron detection efficiency, as calculated from Eq. (2), for each photon beam energy (ϵeffn). Eγ [MeV] \| Ei [MeV] \| Jπii \| Eni [MeV] \| li \| ϵni [%] \| bi \| ϵeffn [%] ---\|---\|---\|---\|---\|---\|---\|--- 12.5 \| 0 \| 5/2+ \| 2.79 \| 1 (p wave) \| 28.78 \| 0.23 \| 39.86 \| 0.9433 \| 1/2+ \| 1.86 \| 1 (p wave) \| 33.87 \| 0.10 \| \| 1.4925 \| 3/2+ \| 1.32 \| 1 (p wave) \| 38.37 \| 0.19 \| \| 1.6950 \| 5/2+ \| 1.12 \| 1 (p wave) \| 40.45 \| 0.06 \| \| 2.1420 \| 5/2+ \| 0.673 \| 1 (p wave) \| 45.74 \| 0.05 \| \| 2.1454 \| 3/2+, 5/2+ \| 0.669 \| 1 (p wave) \| 45.60 \| 0.10 \| \| 2.1811 \| 3/2+ \| 0.63 \| 1 (p wave) \| 46.26 \| 0.10 \| \| 2.4374 \| 1/2+ \| 0.38 \| 1 (p wave) \| 49.81 \| 0.08 \| \| 2.5297 \| 1/2−, 3/2− \| 0.29 \| 0 (s wave) \| 50.60 \| 0.01 \| \| 2.6190 \| 1/2−, 3/2− \| 0.20 \| 0 (s wave) \| 51.55 \| 0.01 \| \| 2.6701 \| 1/2+ \| 0.15 \| 1 (p wave) \| 52.15 \| 0.04 \| \| 2.7046 \| 1/2+ \| 0.12 \| 1 (p wave) \| 52.26 \| 0.03 \| 12.8 \| 0 \| 5/2+ \| 3.09 \| 1 (p wave) \| 27.66 \| 0.28440 \| 37.16 \| 0.9433 \| 1/2+ \| 2.16 \| 1 (p wave) \| 31.84 \| 0.07408 \| \| 1.4925 \| 3/2+ \| 1.61 \| 1 (p wave) \| 35.74 \| 0.14420 \| \| 1.6950 \| 5/2+ \| 1.41 \| 1 (p wave) \| 37.65 \| 0.07771 \| \| 2.1420 \| 5/2+ \| 0.970 \| 1 (p wave) \| 42.27 \| 0.03704 \| \| 2.1454 \| 3/2+, 5/2+ \| 0.966 \| 1 (p wave) \| 42.25 \| 0.07167 \| \| 2.1811 \| 3/2+ \| 0.93 \| 1 (p wave) \| 46.46 \| 0.06981 \| \| 2.4374 \| 1/2+ \| 0.68 \| 1 (p wave) \| 45.92 \| 0.06373 \| \| 2.5297 \| 1/2−, 3/2− \| 0.59 \| 0 (s wave) \| 46.71 \| 0.01074 \| \| 2.6190 \| 1/2−, 3/2− \| 0.50 \| 0 (s wave) \| 48.01 \| 0.00883 \| \| 2.6701 \| 1/2+ \| 0.45 \| 1 (p wave) \| 48.90 \| 0.05549 \| \| 2.7046 \| 1/2+ \| 0.41 \| 1 (p wave) \| 49.35 \| 0.05335 \| \| 2.8421 \| 1/2+ \| 0.28 \| 1 (p wave) \| 50.66 \| 0.04113 \| \| 2.9552 \| 1/2−, 3/2− \| 0.17 \| 0 (s wave) \| 51.80 \| 0.00499 \| \| 3.0640 \| 1/2−, 3/2− \| 0.06 \| 0 (s wave) \| 52.66 \| 0.00283 \| 13.5 \| 0 \| 5/2+ \| 3.78 \| 1 (p wave) \| 25.35 \| 0.32964 \| 32.93 \| 0.9433 \| 1/2+ \| 2.85 \| 1 (p wave) \| 28.48 \| 0.07525 \| \| 1.4925 \| 3/2+ \| 2.30 \| 1 (p wave) \| 30.79 \| 0.10762 \| \| 1.6950 \| 5/2+ \| 2.10 \| 1 (p wave) \| 32.17 \| 0.07133 \| \| 2.1420 \| 5/2+ \| 1.660 \| 1 (p wave) \| 35.24 \| 0.03091 \| \| 2.1454 \| 3/2+, 5/2+ \| 1.659 \| 1 (p wave) \| 35.27 \| 0.05072 \| \| 2.1811 \| 3/2+ \| 1.62 \| 1 (p wave) \| 35.53 \| 0.04801 \| \| 2.4374 \| 1/2+ \| 1.37 \| 1 (p wave) \| 38.07 \| 0.04383 \| \| 2.5297 \| 1/2−, 3/2− \| 1.28 \| 0 (s wave) \| 38.71 \| 0.01244 \| \| 2.6190 \| 1/2−, 3/2− \| 1.19 \| 0 (s wave) \| 39.69 \| 0.00935 \| \| 2.6701 \| 1/2+ \| 1.14 \| 1 (p wave) \| 40.15 \| 0.04146 \| \| 2.7046 \| 1/2+ \| 1.11 \| 1 (p wave) \| 40.43 \| 0.04142 \| \| 2.8421 \| 1/2+ \| 0.97 \| 1 (p wave) \| 42.03 \| 0.04114 \| \| 2.9552 \| 1/2−, 3/2− \| 0.86 \| 0 (s wave) \| 43.52 \| 0.00869 \| \| 3.0640 \| 1/2−, 3/2− \| 0.75 \| 0 (s wave) \| 44.92 \| 0.00680 \| \| 3.1592 \| 3/2+, 5/2+ \| 0.66 \| 1 (p wave) \| 45.77 \| 0.02017 \| \| 3.3876 \| 3/2+, 5/2+ \| 0.43 \| 1 (p wave) \| 48.92 \| 0.01743 \| \| 3.4503 \| 3/2+, 5/2+ \| 0.37 \| 1 (p wave) \| 49.78 \| 0.03078 \| \| 3.5900 \| 1/2−, 3/2− \| 0.23 \| 0 (s wave) \| 50.66 \| 0.00354 \| \| 3.5963 \| 3/2+, 5/2+ \| 0.22 \| 1 (p wave) \| 51.08 \| 0.00348 \| \| 3.7089 \| 3/2+, 5/2+ \| 0.11 \| 1 (p wave) \| 52.06 \| 0.00241 \| \| 3.7200 \| 1/2−, 3/2− \| 0.10 \| 0 (s wave) \| 52.33 \| 0.00229 \| \| 3.7900 \| 1/2−, 3/2− \| 0.03 \| 0 (s wave) \| 52.57 \| 0.00129 \| Table 3: 94Mo(γ,n)93Mo: Photon beam energies (Eγ), energy levels in 89Zr (Ei), energies of emitted neutrons calculated as Eni=(9394)(Eγ−Sn−Ei), orbital angular momentum of the emitted neutrons (li), percent neutron detection efficiency at each neutron energy (ϵni), neutron branching ratio for each neutron energy (bi), and the percent effective neutron detection efficiency, as calculated from Eq. (2), for each photon beam energy (ϵeffn). To assess the uncertainty in the calculation of the neutron branching ratios in TALYS we considered another four sets of nuclear inputs such as: Input-1Level density: Constant-temperature (CT) plus Fermi gas model Koning _et al._ (2008); Optical potential: Koning and Delaroche Koning and Delaroche (2003); $\gamma$ strength: axially-symmetric-deformed Hartree-Fock-Bogoliubov (HFB) plus QRPA model based on the D1M Gogny interaction Martini _et al._ (2016); Goriely _et al._ (2016); Péru and Goutte (2008); Goriely _et al._ (2018) Input-2Level density: HFB plus combinatorial nuclear level model Goriely _et al._ (2008); Optical potential: Koning and Delaroche Koning and Delaroche (2003); $\gamma$ strength: Generalized Lorentzian (GLO) model Kopecky and Uhl (1990); Capote _et al._ (2009) Input-3Level density: HFB plus combinatorial nuclear level model Goriely _et al._ (2008); Optical potential: semi-microscopic neutron-nucleus spherical optical model potential from the nuclear matter approach of Jeukenne, Lejeune, and Mahaux (JLM) Bauge _et al._ (1998, 2001); $\gamma$ strength: axially-symmetric-deformed Hartree-Fock-Bogoliubov (HFB) plus QRPA model based on the D1M Gogny interaction Martini _et al._ (2016); Goriely _et al._ (2016); Péru and Goutte (2008); Goriely _et al._ (2018) Input-4Level density: HFB plus combinatorial nuclear level model Goriely _et al._ (2008); Optical potential: semi-microscopic neutron-nucleus spherical optical model potential from the nuclear matter approach of Jeukenne, Lejeune, and Mahaux (JLM) Bauge _et al._ (1998, 2001); $\gamma$ strength: Generalized Lorentzian (GLO) model Kopecky and Uhl (1990); Capote _et al._ (2009). We obtained very similar values for the neutron branching ratios with differences varying between less than 1$\%$ and $\sim$3$\%$. ### Beam-induced background measurements Accurate neutron counting in the ${}^{3}$He counters requires the ability to distinguish ($\gamma$,n) events originating in the target from beam-induced background events. That is particularly important for the ($\gamma$,n) reaction cross sections measured at the neutron emission threshold where the photoneutron cross sections are very small but astrophysically relevant. For the ${}^{94}$Mo($\gamma$,n) reaction cross section measurements, the ${}^{90}$Zr target with an atomic mass number $Z$ = 40, close to the atomic mass number of $Z$ = 42 for Mo but with a higher $S_{n}$ of 11.97 MeV, was used to mimic the $\gamma$-ray induced background in the ${}^{3}$He counters caused by Compton scattering and pair production from the ${}^{94}$Mo target. Because ${}^{nat}$H has a deuterium ($S_{n}$ = 2.225 MeV) abundance of 0.016% and the d($\gamma$,n) reaction cross section peaks at 2.5 mb, a significant amount of H needs to be in the path of the $\gamma$-ray beam for this background to be measurable. The polyethylene moderator of the INVS, however, cannot be removed as it is an integral part of the detector. Thus, $\gamma$-ray beam induced-neutron background measurements with the ${}^{90}$Zr target were carried out at photon beam energies corresponding to the cross section measurements of the ${}^{94}$Mo($\gamma$,n) reaction. At the neutron emission threshold ($S_{n}$ = 9.68 MeV), the background was about 25% of the total counting rate of the INVS detector. Once the rates of the beam-induced background were measured on the ${}^{90}$Zr target, the corresponding background rates for the ${}^{94}$Mo($\gamma$,n) reaction cross section measurements were determined by scaling the rates with a factor of $\sim$0.6 which comes from target thickness normalization. For the ${}^{90}$Zr($\gamma$,n) reaction cross section measurements, the beam-induced background rates registered in the INVS detector at $\gamma$-ray beam energies below the ${}^{90}$Zr neutron emission threshold were extrapolated by a linear fit to the energies above the threshold at which the ${}^{90}$Zr($\gamma$,n) reaction cross section measurements were performed. In addition to the beam-induced background runs, empty target runs were also carried out to account for possible neutron-induced background by the INVS detector itself. At $\gamma$-ray beam energies close to the neutron threshold, the empty target rates, of about 3% of the total INVS rate, were insignificant compared to the rates recorded for the $\gamma$-ray-beam-induced background on the ${}^{90}$Zr target. ## III Data analysis and results Under the assumption of a monoenergetic $\gamma$-ray beam, the photoneutron reaction cross section as a function of beam energy may be written as: \[\sigma_{(\gamma,n)}(E_{\gamma})=(R_{n}-R_{bkgd})/(R_{\gamma}\cdot N_{t}\cdot f \cdot\epsilon^{eff}_{n}(E_{\gamma})),\] (3) where $R_{n}$ is the rate of total number of detected neutrons, $R_{bkgd}$ is the rate of background events, $R_{\gamma}$ is the rate of the incident $\gamma$-ray beam, $N_{t}$ is the number of atoms in the target per unit area, $f$ is the thick-target correction factor calculated as $f=(1-e^{-\mu d})/(\mu d)$ with the linear attenuation coefficient of photons ($\mu$) and the target thickness ($d$), and $\epsilon^{eff}_{n}$ is the effective neutron detection efficiency as calculated from Eq. (2). $R_{\gamma}$ was determined as the ratio between the rate measured in the paddle detector and the detection efficiency of the paddle detector. Contributions to the systematic experimental uncertainties of the cross section measurements come mainly from target thickness (2%), the error in the simulation of the ${}^{3}$He counter array’s neutron detection efficiency (3%), and the error in the simulation of the efficiency of the paddle detector (4%). In Tables IV and V the experimental values are presented for the two ($\gamma$,n) reaction cross sections obtained using Eq. (3). Eγ[MeV] \| σEγ[MeV] \| σ(γ,n)[mb] \| η = ϵn0ϵeffn = σ(γ,n)σ(γ,n0) ---\|---\|---\|--- 11.75 \| 0.21 \| 0.01 ± 0.01 \| 1 12 \| 0.23 \| 0.11 ± 0.01 \| 1 12.1 \| 0.21 \| 0.14 ± 0.02 \| 1 12.2 \| 0.22 \| 0.50 ± 0.03 \| 1 12.4 \| 0.22 \| 2.28 ± 0.12 \| 1 12.5 \| 0.23 \| 4.42 ± 0.24 \| 1 12.8 \| 0.23 \| 9.67 ± 0.52 \| 0.88 13 \| 0.22 \| 12.66 ± 0.68 \| 0.88 13.5 \| 0.24 \| 20.94 ± 1.13 \| 0.85 Table 4: Experimental cross sections for the 90Zr(γ,n) reaction, as determined from Eq. (3), along with their uncertainties. Measurements were performed for a Gaussian full-energy peak of the γ-ray beams with mean Eγ and spread σEγ. The η ratio in the last column is the ratio between the neutron detection efficiency of the (γ,n0) channel and the effective neutron efficiency, which is also the ratio between the measured σ(γ,n) cross section and the σ(γ,n0) cross section that would correspond to the detection of neutrons emitted only when the ground state is populated directly. See TABLE I for the corresponding values of ϵn0 and ϵeffn. Eγ[MeV] \| σEγ[MeV] \| σ(γ,n)[mb] \| η = ϵn0ϵeffn = σ(γ,n)σ(γ,n0) ---\|---\|---\|--- 9.5 \| 0.18 \| 0.28 ± 0.02 \| 1 9.6 \| 0.17 \| 1.21 ± 0.07 \| 1 9.65 \| 0.17 \| 2.51 ± 0.14 \| 1 9.7 \| 0.17 \| 2.97 ± 0.16 \| 1 9.75 \| 0.17 \| 4.50 ± 0.24 \| 1 9.8 \| 0.17 \| 4.93 ± 0.27 \| 1 9.85 \| 0.17 \| 6.28 ± 0.34 \| 1 9.95 \| 0.16 \| 7.83 ± 0.42 \| 1 10 \| 0.19 \| 8.44 ± 0.46 \| 1 10.2 \| 0.17 \| 10.11 ± 0.55 \| 1 10.5 \| 0.17 \| 11.77 ± 0.63 \| 1 10.8 \| 0.17 \| 13.06 ± 0.70 \| 0.89 11 \| 0.17 \| 14.53 ± 0.78 \| 0.86 11.5 \| 0.24 \| 17.47 ± 0.94 \| 0.80 11.65 \| 0.25 \| 18.73 ± 1.01 \| 0.78 11.8 \| 0.22 \| 20.63 ± 1.11 \| 0.79 11.95 \| 0.23 \| 22.61 ± 1.22 \| 0.79 12.25 \| 0.22 \| 24.20 ± 1.30 \| 0.71 12.5 \| 0.23 \| 27.86 ± 1.50 \| 0.72 12.8 \| 0.23 \| 32.39 ± 1.74 \| 0.74 13.5 \| 0.24 \| 48.64 ± 2.62 \| 0.77 Table 5: Experimental cross sections for the 94Mo(γ,n) reaction, determined from Eq. (3), along with their uncertainties (same as TABLE IV). See TABLES II and III for the corresponding values of ϵn0 and ϵeffn. As mentioned previously, the HI$\gamma$S $\gamma$-ray beam had an energy width in the range of 4% - 5% (FWHM). Hence, the experimental cross sections determined from Eq. (2) do not represent cross section values at single energies, but rather cross section integrated over an energy range defined by the width of the $\gamma$-ray beam energy profile. The effects of this convolution are particularly significant near the neutron emission threshold, where the cross section changes rapidly and the beam energy distribution extends both above and below the reaction threshold. Since this is the energy region most relevant in astrophysics, the energy window of effective stellar burning for photoneutron reactions, it is important to deconvolve the effects of the finite $\gamma$-ray beam energy distribution to recover the photoneutron cross section. An iterative fitting procedure, dubbed ICARUS (Iterative Code for Automatically Resolving and Unfolding Spectral effects) has been developed for finding an analytical excitation function that when convolved with the $\gamma$-ray beam energy profile will reproduce best the experimental cross sections determined from Eq. (3). ICARUS takes an arbitrary, user defined function with an arbitrary number of fit parameters to represent the photoneutron reaction cross section. ICARUS then convolves that analytical function with the gaussian energy spectrum of the HI$\gamma$S photon beam to produce the effective cross section, or yield, that is measured experimentally. The fit parameters are then varied to minimize the $\chi^{2}$-value of the ICARUS calculated yields, compared with the experimentally measured yields. In the case of the ${}^{94}$Mo($\gamma$,n) reaction, a Gaussian $\gamma$-ray beam energy profile with experimentally measured mean and width values was convolved with an ICARUS excitation function that was a 8-parameter function described as a product between the threshold behavior of the ($\gamma$,n) reaction cross section from Ref. Voght _et al._ (2002) and a fifth-degree polynomial function as follows: \[\begin{split}\sigma_{(\gamma,n)}^{ICARUS}(E_{\gamma})=\sigma_{0}[ (E_{\gamma}-S_{n})/S_{n}]^{p_{1}}\cdot[p_{2}\\ +p_{3}\cdot(E_{\gamma}-S_{n})\\ +p_{4}\cdot(E_{\gamma}-S_{n})^{2}+p_{5}\cdot(E_{\gamma}-S_{n})^{3 }\\ +p_{6}\cdot(E_{\gamma}-S_{n})^{4}+p_{7}\cdot(E_{\gamma}-S_{n})^{5 }].\end{split}\] (4) The best fit values of the $\sigma_{0}$ and $p_{i}$ (i = 1,7) parameters of the analytical cross section function from Eq. (4) are $-$ 30.59 mb, 0.585, 2.961, -3.215, 2.693, -1.098, 0.226, -0.017, respectively. Figure 4 shows the ICARUS fitting results for the experimental excitation function of the ${}^{94}$Mo($\gamma$,n) reaction. The ICARUS fitting procedure was not applicable in case of the ${}^{90}$Zr($\gamma$,n) reaction due to the scarcity of the experimental data points. <figure><img src="content_image/1808.04937/x4.png"><figcaption>Figure 4: (Color online) ICARUS fitting plot for the excitation function ofthe 94Mo(γ,n) reaction. The horizontal error bars represent the measured γ-raybeam energy widths.</figcaption></figure> The _ICARUS excitation function for the ${}^{94}$Mo($\gamma$,n) reaction_ corresponding to the experimental $\gamma$-ray energies and _the experimental excitation function for ${}^{90}$Zr($\gamma$,n) reaction determined from Eq. (3)_ are shown in Figure 5 and 6, respectively, in comparison with the previous measurements carried out with quasi-monochromatic laser-Compton scattering photons Utsunomiya _et al._ (2013) and with quasi-monochromatic annihilation photons Beil _et al._ (1974); Berman _et al._ (1967); Leprêtre _et al._ (1971). <figure><img src="content_image/1808.04937/x5.png"><figcaption>Figure 5: (Color online) ICARUS excitation function for 94Mo(γ,n) of this workcompared with the previous measurements Utsunomiya _et al._ (2013); Beil _etal._ (1974).</figcaption></figure> <figure><img src="content_image/1808.04937/x6.png"><figcaption>Figure 6: (Color online) Excitation function for 90Zr(γ,n) of this workcompared with the previous measurements Berman _et al._ (1967); Leprêtre _etal._ (1971).</figcaption></figure> In the case of the ${}^{94}$Mo($\gamma$,n) reaction, Fig. 5 shows that there is a good agreement between the present results and the previous results by Ustunomiya $et$$al.$Utsunomiya _et al._ (2013) and by Beil $et$$al.$Beil _et al._ (1974) for photon beam energies below 10.8 MeV. However above that energy, when neutrons emitted from excited states in ${}^{93}$Mo that $\gamma$ decay to the ground state contribute to the measured cross sections, present results start to deviate from previous work. Similarly, in the case of the ${}^{90}$Zr($\gamma$,n) reaction, Fig. 6 shows that the present results are in good agreement with the results by Berman $et$$al.$Berman _et al._ (1967) for photon beam energies below 12.8 MeV, whereas above 12.8 MeV when neutrons emitted from excited states in ${}^{89}$Zr that $\gamma$ decay to the ground state contribute to the measured cross sections, the agreement worsens. However, the results of Lepr$\hat{e}$tre $et$$al.$Leprêtre _et al._ (1971) show outstanding discrepancies both with our results and with the results of Ref. Berman _et al._ (1967) for all photon beam energies. Berman $et$$al.$Berman _et al._ (1987) reviewed the inconsistencies between the data of Refs. Berman _et al._ (1967) and Leprêtre _et al._ (1971), notable not only for ${}^{90}$Zr but also for a few other cases where results disagree in the GDR peak height by 15% or more. Particularly for ${}^{90}$Zr, the previous results of Ref. Berman _et al._ (1967) were confirmed, which led to the conclusion in Ref. Berman _et al._ (1987) that there was an error for the dataset of Ref. Leprêtre _et al._ (1971) either in the photon flux determination or in the neutron detection efficiency or in both. In a more recent data evaluation, Varlamov $et$$al.$Varlamov _et al._ (2009) stated that the incorrectness of the special procedure used in the experiments of Ref. Leprêtre _et al._ (1971) to sort photoneutrons in multiplicity is the reason behind the observed discrepancies. This reasoning is used as well in the current EXFOR database [48]. Note that the discrepancies between the data of Refs. Berman _et al._ (1967) and Leprêtre _et al._ (1971) are currently under review within the framework of the IAEA Coordinated Research Project entitled “Updating the Photoneutron Data Library and Generating a Reference Database for Photon Strength Functions”. ## IV Statistical model calculations The photoneutron reaction cross sections of the present work are now compared with theoretical calculations obtained with the TALYS nuclear reaction code Koning and Rochman (2012) and two different models of the $\gamma$SF, namely the Generalized Lorentzian (GLO) model Kopecky and Uhl (1990); Capote _et al._ (2009) and the axially-symmetric-deformed Hartree-Fock-Bogoliubov (HFB) plus QRPA model based on the D1M Gogny interaction Martini _et al._ (2016); Goriely _et al._ (2016); Péru and Goutte (2008); Goriely _et al._ (2018). The D1M$+$QRPA model includes phenomenologically the impact of multi-particle multi-hole configurations as well as phonon coupling and has proven its capacity to reproduce experimental data relatively well Martini _et al._ (2016); Goriely _et al._ (2016, 2018). Both the GLO and D1M+QRPA models are standard inputs in TALYS reaction code and are classically used for practical applications. Since they are based on fundamentally different physics, they can reflect the existing uncertainties affecting the $\gamma$SF, but also the impact of such uncertainties on reaction cross sections and astrophysical rates. The HFB plus combinatorial nuclear level density model Goriely _et al._ (2008) is used for the present photoneutron reaction cross section calculations. ### Cross section calculations and comparison with experimental results The D1M$+$QRPA calculation has been renormalized, as detailed in Refs. Martini _et al._ (2016); Goriely _et al._ (2016, 2018), in order to reproduce the present data. As seen in Fig. 7, this leads, however, to some overestimate of the data in the vicinity of 13 MeV for the ${}^{90}$Zr($\gamma$,n) reaction, while the traditional GLO model, adjusted to the former data Berman _et al._ (1967); Leprêtre _et al._ (1971), strongly overpredicts the present data in the vicinity of 13 MeV and above. The large GDR width adopted in the GLO model to reproduce the data of Ref. Leprêtre _et al._ (1971) in the 13-15 MeV region may be questionable, especially in view of our new low-energy measurements below 14 MeV. <figure><img src="content_image/1808.04937/x7.png"><figcaption>Figure 7: (Color online) Comparison between the present (γ,n) photoneutronreaction cross sections as a function of the γ-ray beam energy and theprevious data Berman _et al._ (1967); Leprêtre _et al._ (1971) for 90Zr.Also included are the predictions obtained with the D1M+QRPA E1 and M1strengths (solid line) and with the GLO model (dotted line).</figcaption></figure> Good agreement between experimental and theoretical photoneutron reaction cross sections is obtained for the ${}^{94}$Mo($\gamma$,n) reaction. In particular, the present data agree fairly well with previous measurements Utsunomiya _et al._ (2013); Beil _et al._ (1974) in the 10-11 MeV range. Fig. 8 shows both the GLO and D1M$+$QRPA models adjusted to reproduce experimental photoneutron cross sections. <figure><img src="content_image/1808.04937/x8.png"><figcaption>Figure 8: (Color online) Comparison between the present (γ,n) photoneutronreaction cross sections as a function of the γ-ray beam energy and theprevious data Utsunomiya _et al._ (2013); Beil _et al._ (1974) for 94Mo.Also included are the predictions obtained with the D1M+QRPA E1 and M1strengths (solid line) and with the GLO model (dotted line).</figcaption></figure> In Fig. 9, we combine experimental information on the $\gamma$SF below and above the neutron separation energy and compare them with the D1M+QRPA and GLO $\gamma$SF. The $\gamma$SF is connected to the photoneutron cross section through \[f(E_{\gamma})=\frac{1}{3\pi^{2}\hbar^{2}c^{2}}\frac{\sigma_{\gamma}(E_{\gamma} )}{E_{\gamma}},\] (5) where the constant $1/3\pi^{2}\hbar^{2}c^{2}=8.67\times 10^{-8}{\rm mb^{-1}MeV^{-2}}$. Note that Eq. 5 holds only when the neutron emission channel dominates over the electromagnetic de-excitation, _i.e._ a few hundred keV above the neutron threshold. For this reason, photodata in this range have not been included in Fig. 9. Both the $E1$ and $M1$ theoretical components are shown separately, as well as the total dipole $E1+M1$ strength. Although, the spin-flip $M1$ strengths are seen to be quite different, the total $\gamma$SF do not differ significantly, _i.e._ typically within less than a factor of 2, even below the neutron separation energy. None of the models are able to reproduce the strong ${}^{90}$Zr and ${}^{94}$Mo low-lying strengths obtained by ($\gamma$,$\gamma^{\prime}$) bremsstrahlung experiments Schwengner _et al._ (2008); Rusev _et al._ (2009). In contrast, the ${}^{94}$Mo dipole strength below 8 MeV extracted with the Oslo method from neutron pickup (${}^{3}$He,$\alpha\gamma$) and inelastic scattering (${}^{3}$He,${}^{3}$He${}^{\prime}$) reactions Utsunomiya _et al._ (2013); Guttormsen _et al._ (2005) agrees relatively well with theory, in particular with D1M+QRPA. Note that the experimental data below 3 MeV is associated with the $M1$ de-excitation strength that is not included here, neither in the GLO nor in the D1M+QRPA photoabsorption description Goriely _et al._ (2018). Discrepancies between the bremsstrahlung and Oslo data are still being investigated, in particular within the above-mentioned IAEA Coordinated Research Project. <figure><img src="content_image/1808.04937/x9.png"><figcaption>Figure 9: (Color online) Comparison between experimental Berman _et al._(1967); Leprêtre _et al._ (1971); Utsunomiya _et al._ (2013); Beil _et al._(1974); Schwengner _et al._ (2008); Rusev _et al._ (2009); Guttormsen _etal._ (2005) and theoretical γ-ray strength function as a function of the γ-rayenergy for (a) 90Zr and (b) 94Mo. The predictions correspond to the D1M+QRPAE1 (red dashed line), M1 (red dotted line) and E1+M1 (red solid line)strengths and the GLO E1 (blue dashed line), M1 (blue dotted line) and E1+M1(blue solid line) strengths.</figcaption></figure> ### Stellar reaction rate calculations Nucleosynthesis investigations require the use of stellar rates for thermal population of excited states in the target. Stellar photoneutron reaction rates are calculated in the TALYS code from the expression \[\lambda_{(\gamma,n)}^{\ast}(T)=\frac{\sum_{\mu}(2J^{\mu}+1) \lambda_{(\gamma,n)}^{\mu}(T)exp(-E^{\mu}/kT)}{\sum_{\mu}(2J^{\mu }+1)exp(-E^{\mu}/kT)},\] (6) where $J_{\mu}$ represents the levels of the target nucleus, $\mu$ labels the thermally populated state, and $E^{\mu}$ stands for the excitation energy of that state. Photoneutron rates $\lambda_{(\gamma,n)}^{\mu}$(T) for individual states are found from the integral of a Planck black-body spectrum $n(E_{\gamma},T)$, which describes the energy distribution of the stellar photons, and the associated photoneutron emission cross section \[\lambda_{(\gamma,n)}^{\mu}(T)=\int_{0}^{\infty}cn_{\gamma}(E,T)\sigma_{(\gamma ,n)}^{\mu}(E)dE,\] (7) where $c$ is the speed of light. In Fig. 10 are shown the resulting ${}^{90}$Zr($\gamma$,n)${}^{89}$Zr and ${}^{94}$Mo($\gamma$,n)${}^{93}$Mo stellar photoneutron rates as a function of the temperature in a typical range of interest for the $p$-process nucleosynthesis Arnould and Goriely (2003). Also shown in Fig. 10, are the competing ${}^{90}$Zr($\gamma$,p)${}^{89}$Y and ${}^{94}$Mo($\gamma$,$\alpha$)${}^{90}$Zr stellar photoreaction rates which dominate the photoneutron channel at temperatures below $T\simeq 4\times 10^{9}$K and $T=2.5\times 10^{9}$K, respectively. Note that the ${}^{90}$Zr($\gamma$,$\alpha$) and ${}^{94}$Mo($\gamma$,p) channels are negligible with respect to the other channels mentioned above. <figure><img src="content_image/1808.04937/x10.png"><figcaption>Figure 10: (Color online) (a) 90Zr(γ,n)89Zr (circles) and 90Zr(γ,p)89Y(squares) stellar reaction rates, as a function of the temperature, obtainedwith the D1M+QRPA (red solid lines) or the GLO (blue dotted lines) γSF shownin Fig. 9. (b) Same for 94Mo(γ,n)93Mo and 94Mo(γ,α)90Zr, respectively.</figcaption></figure> In this temperature range, we obtain a ${}^{94}$Mo($\gamma$,n)${}^{93}$Mo stellar reaction rate with the D1M$+$QRPA dipole strength about 40% higher than with the GLO model, whereas for ${}^{90}$Zr($\gamma$,n)${}^{89}$Zr, a factor of 5 lower is obtained with D1M$+$QRPA adjusted to the present experimental data. This latter case shows how sensitive the stellar photoneutron reaction rate of astrophysical interest can be to experimental data in the vicinity of the neutron emission threshold, but also that even if the $\gamma$SF, hence the total photoabsorption cross sections, differ only by less than a factor of 2 (Figs. 9-10), the partial ($\gamma,n$) rates, strongly dominated by another emission channel, may differ by as much as a factor of 5. In the ${}^{94}$Mo case, the ($\gamma$,$\alpha$) rates obtained with both $\gamma$SF models are quite similar, and larger differences are again found on the photoneutron channel, the D1M+QRPA predictions giving this time higher rates. For a typical temperature of 2.5 GK for the $p$-process nucleosynthesis the ($\gamma$,n) reaction rates on the thermalized ${}^{90}$Zr and ${}^{94}$Mo are estimated with the D1M+QRPA dipole strength to be about 480 times and 200 times, respectively, larger than the rates on ${}^{90}$Zr and ${}^{94}$Mo in their ground state. Clearly, at these temperatures, transitions from and to the ground state (which are measured in the laboratory) contribute only to a small fraction of the stellar cross sections Rauscher (2011). However, they are relevant for constraining HF statistical model parameters, such as the $\gamma$SF, as demonstrated with the present results especially in the case of the ${}^{90}$Zr($\gamma$,n) reaction. ## V Conclusions High-precision measurements of the photoneutron reaction cross sections on the nuclei ${}^{94}$Mo and ${}^{90}$Zr have been conducted from the respective neutron emission thresholds up to 13.5 MeV. Beams of high intensity quasi-monochromatic $\gamma$-rays from laser Compton scattering at the HI$\gamma$S facility were used. The new experimental cross sections were accurately measured near the neutron emission threshold which is where the photoneutron reaction cross sections are very small but astrophysically relevant. In order to constrain the $\gamma$SF in the A $\approx$ 90 mass region, the measured cross sections were compared with predictions of Hauser-Feshbach statistical model calculations using two different dipole $\gamma$SF models $-$ GLO model and HFB$+$QRPA model. For the ${}^{94}$Mo($\gamma$,n)${}^{93}$Mo reaction the resulting stellar reaction rate, calculated with the D1M$+$QRPA dipole strength, was found to be about 40% higher than the stellar reaction rate calculated with the GLO model. In contrast, for the ${}^{90}$Zr($\gamma$,n)${}^{89}$Zr reaction, the stellar reaction rate calculated with the D1M$+$QRPA dipole strength and adjusted to the new experimental data was found to be a factor of 5 lower than the stellar reaction rate calculated with the GLO model. Hence, the present results show how sensitive the stellar photoneutron reaction rates of astrophysical interest are to experimental data in the vicinity of the neutron emission threshold. Considering the very large number of nuclear reactions involved in the production of a single $p$-nucleus, measurements of photodisintegration reaction cross sections at astrophysically relevant energies on nuclei located as close as possible to the $p$-process path will put the nucleosynthesis calculations on a firmer ground. Therefore the present results on ${}^{94}$Mo, one of the most abundant of the $p$-nuclei and currently underproduced in all of the existing astrophysical models, and on ${}^{90}$Zr, a neutron magic nucleus known until recently as a genuine $s$-process nuclide, may have a significant impact on the efforts of understanding the $p$-process nucleosynthesis. ###### Acknowledgements. We thank C. Travaglio for her support and stimulating discussions in the early stage of this research. We would like to acknowledge the contributions of W. R. Zimmerman, M. Bhike, W. Tornow, and M. McCleskey during the collection of the data, as well as the staff at the HI$\gamma$S facility for their help during the experimental setup and for the production of the $\gamma$-ray beam. Additionally, we would like to thank R. Schwengner for providing the ${}^{94}$Mo and ${}^{90}$Zr targets, M. Starnes for machining the Al collimator, J. A. Gallant for preparing the experimental figures, T. M.-R. Chu and J. E. Mayer for assisting in the data reduction. Figs. 4$-$6 were created using R$:$ A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. R Core Team (2017)$;$ URL https://www.R-project.org/ while Fig. 3 was created using the software processing; URL https://www.processing.org/. AB acknowledges support in part by the Research Corporation for Science Advancement, Grant No. 22662, Office of Sponsored Programs at James Madison University (JMU), Grant No. 100672, and by the Department of Physics and Astronomy at JMU. JAS acknowledges that this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. HJK acknowledges support of the U.S. Department of Energy, Office of Science, Office of Nuclear Physics through grants DE-FG02-97ER41033 and DE-FG02-97ER41041. SG acknowledges the support of the FRS-FNRS. The theoretical work was performed within the IAEA CRP on “Updating the Photonuclear data Library and Generating a Reference Database for Photon Strength Functions” (F41032). Last but not least, we highly appreciated the thorough review of the anonymous referee. ## References * Burbidge _et al._ (1957)M. E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and F. Hoyle, Rev. Mod. Phys. 29, 547 (1957). * Arnould (1976)M. Arnould, Astron. Astrophys. 46, 17 (1976). * Woosley and Howard (1978)S. E. Woosley and W. M. Howard, The Astrophys. J. Suppl. S. 36, 285 (1978). * Lambert (1992)D. L. Lambert, Astron. Astrophys. Rev. 3, 201 (1992). * Arnould and Goriely (2003)M. Arnould and S. Goriely, Phys. Rep. 384, 1 (2003). * Rayet _et al._ (1995)M. Rayet, M. Arnould, M. Hashimoto, N. Prantzos, and K. Nomoto, Astron. Astrophys. 298, 517 (1995). * Rauscher and Thielemann (2000)T. Rauscher and F.-K. Thielemann, At. Data Nucl. Data Tables 75, 1 (2000). * Goriely _et al._ (2002)S. Goriely, J. José, M. Hernanz, M. Rayet, and M. Arnould, Astron. Astrophys. 383, L27 (2002). * Fröhlich _et al._ (2006)C. Fröhlich, G. Martínez-Pinedo, M. Liebendörfer, F.-K. Thielemann, E. Bravo, W. R. Hix, K. Langanke, and N. T. Zinner, Phys. Rev. Lett. 96, 142502 (2006). * Travaglio _et al._ (2011)C. Travaglio, F. K. Röpke, R. Gallino, and W. Hillebrandt, Astroph. J. 739, 93 (2011). * Goriely and Khan (2002)S. Goriely and E. Khan, Nucl. Phys. A706, 217 (2002). * Tsoneva _et al._ (2015)N. Tsoneva, S. Goriely, H. Lenske, and R. Schwengner, Phys. Rev. C 91, 044318 (2015). * Martini _et al._ (2016)M. Martini, S. Péru, S. Hilaire, S. Goriely, and F. Lechaftois, Phys. Rev. C 94, 014304 (2016). * Goriely _et al._ (2016)S. Goriely, S. Hilaire, S. Péru, M. Martini, I. Deloncle, and F. Lechaftois, Phys. Rev. C 94, 044306 (2016). * Goriely _et al._ (2018)S. Goriely, S. Hilaire, S. Péru, and K. Sieja, Phys. Rev. C 98, 014327 (2018). * Utsunomiya _et al._ (2009)H. Utsunomiya, S. Goriely, M. Kamata, T. Kondo, O. Itoh, H. Akimune, T. Yamagata, H. Toyokawa, Y.-W. Lui, S. Hilaire, and A. J. Koning, Phys. Rev. C 80, 055806 (2009). * Tonchev _et al._ (2010)A. P. Tonchev, S. L. Hammond, C. R. Howell, C. Huibregtse, A. Hutcheson, J. H. Kelley, E. Kwan, R. Raut, G. Rusev, W. Tornow, T. Kawano, D. J. Vieira, and J. B. Wilhelmy, Phys. Rev. C 82, 054620 (2010). * Utsunomiya _et al._ (2013)H. Utsunomiya, S. Goriely, T. Kondo, C. Iwamoto, H. Akimune, T. Yamagata, H. Toyokawa, H. Harada, F. Kitatani, Y.-W. Lui, A. C. Larsen, M. Guttormsen, P. E. Koehler, S. Hilaire, S. Péru, M. Martini, and A. J. Koning, Phys. Rev. C 88, 015805 (2013). * Raut _et al._ (2013)R. Raut, A. P. Tonchev, G. Rusev, W. Tornow, C. Iliadis, M. Lugaro, J. Buntain, S. Goriely, J. H. Kelley, R. Schwengner, A. Banu, and N. Tsoneva, Phys. Rev. Lett. 111, 112501 (2013). * Filipescu _et al._ (2014)D. M. Filipescu, I. Gheorghe, H. Utsunomiya, S. Goriely, T. Renstrom, H.-T. Nyhus, O. Tesileanu, T. Glodariu, T. Shima, K. Takahisa, S. Miyamoto, Y.-W. Lui, S. Hilaire, S. Péru, M. Martini, and A. J. Koning, Phys. Rev. C 90, 064616 (2014). * Sauerwein _et al._ (2014)A. Sauerwein, K. Sonnabend, M. Fritzsche, J. Glorius, E. Kwan, N. Pietralla, C. Romig, G. Rusev, D. Savran, L. Schnorrenberger, A. P. Tonchev, W. Tornow, and H. R. Weller, Phys. Rev. C 89, 035803 (2014). * Nyhus _et al._ (2015)H.-T. Nyhus, T. Renstrom, H. Utsunomiya, S. Goriely, D. M. Filipescu, I. Gheorghe, O. Tesileanu, T. Glodariu, T. Shima, K. Takahisa, S. Miyamoto, Y.-W. Lui, S. Hilaire, S. Peru, M. Martini, L. Siess, and A. J. Koning, Phys. Rev. C 91, 015808 (2015). * Renstrøm _et al._ (2018)T. Renstrøm, H. Utsunomiya, H. T. Nyhus, A. C. Larsen, M. Guttormsen, G. M. Tveten, D. M. Filipescu, I. Gheorghe, S. Goriely, S. Hilaire, Y.-W. Lui, J. E. Midtbo, S. Péru, T. Shima, S. Siem, and O. Tesileanu, Phys. Rev. C 98, 054310 (2018). * Utsunomiya _et al._ (2018)H. Utsunomiya, T. Renstrøm, G. M. Tveten, S. Goriely, S. Katayama, T. Ari-izumi, D. Takenaka, D. Symochko, B. V. Kheswa, V. W. Ingeberg, T. Glodariu, Y.-W. Lui, S. Miyamoto, A. C. Larsen, J. E. Midtbo, A. Gorgen, S. Siem, L. C. Campo, M. Guttormsen, S. Hilaire, S. Péru, and A. J. Koning, Phys. Rev. C 98, 054619 (2018). * Travaglio _et al._ (2015)C. Travaglio, R. Gallino, T. Rauscher, F. K. Röpke, and W. Hillebrandt, Astroph. J. 799, 54 (2015). * Weller _et al._ (2009)H. R. Weller _et al._, Prog. Part. Nucl. Phys. 62, 257 (2009). * Pywell _et al._ (2009)R. E. Pywell, O. Mavrichi, W. A. Wurtz, and R. Wilson, Nucl. Instrum. Methods Phys. Res. Sect. A 606, 517 (2009). * GEANT4 (2003)GEANT4, Nucl. Instrum. Methods A 503, 250 (2003). * Birenbaum _et al._ (1985)Y. Birenbaum, S. Kahane, and R. Moreh, Phys. Rev. C 32, 1825 (1985). * Bernabei _et al._ (1986)R. Bernabei, A. Incicchitti, M. Mattioli, P. Picozza, D. Prosperi, L. Casano, S. d’Angelo, M. P. D. Pascale, C. Schaerf, G. Giordano, G. Matone, S. Frullani, and B. Girolami, Phys. Rev. Lett. 57, 1542 (1986). * Schiavilla (2005)R. Schiavilla, Phys. Rev. C 72, 034001 (2005). * Arnold _et al._ (2011)C. W. Arnold, T. B. Clegg, H. J. Karwowski, G. C. Rich, J. R. Tompkins, and C. R. Howell, Nucl. Instrum. Methods Phys. Res. Sect. A 647, 55 (2011). * Koning and Rochman (2012)A. J. Koning and D. Rochman, Nucl. Data Sheets 113, 2841 (2012). * Péru and Goutte (2008)S. Péru and H. Goutte, Phys. Rev. C 77, 044313 (2008). * Goriely _et al._ (2008)S. Goriely, S. Hilaire, and A. J. Koning, Phys. Rev. C 78, 064307 (2008). * Koning and Delaroche (2003)A. J. Koning and J. P. Delaroche, Nucl. Phys. A 713, 231 (2003). * Capote _et al._ (2009)R. Capote _et al._, Nucl. Data Sheets 110, 3107 (2009). * Koning _et al._ (2008)A. J. Koning, S. Hilaire, and S. Goriely, Nucl. Phys. A 810, 13 (2008). * Kopecky and Uhl (1990)J. Kopecky and M. Uhl, Phys. Rev. C 41, 1941 (1990). * Bauge _et al._ (1998)E. Bauge, J. P. Delaroche, and M. Girod, Phys. Rev. C 58, 1118 (1998). * Bauge _et al._ (2001)E. Bauge, J. P. Delaroche, and M. Girod, Phys. Rev. C 63, 024607 (2001). * Voght _et al._ (2002)K. Voght _et al._, Nucl. Phys. A 707, 241 (2002). * Beil _et al._ (1974)H. Beil, R. Bergére, P. Carlos, A. Leprêtre, A. de Miniac, and A. Veyssiére, Nucl. Phys. A 227, 427 (1974). * Berman _et al._ (1967)B. L. Berman, J. T. Caldwell, R. R. Harvey, M. A. Kelly, R. L. Bramblett, and S. C. Fultz, Phys. Rev. 162, 1098 (1967). * Leprêtre _et al._ (1971)A. Leprêtre, H. Beil, R. Bergére, P. Carlos, and A. Veyssiére, Nucl. Phys. A 175, 609 (1971). * Berman _et al._ (1987)B. L. Berman, R. E. Pywell, S. S. Dietrich, M. N. Thompson, K. G. McNeill, and J. W. Jury, Phys. Rev. C 36, 1286 (1987). * Varlamov _et al._ (2009)V. V. Varlamov, N. N. Peskov, and M. E. Stepanov, Phys. of At. Nuclei 72, 214 (2009). * (48)“http://www-nds.iaea.org/exfor,” . * Schwengner _et al._ (2008)R. Schwengner, G. Rusev, N. Tsoneva, N. Benouaret, R. Beyer, M. Erhard, E. Grosse, A. R. Junghans, J. Klug, K. Kosev, H. Lenske, C. Nair, K. D. Schilling, and A. Wagner, Phys. Rev. C 78, 064314 (2008). * Rusev _et al._ (2009)G. Rusev, R. Schwengner, R. Beyer, M. Erhard, E. Grosse, A. R. Junghans, K. Kosev, C. Nair, K. D. Schilling, A. Wagner, F. Dönau, and S. Frauendorf, Phys. Rev. C 79, 061302 (2009). * Guttormsen _et al._ (2005)M. Guttormsen, R. Chankova, U. Agvaanluvsan, E. Algin, L. A. Bernstein, F. Ingebretsen, T. Lönnroth, S. Messelt, G. E. Mitchell, J. Rekstad, A. Schiller, S. Siem, A. C. Sunde, A. Voinov, and Ødegård, Phys. Rev. C 71, 044307 (2005). * Rauscher (2011)T. Rauscher, Astroph. J. 738, 143 (2011).
1212.2132	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 19475, "num_imgs": 2, "llama3_tokens_count": 5734 }	[ "content_image/1212.2132/doublebox2.png", "content_image/1212.2132/Contour-change.png" ]	# An Overview of Maximal Unitarity at Two Loops Henrik Johansson Theory Group, Physics Department, CERN CH–1211 Geneva 23, Switzerland Institut de Physique Théorique, CEA–Saclay, F–91191 Gif-sur-Yvette cedex, France Kasper J. Larsen NIKHEF, Science Park 105, NL–1098 XG Amsterdam, The Netherlands ###### Abstract: We discuss the extension of the maximal-unitarity method to two loops, focusing on the example of the planar double box. Maximal cuts are reinterpreted as contour integrals, with the choice of contour fixed by the requirement that integrals of total derivatives vanish on it. The resulting formulæ, like their one-loop counterparts, can be applied either analytically or numerically. ## 1 Introduction Amplitudes are the basic building blocks for physics predictions in QCD. Predictions of differential cross sections are essential to controlling backgrounds to new physics at the Large Hadron Collider (LHC). Because of their strong dependence on the unphysical renormalization and factorization scales, leading-order (LO) predictions are not quantitatively reliable. Next-to-leading order (NLO) calculations give the first quantitative predictions of processes involving QCD. NLO calculations require one-loop amplitudes, in addition to other ingredients. Recent years have seen a revolution in our ability to calculate these amplitudes, thanks to maximal unitarity [1, 2] and other developments [4] such as the Ossola–Papadopoulos–Pittau [3] decomposition. For some processes, such as $gg\to W^{+}W^{-}$ and $gg\to ZZ$, the tree-level amplitude vanishes, and accordingly the one-loop amplitude furnishes only an LO prediction. Such subprocesses are nominally higher order in the strong coupling, of ${\cal O}(\alpha_{s}^{2})$, compared to ${\cal O}(\alpha_{s}^{0})$ for the basic $q\bar{q}\to W^{+}W^{-},ZZ$ subprocesses. This is however partly compensated by the much larger gluon densities, so that they merit computation. To compute these subprocesses to NLO, one needs two-loop amplitudes. Two-loop amplitudes are also needed for NNLO calculations, which in turn will be needed for future precision physics at the LHC. Such calculations will also be useful in providing honest uncertainty estimates for existing NLO predictions. ## 2 On-Shell Methods Traditional Feynman-diagrammatic methods suffer from an explosion in the number of diagrams, and an even greater explosion in the number of terms, as the number of external legs or the number of loops increases. Yet many results for amplitudes, especially in the ${\cal N}=4$ supersymmetric gauge theory, are extremely compact; and all known loop results in gauge theories are vastly more compact than would be suggested by the number of diagrams. This reflects the vast redundancy present in Feynman diagrams, due to explicit handling of non-physical states, and the resulting gauge dependence of intermediate quantities. On-shell methods use only information from physical, on-shell, states to compute amplitudes, thereby avoiding throughout the computation of gauge-variant quantities which must cancel at the end. This makes calculations simpler and made possible new NLO calculations at high multiplicity, such as those of $W,Z\,+\,4$ jets and $W\,+\,5$ jets [5]. On-shell methods make use of general properties of amplitudes to derive tools for computations: tree-level factorization leads to the Britto–Cachazo–Feng–Witten on-shell recursion relations for tree amplitudes [6]; the unitarity of the $S$-matrix gives rise to the unitarity [7] and generalized-unitarity methods; and the presence of an underlying field theory allows for a representation in terms of an integral basis. The formalism can be summarized in the following equation, \[\textrm{Amplitude}=\sum_{j\in\textrm{Basis}}c_{j}\textrm{Int}_{j}+\textrm{ Rational}\,,\] (1) where the sum is taken over a basis of integrals, and the coefficients $c_{j}$ as well as the remaining rational terms are rational functions of spinor variables. For analytic calculations, having a basis of linearly-independent integrals simplifies calculations but is not strictly essential. For numerical calculations, it is essential. ## 3 Integral Basis <figure><img src="content_image/1212.2132/doublebox2.png"><figcaption>Figure 1: The double-box integral</figcaption></figure> At one loop, the basis for computations in massless gauge theories consists of box, triangle, and bubble integrals, where all internal lines are massless, and external legs may be massive or massless. In general, we must distinguish between two different notions of basis: a “$D$-dimensional basis,” which keeps terms to all orders in the dimensional regulator $\epsilon$, and a “regulated four-dimensional basis,” which keeps only terms through ${\cal O}(\epsilon^{0})$. Because some integrals are linearly independent only at ${\cal O}(\epsilon)$, the latter basis is more compact. We will implicitly be using this latter basis in these Proceedings. The planar part of the basis at two loops will contain integrals with up to eight propagators [8]. Beyond one loop, some basis integrals will necessarily contain irreducible numerators, numerators which cannot be written as linear combinations of inverse propagators. For the double boxes we consider here and shown in fig. 1, for example, there are two basis integrals when all external legs are massless, or when one external leg is massive. ## 4 Maximal Unitarity In the basic unitarity method at one loop, we sew together two tree amplitudes with a phase-space integral, and promote the positive-energy on-shell delta functions to off-shell propagators. The resulting object will yield an integral containing the correct contribution to the target amplitude in the channel in which we have performed the sewing. We must still reduce the resulting integral symbolically in order to separate the contributions from different basis integrals, and find their respective coefficients. Finally, we must merge contributions from all channels. This sewing procedure inverts the procedure of cutting a one-loop amplitude, in which we replace a pair of propagators surrounding the given channel by positive-energy delta functions. This isolates all integrals containing those two propagators. There are of course many such integrals: various boxes and triangles, and a bubble integral as well. In order to isolate a smaller number of integrals, we must cut more propagators. This is possible; indeed at one loop, we have four degrees of freedom in the loop momentum (ignoring the $(-2\epsilon)$-dimensional components), and so we can imagine cutting four propagators at once [2]. There is, however, a subtlety involved. We might imagine replacing the four propagators in the box integral, \[\int{d^{4-2\epsilon}\ell\over(2\pi)^{4-2\epsilon}}\;{1\over\ell^{2}(\ell-k_{1} )^{2}(\ell-K_{12})^{2}(\ell+k_{4})^{2}}\,,\] (2) by positive-energy delta functions, \[\int{d^{4-2\epsilon}\ell\over(2\pi)^{4-2\epsilon}}\;{\delta^{(+)}\bigl{(}\ell^ {2}\bigr{)}\delta^{(+)}\bigl{(}(\ell-k_{1})^{2}\bigr{)}\delta^{(+)}\bigl{(}( \ell-K_{12})^{2}\bigr{)}\delta^{(+)}\bigl{(}(\ell+k_{4})^{2}\bigr{)}}\,.\] (3) These delta functions instruct us to solve the simultaneous equations, \[\ell^{2}=0\,,\quad-2\ell\cdot k_{1}+k_{1}^{2}=0\,,\quad-2\ell\cdot k_{2}+K_{12 }^{2}-k_{1}^{2}=0\,,\quad 2\ell\cdot k_{4}+k_{4}^{2}=0\,,\] (4) which are linear combinations of the delta-function arguments. Let us examine the special case when legs 1, 2, and 4 are massless; we can then solve the first, second, and last equations by setting, \[\ell^{\mu}=\frac{\xi}{2}\left\langle\smash{1}{}^{-}\right\|{\mu}\left\|\smash{4} {}^{-}\right\rangle\,,\] (5) and then solve for $\xi$, \[\xi=-\frac{\left[1\,2\right]}{\left[2\,4\right]}\,,\] (6) using the third equation in eq. (4). Similarly, we see that there is a second solution, (7) The subtlety arises from the fact that for generic external momenta, these solutions are complex. The domain of the delta functions, on the other hand, is real; taken literally, the delta functions would yield zero! Cutting both sides of eq. (1) would then give us the equation $0=0$, which is true but not very useful. This is the same issue that arises in straightforward interpretations of delta functions in the connected picture for twistor-string amplitudes [9]. To find a solution to this subtlety, we may note [10] that contour integration behaves very much like integration over a delta function, \[\oint_{C(z_{0})}\hskip-5.690551ptdz\;\frac{{\rm Poly}_{1}(z)}{{\rm Poly}_{2}(z )-a}=\frac{{\rm Poly}_{1}(z_{0})}{{\rm Poly}_{2}^{\prime}(z_{0})}\,,\] (8) where $z_{0}$ is defined by the equation ${\rm Poly_{2}}(z_{0})=a$ (with multiciplicity one). We could _define_ the desired delta function as follows, \[\int dz\;{\rm Poly_{1}}(z)\delta({\rm Poly}_{2}(z)-a)\equiv\oint_{C(z_{0})} \hskip-5.690551ptdz\;\frac{{\rm Poly}_{1}(z)}{{\rm Poly}_{2}(z)-a}\,.\] (9) There is one significant difference from ordinary delta function integration: there is no absolute value around the derivative in the denominator on the right-hand side of eq. (8), so that the result remains an analytic function. <figure><img src="content_image/1212.2132/Contour-change.png"><figcaption>Figure 2: Cutting as contour replacement</figcaption></figure> That is, we must reinterpret cutting propagators as contour _replacement_: instead of replacing the propagators by delta functions, we replace the original contours of integration, along the real axes of the now-complexified loop momenta $\ell^{\mu}$, by contours surrounding the _global poles_, that is the simultaneous solutions to eqs. (4), in $\mathbb{C}^{4}$. The contour in the case of the one-loop box is a product of four circles, that is a four-torus $T^{4}$. The replacement is illustrated schematically in fig. 2, with contours ${\cal C}_{1,2}$ encircling the two global poles. We stress that this is _not_ a contour deformation leaving the value of the integral unchanged; it is a replacement, changing the value of the integral and ultimately allows us to derive an equation for the coefficient of the one-loop box. This reinterpretation raises two new problems, however. We have to choose $T^{4}$ contours surrounding two global poles, around which we could wind an arbitrary number of times (even a fractional number of times). How should we choose the contour? Also, replacing the contour by such an arbitrary winding can break integral identities. For example, the identity, \[0=\int{d^{4-2\epsilon}\ell\over(2\pi)^{4-2\epsilon}}\;{\varepsilon(\ell,k_{1}, k_{2},k_{4})\over\ell^{2}(\ell-k_{1})^{2}(\ell-K_{12})^{2}(\ell+k_{4})^{2}}\,,\] (10) is spoiled if we choose the contour to encircle just one of the global poles. Remarkably, these two problems cancel each other out. If we take a general contour, ${\cal C}=a_{1}{\cal C}_{1}+a_{2}{\cal C}_{2}$, we find that the integral in eq. (10) takes the value, \[(a_{1}-a_{2})f(k_{1},k_{2},k_{4})\,,\] (11) so that it will still vanish if $a_{1}=a_{2}$. This fixes the contour up to an overall irrelevant constant which will cancel out of eq. (1). Applying the cutting via contour replacement to our basic equation (1), we can derive a formula for the coefficient of the corresponding one-loop box. The resulting equation is the same as the one obtained by Britto, Cachazo, and Feng (BCF) [2]. Let us now apply these ideas to the double-box integral (fig. 1). In the same way that this derivation can be seen as a generalization to two loops of the formalisms of BCF and Forde [11], recent work on two-loop integrands by Mastrolia, Mirabella, Ossola, and Peraro [12] and by Badger, Frellesvig, and Zhang [13, 14] can be seen as the two-loop generalization of the Ossola–Papadopoulos–Pittau construction [3]. The maximal cut in the double box involves cutting seven propagators. Each solution has one continuous degree of freedom $z$. The number of distinct solutions depends on the number and configuration of external masses. When all four external legs are massless; when one external leg is massive; when two diagonally-opposite legs (for example, legs 1 and 3) are massive; or when two long-edge legs (for example, legs 1 and 4) are massive, there are six solutions. We will call these configurations ‘class (c)’. When two short-edge legs (for example, legs 1 and 2) are massive, or when three legs are massive, there are four solutions. We will label these configurations ‘class (b)’. (This classification is explained in ref. [15].) Performing the corresponding contour integrals gives rise to a Jacobian factor (the analog for the maximal cut in the double box of the ${\rm Poly}_{2}^{\prime}(z_{0})$ factor in eq. (8)). Here, the resulting Jacobian is a function of $z$; it has poles in $z$, so that we can choose a contour for the $z$ integration as well, thereby obtaining a global pole. After identifying different parametrizations of the same global pole arising from different solutions to the heptacut equations, and taking into account poles that arise in the loop momenta $\ell_{1,2}$ as well as in the Jacobian, we see that there are always eight global poles, independent of the number and configuration of external masses [15]. Unlike the number of global poles, the number of independent basis integrals does depend on the number and configuration of external masses. In class (c), there are two basis integrals; in class (b), there are three. We would like to construct independent “projectors”, which give formulæ for the coefficients of each of these basis integrals. Each projector will be a linear combination of contours around the global poles. How should we choose them? We again impose the requirement, analogous to eq. (10), that all vanishing loop integrals continue to vanish on the chosen contours. As at one loop, there are vanishing integrals where a Levi-Civita tensor is inserted into the numerator of the scalar integral. There are five possible integrals, which give rise to four independent constraints on the contours. In addition, there are integration-by-parts (IBP) identities [16, 17] which give 20 linear relations in class (c) between the 22 integrals with different powers of the two irreducible numerators $\ell_{1}\cdot k_{4}$ and $\ell_{2}\cdot k_{1}$. Not all of the resulting constraints on the contours are independent; in class (c), we find two independent constraints, while in class (b) we have only 19 IBP equations, which reduce to a single constraint. We work here to leading order in $\epsilon$, leaving higher-order terms in the coefficients to future work. The constraints leave us with two independent contours for the two master integrals in class (c), and three independent contours for the three master integrals in class (b). We can obtain a projector for any given integral by imposing the further constraint that the other integrals vanish on the contour, and that it reproduces the integral itself with unit coefficient. The formulæ for the coefficients of double boxes take the following form, \[c=i\sum_{j=1}^{8}a_{j}\oint_{T^{8}({\cal G}_{j})}d^{8}v_{a,b}\;J_{\oint\!,j} \sum_{{\rm particles\atop\rm helicities}}\prod_{p=1}^{6}A^{(0)}_{p}(v_{a,b})\,,\] (12) where $A^{(0)}$ are tree-level amplitudes in the gauge theory, and the $a_{j}$ are weights (or winding numbers) for the different global poles. For example, for the one-mass double box (with $m_{1}^{2}\neq 0$) the weights $a_{j}$ for the two basis integrals, $I[1]$ and $I[\ell_{1}\cdot k_{4}]$, are, \[(a_{j})=\frac{1}{4}\left(1,1,1,1,0,0,1,1\right)\,,\] (13) and \[(a_{j})=\frac{m_{1}^{2}-s_{12}}{2s_{12}s_{14}}\left(1,1,1,1,-2,-2,3,3\right)\,,\] (14) respectively. The solutions and global poles are given in terms of a parametrization of the loop momenta given in ref. [18], where the reader may also find complete formulæ for the solutions, global poles, and projectors for classes (b) and (c). This work is supported by the European Research Council under Advanced Investigator Grant ERC–AdG–228301. ## References * [1] Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B 513, 3 (1998) [hep-ph/9708239]. * [2] R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 725, 275 (2005) [hep-th/0412103]. * [3] G. Ossola, C. G. Papadopoulos and R. Pittau, Nucl. Phys. B 763, 147 (2007) [hep-ph/0609007]. * [4] C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, Phys. Lett. B 645, 213 (2007) [hep-ph/0609191]; JHEP 0703, 111 (2007) [hep-ph/0612277]; W. T. Giele, Z. Kunszt and K. Melnikov, JHEP 0804, 049 (2008) [arXiv:0801.2237 [hep-ph]]; C. F. Berger and D. Forde, Ann. Rev. Nucl. Part. Sci. 60, 181 (2010) [arXiv:0912.3534 [hep-ph]]; R. K. Ellis, W. T. Giele and Z. Kunszt, JHEP 0803, 003 (2008) [0708.2398 [hep-ph]]; C. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, H. Ita, D. A. Kosower and D. Maître, Phys. Rev. D 78, 036003 (2008) [0803.4180 [hep-ph]]; G. Ossola, C. G. Papadopoulos and R. Pittau, JHEP 0803, 042 (2008) [arXiv:0711.3596 [hep-ph]]; P. Mastrolia, G. Ossola, C. G. Papadopoulos and R. Pittau, JHEP 0806, 030 (2008) [arXiv:0803.3964 [hep-ph]]; W. T. Giele and G. Zanderighi, JHEP 0806, 038 (2008) [arXiv:0805.2152 [hep-ph]]; R. K. Ellis, W. T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, JHEP 0901, 012 (2009) [arXiv:0810.2762 [hep-ph]]. * [5] C. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg, H. Ita, D. A. Kosower, and D. Maître, Phys. Rev. Lett. 106, 092001 (2011) [arXiv:1009.2338 [hep-ph]]; H. Ita, Z. Bern, L. J. Dixon, F. Febres Cordero, D. A. Kosower and D. Maître, Phys. Rev. D 85, 031501 (2012) [arXiv:1108.2229 [hep-ph]]; Z. Bern, G. Diana, L. J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg, S. Hoeche, H. Ita, D. A. Kosower, D. Maître, and K. Ozeren, these proceedings. * [6] R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. 94, 181602 (2005) [hep-th/0501052]. * [7] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425, 217 (1994) [hep-ph/9403226]; Nucl. Phys. B 435, 59 (1995) [hep-ph/9409265]. * [8] J. Gluza, K. Kajda and D. A. Kosower, Phys. Rev. D 83, 045012 (2011) [arXiv:1009.0472 [hep-th]]. * [9] R. Roiban, M. Spradlin and A. Volovich, Phys. Rev. D 70, 026009 (2004) [hep-th/0403190]. * [10] C. Vergu, Phys. Rev. D 75, 025028 (2007) [hep-th/0612250]. * [11] D. Forde, Phys. Rev. D 75, 125019 (2007) [0704.1835 [hep-ph]]. * [12] P. Mastrolia and G. Ossola, JHEP 1111, 014 (2011) [arXiv:1107.6041 [hep-ph]]; P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, arXiv:1205.7087 [hep-ph]; P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, arXiv:1209.4319 [hep-ph]. * [13] S. Badger, H. Frellesvig and Y. Zhang, JHEP 1204, 055 (2012) [arXiv:1202.2019 [hep-ph]]; JHEP 1208, 065 (2012) [arXiv:1207.2976 [hep-ph]]. * [14] Y. Zhang, JHEP 1209, 042 (2012) [arXiv:1205.5707 [hep-ph]]. * [15] S. Caron-Huot and K. J. Larsen, JHEP 1210, 026 (2012) [arXiv:1205.0801 [hep-ph]]. * [16] F. V. Tkachov, Phys. Lett. B 100, 65 (1981); K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192, 159 (1981). * [17] S. Laporta, Phys. Lett. B 504, 188 (2001) [hep-ph/0102032]; S. Laporta, Int. J. Mod. Phys. A 15, 5087 (2000) [hep-ph/0102033]. * [18] H. Johansson, D. A. Kosower and K. J. Larsen, arXiv:1208.1754 [hep-th].
1604.06403	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 47657, "num_imgs": 6, "llama3_tokens_count": 12756 }	[ "content_image/1604.06403/x1.png", "content_image/1604.06403/x3.png", "content_image/1604.06403/x5.png", "content_image/1604.06403/x7.png", "content_image/1604.06403/x9.png", "content_image/1604.06403/x10.png" ]	# Classical nature of ordered quantum phases and origin of spontaneous symmetry breaking M. Cianciaruso School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy S. M. Giampaolo International Institute of Physics, UFRN, Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy L. Ferro Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy W. Roga Department of Physics, University of Strathclyde, John Anderson Building, 107 Rottenrow, Glasgow G4 0NG, United Kingdom Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy G. Zonzo Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy M. Blasone Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy F. Illuminati Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy Consiglio Nazionale delle Ricerche, Istituto di Nanotecnologia, Rome Unit, I-00195 Roma, Italy April 21, 2016 ###### Abstract We analyse the nature of spontaneous symmetry breaking in complex quantum systems by investigating the long-standing conjecture that the maximally symmetry-breaking quantum ground states are the most classical ones corresponding to a globally ordered phase. We make this argument quantitatively precise by comparing different local and global indicators of classicality and quantumness, respectively in symmetry-breaking and symmetry-preserving quantum ground states. We first discuss how naively comparing local, pairwise entanglement and discord apparently leads to the opposite conclusion. Indeed, we show that in symmetry-preserving ground states the two-body entanglement captures only a modest portion of the total two-body quantum correlations, while, on the contrary, in maximally symmetry-breaking ground states it contributes the largest amount to the total two-body quantum correlations. We then put to test the conjecture by looking at the global, macroscopic correlation properties of quantum ground states. We prove that the ground states which realize the maximum breaking of the Hamiltonian symmetries, associated to a globally ordered phase, are the only ones that: I) are always locally convertible, i.e. can be obtained from all other ground states by only applying LOCC transformations (local operations and classical communication), while the reverse is never possible; II) minimize the monogamy inequality on the globally shared, macroscopic bipartite entanglement. pacs: 03.67.Mn, 03.65.Ud, 75.10.Pq, 05.30.Rt † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction In the study of collective quantum phenomena, the understanding of the globally ordered phases associated to local order parameters relies on the key concept of spontaneous symmetry breaking Goldstone _et al._ (1962). The latter is required to explain the existence of locally inequivalent ground states that are not eigenstates of one or more symmetry operators for the corresponding many-body Hamiltonian Sachdev (2000). In recent years, knowledge of quantum phase transitions has been sharpened by the application of methods and techniques originally developed in the field of quantum information Amico _et al._ (2008); Verstraete _et al._ (2008). Various types of quantum phase transitions have been indeed characterized by identifying the singular points in the derivatives of different measures of bipartite Osterloh _et al._ (2002); Osborne and Nielsen (2002) and multipartite entanglement Giampaolo and Hiesmayr (2013, 2014). Moreover, different ordered phases have been identified by looking at the factorization properties of different ground states Giampaolo _et al._ (2008, 2009, 2010) or by studying the behavior of the ground-state fidelity under local or global variations of the Hamiltonian parameters Zanardi and Paunković (2006); Zanardi _et al._ (2007). Efforts have been devoted to the investigation of the behavior of the bipartite concurrence Osterloh _et al._ (2006), multipartite entanglement de Oliveira _et al._ (2006); Giampaolo and Hiesmayr (2013, 2014), and quantum discord Amico _et al._ (2012); Tomasello _et al._ (2012) for some specific symmetry-breaking ground states. However, on the whole, the complete understanding of the physical mechanism that selects the symmetry-breaking ground states in the thermodynamic limit remains an open problem Bratteli and Robinson (2012); Arodz _et al._ (2012). In complete analogy with the case of classical phase transitions driven by temperature, the common explanation of this phenomenon invokes the unavoidable presence of some local, however small, perturbing external field that selects one of the maximally symmetry-breaking ground states (MSBGSs) among all the elements of the quantum ground space. Crucially, in this type of reasoning it is assumed that the MSBGSs are the most classical ones and thus the ones that are selected in real-world situations, under the effect of decoherence that quickly destroys macroscopic coherent superpositions. At first glance, this notion appears to be obvious. For instance, in the paradigmatic case of the quantum Ising model, the ground space of the ferromagnetic phase at zero transverse field $h$ is spanned by two orthogonal product states $\|0\rangle^{\otimes N}$ and $\|1\rangle^{\otimes N}$ which are in the same class of pointer states of the typical decoherence argument, while the symmetric states $\Psi_{\pm}=1/\sqrt{2}(\|0\rangle^{\otimes N}\pm\|1\rangle^{\otimes N})$ realize macroscopic coherent superpositions (Schroedinger cats) that are not stable under decoherence Zurek (2003); van Wezel (2008). Therefore, at zero transverse field $h$, the situation is very clear: the only stable states are those that maximally break the symmetry of the Hamiltonian, and at the same time, those that feature vanishing macroscopic total correlations, including entanglement, between spatially separated regions. On the other hand, as we turn on the external field $h$, we have a whole range of values where, before a critical value $h=h_{c}$ is reached, there is a magnetic order associated to spontaneous symmetry breaking Barouch and McCoy (1971), and the decoherence argument applies within the entire, globally ordered phase. This means that, again, the only stable states are those that maximally break the Hamiltonian symmetry. However, now the symmetry-breaking states are entangled, and their mixed-state reductions on arbitrary subsystems possess in general nonvanishing pairwise entanglement Osborne and Nielsen (2002); Osterloh _et al._ (2002); Amico _et al._ (2008), as well as pairwise quantum Tomasello _et al._ (2012); Campbell _et al._ (2013) and classical correlations Barouch and McCoy (1971). It is thus now unclear if and in what sense the MSBGSs are the most classical among all quantum ground states. Indeed, as we shall see in the following, the symmetry-breaking ground states can be, in general, locally more entangled than symmetric ground states (see also Ref. Osterloh _et al._ (2006)). On the other hand, it is always implicitly assumed that such symmetry-breaking states are not macroscopically correlated, while their symmetric superpositions are, in complete analogy with the case $h=0$. Although this is a very plausible picture, a rigorous proof has never been provided, due to the mathematical difficulties in dealing with measures of macroscopic entanglement and correlations based on the von Neumann entropy; see, e.g., the difficulties in proving the boundary (_area_) law in generic gapped systems Hastings (2007); Brandão and Horodecki (2013). The symmetry-breaking states obey the boundary law for entanglement Holzhey _et al._ (1994); Vidal _et al._ (2003); Korepin (2004), while the macroscopic correlations featured by the superposition of two different symmetry broken sectors are of order one. The task is then, to identify quantities that are able to distinguish the presence of macroscopic entanglement and quantum correlations, among all possible sources of entanglement and correlations. In the present work we promote such qualitative picture to an explicit quantitative investigation on the nature of globally ordered quantum phases and the origin of spontaneous symmetry breaking, and we carry it out by comparing various quantifiers of local and global quantum correlations in symmetry-breaking and symmetry-preserving quantum ground states. We will first compare measures of local, pairwise quantum correlations and show that in symmetry-preserving ground states the two-body entanglement captures only a modest portion of the local, two-body quantum correlations, while in maximally symmetry-breaking ground states it accounts for the largest contribution. Next, we will introduce (see below) proper criteria and quantifiers of the degree of classicality of quantum states with respect to their global contents of macroscopic entanglement and quantum correlations. Finally, we will show that, within the quantum ground space corresponding to macroscopically ordered phases with nonvanishing local order parameters, the MSBGSs are the most classical ground states in the sense that they are the only quantum ground states that satisfy the following two criteria for each set of Hamiltonian parameters consistent with an ordered quantum phase in the thermodynamic limit: * _Local convertibility_ – All global ground states are convertible into MSBGSs applying only local operations and classical communication (LOCC transformations), while the reverse transformation is impossible; * _Entanglement distribution_ – The MSBGSs are the only global ground states that minimize the residual tangle between a dynamical variable and the remainder of a macroscopic quantum system. Stated otherwise, the MSBGSs are the only ground states that satisfy the monogamy inequality – a strong constraint, with no classical counterpart, on the shared bipartite entanglement between all components of a macroscopic quantum system – at its minimum among all other possible ground states, and thus minimize the macroscopic multipartite entanglement as measured by the residual tangle. Verification of these two features amounts to proving that the mechanism of spontaneous symmetry breaking selects the most classical ground states associated to globally ordered phases of quantum matter with nonvanishing local order parameters. Our results are of general validity for all systems that belong to the same universality class of exactly solvable models that are standard prototypes for quantum phase transitions associated to spontaneous symmetry breaking, such as the $XY$ quantum spin models Sachdev (2000). The paper is organized as follows. In Section II we recall the main features of the one-dimensional $XY$ models in transverse field with periodic boundary conditions. In Section III we perform the comparison between entanglement and discord for spin pairs in infinite $XY$ chains (thermodynamic limit), respectively in symmetry-preserving and MSBGSs. In Section IV we compare global (as opposed to pairwise) measures of classicality and quantumness, such as local convertibility and entanglement distribution, for symmetry-breaking and symmetry-preserving quantum ground states. Conclusions and outlook are discussed in Section V. ## II $Xy$ Models The one-dimensional spin-$1/2$$XY$ Hamiltonian with ferromagnetic nearest-neighbor interactions in a transverse field with periodic boundary conditions reads Lieb _et al._ (1961); Pfeuty (1970); Barouch _et al._ (1970); Barouch and McCoy (1971); Johnson and McCoy (1971): \[H\!=\!-\!\sum_{i=1}^{N}\!\!\left[\!\left(\!\frac{1+\gamma}{2}\!\right)\!\sigma _{i}^{x}\sigma_{i+1}^{x}\!+\!\left(\!\frac{1-\gamma}{2}\!\right)\!\sigma_{i}^{ y}\sigma_{i+1}^{y}\!+\!h\sigma_{i}^{z}\right]\!\;,\] (1) where $\sigma_{i}^{\mu}$, $\mu=x,y,z$, are the Pauli spin-$1/2$ operators acting on site $i$, $\gamma$ is the anisotropy parameter in the $xy$ plane, $h$ is the transverse magnetic field, and the periodic boundary conditions $\sigma_{N+1}^{\mu}\!\equiv\!\sigma_{1}^{\mu}$ ensure the invariance under spatial translations. For this class of models, the phase diagram can be determined exactly in great detail Lieb _et al._ (1961); Barouch _et al._ (1970). In the thermodynamic limit, for any $\gamma\!\in\!(0,1]$, a quantum phase transition occurs at the critical value $h_{c}=1$ of the transverse field. For $h\!<\!h_{c}\!=\!1$ the system is ferromagnetically ordered and is characterized by a twofold ground-state degeneracy such that the $\mathbb{Z}_{2}$ parity symmetry under inversions along the spin-$z$ direction is broken by some elements of the ground space. Given the two symmetric ground states, the so-called even $\|e\rangle$ and odd $\|o\rangle$ states belonging to the two orthogonal subspaces associated to the two possible distinct eigenvalues of the parity operator, any symmetry-breaking linear superposition of the form \[\|g(u,v)\rangle=u\|e\rangle+v\|o\rangle\;\] (2) is also an admissible ground state, with the complex superposition amplitudes $u$ and $v$ constrained by the normalization condition $\|u\|^{2}\!+\!\|v\|^{2}\!=\!1$. Taking into account that the even and odd ground states are orthogonal, the expectation values of operators that commute with the parity operator are independent of the superposition amplitudes $u$ and $v$. On the other hand, spin operators that do not commute with the parity may have nonvanishing expectation values on such linear combinations and hence break the symmetry of the Hamiltonian (1). Consider observables $O_{S}$ that are arbitrary products of spin operators and anti-commute with the parity. Their expectation values in the superposition ground states (2) are of the form \[\langle g(u,v)\|O_{S}\|g(u,v)\rangle=uv^{}\langle o\|O_{S}\|e\rangle+vu^{} \langle e\|O_{S}\|o\rangle\;.\] (3) Both $\langle o\|O_{S}\|e\rangle$ and $\langle e\|O_{S}\|o\rangle$ are real and independent of $u$ and $v$ and hence the expectation (3) is maximum for $u\!=\!\pm v\!=\!1\!/\!\sqrt{2}$ Barouch _et al._ (1970). These are the values of the superposition amplitudes that realize the maximum breaking of the symmetry and identify the order parameter as well as the MSBGSs. Besides the quantum critical point, there exists another relevant value of the external magnetic field, that is $h_{f}=\sqrt{1-\gamma^{2}}$, the _factorizing field_. Indeed, at this value of $h$, the system admits a two-fold degenerate, completely factorized ground state Kurmann _et al._ (1982); Roscilde _et al._ (2005); Giampaolo _et al._ (2008, 2009, 2010). In order to discuss the entanglement and discord-type correlations of quantum ground states, we consider arbitrary bipartitions $(A\|B)$ such that subsystem $A=\{i_{1},\ldots,i_{L}\}$ is any subset made of $L$ spins, and subsystem $B$ is the remainder. Given any global ground state of the total system, the reduced density matrix $\rho_{A}$ ($\rho_{B}$) of subsystem $A$ ($B$) can be expressed in general in terms of the $n$-point correlation functions Osborne and Nielsen (2002): \[\rho_{A}\!(u,v)\!=\!\frac{1}{2^{L}}\!\!\!\!\!\!\!\ \sum_{\mu_{1},\ldots,\mu_{L }}\!\!\!\!\!\langle g(u,v)\!\|\sigma_{i_{1}}^{\mu_{1}}\!\cdots\!\sigma_{i_{L}}^ {\mu_{L}}\!\|g(u,v)\!\rangle\sigma_{i_{1}}^{\mu_{1}}\!\cdots\!\sigma_{i_{L}}^{ \mu_{L}}\,,\] (4) and analogously for $\rho_{B}$. All expectations in Eq. (4) are associated to spin operators that either commute or anti-commute with the parity along the spin-$z$ direction. Therefore the reduced density matrix $\rho_{A}$ can be expressed as the sum of a symmetric part $\rho_{A}^{(s)}$, i.e. the reduced density matrix obtained from $\|e\rangle$ or $\|o\rangle$, and a traceless matrix $\rho_{A}^{(a)}$ that includes all the terms that are nonvanishing only in the presence of a breaking of the symmetry: \[\rho_{A}(u,v)=\rho_{A}^{(s)}+(uv^{}+vu^{})\rho_{A}^{(a)}\;.\] (5) Both $\rho_{A}^{(s)}$ and $\rho_{A}^{(a)}$ are independent of the superposition amplitudes $u$ and $v$, while the reduced density matrix depends on the choice of the ground state. This implies that the elements of the ground space are not locally equivalent. Explicit evaluation of expectations and correlations in symmetry-breaking ground states in the thermodynamic limit is challenging even when the exact solution for the symmetric elements of the ground space is available. We will now sketch a method that allows to overcome this difficulty and whose general validity is not in principle restricted to the particular model considered. In order to obtain $\rho_{A}^{(s)}$ it is sufficient to transform the spin operators in fermionic ones and then apply Wick’s theorem. Such algorithm cannot be applied to spin operators $O_{A}$, acting on subsystem $A$, that anti-commute with the parity. In order to treat this case we first introduce the symmetric operator $O_{A}O_{A+r}$, for which, by applying the previous procedure, we can evaluate $\langle e\|O_{A}O_{A+r}\|e\rangle$. Then, the desired expectation $\langle e\|O_{A}\|o\rangle$ can be computed by exploiting the property of asymptotic factorization of products of local operators at infinite separation Barouch _et al._ (1970); Sachdev (2000); Bratteli and Robinson (2012) that yields $\langle e\|O_{A}\|o\rangle=\sqrt{\lim\limits_{r\to\infty}\langle e\|O_{A}O_{A+r}\| e\rangle}$, where the root’s sign is fixed by imposing positivity of the density matrix $\rho_{A}(u,v)$. Having obtained the exact reduced density matrix $\rho_{A}(u,v)$ for all possible subsystems $A$ and superposition amplitudes $u$ and $v$, we are equipped to investigate the nature of quantum ground states with respect to their properties of classicality and quantumness. ## III Two-body quantum correlations In this Section we analyze the behavior of one-way discord-type correlations and entanglement between any two spins for different ground states. One-way discord-type correlations are properties of quantum states more general than entanglement. Operationally, they are defined in terms of state distinguishability with respect to the so-called _classical-quantum_ states. The latter are quantum states that, besides being separable, i.e. not entangled, remain invariant under the action of at least one nontrivial local unitary operation. In geometric terms, a _bona fide_ measure of quantum correlations must quantify how much a quantum state _discords_ from classical-quantum states and must be invariant under the action of all local unitary operations. A computable and operationally well defined geometric measure of quantum correlations is then the _discord of response_ Roga _et al._ (2014); Giampaolo _et al._ (36). The pairwise discord of response $D_{R}$ for a two-spin reduced density matrix is defined as: \[D_{R}(\rho_{ij}^{(r)}(u,v))\equiv\frac{1}{2}\min_{U_{i}}d_{x}\left(\rho_{ij}^{ (r)}(u,v),\tilde{\rho}_{ij}^{(r)}(u,v)\right)^{2}\,,\] (6) where $\rho_{ij}^{(r)}(u,v)$ is the state of two spins $i$ and $j$ at a distance $r$, obtained by taking the partial trace of the ground state $\|g(u,v)\rangle$ with respect to all other spins in the system, $\tilde{\rho}_{ij}^{(r)}(u,v)\!\equiv\!U_{i}\rho_{ij}^{(r)}(u,v)U_{i}^{\dagger}$ is the two-spin state transformed under the action of a local unitary operation $U_{i}$ acting on spin $i$, and $d_{x}$ is any well-behaved, contractive distance (e.g. Bures, trace, Hellinger) of $\rho_{ij}^{(r)}$ from the set of locally unitarily perturbed states, realized by the least-perturbing operation in the set. The trivial case of the identity is excluded by considering only unitary operations with _harmonic_ spectrum, i.e. the fully non-degenerate spectrum on the unit circle with equispaced eigenvalues. For pure states the discord of response reduces to an entanglement monotone, whose convex-roof extension to mixed states is the so-called _entanglement of response_ Giampaolo and Illuminati (2007); Monras _et al._ (2011); Gharibian (2012). Therefore, the entanglement and the discord of response quantify different aspects of bipartite quantum correlations via two different uses of local unitary operations. The discord of response arises by applying local unitaries directly to the generally mixed state, while the entanglement of response stems from the application of local unitaries to pure states. By virtue of their common origin, it is thus possible to perform a direct comparison between these two quantities. In terms of the trace distance, which will be relevant in the following, the two-qubit entanglement of response is simply given by the squared concurrence Wootters (1998); Roga _et al._ (2014), whereas the two-qubit discord of response relates nicely to the trace distance-based geometric discord Nakano _et al._ (2013), whose closed formula is known only for a particular class of two-qubit states Ciccarello _et al._ (2014), although it can be computed for a more general class of two-qubit states through a very efficient numerical optimization. ### Symmetry-preserving ground states <figure><img src="content_image/1604.06403/x1.png"><figcaption>Figure 1: Nearest-neighbor trace distance-based discord of response (upperpanel) and nearest-neighbor trace distance-based entanglement of response(lower panel) for symmetry-preserving ground states, in the thermodynamiclimit, as functions of the external field h, and for different values of theanisotropy γ. Solid blue curve: γ=0.2; dashed red curve: γ=0.4; dot-dashedgreen curve: γ=0.6; double-dot-dashed black curve: γ=0.8; dotted orange curve:γ=1. In the lower panel, to each of these curves, there corresponds a verticalline denoting the associated factorizing field hf. In the upper panel, thesolid vertical line denotes the critical field hc=1.</figcaption></figure> We first compare the two-body entanglement of response and the two-body discord of response in symmetry-preserving ground states. For two neighboring spins, these two quantities are plotted in Fig. 1 as functions of the external field $h$ and for different values of the anisotropy $\gamma$. For any intermediate value of $\gamma$, the nearest-neighbor entanglement of response $E_{1}$ exhibits the following behavior. If $h<h_{f}$, $E_{1}$ decreases until it vanishes at the factorizing field $h=h_{f}$. Otherwise, if $h>h_{f}$, $E_{1}$ first increases until it reaches a maximum at some value of $h$ higher than the critical point $h_{c}=1$, then it decreases again until it vanishes asymptotically for very large values of $h$ in the paramagnetic phase (saturation). Overall, $E_{1}$ features two maxima at $h=0$ and $h>h_{c}$ and two minima at $h=h_{f}$ (factorization) and $h\rightarrow\infty$ (saturation). For the Ising model ($\gamma=1$) the point $h=0$ corresponds instead to a minimum, since it coincides with the factorizing field $h_{f}=\sqrt{1-\gamma^{2}}$. On the other hand, regardless of the value of $\gamma$, the nearest-neighbor discord of response $Q_{1}$ always features a single maximum. Depending on the value of $\gamma$ such maximum can be either in the ordered phase $h<h_{c}$ or in the disordered (paramagnetic) phase $h>h_{c}$, moving towards higher values of $h$ with increasing $\gamma$. Remarkably, $Q_{1}$ never vanishes at the factorizing field, except in the extreme case of $\gamma=1$. Indeed, at the factorizing field $h=h_{f}$ and for any $\gamma\neq 0,1$, the symmetry-preserving ground state is not completely factorized but rather is a coherent superposition with equal amplitudes of the two completely factorized MSBGSs. Consequently, while the two-body entanglement of response must vanish in accordance with the convex roof extension, the two-body discord of response remains always finite. <figure><img src="content_image/1604.06403/x3.png"><figcaption>Figure 2: Two-body trace distance-based discord of response (upper panel) andtwo-body trace distance-based entanglement of response (lower panel) forsymmetry-preserving ground states, in the thermodynamic limit, as functions ofthe external field h, in the case of γ=0.4, for different inter-spin distancesr. Solid blue curve: r=2; dashed red curve: r=3; dot-dashed green curve: r=8;dotted black curve: r=∞. In both panels, the two solid vertical linescorrespond, respectively, to the factorizing field (left) and to the criticalfield (right).</figcaption></figure> When increasing the inter-spin distance $r$, the pairwise entanglement of response $E_{r}$ and discord of response $Q_{r}$ behave even more differently (see Fig. 2). Due to the monogamy of the squared concurrence Coffman _et al._ (2000); Osborne and Verstraete (2006), $E_{r}$ dramatically drops to zero as $r$ increases, except in a small region around the factorizing field $h=h_{f}$ that gets smaller and smaller as $r$ increases, in agreement with the findings of Ref. Amico _et al._ (2006). On the other hand, while in the disordered and critical phases $Q_{r}$ vanishes as $r$ increases, in the ordered phase $Q_{r}$ survives even in the limit of infinite $r$. Indeed, in both the disordered and critical phases, and when $r$ goes to infinity, the only non-vanishing one-body and two-body correlation functions in the symmetry-preserving ground states are $\langle\sigma_{i}^{z}\rangle$ and $\langle\sigma_{i}^{z}\sigma_{i+r}^{z}\rangle$, so that the two-body reduced state can be written as a classical mixture of eigenvectors of $\sigma_{i}^{z}\sigma_{i+r}^{z}$. On the other hand, in the ordered phase, also the two-body correlation function $\langle\sigma_{i}^{x}\sigma_{i+r}^{x}\rangle$ appears, while $\langle\sigma_{i}^{x}\rangle$ vanishes due to symmetry preservation, thus preventing the two-body marginal of the symmetry-preserving ground state from being a mixture of classical states. ### Maximally symmetry-breaking ground states <figure><img src="content_image/1604.06403/x5.png"><figcaption>Figure 3: Nearest-neighbor trace distance-based discord of response (upperpanel) and nearest-neighbor trace distance-based entanglement of response(lower panel) in MSBGSs as functions of the external field h, for differentvalues of the anisotropy γ. Solid blue curve: γ=0.2; dashed red curve: γ=0.4;dot-dashed green curve: γ=0.6; double-dot-dashed black curve: γ=0.8; dottedorange curve: γ=1. In both panels, to each of these curves, there correspondsa vertical line denoting the associated factorizing field hf. The rightmostvertical line denotes the critical point.</figcaption></figure> <figure><img src="content_image/1604.06403/x7.png"><figcaption>Figure 4: Two-body trace distance-based discord of response (upper panel) andtwo-body trace distance-based entanglement of response (lower panel) in MSBGSsas functions of the external field h, at γ=0.4, for different inter-spindistances r. Solid blue curve: r=2; dashed red curve: r=3; dot-dashed greencurve: r=8; dotted black curve: r=∞. In both panels, the two solid verticallines correspond, respectively, to the factorizing field (left) and to thecritical field (right).</figcaption></figure> In this section we move the focus of the comparison between two-body entanglement of response and discord of response from symmetry-preserving to MSBGSs. Spontaneous symmetry breaking manifests itself in the thermodynamic limit, in the ordered phase $h<h_{c}=1$ and for any non zero anisotropy $\gamma$, so that hereafter we will restrict the region of the phase space under investigation accordingly. Fig. 3 shows that, as soon as symmetry breaking is taken into account, only the discord of response is affected by symmetry breaking at the critical point $h_{c}=1$. In fact, according to Ref. Osterloh _et al._ (2006), the concurrence and, consequently, the two-body entanglement of response, attain the same value for any $h\geq h_{f}$ both in the symmetry-preserving and MSBGSs. Otherwise, if $h<h_{f}$, there is a slight enhancement in the pairwise entanglement of response in the MSBGSs compared to the corresponding symmetry-preserving ones. Conversely, in general, the pairwise discord of response undergoes a dramatic suppression in the entire ordered phase $h<h_{c}$ when moving from symmetry-preserving to MSBGSs. Considering the dependence on the inter-spin distance, we observe that the pairwise discord of response loses its long-range nature when moving from symmetry-preserving to MSBGSs (see Fig. 4). More precisely, both the pairwise entanglement of response and the pairwise discord of response vanish asymptotically with increasing inter-spin distance. In the case of the pairwise entanglement of response, this result is again due to the monogamy of the squared concurrence Coffman _et al._ (2000); Osborne and Verstraete (2006). In the case of the pairwise discord of response, it is instead due to the fact that not only the correlation function $\langle\sigma_{i}^{x}\sigma_{i+r}^{x}\rangle$ but also $\langle\sigma_{i}^{x}\rangle$ and $\langle\sigma_{i}^{x}\sigma_{i+r}^{z}\rangle$ are nonvanishing in the limit of infinite inter-spin distance $r$. This feature allows to write any two-spin reduced density matrix obtained from the MSBGSs as a classical mixture of eigenvectors of $O_{i}O_{i+r}$, where $O_{i}$ is an Hermitian operator defined on the $i$-th site as $O_{i}=\cos\beta\sigma_{i}^{z}+\sin\beta\sigma_{i}^{x}$ with $\tan\beta=\frac{\langle\sigma_{i}^{x}\rangle}{\langle\sigma_{i}^{z}\rangle}$. Overall, the quantum correlations between any two spins decrease significantly in the entire ordered phase when symmetry breaking is taken into account, and are almost entirely made up by contributions due to entanglement. In particular, at the factorizing field $h_{f}$, both the entanglement of response and the discord of response vanish. Indeed, we recall that the factorizing field $h_{f}$ owes its name to the two MSBGSs that are completely separable (product) at such value of the external magnetic field. ## IV Global properties: local convertibility and many-body entanglement sharing We now investigate the nature of quantum ground states in the ordered phase with respect to the properties of local convertibility of the global ground states and the many-body entanglement distribution. ### Local convertibility of many-body quantum ground states We begin by studying the property of local convertibility of quantum ground states in an ordered phase. In general, given two pure bipartite quantum states, $\|{\psi_{1}}\rangle$ and $\|{\psi_{2}}\rangle$, we say that $\|{\psi_{1}}\rangle$ is locally convertible into $\|{\psi_{2}}\rangle$ if $\|{\psi_{1}}\rangle$ can be transformed into $\|{\psi_{2}}\rangle$ by using only local quantum operations and classical communication (LOCC), and the aid of an ancillary entangled system Jonathan and Plenio (46, 47). This concept of local convertibility can be formalized in terms of the entire hierarchy of the Rényi entanglement entropies $S_{\alpha}(\rho_{A})=\frac{1}{1-\alpha}\log_{2}\left[Tr(\rho_{A}^{\alpha})\right]$ of the reduced density operator of subsystem $A$, which provides a complete characterization of the entanglement spectrum and its scaling behavior in different quantum phases Giampaolo _et al._ (48). In a many-body setting, the necessary and sufficient conditions for a bipartite global state $\|{\psi_{1}}\rangle$ to be locally convertible to another global state $\|{\psi_{2}}\rangle$ is that the inequality $S_{\alpha}(\psi_{1})\geq S_{\alpha}(\psi_{2})$ holds for all bipartitions and all $\alpha>0$ Turgut (2007). Local convertibility has been recently applied to the characterization of topological order and the computational power of different quantum phases Hamma _et al._ (2013); Cui _et al._ (2012, 2013). <figure><img src="content_image/1604.06403/x9.png"><figcaption>Figure 5: Behavior of the Rényi entropies Sα(ρA) as functions of the differentground states in the ordered phase, h<hc, in the case of a subsystem Aℓ madeof ℓ contiguous spins. Each line stands for a different value of α. Blackdotted line: α=0.5. Green solid line: α→1+ (von Neumann entropy). Blue dot-dashed line: α=3. Red dashed line: α→∞. The different ground states areparameterized by the superposition amplitudes u=cos(θ) and v=sin(θ). The twovertical lines correspond to the two MSBGSs, respectively obtained for θ=π/4and θ=3π/4. The Hamiltonian parameters are set at the intermediate valuesγ=0.5 and h=0.5. Analogous behaviors are observed for different values of theanisotropy and external field.</figcaption></figure> <figure><img src="content_image/1604.06403/x10.png"><figcaption>Figure 6: Behavior of the Rényi entropies Sα(ρA) as functions of the differentground states in the ordered phase, h<hc, in the case of a subsystem Ar madeby two spins, for different inter-spin distances r. Each line stands for adifferent value of α. Black dotted line: α=0.5. Green solid line: α→1+ (vonNeumann entropy). Blue dot-dashed line: α=3. Red dashed line: α→∞. Thedifferent ground states are parameterized by the superposition amplitudesu=cos(θ) and v=sin(θ). The two vertical lines correspond to the two MSBGSs,respectively obtained for θ=π/4 and θ=3π/4. The Hamiltonian parameters are setat the intermediate values γ=0.5 and h=0.5. Analogous behaviors are observedfor different values of the anisotropy and external field.</figcaption></figure> It was previously shown that symmetric ground states are always locally convertible among themselves for $h_{f}<h<h_{c}$, and never for $h<h_{f}<h_{c}$ Giampaolo _et al._ (48). Here, thanks to the general methods developed in Section II, we are able to investigate the local convertibility property of _all_ quantum ground states in the ordered phase. In Fig. 5 we report the behavior of the Rényi entropies $S_{\alpha}$ as functions of the different ground states for a bipartition of the system in which subsystem $A$ is made of $\ell$ contiguous spins, while in Fig. 6 we report it for subsystem $A$ made of two spins with various inter-spin distances. We observe that the MSBGSs are the ground states characterized by the smallest value of all Rényi entropies, independently of the size $\ell$ of the subsystem and of the inter-spin distance $r$. Therefore, all elements in the ground space are always locally convertible to a MSBGS, while the opposite is impossible. This first quantitative criterion for classicality is thus satisfied only by MSBGSs. ### Many-body entanglement distribution We now compare symmetry-breaking and symmetry-preserving ground states with respect to entanglement distribution. The monogamy inequality quantifies in a simple and direct way the limits that are imposed on how bipartite entanglement may be shared among many parties Coffman _et al._ (2000); Osborne and Verstraete (2006). For a given many-body system of $N$$1/2$-spins it reads: \[\tau(i\|N-1)\geq\sum_{j\neq i}\tau(i\|j)\;\;\;\;,\;\;\;\forall\;i\;.\] (7) In the above expression, $\tau=C^{2}$ is known as the tangle, where $C$ is the concurrence Hill and Wootters (1997); Wootters (1998); the sum in the r.h.s. runs over all $N-1$ spins excluding spin $i$. The l.h.s. quantifies the bipartite entanglement between one particular, arbitrarily chosen, spin in the collection (reference spin $i$) and all the remaining $N-1$ spins. The r.h.s. is the sum of all the pairwise entanglements between the reference spin and each of the remaining $N-1$ spins. The inequality implies that entanglement cannot be freely distributed among multiple quantum parties $N\geq 3$, a constraint of quantum origin with no classical counterpart. The residual tangle $\tilde{\tau}$ is the positive semi-definite difference between the l.h.s and the r.h.s in Eq. (7). It measures the amount of entanglement not quantifiable as elementary bipartite spin-spin entanglement. Its minimum value compatible with monogamy provides yet another quantitative criterion for classicality. Specializing, for simplicity but without loss of generality, to translationally-invariant $XY$ spin systems in magnetically ordered phases, since the expectation value of $\sigma_{i}^{y}$ vanishes on every element of the ground space, the expressions of the tangle $\tau$ and the residual tangle $\tilde{\tau}$ for any arbitrarily chosen spin in the chain read, respectively, \[\tau = 1-m_{z}^{2}-(u^{}v+v^{}u)^{2}m_{x}^{2}\;,\] (8) \[\tilde{\tau} = \tau-2\sum_{r=1}^{\infty}C_{r}^{2}(u,v)\geq 0\;,\] (9) where $m_{z}\!=\!\langle e\|\sigma_{i}^{z}\|e\rangle\!=\!\langle o\|\sigma_{i}^{z}\|o\rangle$ is the on-site magnetization along $z$, the order parameter $m_{x}\!=\!\langle e\|\sigma_{i}^{x}\|o\rangle\!=\!\sqrt{\lim\limits_{r\to\infty} \langle e\|\sigma_{i}^{x}\sigma_{i+r}^{x}\|e\rangle}$, and $C_{r}(u,v)$ stands for the concurrence between two spins at a distance $r$ when the system is in any one of the possible ground states $\|g(u,v)\rangle$, Eq. (2). As already mentioned, by comparing the symmetric ground states with the MSBGSs, the spin-spin concurrence is larger in the MSBGSs Osterloh _et al._ (2006) if $h<h_{f}<h_{c}$, where $h_{f}=\sqrt{1-\gamma^{2}}$ is the factorizing field, while for $h_{f}<h<h_{c}$ they are equal. In fact, we have verified that these two results are much more general. We have compared all ground states (symmetric, partially symmetry breaking, and MSBGSs) and we have found that for $h<h_{f}<h_{c}$ the spin-spin concurrences are maximum in the MSBGSs for all values of the inter-spin distance $r$, while for $h_{f}<h<h_{c}$ and for all values of $r$ they are independent of the superposition amplitudes $u$ and $v$ and thus acquire the same value irrespective of the chosen ground state. Finally, it is immediate to see that the third term in the r.h.s. of Eq. (8) is maximized by the two MSBGSs. Collecting all these results, it follows that the many-body, macroscopic multipartite entanglement, as quantified by the residual tangle, is minimized by the two MSBGSs and therefore also this second quantitative criterion for classicality is satisfied only by the MSBGSs among all possible quantum ground states. ## V Conclusions and outlook In the present work we have investigated the classical nature of globally ordered phases associated to nonvanishing local order parameters and spontaneous symmetry breaking. We have put on quantitative grounds the long-standing conjecture that the maximally symmetry-breaking ground states (MSBGSs) are macroscopically the most classical ones among all possible ground states. We have proved the conjecture by introducing and verifying two independent quantitative criteria of macroscopic classicality. The first criterion states that all global ground states in the thermodynamic limit are locally convertible to MSBGSs, i.e. by applying only local operations and classical communication (LOCC transformations), while the opposite is impossible. The second criterion states that the MSBGSs are the ones that satisfy at its minimum the monogamy inequality for globally shared bipartite entanglement and thus minimize the macroscopic multipartite entanglement as quantified by the residual tangle. We have thus verified that, according to these two criteria, the MSBGSs are indeed the most classical ones among all possible quantum ground states. These findings lend a strong quantitative support to the intuitive idea that the physical mechanism which selects the MSBGSs among all possible ground states at the macroscopic level is due to the unavoidable presence of environmental perturbations, such as local fields, which in real-world experiments necessarily drive the system onto the most classical among the possible ground states via decoherence. This reasoning is strengthened by the fact that local perturbations may be described by LOCC transformations and for each set of parameters consistent with a globally ordered phase all quantum ground states are always locally convertible into the MSBGSs. The above conclusions are further strengthened by the results appeared recently in Ref. Hamma _et al._ (2016), where it is shown that the MSBGSs of quantum many-body Hamiltonians have vanishing total correlations between macroscopically separated regions, as measured by the quantum mutual information. _Acknowledgments_ - The authors acknowledge financial support from the Italian Ministry of Scientific and Technological Research under the PRIN 2010/2011 Research Fund, and from the EU FP7 STREP Project EQuaM, Grant Agreement No. 323714. SMG acknowledges financial support from the Austrian Science Foundation, Grant FWF-P23627-N16. ## References * Goldstone _et al._ (1962)J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962). * Sachdev (2000)S. Sachdev, _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, 2000). * Amico _et al._ (2008)L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008). * Verstraete _et al._ (2008)F. Verstraete, V. Murg, and J. I. Cirac, Advances in Physics 57, 143 (2008). * Osterloh _et al._ (2002)A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002). * Osborne and Nielsen (2002)T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002). * Giampaolo and Hiesmayr (2013)S. M. Giampaolo and B. C. Hiesmayr, Phys. Rev. A 88, 052305 (2013). * Giampaolo and Hiesmayr (2014)S. M. Giampaolo and B. C. Hiesmayr, New Journal of Physics 16, 093033 (2014). * Giampaolo _et al._ (2008)S. M. Giampaolo, G. Adesso, and F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008). * Giampaolo _et al._ (2009)S. M. Giampaolo, G. Adesso, and F. Illuminati, Phys. Rev. B 79, 224434 (2009). * Giampaolo _et al._ (2010)S. M. Giampaolo, G. Adesso, and F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010). * Zanardi and Paunković (2006)P. Zanardi and N. Paunković, Phys. Rev. E 74, 031123 (2006). * Zanardi _et al._ (2007)P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. 99, 100603 (2007). * Osterloh _et al._ (2006)A. Osterloh, G. Palacios, and S. Montangero, Phys. Rev. Lett. 97, 257201 (2006). * de Oliveira _et al._ (2006)T. R. de Oliveira, G. Rigolin, and M. C. de Oliveira, Phys. Rev. A 73, 010305 (2006). * Amico _et al._ (2012)L. Amico, D. Rossini, A. Hamma, and V. E. Korepin, Phys. Rev. Lett. 108, 240503 (2012). * Tomasello _et al._ (2012)B. Tomasello, D. Rossini, A. Hamma, and L. Amico, International Journal of Modern Physics B 26, 1243002 (2012). * Bratteli and Robinson (2012)O. Bratteli and D. W. Robinson, _Operator Algebras and Quantum Statistical Mechanics: Volume 1: C-and W-Algebras. Symmetry Groups. Decomposition of States_ (Springer Science & Business Media, 2012). * Arodz _et al._ (2012)H. Arodz, J. Dziarmaga, and W. H. Zurek, _Patterns of symmetry breaking_, Vol. 127 (Springer Science & Business Media, 2012). * Zurek (2003)W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003). * van Wezel (2008)J. van Wezel, Phys. Rev. B 78, 054301 (2008). * Barouch and McCoy (1971)E. Barouch and B. M. McCoy, Phys. Rev. A 3, 786 (1971). * Campbell _et al._ (2013)S. Campbell, J. Richens, N. L. Gullo, and T. Busch, Phys. Rev. A 88, 062305 (2013). * Hastings (2007)M. B. Hastings, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007). * Brandão and Horodecki (2013)F. G. Brandão and M. Horodecki, Nature Physics 9, 721 (2013). * Holzhey _et al._ (1994)C. Holzhey, F. Larsen, and F. Wilczek, Nuclear Physics B 424, 443 (1994). * Vidal _et al._ (2003)G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003). * Korepin (2004)V. E. Korepin, Phys. Rev. Lett. 92, 096402 (2004). * Lieb _et al._ (1961)E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961). * Pfeuty (1970)P. Pfeuty, Annals of Physics 57, 79 (1970). * Barouch _et al._ (1970)E. Barouch, B. M. McCoy, and M. Dresden, Phys. Rev. A 2, 1075 (1970). * Johnson and McCoy (1971)J. D. Johnson and B. M. McCoy, Phys. Rev. A 4, 2314 (1971). * Kurmann _et al._ (1982)J. Kurmann, H. Thomas, and G. Müller, Physica A: Statistical Mechanics and its Applications 112, 235 (1982). * Roscilde _et al._ (2005)T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 94, 147208 (2005). * Roga _et al._ (2014)W. Roga, S. M. Giampaolo, and F. Illuminati, Journal of Physics A: Mathematical and Theoretical 47, 365301 (2014). * Giampaolo _et al._ (2013a)S. M. Giampaolo, A. Streltsov, W. Roga, D. Bruß, and F. Illuminati, Phys. Rev. A 87, 012313 (2013a). * Giampaolo and Illuminati (2007)S. M. Giampaolo and F. Illuminati, Phys. Rev. A 76, 042301 (2007). * Monras _et al._ (2011)A. Monras, G. Adesso, S. M. Giampaolo, G. Gualdi, G. B. Davies, and F. Illuminati, Phys. Rev. A 84, 012301 (2011). * Gharibian (2012)S. Gharibian, Phys. Rev. A 86, 042106 (2012). * Wootters (1998)W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). * Nakano _et al._ (2013)T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013). * Ciccarello _et al._ (2014)F. Ciccarello, T. Tufarelli, and V. Giovannetti, New Journal of Physics 16, 013038 (2014). * Coffman _et al._ (2000)V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000). * Osborne and Verstraete (2006)T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006). * Amico _et al._ (2006)L. Amico, F. Baroni, A. Fubini, D. Patanè, V. Tognetti, and P. Verrucchi, Phys. Rev. A 74, 022322 (2006). * Jonathan and Plenio (1999a)D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566 (1999a). * Jonathan and Plenio (1999b)D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 1455 (1999b). * Giampaolo _et al._ (2013b)S. M. Giampaolo, S. Montangero, F. Dell’Anno, S. De Siena, and F. Illuminati, Phys. Rev. B 88, 125142 (2013b). * Turgut (2007)S. Turgut, Journal of Physics A: Mathematical and Theoretical 40, 12185 (2007). * Hamma _et al._ (2013)A. Hamma, L. Cincio, S. Santra, P. Zanardi, and L. Amico, Phys. Rev. Lett. 110, 210602 (2013). * Cui _et al._ (2012)J. Cui, M. Gu, L. C. Kwek, M. F. Santos, H. Fan, and V. Vedral, Nature Communications 3, 812 (2012). * Cui _et al._ (2013)J. Cui, L. Amico, H. Fan, M. Gu, A. Hamma, and V. Vedral, Phys. Rev. B 88, 125117 (2013). * Hill and Wootters (1997)S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997). * Hamma _et al._ (2016)A. Hamma, S. M. Giampaolo, and F. Illuminati, Phys. Rev. A 93, 012303 (2016).
1609.09037	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 72840, "num_imgs": 5, "llama3_tokens_count": 19361 }	[ "content_image/1609.09037/x1.png", "content_image/1609.09037/x3.png", "content_image/1609.09037/x7.png", "content_image/1609.09037/x11.png", "content_image/1609.09037/x12.png" ]	# Control of Charging of Electric Vehicles through Menu-Based Pricing Arnob Ghosh and Vaneet Aggarwal, The authors are with the School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, email: {ghosh39,vaneet}@purdue.edu. This work was supported in part by the U.S. National Science Foundation under grant CCF-1527486. This paper was presented in part at the IEEE International Conference on Communications, Paris, France, May 2017 [1]. ###### Abstract We propose an online pricing mechanism for electric vehicle (EV) charging. A charging station decides prices for each arriving EV depending on the energy and the time within which the EV will be served (i.e. deadline). The user selects either one of the contracts by paying the prescribed price or rejects all depending on their surpluses. The charging station can serve users using renewable energy and conventional energy. Users may select longer deadlines as they may have to pay less because of the less amount of conventional energy, however, they have to wait a longer period. We consider a _myopic_ charging station and show that there exists a pricing mechanism which jointly maximizes the social welfare and the profit of the charging station when the charging station knows the utilities of the users. However, when the charging station does not know the utilities of the users, the social welfare pricing strategy may not maximize the expected profit of the charging station and even the profit may be $0$. We propose a fixed profit pricing strategy which provides a guaranteed fixed profit to the charging station and can maximize the profit in practice. We empirically show that our proposed mechanism reduces the peak-demand and utilizes the limited charging spots in a charging station efficiently. Electric vehicle charging, menu-based pricing, energy harvesting, myopic strategy, social welfare. ## I Introduction Electric Vehicles (EVs) have several advantages over the traditional gasoline powered vehicles. For example, EVs are more environment friendly and more energy efficient. Thus, regulators (e.g. Federal Energy Regulator Commission (FERC)) are providing incentives to the consumers to switch to electric vehicles. However, the successful deployment of charging stations crucially depends on the profit of the charging stations and how efficiently the resources are used for charging the electric vehicles. Without profitable charging stations, the wide deployment of the EVs will remain a distant dream. On the other hand, because of the environment-friendly nature of the electric vehicles, it is also important for the regulators to increase the user (or, consumer) surplus to provide an incentive for the users. Hence, selecting a price is an imperative issue for the charging stations. The charging station may have limited charging spots or renewable energy harvesting devices. Hence, intelligent allocation of the resources among the EVs is a key component for fulfilling the potential of EVs’ deployment. We propose a menu based pricing scheme for charging an EV. Whenever an EV arrives, the charging station offers a variety of contracts $(l,t_{dead})$ at price $p_{l,t_{dead}}$ to the user where the user will be able to use up to $l$ units of energy within the deadline $t_{dead}$ for completion. The EV user either accepts one of the contracts by paying the specified price or rejects all of those based on its payoff. We assume that the user gets an utility for consuming $l$ amount of energy with the deadline $t_{dead}$. The payoff of the user (or, user’s surplus) for a contract is the difference between the utility and the price paid for one contract. The user will select the option which fetches the highest payoff. The various advantages of the above pricing scheme should be noted. First, it is an online pricing scheme. It can be adapted for each arriving user. Second, since the charging station offers prices for different levels of energy and the deadline, the charging station can prioritize one contract over the others depending on the energy resources available. Favorable prices for shorter deadlines can attract users to vacate the charging stations early and only use it when it is necessary. Third, the user’s decision is much simplified. She only needs to select one of the contracts (or, reject all) and will receive the prescribed amount within the prescribed deadline. Fourth, the pricing mechanism is inherently _individual rational¹_, _incentive compatible²_, and _truthful³_. [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] We consider that the charging station is equipped with renewable energy harvesting devices and a storage device for storing energies. The charging station may also buy conventional energy from the market to fulfill the contract of the user if required. Hence, if a new user accepts the contract $(l,t_{dead})$, a cost is incurred to the charging station. This cost may also depend on the existing EVs and their resource requirements. Hence, the charging station needs to find the optimal cost for fulfilling each contract. We show that obtaining that cost is equivalent to solve a _linear programming_ problem. We consider two optimization problems--i) social welfare⁴ maximization, and ii) the EV charging station’s profit maximization. We first propose a pricing scheme which maximizes the social welfare _irrespective of whether the charging station is aware of the utilities of the users or not_. The pricing scheme is simple to compute, as the charging station selects a price which is equal to the marginal cost for fulfilling a certain contract for a new user (Theorem 2, Corollary 1). However, the above pricing scheme only provides _zero_ profit to the charging stations. Thus, such a pricing scheme may not be useful to the charging station. We show that when a charging station is _clairvoyant_ (i.e., the charging station knows the utilities of the users), there exists a pricing scheme which satisfies both the objectives (Theorem 3). Though in the above pricing mechanism, the user’s surplus becomes $0$. Thus, a _clairvoyant_ charging station may not be beneficial for the user’s surplus. [FOOTNOTE:4][ENDFOOTNOTE] The charging station may not know the exact utilities of the users, however, it may know the distribution function⁵ from where it is drawn. We investigate the existence of a pricing mechanism which maximizes the ex-post social welfare i.e. maximizes the social welfare for every possible realization of the utility function. In the scenario where the charging station does not know the exact utilities of the users, we show that there _may not exist a pricing strategy which simultaneously maximizes the ex-post social welfare and the expected profit_. One has to give away the ex-post social welfare maximization in order to achieve expected profit maximization. Thus, unlike when the charging station _is clairvoyant_, there may not exist a pricing strategy which simultaneously satisfies both the objectives when the exact utilities are unknown. We propose a pricing strategy which can fetch the highest possible profit to the charging station under the condition that it maximizes the ex-post social welfare (Theorem 4). Above pricing strategy provides a _worst case_ maximum profit to the charging station. We show that such a pricing strategy can fetch a higher profit when the charging station can harvest a large amount of renewable energy. However, the profit only increases up to a certain threshold, beyond that threshold harvested energy has no effect on the profit. [FOOTNOTE:5][ENDFOOTNOTE] Since the above pricing strategy may not yield the _maximum expected_ profit to the charging station, we have to relax the constraint the social welfare to be maximized in order to yield a higher profit to the charging station. Whether a contract will be selected by the user does not depend on the price of the contract, but also the prices of other contracts. Thus, achieving a pricing scheme which maximizes the expected profit is difficult because of the discontinuous nature of the profits. We propose a pricing strategy which yields a fixed (say, $\beta$) amount of profit to the charging station. We show that the above pricing strategy also maximizes the social welfare with the desired level of probability for a suitable choice of $\beta$ (Theorem 5). Hence, such a pricing scheme is also attractive to the regulators. Further, we show that a suitable choice of $\beta$ can maximize the profit of the charging station for a class of utility functions (Theorem 7). Finally, we, empirically provide insights how a trade-off between the profit of the charging station and the social welfare can be achieved for various pricing schemes (Section VI). We also show that how our pricing scheme can increase greater utilization of the resources and result in a lower number of charging spots compared to the existing ones. _The proofs are deferred to the technical report[2] owing to the space constraint._ Related Literature: _To the best our knowledge this is the first attempt to consider contract based online pricing for controlling both the energy and deadline of the EVs._ However, other pricing mechanisms to control the charging pattern of EV in a _residential_ charging in a day-ahead market are proposed[3, 4, 5, 6, 7, 8]. In contrast to the residential charging, in a commercial or workplace charging station, users do not have control of charging the car at each instance. They need a certain amount of energy within a deadline. In our proposed mechanism, the charging station selects different prices for different options and the user only needs to select a contract, it does not need to control the charging pattern at each instance. Optimal pricing for a day-ahead demand response program have been studied [9, 10]. However, we need an online pricing mechanism in the EV charging station. Since the charging station is unaware of the future arrivals of the users and has limited renewable energy, determining the optimal pricing in such a setting is more challenging. The menu-based pricing is an online pricing mechanism and can enhance the efficient usage of the resources by controlling the deadline. Unlike in the demand response program, the users also do not need to control the demand for each instance in our menu-based pricing. [11] proposed an online VCG auction mechanism. However, in [11] the user’s payment is determined at the end of the day, and thus users are not sure how much they have to pay beforehand. Hence, it may not be preferred by the users. In contrast, in our mechanism the users select one of the contracts and pay the prescribed price beforehand. In [12, 13, 14, 15] online scheduling algorithms have been proposed for charging EVs. The main focus of these papers was scheduling, they did not consider the optimal pricing approach for the charging station which we did. Further, unlike in [12, 13, 14, 15], in our menu-based pricing scheme, the charging station can control the energy requirement and the deadline of the users by selecting the prices to the users. Hence, a greater flexibility can be achieved. Additionally, [12, 15] did not guarantee that the energy demand will be fulfilled. In contrast, in our model once a user opts for an option, the EV charging station always fulfills the request of users. In the deadline differentiated pricing [16, 17, 18, 19], each user’s total energy consumption is fixed, however the user can specify the deadline. On the contrary, in our proposed menu-based pricing mechanism each user can jointly choose _any pair_ of energy level and deadline. The deadline differentiated pricing is suited for a day-ahead offline setting where an equilibrium price is attained for a specific set of pre-determined decision of the users. However, the users’ utilities and thus optimal decision may change in real time and thus, the deadline differentiated pricing may not be suitable for online setting. Our menu-based pricing approach is online, where the price menu is adapted for each arriving user. Further, [18] assumed that the price setter knows the utilities of the users. [16, 19] assumed that the utility functions are strictly concave, and [17] put some restrictions on the utility functions to achieve optimality. However, such assumptions are not necessary for our approach. ## II Model We consider a charging station which wants to select a pricing strategy in order to maximize its payoff over a certain period of time $T$ (e.g. one day). Suppose that user $k=1,\ldots,K$ arrives at the charging station at time $t_{k}$. The charging station decides a price menu or a contract $p_{k,l,t}$ to user $k$ for different energy levels $l\in\{1,\ldots,L\}$ and deadline $t\in\{t_{k}+1,\ldots,T\}$ (Fig. 2).⁶ It is needless to say that we can discretize the time and energy at any level that one may want⁷, however, the computational cost will increase. User $k$ has to decide $l$ and $t$ based on the menu; if she decided to accept any option on the menu, she has to pay the prescribed price $p_{k,l,t}$. The user can decide not to accept any price too (Fig 2). The EV may not be charged continuously i.e. preemption is allowed. A preempted battery of the EV can be resumed charging from the previous battery level upon preemption. [FOOTNOTE:6][ENDFOOTNOTE] [FOOTNOTE:7][ENDFOOTNOTE] <figure><img src="content_image/1609.09037/x1.png"><figcaption>Fig. 1: The trading model: Charging station offers a menu of contracts, thearriving user decides either one of them or rejects all.</figcaption></figure> ### _User’s utilities_ If user $k$ selects the price option $p_{k,l,t}$, it will get an utility $u_{k,l,t}$. Hence, its _surplus_ or payoff will be $u_{k,l,t}-p_{k,l,t}$. If the user rejects all the options then its utility is $0$ (Fig. 2). We assume that the realized value $u_{k,l,t}$ is drawn from a distribution function of the random variable $U_{k,l,t}$. The random variables $U_{k,l,t}$ need not be independent, in fact, they can be generated from a joint distribution. In practice, there is a correlation of the utilities among different deadlines and charging amount. For example, $U_{k,l_{1},t}\geq U_{k,l,t}$ if $l_{1}>l$ as a higher amount of energy for a fixed deadline should induce higher utility to a user. Similarly, $U_{k,l,t_{1}}\leq U_{k,l,t_{2}}$ if $t_{1}>t_{2}$ since for a similar level of charge, the user will prefer the smaller deadline menu as it will give the user more flexibility. On the other hand, a user who wants to park a long time may not mind a longer deadline. Thus, we do not make any a priori assumptions on the utility functions since they can be different for different users. We assume that the car vacates the charging spot once it exceeds its prescribed deadline⁸. [FOOTNOTE:8][ENDFOOTNOTE] ### _The charging Station_ #### Ii-B1 Hybrid Energy Source We assume that the charging station can obtain energies for fulfilling the charging request both from the renewable sources and conventional sources (Fig. 2). The charging station can buy a conventional energy $q_{t}$ at a price $c_{t}$ for usage during the interval $[t,t+1)$. We do not assume any specific type of pricing schemes for buying conventional energy, however, we assume that $c_{t}$ is known. If the real time pricing is used, then we consider $c_{t}$ as the expected real-time price at time $t$. The charging station is also equipped with an energy harvesting device and a storage capacity of $B_{max}$ (Fig. 2). The harvesting device harvests $\bar{E}^{t}$ amount of energy between $[t,t+1)$. We assume that the marginal cost to harvest renewable energy is $0$. The amount of energy that the charging station uses from the storage for the time $[t,t+1)$ be $r_{t}$. #### Ii-B2 System Constraints The charging station must procure the $l$ amount of energy for user $k$ by time $d_{k}$ if the user accepts the price menu $p_{k,l,d_{k}}$. Let $q_{k,t}$ be the conventional energy and $r_{k,t}$ be the energy from the storage device used by the charging station to charge user $k$ for time interval $[t,t+1)$. Then, we must have the following constraint \[\sum_{t=t_{k}}^{d_{k}-1}(r_{k,t}+q_{k,t})\geq l\] (1) Suppose that initially, there is a set of users $\mathcal{K}_{0}$ already present in the charging station at time $t_{k}$. Now, the user $i\in\mathcal{K}_{0}$ has a deadline of $w_{i}$ and additional demand $N_{i}$. The charging station must have to satisfy the demand of those users. Thus, the charging station must also satisfy \[\sum_{t=t_{k}}^{w_{i}-t_{k}-1}(r_{i,t}+q_{i,t})\geq N_{i},\forall i \in\mathcal{K}_{0}\] (2) We also assume that the charging station has one kind of charging equipment (either slow charging or fast charging) and there is a maximum rate constraint ($R_{max}$). Hence, \[r_{k,t}+q_{k,t}\leq R_{max},\quad r_{i,t}+q_{i,t}\leq R_{max} \forall i\in\mathcal{K}_{0}\quad\forall t.\] (3) Since the total energy $r_{t}$ used for charging from the storage device and the amount of conventional energy, $q_{t}$ bought from the market, thus, \[r_{k,t}+\sum_{i\in\mathcal{K}_{0}}r_{i,t}=r_{t},\quad q_{k,t}+ \sum_{i\in\mathcal{K}_{0}}q_{i,t}\leq q_{t}\quad\forall t\] (4) Note that the charging station may store the unused conventional energy bought from the market i.e. $q_{t}-q_{k,t}-\sum_{i\in\mathcal{K}_{0}}q_{i,t}$ is stored in the storage device. The charging station may buy an additional amount of conventional energy at time $t$, if the future prices are higher. Let the battery level at time $t+1$ be $B^{t+1}$ and $B_{0}$ be the initial battery level. The charging station also wants to keep the battery level at the end of the day as $B_{0}$. If the final battery level need not match the initial level, our pricing approach can be easily extended to that scenario. If the battery can not hold the excessive energy, then it is wasted. Let use denote the wasted energy for time $[t,t+1)$ be $D_{t}\geq 0$, ⁹then [FOOTNOTE:9][ENDFOOTNOTE] \[0\leq B^{t+1}\leq B_{max},B^{t_{k}}=B_{0}\quad B^{T}=B_{0}.\] (5) The constraints in (4) and (5) put a bound on the maximum amount of energy can be used for charging. ## III Problem Formulation The profit of the charging station inherently depends on whether the user will accept that menu or not. Hence, before how the charging station will select $p_{k,l,t}$ for user $k$, we delve into the decision process of the users. ### _User’s decision_ A user selects at most one of the price menus in order to maximize its payoff or surplus. We assume that the user is a _price taker_. Thus, for a menu of prices $p_{k,l,t}$, the user $k$_selects_$A_{k,l,t}\in[0,1]$ such that it maximizes the following \[\text{maximize}\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}A_{k,l,t}(u_{k,l ,t}-p_{k,l,t})\] \[\text{subject to }\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}A_{k,l,t}\leq 1\] (6) Note from the formulation in (III-A) the maximum is achieved when $A_{k,l,t}=1$ for the contract which maximizes the user $k$’s payoff (i.e., $\max_{i,j}\{u_{k,i,j}-p_{k,i,j}\}=u_{k,l,t}-p_{k,l,t}$.) and is $0$ otherwise. If such a solution is not unique, any convex combination of these solutions is also optimal since a user can select any of the maximum payoff contracts. We denote the decision as $A_{k,l,t}(\mathbf{p}_{k})$. Note that the decision whether to accept the menu $p_{k,l,t}$ not only depend on the price $p_{k,l,t}$ but also other price menus i.e. $p_{k,i,j}$ where $i\in\{1,\ldots,L\}$ and $j\in\{t_{k}+1,\ldots,T\}$ as the user only selects the price menu which is the most favorable to himself. Note that if the maximum payoff that user gets among all the price menus (or, contracts) is negative, then the user will not charge i.e. $A_{k,l,t}=0$ for all $l$ and $t$. We also assume that if there is a tie between charging and not charging, then the user will decide to charge i.e. if the maximum payoff that user can get is $0$, then the user will decide to charge.¹⁰ [FOOTNOTE:10][ENDFOOTNOTE] ### _Myopic Charging Station_ Since the users arrive for the charging request at any time throughout the day, the charging station does not know the exact arrival times for the future vehicles. We consider that the charging station is myopic or near-sighted i.e. it selects its price for user $k$ without considering the future arrival process of the vehicles. However, it will consider the cost incurred to charge the existing EVs. Note that as the number of existing users increases, the marginal cost can increase to fulfill a contract for an arriving user, hence, such a pricing strategy may not maximize the payoff in a long run. We, later show that _a myopic pricing strategy is optimal in the case the marginal cost of fulfilling a demand of a new user is independent of the number of existing users._ In practice, the charging station often has fixed number of charging spots, thus, the charging station may want to select high prices for user $k$, in order to make the charging spots available for the users who can pay more but only will arrive in future¹¹. However, such a pricing strategy is against the first come first serve basis which is the current norm for charging a vehicle. Our approach considers a fair allocation process, where the charging station serves users based on the first come first serve basis. Later in Section VI, we show that since the charging station can control the time spent by an EV through pricing, our approach results into a lower number of charging spots compared to the existing pricing mechanism. [FOOTNOTE:11][ENDFOOTNOTE] ### _Charging Station’s Decisions and cost_ Note that if the user $k$ accepts the menu $(l,d_{k})$. Then, the charging station needs to allocate resources among the EVs in order to minimize the total cost of fulfilling the contract. First, we introduce some notations which we use throughout. Definition 1.: _The charging station has to incur the cost $v_{l,d_{k}}$ for fulfilling the contracts of existing customers and the contract $(l,d_{k})$ of the new user $k$, where $v_{l,d_{k}}$ is the value of the following linear optimization problem:_ \[\mathcal{P}_{l,d_{k}}: \text{min }\sum_{t=t_{k}}^{T-1}c_{t}q_{t}\] (7) \[\text{subject to }(\ref{eq:setofaccepted}),(\ref{eq:deadline}),( \ref{eq:rate_constraint}),(\ref{eq:energy}),(\ref{eq:batterylevel})\] \[\text{var: }r_{k,t},q_{k,t},q_{t},r_{t},D_{t}\quad\geq 0\] Note that our model can also incorporate time varying, strictly increasing convex costs $C_{t}(\cdot)$¹². Since $\mathcal{P}_{l,d_{k}}$ is a linear optimization problem, it is easy to compute $v_{l,d_{k}}$. Also, note that if the above problem is infeasible for some $l$ and $t$, then we consider $v_{l,t}$ as $\infty$. We assume that the prediction $\bar{E}^{t}$ is perfect for all future times and known to the charging station¹³. [FOOTNOTE:12][ENDFOOTNOTE] [FOOTNOTE:13][ENDFOOTNOTE] Definition 2.: _Let $v_{-k}$ be the amount that the charging station has to incur to satisfy the requirements of the existing EVs if the new user does not opt for any of the price menus._ If user $k$ does not accept any price menu, then the charging station still needs to satisfy the demand of existing users i.e. the charging station must solve the problem $\mathcal{P}_{l,t}$ with $q_{k,t}=r_{k,t}=0$. $v_{-k}$ is the value of that optimization problem. Thus, from Definitions 1 and 2 we can visualize $v_{l,d_{k}}-v_{-k}$ as the additional cost or marginal cost to the charging station when the user $k$ accepts the price menu $p_{k,l,d_{k}}$. It is easy to discern that $v_{l,d_{k}}-v_{-k}$ _is non-negative for any $d_{k}$ and $l$._ ### _Profit of the charging station_ Now, we discuss the profit of the charging station based on its pricing strategies. Note that if all the spots are occupied then, the charging station can not accommodate a new user. Thus, we consider the scenario where a charging spot is available. #### Iii-D1 Pricing with Perfect Foresight First, we consider the scenario where the charging station has a perfect foresight of the utility of the user i.e. the charging station is clairvoyant and has a perfect knowledge about the user’s utility. Note that if the user $k$ selects the price menu $p_{k,l,t}$, then the charging station has to pay $v_{l,t}$ amount (Definition 1). Thus, the charging station has to pay additional amount $v_{l,t}-v_{-k}$ (Definition 2) when the user selects the price menu $p_{k,l,t}$. Thus, the profit of the charging station is \[\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}(p_{k,l,t}-v_{k,l,t}+v_{-k})A_{ k,l,t}(\mathbf{p}_{k})\] (8) Note that here the charging station selects a price $p_{k,l,t}$ to maximize its own profit. $A_{k,l,t}>0$ only if $p_{k,l,t}$ gives the highest payoff for the user $k$ as discussed in Section III-A. #### Iii-D2 Prediction Based Pricing In practice the charging station may not know the exact realization of the utility function of the users. Thus, it can only use predictions of the utility function in order to select the price menu. Here, we consider such a scenario where the charging station does not know the exact utilities of the user. We assume that the charging station knows the statistic of the user’s utility. Let $R_{k,l,t}$ be the event that the price menu $p_{k,l,t}$ is selected, hence, the profit of the charging station is \[\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}\mathbb{E}[(p_{k,l,t}-v_{l,t}+v _{-k})\mathbbm{1}_{R_{k,l,t}}]\] \[=\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}(p_{k,l,t}-v_{l,t}+v_{-k})\Pr( R_{k,l,t})\] (9) The indicator variable $R_{k,l,t}$ in (9) denotes the event that the contract $(l,t)$ is being chosen by the user $k$. The expectation is taken over the joint distribution of $U_{k,l,t}$. _The expected profit maximization problem for the charging station is to maximize the above objective over_$p_{k,l,t}$. We assume that the utilities are distributed according to some continuous distribution¹⁴. Hence, $\Pr(R_{l,t})$ is given by [FOOTNOTE:14][ENDFOOTNOTE] \[\Pr(R_{k,l,t})=\Pr(U_{k,l,t}-p_{k,l,t}\geq(\max_{i,j}\{U_{k,i,j}- p_{i,j}\})^{+})\] Thus, $\Pr(R_{k,l,t})$ not only depends on $p_{k,l,t}$, but also prices for other menus. ### _Objectives_ We consider that the charging station decides the price menus in order to fulfill one of the two objectives (or, both)–i) Social Welfare Maximization and ii) its profit maximization. #### Iii-E1 Social Welfare The social welfare is the sum of user surplus and the profit of the charging station. As discussed in Section III-A for a certain realized values $u_{k,l,t}$ if the user $k$ selects the price menu $p_{k,l,t}$, then its surplus is $u_{k,l,t}-p_{k,l,t}$, otherwise, it is $0$. As discussed in Section III-D the profit of the charging station is $p_{k,l,t}-v_{l,t}+v_{k}$ for a given price $p_{k,l,t}$ if the user selects the menu, otherwise it is $0$. Hence, the social welfare maximization problem is to select the price menu $p_{k,l,t}$ which will maximize the following \[\mathcal{P}_{perfect}: \text{maximize}\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}(u_{k,l,t}-v_{l, t}+v_{-k})A_{k,l,t}(\mathbf{p}_{k})\] \[\text{var }:p_{k,l,t}\geq 0.\] (10) Recall that in order to find $v_{l,t}$ we have to solve $\mathcal{P}_{l,t}$(cf. (7)) which is a constrained optimization problem. Since EV is expected to increase the social value such as providing a cleaner environment, and higher energy efficiency, hence, it is important for a regulator (e.g. FERC) whether there exists a pricing strategy which maximizes the social welfare of the system. If the charging station is operated by the regulator or some government agency, then the main objective is indeed maximizing the social welfare or user surplus is maintained. _Ex-ante and Ex-post Maximization_: When the charging station is unaware of the utilities of the users, then two options are considered–i) decides a price and hopes that it will maximize the social welfare for the realized values of utilities (_ex-post_ maximization), or ii) decides a price and hopes that it will maximize the social welfare in an expected sense (_ex-ante_ maximization). Thus, the _ex-ante_ maximization does not guarantee that the social welfare will be maximized for every realization of the random variables $U_{k,l,t}$. However, in the _ex-post_ maximization, the social welfare is maximized for each realization of the random variables. Thus, _ex-post_ maximization is a stronger concept of maximization (and thus, more desirable) and it is not necessary that there exist pricing strategies which maximize the ex-post social welfare. However, we show that in our setting there exist pricing strategies which maximize the ex-post social welfare. Note that _ex-post_ social welfare maximization is the same as (III-E1). #### Iii-E2 Profit Maximization Social welfare maximization does not guarantee that the charging station may get a positive profit. It is important for the wide-scale deployment of the charging stations, the charging station must have some profit. Further, if the charging station is operated by a private entity its objective is indeed to maximize the profit. When the charging station is clairvoyant, then the charging station wants to maximize the profit given in (8) by selecting $p_{k,l,t}$. On the other hand when the charging station does not know the user’s utility, then it wants to maximize the expected payoff given in (III-D2) by selecting $p_{k,l,t}$. #### Iii-E3 Separation Problem Note that in order to select optimal $p_{k,l,t}$, the charging station has to obtain $v_{l,t}$ and $v_{-k}$ (Definitions 1 & 2). However, $v_{l,t}$ and $v_{-k}$ do not depend on $p_{k,l,t}$. Hence, we can separate the problem–first the charging station finds $v_{l,t}$ and $v_{k}$, and then it will select $p_{k,l,t}$ to fulfill the objective. _We now focus on finding optimal_$p_{k,l,t}$. ## IV Results: Social Welfare Maximization First, we state the optimal values of the social welfare for any given realization of the user’s utilites. Next, we state a pricing strategy which attains the above optimal value. Note that if $u_{k,l,t}-v_{l,t}+v_{-k}<0$ for each $l$ and $t$, then the social welfare is maximized when the user $k$ does not charge or equivalently, the price $p_{k,l,t}$ is very high for each $l$ and $t$. In this case, the optimal value of social welfare is $0$. On the other hand if $u_{k,l,t}-v_{l,t}\geq-v_{-k}$ for some $l$ and $t$, then the social welfare is maximized when the user $k$ charges its car. If the user accepts the price menu $p_{k,l,t}$, then the social welfare is $u_{k,l,t}-v_{l,t}+v_{-k}$. Thus, the maximum social welfare in the above scenario is $\max_{l,t}(u_{k,l,t}-v_{l,t}+v_{-k})$. Hence- Theorem 1.: _The maximum value of social welfare is $\max\{\max_{l,t}(u_{k,l,t}-v_{l,t}+v_{-k}),0\}$._ Note that even though the maximum value of social welfare is unique (as in Theorem 1), the optimal pricing strategy is not unique. In the following, we give one possible pricing strategy that achieves the optimal social welfare. Theorem 2.: _The pricing strategy $p_{k,l,t}=v_{l,t}-v_{-k}$ maximizes the social welfare._ Note that in the pricing strategy, the charging station does not need to know the utility of the users. It optimizes the social welfare for each possible realization of the utility functions. Hence, we obtain Corollary 1.: _The pricing strategy $p_{k,l,t}=v_{l,t}-v_{-k}$ maximizes the ex-post social welfare._ Though the pricing strategy maximizes the social welfare, the above pricing strategy does not provide any positive profit to the charging station. Thus, the charging station may not prefer this pricing strategy as it will not have any incentive to provide the charging spots. Also note that the pricing strategy also maximizes the social welfare in the long run when the additional cost of fulfilling a contract (i.e. $v_{l,t}-v_{-k}$) does not depend on the existing users in the charging station. The condition that $v_{l,t}-v_{-k}$ is independent of the existing EVs in the charging station is satisfied if either all demand can be fulfilled using renewable energy or there is no renewable energy generation and the conventional energy is bought at a flat rate. Hence, in the _two above extreme cases, the myopic pricing strategy is also optimal in the long run_. ## V Profit Maximization of the charging station ### _Charging station with perfect foresight_ We now provide a price strategy which maximizes the profit of the charging station and also the social welfare when _the charging station is clairvoyant_. Recall that the profit of the charging station is given by (8). First, we introduce a notation. Definition 3.: _Let $(l^{},t^{})=$argmax${}_{l,t}\{u_{k,l,t}-v_{l,t}\}$._ Theorem 3.: _Set $p_{k,l,t}=v_{l,t}-v_{-k}+(u_{k,l^{},t^{}}-v_{l^{},t^{}}+v_{-k})^{+}$ where $(l^{},t^{})$ is defined in Definition 3. Such a pricing strategy maximizes the profit as well as the social welfare._ The above pricing strategy is an example of _value-based_ pricing strategy where prices are set depending on the valuation or the utility of the users [20]. The user’s utility dependent pricing strategy is also proposed in smart grids in some recent papers [7, 21]. In contrast, the price strategy stated in Theorem 2 is an example of _cost-based_ pricing strategy where the prices only depend on the costs. If the utilities of the users are same, the pricing strategy becomes similar to a time-dependent pricing scheme, which is prevalent in practice. In the value-based pricing strategy, the user surplus decreases, in fact it is¹⁵$0$ in our case. Thus all the user surplus is transferred as the profit of the charging station. Hence, when the charging station is clairvoyant, then the pricing strategy which maximizes the profit of the charging station and it does not entail any positive _user surplus_. [FOOTNOTE:15][ENDFOOTNOTE] _Note that there can be other pricing strategies which simultaneously maximize the social welfare and the profit._ For example, if $p_{k,l,t}$ is $\infty$ for all $(l,t)\neq(l^{},t^{})$ and $p_{k,l^{},t^{}}=v_{l^{},t^{}}-v_{-k}+(u_{k,l^{},t^{}}-v_{l^{},t^{}}+v_ {-k})^{+}$, then it also maximizes the profit of the charging station. Thus, in this scenario, it can give only one possible contract to the EVs. Though the joint profit maximizing and social welfare pricing strategy may not be unique, the profit of the charging station is the unique and is given by \[(u_{k,l^{},t^{}}-v_{l^{},t^{}}+v_{-k})^{+}\] (11) ### _Prediction based pricing_ #### V-B1 Maximum Profit under ex-post social welfare maximization Note from Theorem 3 that the profit maximization pricing strategy which maximizes the social welfare requires that the charging station has the complete information of the utilities of the users. Hence, such a pricing strategy can not be implemented when the charging station does not know the exact utilities of the users. Note from (III-D2) that the profit maximization is a difficult problem as the user will select one menu inherently depends on the prices selected for other menus. For example, if the price selected for a particular contract is high, the user will be reluctant to take that compared to a lower price one. The profit is a discontinuous function of the prices and thus, the problem may not be convex even when the marginal distribution of the utilities are concave. We have already seen (Corollary 1) that a pricing strategy which can maximize the ex-post social welfare, however, it does not give any positive profit. We now show that there exists a pricing strategy which may provide better profit to the charging station while maximizing the ex-post social welfare. First, we introduce a notation which we use throughout. Definition 4.: _Let $L_{k,l,t}$ be the lowest end-point of the marginal distribution of the utility $U_{k,l,t}$._ Theorem 4.: _Consider the pricing strategy:_ \[p_{k,l,t}=v_{l,t}-v_{-k}+(\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\} )^{+}.\] (12) _The pricing strategy maximizes the ex-post social welfare._ _The profit is $(\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\})^{+}$._ The pricing strategy maximizes the ex-post social welfare similar to Corollary 1. This is also the _maximum possible profit that the charging station can have under the condition that it maximizes the ex-post social welfare with probability $1$_. However, it may not maximize the expected profit of the charging station or in other words, the pricing strategy which maximizes the expected profit needs not maximize the ex-post social welfare. Hence, unlike in the scenario where the charging station is clairvoyant (Theorem 3) _there may not exist a profit maximization strategy which is also a social welfare maximizer when the charging station is unaware of the utilities_. Note that the user surplus is _not_$0$, hence, uncertainty regarding the user’s utility functions is required for a positive consumer’s surplus. Also, note the similarity with Theorem 3. If the user knows the utility, then $L_{k,l,t}=u_{k,l,t}$ as there is no uncertainty and we get back the pricing strategy stated in Theorem 3. Note that if $\max_{l,t}(L_{k,l,t}-v_{l,t}+v_{-k})>0$, then such a pricing strategy gives a positive profit to the charging station. If the charging station has large storage or large renewable energy harvesting devices, then, the cost $v_{l,t}-v_{-k}$ will be lower and thus, the charging station can get a higher profit. It also increases the user surplus, as the price set by the charging station decreases. Thus, the impact of higher degrees of renewable energy integration for the charging station increases both the profit of the charging station and the user surplus. The above illustrates the importance of the storage and harvesting energy devices in the charging station. The regulator (e.g. FERC) can also provide incentives to the charging station to set up those devices as the pricing strategy increases profit to the charging station as well as the ex-post social welfare. In the extreme, when $v_{l,t}=0$ for all $l$ and $t$, then the profit of the charging station becomes maximum. However, further decreasing $v_{l,t}$ will not have any effect on the profit of the charging station as well as the user surplus, thus, it also shows the investment that the charging station needs to make for storage and renewable energy harvesting devices. Also note that the users which have higher utilities i.e. $L_{k,l,t}$ is higher, it will give more profits to the charging station. The charging station needs to know the lowest end-points of the support set of the utilities unlike in Corollary 1. However, the charging station does not need to know the exact distribution functions of the utilities. The lowest end-point can be easily obtained from the historical data. For example, $L_{k,l,t}$ may be computed by the lowest possible price that the user accepts for the energy level $l$ and the deadline $t$. #### V-B2 Guaranteed positive profit to the Charging station In Theorem 4 the charging station only has a positive profit if $\max_{l,t}\{L_{k,l,t}-v_{l,t}+v_{k}\}>0$. In the case, the above condition is not satisfied, then the charging station’s profit will be $0$. Naturally, the question is whether there exists a pricing strategy which gives a guaranteed positive profit to the charging station without decreasing the social welfare much. In the following we provide such a pricing strategy. First note that by the continuity of the joint distribution function we have the following Lemma 1.: _Let for each $\epsilon>0$, there exists a $\delta>0$ such that_ \[\Pr(\max_{l,t}\{U_{k,l,t}-v_{l,t}+v_{-k}\}\geq 0)\] \[\leq\epsilon+\Pr(\max_{l,t}\{U_{k,l,t}-v_{l,t}+v_{-k}-\delta\} \geq 0)\] (13) Theorem 5.: _Fix an $\epsilon>0$. Now, consider the pricing strategy_ \[p_{k,l,t}=v_{l,t}-v_{-k}+\delta(\epsilon)\] (14) _where $\delta(\epsilon)$ is the $\delta$ which satisfies the Lemma 1._ _Then such a pricing strategy maximizes the ex-post social welfare with probability $1-\epsilon$._ _Outline of proof_: First, note that adding a constant does not change the optimal solution. Hence, if $(l^{},t^{})=\text{argmax}_{l,t}(u_{k,l,t}-v_{l,t}+v_{-k})$, then $(l,^{},t^{})$ is also optimal for price strategy in (14). The rest of the proof follows from Lemma 1. Lemma 1 entails that there exists some $\delta>0$ such that it will ensure that the price strategy is off from the social welfare maximizer pricing strategy by at most $1-\epsilon$ in probability. ∎ Note that the pricing strategy stated in (14) gives a positive profit of $\delta(\epsilon)$ amount irrespective of the menu selected by the user. Note that _the assumption of a continuous distribution is key_. If the distributions are discrete, then $\delta(\epsilon)$ may be $0$. Hence, the charging station may get zero profit. The expected profit of the charging station for the above pricing strategy can be readily obtained– Theorem 6.: _Suppose that $p_{k,l,t}=v_{l,t}-v_{-k}+\beta$, then the expected profit of the charging station is $\beta\max_{l,t}\{\Pr(U_{k,l,t}\geq v_{l,t}-v_{-k}+\beta)\}$._ _Outline of the Proof_: Note that if a user selects any of the contracts, then the charging station’s profit is $\beta$. Hence, the charging station’s expected profit is $\beta$ times the probability that at least one of the contracts will be accepted.∎ The regulator such as FERC can select $\beta$ judiciously to trade off between the profit of the charging station and the user surplus. We empirically study the effect of $\beta$ in Section VI. Now, we provide an example where the above pricing strategy can also be a profit maximizing for a suitable choice of $\beta$. First, we introduce a notation Definition 5.: _Let $\zeta=\max\{\gamma\|\gamma=\text{argmax}_{\beta\geq 0}\beta\{\max_{i,j}\Pr(U_{k ,i,j}>\beta+v_{i,j}-v_{-k})\}\}$._ Note that since $U_{k,l,t}$ is bounded and the probability distribution is continuous, thus, $\zeta$ exists. Note from Theorem 6 that $\zeta$ corresponds to the $\beta$ for which the charging station can get the maximum possible profit when the prices are of the form $p_{k,l,t}=v_{l,t}-v_{-k}+\beta$. Now consider a class of widely seen utility functions Assumption 1.: _Suppose that the utility function $U_{k,l,t}=(Y_{k,l,t}+X_{k})^{+}$ for all $l$ & $t$; $Y_{k,l,t}$ is a constant and known to the charging station, however, $X_{k}$ is a random variable and whose realized value is not known to the charging station._ In the above class of utility function, the uncertainty is only regarding the realized value of the random variable $X_{k}$. Note that $X_{k}$ is independent of $l$ and $t$, hence,$X_{k}$ is considered to be an additive white noise. It is important to note that we do not put any assumption whether $X_{k}$ _should be drawn from a continuous or discrete distribution._ However, if the distribution is discrete, we need the condition that $\zeta$ must exist. Theorem 7.: _Consider the pricing strategy $p_{k,l,t}=v_{l,t}-v_{-k}+\zeta$; where $\zeta$ is defined in Definition 5,_ _The pricing strategy maximizes the expected profit of the charging station (given in (III-D2)) when the utility functions are of the form given in Assumption 1._ _Remark_: The above result is surprising. It shows that a simple pricing mechanism such as fixed profit can maximize the expected payoff for a large class of utility functions. However, if the utilities do not satisfy Assumption 1 then, the above pricing strategy may not be optimal. ### _The pricing algorithm_ 1. User $k$ comes at time $t_{k}$. 2. The charging station solves the linear programming problem $\mathcal{P}_{l,t}$ (eq. (7)) and finds the additional cost $v_{l,t}-v_{-k}$ for fulfilling the contract $(l,t)$ for user $k$ for each $l$ and $t$. 3. The charging station selects the price $p_{k,l,t}=v_{l,t}-v_{-k}+\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}^{+}+\beta.$ where $\beta\geq 0$. 4. The user selects the contract which maximizes its payoff (eq.(6)). Note that when $\beta=0$ gives the worst possible payoff to the charging station as discussed before. The charging station needs to solve the linear programming problem $\mathcal{P}_{l,t}$. The linear programing problem can be efficiently solved by many solvers such as MOSEK, CPLEX, Simplex, CVX, and Linprog tool of MATLAB. ## VI Simulation Results We numerically study and compare various pricing strategies presented in this paper. We evaluate the profit of the charging station and the user’s surplus achieved in those pricing strategies. We also show that our mechanism requires less charging spots compared to our nearest pricing model. ### _Simulation Setup_ Similar to [22], the user’s utility for energy $x$ is taken to be of the form $\min\{-ax^{2}+bx,\dfrac{b^{2}}{4a}\}$. Thus, the user’s utility is a strictly increasing and concave function in the amount energy consumed $x$. The quadratic utility functions for EV charging have also been considered in [23]. Note that the user’s desired level of charging is $b/(2a)$. _We assume that $b/(2a)$ is a random variable_. [24] shows that in a commercial charging station, the average amount of energy consumed per EV is $6.9$kWh with standard deviation $4.9$kWh. We thus consider that $b/(2a)$ is a truncated Gaussian random variable with mean $6.9$kWh and standard deviation $4.9$kWh in the interval $[2,20]$. We assume $a$ is a uniform random variable in the interval $[1/20,1/8]$. From [24], the deadline or the time spent by an electric vehicle in a commercial charging is distributed with an exponential distribution with mean $2.5$ hours. Thus, we also consider the preferred deadline ($T_{pref}$) of the user to be an exponentially distributed random variable with mean $2.5$. The user strictly prefers a lower deadline. Hence, we assume that the utility is a convex decreasing function of the deadline [12]. The utility of the user after the preferred deadline is considered to be $0$. Hence, the user’s utility is \[U_{k,l,t}=\min\{-al^{2}+lb,b^{2}/4a\}\times\] \[(\exp(T_{pref}-t-t_{k})-1)^{+}/(\exp(T_{pref}-t_{k})-1)\] (15) The arrival process of electric vehicles is considered to be a Poisson arrival process. However, the arrival rates vary over time. For example, during the peak-hours (8 am to 5pm) the arrival rate is higher compared to the off-peak hours. We, thus, consider a non-homogeneous Poisson process with the arrival rate is $15$ ($5$, resp.) vehicles per hour during the peak period (off-peak period, resp.). We also assume that the maximum charging rate $R_{max}$ is $3.3$ Kw. We assume that the renewable energy is harvested according to a truncated Gaussian distribution with mean $2$ and variance $2$ per hour. The storage unit is assumed to be of capacity $20$kW-h. Initial battery level is assumed to be $0$ i.e. it is fully discharged. The prices for the conventional energy is assumed to be governed by Time-of-Use (ToU) time scale. Thus, the cost of buying the conventional energy varies over time. ### _Results_ We consider the scenario where the charging station is unaware of the exact utilities of the users, however, it knows the distribution function. We consider the pricing strategy that we have introduced in Section V-B– \[p_{k,l,t}=v_{l,t}-v_{-k}+\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}^ {+}+\beta.\] Recall from Definition 4 that $L_{k,l,t}$ is the lowest end-point of the utility $U_{k,l,t}.$ We study the impact of $\beta$. #### Vi-B1 Effect on Percentage of the users admitted Fig. 6 shows that as $\beta$ increases the number of admitted users decreases. However, the decrement is slow initially. As $\beta$ becomes larger than a threshold, the price selected to the users becomes very large, and thus, a fewer number of EVs are admitted. #### Vi-B2 Effect of $\beta$ on User’s Surplus and Profit of the charging station The total surpluses of the users decreases with increase in $\beta$ (Fig. 6) as the user pays larger price when $\beta$ increases. The user surplus is maximum at $\beta=0$. The decrement of total users’ surplus is not significant with $\beta$ for $\beta<1.6$. However, as $\beta>1.6$, it decreases rapidly. For $\beta<1.6$, the number of users served does not decrease much with $\beta$. Hence, the total users’ surpluses decrease slowly. As $\beta$ increases the profit increases initially (Fig. 6). However, as $\beta>3$, the number of users served decreases rapidly, hence, the profit also drops. At high values of $\beta$ both users’ surpluses and the profit decrease significantly. Low values of $\beta$ give high users’ surpluses, however, the profit is low. $\beta\in[0.8,1.6]$ is the best candidate for the balance between profit and users’ surpluses. <figure><img src="content_image/1609.09037/x3.png"><figcaption>Fig. 3: Variation of the percentage of EVs admitted with β.</figcaption></figure> #### Vi-B3 Effect on the average deadline Our analysis shows that users spend more time in the charging station with the increase in $\beta$ (Fig. 6). As $\beta$ increases, the users which have preferences for lower deadlines have to pay more; since the cost of fulfilling lower deadline contracts is high. Hence, those users are reluctant to accept the contract. Thus, the accepted users spend more time in the charging station. Though the increment of the average time spent by an EV is not exponential with $\beta$. The average time spent by an EV is $2.5$ hours for $\beta=1.2$ which is in accordance with the average time spent by an EV [24]. <figure><img src="content_image/1609.09037/x7.png"><figcaption>Fig. 7: Variation of the maximum of the number of active users with β andcomparison with the differentiated pricing scheme proposed in [16, 17] (indotted line).</figcaption></figure> #### Vi-B4 Effect on the maximum number of active users Since the average time spent by users in the charging station increases with $\beta$ and the number of admitted users are almost the same for $\beta\leq 1.2$, hence the number of active users increases initially as $\beta$ increases (Fig. 10). Though the maximum never reaches beyond $22$ for any value of $\beta$. However, when $\beta>1.2$, the number of active users decreases with $\beta$. #### Vi-B5 Advantages of our proposed mechanism Fig. 10 shows that our pricing algorithm requires less charging spots compared to the differentiated pricing mechanism [16, 17] closest to our proposed approach. Similar to [16, 17] the users select the amount of energy to be consumed for each time period based on the price set by the charging station. We assume that the user will not charge beyond the preferred deadline and before the arrival time. In [16, 17] the EVs tend to spend more time as it reduces the cost¹⁶ and thus, the maximum of the number of EVs present at any time is also higher (Fig. 10) compared to our proposed mechanism.¹⁷ In our proposed mechanism, the charging station controls the time spent by an EV through pricing and results into lower charging spots. [FOOTNOTE:16][ENDFOOTNOTE] [FOOTNOTE:17][ENDFOOTNOTE] #### Vi-B6 Effect on the average energy As $\beta$ increases users only with higher utilities should accept the contracts. Thus, the average charging amount for each EV should increase with $\beta$. However, Fig. 10 shows that for $\beta\leq 0.8$, the average energy consumed by each EV decreases with the increase in $\beta$. The apparent anomaly is due to the fact that the users with higher demand but with smaller deadline preferences, may have to pay more because of the increase in the price to fulfill the contract as $\beta$ increases. Hence, such users will not accept the offers which result into initial decrement of the average energy consumption with the increase in $\beta$. However, as $\beta$ becomes large, only the users with higher demand accept the offers, hence, the average energy consumption increases. However, the increment is only linear. In fact for $\beta=2$, the average energy consumption per EV is around $6.9$ kW-h. #### Vi-B7 Effect on the Cost of the EV charging station The cost of the EV charging station decreases with the increase in $\beta$ (Fig. 10). Since the time spent by users increases and thus, the demand of the users can be met through renewable energies. The charging station buys a lower amount of conventional energies which results in lower cost for the charging station. When $\beta\leq 1.6$, the number of admitted users decreases _sub-linearly_, still the cost decreases _linearly_. Hence, the FERC will prefer this setting as it decreases the cost without decreasing the admitted users much. #### Vi-B8 Effect on the price selected by the charging station The price is higher during the peak period when the arrival rates is higher and the time-of-use price is high (Fig. 10). Hence, the pricing mechanism is consistent with the FERC’s objective of selecting higher prices during the peak time to flatten the demand curve. A new price is selected when an EV arrives. As $\beta$ decreases the admitted users is higher, hence the price variation is also higher as $\beta$ decreases. Also note that when the number of active users is large, serving additional user can be significant and thus, the price is also high. <figure><img src="content_image/1609.09037/x11.png"><figcaption>Fig. 11: Left-hand figure shows the amount of energy drawn from the battery ofthe charging station at various times for different values of β. The right-hand figure shows the amount of energy drawn from the grid at various timesfor different values of β.</figcaption></figure> #### Vi-B9 Impact on the energy drawn from the grid and the storage of the charging station Fig. 11 shows that as $\beta$ increases the energy bought from the grid decreases. This is because the number of accepted users decreases with $\beta$. The energy used from the battery also decreases as $\beta$ increases. Note that by selecting a $\beta$ the charging station can also limit the peak energy consumption from the grid. Fig. 11 also shows that when no menu based pricing is applied i.e., the EVs are charged as soon as they arrive, then, the peak energy consumption from the grid is very high. Even $\beta=0$ lowers the energy consumption from the grid significantly. The energy used from the battery of the charging station is also low when there is no menu-based pricing. _The above shows the usefulness of the menu-based pricing in reducing the peak-energy consumption and efficient use of the renewable energy._ <figure><img src="content_image/1609.09037/x12.png"><figcaption>Fig. 12: In the left-hand figure, we consider a∼U[1/40,1/16]. In the right-hand figure, we consider a∼U[1/10,1/4]. We show the consumer surplus andprofit of the charging station for β=0,1.2,2.</figcaption></figure> #### Vi-B10 Impact of $a$ Fig. 12 shows that as $a$ increases, the profit and the user’s surpluses both decrease. Note that as $a$ increases, the utility decreases, and the preferred energy $\dfrac{b}{2a}$ also decreases, hence, the profit and the user’s surplus both decrease. ## VII Conclusions and Future Works We propose an online menu-based pricing mechanism for EV charging. Specifically, we consider that the charging station will offer a price to each arriving user for a plethora of options; the user selects either one of them or rejects all. We show that there exists a _prior-free_ pricing strategy which maximizes the ex-post social welfare. We characterize the maximum possible profit that the charging station can get while maximizing the _ex-post_ social welfare. The charging station only needs to know the lower end-points of the utilities to implement the pricing strategy. The profit increases if the renewable energy penetration increases or the storage capacity of the charging station increases. However, the increment is bounded. The charging station can not simultaneously maximize the profit and the ex-post social welfare unless it is _clairvoyant_. We propose a fixed profit pricing scheme which provides a fixed profit to the charging station. The above can also maximize the expected profit of the charging station under some assumptions which frequently arise in practice. Numerical evaluation suggests that the menu-based pricing scheme can reduce the peak-demand and utilize the limited number of charging spots more efficiently compared to the baseline approaches. Following this work, we have considered the case where the EVs can inject energies by discharging via a Vehicle-to-Grid (V2G) service which can enhance the profits of the charging station [25]. We considered that the EV charging station is myopic which does not consider the future arrival process while selecting an optimal price for an incoming EV. In future we consider the case where the charging station knows the statistics of the future arrival process of the EVs and selects price accordingly. We also considered that the charging station has the only type of charger (either fast or slow), the characterization of prices when the charging station selects prices for different chargers is also left for the future. Considering stochastic pattern of energy harvesting is an important next step. Finally, the consideration of the multiple charging stations which set prices in a competitive manner also constitutes a future research direction. ## References * [1] A. Ghosh and V. Aggarwal, “Control of charging of electric vehicles through menu-based pricing under uncertainty,” in _2017 IEEE International Conference on Communications (ICC)_, May 2017. * [2] ——, “Control of electric vehicles through menu based pricing,” _CoRR_, vol. abs/1609.09037, 2016. * [3] N. Y. Soltani, S. J. Kim, and G. B. Giannakis, “Real-time load elasticity tracking and pricing for electric vehicle charging,” _IEEE Transactions on Smart Grid_, vol. 6, no. 3, pp. 1303–1313, May 2015. * [4] Z. Liu, Q. Wu, S. Oren, S. Huang, R. Li, and L. Cheng, “Distribution locational marginal pricing for optimal electric vehicle charging through chance constrained mixed-integer programming,” _IEEE Transactions on Smart Grid_, vol. PP, no. 99, pp. 1–1, 2016. * [5] M. Alizadeh, H. T. Wai, M. Chowdhury, A. Goldsmith, A. Scaglione, and T. Javidi, “Optimal pricing to manage electric vehicles in coupled power and transportation networks,” _IEEE Transactions on Control of Network Systems_, vol. PP, no. 99, pp. 1–1, 2016. * [6] S. G. Yoon, Y. J. Choi, J. K. Park, and S. Bahk, “Stackelberg-game-based demand response for at-home electric vehicle charging,” _IEEE Transactions on Vehicular Technology_, vol. 65, no. 6, pp. 4172–4184, June 2016. * [7] S. Bhattacharya, K. Kar, J. H. Chow, and A. Gupta, “Extended second price auctions with elastic supply for pev charging in the smart grid,” _IEEE Transactions on Smart Grid_, vol. 7, no. 4, pp. 2082–2093, July 2016. * [8] S. Zou, Z. Ma, and X. Liu, “Distributed efficient charging coordinations for electric vehicles under progressive second price auction mechanism,” in _52nd IEEE Conference on Decision and Control_, Dec 2013, pp. 550–555. * [9] C. Joe-Wong, S. Sen, S. Ha, and M. Chiang, “Optimized day-ahead pricing for smart grids with device-specific scheduling flexibility,” _IEEE Journal on Selected Areas in Communications_, vol. 30, no. 6, pp. 1075–1085, July 2012. * [10] N. Li, L. Chen, and S. H. Low, “Optimal demand response based on utility maximization in power networks,” in _2011 IEEE Power and Energy Society General Meeting_, July 2011, pp. 1–8. * [11] E. H. Gerding, V. Robu, S. Stein, D. C. Parkes, A. Rogers, and N. R. Jennings, “Online mechanism design for electric vehicle charging,” in _The 10th International Conference on Autonomous Agents and Multiagent Systems - Volume 2_, ser. AAMAS ’11. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems, 2011, pp. 811–818. [Online]. Available: http://dl.acm.org/citation.cfm?id=2031678.2031733 * [12] S. Chen, Y. Ji, and L. Tong, “Large scale charging of electric vehicles,” in _2012 IEEE Power and Energy Society General Meeting_, July 2012, pp. 1–9. * [13] Q. Huang, Q. S. Jia, Z. Qiu, X. Guan, and G. Deconinck, “Matching ev charging load with uncertain wind: A simulation-based policy improvement approach,” _IEEE Transactions on Smart Grid_, vol. 6, no. 3, pp. 1425–1433, May 2015. * [14] Y. Xu and F. Pan, “Scheduling for charging plug-in hybrid electric vehicles,” in _2012 IEEE 51st IEEE Conference on Decision and Control (CDC)_, Dec 2012, pp. 2495–2501. * [15] Z. Yu, Y. Xu, and L. Tong, “Large scale charging of electric vehicles: A multi-armed bandit approach,” in _53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)_, 2015, pp. 389–395. * [16] E. Bitar and S. Low, “Deadline differentiated pricing of deferrable electric power service,” in _2012 IEEE 51st IEEE Conference on Decision and Control (CDC)_, Dec 2012, pp. 4991–4997. * [17] E. Bitar and Y. Xu, “Deadline differentiated pricing of deferrable electric loads,” _IEEE Transactions on Smart Grid_, vol. 8, no. 1, pp. 13–25, Jan 2017. * [18] A. Nayyar, M. Negrete-Pincetic, K. Poolla, and P. Varaiya, “Duration-differentiated energy services with a continuum of loads,” _IEEE Transactions on Control of Network Systems_, vol. 3, no. 2, pp. 182–191, June 2016. * [19] F. Salah and C. M. Flath, “Deadline differentiated pricing in practice: marketing ev charging in car parks,” _Computer Science - Research and Development_, vol. 31, no. 1, pp. 33–40, 2016. * [20] A. Hinterhuber, “Towards value-based pricing—an integrative framework for decision making,” _Industrial Marketing Management_, vol. 33, no. 8, pp. 765 – 778, 2004. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0019850103001524 * [21] M. H. Yaghmaee, M. S. Kouhi, and A. L. Garcia, “Personalized pricing: A new approach for dynamic pricing in the smart grid,” in _2016 IEEE Smart Energy Grid Engineering (SEGE)_, Aug 2016, pp. 46–51. * [22] W. Shuai, P. Maillé, and A. Pelov, “Charging electric vehicles in the smart city: A survey of economy-driven approaches,” _IEEE Transactions on Intelligent Transportation Systems_, vol. 17, no. 8, pp. 2089–2106, Aug 2016. * [23] L. Gan, U. Topcu, and S. H. Low, “Optimal decentralized protocol for electric vehicle charging,” _IEEE Transactions on Power Systems_, vol. 28, no. 2, pp. 940–951, May 2013. * [24] “Evaluating electric vehicle charging impacts and customer charging behaviors- experiences from six smart grid investment grant projects,” https://www.smartgrid.gov/files/B3_revised_master-12-17-2014_report.pdf, December, 2014. * [25] A. Ghosh and V. Aggarwal, “Menu-Based Pricing for Charging of Electric Vehicles with Vehicle-to-Grid Service,” _ArXiv e-prints_, vol. 1612.00106, Nov. 2016. \begin{tabular}{c c} & Arnob Ghosh received the B.E. degree in Electronics and Telecommunications from Jadavpur University, Kolkata, India in 2011; the PhD and M.S. degree in Electrical Engineering from the University of Pennsylvania in 2016 and 2013 respectively. He is currently a Postdoctoral Research Associate in the Industrial Engg. Department of Purdue University. His research interests are smart grid, economic aspects of spectrum sharing in wireless network, and game theoretic approach for resource allocations in wireless network. He served as a reviewer in _IEEE Transactions on Signal Processing_, _ACM/IEEE Transactions on Networking_, and _IEEE Transactions on Communication_. \\ \end{tabular} \begin{tabular}{c c} & Vaneet Aggarwal (S’08 - M’11 - SM’15) received the B.Tech. degree in 2005 from the Indian Institute of Technology, Kanpur, India, and the M.A. and Ph.D. degrees in 2007 and 2010, respectively from Princeton University, Princeton, NJ, USA, all in Electrical Engineering. He is currently an Assistant Professor at Purdue University, West Lafayette, IN. Prior to this, he was a Senior Member of Technical Staff Research at AT\&T Labs-Research, NJ, and an Adjunct Assistant Professor at Columbia University, NY. His current research interests are in communications and networking, machine learning, and smart grids. Dr. Aggarwal was the recipient of Princeton University’s Porter Ogden Jacobus Honorific Fellowship in 2009. In addition, he received AT\&T Key Contributor award in 2013, AT\&T Vice President Excellence Award in 2012, and AT\&T Senior Vice President Excellence Award in 2014. He is serving on the editorial board of the _IEEE Transactions on Communications_ and the _IEEE Transactions on Green Communications and Networking._ \\ \end{tabular} ### _Proof of Theorem 2_ Note that the user $k$ only selects the menu which fetches the highest payoff. Hence, the payoff of user $k$ is \[(\max_{l,t}\{u_{k,l,t}-p_{k,l,t}\})^{+}\] (16) Since $p_{k,l,t}=v_{l,t}-v_{-k}$, thus, the usersurplus is \[(\max_{l,t}(u_{k,l,t}-v_{l,t}+v_{-k}))^{+}\] (17) The charging station’s profit is $p_{k,l,t}-v_{l,t}$ if the user selects the menu $(l,t)$ and $-v_{-k}$ if the user does not select any price option. Since $p_{k,l,t}=v_{l,t}-v_{-k}$, thus, the charging station’s profit is $-v_{-k}$ irrespective of the decision of the user. Hence, the social welfare is \[\max\{\max_{l,t}(u_{k,l,t}-v_{l,t}),v_{-k}\}\] (18) Hence, under the pricing strategy, the value of the social welfare is the same as in Theorem 1.∎ ### _Proof of Theorem 3_ First, we show that the pricing strategy maximizes the profit of the charging station. From (8), the highest possible profit of the charging station is \[\max\{u_{k,l^{},t^{}}-v_{l^{},t^{}},-v_{-k}\}\] (19) If the price is selected as stated in the theorem, then, if $u_{k,l^{},t^{}}-v_{l^{},t^{}}<-v_{-k}$, then, the profit of the charging station is $-v_{-k}$. On the other hand, if $u_{k,l^{},t^{}}-v_{l^{},t^{}}\geq-v_{-k}$, then the price $p_{k,l,t}$ to user $k$ is \[p_{k,l,t}=v_{l,t}+u_{k,l^{},t^{}}-v_{l^{},t^{}}\] (20) User will select a price $p_{k,l,t}$ if $u_{k,l,t}-p_{k,l,t}\geq u_{k,i,j}-p_{k,i,j}$ for all $i,j$ and $u_{k,l,t}\geq p_{k,l,t}$. Since $u_{k,l^{},t^{}}-v_{l^{},t^{}}\geq u_{k,l,t}-v_{l,t}$, thus, the user will only select the price menu $p_{k,l^{},t^{}}$. Note that at only $p_{k,l^{},t^{}}$ the user surplus is $0$, at other prices it is less than or equal to $0$. Thus, the profit of the charging station is $u_{k,l^{},t^{}}-v_{l^{},t^{}}$. Now, we show that such a pricing scheme also maximizes the social welfare. Note that in the above pricing strategy, the user’s surplus is always $0$. On the other hand, the profit that the charging station makes is $\max\{\max_{l,t}(u_{k,l,t}-v_{l,t}),-v_{-k}\}$. Hence, the pricing strategy obtains the optimal value of social welfare by Theorem 1.∎ ### _Proof of Theorem 4_ If $\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}\leq 0$, then the pricing strategy is the same as in Theorem 1 which we already proved to be ex-post social welfare maximizer. Hence, we consider the case when $\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}>0$. Now, we show that for every possible realization $u_{k,l,t}$ such a pricing strategy will maximize the social welfare. Note that $\max_{l,t}\{u_{k,l,t}-v_{l,t}+v_{-k}\}>0$ since $\max_{l,t}\{L_{k,l,t}-v_{l,t}+v_{-k}\}>0$ and $L_{k,l,t}$ is the lowest end-point of the distribution of $U_{k,l,t}$. Hence by Theorem 1 the maximum value of the ex-post social welfare is $\max_{l,t}\{u_{k,l,t}-v_{l,t}\}$. Now, we show that the pricing strategy defined in (12) will give rise the above optimal value of social welfare. Note that the user’s surplus is \[(\max_{i,j}\{u_{k,i,j}-p_{k,i,j}\})^{+}\] (21) First, we show that $\max_{i,j}\{u_{k,i,j}-p_{k,i,j}\}\geq 0$ if $\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}>0$ and $p_{k,i,j}$ is given by (12). Suppose not, i.e. $\max_{i,j}\{u_{k,i,j}-p_{k,i,j}\}<0$. Let $(l^{},t^{})=\text{argmax}_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}$. Then, $p_{k,l^{},t^{}}=L_{k,l^{},t^{}}$. Since $u_{k,l^{},t^{}}\geq L_{k,l^{},t^{}}$, hence, $u_{k,l^{},t^{}}-p_{k,l^{},t^{}}>0$ which leads to a contradiction. Thus, $\max_{i,j}\{u_{k,i,j}-p_{k,i,j}\}>0$. Thus, the user surplus is $\max_{i,j}\{u_{k,i,j}-p_{k,i,j}\}$. Since $p_{k,l,t}=v_{l,t}-v_{-k}+r$ where $r=\max_{i,j}\{L_{k,i,j}-v_{i,j}+v_{-k}\}$ is constant and independent of index $l$ and $t$. Hence, the user will select the price menu $p_{k,l^{},t^{}}$ such that $(l^{},t^{})=\text{argmax}_{l,t}\{u_{k,l,t}-v_{l,t}\}$ . The profit of the charging station is $p_{k,l^{},t^{}}-v_{l^{},t^{}}$. Thus, the ex-post social welfare is \[u_{k,l^{},t^{}}-p_{k,l^{},t^{}}+p_{k,l^{},t^{}}-v_{l^{},t ^{}}\] \[=u_{k,l^{},t^{}}-v_{l^{},t^{}}\] \[=\max_{l,t}\{u_{k,l,t}-v_{l,t}\}\] (22) Thus, for each realized values of the utilities $u_{k,l,t}$, the pricing strategy (12) provides the maximum social welfare. Hence, the result follows.∎ ### _Proof of Theorem 5_ Suppose that $u_{k,l,t}$ be the realized values of $U_{k,l,t}$. Note that if $\max_{l,t}\{u_{k,l,t}-p_{k,l,t}\}\geq 0$ where $p_{k,l,t}$ is given by (14) then the user selects the price menu $p_{k,l^{},t^{}}$ where \[(l^{},t^{})=\text{argmax }_{l,t}\{u_{k,l,t}-v_{l,t}\}.\] (23) Hence, the social welfare is $\max_{l,t}\{u_{k,l,t}-v_{l,t}\}$ which is the same as the social welfare by Theorem 1. On the other hand if $\max_{l,t}\{u_{k,l,t}-p_{k,l,t}\}<00$, then the social welfare is $-v_{-k}$. Hence, by Theorem 1, the only cases where the social welfare is not maximized when $\max_{l,t}\{u_{k,l,t}-p_{k,l,t}\}<0$, however, $\max_{l,t}\{u_{k,l,t}-v_{l,t}+v_{-k}\}\geq 0$. However, by the definition of $\delta(\epsilon)$ such a scenario can only occur with probability $\epsilon$. Hence, the result follows. ∎ ### _Proof of Theorem 7_ Suppose the statement is false. Without loss of generality, assume that $p_{k,l,t}=v_{l,t}-v_{-k}+\alpha_{l,t}$ where $\alpha_{l,t}\neq\alpha$ for some $l$ and $t$ achieves a strictly higher expected payoff than the pricing strategy $p_{k,l,t}=v_{l,t}-v_{-k}+\alpha$. The expected profit of the charging station for pricing strategy $p_{k,l,t}=v_{l,t}-v_{-k}+\alpha_{l,t}$ is given by \[\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}(p_{k,l,t}-v_{l,t}+v_{-k})\Pr(R _{l,t})-v_{-k}=\sum_{l=1}^{L}\sum_{t=t_{k}+1}^{T}\alpha_{l,t}\Pr(R_{l,t})-v_{-k}\] (24) Now, we evaluate the expression $\Pr(R_{l,t})$. The user $k$ will select the menu $p_{k,l,t}$ with a positive probability if $Y_{k,l,t}+X_{k}\geq v_{l,t}-v_{-k}+\alpha_{l,t}$ and for every $(i,j)\neq(l,t)$, \[\alpha_{l,t}-\alpha_{i,j}\leq Y_{k,l,t}-v_{l,t}-Y_{k,i,j}+v_{i,j}\] (25) Since $Y_{k,l,t},Y_{k,i,j},v_{l,t},v_{i,j}$ are fixed, hence, the above inequality is either satisfied or not satisfied with probability $1$. More specifically, the user selects the menu $(l,t)$ if $Y_{k,l,t}+X_{k}\geq v_{l,t}-v_{-k}+\alpha_{l,t}$ and \[Y_{k,l,t}-v_{l,t}-\alpha_{l,t}\geq\max_{i,j}(Y_{k,i,j}-v_{i,j}- \alpha_{i,j})\] (26) Without loss of generality, assume that $\alpha_{l_{1},t_{1}}$ be the maximum value for which the above inequality is satisfied i.e. \[\alpha_{l_{1},t_{1}}=\max\{\alpha_{l,t}:Y_{k,l,t}-v_{l,t}-\alpha_ {l,t}\geq\max_{i,j}(Y_{k,i,j}-v_{i,j}-\alpha_{i,j})\}\] (27) The random variable $X_{k}$ only affects the probability whether $Y_{k,l,t}+X_{k}\geq v_{l,t}-v_{-k}+\alpha_{l_{1},t_{1}}$ or not. Hence, the charging station’s expected profit is upper bounded by \[\alpha_{l_{1},t_{1}}\Pr(X_{k}\geq v_{l_{1},t_{1}}-v_{-k}+\alpha_{ l_{1},t_{1}}-Y_{k,l_{1},t_{1}})\] (28) Note that by the definition of $\alpha$ (Definition 5), \[\alpha\max_{l,t}\Pr(Y_{k,l,t}+X_{k}\geq v_{l,t}-v_{-k}+\alpha)\] \[\geq\alpha_{l_{1},t_{1}}\Pr(X_{k}\geq v_{l_{1},t_{1}}-v_{-k}+ \alpha_{l_{1},t_{1}}-Y_{k,l_{1},t_{1}})\] (29) However by Theorem 6 the expected payoff of the charging station when it selects the price $v_{l,t}-v_{-k}+\alpha$ is given by the expression in the left hand of the expression in (-E). Hence, this leads to a contradiction. Thus, the result follows.∎
1608.00357	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 93886, "num_imgs": 0, "llama3_tokens_count": 35292 }	[]	# A generalization of the simulation theorem for semidirect products Sebastián Barbieri and Mathieu Sablik ###### Abstract We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed $\mathbb{Z}^{d}$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^{2}$. Let $H$ be a finitely generated group and $G=\mathbb{Z}^{2}\rtimes H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,f)$ where $Y$ is a Cantor set, there exists a $G$-subshift of finite type $(X,\sigma)$ such that the $H$-subaction of $(X,\sigma)$ is an extension of $(Y,f)$. In the case where $f$ is an expansive action of a recursively presented group $H$, a subshift conjugated to $(Y,f)$ can be obtained as the $H$-projective subdynamics of a $G$-sofic subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable. ## 1 Introduction A dynamical system is a tuple $(X,T)$ where $X$ is a set and $T:X\to X$ is a map which describes the evolution of points of $X$ in time. In the case where $T$ is bijective one can describe $T$ as a $\mathbb{Z}$-action by associating $(n,x)\to T^{n}(x)$. This can be generalized to a set of bijective maps $T_{1},\dots,T_{n}$ which satisfy some set of relations $R$ –for instance, the relation $T_{1}\circ T_{2}=T_{2}\circ T_{1}$ which indicates $T_{1}$ and $T_{2}$ commute–. These actions and their relations can be expressed by the group action $\mathcal{T}:G\times X\to X$ where $G\cong\langle T_{1},\dots,T_{n}\mid R\rangle$ and $\mathcal{T}(T_{i_{1}}\circ\dots\circ T_{i_{k}},x)=T_{i_{1}}\circ\dots\circ T_{ i_{k}}(x)$. More than often dynamical systems arising from group actions are difficult to study, and a fruitful technique is to look at their subactions, that is, the restriction of the group action to a particular subgroup. For instance, see the study of expansive subdynamics of $\mathbb{Z}^{d}$ actions [6, 10]. It is thus appealing to ask the following question: What systems can be obtained as subactions of a class of dynamical systems? An interesting class is the one of subshifts of finite type (SFT), that is, the sets of colorings of a group along with the shift action which are defined by a finite number of forbidden patterns. For the class of $\mathbb{Z}^{d}$-SFTs there is still no characterization of which dynamical systems can arise as their subactions, nevertheless, it has been proven by Hochman [12] that every $\mathbb{Z}^{d}$-action over a cantor set $T:\mathbb{Z}^{d}\times X\to X$ which is effectively closed – meaning that it can be described with a Turing machine– admits an almost trivial isometric extension which can be realized as the subaction of a $\mathbb{Z}^{d+2}$-SFT. This result has subsequently been improved for the expansive case independently in [3] and [9] showing that every effectively closed subshift can be obtained as the projective subdynamics of a sofic $\mathbb{Z}^{2}$-subshift. These kind of results yield powerful techniques to prove properties about the original systems. An example is the characterization of the set of entropies of $\mathbb{Z}^{2}$-SFTs [13] as the set of right recursively enumerable numbers. In this article we extend Hochman’s result to the case of group actions for groups which are of the form $G=\mathbb{Z}^{2}\rtimes_{\varphi}H$ for some finitely generated group $H$ and an homomorphism $\varphi:H\to\text{Aut}(\mathbb{Z}^{2})$. More specifically we prove the following result. Theorem 3.1.: _For every $H$-effectively closed dynamical system $(X,f)$ there exists a $(\mathbb{Z}^{2}\rtimes H)$-SFT whose $H$-subaction is an extension of $(X,f)$._ We remark the strong gap which occurs when passing from $\mathbb{Z}$-SFTs to the multidimensional case. For instance, $\mathbb{Z}$-SFTs contain periodic points, have regular languages and the possible set of entropies they can have is reduced to logarithms of Perron numbers [17]. In the other hand multidimensional SFTs can be strongly aperiodic [5, 20, 16, 15], can be composed uniquely of non-computable points [11, 19] and their entropies are not even computable [13]. Most of these differences can be put into evidence with simulation theorems by the fact that multidimensional SFTs can be projected onto effectively closed subshifts in one dimension. Our Theorem 3.1 allows analogously to extend properties of effectively closed subshifts in general groups $H$ and show that they also appear in SFTs when the group is replaced by $\mathbb{Z}^{2}\rtimes H$. This is a powerful tool to construct examples of groups with admit subshifts of finite type with some desired property which is easier to realize in an effective subshift. Readers who are not familiar with computability or the embedding of Turing machine computations in subshifts of finite type will be reassured by the fact that in the proof all of those aspects are hidden in black boxes. Namely, we use the result of [3, 9] that every effectively closed $\mathbb{Z}$-subshift is the projective subdynamics of a sofic $\mathbb{Z}^{2}$-subshift whose vertical shift action is trivial. We also make use of a theorem of Mozes [18] which states that subshifts arising from two-dimensional substitutions are sofic. In the case when $H$ is a recursively presented group, Theorem 3.1 can be presented in a purely symbolic dynamics fashion for expansive actions, namely we show: Theorem 4.2.: _Let $X$ be an effectively closed $H$-subshift. Then there exists a sofic $(\mathbb{Z}^{2}\rtimes H)$-subshift $Y$ such that its $H$-projective subdynamics $\pi_{H}(Y)$ is $X$._ It is known that every $\mathbb{Z}$-SFT contains a periodic configuration [17]. However, it was shown by Berger [5] that there are $\mathbb{Z}^{2}$-SFTs which are strongly aperiodic, that is, such that the shift acts freely on the set of configurations. This result has been proven several times with different techniques [20, 16, 15] giving a variety of constructions. However, it remains an open question which is the class of groups which admit strongly aperiodic SFTs. Amongst the class of groups that do admit strongly aperiodic SFTs are: $\mathbb{Z}^{d}$ for $d>1$, hyperbolic surface groups [8], Osin and Ivanov monster groups [14], and the direct product $G\times\mathbb{Z}$ for a particular class of groups $G$ which includes Thompson’s $T$ group and $\text{PSL}(\mathbb{Z},2)$ [14]. It is also known that no group with two or more ends can contain strongly aperiodic SFTs [7] and that recursively presented groups which admit strongly aperiodic SFTs must have decidable word problem [14]. As an application of Theorem 3.1 we present a new class of groups which admit strongly aperiodic SFTs, that is: Theorem 4.3.: _Every semidirect product $\mathbb{Z}^{2}\rtimes H$ where $H$ is finitely generated and has decidable word problem admits a non-empty strongly aperiodic SFT._ Amongst this new class of groups which admit strongly aperiodic SFTs, we remark the Heisenberg group which admits a presentation $\mathcal{H}\cong\mathbb{Z}^{2}\rtimes\mathbb{Z}$. ## 2 Preliminaries Consider a group $G$ and a compact topological space $(X,\mathcal{T})$. The tuple $(X,f)$ where $f:G\times X\to X$ is a left $G$ action by homeomorphisms is called a $G$-flow (or $G$-dynamical system). Let $(X,f)$, $(X^{\prime},f^{\prime})$ be two $G$-flows. We say $\phi:X\to X^{\prime}$ is a _morphism_ if it is continuous and $\phi\circ f_{g}=f^{\prime}_{g}\circ\phi$ for all $g\in G$. A surjective morphism $\phi:X\twoheadrightarrow X^{\prime}$ is a _factor_ and we say that $(X^{\prime},f^{\prime})$ is a _factor_ of $(X,f)$ and that $(X,f)$ is an _extension_ of $(X^{\prime},f^{\prime})$. When $\phi$ is a bijection and its inverse is continuous we say it is a _conjugacy_ and that $(X,f)$ is _conjugated_ to $(X^{\prime},f^{\prime})$. In what follows, we consider only cantor sets with the product topology and finitely generated groups. Without loss of generality, we consider actions over closed subsets of $\{0,1\}^{\mathbb{N}}$. Let $G$ be a group generated by a finite set $S$. A _$G$-effectively closed flow_ is a $G$-flow $(X,f)$ where: 1. $X\subset\{0,1\}^{\mathbb{N}}$ is a closed effective subset: $X=\{0,1\}^{\mathbb{N}}\setminus\bigcup_{i\in I}{[w_{i}]}$ where $\{w_{i}\}_{i\in I}\subset\{0,1\}^{}$ is a recursively enumerable language. That means that $X$ is the complement of a union of cylinders which can be enumerated by a Turing machine. 2. $f$ is an effectively closed action: there exists a Turing machine which on entry $s\in S$ and $w\in\{0,1\}^{}$ enumerates a sequence of words $(w_{j})_{j\in J}$ such that $f_{s}^{-1}([w])=\{0,1\}^{\mathbb{N}}\setminus\bigcup_{j\in J}[w_{j}]$. The idea behind the definition is the following: There is a Turing machine $T$ which given a word $g\in S^{}$ representing an element of $G$ and $n$ coordinates of $x\in X\subset\{0,1\}^{\mathbb{N}}$ returns an approximation of the preimage of $x$ by $f_{g}$. Let $\mathcal{A}$ be a finite alphabet and $G$ a finitely generated group. The set $\mathcal{A}^{G}=\{x:G\to\mathcal{A}\}$ equipped with the left group action $\sigma:G\times\mathcal{A}^{G}\to\mathcal{A}^{G}$ given by: $(\sigma_{h}(x))_{g}=x_{h^{-1}g}$ is the $G$_-full shift_. The elements $a\in\mathcal{A}$ and $x\in\mathcal{A}^{G}$ are called _symbols_ and _configurations_ respectively. We endow $\mathcal{A}^{G}$ with the product topology, therefore obtaining a compact metric space. The topology is generated by the metric ${d(x,y)=2^{-\inf\{\|g\|\;\mid\;g\in G:\;x_{g}\neq y_{g}\}}}$ where $\|g\|$ is the length of the smallest expression of $g$ as the product of some fixed set of generators. This topology is also generated by a clopen basis given by the _cylinders_$[a]_{g}=\{x\in\mathcal{A}^{G}\|x_{g}=a\in\mathcal{A}\}$. A _support_ is a finite subset $F\subset G$. Given a support $F$, a _pattern with support $F$_ is an element $P$ of $\mathcal{A}^{F}$, i.e. a finite configuration and we write $supp(P)=F$. We also denote the cylinder generated by $P$ centered in $g$ as $[P]_{g}=\bigcap_{h\in F}[P_{h}]_{gh}$. If $x\in[P]_{g}$ for some $g\in G$ we write $P\sqsubset x$. A subset $X$ of $\mathcal{A}^{G}$ is a _$G$-subshift_ if it is $\sigma$-invariant – $\sigma(X)\subset X$ – and closed for the cylinder topology. Equivalently, $X$ is a $G$-subshift if and only if there exists a set of forbidden patterns $\mathcal{F}$ that defines it. \[X=X_{\mathcal{F}}:={\mathcal{A}^{G}\setminus\bigcup_{P\in\mathcal{F},g\in G}[P ]_{g}}.\] That is, a $G$-subshift is a subset of $\mathcal{A}^{G}$ which can be written as the complement of a union of cylinders. If the context is clear enough, we will drop the $G$ and simply refer to a subshift. A subshift $X\subseteq\mathcal{A}^{G}$ is _of finite type_ – SFT for short – if there exists a finite set of forbidden patterns $\mathcal{F}$ such that $X=X_{\mathcal{F}}$. A subshift $X\subseteq\mathcal{A}^{G}$ is _sofic_ if there exists a subshift of finite type $Y\subset\mathcal{A}^{\prime G}$ and a factor $\phi:Y\twoheadrightarrow X$. A subshift is effectively closed if there exists a recursively enumerable coding of a set of forbidden patterns $\mathcal{F}$ such that $X=X_{\mathcal{F}}$. More details can be found in [2] or in Section 4. Any $G$-flow over a cantor set can be seen as a subshift over an infinite alphabet: Indeed, $(X,f)$ can be seen as $Y\subset(\{0,1\}^{\mathbb{N}})^{G}$ equipped with the shift action such that $x\in Y$ if and only if $\forall g\in G$$x_{g}=f_{g}(x_{1_{G}})$. In this setting, effectively closed $G$-flows correspond to effectively closed subshifts in this infinite alphabet. Let $H\leq G$ be a subgroup and $(X,f)$ a $G$-flow. The _$H$-subaction_ of $(X,f)$ is $(X,f^{H})$ where $f^{H}:H\times X\to X$ is the restriction of $f$ to $H$, that is $\forall h\in H,(f^{H})_{h}(x)=f_{h}(x)$. In the case of a subshift $X\subset\mathcal{A}^{G}$ there is also the different notion of projective subdynamics. The _$H$-projective subdynamics_ of $X$ is the set $\pi_{H}(X)=\{y\in\mathcal{A}^{H}\mid\exists x\in X,\forall h\in H,y_{h}=x_{h}\}$. ## 3 Simulation Theorem The purpose of this section is to prove our main result. Theorem 3.1.: _Let $H$ be finitely generated group and $G=\mathbb{Z}^{2}\rtimes H$. For every $H$-effectively closed flow $(X,f)$ there exists a $G$-SFT whose $H$-subaction is an extension of $(X,f)$._ We begin by introducing some general constructions. The general schema of the proof is the following: First we construct for each non-zero vector $v\in(\mathbb{Z}/3\mathbb{Z})^{2}$ a substitution $\texttt{s}_{v}$ which encodes countable copies of $\mathbb{Z}^{2}$ as lattices with the property that any automorphism $\varphi\in\text{Aut}(\mathbb{Z}^{2})$ sends each of the lattices of $\texttt{s}_{v}$ to those of $\texttt{s}_{\tilde{\varphi}(v)}$ where $\tilde{\varphi}\in\text{Aut}((\mathbb{Z}/3\mathbb{Z})^{2})$ is the automorphism obtained by projecting each component to $\mathbb{Z}/3\mathbb{Z}$. This structure allows us to pair lattices of $\mathbb{Z}^{2}$ when moving in $G$ by elements of $H$. Then we encode the elements of $X$ and the $H$-flow $f$ in an effective Toeplitz $\mathbb{Z}$-subshift. We do so in a way that the projections of the $n$-th order lattice to the line in the previous construction always matches with the symbol $x_{n}$. For technical reasons of matching all possible lattices, we do this coding in two different ways. Afterwards, we extend the Toeplitz subshift to a $\mathbb{Z}^{2}$-subshift by repeating rows (or columns). Using a known simulation theorem we obtain that this object is a sofic $\mathbb{Z}^{2}$-subshift from which we extract an SFT extension. Finally, we extend this construction to $G$ by adding local rules that ensure that if a $\mathbb{Z}^{2}$-coset codes the point $x\in X$ then the coset of $\mathbb{Z}^{2}$ given by the action of $h\in H$ codes $f_{h}(x)$. This set of rules is coded as a finite amount of forbidden patterns. Finally, we define the factor code, and show that it satisfies the required properties. ### A substitution which encodes an action of $\text{Aut}((\mathbb{Z}/p\mathbb{Z})^{2})$. Let $p\in\mathbb{N}$. We define a substitution over a two symbol alphabet which generates a sofic $\mathbb{Z}^{2}$-subshift encoding translations of $p^{m+1}\mathbb{Z}^{2}$ for $m\in\mathbb{N}$. In the proof of the simulation theorem we will only use the case where $p=3$, but we prefer to proceed here with more generality. Let $v\in(\mathbb{Z}/p\mathbb{Z})^{2}\setminus\{(0,0)\}$ and \(\mathcal{A}=\{\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}},\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\}\). The $\mathbb{Z}^{2}$-substitution $\texttt{s}_{v}:\mathcal{A}\to\mathcal{A}^{\{0,\dots,p-1\}^{2}}$ is defined by: \[\texttt{s}_{v}(\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}})_{z}=\begin{cases}\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\mbox{\ \ \ if }z=v\\ \leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter\hbox to 0. 0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\mbox{ \ \ \ \ in the contrary case. }\end{cases}\] \[\texttt{s}_{v}(\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}})_{z}=\begin{cases}\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\mbox{\ \ \ if }z\in\{(0,0),v\}\\ \leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter\hbox to 0. 0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\mbox{ \ \ \ \ in the contrary case. }\end{cases}\] As an example, if $p=3$ and $v=(1,1)$ we get the following: $\texttt{s}_{v}(\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss}{ }\endpgfpicture}})=$ $\texttt{s}_{v}(\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss}{ }\endpgfpicture}})=$ In this example we obtain that the patterns $\texttt{s}_{v}^{3}(\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}})$ and $\texttt{s}_{v}^{4}(\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}})$ are: [FIGURE:S3.SS1.fig1][ENDFIGURE] To a substitution $\texttt{s}_{v}$ we associate the subshift ${\texttt{Sub}}_{v}$ defined as the set of $\mathbb{Z}^{2}$-configurations such that every subpattern appears in some iteration of the substitution $\texttt{s}_{v}$. \[\texttt{Sub}_{v}=\{x\in\{\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}},\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\}^{\mathbb{Z}^{2}}\mid\forall P\sqsubset x,\exists n\in \mathbb{N}:P\sqsubset\texttt{s}_{v}^{n}(\leavevmode\hbox to11.63pt{\vbox to 11.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{} \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}})\}\] We remark the following properties of these objects: 1. $s_{v}$ is a primitive substitution, and thus ${\texttt{Sub}}_{v}$ is a minimal subshift. 2. \(\forall a\in\{\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}{}\pgfsys@moveto{ -3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt} \pgfsys@lineto{-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt} \pgfsys@lineto{3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.41433 1pt}{3.414331pt}\pgfsys@stroke\pgfsys@invoke{ } {}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}},\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{ }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\}\), $n\in\mathbb{N}$, $\texttt{s}_{v}^{n+1}(a)_{p^{n}v}=\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$. 3. By Mozes Theorem [18]$\texttt{Sub}_{v}$ is a $\mathbb{Z}^{2}$-sofic subshift. 4. $\texttt{s}_{v}$ has unique derivation. This implies by Mozes’s results that there is an almost 1-1 SFT extension for $\texttt{Sub}_{v}$. 5. Putting together the unique derivation and the second property we obtain the following: $\forall z\in\texttt{Sub}_{v}$ and $\forall n\in\mathbb{N}$ there exists a unique $(i_{n},j_{n})\in P_{n}:=[0,p^{n+1}-1]^{2}\cap\mathbb{Z}^{2}$ such that $\forall m\leq n$ then $(i_{n},j_{n})+p^{m}v+p^{m+1}\mathbb{Z}^{2}$ is composed completely of black squares. We denote each of these sets by $B_{m}(z)$ – The lattice of black squares of level $m$. These sets are all disjoint and cover every black square in $\texttt{Sub}_{v}$ with the possible exception of at most one. We denote this degenerated lattice by $B_{\infty}(z)$, and note that it can either be empty or contain a single position. 6. Let $z\in\texttt{Sub}_{v}$ and $\varphi\in\text{Aut}(\mathbb{Z}^{2})$. We can identify $\varphi$ as an invertible matrix $A_{\varphi}\in GL(\mathbb{Z},2)$ and construct $A_{\widetilde{\varphi}}\in\mathcal{M}(\mathbb{Z}/p\mathbb{Z},2)$ by reducing every entry of this matrix modulo $p$. As $\det(A_{\varphi})\in\{-1,1\}$ then $\det(A_{\widetilde{\varphi}})\in\{1,p-1\}$. Therefore $A_{\widetilde{\varphi}}\in GL(\mathbb{Z}/p\mathbb{Z},2)$ and it is identified as an automorphism $\widetilde{\varphi}\in\text{Aut}((\mathbb{Z}/p\mathbb{Z})^{2})$. With this in mind, we obtain the following relation: we have that for $m\leq n$ then $B_{m}(z)=(i_{n},j_{n})+p^{m}v+p^{m+1}\mathbb{Z}^{2}$. Therefore: \[\varphi(B_{m}(z)) =\varphi((i_{n},j_{n}))+p^{m}\varphi(v)+p^{m+1}\varphi(\mathbb{Z} ^{2})\] \[=\varphi((i_{n},j_{n}))+p^{m}A_{\varphi}(v)+p^{m+1}\mathbb{Z}^{2}\] \[=\varphi((i_{n},j_{n}))+p^{m}(A_{\widetilde{\varphi}}+p(\frac{A_{ \varphi}-A_{\widetilde{\varphi}}}{p})(v)+p^{m+1}\mathbb{Z}^{2}\] \[=\varphi((i_{n},j_{n}))+p^{m}A_{\widetilde{\varphi}}(v)+p^{m+1}(( \frac{A_{\varphi}-A_{\widetilde{\varphi}}}{p})(v)+\mathbb{Z}^{2})\] \[=\varphi((i_{n},j_{n}))+p^{m}\widetilde{\varphi}(v)+p^{m+1} \mathbb{Z}^{2}\] This means that for fixed $n$ all lattices of size $m\leq n$ are sent to lattices appearing in $\texttt{Sub}_{\widetilde{\varphi}(v)}$. Making $n$ go to infinity and reasoning by compactness we conclude that $\forall z\in\texttt{Sub}_{v}$ there exists $z^{\prime}\in\texttt{Sub}_{\widetilde{\varphi}(v)}$ such that the image of $(B_{m}(z))_{m\in\mathbb{N}}$ under $\varphi$ is $(B_{m}(z^{\prime}))_{m\in\mathbb{N}}$. We shall use these lattices to encode elements of $\{0,1\}^{\mathbb{N}}$ belonging to our $H$-flow $(X,f)$. In order to do this, we need to define a subshift which forces to match these lattices to actual values from $X$ and to code the action of $f$. ### Encoding configurations in Toeplitz sequences. Consider $p\geq 3,q\in\{1,\dots,p-1\}$ and the application $\Psi_{q}:\{0,1\}^{\mathbb{N}}\to\{0,1,\$\}^{\mathbb{Z}}$ given by: \[\Psi_{q}(x)_{j}=\begin{cases}x_{n}\mbox{\ \ \ if }j=qp^{n}\mod{p^{n+1}}\\ \$\mbox{ \ \ \ \ in the contrary case. }\end{cases}\] The idea behind this encoding is to match for each $m\in\mathbb{N}$ the projection of the lattice $B_{m}(x)$ to the symbol $x_{m}$. We need to do this for every possible choice of $q$ as the projections of the lattice associated to $v=(1,1)$ are different than the ones for $v=(2,2)$ for example. Every configuration $x\in\{0,1\}^{\mathbb{N}}$ is encoded in a Toeplitz sequence $\Psi_{q}(x)$. We begin this section by studying the structure of $\Psi_{q}(x)$. First notice that $\Psi_{q}(x)\|_{q+p\mathbb{Z}}\equiv x_{0}$ and $\forall q^{\prime}\in\{1,\dots p-1\}\setminus\{q\}$ we have that $\Psi_{q}(x)_{q^{\prime}+p\mathbb{Z}}\equiv\$$. Indeed, as $q^{\prime}+pk\neq 0\mod{p}$ thus $q^{\prime}+pk\neq p^{i}\mod{p^{i+1}}$. Also, if $i\geq 1$ and $\Psi_{q}(x)_{j}=x_{i}$ then $\Psi_{q}(x)_{j+q}=x_{0}$ as $j=p^{i}\mod{p^{i+1}}\implies j=0\mod p$. This means that every $x_{0}$ is a special coordinate in a string of $p-1$ symbols where every other symbol is $\$$ and every $x_{i}$ with $i\geq 1$ is necessarily followed by such string. As $p\geq 3$ the lattice of $x_{0}$ can be recognized as they are the only symbols which are preceded by $q-1$ symbols $\$$ and followed by $p-q-1$ symbols $\$$ and at least one of these two values is positive. For $x=(x_{i})_{i\in\mathbb{N}}\in\{0,1\}^{\mathbb{N}}$ let $\sigma(x)\in\{0,1\}^{\mathbb{N}}$ be defined by $\sigma(x)_{i}=x_{i+1}$ (we shall use the same notation as in the case of the group shift action, though in this case it’s a one-sided $\mathbb{N}$-action). We define also for $k\in\{0,\dots,p-1\}$ the transformation $\Omega_{k}:\{0,1,\$\}^{\mathbb{Z}}\to\{0,1,\$\}^{\mathbb{Z}}$ by $(\Omega_{k}(y))_{j}=y_{jp+k}$. Proposition 3.2.: _Let $x\in\{0,1\}^{\mathbb{N}}$ and $y\in\overline{\text{Orb}_{\sigma}(\Psi_{q}(x))}$. There exists a unique $k_{0}\in\{0,\dots p-1\}$ such that:_ \[\Omega_{k_{0}}(y)\in\overline{\text{Orb}_{\sigma}(\Psi_{q}(\sigma(x)))}.\] Proof.: The application $\Omega_{k}$ is clearly continuous in the product topology as fixing $y$ in the interval $\mathbb{Z}\cap[-lp,lp-1]$ for $l\geq 1$ necessarily fixes $\Omega_{k}(y)$ in the interval $\mathbb{Z}\cap[-l,l-1]$. Let $y\in\overline{\text{Orb}_{\sigma}(\Psi_{q}(x))}$. As $\Psi_{q}(x)\|_{q+p\mathbb{Z}}\equiv x_{0}$ we can deduce by compactness that there exists $k^{\prime}\in\{1,\dots,p\}$ such that $y\|_{k^{\prime}+p\mathbb{Z}}\equiv x_{0}$. Using the argument that every $x_{0}$ is in a string of $p-1$ symbols which repeats recurrently we can choose $k_{0}:=k^{\prime}-q\mod p$ which satisfies that $x_{k_{0}+1,\dots k_{0}+p-1}$ is this string. Consider a sequence $(\sigma_{z_{i}}(\Psi_{q}(x)))_{i\in\mathbb{N}}\to y$. Without loss of generality we can ask that $z_{i}\in p\mathbb{Z}-k_{0}$, if not it suffices to eliminate a finite number of terms. We get that \[\Omega_{k_{0}}(\sigma_{k_{0}+pl}(\Psi_{q}(x))) =\Omega_{0}(\sigma_{pl}(\Psi_{q}(x)))\] \[=\sigma_{l}\Omega_{0}(\Psi_{q}(x))\] \[=\sigma_{l}(\Psi_{q}(\sigma(x)))\in\text{Orb}(\Psi_{q}(\sigma(x)))\] As $\Omega_{k}$ is continuous, we obtain that $\Omega_{k_{0}}(y)\in\overline{\text{Orb}_{\sigma}(\Psi_{q}(\sigma(x)))}$. ∎ Example.: _For $p=3$, $q=1$ and $x=x_{0}x_{1}x_{2}\dots$ we obtain that:_ \[\Psi_{q}(x)\|_{\{0,\dots,30\}}=\$x_{0}\$x_{1}x_{0}\$\$x_{0}\$x_{2}x_{0}\$x_{1}x _{0}\$\$x_{0}\$\$x_{0}\$x_{1}x_{0}\$\$x_{0}\$x_{3}x_{0}\$x_{1}\] \[\Omega_{0}(\Psi_{q}(x))\|_{\{0,\dots,10\}}=\$x_{1}\$x_{2}x_{1}\$\$x_{1}\$x_{3}x _{1}=\Psi_{q}(\sigma(x))\|_{\{0,\dots,10\}}\] \[\Omega_{0}^{2}(\Psi_{q}(x))\|_{\{0,\dots,3\}}=\$x_{2}\$x_{3}=\Psi_{q}(\sigma^{2 }(x))\|_{\{0,\dots,3\}}\] The previous proposition actually shows that $x$ can be decoded from any element of the closure of the orbit of $\Psi_{q}(x)$ under the shift action. That necessarily implies that the orbits are disjoint. Proposition 3.3.: _Let $x,x^{\prime}\in X$. If $x\neq x^{\prime}$ then $\overline{\text{Orb}_{\sigma}(\Psi_{q}(x))}\cap\overline{\text{Orb}_{\sigma}( \Psi_{q}(x^{\prime}))}=\emptyset$_ Proof.: Let $y\in\overline{\text{Orb}_{\sigma}(\Psi_{q}(x))}\cap\overline{\text{Orb}_{ \sigma}(\Psi_{q}(x^{\prime}))}$. Using Proposition 3.2 we can find $k_{0}$ such that $\Omega_{k_{0}}(y)\in\overline{\text{Orb}_{\sigma}(\Psi_{q}(\sigma(x)))}$. Moreover, we get that $x_{0}=x^{\prime}_{0}=y_{k_{0}}$. Iterating this procedure we obtain that $\forall i\in\mathbb{N}$ then $x_{i}=x^{\prime}_{i}$ and thus $x=x^{\prime}$. ∎ Before continuing, let’s draw the attention to the structure of the subshift $\overline{\text{Orb}_{\sigma}(\Psi_{q}(x))}$. Every element here encodes the structure of $x$ by repeating its $n$-th coordinate in gaps of size $p^{n+1}$. Therefore, every non $\$$ element appears periodically with at most one exception – a position obtained by compactness – which we denote by $x_{\infty}$. This point may take any value if both $0$ and $1$ appear infinitely often in $x$ but is restricted if $x$ is eventually constant. This point is analogous to the lattice $B_{\infty}(z)$ appearing in the substitution we defined before. Let $(X,f)$ be an $H$-flow and $p\geq 3$. We use the encoding $\Psi_{q}$ defined above to construct a $\mathbb{Z}$-subshift $\texttt{Top}(X,f)$ which encodes the points of $x$ and the action of $f$ around a unit ball in $H$. Formally, let $S\subset H$ be a finite set such that $1_{H}\in S$ and $\langle S\rangle=H$. $\texttt{Top}(X,f)\subset(\{0,1,\$\}^{(p-1)\|S\|})^{\mathbb{Z}}$ is the $\mathbb{Z}$-subshift given by: \[\texttt{Top}(X,f):=\bigcup_{x\in X}\left(\overline{\text{Orb}_{\sigma}\left( \Psi_{q}(f_{s}(x))_{(q,s)\in\{1,\dots,p-1\}\times S}\right)}\right)\] Elements of $\texttt{Top}(X,f)$ are $(p-1)\|S\|$-tuples which encode elements of the $\mathbb{Z}$-orbit of each $\Psi_{q}(f_{s}(x))$. The idea behind this construction is to let each $q$-row code an element $x\in X$ and its image $f_{s}(x)$ for each $s\in S$. Given $y\in\texttt{Top}(X,f)$ we denote the projection to the $q,s$-th layer by $\texttt{Layer}_{q,s}(y)\in\{0,1,\$\}^{\mathbb{Z}}$. We need to do this for every possible $q$ just for technical reasons, as we’ll need to match every possible lattice in the substitution defined above. For all practical purposes, just one coordinate $q$ carries all the information we need to code. Proposition 3.4.: _If $(X,f)$ is an effectively closed $H$-flow then $\texttt{Top}(X,f)$ is an effectively closed $\mathbb{Z}$-subshift._ Proof.: $\texttt{Top}(X,f)$ is clearly shift invariant. It is closed as $X$ is closed and thus a diagonal argument allows to extract convergent subsequences. A set of forbidden patterns defining $\texttt{Top}(X,f)$ is the following. We consider for $n\in\mathbb{N}$ all words of length $p^{n+1}$ over the alphabet $\{\$,0,1\}^{\|S\|(p-1)}$ which do not appear in any configuration of $\texttt{Top}(X,f)$. As this is an increasing sequence of forbidden patterns it is enough to define $\texttt{Top}(X,f)$. This set of forbidden words is recursively enumerable. The following algorithm accepts a set of forbidden patterns defining $\texttt{Top}(X,f)$. Let the input be a word of length $p^{n}$ for $n\in\mathbb{N}$. The structure of $\texttt{Top}(X,f)$ makes it possible to recognize algorithmically all gaps in every layer (formally the algorithm checks that each substring of $p$ contiguous symbols is a cyclic permutation of $a\$^{q-1}b\$^{p-q-1}$ for some $a\in\{0,1,\$\}$ and $b\in\{0,1\}$). Then if this stage is passed, it computes $k_{0}$ from Proposition 3.2 for each layer, checks that $b$ is the same symbol throughout the word. Finally it checks that $k_{0}$ is the same in every layer (thus the layers are aligned). Then it applies $\Omega_{k_{0}}$ to this string obtaining a word of length $p^{n-1}$. The algorithm is repeated until reaching a word of length $0$. If at any stage a check fails, the word is accepted as forbidden. The previous stage recognizes all words that haven’t got the correct structure. After that stage ends, we can use the same algorithm and the function $\Omega_{k}$ to decode $n$ coordinates $x_{0}x_{1}\dots x_{n-1}$ for each pair $(q,s)$ and check for every $s$ that the word is the same independently of $q$. If this stage is passed we end up with $\|S\|$ words which depend only on $s$ and we denote them by $(w_{s})_{s\in S}$. Here we run two recognition algorithms in parallel. One searches for a cylinder $[w_{s}]\not\subset X$ and the other searches if $[w_{1_{H}}]\not\subset f^{-1}_{s}([w_{s}])$. If any of these two searches succeed at a certain step then the algorithm returns that the pattern is forbidden. These two last algorithms do exists as $(X,f)$ is an effectively closed $H$-flow. ∎ The subshift $\texttt{Top}(X,f)$ is the ingredient of the proof which allows us to simulate points $x\in X$ and their images under the generators of $H$ in a sofic $\mathbb{Z}^{2}$-subshift which contains this information. The next step is to put one of these configurations in each $\mathbb{Z}^{2}$-coset of $\mathbb{Z}^{2}\rtimes_{\varphi}H$ and force by local rules that the shift action by $(0,h)$ yields the $\mathbb{Z}^{2}$-coset where the point $f_{h}(x)$ is codified. The obvious obstruction to this idea is the fact that the action under $(0,h)$ in a semidirect product disturbs the adjacency relations in a coset if the automorphism $\varphi_{h}$ isn’t trivial. The way to go around this obstruction is to use the lattices given by the layer $\texttt{Sub}_{v}$ which are invariant under automorphisms. We specify how these two elements go together in the next subsection. ### Proof of Theorem 3.1 Denote $\varphi:H\to\text{Aut}(\mathbb{Z}^{2})$ a group homomorphism such that $G=\mathbb{Z}^{2}\rtimes_{\varphi}H$ is given by: \[(n_{1},h_{1})\cdot(n_{2},h_{2})=(n_{1}+\varphi_{h_{1}}(n_{2}),h_{1}h_{2})\] To make notations shorter, we write $\vec{0}=(0,0)\in\mathbb{Z}^{2}$ throughout the whole proof. Let $S$ be a finite set of generators of $H$ where $1_{H}\in S$ , $\|S\|=d$ and let’s fix the parameter $p=3$ which is used to construct $\texttt{Top}(X,f)$ (which contains thus $2d$ layers) and the substitutions $\texttt{Sub}_{v}$ for $v\in(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}$. Consider the following two $\mathbb{Z}^{2}$-subshifts. \[\texttt{Top}(X,f)^{H}\subseteq(\{0,1,\$\}^{2d})^{\mathbb{Z}^{2}}\] \[\texttt{Top}(X,f)^{V}\subseteq(\{0,1,\$\}^{2d})^{\mathbb{Z}^{2}}\] Where $x\in\texttt{Top}(X,f)^{H}$ is the subshift whose projection to $(\mathbb{Z},0)$ belongs to $\texttt{Top}(X,f)$ and any vertical strip is constant. Analogously $x\in\texttt{Top}(X,f)^{V}$ is the subshift whose projection to $(0,\mathbb{Z})$ belongs to $\texttt{Top}(X,f)$ and any horizontal strip is constant. Formally: $x\in\texttt{Top}(X,f)^{H}$ if $\forall i,j\in\mathbb{Z}$ then $x_{i,j}=x_{i,j+1}$ and $(x_{(i,0)})_{i\in\mathbb{Z}}\in\texttt{Top}(X,f)$. An analogous definition can be given for $\texttt{Top}(X,f)^{V}$. Proposition 3.4 says that $\texttt{Top}(X,f)$ is an effective $\mathbb{Z}$-subshift and therefore $\texttt{Top}(X,f)^{H}$ and $\texttt{Top}(X,f)^{V}$ are sofic $\mathbb{Z}^{2}$-subshifts by the simulation theorem proven in [3, 9]. Next we are going to put these subshifts together with the substitution layers to create a rich structure in each $\mathbb{Z}^{2}$-coset. Let $\Pi(X,f)\subset\texttt{Top}(X,f)^{H}\times\texttt{Top}(X,f)^{V}\times \bigotimes_{v\in(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}}\texttt{Sub}_ {v}$ be the $\mathbb{Z}^{2}$-subshift defined by forbidding the following symbols in the product alphabet. In order to describe the forbidden symbols correctly, we introduce the following notation: For $y\in\Pi(X,f)$ we denote by $\texttt{Layer}^{H}_{q,s}(y)$ and $\texttt{Layer}^{V}_{q,s}(y)$ the projections to the first and second layers in the $(q,s)$ coordinate respectively and for $v\in(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}$ we denote by $\texttt{Sub}_{v}(y)$ the projection to the corresponding substitutive layer. 1. $\forall(i,j)\in\mathbb{Z}^{2}$ and $(a,b)\in(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}$ the following is satisfied. If $a\neq 0$ then $(\texttt{Sub}_{(a,b)}(y))_{(i,j)}=\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ if and only if $(\texttt{Layer}^{H}_{a,1_{H}}(y))_{(i,j)}\in\{0,1\}$. Analogously, if $b\neq 0$ then then $(\texttt{Sub}_{(a,b)}(y))_{(i,j)}=\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ if and only if $(\texttt{Layer}^{V}_{b,1_{H}}(y))_{(i,j)}\in\{0,1\}$. 2. If $(\texttt{Sub}_{(1,1)}(y))_{(i,j)}=\leavevmode\hbox to11.63pt{\vbox to11.63pt{ \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ then $\forall s\in S$$(\texttt{Layer}^{H}_{1,s}(y))_{(i,j)}=(\texttt{Layer}^{V}_{1,s}(y))_{(i,j)}$. The $\mathbb{Z}^{2}$-subshift $\Pi(X,f)$ is sofic. Indeed, all the component are sofic subshifts and the added rules are local (we just forbid symbols in the product alphabet). In what follows we use the following notation: for a configuration $x\in\mathcal{A}^{G}$, $A\subset G$ and $a\in\mathcal{A}$ we write $x\|_{A}\equiv a$ if $\forall g\in A$ then $x_{g}=a$. Recall that we denote by $B_{m}(z)$ the $m$-th lattice of black squares in a configuration $z$ in a substitutive layer. Claim 3.1.: _Let $y\in\Pi(X,f)$, $(a,b)\in(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}$ and $z=\texttt{Sub}_{(a,b)}(y)$. Suppose that $\texttt{Layer}^{H}_{a,1_{H}}(y)$ is given by $x\in X$. Then:_ _If_ $a\neq 0$ _then_ $\forall m\in\mathbb{N},\forall s\in S$_:_ $\texttt{Layer}^{H}_{a,s}(y)\|_{B_{m}(z)}\equiv f_{s}(x)_{m}$__ * _If_ $b\neq 0$ _then_ $\forall m\in\mathbb{N},\forall s\in S$_:_ $\texttt{Layer}^{V}_{b,s}(y)\|_{B_{m}(z)}\equiv f_{s}(x)_{m}$__ * _The configurations in the layers_ $\texttt{Top}(X,f)^{H}$ _and_ $\texttt{Top}(X,f)^{V}$ _are defined by the same_ $x\in X$_._ Proof.: Let $a\neq 0$. It suffices to show this property for $s=1_{H}$ as the definition of $\texttt{Top}(X,f)$ forces the configurations to be aligned. The lattice $B_{0}(z)$ has the form $(i_{0},j_{0})+(a,b)+3\mathbb{Z}^{2}$, therefore its projection in the horizontal coordinate is $k_{0}+3\mathbb{Z}$ for $k_{0}=i_{0}+a\mod{3}$. Using the structure of $\Psi_{a}(x)$ there are three possibilities for $3$-lattices: One contains uniformly the symbol $x_{0}$, another contains only the symbol $\$$ and the third one contains $\Psi_{a}(\sigma(x))$ by proposition 3.2. The first rule of $\Pi(X,f)$ rules out the second and third possibility because there would be $\$$’s matched with . Therefore $\texttt{Layer}^{H}_{a,1_{H}}\|_{B_{0}(z)}\equiv x_{0}$. Inductively, let $B_{m}(z)=(i_{m},j_{m})+(a,b)3^{m}+3^{m+1}\mathbb{Z}^{2}$ and suppose $\forall m^{\prime}<m$$\texttt{Layer}^{H}_{a,1_{H}}\|_{B_{m^{\prime}}(z)}\equiv x_{m^{\prime}}$. Note that for $m^{\prime}$ the projection to the horizontal layer is $k_{m^{\prime}}+3^{m^{\prime}+1}\mathbb{Z}$ for $k_{m^{\prime}}:=i_{m}+a3^{m^{\prime}}\mod{3^{m^{\prime}+1}}$. Using iteratively the previous argument and applying the function $\Omega_{k_{m^{\prime}}}$ defined in 3.2 we end up with three possibilities for $3^{m}$-lattices (that is, the value of $k_{m^{\prime}}$), and again the first rule of $\Pi(X,f)$ rules out two of them, yielding $\texttt{Layer}^{H}_{a,1_{H}}\|_{B_{m}(z)}\equiv x_{m}$. Suppose the configuration in $\texttt{Top}(X,f)^{V}$ is given by $x^{\prime}\in X$. For $b$ the proof is analogous and we get that $b\neq 0$ implies that $\forall m\in\mathbb{N},\forall s\in S$: $\texttt{Layer}^{V}_{b,s}\|_{B_{m}(z)}\equiv f_{s}(x^{\prime})_{m}$. Now set $(a,b)=(1,1)$. The second rule of $\Pi(X,f)$ implies that $\forall s\in S,m\in\mathbb{N}$ then $(\texttt{Layer}^{H}_{1,s}(y))\|_{B_{m}(z)}=(\texttt{Layer}^{V}_{1,s}(y))\|_{B_{m }(z)}$. Using the previous two properties we conclude that $\forall s\in S,m\in\mathbb{N}$ we have $f_{s}(x)_{m}=f_{s}(x^{\prime})_{m}$. Using $s=1_{H}$ yields $x=x^{\prime}$ hence proving the second and third statement. ∎ From Claim 3.1 we obtain that each configuration $y\in\Pi(X,f)$ contains the information of a single $x\in X$. We can thus define properly the decoding function $\Upsilon:\Pi(X,f)\to X$ such that $\Upsilon(y)=x$ if and only if $\forall m\in\mathbb{N}$: $\texttt{Layer}^{H}_{1,1_{H}}(y)\|_{B_{m}(\texttt{Sub}_{(1,1)}(y))}\equiv x_{m}$. Consider the set of forbidden patterns $\mathcal{F}$ defining $\Pi(X,f)$. Each of these patterns has a finite support $F\subset\mathbb{Z}^{2}$. We extend those patterns to patterns in $G=\mathbb{Z}^{2}\rtimes_{\varphi}H$ by associating $d\in F\to(d,1_{H})\in G$. Therefore every pattern $P\in\mathcal{F}$ with support $F\subset\mathbb{Z}^{2}$ is embedded into a pattern $\widetilde{p}$ with support $(F,1_{H})\subset G$. We consider the set $\widetilde{\mathcal{F}}=\{\widetilde{p}\mid p\in\mathcal{F}\}$ and we define $\texttt{Final}(X,f)$ as the subshift over the same alphabet as $\Pi(X,f)$ defined by the set of forbidden patterns $\widetilde{\mathcal{F}}\cup\mathcal{G}$ where $\mathcal{G}$ is defined as follows: For each $s\in S$ consider $\varphi_{s^{-1}}$ the automorphism associated to $s^{-1}$ and $(a,b)=\widetilde{\varphi}_{s^{-1}}(1,1)$. We put in $\mathcal{G}$ all the patterns $P$ with support $\{(\vec{0},1_{H}),(\vec{0},s^{-1})\}$ which satisfy that $\texttt{Sub}_{(a,b)}(P_{(\vec{0},1_{H})})=\leavevmode\hbox to11.63pt{\vbox to 11.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{} \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ but either: * $\texttt{Sub}_{(1,1)}(P_{(\vec{0},s^{-1})})\neq\leavevmode\hbox to11.63pt{\vbox to 11.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{} \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ or * $\texttt{Sub}_{(1,1)}(P_{(\vec{0},s^{-1})})=\leavevmode\hbox to11.63pt{\vbox to 11.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{} \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{} \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t o 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ and * If $a\neq 0$ then $\texttt{Layer}^{H}_{a,s}(P_{(\vec{0},1_{H})})\neq\texttt{Layer}^{H}_{1,1_{H}}( P_{(\vec{0},s^{-1})})$ or * If $b\neq 0$ then $\texttt{Layer}^{V}_{b,s}(P_{(\vec{0},1_{H})})\neq\texttt{Layer}^{V}_{1,1_{H}}( P_{(\vec{0},s^{-1})})$. In simpler words: we force that every in layer $\texttt{Sub}_{(a,b)}$ of the $(\mathbb{Z}^{2},1_{H})$-coset must be matched with a in $\texttt{Sub}_{(1,1)}$ in the $(\mathbb{Z}^{2},s^{-1})$-coset and that if $a\neq 0$ then the symbol in $(\vec{0},1_{H})$ in $\texttt{Layer}^{H}_{a,s}$ is the same as the symbol in $(\vec{0},s^{-1})$ in $\texttt{Layer}^{H}_{1,1_{H}}$. If $b\neq 0$ we impose that the symbol in $(\vec{0},1_{H})$ in $\texttt{Layer}^{V}_{b,s}$ is the same as the symbol in $(\vec{0},s^{-1})$ in $\texttt{Layer}^{V}_{1,1_{H}}$. Before continuing let’s translate $\widetilde{\mathcal{F}}\cup\mathcal{G}$ into properties of $\texttt{Final}(X,f)$. In order to do that properly, for $y\in\texttt{Final}(X,f)$ we denote by $\pi(y)$ the $\mathbb{Z}^{2}$-configuration such that $\forall(i,j)\in\mathbb{Z}^{2}$$\pi(y)_{(i,j)}=y_{((i,j),1_{H})}$. Claim 3.2.: $\texttt{Final}(X,f)$ _satisfies the following properties:_ * $\texttt{Final}(X,f)$ _is a sofic_ $G$_-subshift._ * _Let_ $y\in\texttt{Final}(X,f)$_. Then_ $\pi(y)\in\Pi(X,f)$_._ * _If_ $\Upsilon(\pi(y))=x$ _then_ $\forall h\in H$_,_ $\Upsilon(\pi(\sigma_{(\vec{0},h)}(y)))=f_{h}(x)$_._ Proof.: As $\Pi(X,f)$ is sofic, it admits an SFT extension $\phi:\widehat{\Pi}(X,f)\twoheadrightarrow\Pi(X,f)$. By embedding as above a finite list of forbidden patterns defining $\widehat{\Pi}(X,f)$ into $G$ we obtain a $G$-SFT extension of $X_{\widetilde{\mathcal{F}}}$. Adding to this list of forbidden patterns the pullback of the finite list of forbidden patterns $\mathcal{G}$ under the local code $\Phi$ defining $\phi$ we obtain an SFT extension $\widehat{\texttt{Final}}(X,f)$ of $\texttt{Final}(X,f)$. The second property comes directly from the definition of $\texttt{Final}(X,f)$ as it contains an embedding of every forbidden pattern defining $\Pi(X,f)$. Note that as $G$ is not necessarily abelian, it may happen that $y\|_{(\mathbb{Z}^{2},h)}$ seen as a $\mathbb{Z}^{2}$-configuration does not belong to $\Pi(X,f)$ for some $h\in H$, but $\pi(\sigma_{(\vec{0},h^{-1})}(y))$ always does. Let’s prove the third property: We claim that it suffices to prove the property for $s\in S$. Indeed, given $h\in H$, as $H=\langle S\rangle$ there exists a minimal length word representing $h$. If $h=1_{H}$ the result is immediate. If not, then $h=sh^{\prime}$ for some $h^{\prime}\in H$ having a shorter word representation. Suppose this third property holds for all words of strictly smaller length and define $y^{\prime}=\sigma_{(\vec{0},h^{\prime})}(y)$ we have that $\Upsilon(\pi(y^{\prime}))=f_{h^{\prime}}(x)=x^{\prime}$, so: \[\Upsilon(\pi(\sigma_{(\vec{0},h)}(y)))=\Upsilon(\pi(\sigma_{(\vec{0},s)}(y^{ \prime})))=f_{s}(x^{\prime})=f_{s}(f_{h^{\prime}}(x))=f_{h}(x).\] It suffices therefore to prove the property for $s\in S$. Let’s denote $y^{\prime}=\sigma_{(\vec{0},s)}(y)$ and let $\Upsilon(\pi(y))=x$ and $\Upsilon(\pi(y^{\prime}))=x^{\prime}$. We want to prove that $x^{\prime}=f_{s}(x)$. Let $\widetilde{\varphi}_{s^{-1}}(1,1)=(a,b)$ and suppose that $a\neq 0$ (if $a=0$ then $b\neq 0$ and the argument is analogous). Let $m\in\mathbb{N}$, using Claim 3.1 we obtain \[\texttt{Layer}^{H}_{1,1_{H}}(y^{\prime})\|_{(B_{m}(\texttt{Sub}_{(1,1)}(y^{ \prime})),1_{H})}\equiv x^{\prime}_{m}\] \[\texttt{Layer}^{H}_{a,s}(y)\|_{(B_{m}(\texttt{Sub}_{(a,b)}(y)),1_{H})}\equiv f_ {s}(x)_{m}.\] Using the forbidden patterns $\mathcal{G}$ results in \[\texttt{Sub}_{(1,1)}(y)\|_{(B_{m}(\texttt{Sub}_{(a,b)}(y)),s^{-1})}\equiv \leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter\hbox to 0. 0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}\] \[\texttt{Layer}^{H}_{1,1_{H}}(y)\|_{(B_{m}(\texttt{Sub}_{(a,b)}(y)),s^{-1})} \equiv f_{s}(x)_{m}.\] Finally, developing the action on $y^{\prime}$ yields \[y^{\prime}\|_{(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime})),1_{H})} =\sigma_{(\vec{0},s)}(y)\|_{(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime} )),1_{H})}\] \[=y\|_{(\vec{0},s^{-1})(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime})),1_{ H})}\] \[=y\|_{(\varphi_{s^{-1}}(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime}))),s ^{-1})}.\] Using the results from Section 3.1 we obtain that $B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime})$ is of the form $(i_{m},j_{m})+(1,1)3^{m}+3^{m+1}\mathbb{Z}^{2}$ and thus $\varphi_{s^{-1}}(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime}))$ is $\varphi_{s^{-1}}(i_{m},j_{m})+(a,b)3^{m}+3^{m+1}\mathbb{Z}^{2}$. Which is $B_{m}(z)$ for some $z\in\texttt{Sub}_{(a,b)}$. As we have $\forall m\in\mathbb{N}$ that $\texttt{Sub}_{(1,1)}(y^{\prime})\|_{(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime})),1_ {H})}\equiv\leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter \hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ and $\texttt{Sub}_{(1,1)}(y)\|_{(B_{m}(\texttt{Sub}_{(a,b)}(y)),s^{-1})}\equiv \leavevmode\hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter\hbox to 0. 0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ we conclude that $\varphi_{s^{-1}}(B_{m}(\texttt{Sub}_{(1,1)}(y^{\prime}))=B_{m}(\texttt{Sub}_{( a,b)}(y))$. Therefore, \[\texttt{Layer}^{H}_{1,1_{H}}(y^{\prime})\|_{(B_{m}(\texttt{Sub}_{(1,1)}(y^{ \prime})),1_{H})}=\texttt{Layer}^{H}_{1,1_{H}}(y)\|_{(B_{m}(\texttt{Sub}_{(a,b) }(y)),s^{-1})}.\] Which yields $x^{\prime}_{m}=f_{s}(x)_{m}$. As $m\in\mathbb{N}$ is arbitrary $x^{\prime}=f_{s}(x)$.∎ Finally we are ready to finish the proof. Consider again the SFT extension $\widehat{\texttt{Final}}(X,f)$ of $\texttt{Final}(X,f)$, the factor map $\phi:\widehat{\texttt{Final}}(X,f)\twoheadrightarrow\texttt{Final}(X,f)$ and the subaction $(\widehat{\texttt{Final}}(X,f),\sigma^{H})$. Proposition 3.5.: $\Upsilon\circ\pi\circ\phi$ _is a factor map from $(\widehat{\texttt{Final}}(X,f),\sigma^{H})$ to $(X,f)$._ Proof.: As $\phi:\widehat{\texttt{Final}}(X,f)\twoheadrightarrow\texttt{Final}(X,f)$ it suffices to show that $\Upsilon\circ\pi$ is a factor map from $(\texttt{Final}(X,f),\sigma^{H})$ to $(X,f)$. Let $y\in\texttt{Final}(X,f)$. Following Claim 3.2 we have $\pi(y)\in\Pi(X,f)$ and thus $\Upsilon(\pi(y))\in X$. Moreover, setting $\Upsilon(\pi(y))=x$ yields $\forall h\in H$ that $\Upsilon(\sigma_{(\vec{0},h)}(y))=f_{h}(x)$. This implies \[\forall h\in H:(\Upsilon\circ\pi)\circ\sigma_{(\vec{0},h)}=f_{h}\circ(\Upsilon \circ\pi).\] Also, both $\Upsilon$ and $\pi$ are clearly continuous, therefore, it only remains to show that $\Upsilon\circ\pi$ is surjective. Let $x\in X$, we construct a configuration $y^{}\in\texttt{Final}(X,f)$ such that $\Upsilon(\pi(y^{}))=x$. In order to do this, we begin by constructing a sequence of configurations $(y^{h})_{h\in H}$ which belong to $\Pi(X,f)$. For $(a,b)\in(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}$ let $z_{(a,b)}\in\texttt{Sub}_{(a,b)}$ defined by $(i_{n},j_{m})=0$ for all $m\in\mathbb{N}$. Said otherwise, $B_{m}(z_{(a,b)})=(a,b)3^{m}+3^{m+1}\mathbb{Z}^{2}$ for $m\in\mathbb{N}$ and $B_{\infty}(z_{(a,b)})=\emptyset$. We define $y^{h}\in\Pi(X,f)$ by $\texttt{Sub}_{(a,b)}(y^{h})=z_{(a,b)}$ and $\forall(i,j)\in\mathbb{Z}^{2}$, $s\in S$, $a,b\in\{1,2\}$ then $\texttt{Layer}^{H}_{a,s}(y^{h})_{(i,j)}=\Psi_{a}(f_{s}(f_{h}(x)))_{i}$ and $\texttt{Layer}^{V}_{b,s}(y^{h})_{(i,j)}=\Psi_{b}(f_{s}(f_{h}(x)))_{j}$. It can easily be verified that for each $h\in H$ the configuration $y^{h}\in\Pi(X,f)$. Finally, we define $y^{}$ as follows: \[(y^{})_{((i,j),h)}=(y^{h^{-1}})_{\varphi_{h^{-1}}(i,j)}.\] As $\varphi_{1_{H}}(i,j)=(i,j)$ then $\pi(y^{})=y^{1_{H}}$ and thus $\Upsilon(\pi(y^{}))=f_{1_{H}}(x)=x$. It suffices to show that $y^{}\in\texttt{Final}(X,f)$. This comes down to showing that no patterns in $\mathcal{F}$ or $\mathcal{G}$ appear in $y^{}$. Suppose a pattern $P\in\mathcal{F}$ appears at position $g=((i,j),h)$, that is $y^{}\in[P]_{g}\iff\sigma_{g^{-1}}(y^{})\in[P]_{1_{G}}$. As $P$ has a support contained in $(\mathbb{Z}^{2},1_{H})$ then $\pi(\sigma_{g^{-1}}(y^{}))\notin\Pi(X,f)$. But \[\sigma_{g^{-1}}(y^{})_{((i^{\prime},j^{\prime}),1_{H})} =(y^{})_{g((i^{\prime},j^{\prime}),1_{H})}\] \[=(y^{})_{((i,j)+\varphi_{h}(i^{\prime},j^{\prime}),h)}\] \[=(y^{h^{-1}})_{(i^{\prime},j^{\prime})+\varphi_{h^{-1}}(i,j)}\] \[=(\sigma_{-\varphi_{h^{-1}}(i,j)}(y^{h^{-1}}))_{(i^{\prime},j^{ \prime})}.\] Therefore $\pi(\sigma_{g^{-1}}(y^{}))=\sigma_{-\varphi_{h^{-1}}(i,j)}(y^{h^{-1}})\in\Pi( X,f)$ which is a contradiction. Hence $y^{}$ does not contain any pattern from $\mathcal{F}$. It remains to show it contains no patterns in $\mathcal{G}$. Recall that patterns $P\in\mathcal{G}$ have support $\{(\vec{0},1_{H}),(\vec{0},s^{-1})\}$ for $s\in S$. Let $g=((i,j),h)$ such that $\sigma_{g^{-1}}(y^{})\in[P]_{1_{G}}$. Then $\sigma_{g^{-1}}(y^{})_{(\vec{0},1_{H})}=\sigma_{-\varphi_{h^{-1}}(i,j)}(y^{h^ {-1}})_{\vec{0}}$ and \[\sigma_{g^{-1}}(y^{})_{(\vec{0},s^{-1})} =(y^{})_{((i,j),h)(\vec{0},s^{-1})}\] \[=(y^{})_{((i,j),hs^{-1})}\] \[=(y^{sh^{-1}})_{\varphi_{(sh^{-1})}(i,j)}\] \[=(\sigma_{-(\varphi_{(sh^{-1})}(i,j))}(y^{sh^{-1}}))_{\vec{0}}.\] Let $m\in\mathbb{N}$ and denote $(a,b)=\widetilde{\varphi}_{s^{-1}}(1,1)$. By definition $B_{m}(\texttt{Sub}_{(a,b)}(y^{h})=(a,b)3^{m}+3^{m+1}\mathbb{Z}^{2}$ therefore, \[B_{m}(\texttt{Sub}_{(a,b)}(\sigma_{-\varphi_{h^{-1}}(i,j)}(y^{h}))=(a,b)3^{m}+ \varphi_{h^{-1}}(i,j)+3^{m+1}\mathbb{Z}^{2}\] In the other hand, \[B_{m}(\texttt{Sub}_{(1,1)}(\sigma_{-(\varphi_{(sh^{-1})}(i,j))}(y^{sh^{-1}}))= (1,1)3^{m}+\varphi_{(sh^{-1})}(i,j)+3^{m+1}\mathbb{Z}^{2}.\] So, if $\texttt{Sub}_{(a,b)}(\sigma_{g^{-1}}(y^{}))_{(\vec{0},1_{H})}=\leavevmode \hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{ \pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$ then $\vec{0}\in(a,b)3^{m}+\varphi_{h^{-1}}(i,j)+3^{m+1}\mathbb{Z}^{2}$ for some $m\in\mathbb{N}$. Applying $\varphi_{s}$ at both sides we obtain: \[\varphi_{s}(\vec{0})=\vec{0} \in\varphi_{s}(a,b)3^{m}+\varphi_{(sh^{-1})}(i,j)+3^{m+1}\mathbb{ Z}^{2}\] \[=\widetilde{\varphi}_{s}(a,b)3^{m}+\varphi_{(sh^{-1})}(i,j)+3^{m+ 1}\mathbb{Z}^{2}\] \[=(1,1)3^{m}+\varphi_{(sh^{-1})}(i,j)+3^{m+1}\mathbb{Z}^{2}\] \[=B_{m}(\texttt{Sub}_{(1,1)}(\sigma_{-(\varphi_{(sh^{-1})}(i,j))}( y^{sh^{-1}})).\] Implying that $\texttt{Sub}_{(1,1)}(\sigma_{g^{-1}}(y^{}))_{(\vec{0},s^{-1})}=\leavevmode \hbox to11.63pt{\vbox to11.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{ \pgfsys@beginscope{}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{} \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{} {}{} {}{}{}{}{}{}\pgfsys@beginscope{}\pgfsys@setlinewidth{0.8pt}{}\definecolor[ named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}{}{}\pgfsys@moveto{- 3.414331pt}{-3.414331pt}\pgfsys@moveto{-3.414331pt}{-3.414331pt}\pgfsys@lineto {-3.414331pt}{3.414331pt}\pgfsys@lineto{3.414331pt}{3.414331pt}\pgfsys@lineto{ 3.414331pt}{-3.414331pt}\pgfsys@closepath\pgfsys@moveto{3.414331pt}{3.414331pt }\pgfsys@fillstroke\pgfsys@invoke{ } {}{}{}\pgfsys@endscope {}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss} \endpgfpicture}}$. Moreover, if either $a$ is non-zero (the $b\neq 0$ case is analogous), then, using the previous computation we get: \[\texttt{Layer}^{H}_{a,s}(\sigma_{g^{-1}}(y^{}))_{(\vec{0},1_{H})}=f_{s}(f_{h^ {-1}}(x))_{m}\] \[\texttt{Layer}^{H}_{1,1_{H}}(\sigma_{g^{-1}}(y^{}))_{(\vec{0},1_{H})}=f_{sh^{ -1}}(x)_{m}.\] So no patterns from $\mathcal{G}$ appear, yielding $y^{}\in\texttt{Final}(X,f)$.∎ Proposition 3.5 concludes the proof of Theorem 3.1. ## 4 Consequences and remarks In this last section we explore some consequences of our simulation theorem. The first one is the case of expansive actions. Here we show that as long as the group is recursively presented, the action can be presented in a convenient way that allows us to replace the subaction by the projective subdynamics. The second is an application of this theorem to produce non-empty strongly aperiodic subshifts in a class of groups where this fact whas previously unknown, answering a question posed by Sahin in a symbolic dynamics workshop held in Chile in december 2014. We also extend a Theorem of Jeandel [14] to the existence of effectively closed strongly aperiodic flows in general. We close this section by remarking that the technique used to prove Theorem 3.1 is valid in an even larger class (namely, simulation in $\mathbb{Z}^{d}\rtimes G$) and with a discussion on the size of the extension. Indeed, in Hochman’s article [12] the subaction is shown to be an almost trivial isometric extension. We dedicate the last part of this section to informally discuss the size of the factor in our construction and how a similar result could be obtained. ### The simulation theorem for expansive effective flows Before presenting the simulation theorem for expansive actions, we must define with more detail effectively closed subshifts in groups. A longer survey of these concepts can be found in [2]. Given a group $G$ generated by $S$ and a finite alphabet $\mathcal{A}$ a _pattern coding_$c$ is a finite set of tuples $c=(w_{i},a_{i})_{i\in I}$ where $w_{i}\in S^{}$ and $a_{i}\in\mathcal{A}$. A set of pattern codings $\mathcal{C}$ is said to be recursively enumerable if there is a Turing machine which takes as input a pattern coding $c$ and accepts it if and only if $c\in\mathcal{C}$. A subshift $X\subset\mathcal{A}^{G}$ is _effectively closed_ if there is a recursively enumerable set of pattern codings $\mathcal{C}$ such that: \[X=X_{\mathcal{C}}:=\bigcap_{g\in G,c\in\mathcal{C}}\left(\mathcal{A}^{G} \setminus\bigcap_{(w,a)\in c}[a]_{gw}\right).\] With this formal concept in hand, we show the following lemma: Lemma 4.1.: _For every finitely generated group, any $G$-subshift which is the factor of an effectively closed $G$-flow is itself effectively closed._ Proof.: Let $G$ be generated by the finite set $S\subset G$, $(X,f)$ an effectively closed $G$-flow over a Cantor set, $(Y,\sigma)$ a $G$-subshift and $\phi:(X,f)\twoheadrightarrow(Y,\sigma)$ a factor. Recall that $X\subset\{0,1\}^{\mathbb{N}}$ and $Y\subset\mathcal{A}^{G}$ for some finite $\mathcal{A}$. As both $X$ and $Y$ are compact, $\phi$ is uniformly continuous. Therefore for each $a\in\mathcal{A}$ then $\phi^{-1}([a])=W_{a}$ where $W_{a}$ is a clopen set depending on a finite number of coordinates. For any pattern coding $c$: \[\phi^{-1}\left(\bigcap_{(w,a)\in c}[a]_{w}\right)=\bigcap_{(w,a)\in c}\phi^{-1 }(\sigma_{w^{-1}}([a]))=\bigcap_{(w,a)\in c}f_{w^{-1}}(\phi^{-1}([a]))\] Therefore, \[Y\cap\bigcap_{(w,a)\in c}[a]_{w}=\emptyset\implies X\cap\bigcap_{(w,a)\in c}f_ {w^{-1}}(W_{a})=\emptyset.\] As $(X,f)$ if effectively closed, there is a Turing machine which can approximate the set $\bigcap_{(w,a)\in c}f_{w^{-1}}(W_{a})$ as each $W_{a}$ is just a finite union of a finite intersection of cylinders and $w^{-1}\in S^{}$. Also, for each partial approximation we can use the enumeration cylinders defining the complement of $X$ to recognize if the intersection is empty, using these tools we can construct a Turing machine recognizing a maximal set of forbidden pattern codings defining $Y$.∎ Theorem 4.2.: _Let $H$ be a finitely generated and recursively presented group._ 1. _Let_ $(X,f)$ _be an effectively closed expansive_ $H$_-flow over a Cantor set. Then there exists a_ $(\mathbb{Z}^{2}\rtimes H)$_-sofic subshift_ $Y$ _such that its_ $H$_-projective subdynamics_ $\pi_{H}(Y)$ _is conjugated to_ $(X,f)$_._ 2. _Let_ $Z$ _be an effectively closed_ $H$_-subshift. Then there exists a sofic_ $(\mathbb{Z}^{2}\rtimes H)$_-subshift_ $Y$ _such that its_ $H$_-projective subdynamics_ $\pi_{H}(Y)$ _is_ $Z$_._ Proof.: Let $S\subset H$ be a finite set such that $\langle S\rangle=H$. Consider a recursive bijection $\varphi:\mathbb{N}\to S^{}$ where $S^{}$ is the set of all words on $S$ and $\varphi(0)=\epsilon$ the empty word. As $H$ is recursively presented, then its word problem $\texttt{WP}(H)=\{w\in S^{}\mid w=1_{H}\}$ is recursively enumerable and thus there is a Turing machine $T$ which accepts all pairs $(n,n^{\prime})\in\mathbb{N}^{2}$ such that $\varphi(n)=\varphi(n^{\prime})$ as elements of $H$. In the first case as $(X,f)$ is an expansive $H$-action on a closed subset of the totally disconnected space $\{0,1\}^{\mathbb{N}}$, it is conjugated to a subshift $Z\subset\mathcal{A}^{H}$. Furthermore, using Lemma 4.1 we obtain that $Z$ is an effectively closed $H$-subshift. In the second case we just work with $(Z,\sigma)$ from the beginning. The argument is the same for both cases. For simplicity, let’s first suppose $\mathcal{A}=\{0,1\}$ and tackle the general case later. Consider the map $\rho:Z\to\{0,1\}^{\mathbb{N}}$ where $\rho(z)_{n}=z_{\varphi(n)}$ where $\varphi(n)\in S^{}$ is identified as an element of $H$. Consider the set $\Omega=\rho(Z)$ and the $H$-action $f^{\prime}:H\times\Omega\to\Omega$ defined as $f^{\prime}_{h}(\rho(z))=\rho(\sigma_{h}(z))$. Clearly $\rho$ is a conjugacy between $(Z,\sigma)$ and $(\Omega,f^{\prime})$. We claim that $(\Omega,f^{\prime})$ is an effectively closed $H$-flow. Indeed, let $w\in\{0,1\}^{}$. A Turing machine which accepts $w$ if and only if $[w]\in\{0,1\}^{\mathbb{N}}\setminus\Omega$ is given by the following scheme: for each pair $(n,n^{\prime})$ in the support of $w$ run $T$ in parallel. if $T$ accepts for a pair such that $w_{n}\neq w_{n^{\prime}}$ then accept $w$ (this means that $w$ did not codify a configuration in $\mathcal{A}^{\mathbb{Z}}$ as two words codifying different group elements have different symbols). Also, in parallel, use the algorithm recognizing a maximal set of forbidden patterns for $Z$ over the pattern coding $c_{w}=(\varphi(n),w_{n})_{n\leq\|w\|}$. This eliminates all $w$ which codify configurations containing forbidden patterns in $Z$. For $f_{s}^{-1}[w]$ just note that the application $n\to\varphi(s^{-1}\varphi^{-1}(n))$ is recursive, thus $f_{s}^{-1}[w]$ can be calculated. As $(\Omega,f^{\prime})$ is effectively closed, using Theorem 3.1 we can construct the sofic subshift $\texttt{Final}(\Omega,f^{\prime})$. Instead of applying $\Upsilon\circ\pi$ to the $H$-subaction consider the local function $\Phi:\{0,1,\$\}^{3}\to\{0,1\}$ defined as follows: Let $u\in\{0,1,\$\}^{3}$. If $u$ contains the word $0\$$, then $\Phi(u)=0$, otherwise $\Phi(u)=1$. Notice that $\phi:\{0,1,\$\}^{\mathbb{Z}}\to\{0,1\}^{\mathbb{Z}}$ defined by $\phi(y)_{n}=\Phi(\sigma_{-n}(y))$ satisfies that for any $x\in\{0,1\}^{\mathbb{N}}$ then $\phi(\Psi_{1}(x))=(x_{0})^{\mathbb{Z}}$. Indeed, every substring of size $3$ is a cyclic permutation of $ax_{0}\$$ for some $a\in\{0,1,\$\}$. Now extend $\Phi$ to the support $F=((\{0,1,2\},0),1_{H})$ and let $\phi:\{0,1\$\}^{G}\to\{0,1\}^{G}$ be defined by $\phi(y)_{(z,h)}=\Phi(\sigma_{(z,h)^{-1}}(y)\|_{F})$ and recall that $\texttt{Layer}^{H}_{1,1_{H}}$ is the projection to the layer containing in each horizontal strip a codification of $\Psi_{1}(x)$ for some $x\in\Omega$. Therefore, we can obtain a sofic $G$-subshift \[\texttt{Proj}(\Omega,f^{\prime})=\phi(\texttt{Layer}^{H}_{1,1_{H}}(\texttt{ Final}(\Omega,f^{\prime})))\] We claim that $\pi_{H}(\texttt{Proj}(\Omega,f^{\prime}))=(Z,\sigma)$. Note that $\forall y\in\texttt{Proj}(\Omega,f^{\prime})$ then $y_{(z,h)}=(f^{\prime}_{h^{-1}}(x))_{0}$. This means the $H$-projective subdynamics of $y$ is the configuration $\widetilde{x}\in\mathcal{A}^{H}$ where $\widetilde{x}_{h}=(f^{\prime}_{h^{-1}}(x))_{0}$. We claim $\widetilde{x}\in Z$. Indeed, as $x\in\Omega$ then there exists $z\in Z$ such that $x=\rho(z)$. By definition of $\rho$ and the fact that $\varphi(0)=\epsilon$ we have \[\widetilde{x}_{h}=(f^{\prime}_{h^{-1}}(x))_{0}=(\rho(\sigma_{h^{-1}}(z)))_{0}= (\sigma_{h^{-1}}(z))_{1_{H}}=z_{h}.\] Conversely, every $z\in Z$ is realized by some $\rho(z)\in\Omega$, and there exists $y\in\texttt{Final}(\Omega,f^{\prime})$ such that $\Upsilon(\pi(y))=\rho(z)$. Thus $z_{h}=\phi(\texttt{Layer}^{H}_{1,1_{H}}(y))_{(0,h)}$ henceforth $z\in\pi_{H}(\texttt{Proj}(\Omega,f^{\prime})).$ Therefore the $H$-projective subdynamics of $\texttt{Proj}(\Omega,f^{\prime})$ is $(Z,\sigma)$ which is conjugated to $(X,f)$. In the case of a bigger $\mathcal{A}$, we can code each $a\in\mathcal{A}$ as a word in $\{0,1\}^{k}$ and redefine $\rho$ such that for $z\in Z$ then $\rho(z)_{n}=(z_{\varphi(\lfloor n/k\rfloor)})_{n\mod{k}}$. Everything stays the same in the construction except that the factor $\phi$ has support $\{0,\dots,3^{k}-1\}$ instead of $\{0,1,2\}$. it recovers from $\Psi_{1}(x)$ the first values $x_{0},\dots,x_{k-1}$ and uses them to decode the corresponding $a\in\mathcal{A}$.∎ ### Existence of strongly aperiodic SFT in a class of groups obtained by semidirect products Next we show how these previous theorems can be applied to produce strongly aperiodic subshifts of finite type. We say a $G$-subshift $(X,\sigma)$ is _strongly aperiodic_ if the shift action is free, that is, $\forall x\in X,\sigma_{g}(x)=x\implies g=1_{G}$. Theorem 4.3.: _Let $H$ be a finitely generated group and $(X,f)$ a non-empty effectively closed $H$-flow which is free. Then $G\cong\mathbb{Z}^{2}\rtimes H$ admits a non-empty strongly aperiodic SFT._ Proof.: We begin by recalling the following general property of factors. Suppose there is a factor $\phi:(X,f)\twoheadrightarrow(Y,f^{\prime})$. and let $x\in X$ such that $f_{g}(x)=x$. Then $f^{\prime}_{g}(\phi(x))=\phi(f_{g}(x))=\phi(x)\in Y$. This means that if $f^{\prime}$ is a free action then $f$ is also a free action. By Theorem 3.1 we can construct the $(\mathbb{Z}^{2}\rtimes H)$-SFT $\widehat{\texttt{Final}}(X,f)$ such that $(\widehat{\texttt{Final}}(X,f),\sigma_{H})$ is an extension of $(X,f)$ via the factor $\phi_{1}=\Upsilon\circ\pi\circ\phi$. We also consider the factor $\phi_{2}=\texttt{Sub}_{(1,1)}\circ\phi$ which sends $\widehat{\texttt{Final}}(X,f)$ first to $\texttt{Final}(X,f)$ and then to its $\texttt{Sub}_{(1,1)}$ layer. Let $y\in\widehat{\texttt{Final}}(X,f)$ and $(z,h)\in\mathbb{Z}^{2}\rtimes H$ such that $\sigma_{(z,h)}(y)=y$. This implies that $\phi_{2}(y)=\sigma_{(z,h)}(\phi_{2}(y))=\sigma_{(z,1_{H})}(\sigma_{(0,h)}(\phi _{2}(y)))$. As we have seen in the proof of Theorem 3.1, the action $\sigma_{(0,h)}$ leaves the lattices $(B_{m})_{m\in\mathbb{N}}$ of $\texttt{Sub}_{(1,1)}$ invariant in the $(\mathbb{Z}^{2},1_{H})$-coset. Let $M>\|\|z\|\|^{2}$. Then $\sigma_{(z,0)}$ does not leave invariant the lattice $B_{m}$. This implies that $z=\vec{0}$. Therefore, $\sigma_{(\vec{0},h)}(y)=y$. Applying $\phi_{1}$ we obtain that $f_{h}(y)=y$, and thus $h=1_{H}$. Therefore $(z,h)=(\vec{0},1_{H})$ and $\widehat{\texttt{Final}}(X,f)$ is strongly aperiodic. It is non-empty as $X\neq\emptyset$. ∎ Theorem 4.3 allows us to produce strongly aperiodic subshifts in many groups, we state this in a corollary. Corollary 4.4.: _Let $H$ be a finitely generated group with decidable word problem, then $\mathbb{Z}^{2}\rtimes H$ admits a non-empty strongly aperiodic SFT._ Proof.: In [4] an effectively closed $H$-subshift is constructed for all finitely generated groups $H$ with decidable word problem. One way to construct this object is as follows: a Theorem [1] of Alon, Grytczuk, Haluszczak and Riordan uses Lovász local lemma to show that every finite regular graph of degree $\Delta$ can be vertex-colored with at most $(2e^{16}+1)\Delta^{2}$ colors such that the sequence of colors in any non-intersecting path does not contain a square. Using compactness arguments this can be extended to Cayley graphs $\Gamma(H,S)$ of finitely generated groups where the bound takes the form $2^{19}\|S\|^{2}$ colors where $\|S\|$ is the cardinality of a set of generators of $H$. One can also show that the set of square-free vertex-coloring of $\Gamma(H,S)$ yields a strongly aperiodic subshift. In the case where $G$ has decidable word problem, a Turing machine can construct the sequence of balls $B(1_{G},n)$ and enumerate a codification of all patterns containing a square colored path. Therefore we obtain an effectively closed, strongly aperiodic and non-empty subshift. Using the fact that $G$ is recursively presented one can do the coding of theorem 4.2 to obtain a free non-empty effectively closed $H$-flow $(\Omega,f^{\prime})$. Applying Theorem 4.3 concludes the proof.∎ We remark that this corollary is an alternative proof to a construction done by Ugarcovici, Sahin and Schraudner in 2014 showing that the discrete Heisenberg group $\mathcal{H}$ admits non-empty strongly aperiodic SFTs. This falls directly from our theorem as $\mathcal{H}\cong\mathbb{Z}^{2}\rtimes_{\varphi}\mathbb{Z}$ for $\varphi(1)=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$. In their proof they use a similar trick using as a base the Robinson tiling [20]. They use the lattices of crosses in this object to match the different $(\mathbb{Z}^{2},0)$-cosets correctly to force a trivial action in the $\mathbb{Z}$ direction and use a counter machine to create aperiodicity in the other direction. In our construction the Robinson tiling got replaced by the substitutive subshifts $\texttt{Sub}_{(a,b)}$ which are able to match correctly the cosets of any possible automorphism and the counter machine by the simulation of the free $H$-flow. We also remark that Corollary 4.4 answers some open questions in their talk asking the same property for the Flip, Sol groups and the powers of the Heisenberg group. The only case it does not solve is the one of their two-dimensional Baumslag Solitar group, as the matrix they used to define it is not invertible and can not be expressed as a semidirect product. A theorem of Jeandel [14] says that for recursively presented groups $G$, the existence of a non-empty strongly aperiodic subshift $X\subset\mathcal{A}^{G}$ implies that the word problem of $G$ is decidable. We can extend this to the case of arbitrary flows. This gives a deep relation between computability and dynamical properties. Corollary 4.5.: _Let $H$ be a recursively presented and finitely generated group. There exists a strongly aperiodic effectively closed $H$-flow if and only if the word problem of $H$ is decidable._ Proof.: If the word problem of $H$ is decidable, we can use the effectively closed subshift constructed in [4] as an example. Conversely, Jeandel’s result implies that if a recursively presented group admits a non-empty effectively closed and strongly aperiodic subshift then it’s word problem is decidable. Using Theorem 4.3 we can construct a strongly aperiodic subshift from any free effectively closed $H$-flow. Therefore the word problem of $H$ is decidable. ∎ ### A generalization and comments on the size of the extension In this last portion we want to make explicit that the technique used in the proof of Theorem 3.1 can be easily be generalized to the following context Theorem 4.6.: _Let $H$ be finitely generated group, $d\geq 2$ and $G=\mathbb{Z}^{d}\rtimes H$. For every $H$-effectively closed flow $(X,f)$ there exists a $G$-SFT whose $H$-subaction is an extension of $(X,f)$._ Indeed, instead of considering vectors in $(\mathbb{Z}/3\mathbb{Z})^{2}\setminus\{\vec{0}\}$ we use $v\in(\mathbb{Z}/3\mathbb{Z})^{d}\setminus\{\vec{0}\}$ and $d$-dimensional substitutions $\texttt{s}_{v}$ defined analogously. The subshifts generated by these substitutions carry $\mathbb{Z}^{d}$-lattices and the configurations $z\in\texttt{Sub}_{v}$ can be described in the same way as before by lattices $B_{m}(z)$. The Toeplitz construction $\texttt{Top}(X,f)$ stays the same but instead of just constructing $\texttt{Top}(X,f)^{H}$ and $\texttt{Top}(X,f)^{V}$ we construct $\texttt{Top}(X,f)^{e_{i}}$ for every canonical vector $\{e_{i}\}_{i\leq d}$ where the $\langle e_{i}\rangle$-projective subdynamics yields $\texttt{Top}(X,f)$ and the configurations are extended periodically everywhere else. The rest of the construction translates directly to this setting. We also want to remark the following: Hochman’s theorem gives further information about the extension. Formally, given an effectively closed $\mathbb{Z}^{d}$-flow. A $\mathbb{Z}^{d+2}$-SFT is constructed such that the $\mathbb{Z}^{d}$-subaction is an _almost trivial isometric extension_ (ATIE) of the $\mathbb{Z}^{d}$-flow. An extension $(Z,f_{Z})\twoheadrightarrow(Y,f_{Y})$ is an ATIE if we can interpolate a factor \[(Z,f_{Z})\twoheadrightarrow(Y,f_{Y})\times(W,f_{W})\twoheadrightarrow(Y,f_{Y})\] such that $(W,f_{W})$ is an isometric action of a totally disconnected space, $(Y,f_{Y})\times(W,f_{W})\twoheadrightarrow(Y,f_{Y})$ is the projection of the first coordinate $(Z,f_{Z})\twoheadrightarrow(Y,f_{Y})\times(W,f_{W})$ is almost everywhere $1-1$, that is, it satisfies that the set of points with unique preimage has measure 1 under any invariant Borel probability measure. The idea behind the notion of ATIE is of an extension which is in a certain sense “small”. It consists basically on adding a simple system $(W,f_{W})$ as a product and then considering a measure equivalent action as the extension. Many properties such as the topological entropy (at least for $\mathbb{Z}^{d}$-actions) are preserved by taking ATIEs. In our construction the only obstruction towards obtaining an ATIE is the use of the simulation theorem of effectively closed $\mathbb{Z}$-subshifts as projective subactions of sofic $\mathbb{Z}^{2}$-subshifts. This theorem in its current state does not yield an almost everywhere $1-1$ extension. The rest of the proof can be adapted to obtain an ATIE, for instance, the substitutive layers can be coupled in a single substitution to avoid the degree of freedom when either $a$ or $b$ are zero. Furthermore, the substitutive layers and the Toeplitz structure can be factorized in the isometric action as they are invariant under the $H$-subaction. Therefore, the maps $\Upsilon\circ\pi$ do not pose obstructions to obtaining an ATIE. Everything that remains is the factor $\phi:\widehat{\texttt{Final}}(X,f)\twoheadrightarrow\texttt{Final}(X,f)$. Here the substitutive layers don’t present a problem as they come from a primitive substitution with unique derivation and thus Mozes’s theorem [18] gives the almost $1-1$ SFT extension. The only thing that remains is the aforementioned almost $1-1$ SFT extension for $\texttt{Top}(X,f)^{H}$ and $\texttt{Top}(X,f)^{V}$ that could be obtained by refining that simulation theorem. ## References [1] N. Alon, J. Grytczuk, M. Haluszczak, and O. Riordan. Nonrepetitive colorings of graphs. _Random Structures & Algorithms_, 21(3-4):336–346, 2002. * [2] N. Aubrun, S. Barbieri, and M Sablik. A notion of effectiveness for subshifts on finitely generated groups. arXiv:1412.2582. * [3] N. Aubrun and M. Sablik. Simulation of effective subshifts by two-dimensional subshifts of finite type. _Acta Applicandae Mathematicae_, 126:35–63, 2013. * [4] S. Aubrun, N. Barbieri and S. Thomassé. Realization of aperiodic subshifts and densities in groups. _arXiv:1507.03369_, 2015. * [5] R. Berger. _The Undecidability of the Domino Problem_. American Mathematical Society, 1966. * [6] M Boyle and D Lind. Expansive subdynamics. _Transactions of the American Mathematical Society_, 349(1):55–102, 1997. * [7] D. B. Cohen. The large scale geometry of strongly aperiodic subshifts of finite type. _ArXiv e-prints_, December 2014. * [8] D. B. Cohen and C. Goodman-Strauss. Strongly aperiodic subshifts on surface groups. _ArXiv e-prints_, October 2015. * [9] B. Durand, A. Romashchenko, and A. Shen. Effective closed subshifts in 1d can be implemented in 2d. In _Fields of Logic and Computation_, volume 6300, pages 208–226. Springer Berlin / Heidelberg, 2010. * [10] M Einsiedler, D Lind, R Miles, and T Ward. Expansive subdynamics for algebraic $\mathbb{Z}^{d}$-actions. _Ergodic theory and dynamical systems_, 21(06):1695–1729, 2001. * [11] W. Hanf. Nonrecursive tilings of the plane. i. _The Journal of Symbolic Logic_, 39(2):283–285, 1974. * [12] M. Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. _Inventiones Mathematicae_, 176(1):131–167, 2009. * [13] M. Hochman and T. Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. _Annals of Mathematics_, 171(3):2011–2038, 2010. * [14] E. Jeandel. Aperiodic Subshifts of Finite Type on Groups. _ArXiv e-prints_, January 2015. * [15] E. Jeandel and M. Rao. An aperiodic set of 11 Wang tiles. _ArXiv e-prints_, June 2015. * [16] J. Kari. A small aperiodic set of wang tiles. _Discrete Mathematics_, 160:259 – 264, 1996. * [17] D. Lind and B. Marcus. _An Introduction to Symbolic Dynamics and Coding_. Cambridge University Press, 1995. * [18] S. Mozes. Tilings, substitution systems and dynamical systems generated by them. _Journal d’Analyse Mathématique_, 53(1):139–186, 1989. * [19] D. Myers. Nonrecursive tilings of the plane. ii. _The Journal of Symbolic Logic_, 39(2):286–294, 1974. * [20] R. Robinson. Undecidability and nonperiodicity for tilings of the plane. _Inventiones Mathematicae_, 12:177–209, 1971.
1604.08541	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 43590, "num_imgs": 10, "llama3_tokens_count": 13331 }	[ "content_image/1604.08541/x1.png", "content_image/1604.08541/x2.png", "content_image/1604.08541/x3.png", "content_image/1604.08541/x4.png", "content_image/1604.08541/x5.png", "content_image/1604.08541/x6.png", "content_image/1604.08541/x7.png", "content_image/1604.08541/x8.png", "content_image/1604.08541/x9.png", "content_image/1604.08541/x10.png" ]	# Calculation of differential cross section for dielectronic recombination with two-electron uranium K. N. Lyashchenko laywer92@mail.ru Department of Physics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia ITMO University, Kronverkskii ave. 49, 197101, Petergof, St. Petersburg, Russia O. Yu. Andreev Department of Physics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia February 27, 2024 ###### Abstract Calculation of the differential cross section for the dielectronic recombination with two-electron uranium within the framework of QED is presented. The polarization of the emitted photon is investigated. The contributions of the Breit interaction and the interference of the photon multipoles are studied. electron recombination, dielectronic recombination, QED, highly charged ions, three-electron ions pacs: 31.10.+z, 31.15.ac, 31.30.J-, 34.70.+e ## I Introduction Dielectronic recombination with few-electron (in particular, with two-electron) highly charged ions is of interest both from experimental and theoretical points of view. Few-electron highly charged ions are relatively simple systems which allow precise theoretical description within the framework of quantum electrodynamics (QED). Dielectronic recombination is a resonant process where electron recombination with an ion is performed via formation of an autoionizing (doubly excited) state of the ion. Interelectron interaction plays a crucial role in formation of the doubly excited states. Accordingly, dielctronic recombination presents a tool for detailed investigation of dynamic electron correlations, in particular, for investigation of the Breit interaction. Dielectronic recombination with two-electron ion of Fe was studied experimentally and theoretically in work Beiersdorfer et al. (1992). Measurements of the radiative and Auger decay rates for K-shell vacancies in highly charged Fe ions were presented in Steinbrügge et al. (2015). The linear polarization measurements of x-rays emitted due to dielectronic recombination into highly charged Kr ions were recently presented in Shah et al. (2015). Measurements of the dielectronic recombination resonant strengths of highly charged ions (in particular, two-electron) Xe ions were performed in work Yao et al. (2010). Experimental investigation of dielectronic recombination with two- and three-electron ions of Pr, Ho and Au is reported in Hu et al. (2014). In particular, dielectronic recombination strengths are measured and calculated. Experimental study of the dielectronic recombination with three-electron U ions is presented in Trotsenko et al. (2015). Calculation of the transition rates for the doubly excited states of a three-electron ion of uranium was reported in Natarajan (2015). The influence of the Breit and QED effects on the radiative transition parameters is analyzed in detail. Dielectronic recombination with two-electron uranium ion was also studied theoretically in Chen et al. (2015) In particular, the linear polarization and angular distribution of the x-ray photoemission was studied, contribution of the magnetic quadrupoles was investigated. Dielectronic recombination with one-electron uranium was studied experimentally and theoretically. The experimental study of the full cross section of was performed in Bernhardt et al. (2011). The corresponding calculations of the dielectronic recombination are presented in Karasiov et al. (1992); Zakowicz et al. (2004); Yu. Andreev et al. (12); Bernhardt et al. (2011); Lyashchenko and Andreev (2015). The electron recombination with highly charged ions presents a tool for investigation of the Breit interaction. The recent studies showed that the Breit interaction may give important and even dominant contribution to the cross section of the dielectronic recombination with few-electron highly charged ions Lyashchenko and Andreev (2015); Nakamura et al. (2008); Fritzsche et al. (2009); Matula et al. (2011); Bernhardt et al. (2011); Hu et al. (2012); Shah et al. (2014). The considered process of electron recombination with a two-electron uranium can be described as \[e^{-}+U^{90+}(1s1s) \to U^{89+}(r)+\gamma\,,\] (1) where $r$ denotes a singly excited state: $(1s1s2s)$ or $(1s1s2p)$. In the initial state of the system the two-electron ion is assumed to be in its ground state. The first emitted photon ($\gamma$) can be registered in the experiment. If the energy of the initial state is close to the energy of a doubly excited state, the cross section shows resonances. The resonances are explained by the dielectronic recombination which can be written as \[e^{-}+U^{90+}(1s1s) \to U^{89+}(d)\to U^{89+}(r)+\gamma\to\ldots\,,\] (2) where $d$ is a doubly excited state: $(1s2s2s)$, $(1s2s2p)$ or $(1s2p2p)$. To study the cross section of the dielectronic recombination with highly charged ions, the QED calculations of the radiative transitions amplitudes between three-electron configurations are necessary. Such calculations can be performed with employment of various methods Johnson et al. (1995); Shabaev (2002); Indelicato et al. (2004); Yu. Andreev et al. (2008, 23, 12). In the present paper the line-profile approach was used Yu. Andreev et al. (2008). We present calculations of the full and differential cross section for the dielectronic recombination with two-electron uranium within the framework of QED. The polarization of the emitted photon is investigated. The contributions of the Breit interaction and the interference of the photon multipoles are studied. At the end, we will estimate the contribution of the three electron recombination for the considered collision system. ## II Method of calculation The present calculations are based on the QED approach already applied for calculation of the cross section of electron recombination with one-electron ions Yu. Andreev et al. (12); Lyashchenko and Andreev (2015) – the line-profile approach (LPA) Yu. Andreev et al. (2008). To describe the electron recombination with two-electron ions, the line-profile approach was generalized to three-electron system. The incident electron is considered as an electron with certain momentum ${\bm{p}}$ and polarization $\mu$ and is described by wave function $\psi_{{\bm{p}},\mu}$. Wave function $\psi_{njlm}$ describes a bound electron with the corresponding quantum numbers ($n$,$j$,$l$ and $m$ are the principle quantum number, total angular momentum, parity and projection of $j$, respectively). Employing expansion of the wave function $\psi_{{\bm{p}},\mu}$ in series over the wave functions of electron with certain energy ($\varepsilon$), total angular momentum ($j$), its projection ($m$), and parity ($l$) the wave function of the incident electron can be written as Akhiezer and Berestetskii (1965) \[\psi_{{\bm{p}},\mu}({\bm{r}}) = \int d\varepsilon\sum_{jlm}a_{{{\bm{p}}}\mu,\varepsilon jlm}\psi_ {\varepsilon jlm}({\bm{r}})\,,\] (3) \[a_{{{\bm{p}}}\mu,\varepsilon jlm} = \frac{(2\pi)^{3/2}}{\sqrt{p\epsilon}}i^{l}e^{i\phi_{jl}}(\Omega^{ +}_{jlm}({\bm{p}}),\upsilon_{\mu}({\bm{p}}))\delta(\epsilon-\varepsilon)\,,\] (4) where $\Omega_{jlm}({\bm{p}})$ is the spherical spinor and $\upsilon_{\mu}({\bm{p}})$ is the spinor with certain projection on the electron momentum (${\bm{p}}$), the phase $\phi_{jl}$ is the Coulomb phase shift. The relativistic units are used throughout unless otherwise stated. In the line-profile approach the initial, final and intermediate three-electron states are expanded is series of three-electron functions in the $j$-$j$ coupling scheme. The following algorithm is employed for composition of the basis set of three-electron wave functions in the $j$-$j$ coupling scheme. Firstly, we select two electrons with closest energies and a compose two-electron wave function in the $j$-$j$ coupling scheme \[\Psi^{(0)}_{j_{12}m_{12}n_{1}j_{1}l_{1}n_{2}j_{2}l_{2}} = N\sum\limits_{m_{1}m_{2}}C^{j_{1}j_{2}}_{j_{12}m_{12}}(m_{1}m_{2 })\det\{\psi_{n_{1}j_{1}l_{1}m_{1}},\psi_{n_{2}j_{2}l_{2}m_{2}}\}\,,\] (5) where $N$ – normalization constant (equal to $1/\sqrt{2}$ for nonequivalent electrons and $1/2$ for equivalent electrons), $C^{j_{1}j_{2}}_{j_{12}m_{12}}(m_{1}m_{2})$ – Clebsch-Gordan coefficients. The electrons are represented by the quantum numbers $n_{i}$,$j_{i}$,$l_{i}$ and $m_{i}$ ($n_{i}$ denotes the principle quantum number for bound electron and the energy for electron from the continuum part of the spectrum, $j_{i}$ is the total angular momentum, $l_{i}$ is the parity, $m_{i}$ is the projection of the total angular momentum, $i=1,2$ denotes the first and second electrons, respectively). Then, we compose three-electron wave functions with certain angular momentum ($J$) and its projection $M$ \[\Psi^{(0)}_{JMj_{12}n_{1}j_{1}l_{1}n_{2}j_{2}l_{2}n_{3}j_{3}l_{3}} = N\sum\limits_{m_{1}m_{2}m_{12}m_{3}}C^{j_{12}j_{3}}_{JM}(m_{12}m _{3})C^{j_{1}j_{2}}_{j_{12}m_{12}}(m_{1}m_{2})\det\{\psi_{n_{1}j_{1}l_{1}m_{1} },\psi_{n_{2}j_{2}l_{2}m_{2}},\psi_{n_{3}j_{3}l_{3}m_{3}}\}\,,\] (6) where $N$ – normalization constant (equal to $1/\sqrt{3!\,2}$ for tree-electron states with two equivalent electrons and to $1/\sqrt{3!}$ for states without equivalent electrons). If three-electron configurations contain three equivalent electrons, the coefficients of fractional parentage ($\langle j_{1}j_{2}[j_{12}]j_{3}J\}j_{1}j_{2}j_{3}\gamma J\rangle$) are to be calculated for composing the three-electron wave functions in $j$-$j$ coupling scheme Sobelman (1963); Veselov and Labzowsky (1986) \[\Psi^{(0)}_{JM\gamma n_{1}j_{1}l_{1}n_{2}j_{2}l_{2}n_{3}j_{3}l_{3}} = \sum\limits_{j_{12}}\langle j_{1}j_{2}[j_{12}]j_{3}J\}j_{1}j_{2}j _{3}\gamma J\rangle\Psi^{(0)}_{JMj_{12}n_{1}j_{1}l_{1}n_{2}j_{2}l_{2}n_{3}j_{3 }l_{3}}\,.\] (7) Here, the quantum number $\gamma$ denotes repeating terms of the electronic structure (if necessary). The initial state of the system (a two-electron ion in its ground $(1s1s)$ state and an incident electron) can be described by the wave function \[\Psi^{{\rm ini}} = \frac{1}{\sqrt{3!}}\det\{\psi_{n_{1}j_{1}l_{1}m_{1}},\psi_{n_{2}j _{2}l_{2}m_{2}},\psi_{{\bm{p}},\mu}\}\,.\] (8) The wave function $\psi_{{\bm{p}},\mu}$ describes the incident electron with momentum ${\bm{p}}$ and polarization $\mu$, the wave function $\psi_{njlm}$ describes a bound electron with the corresponding quantum numbers ($n$,$j$,$l$ and $m$). Employing expansion Eq. (3) the wave function of the initial state can be presented as a linear combination of the following determinants \[\Psi^{(0)}_{n_{1}j_{1}l_{1}m_{1}n_{2}j_{2}l_{2}m_{2}\varepsilon j _{3}l_{3}m_{3}} = \frac{1}{\sqrt{3!}}\det\{\psi_{n_{1}j_{1}l_{1}m_{1}},\psi_{n_{2}j _{2}l_{2}m_{2}},\psi_{\varepsilon j_{3}l_{3}m_{3}}\}\,.\] (9) Accordingly, the wave function of the initial state can be expanded into series of three-electron functions with certain total momentum (in the $j$-$j$ coupling scheme) \[\Psi^{{\rm ini}} = \sum\limits_{JMj_{12}n_{1}j_{1}l_{1}n_{2}j_{2}l_{2}j_{3}l_{3}} \int d\varepsilon\,\langle\Psi^{(0)}_{JMj_{12}n_{1}j_{1}l_{1}n_{2}j_{2}l_{2} \varepsilon j_{3}l_{3}}\,\|\,\Psi^{{\rm ini}}\rangle\,\Psi^{(0)}_{JMj_{12}n_{1} j_{1}l_{1}n_{2}j_{2}l_{2}\varepsilon j_{3}l_{3}}\,.\] (10) The final state ($(1s1s2s)$, $(1s1s2p)$ tree-electron states) can be written as a tree-electron configuration in the $j$-$j$ coupling scheme \[\Psi^{{\rm fin}} = \Psi^{(0)}_{JMj_{12}n_{1}j_{1}l_{1}n_{2}j_{2}l_{2}n_{3}j_{3}l_{3} }\,.\] (11) To describe highly charged ion within the framework of QED, the line-profile approach (LPA) was employed Yu. Andreev et al. (2008, 12). The interaction with the quantized electromagnetic and electron-positron fields leads to various correction to the amplitude: interelectron interaction correction, electron self-energy and vacuum polarization corrections. Within the line-profile approach the system is considered to be enclosed into a sphere of a large radius $R\to\infty$. Then, the incident electron can be described by a wave function normalized to unit which corresponds to a artificial bound electron state ($e_{R}$). For calculation of the amplitudes of the transitions between bound states the standard QED perturbation theory for the quasidegenerate states can be applied Shabaev (2002); Lindgren et al. (2004, 2014); Yu. Andreev et al. (2008). Within the LPA we introduce the set of three-electron configurations ($g$) which includes all the three-electron configurations composed by $1s,2s,2p,3s,3p,3d$ electrons and the electron $e_{R}$ describing the incident electron. There is also introduced the matrix $V$ which is defined by the one and two-photon exchange, electron self-energy and vacuum polarization matrix elements. Matrix $V=V^{(0)}+\Delta V$ is considered as a block matrix \[V = \left[\begin{array}[]{cc}V_{11}&V_{12}\\ V_{21}&V_{22}\\ \end{array}\right]\,=\,\left[\begin{array}[]{cc}V^{(0)}_{11}+\Delta V_{11}& \Delta V_{12}\\ \Delta V_{21}&V^{(0)}_{22}+\Delta V_{22}\\ \end{array}\right]\,.\] (12) Matrix $V_{11}$ is defined on set $g$, which contains configurations mixing with the reference state $n_{g}$$\in$$g$. Matrix $V^{(0)}$ is a diagonal matrix: sum of the one-electron Dirac energies. Matrix $\Delta V_{11}$ is not a diagonal matrix, but it contains a small parameter of the QED perturbation theory. Matrix $V_{11}$ is a finite-dimensional matrix and can be diagonalized numerically. Then, the standard perturbation theory can be applied for the diagonalization of the matrix $V$. The amplitude of the electron recombination is written as a matrix element of the photon emission operator ($\Xi^{(0)}$) with the bra and ket vectors given by the eigenfunctions of the matrix $V$: $\Psi^{\rm fin}$ and $\Psi^{\rm ini}$ corresponding to the final and initial states of the system, respectively, \[U_{if} = \langle\Psi^{\rm fin}\|\Xi\|\Psi^{\rm ini}\rangle\,.\] (13) The operator $\Xi$ is derived within the QED perturbation theory order by order Yu. Andreev et al. (2008, 12). The operator $\Xi$ can be represented by its matrix elements, in the zeroth order of the perturbation theory it reads \[\Xi^{(0)}_{u_{1}u_{2}u_{3}d_{1}d_{2}d_{3}} = eA^{(k,\lambda)}_{u_{1}d_{1}}\delta_{u_{2}d_{2}}\delta_{u_{3}d_ {3}}+\delta_{u_{1}d_{1}}eA^{(k,\lambda)}_{u_{2}d_{2}}\delta_{u_{3}d_{3}}+ \delta_{u_{1}d_{1}}\delta_{u_{2}d_{2}}eA^{(k,\lambda)}_{u_{3}d_{3}}\,,\] (14) where $u_{1}$, $u_{2}$, $u_{3}$, $d_{1}$, $d_{2}$, $d_{3}$ are one-electron states with certain total angular momentum and parity, the one-electron matrix elements $A^{(k,\lambda)}_{ud}$ are defined as \[A^{(k,\lambda)}_{ud} = \int d^{3}{{\bm{r}}}\,\overline{\psi}_{u}({{\bm{r}}})\gamma^{\nu} A^{(k,\lambda)}_{\nu}({\bm{r}})\psi_{d}({\bm{r}})\,,\] (15) where $\gamma^{\nu}$ are Dirac gamma matrices. $A^{(k,\lambda)}_{\nu}=(A_{0}^{(k,\lambda)},{\bm{A}}^{(k,\lambda)})$ is the emitted photon wave function. We use a gauge in which $A_{0}^{(k,\lambda)}=0$. ${\bm{A}}^{(k,\lambda)}$ reads as \[{{\bm{A}}}^{(k,\lambda)}({{\bm{r}}}) = \sqrt{\frac{2\pi}{\omega}}e^{i{\bm{k}}{\bm{r}}}{\bm{e}}^{(\lambda )}\,,\] (16) $\omega$ and ${\bm{k}}$ are frequency and momentum of the photon, respectively. Employing the multipole expansion we can write Labzowsky (1996) \[{{\bm{A}}}^{(k,\lambda)} = \sqrt{\frac{2\pi}{\omega}}\sum_{j_{0}l_{0}m_{0}}i^{l_{0}}g_{l_{0} }(\omega r)({\bm{e}}^{(\lambda)},{{\bm{Y}}}_{j_{0}l_{0}m_{0}}^{}({\bm{k}})){ \bm{Y}}_{j_{0}l_{0}m_{0}}({\bm{r}})\,,\] (17) where $g_{l_{0}}(x)=4\pi j_{l_{0}}(x)$ and $j_{l_{0}}(x)$ is the spherical Bessel function, ${\bm{Y}}_{j_{0}l_{0}m_{0}}$ – vector spherical harmonics, ${\bm{e}}^{(\lambda)}$ – vector of photon polarization. We consider the linear polarization of the photon \[{\bm{e}}_{1} = \frac{[{\bm{p}}\times{\bm{k}}]}{\|[{\bm{p}}\times{\bm{k}}]\|}\,, \qquad{\bm{e}}_{2}\,=\,\frac{[{\bm{e}}_{1}\times{\bm{k}}]}{\|[{\bm{e}}_{1} \times{\bm{k}}]\|}\] (18) and the circular polarization of the photon \[{\bm{e}}_{+} = \frac{1}{\sqrt{2}}({\bm{e}}_{1}+i{\bm{e}}_{2})\,,\qquad{\bm{e}}_{ -}\,=\,\frac{1}{\sqrt{2}}({\bm{e}}_{1}-i{\bm{e}}_{2})\,.\] (19) The $z$ axis is defined by the incident electron momentum ${\bm{p}}$. Accordingly, the vectors ${\bm{p}}$, ${\bm{k}}$ and the polarization vectors look like \[\frac{{\bm{p}}}{\|{\bm{p}}\|} = \left(\begin{array}[]{c}0\\ 0\\ 1\\ \end{array}\right)\,,\qquad\frac{{\bm{k}}}{\|{\bm{k}}\|}\,=\,\left(\begin{array} []{c}\sin\theta\cos\phi\\ \sin\theta\sin\phi\\ \cos\theta\\ \end{array}\right)\,,\] (20) \[{\bm{e}}_{1} = \left(\begin{array}[]{c}-\sin\phi\\ \cos\phi\\ 0\\ \end{array}\right)\,,\qquad{\bm{e}}_{2}\,=\,\left(\begin{array}[]{c}\cos\theta \cos\phi\\ \cos\theta\sin\phi\\ -\sin\theta\\ \end{array}\right)\,,\] (21) respectively. In the electron-ion collision process has an axial symmetry and it not depend from angle $\phi$. The operator $\Xi$ in the first order of the perturbation theory gives a small contribution and is omitted in the present calculation. ## III Results We have studied the process of electron recombination with two-electron uranium being initially in its ground state. The process is considered in the rest frame of the uranium ion. We investigated regions of the incident electron energy where the role of dielectronic recombination is prominent. We restricted ourselves to the consideration of four low lying energy regions, in particular, the regions where the energy of the initial state ($(1s1s)$ plus incident electron $e$) is close to the energies of doubly excited states $(1s2s2s)$, $(1s2s2p)$, $(1s2p2p)$. Accordingly, we performed the calculations for the following resonance regions of the incident electron (kinetic) energy: [$63.03$,$63.075$] keV, [$63.075$,$63.45$] keV, [$67.25$,$68.00$] keV and [$71.95$,$72.15$] keV. The total cross section of electron recombination with two-electron uranium is presented as a function of the kinetic energy of the incident electron in Fig. 1. The left four graphs represent the exact QED calculation of the total cross section for the four resonance regions, respectively. The right ones represent the calculation of the total cross section with disregard of the Breit interelectron interaction (in the Feynman graphs representing to the one- and more photon exchange). The graphs reveal a large contribution of the Breit interaction to the cross section. We note that the relative contribution of the Breit interaction to the cross section for dielectronic recombination with two-electron uranium ions is much larger than that for dielectronic recombination with one-electron ions (see Bernhardt et al. (2011); Lyashchenko and Andreev (2015)). The importance of the Breit interaction is explained by large sensitivity of the widths of three-electron energy levels to the Breit interaction. Within the framework of the standard QED theory, the energy shift of energy levels (due to the interaction with quantized electromagnetic and electron-positron fields) is commonly written as $\Delta{E}=\mathop{\rm Re}\nolimits\{\Delta{E}\}-i\frac{\Gamma}{2}$Low (1952); Shabaev (2002); Lindgren et al. (2004); Yu. Andreev et al. (2008), where $\mathop{\rm Re}\nolimits\{\Delta{E}\}$ is a correction to the energy, $\Gamma$ defines the width of the energy level. For the one- and two-electron configurations the major contribution to the width of energy level is given by the electron self-energy Feynman graph. However, for three-electron configurations the contribution of the electron self-energy graph can be considerably canceled by the contribution of the Breit part of the one-photon exchange graph. For example, $(1s1s2s)$ configuration is the ground state of tree-electron ion, accordingly, contribution of the imaginary part of the electron self-energy graph is completely canceled by contribution of the Breit part of the one-photon exchange graph. It can be referred as a realization of the Pauli exclusion principle Labzowsky (1996). The energies and widths of the considered doubly excited states are very sensitive to the Breit interaction. In Table 1 we present data which show the role of the Breit interaction for the doubly excited states. Doubly excited states are specified in the first column. In the second and the third columns ( ’$V$ Coulomb’ and ’$V$ Coulomb+Breit’) presented are values of the diagonal matrix elements of matrix $V$ (see Eq. (12)); these data are given by summation of the diagonal matrix elements of the corresponding Feynman graphs (electron self-energy, vacuum polarization, one-photon exchange and part of the two-photon exchange graphs). The column ’$V$ Coulomb+Breit’ contains results of the exact QED calculation, the column ’$V$ Coulomb’ presents results of the calculation with disregard of the Breit part of photon exchange graphs. These data have no clear physical meaning; however, they demonstrate a strong cancellation of the imaginary parts of the electron self-energy and the Breit part of the photon exchange corrections for some of the configurations. The next three columns present results of calculation of the energies and widths of the doubly excited states performed within the exact QED approach (column ’Coulomb+Breit (with retardation)’), with disregard of retardation in the Breit interaction (column ’Coulomb+Breit (without retardation)’) and with complete neglect of the Breit interaction (column ’Coulomb’). The corresponding resonance kinetic energies of the incident electron are also given. These data demonstrate the importance of the Breit interaction for the energies and widths of the doubly excited states what explains the large contribution of the Breit interaction to the cross section for the dielectronic recombination with two-electron uranium ions which is seen in Fig. 1. The process of dielectronic recombination proceeds via formation of doubly excited states. Peaks of the cross section correspond to these doubly excited states. In order to study the individual contributions of the doubly excited states, we performed calculations of the cross section where only fixed doubly excited states were taken into account. The results are presented in Fig. 2. Plots in the left column show individual contribution of the doubly excited states. Plots in the right column show separate contribution of resonant channel (electron capture via formation of doubly excited states, i.e., dielectronic recombination) and nonresonant channel – radiative electron capture (REC). We note that partition of the electron recombination into resonant and nonresonant channels is ambiguous. Results of the full calculation of the cross section are given with a mark (Full) in Fig. 2. These plots also show interference between the dielectronic recombination and the radiative electron capture. Results of calculation of the differential cross section (in barn/str) \[\sigma^{\prime} \equiv d\sigma/d\Omega\] (22) as a function of the kinetic energy of incident electron are given in Fig. (3). The left plots present the exact QED calculation, the right plots present results of calculation with disregard of the Breit interaction. As a consequence of large contribution of the Breit interaction to the total cross section (see Fig. 1), the Breit interaction is also important for the differential cross section of electron capture by two-electron uranium ion. In the present calculation the multipole expansion of the emitted photon wave function was employed (see Eq. (17)). The multipoles up to $j_{0}=9$ were taken into account. Investigation of contribution of the higher multipoles of emitted photon is presented in Fig. 4. The upper plot presents the total cross section. The red curve in the upper plot corresponds to calculation of the total cross section within dipole approximation where only terms with $j_{0}=1$ are taken into account in the multipole expansion Eq. (17). The black curve in the upper plot corresponds to the full calculation ($j_{0}\leq 9$). It is seen that contribution of the higher multipoles to the total cross section is insignificant. The lower plot presents investigation of the differential cross section: there is a relative difference between differential cross section calculated in dipole approximation $\sigma^{\prime(j_{0}=1)}$ and the full calculation $\sigma^{\prime}$ (see Eq. (22)) \[\delta\sigma = \frac{\sigma^{\prime(j_{0}=1)}-\sigma^{\prime}}{\sigma^{\prime}}\,.\] (23) In spite of a small contribution of the higher multipoles to total cross section ($<5\%$), they play a significant role for the differential cross section. Our calculations show, that mainly due to the interference between $E1$, $M1$ and $E2$, $M2$ emitted photons, $\delta\sigma$ may reach up to $66\%$ in regions between the peaks. We have also studied the contribution of the various polarizations of the incident electron and emitted photon. Polarization of the initial state is defined by polarization of the incident electron (projection of the spin to the direction of momentum) which can be equal to $\mu=\pm 1/2$. Different polarizations of the incident electron give equal contributions to the cross section, while the summation over the photon polarizations is performed; however, they give different contributions if the polarization of the emitted photon is fixed. In Fig. 5 we present results of calculation of the differential cross section $\sigma^{\prime}_{-+}$, where the incident electron has polarization $\mu=-1/2$ and emitted photon has polarization ${\bm{e}}_{+}$ (see Eq. (19)). The result of the calculation of the differential cross section $\sigma^{\prime}_{--}$ (the polarization of incident electron is $\mu=-1/2$, and the photon polarization is ${\bm{e}}_{-}$) can be obtained from Fig. 5 by inversion of the polar axis. Differential cross sections with different polarizations are connected by the following condition \[\sigma^{\prime}_{\mu,-} = \sigma^{\prime}_{-\mu,+}\,.\] (24) To investigate the polarizations of emitted photon, we calculated the Stokes parameters. We have calculated the Stokes parameters for incident electrons with polarization $\mu=-1/2$; the results of the calculations are presented in Figs. 6-8. The Stokes parameters $P_{1}$, and $P_{2}$ for different linear polarizations of the photon are given in Figs. 6, 7 \[P_{1} = \frac{\sigma^{\prime}_{0^{\circ}}-\sigma^{\prime}_{90^{\circ}}}{ \sigma^{\prime}_{0^{\circ}}+\sigma^{\prime}_{90^{\circ}}}\,,\] (25) \[P_{2} = \frac{\sigma^{\prime}_{45^{\circ}}-\sigma^{\prime}_{135^{\circ}}} {\sigma^{\prime}_{45^{\circ}}+\sigma^{\prime}_{135^{\circ}}}\,,\] (26) where $\sigma^{\prime}_{0^{\circ}}$, $\sigma^{\prime}_{90^{\circ}}$ are the differential cross section for emission of photon with polarization vector laid or orthogonal to the (${\bm{p}},{\bm{k}}$) plane, respectively, and $\sigma^{\prime}_{45^{\circ}}$, $\sigma^{\prime}_{135^{\circ}}$ are the differential cross sections for emission of photon with polarization vector at $45^{\circ}$ and $135^{\circ}$ to the (${\bm{p}},{\bm{k}}$) plane, respectively. $P_{1}$ and $P_{2}$ equal to zero at angles $0^{\circ}$ and $180^{\circ}$ (see Eq. (21)). The Stokes parameter ($P_{3}$) describing the circular polarization Eq. (19) is presented in Fig. (8) \[P_{3} = \frac{\sigma^{\prime}_{+}-\sigma^{\prime}_{-}}{\sigma^{\prime}_{+ }+\sigma^{\prime}_{-}}\,,\] (27) where $\sigma^{\prime}_{\pm}\equiv d\sigma_{\pm}/d\Omega$ are the differential cross section for emission of the photon with the corresponding chirality. Figs. 6-8 show that Stokes parameters also very sensitive to Breit interaction. For nonpolarized incident electrons the corresponding polarizations of the emitted photon give equal contributions to the differential cross sections $\sigma^{\prime}_{45^{\circ}}$, $\sigma^{\prime}_{135^{\circ}}$ and $\sigma^{\prime}_{-}$, $\sigma^{\prime}_{+}$, respectively. Accordingly, the Stokes parameters $P_{2}$ and $P_{3}$ are equal to zero. The parameter $P_{1}$ is independent of the polarization of the incident electron and is the same for the polarized and nonpolarized incident electrons. For addition characteristics of angular distribution we present the differential cross section with photon emission angles $0^{\circ}$ ($\sigma^{\prime}(\theta=0^{\circ})$), $90^{\circ}$ ($\sigma^{\prime}(\theta=90^{\circ})$) in Fig. 9 and asymmetry parameter Shah et al. (2014) \[A = \frac{\sigma^{\prime}(\theta=90^{\circ})}{\sigma^{\prime}(\theta= 0^{\circ})}\] (28) in Fig. (10), respectively. The calculations show that the emission to $90^{\circ}$ dominates over emission to $0^{\circ}$, particularly in regions of the resonance energies. Far from the resonance regions the photon is emitted mainly to $90^{\circ}$ angle what results in corresponding growth of the parameter $A$. We would like to note that we have also investigated the contribution of tri-electronic recombination to the process of electron capture by two-electron uranium ion initially being in its ground state. We performed calculations of the cross section for regions of the incident electron energy where the contribution of the triply excited states $((2s2s)_{0}2p_{1/2})_{1/2}$ and $((2p_{1/2}2p_{1/2})_{0}2s)_{1/2}$ could be significant. It was found that the contributions of these states to the cross section is in $8$-$11$ orders smaller than the corresponding contribution of the (nonresonant) radiative electron capture (REC), and it offers no possibility to detect the tri-electronc recombination in experiment. However, we found that for the process of electron capture with two-electron uranium initially being in a single excited state (for example, $(1s2s)_{0}$), the contribution of the tri-electronic recombination is much larger than the corresponding contribution of the radiative electron capture, which, in principle, makes it possible to experimentally investigate the trielectronic recombination. We have presented QED calculations of the total and differential cross section for dielectronic recombination of nonpolarized and polarized electrons with two-electron uranium initially being in its ground state. We have also investigated contribution of the higher multipoles of the photon wave-function expansion and found that their contributions to the deferential cross section are significant. The polarization of the emitted photons and the photon emission asymmetry are investigated. It was found that contribution of the Breit interaction is very important for the total and differential cross section as well as for various polarization parameters. The contribution of the Breit interaction is very significant, which allows to perform a successful experimental investigation of the Breit interaction in the process of dielectronic recombination with two-electron uranium. \| V \| V \| \| \| ---\|---\|---\|---\|---\|--- \| Coulomb \| Coulomb+Breit \| Coulomb \| Coulomb+Breit \| Coulomb+Breit \| \| (with retardation) \| \| (without retardation) \| (with retardation) Three electron \| ΔE \| Γ \| ΔE \| Γ \| ΔE \| Γ \| ΔE \| Γ \| ΔE \| Γ state \| ϵ \| ϵ \| ϵ \| ϵ \| ϵ (1s(2s2s)0)1/2 \| -198.350 \| 0.000 \| -198.304 \| 0.000 \| -198.377 \| 0.004 \| -198.327 \| 0.004 \| -198.332 \| 0.002 \| 63.366 \| 63.078 \| 63.340 \| 63.058 \| 63.054 (1s(2s2p1/2)1)3/2 \| -198.282 \| 0.031 \| -198.281 \| 0.021 \| -198.291 \| 0.031 \| -198.281 \| 0.031 \| -198.289 \| 0.021 \| 63.433 \| 63.102 \| 63.426 \| 63.105 \| 63.097 (1s(2s2p1/2)0)1/2 \| -198.299 \| 0.032 \| -198.231 \| 0.016 \| -198.357 \| 0.032 \| -198.245 \| 0.032 \| -198.256 \| 0.011 \| 63.416 \| 63.151 \| 63.360 \| 63.140 \| 63.129 (1s(2s2p1/2)1)1/2 \| -198.174 \| 0.032 \| -198.009 \| 0.006 \| -198.131 \| 0.032 \| -198.000 \| 0.032 \| -198.002 \| 0.012 \| 63.541 \| 63.373 \| 63.585 \| 63.385 \| 63.384 (1s(2p1/22p1/2)0)1/2 \| -198.058 \| 0.063 \| -197.962 \| 0.032 \| -198.048 \| 0.059 \| -197.944 \| 0.060 \| -197.954 \| 0.031 \| 63.658 \| 63.421 \| 63.669 \| 63.441 \| 63.431 (1s(2s2p3/2)1)3/2 \| -193.832 \| 0.027 \| -193.795 \| 0.003 \| -193.769 \| 0.027 \| -193.679 \| 0.027 \| -193.901 \| 0.026 \| 67.883 \| 67.588 \| 67.948 \| 67.706 \| 67.485 (1s(2s2p3/2)1)1/2 \| -193.856 \| 0.026 \| -193.821 \| 0.034 \| -193.857 \| 0.026 \| -193.824 \| 0.026 \| -193.822 \| 0.034 \| 67.859 \| 67.561 \| 67.860 \| 67.561 \| 67.563 (1s(2p1/22p3/2)2)5/2 \| -193.745 \| 0.058 \| -193.747 \| 0.022 \| -193.750 \| 0.058 \| -193.730 \| 0.058 \| -193.752 \| 0.022 \| 67.970 \| 67.635 \| 67.967 \| 67.655 \| 67.634 (1s(2p1/22p3/2)1)3/2 \| -193.788 \| 0.058 \| -193.717 \| 0.016 \| -193.801 \| 0.058 \| -193.718 \| 0.058 \| -193.732 \| 0.029 \| 67.928 \| 67.665 \| 67.915 \| 67.667 \| 67.654 (1s(2p1/22p3/2)1)1/2 \| -193.732 \| 0.057 \| -193.690 \| 0.056 \| -193.736 \| 0.057 \| -193.698 \| 0.057 \| -193.695 \| 0.056 \| 67.983 \| 67.692 \| 67.980 \| 67.687 \| 67.691 (1s(2s2p3/2)2)3/2 \| -193.860 \| 0.026 \| -193.788 \| 0.032 \| -193.927 \| 0.026 \| -193.897 \| 0.026 \| -193.686 \| 0.009 \| 67.855 \| 67.594 \| 67.790 \| 67.488 \| 67.700 (1s(2p1/22p3/2)2)3/2 \| -193.723 \| 0.058 \| -193.610 \| 0.041 \| -193.717 \| 0.058 \| -193.594 \| 0.058 \| -193.606 \| 0.029 \| 67.993 \| 67.772 \| 67.999 \| 67.791 \| 67.780 (1s(2p3/22p3/2)2)5/2 \| -189.426 \| 0.053 \| -189.415 \| 0.010 \| -189.426 \| 0.053 \| -189.385 \| 0.053 \| -189.414 \| 0.010 \| 72.289 \| 71.968 \| 72.291 \| 72.001 \| 71.971 (1s(2p3/22p3/2)2)3/2 \| -189.369 \| 0.052 \| -189.324 \| 0.052 \| -189.369 \| 0.052 \| -189.325 \| 0.052 \| -189.324 \| 0.052 \| 72.346 \| 72.058 \| 72.348 \| 72.060 \| 72.061 (1s(2p3/22p3/2)0)1/2 \| -189.334 \| 0.053 \| -189.288 \| 0.027 \| -189.332 \| 0.053 \| -189.269 \| 0.053 \| -189.286 \| 0.027 \| 72.381 \| 72.094 \| 72.385 \| 72.117 \| 72.100 \| \| \| \| \| \| \| \| \| \| Table 1: Comparison of the energies (E−3mc2=ΔE−iΓ2, in keV) of three electronic states and corresponding resonance kinetic energies (ϵ) of the impact electron (on the second line for each state). Second and third columns show the diagonal element of the matrix V (see Eq. (12)). Columns 4-6 show eigenvalues of the matrix V with different electron-electron interaction models. ###### Acknowledgements. The authors are indebted to senior lecture V.V. Bankevich for reading and improving the manuscript. The authors acknowledge the support by RFBR grants 14-02-20188 and the support by St. Petersburg State University through a research grant 11.38.227.2014. The work of K.N.L. was supported by RFBR grant 16-32-00620 and G-RISC P-2016a-8. ## References * Beiersdorfer et al. (1992) P. Beiersdorfer, T. W. Phillips, K. L. Wong, R. E. Marrs, and D. A. Vogel, Phys. Rev. A 46, 3812 (1992). * Steinbrügge et al. (2015) R. Steinbrügge, S. Bernitt, S. W. Epp, J. K. Rudolph, C. Beilmann, H. Bekker, S. Eberle, A. Müller, O. O. Versolato, H.-C. Wille, et al., Phys. Rev. A 91, 032502 (2015). * Shah et al. (2015) C. Shah, H. Jörg, S. Bernitt, S. Dobrodey, R. Steinbrügge, C. Beilmann, P. Amaro, Z. Hu, S. Weber, S. Fritzsche, et al., Phys. Rev. A 92, 042702 (2015). * Yao et al. (2010) K. Yao, Z. Geng, J. Xiao, Y. Yang, C. Chen, Y. Fu, D. Lu, R. Hutton, and Y. Zou, Phys. Rev. A 81, 022714 (2010). * Hu et al. (2014) Z. Hu, Y. Li, X. Han, D. Kato, X. Tong, H. Watanabe, and N. Nakamura, Phys. Rev. A 90, 062702 (2014). * Trotsenko et al. (2015) S. Trotsenko, A. Gumberidze, Y. Gao, C. Kozhuharov, S. Fritzsche, H. F. Beyer, S. Hagmann, P.-M. Hillenbrand, N. Petridis, U. Spillmann, et al., Phys. Scr. T166, 014024 (2015). * Natarajan (2015) L. Natarajan, Phys. Rev. A 92, 012507 (2015). * Chen et al. (2015) Z. B. Chen, J. L. Zeng, H. W. Hu, and C. Z. Dong, Journal of Physics B: Atomic, Molecular and Optical Physics 48, 144005 (2015). * Bernhardt et al. (2011) D. Bernhardt, C. Brandau, Z. Harman, C. Kozhuharov, A. Müller, W. Scheid, S. Schippers, E. W. Schmidt, D. Yu, A. N. Artemyev, et al., Phys. Rev. A 83, 020701 (2011). * Karasiov et al. (1992) V. V. Karasiov, L. N. Labzowsky, A. V. Nefiodov, and V. M. Shabaev, Phys. Lett. A 161, 453 (1992). * Zakowicz et al. (2004) S. Zakowicz, W. Scheid, and N. Grün, J. Phys. B 37, 131 (2004). * Yu. Andreev et al. (2009a) O. Yu. Andreev, L. N. Labzowsky, and A. V. Prigorovsky, Phys. Rev. A 80, 042514 (2009a). * Lyashchenko and Andreev (2015) K. N. Lyashchenko and O. Y. Andreev, Phys. Rev. A 91, 012511 (2015). * Nakamura et al. (2008) N. Nakamura, A. P. Kavanagh, H. Watanabe, H. A. Sakaue, Y. Li, D. Kato, F. J. Currell, and S. Ohtani, Phys. Rev. Lett. 100, 073203 (2008). * Fritzsche et al. (2009) S. Fritzsche, A. Surzhykov, and T. Stöhlker, Phys. Rev. Lett. 103, 113001 (2009). * Matula et al. (2011) O. Matula, S. Fritzsche, F. J. Currell, and A. Surzhykov, Phys. Rev. A 84, 052723 (2011). * Hu et al. (2012) Z. Hu, X. Han, Y. Li, D. Kato, X. Tong, and N. Nakamura, Phys. Rev. Lett. 108, 073002 (2012). * Shah et al. (2014) C. Shah, P. Amaro, R. Steinbrügge, S. Bernitt, Z. Harman, S. Fritzsche, A. Surzhykov, J. R. C. López-Urrutia, and S. Tashenov, in _17th International Conference on the Physics of Highly Charged Ions (HCI 2014). Book of Abstracts._ (2014), p. 174. * Johnson et al. (1995) W. R. Johnson, D. R. Plante, and J. Sapirstein, Adv. At. Mol. Opt. Phys. 35, 255 (1995). * Shabaev (2002) V. M. Shabaev, Phys. Rep. 356, 119 (2002). * Indelicato et al. (2004) P. Indelicato, V. M. Shabaev, and A. V. Volotka, Phys. Rev. A 69, 062506 (2004). * Yu. Andreev et al. (2008) O. Yu. Andreev, L. N. Labzowsky, G. Plunien, and D. A. Solovyev, Phys. Rep. 455, 135 (2008). * Yu. Andreev et al. (2009b) O. Yu. Andreev, L. N. Labzowsky, and G. Plunien, Phys. Rev. A 79, 032515 (2009b). * Akhiezer and Berestetskii (1965) A. I. Akhiezer and V. B. Berestetskii, _Quantum Electrodynamics_ (Wiley Interscience, New York, 1965). * Sobelman (1963) I. I. Sobelman, _Vvedenie v teoriyu atomnyh spektrov [Introduction to the theory of atomic spectra.] (in Russian)_ (Fiz. Mat. Lit., Moscow, 1963). * Veselov and Labzowsky (1986) M. G. Veselov and L. N. Labzowsky, _Teorija atoma. Strojenie elektronnykh obolochek [Theory of atoms. The structure of the electron shells.] (in Russian)_ (Nauka, Moscow, 1986). * Lindgren et al. (2004) I. Lindgren, S. Salomonson, and B. Åsén, Phys. Rep. 389, 161 (2004). * Lindgren et al. (2014) I. Lindgren, S. Salomonson, and J. Holmberg, Phys. Rev. A 89, 062504 (2014). * Labzowsky (1996) L. N. Labzowsky, _Teoriya atoma. Kvantovaya elektrodinamika elektronnyh obolochek i processy izlucheniya [Theory of atoms. Quantum electrodynamics of the electron shells and the processes of radiation] (in Russian)_ (Nauka, Moscow, 1996). * Low (1952) F. Low, Phys. Rev. 88, 53 (1952). <figure><img src="content_image/1604.08541/x1.png"><figcaption>Figure 1: The total cross section of the dielectronic recombination with two-electron uranium as a function of the kinetic energy of the incident electronis presented (in kbarn). The left column correspond to full QED calculation(Coulomb + Breit with retardation) and the right column denote the calculationwith disregard of the Breit interaction. Red dashed vertical lines indicatethe positions of the resonances with the doubly excited states.</figcaption></figure> <figure><img src="content_image/1604.08541/x2.png"><figcaption>Figure 2: Interference figures. The left column corresponds to theinterference of the individual peaks with each other. Colorful dashed curvesrepresent the total cross section for the individual peaks (without REC). Thered solid curves represents the total cross section for dielectronicrecombination (without REC). Vertical lines indicate the positions of theresonances with the doubly excited states. The right column corresponds to theinterference of DR (red dashed curves) and REC (blue dots lines). Black solidcurves denotes full calculation.</figcaption></figure> <figure><img src="content_image/1604.08541/x3.png"><figcaption>Figure 3: The differential cross section (dσ/dΩ, in barn/str) of dielectronicrecombination with two-electron ions of uranium are presented. The graphs inthe left column correspond to the full QED calculation (Coulomb + Breit withretardation) of the differential cross section, the graphs in the right columnpresent the calculation with disregard of the Breit interaction.</figcaption></figure> <figure><img src="content_image/1604.08541/x4.png"><figcaption>Figure 4: The contribution of the higher multipoles to the full anddifferential cross section is presented. The top graph correspond to fullcross section (in kbarn). Red dashed curve represent calculation where justj0=1 (see Eq. (17)) taken into account. Black solid curve denote calculationwith 0<j0<10. The bottom graph represents the relative contribution of thehigher multipoles δσ (see Eq. (23)).</figcaption></figure> <figure><img src="content_image/1604.08541/x5.png"><figcaption>Figure 5: The differential cross section (dσ/dΩ, in barn/str) of dielectronicrecombination with two-electron uranium for polarized incident electron(μ=−1/2) and the photon emission with circular polarization e+ (see Eq. (19))are presented.</figcaption></figure> <figure><img src="content_image/1604.08541/x6.png"><figcaption>Figure 6: The Stokes parameter describing the linear polarization P1 (see Eq.(25)) is presented. The graphs in the left column correspond to the full QEDcalculation (Coulomb + Breit with retardation) of the differential crosssection, the graphs in the right column present the calculation with disregardof the Breit interaction.</figcaption></figure> <figure><img src="content_image/1604.08541/x7.png"><figcaption>Figure 7: The Stokes parameter describing the linear polarization P2 (see Eq.(26)) is presented. The graphs in the left column correspond to the full QEDcalculation (Coulomb + Breit with retardation) of the differential crosssection, the graphs in the right column present the calculation with disregardof the Breit interaction.</figcaption></figure> <figure><img src="content_image/1604.08541/x8.png"><figcaption>Figure 8: The Stokes parameter describing the circular polarization P3 (seeEq. (27)) is presented. The graphs in the left column correspond to the fullQED calculation (Coulomb + Breit with retardation) of the differential crosssection, the graphs in the right column present the calculation with disregardof the Breit interaction.</figcaption></figure> <figure><img src="content_image/1604.08541/x9.png"><figcaption>Figure 9: The comparison of the differential cross section for photon emissionat 0∘ angle (σ′(θ=0∘), black curve, in barn/str) and the differential crosssection for photon emission at 90∘ angle (σ′(θ=90∘), red curve, in barn/str).The graphs in the left column correspond to the full QED calculation (Coulomb+ Breit with retardation) of the differential cross section, the graphs in theright column present the calculation with disregard of the Breit interaction.</figcaption></figure> <figure><img src="content_image/1604.08541/x10.png"><figcaption>Figure 10: The asymmetry parameter A (see Eq. (28)) is presented.The graphs inthe left column correspond to the full QED calculation (Coulomb + Breit withretardation) of the differential cross section, the graphs in the right columnpresent the calculation with disregard of the Breit interaction.</figcaption></figure>
1511.05485	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 37959, "num_imgs": 7, "llama3_tokens_count": 11178 }	[ "content_image/1511.05485/x1.png", "content_image/1511.05485/x2.png", "content_image/1511.05485/x3.png", "content_image/1511.05485/x4.png", "content_image/1511.05485/x5.png", "content_image/1511.05485/x6.png", "content_image/1511.05485/x7.png" ]	# Hydrodynamic interactions of cilia on a spherical body Babak Nasouri Department of Mechanical Engineering, University of British Columbia, Vancouver, B.C., V6T 1Z4, Canada Gwynn J. Elfring Electronic mail: gelfring@mech.ubc.ca Department of Mechanical Engineering, University of British Columbia, Vancouver, B.C., V6T 1Z4, Canada ###### Abstract Microorganisms develop coordinated beating patterns on surfaces lined with cilia known as metachronal waves. For a chain of cilia attached to a flat ciliate, it has been shown that hydrodynamic interactions alone can lead the system to synchronize. However, several microorganisms possess a curve shaped ciliate body and so to understand the effect of this geometry on the formation of metachronal waves, we evaluate the hydrodynamic interactions of cilia near a large spherical body. Using a minimal model, we show that for a chain of cilia around the sphere, the natural periodicity in the geometry leads the system to synchronize. We also report an emergent wave-like behavior when an asymmetry is introduced to the system. ## 1 Introduction To propel themselves in a low-Reynolds-number regime Purcell (1977), many microorganisms use small whip-like extensions, called flagella (when they possess one or two) or cilia (when they possess many) Lodish _et al._ (2000); Brennen and Winet (1977). The motion of cilia is controlled by ATP-fuelled motor proteins which exert a driving force on the cilia by converting chemical energy in the cell Roberts _et al._ (2013). The cyclic motion of each cilium in a chain can form a coordinated pattern of beating, wherein each pair of neighboring cilia are orbiting with a constant, non-zero, phase difference Knight-Jones (1954). As a result of this synchrony, the tips of cilia form a moving wave, known as a metachronal wave Gueron _et al._ (1997). By forming metachronal waves, the microorganism minimizes the required energy for beating Gueron and Levit-Gurevich (1999), which enhances the efficiency of the motion Kim and Netz (2006). In addition to providing a means of locomotion, cilia in the human body filter air flow channels in the lung from the harmful inhaled material Shah _et al._ (2009), and also play a crucial role in breaking the left-right symmetry in human embryonic development Hirokawa, Okada, and Tanaka (2009). Several analytical studies have shown that hydrodynamic interactions alone can lead to synchronization (zero phase difference) or phase-locking (constant non-zero phase difference) for model systems of two flagella Vilfan and Jülicher (2006); Elfring and Lauga (2011); Uchida and Golestanian (2011) or many cilia Cosentino Lagomarsino, Jona, and Bassetti (2003); Niedermayer, Eckhardt, and Lenz (2008). Using minimal models, it was observed that certain conditions, be they elastic deformations of trajectory Niedermayer, Eckhardt, and Lenz (2008), or shape Elfring, Pak, and Lauga (2010), or a certain forcing profile Uchida and Golestanian (2011), may be required to reach such phase-locking or synchronization. Recent experimental studies have also confirmed the hydrodynamic synchronization of micro-scale oscillators in natural systems. Using high-speed video microscopy, it was shown that beating flagella on _Chlamydomonas reinhardtii_Goldstein, Polin, and Tuval (2009); Polin _et al._ (2009) and _Volvox carteri_Brumley _et al._ (2012, 2015), exhibit a synchronization due to hydrodynamic interactions. Similar synchrony was observed in model colloidal systems where two spheres were oscillating on linear Kotar _et al._ (2010); Bruot _et al._ (2011) or circular Box _et al._ (2015) trajectories and each sphere was driven by optical tweezers. In a ciliary array, the distribution of cilia as well as the details of the ciliate body affect the behavior of the dynamical system Golestanian, Yeomans, and Uchida (2011); Elgeti, Winkler, and Gompper (2015); Bruot and Cicuta (2015). Niedermayer _et al._ reported that introducing radial flexibility to the circular trajectory of two orbiting beads leads to synchronization, but that a non-periodic array of such beads cannot reach stable collective phase-locking Niedermayer, Eckhardt, and Lenz (2008). They also showed that marginally-stable metachronal waves are formed only when the cilia are distributed in a periodic fashion. More recently it was observed that an open-ended array of cilia can indeed form robust metachronal waves if the cilia beat perpendicular to the ciliate boundary Brumley _et al._ (2012). It has also been shown that the presence of a large body near an array of linearly oscillating beads is necessary for emergence of metachronal waves Wollin and Stark (2011). The bounding surface restricts the range of hydrodynamic interactions of the beads and leads the system to a collectively phase-locked state. The emerging picture from the literature is that the stability and existence of metachronal waves depends on the geometry of the cilia and ciliate body. Notably however, in many ciliates in nature the cilia are continuously distributed about a closed curved body such as on _paramecia_ or _volvox_, and this imposes a natural periodicity to the dynamical system and mediates the hydrodynamic interactions of the ciliary chains in a way that is yet to be understood. In this paper we investigate the effects of a large curved ciliate body on the hydrodynamic interactions of cilia in a viscous fluid. Following the work of Niedermayer _et al._ who studied interactions of cilia above a flat wall Niedermayer, Eckhardt, and Lenz (2008), we use the discrete-cilia model Blake (1971); Liron and Mochon (1976) where each cilium is replaced by a single sphere and assume that a constant tangential forcing is applied by the dynein motors. We first present an analysis of the interactions of two cilia and then build up our model of a chain of cilia around a large spherical body. We show that the radial flexibility in the trajectories can lead the system to synchronize similar to the case of cilia near a flat boundary Niedermayer, Eckhardt, and Lenz (2008). Furthermore, we show that with this model, the only stable fixed point for a chain of identical cilia is when all cilia are in phase (synchronized). Finally, we demonstrate an emergent wave-like behavior of the cilia in response to an imposed asymmetry in the beating rate of one cilium. ## 2 Motion of a single cilium We model the cilium as a single sphere of radius $\hat{a}$ undergoing a circular orbit, of radius $\hat{R}$, whose center is at distance $\hat{h}$ from the ciliate body as shown in Fig. 1. Dynein motors drive the motion of the cilium and, in a viscous fluid at small scales, this forcing is balanced entirely by the hydrodynamic drag, \[\hat{\mathbf{F}}^{m}+\hat{\mathbf{F}}^{D}=\mathbf{0}.\] (2.1) A simple model of the forcing stipulates a constant tangential driving force Lenz and Ryskin (2006), $\hat{F}^{\text{dr}}$, and an elastic restoring force that keeps the cilia moving along a preferred path (of radius $\hat{R}_{0}$) Niedermayer, Eckhardt, and Lenz (2008), such that \[\hat{\mathbf{F}}^{m}=\hat{F}^{\text{dr}}\mathbf{e}_{\phi}-\hat{k} \left(\hat{R}-\hat{R}_{0}\right)\mathbf{e}_{R},\] (2.2) where $\hat{k}$ is the stiffness of the cilia. The drag force, $\hat{\mathbf{F}}^{D}$, for the rigid body translation of a sphere at velocity $\hat{\mathbf{U}}$ is given by the drag law \[\hat{\mathbf{F}}^{D}=-\hat{\mathbf{R}}_{FU}\cdot\left(\hat{ \mathbf{U}}-\mathcal{F}\left[\hat{\mathbf{u}}^{\infty}\right]\right),\] (2.3) where $\mathbf{u}^{\infty}$ is the background flow and the Faxén operator is $\mathcal{F}=1+\frac{\hat{a}^{2}}{6}\nabla^{2}$Batchelor and Green (1972). The resistance tensor for a sphere moving parallel to a wall is $\hat{\mathbf{R}}_{FU}=6\pi\hat{\mu}\hat{a}\left(\mathbf{I}+{O}(\hat{a}^{3}/ \hat{h}^{3})\right)$Happel and Brenner (1981). In our analysis, we assume the thickness of a cilium is much smaller than its length so that in our minimal model $\hat{a}\ll\hat{h}$, therefore the effect of wall on the hydrodynamic resistance shall be neglected. <figure><img src="content_image/1511.05485/x1.png"><figcaption>Figure 1: A schematic of the motion of a model cilium near a spherical body.The circular trajectory has a radius of ^R and its center is at distance ^hfrom the boundary. The cilium moves with velocity ^U through a fluid withvelocity ^u∞. In this study, ϕ indicate the instantaneous phase of the ciliumand the vectors eϕ and eR show the tangential and radial directions of themotion, respectively.</figcaption></figure> As a starting point, we examine the motion of a single cilium in the absence of other cilia. The background flow field is then zero and the cilium orbits strictly on its preferred circular path. In this case, equation (2.2) leads to a steady state solution \[\dot{\hat{\phi}}_{ss} =\dfrac{\hat{F}^{\text{dr}}}{6\pi\hat{\mu}\hat{a}\hat{R}_{0}} \equiv\hat{\omega},\] (2.4) \[\dot{\hat{R}}_{ss} =0,\quad\hat{R}_{ss}=\hat{R}_{0},\] (2.5) where the over-dot indicates differentiation with respect to time and $\hat{\omega}$ defined as the intrinsic angular velocity of the cilium. Using the reported values for the bending rigidity of a cilium Okada _et al._ (1999); Camalet and Jülicher (2000), Niedermayer _et al._Niedermayer, Eckhardt, and Lenz (2008) noted that radial relaxation is much faster than a period of rotation, namely $\hat{F}^{dr}/\hat{k}\hat{R}_{0}\ll 1$, and so a quasi-static assumption may be employed for the radial dynamics of the system. ## 3 Interactions of two cilia <figure><img src="content_image/1511.05485/x2.png"><figcaption>Figure 2: A system of two cilia around a spherical body of radius ^A. In thisfigure, ^d12 is the distance and θ12 is the angle between center of thetrajectories.</figcaption></figure> We now consider a system of two cilia around a spherical body, with trajectories centered at a distance $\hat{d}_{12}$ as shown in Fig. 2. The velocity of each cilium can be written as $\hat{\mathbf{U}}_{i}=\hat{R}_{i}\dot{\hat{\phi}}_{i}\mathbf{e}_{\phi_{i}}+\dot {\hat{R}}_{i}\mathbf{e}_{R_{i}}$, where $i\in\{1,2\}$. The motion of each cilium in this case is affected by the background flow field induced by the other cilium. The ciliate body is considerably larger than the thickness of a cilium, $\hat{a}\ll\hat{A}$, where $\hat{A}$ is the radius of the spherical body. We also assume the cilia are far apart from one another ($\hat{d}_{12}\gg\hat{h},\hat{R_{0}}$) Okada _et al._ (1999), so that far-field approximations for the induced flow fields may be employed. Under these assumptions one can model the flow field due to the motion of a sphere by a point force (or Stokeslet) to leading order while the no-slip boundary condition on the surface of the spherical body is satisfied by an image Stokeslet set in the body. The background flow field on cilium (1), induced by cilium (2) is \[\hat{\mathbf{u}}^{\infty}(\hat{\mathbf{x}}_{1})=\dfrac{1}{8\pi \hat{\mu}}\left(\hat{\mathbf{G}}(\hat{\mathbf{x}}_{1},\hat{\mathbf{x}}_{2})+ \hat{\mathbf{G}}^{}(\hat{\mathbf{x}}_{1},\hat{\mathbf{x}}^{}_{2})\right) \cdot\hat{\mathbf{F}}^{m}_{2},\] (3.1) where $\hat{\mathbf{F}}^{m}_{2}$ refers to the driving force of the cilium (2), $\hat{\mathbf{x}}_{1}$ and $\hat{\mathbf{x}}_{2}$ indicate the location of each cilium, $\hat{\mathbf{G}}$ is the Oseen tensor and $\hat{\mathbf{G}}^{}$ is the Blake’s solution for the image Stokeslet Happel and Brenner (1981); Blake (1971), at a point $\hat{\mathbf{x}}^{}_{2}=(\hat{A}^{2}/\|\hat{\mathbf{x}}_{2}\|^{2})\hat{\mathbf{ x}}_{2}$ located to satisfy the no-slip condition at the spherical boundary Spagnolie _et al._ (2015). Before going further, we non-dimensionalize all equations by scaling lengths by the radius of the spherical body, $\hat{A}$, and rates by the average angular velocity of the two cilia, $\bar{\hat{\omega}}$. We drop the notation $(~{}\hat{}~{})$ for the dimensionless quantities defined by $\hat{a}=\hat{A}a$, $\hat{R}_{j}=\hat{A}R_{j}$, $\hat{h}=\hat{A}h$, $\hat{d}_{12}=\hat{A}d_{12}$, $\hat{\omega}_{j}=\bar{\hat{\omega}}\omega_{j}$ and $\dot{\hat{\phi}}_{j}=\bar{\hat{\omega}}\dot{\phi}_{j}$, $\hat{\mathbf{U}}_{j}=(\hat{A}\bar{\hat{\omega}})\mathbf{U}_{j}$, $\dot{\hat{R}}_{j}=(\hat{A}\bar{\hat{\omega}})\dot{R}_{j}$, $\hat{t}=(1/\bar{\hat{\omega}})t$ and $\hat{\mathbf{R}}_{FU}=(\hat{k}/\bar{\hat{\omega}}){\mathbf{R}}_{FU}$. A dimensionless parameter $\kappa=\hat{k}/6\pi\hat{\mu}\hat{a}\bar{\hat{\omega}}\gg 1$ which indicates the ratio of the elastic restoring force to the hydrodynamic drag force then naturally arises. We assume the dimensionless length scales are ordered as follows, $a\ll\{h,R_{0}\}\ll 1$ and take $R_{0}=O(h)$Okada _et al._ (1999). Since the radius of the trajectory is small compared to the scale of the body, we may write $d_{12}=2\sin(\|\theta_{12}\|/2)+O(h)$. In these limits, by using the description for background flow field in equation (3.1), the motion equation (2.2) can be solved for case of two neighboring cilia, asymptotically. The evolution equations are then to leading order \[\dot{\phi}_{1} =\omega_{1}+\rho\omega_{2}S_{12}-\dfrac{\rho\omega_{1}\omega_{2}} {\kappa}L_{12},\] (3.2) \[\dot{\phi}_{2} =\omega_{2}+\rho\omega_{1}S_{21}-\dfrac{\rho\omega_{1}\omega_{2}} {\kappa}L_{21},\] (3.3) \[R_{1} =R_{0}+\dfrac{\rho R_{0}\omega_{2}}{\kappa}L_{12},\] (3.4) \[R_{2} =R_{0}+\dfrac{\rho R_{0}\omega_{1}}{\kappa}L_{21},\] (3.5) where $\rho=9ah^{2}/8$ is the strength of the hydrodynamic interactions dictated by the functions \[S_{ij} =\frac{4}{d_{ij}^{3}}\left[\Theta_{ij}\cos\Delta_{ij}+\Phi_{ij} \cos\varphi_{ij}\right],\] (3.6) \[L_{ij} =\frac{4}{d_{ij}^{3}}\left[\Theta_{ij}\sin\Delta_{ij}+\Phi_{ij} \sin\varphi_{ij}\right].\] (3.7) Here we’ve defined phase difference $\Delta_{ij}=\phi_{i}-\phi_{j}$ and sum $\varphi_{ij}=\phi_{i}+\phi_{j}$ while the distance between cilium is given by $d_{ij}=2\sin(\|\theta_{ij}\|/2)$ where $\theta_{ij}$ is the angle between the cilia $i$ and $j$. Finally the functions \[\Theta_{ij}=\dfrac{\cos\|\theta_{ij}\|+\sin(\|\theta_{ij}\|/2)}{1+ \sin(\|\theta_{ij}\|/2)},\] (3.8) \[\Phi_{ij}=\dfrac{\cos\|\theta_{ij}\|-\sin(\|\theta_{ij}\|/2)}{1+\sin( \|\theta_{ij}\|/2)},\] (3.9) capture the effect of the geometry of the spherical body on the hydrodynamic interactions, as shown in Fig. 3. <figure><img src="content_image/1511.05485/x3.png"><figcaption>Figure 3: Geometric terms (a) Θ and (b) Φ as functions of the angle betweencilia.</figcaption></figure> We observe that as expected hydrodynamic interactions above a wall scale as ${O}(d_{ij}^{-3})$Vilfan and Jülicher (2006); Lenz and Ryskin (2006), but now, due to the spherical shape of the ciliate body, both the relative position, $\Delta$ and average position $\varphi$, of the two cilia around the ciliate body affect the hydrodynamic interactions as well by way of the geometric functions $\Theta$ and $\Phi$ respectively. The background flow velocity on each cilium (induced by the other) directly impacts both the angular velocity of each cilium, through the function $S_{ij}$, as well as its radial position via the function $L_{ij}$, which in turn affects the phase-speed as well. Taking the difference of (3.2) and (3.3) we obtain an evolution equation for the phase difference \[\dot{{\Delta}}_{12}=\Delta\omega(1-\rho S_{12})-2(\rho/\kappa) \omega_{1}\omega_{2}{L}_{12},\] (3.10) where $\Delta\omega=\omega_{1}-\omega_{2}$. We see the phase difference evolves due to a difference in the intrinsic phase-speed, due to hydrodynamic interactions directly but also indirectly because of elastic radial displacements. To illustrate the latter point, let us assume cilium (2) is ahead by a positive phase difference of $\Delta_{12}$, as shown in Fig. 4. In this case, the background flow drives a contraction of the orbit for cilium (1) ($R_{1}<R_{0}$) and expansion of the orbit for cilium (2) ($R_{2}>R_{0}$). Since the internal driving forces of the cilia are constant, the changes in trajectories speed up cilium (1) and slow down cilium (2). <figure><img src="content_image/1511.05485/x4.png"><figcaption>Figure 4: The effect of the background flow field on the motion of each cilia.The two cilia are orbiting clockwise, cilium (2) is ahead, thus its inducedflow field pulls cilium (1) to a smaller radius of trajectory which increasesthe instantaneous velocity of cilium (1). On the other hand, the velocity ofcilium (2) decreases as the flow field of cilium (1) pushes cilium (2) to alarger trajectory. In this figure arrows show the flow field induced by eachcilium.</figcaption></figure> If intrinsic velocities are different, $\Delta\omega\neq 0$, for an equilibrium phase difference to arise, this difference must not overwhelm the elasto-hydrodynamic interactions, in other words $\Delta\omega=O(\rho)$ for fixed points in phase-difference. The second term on the right-hand side of equation (3.10) is then $O(\rho^{2})$ and shall be neglected. To provide further clarity we note that the individual phases evolve on a much shorter time scale than the phase-differences, $\dot{\phi}_{i}=O(1)$ while $\dot{\Delta}_{12}=O(\rho)$; hence, we use a multiple scale analysis and average over a period of the short-time scale, $\tau_{{}_{\phi}}=2\pi/\omega_{1}+O(\rho)$ to focus on the long-time behavior of the phase difference (indicated with an overbar). The cycle-averaged evolution equation for the phase-difference is then an Adler equation \[\dot{\bar{\Delta}}_{12}=\Delta\omega-\gamma\Theta_{12}\sin\bar{ \Delta}_{12},\] (3.11) where the synchronization strength in the case of a flat wall $\gamma=8(\rho/d^{3}\kappa)\omega_{1}\omega_{2}$Niedermayer, Eckhardt, and Lenz (2008), is augmented by the geometric term $\Theta_{12}$. If the frequency mismatch is small enough to be balanced by the elasto-hydrodynamic coupling, $\left\|\Delta\omega\right\|<\gamma\Theta_{12}$, a steady-state phase difference emerges given by $\bar{\Delta}^{\text{eq}}_{12}=\sin^{-1}(\Delta\omega/\gamma\Theta_{12})$. In the limiting case of a rigid cilium ($\kappa\rightarrow\infty$) hydrodynamic interactions do not lead to an evolution of phase, and no phase-locking can occur where $\Delta\omega\neq 0$. When the cilia are identical (i.e., $\omega_{1}=\omega_{2}=1$) the phase-locking of the system is guaranteed (if $\theta_{12}\neq\pi$) as equation (3.11) reduces to \[\dot{\bar{\Delta}}_{12}=-\gamma\Theta_{12}\sin\bar{\Delta}_{12},\] (3.12) indicating that the equilibrium phase difference is zero. Unsurprisingly, the evolution equations for phase difference on a spherical body are largely similar to above a flat wall under the assumption that the cilia are much smaller than the ciliate. The difference is that the hydrodynamic interactions are mediated by the geometry of the body through $\Theta_{12}$. We see that when the two cilia are located at the opposite sides of the spherical body ($\theta_{12}=\pi$), radial interactions are completely screened by the ciliate as $\Theta_{12}=0$. We also note that for the angles near zero (and $2\pi$), special care must be used as these limits force $d_{12}\to 0$. To evaluate the system at these angles we can rescale the evolution equations (3.2) and (3.3) with distance $\hat{d}_{12}$, thereby recovering the flat body solution reported in Ref. Niedermayer, Eckhardt, and Lenz (2008) in the limiting case where $\theta_{12}\to 0~{}(\text{or}~{}2\pi)$ and $\hat{A}\rightarrow\infty$. ## 4 Interactions of chain of cilia Now we proceed to the system of $N$ identical cilia around a spherical body where $N\geq 3$. Relying on the linearity of the Stokes equation, the flow field induced by a chain of cilia can be determined by summing the contributions of all the cilia as well as their image points. Following the procedure outlined in the case of two cilia, the evolution equation for cilium (i) in a chain of $N$ cilia, to the leading order, is \[\dot{\phi}_{i} =\omega_{i}+\rho\sum_{j\neq i}^{N}\omega_{j}{S}_{ij}-\dfrac{\rho \omega_{i}}{\kappa}\sum^{N}_{j\neq i}\omega_{j}{L}_{ij},\] (4.1) where $\{i,j\}\in\{1,2,\ldots,N\}$. For simplicity, we shall assume first that all cilia in the chain have the same intrinsic angular velocity, $\omega_{i}=\omega_{j}=1$. The evolution of the phase difference on the long time scale is then \[{\dot{\bar{\Delta}}}_{1i} =4\rho\sum_{m\neq 1}^{N}\sum_{j\neq i}^{N}\left(d_{1m}^{-3}\Theta _{1m}\cos\bar{\Delta}_{1m}-d_{ij}^{-3}\Theta_{ij}\cos\bar{\Delta}_{ij}\right)\] \[-\dfrac{4\rho}{\kappa}\sum_{m\neq 1}^{N}\sum^{N}_{j\neq i}\left(d _{1m}^{-3}\Theta_{1m}\sin\bar{\Delta}_{1m}-d_{ij}^{-3}\Theta_{ij}\sin\bar{ \Delta}_{ij}\right),\] (4.2) where we have set cilium (1) as the reference phase, defining ${{{\Delta}}}_{1i}=\phi_{1}-\phi_{i}$. We note that unlike the case of two identical cilia where the average tangential terms do not contribute to synchronization (because of a pair-wise symmetry), in the case of many cilia the tangential terms ($S_{ij}$) do not vanish and hence contribute to evolution of the phase differences. To further simplify the system, we now consider a chain of cilia which are equally distributed around the body and hence the angle between any two cilia is $\theta_{ij}=2\pi(i-j)/N$. By direct substitution into equation (4), one can show that an equal phase-difference, $\Delta^{\text{eq}}$, between all neighboring cilia is a fixed point of equation (4). Because the system is periodic (in $\theta$), the sum of the phase differences must be an integer multiple of $2\pi$, \[\Delta^{\text{eq}}N=2\pi M,\] (4.3) where $M\in\mathbb{Z}$ for which there are only $N$ unique solutions (due to periodicity in $\phi$), synchronized (when $M=0$) or metachronal waves (when $M\neq 0$). The strength of the interactions between a pair of cilia decays rapidly as their distance increases, due both to the $d_{ij}^{-3}$ term as well as the effect of the angle, thus, we perform a linear stability analysis of these equilibrium states of the system considering only nearest neighbor interactions. Without loss of generality, we assume $-\pi\leq\Delta^{\text{eq}}\leq\pi$. Using Gaussian elimination, we determined the maximum eigenvalues of the Jacobian at $\Delta^{\text{eq}}$, as \[\lambda_{1}=4d_{12}^{-3}\Theta_{12}\Big{[}(\rho/\kappa)( \alpha\text{sgn}[\cos\Delta^{\text{eq}}]-2)\cos\Delta^{\text{eq}} +\alpha\rho\sin\|\Delta^{\text{eq}}\|\Big{]},\] (4.4) where $\alpha(N)\in[0,2)$ is a constant which depends on the number of cilia as shown in Fig. 5. As an example, when $N=3$ the angle between each pair of cilia is $2\pi/3$. In this case $\alpha=0$ hence the system has a stable equilibrium only if $\cos\Delta^{\text{eq}}>0$ and of the possible solutions $\Delta^{\text{eq}}={0,\pm 2\pi/3}$ only $\Delta^{\text{eq}}=0$ is stable. When the cilia are all in-phase, $\Delta^{\text{eq}}=0$, $\lambda_{1}<0$ for any $N$, indicating asymptotic stability of the synchronized state for any number of identical, evenly distributed cilia on a spherical ciliate provided the system has finite flexiblity. In the rigid limit, $\kappa\rightarrow\infty$, the largest eigenvalue is zero which causes a loss of stability of the synchronized state (as shown in numerical simulations). <figure><img src="content_image/1511.05485/x5.png"><figcaption>Figure 5: The value of the coefficient α, which dictates the stability offixed points of a ciliary chain, is shown as a function of the given number ofcilia N.</figcaption></figure> For a system to form metachronal waves, a non-zero equilibrium phase difference between the cilia is required. However, one can show directly that if $\pi/2\leq\|\Delta^{\text{eq}}\|\leq\pi$, $\lambda_{1}>0$ while when $0\leq\|\Delta^{\text{eq}}\|<\pi/2$ for stability one must have the integer \[M<\dfrac{N}{2\pi}\tan^{-1}\left(\dfrac{2-\alpha}{\alpha\kappa} \right),\] (4.5) which is satisfied only by $M=0$ for $\kappa>1$ (and here $\kappa\gg 1$). Thus, all non-zero values of $M$ (metachronal waves) are linearly unstable for any number of identical, evenly distributed cilia on a spherical ciliate. Unlike the reported results for the chain of cilia near a flat boundary Niedermayer, Eckhardt, and Lenz (2008), this system cannot form a stable metachronal wave and all the cilia eventually synchronize. The synchronization of two chains of $N=10$ and $N=15$ cilia from random initial conditions has been illustrated numerically in Fig. 6. <figure><img src="content_image/1511.05485/x6.png"><figcaption>Figure 6: Synchronization of a chain of (a) 10 and (b) 15 identical ciliadistributed uniformly around a spherical body, with the random initial phases.Each line indicates the evolution of the phase difference for each cilium icompared to cilium (1), Δ1i=ϕ1−ϕi, over the time T=t/(κ/ρ). These plots arethe numerical evaluation of equation (4.1) at the characteristic values ofρ=3.6×10−6, κ=100 and ¯^ω=20π rad.s−1 Okada _et al._ (1999); Camalet andJülicher (2000).</figcaption></figure> In real biological examples one can hardly expect perfect symmetry and uniformity in the system so it is important to understand the effect of an imposed asymmetry on the stability of this system. There are several well documented sources of asymmetry, from biochemical noise Goldstein, Polin, and Tuval (2009); Wan and Goldstein (2014), to the different intrinsic properties of a developing cilium Hagiwara, Ohwada, and Takata (2004); Goldstein, Polin, and Tuval (2011) or even the addition of a transverse external flow Guirao and Joanny (2007) which have all been found to spontaneously affect the behavior of a ciliary system. In particular, the beating rate of a developing cilium fluctuates as it grows, which can perturb the equilibrium state of the system Hagiwara, Ohwada, and Takata (2004). To analyze this phenomenon, we impose an asymmetry on the system by increasing the intrinsic velocity of cilium (1) to $1+\Delta\omega$, where we assume $\Delta\omega\ll 1$. The evolution equations for the phase differences are then \[{\dot{\bar{\Delta}}}_{1i}=\Delta\omega +4\rho\sum_{m\neq 1}^{N}\sum_{j\neq i}^{N}\left(d_{1m}^{-3}\Theta _{1m}\cos\bar{\Delta}_{1m}-d_{ij}^{-3}\Theta_{ij}\cos\bar{\Delta}_{ij}\right)\] \[-\dfrac{4\rho}{\kappa}\sum_{m\neq 1}^{N}\sum^{N}_{j\neq i}\left(d _{1m}^{-3}\Theta_{1m}\sin\bar{\Delta}_{1m}-d_{ij}^{-3}\Theta_{ij}\sin\bar{ \Delta}_{ij}\right).\] (4.6) Now, due to the imposed asymmetry, the system no longer converges to a synchronized state where phases are equal. There must be a non-zero equilibrium phase difference between cilium (2) and cilium (1) (for example) to balance the difference in the intrinsic velocities. However, the effect of the imposed asymmetry becomes weaker as the distance from cilium (1) increases and, therefore, the phase difference between a two adjacent cilia decreases. These phase differences form a coordinated system of beating, which is illustrated in Fig. 7 for two sample cases of $N=10$ and $N=15$. <figure><img src="content_image/1511.05485/x7.png"><figcaption>Figure 7: Phase differences of nearby cilia in a chain of 10 and 15 ciliaaround a spherical body when the intrinsic angular velocity of cilium (1) ishigher compared to the other cilia by Δω=10−6 for both cases.</figcaption></figure> These results indicate that the system responds to this asymmetry through an emergent wave-like behavior. Since this asymmetry arises from any developing cilium in the chain, these waves can originate from different parts of the ciliate and vanish once the beating rate of the developing cilium reaches the frequency of the other cilia. Here we should note that unlike metachronal waves which have $\sim 7-10$ cilia per wavelength Machemer (1972); Wollin and Stark (2011), the asymmetry-induced behavior has a characteristic wavelength which spans the entire chain of $N$ cilia. Furthermore, as $N$ increases, the strength of the imposed asymmetry becomes weaker and the equilibrium phase differences of the cilia decrease. Thus, the amplitude of such waves is inversely proportional to the number of cilia in the chain, as shown in Fig. 7. ## 5 Conclusion In this paper, we used a minimal model to capture the dynamics of cilia on a spherically shaped microorganism. We showed that, similar to the case of cilia above a flat wall, elasto-hydrodynamic interactions can lead to synchronization, however here the interactions are additionally mediated due to the geometry of the ciliate body. For a chain of identical cilia uniformly distributed around a spherical boundary, we showed that the natural periodicity in the geometry of the ciliate leads the system to synchronize. We also showed that in this system, metachronal waves are strictly unstable fixed points of the dynamical system unlike in the case of interaction above a flat wall. This result suggests that the geometry of the ciliate plays a crucial role in the behavior of the ciliary chain and it has to be accounted for when analyzing microorganisms with curved bodies and suggests that a natural extension of this analysis would be to investigate a distrubtion of cilia over the whole surface of the ciliate. Our results also suggest that to form stable metachronal waves, rotation and translation of the ciliate Friedrich and Jülicher (2012), elasticity of the cell-internal fibers connecting the cilia Quaranta, Aubin-Tam, and Tam (2015), or motion of the cilia perpendicular to the ciliate body Brumley _et al._ (2012), may be necessary in such microorganisms. We also reported a wave-like response of the system when one of the cilia is intrinsically faster. In this case, the neighboring cilia display stable phase-locking with the faster cilium with a phase difference that decreases with distance from the asymmetry. Although the characteristics of this asymmetry-induced phenomenon do not match metachronal waves, we should note that in real ciliary chains there are likely many cilia of differing lengths or biochemical noise which may lead to more complex dynamics in biological systems. ###### Acknowledgements. The authors thank Professor G.M. Homsy for helpful discussions and support. G.J.E. acknowledges funding from NSERC. ## References * Purcell (1977)E. M. Purcell, “Life at low Reynolds number,” Am. J. Phys. 45, 11 (1977). * Lodish _et al._ (2000)H. Lodish, A. Berk, S. L. Zipursky, P. Matsudaira, D. Baltimore, and J. Darnell, _Cilia and Flagella: Structure and Movement_ (W. H. Freeman, New York, 2000). * Brennen and Winet (1977)C. Brennen and H. Winet, “Fluid mechanics of propulsion by cilia and flagella,” Annu. Rev. Fluid Mech. 9, 339–398 (1977). * Roberts _et al._ (2013)A. J. Roberts, T. Kon, P. J. Knight, K. Sutoh, and S. A. Burgess, “Functions and mechanics of dynein motor proteins,” Nat. Rev. Mol. Cell Biol. 14, 713–726 (2013). * Knight-Jones (1954)E. W. Knight-Jones, “Relations between metachronism and the direction of ciliary beat in Metazoa,” Quart. J. Microsc. Sci. s3-95, 503–521 (1954). * Gueron _et al._ (1997)S. Gueron, K. Levit-Gurevich, N. Liron, and J. J. Blum, “Cilia internal mechanism and metachronal coordination as the result of hydrodynamical coupling,” Proc. Natl. Acad. Sci. 94, 6001–6006 (1997). * Gueron and Levit-Gurevich (1999)S. Gueron and K. Levit-Gurevich, ‘‘Energetic considerations of ciliary beating and the advantage of metachronal coordination,” Proc. Natl. Acad. Sci. 96, 12240–12245 (1999). * Kim and Netz (2006)Y. W. Kim and R. R. Netz, “Pumping fluids with periodically beating grafted elastic filaments,” Phys. Rev. Lett. 96, 158101 (2006). * Shah _et al._ (2009)A. S. Shah, Y. Ben-Shahar, T. O. Moninger, J. N. Kline, and M. J. Welsh, “Motile cilia of human airway epithelia are chemosensory,” Science 325, 1131–1134 (2009). * Hirokawa, Okada, and Tanaka (2009)N. Hirokawa, Y. Okada, and Y. Tanaka, “Fluid dynamic mechanism responsible for breaking the left-right symmetry of the human body: The nodal flow,” Annu. Rev. Fluid Mech. 41, 53–72 (2009). * Vilfan and Jülicher (2006)A. Vilfan and F. Jülicher, “Hydrodynamic flow patterns and synchronization of beating cilia,” Phys. Rev. Lett. 96, 058102 (2006). * Elfring and Lauga (2011)G. J. Elfring and E. Lauga, “Synchronization of flexible sheets,” J. Fluid Mech. 674, 163–173 (2011). * Uchida and Golestanian (2011)N. Uchida and R. Golestanian, “Generic conditions for hydrodynamic synchronization,” Phys. Rev. Lett. 107, 058104 (2011). * Cosentino Lagomarsino, Jona, and Bassetti (2003)M. Cosentino Lagomarsino, P. Jona, and B. Bassetti, “Metachronal waves for deterministic switching two-state oscillators with hydrodynamic interaction,” Phys. Rev. E 68, 021908 (2003). * Niedermayer, Eckhardt, and Lenz (2008)T. Niedermayer, B. Eckhardt, and P. Lenz, “Synchronization, phase locking, and metachronal wave formation in ciliary chains,” Chaos 18, 037128 (2008). * Elfring, Pak, and Lauga (2010)G. J. Elfring, O. S. Pak, and E. Lauga, ‘‘Two-dimensional flagellar synchronization in viscoelastic fluids,” J. Fluid Mech. 646, 505 (2010). * Goldstein, Polin, and Tuval (2009)R. E. Goldstein, M. Polin, and I. Tuval, “Noise and synchronization in pairs of beating eukaryotic flagella,” Phys. Rev. Lett. 103, 168103 (2009). * Polin _et al._ (2009)M. Polin, I. Tuval, K. Drescher, J. P. Gollub, and R. E. Goldstein, ‘‘Chlamydomonas swims with two ”gears” in a eukaryotic version of run-and-tumble locomotion,” Science 325, 487–490 (2009). * Brumley _et al._ (2012)D. R. Brumley, M. Polin, T. J. Pedley, and R. E. Goldstein, “Hydrodynamic synchronization and metachronal waves on the surface of the colonial alga volvox carteri,” Phys. Rev. Lett. 109, 268102 (2012). * Brumley _et al._ (2015)D. R. Brumley, M. Polin, T. J. Pedley, and R. E. Goldstein, “Metachronal waves in the flagellar beating of volvox and their hydrodynamic origin,” J. R. Soc. Interface 12, 20141358 (2015). * Kotar _et al._ (2010)J. Kotar, M. Leoni, B. Bassetti, M. Cosentino Lagomarsino, and P. Cicuta, “Hydrodynamic synchronization of colloidal oscillators,” Proc. Natl. Acad. Sci. 107, 7669–7673 (2010). * Bruot _et al._ (2011)N. Bruot, L. Damet, J. Kotar, P. Cicuta, and M. Cosentino Lagomarsino, “Noise and synchronization of a single active colloid,” Phys. Rev. Lett. 107, 094101 (2011). * Box _et al._ (2015)S. Box, L. Debono, D. B. Phillips, and S. H. Simpson, “Transitional behavior in hydrodynamically coupled oscillators,” Phys. Rev. E 91, 022916 (2015). * Golestanian, Yeomans, and Uchida (2011)R. Golestanian, J. M. Yeomans, and N. Uchida, “Hydrodynamic synchronization at low reynolds number,” Soft Matter 7, 3074 (2011). * Elgeti, Winkler, and Gompper (2015)J. Elgeti, R. G. Winkler, and G. Gompper, “Physics of microswimmers—single particle motion and collective behavior: a review,” Rep. Prog. Phys. 78, 056601 (2015). * Bruot and Cicuta (2015)N. Bruot and P. Cicuta, ‘‘Realizing the physics of motile cilia synchronization with driven colloids,” arXiv:1510.04714 (2015). * Wollin and Stark (2011)C. Wollin and H. Stark, “Metachronal waves in a chain of rowers with hydrodynamic interactions,” Eur. Phys. J. E 34, 42 (2011). * Blake (1971)J. R. Blake, “A spherical envelope approach to ciliary propulsion,” J. Fluid Mech. 46, 199–208 (1971). * Liron and Mochon (1976)N. Liron and S. Mochon, “The discrete-cilia approach to propulsion of ciliated micro-organisms,” J. Fluid. Mech. 75, 593 (1976). * Lenz and Ryskin (2006)P. Lenz and A. Ryskin, “Collective effects in ciliar arrays,” Phys. Biol. 3, 285 (2006). * Batchelor and Green (1972)G. K. Batchelor and J. T. Green, “The hydrodynamic interaction of two small freely-moving spheres in a linear flow field,” J. Fluid Mech. 56, 375 (1972). * Happel and Brenner (1981)J. Happel and H. Brenner, _Low Reynolds Number Hydrodynamics_ (Springer Netherlands, 1981). * Okada _et al._ (1999)Y. Okada, S. Nonaka, Y. Tanaka, Y. Saijoh, H. Hamada, and N. Hirokawa, “Abnormal nodal flow precedes situs inversus in iv and inv mice,” Mol. Cell 4, 459–468 (1999). * Camalet and Jülicher (2000)S. Camalet and F. Jülicher, “Generic aspects of axonemal beating,” New J. Phys. 2, 24–22 (2000). * Spagnolie _et al._ (2015)S. E. Spagnolie, G. R. Moreno-Flores, D. Bartolo, and E. Lauga, “Geometric capture and escape of a microswimmer colliding with an obstacle,” Soft Matter 11, 3396–3411 (2015). * Wan and Goldstein (2014)K. Y. Wan and R. E. Goldstein, “Emergence of synchronized beating during the regrowth of eukaryotic flagella,” Phys. Rev. Lett. 113, 238103 (2014). * Hagiwara, Ohwada, and Takata (2004)H. Hagiwara, N. Ohwada, and K. Takata, “Cell biology of normal and abnormal ciliogenesis in the ciliated epithelium,” in _Int. Rev. Cytol._ (Elsevier BV, 2004) pp. 101–141. * Goldstein, Polin, and Tuval (2011)R. E. Goldstein, M. Polin, and I. Tuval, “Emergence of synchronized beating during the regrowth of eukaryotic flagella,” Phys. Rev. Lett. 107, 148103 (2011). * Guirao and Joanny (2007)B. Guirao and J. Joanny, “Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia,” Biophys. J. 92, 1900–1917 (2007). * Machemer (1972)H. Machemer, “Ciliary activity and the origin of metachrony in paramecium: Effects of increased viscosity,” J. Exp. Biol. 57, 239–259 (1972). * Friedrich and Jülicher (2012)B. M. Friedrich and F. Jülicher, “Flagellar synchronization independent of hydrodynamic interactions,” Phys. Rev. Lett. 109, 138102 (2012). * Quaranta, Aubin-Tam, and Tam (2015)G. Quaranta, M. Aubin-Tam, and D. Tam, “Hydrodynamics versus intracellular coupling in the synchronization of eukaryotic flagella,” Phys. Rev. Lett. 115, 238101 (2015).
0906.0749	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 67278, "num_imgs": 4, "llama3_tokens_count": 17713 }	[ "content_image/0906.0749/x1.png", "content_image/0906.0749/x2.png", "content_image/0906.0749/x3.png", "content_image/0906.0749/x5.png" ]	# Proposal for a Raman X-ray Free Electron Laser Ph. Balcou Université de Bordeaux, Centre Lasers Intenses et Applications (CELIA), CNRS, CEA, 351 Cours de la libération, F–33405 Talence, FRANCE Received: date / Revised version: dateversion 29 mai 2009 Received: date / Revised version: dateversion 29 mai 2009 ###### Abstract A scheme for an X-ray free electron laser is proposed, based on a Raman process occurring during the interaction between a moderately relativistic bunch of free electrons, and twin intense short pulse lasers interfering to form a transverse standing wave along the electron trajectories. In the high intensity regime of the Kapitza-Dirac effect, the laser ponderomotive potential forces the electrons into a lateral oscillatory motion, resulting in a Raman scattering process. I show how a parametric process is triggered, resulting in the amplification of the Stokes component of the Raman-scattered photons. Experimental operating parameters and implementations, based both on LINAC and Laser Wakefield Acceleration techniques, are discussed. pacs: 42.55.VcX- and gamma-ray laser and 41.60.CrFree-electron lasers and 42.65.DrStimulated Raman Scattering ## 1 Introduction Obtaining a laser effect in the extreme Ultraviolet and X-ray ranges has long been a major objective in laser science. The first proposals and attempts started in the late sixties and early seventies with the first contributions of Duguay and Rentzepis [1], and of Jaeglé [2]. After almost thirty years of research coupling the physics of lasers and of plasmas used as active media, numerous lasing lines have been demonstrated and brought to saturation in the extreme ultraviolet and very soft X-ray ranges. In parallel, the progress of ultrashort pulse intense lasers have led to other scenarios : high harmonic generation is now a well established method to use extreme non-linear optics in order to create laser-like radiation in the XUV spectral range. Both high harmonic and soft X-ray lasers from laser/plasma interactions are reviewed in a recent textbook [3]. However, these processes are usually limited to photon energies of few hundreds of eV; the conversion efficiency from laser to XUV pulses drops at higher photon energies, which limits severely the applicability of laser-plasma X-ray lasers and high harmonic generation, in the soft to hard X-ray ranges. Two main paths have been followed in recent years to obtain intense X-ray pulses : X-ray free electron lasers, and incoherent X-ray emission during the interactions between intense lasers, and matter under various phases : solids, clusters, relativistic free electrons. After the pioneering proposals and first developments of Free Electron Lasers [4; 5], it was proposed to extend the concept to the extreme ultraviolet and soft X-ray ranges [6; 7]. This resulted in large scale X-ray free electron projects in the US (LCLS), Japan (SCSS) and Europe (FLASH and TESLA X-FEL) [8]. A huge potential of new applications is expected in many sciences, from physics to biology; the cost and size of these projects are obvious limiting factors to a widespread use of X-ray free electron lasers. _A contrario_, facilities to generate X-ray pulses during the interaction of intense lasers and matter are compact, less costly, but yield mostly incoherent light, with brightnesses smaller by several decades. Of particular interest here is the process of Thomson (or inverse Compton) scattering of laser light, in which photons from a high power laser impinge on a bunch of moderately relativistic electrons , and scatter with an important Doppler shift of $4\gamma^{2}$, $\gamma$ denoting the Lorentz factor, thus appearing in the laboratory frame as X-ray photons, collimated in a small angle [9]. The main advantage of this laser/free electron interaction process is the compacity of the setup : scattering real laser photons, whose wavelength is in the micrometer range, allows to reach X-ray wavelengths with Lorentz factors of typically $10^{2}$, whereas a Lorentz factor of $10^{4}$ or more is required to scatter virtual photons of an undulator, with a period of a few centimeters. Electron energies up to 50 MeV only are therefore required with laser scattering, which can be obtained with a small linac of only few meters, instead of few kilometers necessary to reach multi-GeV energies. Being able to combine both schemes in order to blend their attractive features: compacity of laser scattering, and coherent amplification of a X-ray free-electron laser, would be extremely appealing. As an attempt in this direction, many authors have emphasized that the action of a laser field, propagating in the opposite direction to the relativistic electrons, is extremely similar to that of the magnetic field within an undulator. A laser-undulator free electron laser has therefore been repeatedly proposed [10; 11; 12; 13]. However, the strength parameter $K$ of the laser undulator remains usually very small with most conceivable laser parameters. The gain per oscillation period is then severely reduced with respect to normal undulators, which implies to force the electrons to wiggle a very large number of times N during the amplification. Since the level of mono-energeticity of the electron bunch has to be smaller than $1/2N$ for the Compton free electron laser effect to be effective [14], this scheme would require a quality of mono-energeticity beyond the present state of the art, as well as a remarkably flat intensity profile of the laser pulse, both temporally and spatially. Numerical studies were performed by Bacci _e_t al. [15], that predict a coherent enhancement of extreme UV radiation, considering however a remarkable mono-energeticity of $10^{-4}$, and extremely ambitious laser parameters. In a variant of this scheme, several contributions predict novel phenomena at the onset of quantum effects, expected again only for outstanding qualities of mono-energeticity [19; 20; 21]. Finally, laser wakefield acceleration of electrons is increasingly considered as a potential compact substitute of conventional accelerator technology, at least for extreme UV free electron lasers [17; 18]. A scheme coupling laser acceleration of electrons , and a laser undulator, can open the way to an all-optical Xray free electron laser [16]. However, all these schemes require very stringent parameters of mono-energeticity and emittance of electron bunches, and seem extremely challenging in view of present day electron and laser technologies. We explore here an alternative opportunity to create a compact X-ray FEL, by coupling the physics of free electron lasers, of laser-plasma XUV lasers, and of extreme non-linear optics. By creating artificially a quasi-internal degree of freedom to relativistic free electrons dressed by intense optical lasers, a non-linear Raman scattering process might be switched, leading to exponential amplification of X-ray light. A laser-like beam could then be envisioned, starting either through a SASE process (Self-Amplified Spontaneous Emission), or through the injection of a low intensity, soft X-rays beam from high harmonic generation [22; 23]. The setup considered is first depicted, and the electron dynamics described; this allows to unravel the characteristic emission frequencies of the Raman lines. In sec. 4, the amplification process is modeled analytically, resulting in the calculation of the gain coefficient. Finally, the prospects for an experimental test are discussed, in view of the present state of the art in laser and electron accelerator technologies. A survey of the main relevant parameters with conventional or laser wakefield acceleration systems is presented, along with order-of-magnitude estimates of the laser specifications required to achieve lasing in the X-ray range. ## 2 Principles of a Raman X-ray FEL ### Interaction geometry As schematized in Fig. 1, let us consider the interaction between : i) a bunch of free electrons, as issued either from a linear accelerator or a small storage ring, with a kinetic energy in the range from 10 to 50 MeV, and hence a Lorentz factor from 20 to 100; the propagation axis of the electrons is taken as the conventional $z$-axis. This element is typical of Thomson (inverse Compton) scattering experiments, or of free electron laser, except for the use of smaller electron kinetic energies; ii) a femtosecond or picosecond intense laser system, whose beam is split into two strictly identical parts. The twin beams are made to counter-propagate with respect to one another, along the $x$-axis perpendicular to the electron direction. The polarization vector of the twin beams will be chosen as linear along the $y$ axis (vertical linear polarization, if $xz$ is considered as an horizontal plane). In this configuration, the magnetic field of the twin lasers is along the $z$ axis. <figure><img src="content_image/0906.0749/x1.png"><figcaption>Figure 1: Proposed configuration for a Raman X-ray laser. The color code fromyellow to red indicates the height of the ponderomotive potential due to thelaser standing wave.</figcaption></figure> Both beams are focused along a line, in order to overlap in space and time over the electron path. This superposition of the laser beams along the $z$ axis results in the formation of a standing wave along $x$. The beam intensity along the focal line will have to be controlled to be as constant as possible, after a beam ramp-up segment, and will be given a spatial profile as flat as possible. Such constraints are similar to those encountered for optical parametric chirped pulse amplification systems, and can be fulfilled by means of high quality optical elements and spatial phase control devices, available with present day technologies. This may also ensure that the positions along $x$ of the nodes of the standing wave are constant along the propagation direction $z$. An important point is to synchronize the advance of the electrons, and the illumination by the twin transverse laser beams. Indeed, most studies of laser-plasma soft X-ray lasers [24] display a similar configuration, in which a transverse high intensity laser impinges at $90^{\circ}$ onto a solid surface, thus creating an optically active plasma. In most cases, the duration of the population inversion at each point within the plasma is well below the traversal time of the photons in the amplification region; as a result, the transverse illumination by the laser has to be made to follow the displacement of the X-ray photons along the target. This is achieved thanks to a special optical geometry, in which the energy front of the illuminating laser is decoupled from its phase fronts, by means of diffractive elements [25; 26]. In this ”inhomogeneous wave” geometry (also sometimes referred to as ”traveling wave” geometry), the transverse laser should ideally have an energy front oriented at 45${}^{\circ}$ from the phase fronts, yielding a displacement of the illumination area at exactly the speed of light. Various variants of the optical implementation of the traveling wave are being considered, with an accuracy at the femtosecond level, in order to explore X-ray laser schemes based on innershell pumping [27]. We propose to use such an inhomogeneous wave geometry, in which the inhomogeneous traveling wave is split into two beams, somehow alike the configuration proposed by Pretzler _et al._[28] for an inverted field auto-correlator. The twin beams are subsequently focused along a line in a counter-propagating configuration, as shown in Fig. 2. The optical implementation may require to control precisely the angle between the phase and energy front, resulting in a fine tuning of the advance velocity of the superposition region of the twin beams. <figure><img src="content_image/0906.0749/x2.png"><figcaption>Figure 2: Configuration of the two counter-propagating laser inhomogeneouswaves. The energy front is oriented at 45∘ from the phase fronts, resulting inthe advance of the interference region at velocity c along the z axis.</figcaption></figure> At moderately high laser intensities, the electrons in the bunch will then interact with the standing wave in a non-linear way, as explained now. ### High intensity relativistic Kapitza-Dirac effect : numerical simulation Kapitza and Dirac [29] have shown that electrons interacting with a light standing wave can diffract from this light lattice – thus undergoing the reverse process of light diffraction on a matter density grating. In the low intensity limit, the interaction with the light is a small perturbation to the electron free motion, that induces a momentum transfer of $\pm 2\hbar\mathbf{k}$, where $\mathbf{k}$ is the wavevector of either beam forming the standing wave [30]. Conversely, at high intensities of the order of $10^{13}$ W/cm${}^{2}$ or more for near infra-red lasers, the electron dynamics is modified considerably by the action of the light lattice. Free electrons interacting with a spatially non uniform laser field are indeed submitted to a significant ponderomotive force, ie, a drift force tending to expel the electrons from the regions of highest intensity [31]. The general expression of the ponderomotive force $F_{p}$ is : \[\mathbf{F}_{p}=-\nabla\frac{e^{2}E^{2}}{4m\omega_{0}^{2}},\] where $-e$ is the electron charge, $E$ the local electric field, $m$ the electron rest mass, and $\omega_{0}$ the laser angular frequency. In this case, non-relativistic electrons injected into the standing wave will feel a ponderomotive force deriving from a spatially oscillating potential : \[V_{p}=\frac{e^{2}E_{0}^{2}}{m\omega_{0}^{2}}\sin^{2}(k_{0}x),\] (1) with $k_{0}=\omega_{0}/c$. If the electron transverse kinetic energy is smaller than the maximum of $V_{p}$, it will be trapped within the ponderomotive potential well. In the opposite case, the electron will succeed in going through the light lattice, with a momentum transfer up to several thousands $\hbar\mathbf{k}$ or more. Bucksbaum, Schumacher, and Bashkansky have studied experimentally the Kapitza-Dirac effect in the high intensity regime, using Above-Threshold Ionization as the source of electrons within the standing wave [32]. Giant momentum transfers were indeed observed; importantly, this study concluded on the validity of a classical description of the electron motion in the high intensity regime. The major difference between the experiment of Bucksbaum _et al._, and our proposed X-ray laser scheme, is related to the relativistic velocity of the injected electrons – a situation not considered so far. As a first step to explore the electron dynamics, we first present the results of a full numerical integration of the electron trajectory. <figure><img src="content_image/0906.0749/x3.png"><figcaption>Figure 3: (a) Transverse motion of a relativistic electron trapped laterallyin a laser standing wave. (b) Corresponding laser-induced oscillation. (c)Emission spectrum for an electron in a standing wave (solid line) and in anormal single side illumination (90∘ Thomson scattering), blue dashed line).</figcaption></figure> As a model case, let us consider a 10 MeV electron ($\gamma=20$), with a small initial transverse velocity, and embedded in the standing wave. We calculate the electron motion using the exact equations of special relativity, and considering both the electric and magnetic fields of the incident laser waves [33] : \[\frac{d}{dt}\left[\gamma mc^{2}\right]=\mathbf{v}.\mathbf{E}\] \[\frac{d\mathbf{v}}{dt}=\frac{q}{\gamma m}\left[\mathbf{E}-\frac{ \mathbf{v}}{c^{2}}(\mathbf{v}.\mathbf{E})+\mathbf{v\times B}\right]\] where all dynamical variables are considered in the laboratory frame. The laser parameters considered are those of a Titanium-Sapphire laser, with a wavelength of 800 nm, and an intensity per beam of $10^{18}$ W/cm${}^{2}$. The electron initial transverse velocity along $x$ is 6. 10${}^{5}m.s^{-1}$. In Fig. 3(a), the electron is seen to wiggle along $x$ around the minimum line of the ponderomotive potential, with a period of 55 fs in this specific case. This period is not only longer than the laser period $T_{0}=2.5fs$, and also much longer than the oscillation period of 2.7 fs expected from the non-relativistic potential function (1) (see Eq. 8 below). One should also notice that the electron motion along $x$ is perfectly smooth, even within the time span of the laser cycle $T_{0}$ – the ponderomotive potential can therefore be considered as a tool to model the electron dynamics, even on a time scale smaller than $T_{0}$. Fig. 3(b) shows how the slow wiggling along $x$ modifies the laser-induced oscillation, which appears now modulated at twice the wiggling frequency. Finally, Fig. 3(c) displays the spectrum of the light scattered in the $+z$ direction (solid line), calculated as the squared modulus of the Fourier transform of the acceleration along $y$, and taking into account the Doppler shift. The dashed line shows for comparison the light spectrum calculated for the same electron initial conditions, but assuming one of the twin beams to be suppressed. The Doppler-shifted emission line, characteristic of 90${}^{\circ}$ Thomson scattering, is seen to be split into two Raman components, with an important drop in emission intensity because of the electron trapping close to the potential minimum. An important issue is how relativistic electrons may be injected into the standing wave. Fig. 4 shows few test cases of electrons, chosen at random in an electron bunch, whose normalized emittance is 1 mm.mrad, focused onto a spot of 50 $\mu m$ radius rms. The standing wave is assumed to start with a 3 $mm$ long ramp, corresponding to 10 $ps$, with sinus-square intensity profile, followed by a plateau of constant intensity. In a first step, the electron motions are hardly affected by the standing wave; as the latter increases further, the electrons are seen to get trapped in one of the potential wells, with a gradually decreasing excursion from the minimum until the end of the ramp. The light lattice then acts as a duct, able to confine and guide the electrons up to the end of the illuminated area. In this simple calculation, we do not taken into account any back action of the light field emitted by the wiggling electrons on their trajectories; their oscillations along $x$ remain therefore purely randomly phased up to the end of the interaction region. [FIGURE:S2.F4][ENDFIGURE] ### Collective electron motion under X-ray irradiation We now examine how electrons injected into the light lattice, may be coupled to an external X-ray field, whose frequency corresponds to one of the Raman modes displayed in Fig. 3(c). We therefore add the possibility to take into account an additional electromagnetic field $E_{1}(z,t)=E_{1}^{0}\cos(\omega_{1}(t-z/c))$, where $\omega_{1}$ corresponds to the Stokes mode. The X-ray electric and magnetic fields are simply added to the laser fields in the computation of electron motions. We wish to investigate how this X-ray field may modify the distribution in space of the electrons close to the bottom of the potential wells, at a given time. We consider an initial ensemble of macro-particles, first injected into the light lattice with the same parameters as in Fig. 4, and follow the electrons in time throughout the ramp and the interaction regions. For the sake of simplicity, we will switch on the X-ray field at the end of the ramp region, and keep it constant up to the end of the interaction region. In the current example, we restrict the calculation to a slice of phase-space for the initial injection, ie, we consider only electrons initially close to the axis of the potential well, to within $\lambda_{0}/15$. Taking all electrons at that stage, including the eccentric ones with large amplitude and reduced frequency oscillations, would indeed blur the final figure. Fig. 5(a) displays the final space distribution (z,x) of an ensemble of 1000 such electrons, with a X-ray field amplitude $E_{1}^{0}=10^{10}V/m$, and an interaction region of $75\mu m$. The region of interest is taken here to have a width of half a laser wavelength, which is the period of the light lattice, and a length of two X-ray wavelengths $2.(2\pi c/\omega_{1})$. Fig. 5(b) shows the distribution of electrons with identical initial conditions following the ramp, but subjected to a X-ray field amplitude $E_{1}^{0}=10^{10}V/m$ within the interaction region. <figure><img src="content_image/0906.0749/x5.png"><figcaption>Figure 5: (a) : space distribution of electrons injected into the lightlattice in the conditions of Fig. 4, with no X-ray field. (b) : spacedistribution after the interaction with a y-polarized X-ray field along z, atthe Stokes frequency. Red lines show least-square fits to sine functions.</figcaption></figure> While each electron oscillates in the light potential well, the random character of the injection into the light lattice results in an evenly distributed electron distribution in Fig. 5(a). On the contrary, one notes easily an overall oscillation of the centroid of electron lateral positions in Fig. 5(b), with the period of the X-ray wavelength along $z$. The red line is a least-square fit a a sine function to the electron distribution; this gives an intuitive notion of a collective transverse displacement function. While the detailed process will be unraveled below, it is clear at this stage that the beating between the Doppler-shifted laser frequency, and the Stokes X-ray frequency, is bound to induce a resonant excitation at the Raman frequency, resulting in this collective behaviour. The same calculation at the anti-Stokes frequency gives absolutely similar distributions. These numerical results will now allow us to propose an analytical modeling of the electron dynamics, based on a ponderomotive potential approach, and that will consider that electrons remain confined close to the bottom of the potential wells. ## 3 Analytical description of single electron dynamics ### Analytical description of the single electron dynamics in the light lattice We model the motion of electrons, moving along the $+z$ direction, and injected into the superposition of two transverse counter-propagating lasers : one beam in the $+x$ direction, $\mathbf{E}_{0}^{+}(x,t)=E_{0}\sin(k_{0}x-\omega_{0}t)\mathbf{e}_{y}$, and an identical beam propagating in the $-x$ direction, $E_{0}^{-}(x,t)=E_{0}\sin(k_{0}x+\omega_{0}t)\mathbf{e}_{y}$. $E_{0}$ is the real-valued electric field associated to each of the twin beams, $k_{0}$ and $\omega_{0}$ the laser wavevector and angular frequency respectively, and $\mathbf{e}_{y}$ (resp. $\mathbf{e}_{x}$, $\mathbf{e}_{z}$) will denote the unit polarization vector in the $y$ (resp. $x$,$z$) direction. These twin laser beams interfere to form a standing wave, described as : \[\mathbf{E}_{0}(x,t) =2E_{0}\sin(k_{0}x)\cos(\omega_{0}t)\mathbf{e}_{y}\] (2) \[\mathbf{B}_{0}(x,t) =-2\frac{E_{0}}{c}\cos(k_{0}x)\sin(\omega_{0}t)\mathbf{e}_{z},\] (3) We assume that the standing wave is switched on adiabatically along $z$ (gradual build-up of the laser intensity along the electron trajectory), and that the transverse kinetic energy of the electron is small with respect of the maximum of the ponderomotive potential $V_{p}$. Then each electron undergoes an harmonic oscillatory motion close to the bottom lines of $V_{p}$, with an effective potential given to first order by : \[V_{p}^{0}=\frac{e^{2}E_{0}^{2}}{mc^{2}}x^{2}\] (4) where for simplicity we have considered small displacements around the minimum potential line $x=0$. In this harmonic potential well, a non-relativistic electron oscillates with a frequency $\Omega^{\prime}$: \[\Omega^{\prime}=\frac{\sqrt{2}eE_{0}}{mc}\] (5) Surprisingly, this oscillation frequency is independent from the laser frequency, but varies as the square root of laser intensity. Let us consider now a relativistic electron, of velocity $v$ (Lorentz factor $\gamma=(1-v/c)^{-1/2}\gg 1$), as issued from a linear accelerator. Due to the relativistic mass increase in the laboratory frame, the ponderomotive potential becomes : \[V_{p}=\frac{e^{2}E_{0}^{2}}{\gamma m\omega_{0}^{2}}\sin^{2}(k_{0}x),\] (6) and the transverse equation of motion close to the bottom of the potential well is : \[\gamma mx^{..}+\frac{2e^{2}E_{0}^{2}}{\gamma mc^{2}}x=0\] (7) , which yields an oscillation frequency : \[\Omega=\frac{\sqrt{2}eE_{0}}{\gamma mc},\] (8) in excellent agreement with the numerical values obtained from the exact numerical calculation of section 2.2. One alternative way to obtain the same expression is to transform the standing wave to the electron rest frame, evaluate the oscillation frequency (5) in the ponderomotive potential, and transform the frequency back to the laboratory frame, thus yielding the same expression (8). The position of an arbitrary electron $i$ can hence be described in the laboratory frame as : \[z_{i}^{0}(t) =z_{i}^{0}(0)+vt\] \[x_{i}^{0}(t) =\Delta X_{i}\cos(\Omega t-\Psi_{i})\] \[y_{i}^{0}(t) =y_{i}^{D}(t)+\frac{eE_{0}k_{0}\Delta X_{i}}{\gamma m}\sum_{ \epsilon=\pm 1}\frac{\cos((\omega_{0}+\epsilon\Omega)t-\epsilon\Psi_{i})}{( \omega_{0}+\epsilon\Omega)^{2}}\] where the initial position $z_{i}^{0}(t)$, the vertical drift $y_{i}^{D}(t)$, the excursion $\Delta X_{i}$ and phase $\Psi_{i}$ of the free oscillation along $x$, all result from the initial injection conditions of the electron in the standing wave. We recover in this simple model that the wiggling in the $y$ direction is split into two Raman shifted lines, of frequencies $\omega_{0}\pm\Omega$. The electron oscillation induces light scattering, which, along the electron direction, occurs at frequencies : \[\omega_{1}=\frac{\omega_{0}+\epsilon\Omega}{1-v/c},\] (10) where $\epsilon=+1$ corresponds to the anti-Stokes Raman component, and $\epsilon=-1$ to the Stokes component. The $1-v/c$ factor results from the Doppler shift, and corresponds to a frequency up-shift of $2\gamma^{2}$ in the highly relativistic limit. These analytical values agree again with those displayed on Fig. 3(c). ### Single electron coupling to a Raman scattered wave Let us now consider the coupling between the single electron dynamics in the standing wave, and a Raman scattered X-ray wave $E_{1}$ propagating along $z$: \[\mathbf{E}_{1}(z,t)=E_{1}\cos(\omega_{1}t-k_{1}z)\mathbf{e}_{y}\] (11) where $k_{1}$ is the wave-vector along $+z$ corresponding to the angular frequency $\omega_{1}$ given by Eq. 10. This field is assumed to be polarized along $y$, since it results from the scattering of the $y$-polarized laser beams. The magnetic field $\mathbf{B}_{1}$ of this X-ray wave is therefore directed along $x$. Each electron will see its motion modified by the coupling of the laser standing wave, and of the X-ray wave, via Lorentz forces. Two terms can be distinguished : $E_{1}$ induces a small amplitude wiggling around $y$ that couples to the large magnetic field of the standing wave along $z$, resulting in a Lorentz force along $x$; and $E_{0}$ induces a large wiggling along $y$, that couples to the initially small magnetic field of the X-ray wave, resulting in a second Lorentz force, directed along $z$. It can easily be shown that these two terms have exactly the same magnitude; however, the latter is obviously non resonant, whereas we will show hereunder that the former induces a resonant oscillation of the electron captured within the ponderomotive potential wells. The electric force experienced by electron $i$ due to the X-ray field $E_{1}$ is : \[{\mathbf{F}}_{1}(t) =-eE_{1}\cos\left(\omega_{1}t-k_{1}(z_{i}^{0}+vt)\right)\mathbf{e }_{y}\] (12) \[=-eE_{1}\cos\left((\omega_{0}+\epsilon\Omega)t-\Phi_{i})\right) \mathbf{e}_{y},\] where $\Phi_{i}=k_{1}z_{i}^{0}$. The resulting wiggling velocity \[{\mathbf{v}}_{1}(t)=\frac{-eE_{1}}{\gamma m(\omega_{0}+\epsilon\Omega)}\sin \left((\omega_{0}+\epsilon\Omega)t-\Phi_{i})\right)\mathbf{e}_{y}\] (13) couples to the laser magnetic field to yield a transverse Lorentz force : \[{\mathbf{F}}_{L}(t)=\frac{-eE_{1}E_{0}}{\gamma mc(\omega_{0}+\epsilon\Omega)} \cos(\epsilon\Omega t-\Phi_{i})\mathbf{e}_{x}\] (14) where we have neglected a rapidly oscillating term at angular frequency $2\omega_{0}$. The $x$-motion follows therefore the following equation : \[\ddot{x}+\Omega^{2}x=\frac{-e^{2}E_{1}E_{0}}{\gamma^{2}m^{2}c(\omega_{0}+ \epsilon\Omega)}\cos(\epsilon\Omega t-\Phi_{i})\] (15) The solution is the sum of a freely oscillating motion $x^{0}(t)$ resulting from the injection conditions of the free electron into the standing wave, as given by (3.1), and of a forced term $\delta x$ : \[\delta x(t)=\frac{-e\epsilon E_{1}t}{2^{3/2}\gamma m(\omega_{0}+\epsilon\Omega )}\sin(\epsilon\Omega t-\Phi_{i})\] (16) It is worth to note that its amplitude almost does not depend on the laser field $E_{0}$. We now aim to infer from the electron forced oscillation the work induced by the X-ray field $E_{1}$ onto the electron velocity along $y$ induced by the laser field. Let us start by the anti-Stokes case $\epsilon=+1$. The $y$-motion of the forced electron can be deduced from its $x$-motion as : \[\delta\ddot{y}=\frac{-2eE_{0}}{\gamma m}\sin(k_{0}x(t))cos(\omega_{0}t)\] (17) We assume again that the electron remains close to the bottom of the potential well, so that $sin(k_{0}x(t))=k_{0}x^{0}(t)+k_{0}\delta x(t)$; the $y$ velocity can therefore be approximated by the sum of the velocity of the free $y$ motion of Eq. (3.1), and a forced velocity $\delta\dot{y}$: \[\delta\dot{y}=\frac{-e^{2}E_{0}E_{1}k_{0}t}{2^{3/2}\gamma^{2}m^{2}c(\omega_{0} +\Omega)^{2}}\cos[(\omega_{0}+\Omega)t-\Phi_{i}].\] (18) The average value of the work of the force $-eE_{1}$ per unit time is therefore : \[P_{AS}=<\delta\dot{y}F_{1}>=\frac{e^{3}E_{0}E_{1}^{2}t}{2^{5/2}\gamma^{2}m^{2} c(\omega_{0}+\Omega)^{2}}\] (19) This power is positive, meaning that the electron gains energy, and conversely that the X-ray wave loses energy. This corresponds necessarily to a damped propagation mode for $E_{1}$. If we now turn to the Stokes ($\epsilon=-1$) case, the forced $y$-velocity is : \[\delta\dot{y}=\frac{e^{2}E_{0}E_{1}k_{0}t}{2^{3/2}\gamma^{2}m^{2}c(\omega_{0}- \Omega)^{2}}\cos[(\omega_{0}-\Omega)t-\Phi_{i}].\] (20) resulting in a negative power transfer : \[P_{S}=<\delta\dot{y}F_{1}>=\frac{-e^{3}E_{0}E_{1}^{2}t}{2^{5/2}\gamma^{2}m^{2} c(\omega_{0}-\Omega)^{2}}\] (21) The Stokes scattered X-ray wave will therefore gain energy from the interaction with the forced part of the electron motion. The increase of $E_{1}$ will result in an enhanced forced motion $\delta x$ and $\delta y$, which will increase in turn the power transfer to $E_{1}$. We can therefore expect an exponential amplification of the Stokes wave, that is, to start a stimulated Raman scattering process in the forward direction with respect to the electron beam. ## 4 Analysis of the amplification process The analysis of the previous section was purely based on a kinetic, single electron description. We now turn to a macroscopic description, and aim to set the evolution equation along $z$ of an X-ray field $E_{1}$, coupled to the current density $J_{1}$ induced by the electron oscillations in the laser field, in conditions where the electrons exhibit bunching in the transverse direction $x$. We will therefore introduce a mean electron displacement function $\delta x(z,t)$ (illustrated as a red line in Fig. 5(b) ), that will play in the derivation a role very similar to that of the longitudinal bunching factor of Compton Free Electron Laser theory. From now on, we will focus on the Stokes case. ### Modal analysis We start from the exact propagation equation of the X-ray field $E_{1}$, polarized along the $y$ axis : \[\frac{\partial^{2}E_{1}}{\partial z^{2}}+\Delta_{\perp}E_{1}-\frac{1}{c^{2}} \frac{\partial^{2}E_{1}}{\partial t^{2}}=\frac{1}{\epsilon_{0}c^{2}}\frac{ \partial}{\partial t}J_{1}\] (22) We split the calculation of Eq. (22) in three steps : _i/_ write down the current density $J_{1}$ as a function of the mean displacement $\delta x$; _ii/_ compute the evolution of $\delta x$ for the electrons, subject to a Lorentz force along $x$ induced the laser B-field and the X-ray field $E_{1}$ ; _iii/_ get back to the propagation equation (22), with the newly obtained expression for the current $J_{1}$. As the various fields are all slowing evolving in space and time, we will systematically introduce envelope functions, and make use of the slowly varying envelope approximation (SVEA) along the electron motion. _i/_ The current density $J_{1}(x,z,t)$ can be obtained from : \[\left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial z}\right)J_{1}= \frac{2e^{2}E_{0}}{\gamma m}\sum_{i}\delta(\mathbf{r-r_{i}})\sin(k_{0}x_{i}) \cos(\omega_{0}t),\] (23) where the summation runs over all electrons $i$ in the bunch, and $\mathbf{r}_{i}$ indicates the position of electron $i$. We know from section (3.2) that the electron motion along $x$ has a free and a forced component, $x_{i}(t)=x_{i}^{0}(t)+\delta x_{i}(t)$. The forced part $\delta x$ is identical for all electrons in a same slice in the electron bunch, assumed for the time being to be mono-energetic, and in the same potential well. In contrast, summation over all particles contained in a slice along $z$ brings the total free motion $x_{i}^{0}$ contribution to average out to $0$. This allows us to define a transverse displacement function $\delta x(z,t)$ : \[\delta x(z,t)=\frac{1}{N(z,z+dz)}\sum_{j}{x_{j}(t)},\] (24) where the summation runs over the all $N(z,z+dz)$ electrons contained in the slice between $z$ and $z+dz$ at time $t$, and in the potential well centered at the origin $x=0$. One may note that the transverse displacement of the next potential well, with respect to its center at $x=\lambda_{0}/2$, has the opposite value $-\delta x$; however the laser electric field is also dephased by $\pi$, so that the resulting polarizations at $\omega_{1}$ are in phase for all potential wells. Consideration of $\delta x$ around $x=0$ is therefore well suited to the following derivation. In the small angle approximation, we also simplify $\sin(k_{0}x_{i})$ to $k_{0}\delta x$. In parallel, we reduce the fields $E_{1}(x,z,t)$ and $J_{1}(x,z,t)$ to their transverse average values $E_{1}(z,t)$ and $J_{1}(z,t)$, and introduce the envelope functions $\tilde{E}_{1}$, $\tilde{j}_{1}$ and $\delta\tilde{x}$, such as $E_{1}=\tilde{E}_{1}\exp{i(k_{1}z-\omega_{1}t)}+c.c.$, $J_{1}=\tilde{j}_{1}\exp{i(k_{1}z-\omega_{1}t)}+c.c.$, and $\delta x=\delta\tilde{x}\exp i(k_{1}z-(\omega_{1}-\omega_{0})t)+c.c.$. To keep consistent with this transverse field averaging, we neglect the diffraction term of Eq. (22). The spatial average procedure leads to introduce the electron average number density $\rho$. With these definitions, the envelopes for current density and transverse displacement are related by : \[\tilde{j}_{1}=\frac{ie^{2}\rho E_{0}k_{0}}{\gamma m(\omega_{0}-\Omega)}\delta \tilde{x}\] (25) _ii/_ The transverse displacement function $\delta x$ follows the eulerian analog of Eq. (15) : \[\left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial z}\right)^{2} \delta x+\Omega^{2}\delta x=\frac{-e}{\gamma m}V_{1}\times B_{0}\] (26) where we have introduced the velocity field $V_{1}$ following from : \[\left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial z}\right)V_{1}= \frac{-e\tilde{E}_{1}}{\gamma m}\exp{i(k_{1}z-\omega_{1}t)}+c.c.\] (27) so that, with $V_{1}=\tilde{V}_{1}\exp{i(k_{1}z-\omega_{1}t)}+c.c.$ : \[\tilde{V}_{1}=\frac{-ie\tilde{E}_{1}}{\gamma m(\omega_{0}-\Omega)}.\] (28) The Lorentz force term is therefore \[-eV_{1}\times B_{0}=\frac{-e^{2}E_{0}\tilde{E}_{1}}{\gamma mc(\omega_{0}- \Omega)}\exp{i(k_{1}z-(\omega_{1}-\omega_{0})t)}-c.c.\] (29) where we have dropped non resonant terms at frequency $\omega_{1}+\omega_{0}$ (that would correspond to an excitation at the anti-Stokes frequency), and assumed the magnetic field $B_{0}$ to be constant in the vicinity of the bottom lines of the potential. We now apply the SVEA to the first term of equation (26), resulting in : \[\left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial z} \right)^{2}\delta x =\big{[}-[\omega_{0}-\omega_{1}(1-\frac{v}{c})]^{2}\] \[+2i[\omega_{0}-\omega_{1}(1-\frac{v}{c})] \left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial z} \right)\big{]}\delta\tilde{x}\,e^{i(k_{1}z-(\omega_{1}-\omega_{0})t)}\] \[+c.c.\] If the frequencies fulfill the condition : \[\omega_{1}\left(1-v/c\right)=\omega_{0}-\Omega,\] (30) then the linear term in $\delta\tilde{x}$ cancels out the restoring force of the harmonic potential in Eq. (26), so that : \[2i\Omega\left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial z}\right) \delta\tilde{x}=\frac{-e^{2}E_{0}\tilde{E}_{1}}{\gamma^{2}m^{2}c(\omega_{0}- \Omega)}.\] (31) Note that, while Eq. (10) was the simple result of a Fourier analysis, Eq. (30) should be interpreted as a resonance condition. Use of equations (25) and (31) allows one to evaluate qualitatively the power gained at resonance by the X-ray field, as $-j_{1}.E_{1}=-2\textit{Re}(\tilde{j}_{1}\tilde{E}_{1}^{})$, where $j_{1}$ is the current induced by a mean displacement induced over an interval $\delta L$ : \[-j_{1}.E_{1}\simeq\frac{e^{4}\rho E_{0}^{2}k_{0}\delta L}{\gamma^{3}m^{3}cv \Omega(\omega_{0}-\Omega)^{2}}.2\tilde{E}_{1}.\tilde{E}_{1}^{}>0\] (32) We therefore recover the conclusion of the single electron analysis, showing that the Stokes mode exhibits amplification, while the anti-Stokes mode, described simply by replacing $\Omega$ by $-\Omega$, should be absorbed. _iii/_ We eventually come back to the propagation equation (22), which, under the SVEA, and neglecting diffraction terms, reads : \[2ik_{1}\left(\frac{\partial}{\partial z}+\frac{\partial}{c\partial t}\right) \tilde{E}_{1}=\frac{-i\omega_{1}\tilde{j}_{1}}{\epsilon_{0}c^{2}}\] (33) Combining equations (25), (31), and (33), and considering the process to be stationary, result in a single propagation equation for the X-ray envelope : \[\frac{\partial^{2}\tilde{E}_{1}}{\partial z^{2}}=\frac{e^{4}\rho k_{0}E_{0}^{2 }}{4\epsilon_{0}\gamma^{3}m^{3}c^{2}v\Omega(\omega_{0}-\Omega)^{2}}\tilde{E}_{1}\] (34) that corresponds to an exponential amplification with a gain of : \[g=\sqrt{\frac{e^{3}\rho k_{0}E_{0}}{2^{5/2}\epsilon_{0}\gamma^{2}m^{2}cv( \omega_{0}-\Omega)^{2}}},\] (35) or, in an approximate simpler way : \[g=\sqrt{\frac{e^{3}\rho E_{0}}{2^{3/2}\epsilon_{0}m^{2}c^{3}\omega_{1}}}.\] (36) ### Effect of electron velocity mismatch The energy dispersion of the incident electron bunch is a major concern for X-ray free electron lasers. In particular, all the simulations on the FEL effect with optical undulators, in the Compton regime, demonstrate that a remarkable value of mono-energeticity is required, typically of the order of $10^{-4}$[15] to few $10^{-4}$ for electron energies of few tens of MeV [16]. Indeed, in the Compton regime, amplification occurs throughout the laser undulator length only if $\delta\gamma/\gamma<1/2N$, $N$ being the number of undulator periods over the whole amplification length [33]. The Doppler frequency shift is therefore limited to the emission linewidth due to the finite emission time. This very stringent condition on the electron energy dispersion is obviously one of the major reasons why this optical undulator scheme has not been demonstrated up to now. How the proposed Raman scheme for a X-ray FEL copes with the electron energy dispersion is therefore a major issue; however, a detailed study of Raman amplification with a spread of electron energies is beyond the scope of the present study, leading us to restrict ourselves to discuss the spectral broadening induced the electron energy spread, and the amplification regime between a monochromatic X-ray field, and an out-of-resonance electron population. In general, one has to consider a electron bunch with a distribution of Lorentz factors, with an interval $2\Delta\gamma$ around a central value $\gamma_{0}$, characterized by a density distribution $\rho(\gamma)$. For each velocity component, the deviation $\delta\gamma=\gamma-\gamma_{0}$ from the central value results in a shifted X-ray angular frequency $\delta\omega_{1}$, with $\delta\omega_{1}/\omega_{1}=2\delta\gamma/\gamma$. The spontaneous scattering spectrum is therefore bound to exhibit a Doppler broadening of $4\omega_{1}(\Delta\gamma/\gamma)$. An outcome of this broadening is the possibility to get spectral overlaps between a Doppler down-shifted emission on a anti-Stokes mode, and a Doppler up-shifted emission on the Stokes mode. Assigning the former to electrons of Lorentz factor $\gamma-\Delta\gamma$ and the latter to those with $\gamma+\Delta\gamma$, the overlap condition reads : \[\frac{\omega_{0}-\Omega}{1-v(+\Delta\gamma)/c}=\frac{\omega_{0}+\Omega}{1-v(- \Delta\gamma)/c}\] (37) Developing to first order results in a simple condition to prevent Stokes / anti-Stokes overlaps : \[\Delta\gamma/\gamma<\Omega/2\omega_{0}.\] (38) Another effect due the electron energy spread is that essentially all electrons violate to some degree the resonance condition (30) . We need therefore to evaluate the spectral acceptance of (30). In this aim, we propose to investigate how a monochromatic X-ray field, at the central frequency $\omega_{1}$, interacts with a population of electrons in the bunch, with a Lorentz factor $\gamma^{\prime}=\gamma+\delta\gamma$, and density $\rho^{\prime}$. This simple approach is of course unable to describe the full complexity of the problem, in which each field frequency component is coupled to all electron populations of different velocities, and conversely each electron is coupled to all field frequency components. It may however give interesting insights on electron - field couplings out of the resonance condition. Revisiting the three steps of the gain calculation of section (4.1), one may notice that the major effect of the offset in electron kinetic energy, and hence in velocities, is to modify the expression of the derivatives along the movement by adding a term resulting from the velocity change $\delta v\frac{\partial}{\partial z}$. We will neglect an additional second order effect, namely, the slight change of the resonant oscillation frequency $\Omega$ in the light lattice. The current density versus transverse displacement function becomes: \[\tilde{j}_{1}=\frac{ie^{2}\rho E_{0}k_{0}}{\gamma^{\prime}m(\omega_{0}-\Omega- \Delta\omega)}\delta\tilde{x},\] (39) where we have set $\Delta\omega=k_{1}.\delta v=\omega_{1}\delta\gamma/\beta\gamma^{3}$. The high frequency velocity field is now: \[\tilde{V}_{1}=\frac{-ie\tilde{E}_{1}}{\gamma^{\prime}m(\omega_{0}-\Omega- \Delta\omega)},\] (40) so that the new expression of the Lorentz force field is: \[-eV_{1}\times B_{0}=\frac{-e^{2}E_{0}\tilde{E}_{1}}{\gamma^{\prime}mc(\omega_{ 0}-\Omega-\Delta\omega)}e^{i(k_{1}z-(\omega_{1}-\omega_{0})t)}-c.c.\quad.\] (41) The differential equation for transverse displacement function $\delta x$ includes new terms : \[-\Delta\omega(\Delta\omega+2\Omega)\delta\tilde{x}+ 2i(\Omega+\Delta\omega)\left(\frac{\partial}{\partial t}+v^{ \prime}\frac{\partial}{\partial z}\right)\delta\tilde{x}=\] \[\frac{-e^{2}E_{0}\tilde{E}_{1}}{\gamma^{\prime 2}m^{2}c(\omega_{0 }-\Delta\Omega)}\] Assuming (38) to be valid, let us define $g^{\prime}$ and $g_{I}$ as: \[g^{\prime}=\left[\frac{e^{4}\rho^{\prime}k_{0}E_{0}^{2}}{4 \epsilon_{0}\gamma^{\prime 3}m^{3}c^{2}v^{\prime}(\Omega+\Delta\Omega)(\omega_ {0}-\Omega-\Delta\omega)^{2}}\right]^{1/2},\] \[g_{I}=-\Delta\omega(\Delta\omega+2\Omega)/[4(\Omega+\Delta\omega )v^{\prime}].\] The differential equation for the field envelope $\tilde{E}_{1}$ becomes : \[\frac{\partial^{2}\tilde{E}_{1}}{\partial z^{2}}-2ig_{I}\frac{\partial\tilde{E }_{1}}{\partial z}-g^{\prime 2}\tilde{E}_{1}=0.\] (43) For small values of $\delta\gamma$, $g^{\prime}\simeq g$, and the reduced discriminant $D=g^{2}-g_{I}^{2}$ of this second order differential equation is positive, which yields a complex gain coefficient with a positive real value: \[g(\delta\gamma)=\sqrt{g^{2}-g_{I}^{2}}+ig_{I},\] (44) where the imaginary part $g_{I}$ has the dimension of a wave-vector. In these conditions the field continues to exhibit gain, but with reduced values, and the electron population has a new dispersive effect. When $\delta\gamma$ becomes such that $g_{I}=g^{\prime}$, then the discriminant gets negative, and the gain take purely imaginary values, corresponding to oscillating solutions for $\tilde{E}_{1}$ and $\delta\tilde{x}$, of wavevectors $g_{I}\pm\sqrt{g_{I}^{2}-g^{2}}$, implying a regular exchange of energy between the field generated at $\omega_{1}$ and the population of electrons at $\delta\gamma$, and essentially no net transfer between the field and the electrons at the exit of the interaction region. To first order in $\delta\gamma$, the discriminant vanishes for $\delta\gamma/\gamma=g/k_{0}$. This defines what can be called an homogeneous spread as the relative width $2\delta\gamma/\gamma$ for which electrons contribute to a gain at $\omega_{1}$, and an homogeneous spectral width $\Delta\omega_{1}^{H}/\omega_{1}=4g/k_{0}$. In realistic conditions, the gain length is bound to be much larger than the laser wavelength, implying that the homogeneous width is likely to be smaller than the inhomogeneous Doppler width. In principle, this narrow homogeneous width should allow stimulated Raman scattering even in conditions of large electron energy spread. These elementary considerations will have to be revisited however in more general studies on the effects of electron energy spread. ### Effect of potential anharmonicity and bunch emittance We have so far assumed the electron wiggling to occur very close to the bottom of the ponderomotive potential, so that the harmonic potential approximation could hold. The assumption is valid if the initial transverse transverse velocity of the electrons at the start of the injection process is extremely small, as would result from a very good beam emittance. However, injection calculations from section 2.2, performed in conditions of the currently best achieved values of beam normalized emittance ($\epsilon_{N}=1\mu m$), show that a number of electrons may also depart from this approximation, and therefore display reduced oscillation frequencies in the ponderomotive potential. By analogy with usual lasers, we will consider each electron as occupying a ”site” given by its position in phase space, as resulting from the injection process, and corresponding to a unique trajectory $x_{0}(t)$; the forced transverse motion $\delta x$ follows the equation, extended from Eq. (15) : \[m\gamma\left(\ddot{x}_{0}+\delta\ddot{x}\right)+\frac{e^{2}E_{0}^{2}}{\gamma m \omega_{0}^{2}}\sin\left[2k_{0}(x_{0}+\delta x)\right]=F_{L},\] (45) where $F_{L}$ denotes again the Lorentz force; developing to second order with respect to $\delta x$, one obtains : \[\delta\ddot{x}+J_{0}(2k_{0}x_{0})\Omega^{2}\delta\ddot{x}=F_{L}/\gamma m\] (46) where $x_{0}$ is the maximum excursion of the electron in the potential well, and $J_{0}$ is the zeroth-order Bessel function of the first kind. We have neglected here periodic potential terms for $\delta x$, resulting in a Matthieu-type equation, but bound to average out to zero for many electrons. The oscillation eigenfrequency is then reduced with respect to the harmonic potential value, as : \[\Omega^{\prime}=\sqrt{J}_{0}(2k_{0}x_{0})\Omega,\] (47) The transverse displacement function $\delta\tilde{x}$ can easily be shown to follow : \[\left[\frac{\partial^{2}}{\partial z^{2}}-i\frac{\Delta\Omega^{2}}{2\Omega v}- \frac{e^{4}\rho k_{0}E_{0}^{2}}{4\epsilon_{0}\gamma^{3}m^{3}c^{2}v\Omega( \omega_{0}-\Omega)^{2}}\right]\delta\tilde{x}=0,\] (48) resulting into a modified gain factor : \[g^{\prime}=\sqrt{g^{2}-\frac{\Omega^{2}}{v^{2}}(2k_{0}x_{0})^{2}}\] (49) If the electron population is spread over a large distribution in transverse phase space, then the amplification spectrum is broadened following Eq. (47), with a reduced gain function depending on the frequency $\Omega^{\prime}$, given by Eq. (49). This situation is again typical of an inhomogeneously broadened laser line. The total spectral width of the lasing depends therefore on a combination of Doppler broadening, due to the finite $\delta\gamma/\gamma$ of the electron bunch, and of emittance broadening. The drawback of a reduced small signal gain is counter-balanced by an important advantage, namely, one can expect the scheme to be robust with respect to initial spreads in phase space, either longitudinally (energy spread) or transversally (emittance). ## 5 Experimental perspectives Several important issues have to be worked out to consider an experimental implementation of this Kapitza-Dirac-Raman X-ray free electron laser. We will not attempt to address all issues, but only to give order-of-magnitude parameters, in order to assess the general experimental feasibility of the proposed scheme. ### Implementation possibilities for the laser and electron acceleration systems We consider laser intensities at focus in the range from $10^{15}$ to $10^{18}W/cm^{2}$, and laser wavelengths of typically 800 $nm$ or 1.05 $\mu m$. Longer wavelength lasers, such as mid-infrared (resulting eg from an optical parametric chirped pulse amplification process) or far-infrared (CO${}_{2}$ lasers) may be advantageous, but are currently more difficult to implement. The laser pulse duration should be long enough for the pulse length to be larger that the active region, which corresponds to typical values between few femtoseconds and few hundreds of femtosecond. Trapping of electrons in the $y$-direction in the active area is an important issue. Several solutions can be considered; one may for instance adopt a 4-wave standing wave geometry, thus providing the same trapping in the $y$ direction as in the $x$ direction. It could offer the advantage of adding a degree of liberty to control the polarization of X-ray light, by controlling the polarization and dephasing of the $y$ lasers. A second possible solution would be to purposefully shear one beam with respect to the other in the $y$ direction; the standing wave would then be suppressed on both sides, thus creating lateral potential walls of $V_{p}/4$ (Eq. 6). A third possibility could be to irradiate a specially shaped a third beam along $x$, or to alter in a controlled way one of the two twin beams. Several options seem therefore possible, that have to be investigated. As concerns the electron acceleration setup, one should fully consider the opportunities of the two families of electron accelerators can be considered : conventional RF acceleration, or laser wakefield acceleration. The major advantage of laser acceleration is to provide extremely short bunches of electrons, with a corresponding very high current density. Moreover, synchronization between the laser-accelerated electron bunch, and the transverse twin lasers, can easily be performed with few femtosecond resolution, if the twin beams are derived from the same laser system, or at least from the same laser oscillator, as the intense laser inducing wakefield acceleration. Typical values of electron beam currents can reach 10 $kA$ or more, with good emittance values, and very small bunch transverse sizes, of the order of one to few $\mu m$. This scheme suffers from two potential drawbacks : the stability of the electron bunch after the exit of the accelerating plasma, which is the price to pay for such high current densities; and an important value of $\delta E/E$ , whose best measured values are currently in the few percents range. Drawback $(i)$ can be compensated if one succeeds to get hold of the electron beam in the laser standing wave almost immediately after the exit of the plasma; problem $(ii)$ could be strongly attenuated in the near future, as a number of numerical simulations suggest the possibility to improve mono-energeticity, through an enhanced control of the injection of electrons in the plasma wake. Generally speaking, laser-acceleration of electrons offers extremely promising prospects, especially if the energy dispersion can be reduced experimentally in the per-cent range. On the other hand, conventional RF acceleration in a LINAC is a well-known and more mature technology, with a number of existing systems proposing electron bunches up to few tens of MeV, with energy spreads below 1%, and a normalized emittances down to below 1 $mm.mrad$, especially thanks to the introduction of emittance compensation schemes. Typical peak current values are in the range of 100 $A$; use of magnetic chicanes, such as those set up for the Compton FEL laser projects LCLS and TESLA-XFEL, allows one to reach peak currents up to 3 kA, at the cost of an increased normalized emittance. An inherent difficulty of conventional RF acceleration is the synchronization issue between the incident electron bunch, and the interference region of the twin laser beams. However, the reliability, and control over conventional LINACs are very good, with the possibility to tune the electron energy, and to control the electron focal position and spot size. As a result of this alternative, we now present estimates of experimental parameters in both schemes. ### Prospective implementation parameters \| Laser plasma accelerator \| Low energy LINAC \| Medium energy LINAC ---\|---\|---\|--- Electron energy \| 60 MeV \| 10 MeV \| 155 MeV X-ray photon energy \| 43 keV \| 1.2 keV \| 220 keV Peak current \| 25 kA \| 2 kA \| 100 A Norm. emittance \| 1 mm.mrad \| 2 mm.mrad \| 1 mm.mrad Electron spot size σx \| 1 μm \| 30 μm \| 20 μm Laser wavelength \| 800 nm \| 800 nm \| 1.05 μm Laser pulse duration \| 30 fs \| 30 fs \| 400 fs Laser vertical spot size D \| 3 μm \| 3 μm \| 5 μm Laser intensity \| 1.4 1018 W.cm−2 \| 2.5 1016 W.cm−2 \| 1\. 1016 W.cm−2 Homogeneous gain \| 240 cm−1 \| 4 cm−1 \| 0.08 cm−1 Amplification length \| 420 μm \| 2.5 cm \| 1.1 m Total laser energy EL \| 1.8 J \| 1.1 J \| 360 J Reference \| malka ; davoine \| rosenzweig \| SPARC ; emit1 Table 1: Electron bunch, laser and interaction geometry parameters considered Based on the various technological approaches mentioned, we now suggest a few scaling laws and order-of-magnitude parameters for an experimental implementation. We first reformulate the gain formula, using standard experimental parameters. The beam density $\rho$ is not usually used, but should be deduced from the peak current $I$, and the equivalent electron focal spot $S$: \[I=e\rho Sc,\] (50) where $S=\sigma_{x}^{2}/2$ is the equivalent spot size, if we assume the electron focusing along $x$ and $y$ to be equivalent. The homogeneous gain formula (36) becomes : \[g=\sqrt{\frac{e^{2}IE_{0}}{2^{5/2}\epsilon_{0}\gamma^{2}m^{2}c^{4}S\omega_{0}}}.\] (51) For the sake of simplicity, we will rely on this homogeneous gain formula to discuss prospective experimental parameters; such effects as inhomogeneous broadening due to electron dispersion or finite beam emittance, or the transverse bunching of electrons within the standing wave, will be investigated in a full numerical study. The electron kinetic energy is fixed in a straightforward way by the ratio between the laser and the desired X-ray photon energies, related to first order by $\hbar\omega_{1}=2\gamma^{2}\hbar\omega_{0}$. The electron technology used to accelerate the electrons then provides fixed values for the emittance $\epsilon_{N}$ and mono-energeticity $\delta E/E$, summarized in few cases in table 1. Two main parameters have to be chosen at that stage : the transverse size $D$ of the active region, and the rms radius $\sigma_{x}$ of the electron spot size. At saturation, the X-ray output will be optimized if $D\simeq 2\sigma_{x}$; however, it may prove useful to concentrate on an active region smaller that the electron beam, in order to enhance the gain by concentrating the available laser power into a small volume. In laser wakefield acceleration, and direct injection into a light standing wave, $\sigma_{X}$ is unlikely to be a free parameter; in RF acceleration, there is on the contrary a certain flexibility to choose $\sigma_{x}$ by playing with the $\beta$ parameter of the focusing magnets. In all cases, we will assume for simplicity that the characteristic sizes are the same in the $x$ and $y$ directions. From the beam charge, size and duration, one can easily infer the electron current or density, which, coupled to realistic parameters for the dressing twin laser beams, allows one to deduce an order of magnitude of the small signal gain, in the homogeneous limit, and of the total laser energy required to reach a gain.length product of 10 in the electric field, or equivalently of 20 in X-ray intensity. Table 1 gives the result in the case (i), of an electron bunch resulting from laser wakefield acceleration [36; 35] (60 MeV, column 1), (ii) of an electron bunch issued from a state-of-the-art linear accelerator , with either a small (10 MeV, column 2) or medium (155 MeV, column 3) electron kinetic energy. In the first two cases, we make the assumption that the dressing laser is a Titanium-Sapphire system, with a pulse duration of 30 fs; in the last case, we consider typical parameters of a Neodymium-glass laser, with a pulse duration of 400 fs. The laser intensity is chosen so that the maximum ponderomotive potential of the light lattice is higher that the maximum transverse kinetic energy of the electrons in the bunch, resulting from the normalized emittance $\epsilon^{N}$ and the bunch size $\sigma_{x}$. The normalized emittance defines the rms $\sigma_{v}$ of the transverse velocity distribution : \[\sigma_{v}=\epsilon^{N}/\beta\gamma c\sigma_{x},\] (52) yielding an upper limit for the transverse kinetic energy of : \[K_{\perp}=0.5\gamma m\sigma_{v}^{2}.\] (53) We suggest a criterium of laser intensity to be defined by a ratio between the maximum ponderomotive potential of Eq. (6), expressed with the usual form $0.9I\lambda^{2}$, and $K_{\perp}$: \[0.9.(4I)\lambda^{2}/\gamma>5K_{\perp},\] (54) where the factor of 5 is here largely arbitrary. The factor of 4 originates from the beating between the twin lasers in the interaction region, as shown by Eq. (6). In the first column, we consider conditions of a laser-plasma accelerator predicted by Davoine _et al._[36], with an electron bunch accelerated to 60 MeV, a fairly conservative value, and a laser intensity of 1.4 10${}^{18}$ W/cm${}^{2}$. The corresponding X-ray energy is as high as 43 keV, beyond the upper limit of X-ray photons expected with Compton Free electron laser. One sees that the predicted small signal gain is extremely high, allowing the amplification process to reach saturation over a very short length, which limits the total energy used to values within the current state of the art in laser technology (Joule class short pulse lasers). In the second column, we apply the same procedure to the case of a photo-gun, yielding 10 MeV electrons, assumed to undergo emittance compensation, and beam compression devices, thereby reaching a high peak current of 2 kA, similar to that achieved at higher energies in the TESLA and LCLS projects. While these parameters are obviously very challenging, the corresponding beam brightness of 5 10${}^{14}$ A/(m.rad)${}^{2}$ remains well below the maximum value of 3.75 10${}^{15}$ A/(m.rad)${}^{2}$ predicted by Rosenzweig _et al._[37]. Finally, we take in the third column the characteristics of the SPARC system in Frascati [38; 39], with state-of-the-art emittance control, but peak current of the order of 100 A. The laser energy required to reach saturation is much higher in this case, but remains in the typical parameters for PetaWatt Nd:glass systems, like the VULCAN laser [40]. It shows however that other options can be considered than ultra-short pulse lasers, that may result in X-ray photon energies reaching the hard X-ray range. The values obtained in this table, especially for the required laser energies, should be considered merely as order of magnitudes; indeed, our scaling laws are based on a simple theoretical model, that neglects inhomogeneous broadening and diffraction effects, which will tend to lower the small-signal gain, and on the other hand neglects the increase in electron density in the bottom of the light potential wells, which will have the opposite effect. While more thorough studies are obviously required, these estimates do raise hope that the Raman X-ray laser scheme could be demonstrated with present day laser technology. ## 6 Perspectives and conclusion We have explored the specificities of a novel interaction geometry between a bunch of moderately relativistic electrons, and a standing wave formed by twin high intensity laser beams. We have shown numerically that, in the high intensity regime of Kapitza-Dirac effect, relativistic electrons may get trapped into the minima of the ponderomotive potential, and be guided until the end of the transverse standing wave. The electrons tend to oscillate close to the bottom of the potential wells, resulting into a Raman splitting of the scattered radiation in the forward direction. We have shown numerically and analytically that the Stokes component may be coupled back to the transverse electron motion, thereby triggering a stimulated Raman scattering. This can be considered as a new kind of free electron laser effect, in which the electron bunching is no longer longitudinal but transverse. The scheme seems to display the capability to accept less stringent parameters of bunch mono-energeticity. This specific robustness may be a key to develop X-ray free electron lasers in the interaction between high intensity lasers, and relativistic electron bunches. Many aspects of the proposed scheme remain however to be studied, both theoretically and experimentally, in order to ascertain its feasibility and its real potential for applications : electron injection regime, space charge effects, electron recoil effects, broadening mechanisms, effect of $y$-trapping on the electron dynamics and X-ray wave amplification, Stokes - anti-Stokes couplings, saturation, coherence properties, possibility of X-ray injection, more complex standing wave patterns… From an experimental point of view, several bottlenecks need to be solved, especially concerning the implementation of the inhomogeneous wave, and the synchronization between the electron bunch and the laser standing wave. The possibility to couple this scheme to setups of laser-plasma wakefield acceleration should be especially considered. If a number of positive answers for all these pending physics issues are obtained, and robust implementation schemes are designed, then this scheme may hold the potential to provide compact sources of intense coherent X-ray radiation, with a large number of potential applications in science, from physics to medicine, and technology. ## 7 Acknowledgements The author would like to acknowledge fruitful discussions with N. Artemiev, V. Malka, and V. Tikhonchuk. ## References * (1) M.A. Duguay and Rentzepis, Appl. Phys. Lett. 10 (1967) 350. * (2) P. Jaeglé _e_t al. Phys. Lett. A 36 (1971) 167. * (3) P. Jaeglé, _Coherent sources of XUV radiation_ (Springer Verlag, Berlin 2006) * (4) J. M. J. Madey, J. Appl. Phys. 42, (1971) 1906. * (5) D.A.G. Deacon, L.R. Elias, J.M.J. Madey, G.J. Ramian, H.A. Schwettman, and T.I Smith, Phys. Rev. Lett. 38 (1977) 892. * (6) J. B. Murphy and C. Pellegrini, J. Opt. Soc. Am. B 2, (1985) 259-264. * (7) C. Pellegrini and J. Stöhr, Nucl. Inst. Meth. A 500, (2003) 33-40. * (8) V. Ayvazyan _et al._, Eur. Phys. J. D 37, (2006) 297 303. * (9) P. Sprangle, A. Ting, E. Esarey, and A. Fisher, J. Appl. Phys. 72, (1992) 5032. * (10) P. Sprangle and A. T. Drobot, J. Appl. Phys. 50, (1979) 2652. * (11) P. Dobiasch, P. Meystre, and M. O. Scully, IEEE J. Quant. El. 19, (1983) 1812. * (12) J. Gea-Banacloche, G.T. Moore, R.R. Schlicher, M.O. Scully, and H. Walther, IEEE J. Quantum Electron. 23 (1987) 1558. * (13) J.C. Gallardo, R.C. Fernow, R. Palmer, and C. Pellegrini, IEEE J. Quant. El. 24, (1988) 1557. * (14) E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, _The physics of free electron lasers_ (Springer Verlag, Berlin 2000). * (15) A. Bacci, M. Ferrario, C. Maroli, V. Petrillo, and L. Serafini, Phys. Rev. ST - AB 9, (2006) 060704. * (16) V. Petrillo, L. Serafini, and P. Tomassini, Phys. Rev. ST - AB 11, (2008) 070703. * (17) F. Grüner _et al._, Appl. Phys. B 86, 431 (2007). * (18) K. Nakajima, Nature Physics 4, (2008) 92. * (19) R. Bonifacio, N. Piovella, M.M. Cola, L. Volpe, Nucl. Instr. Meth. A 577 (2007) 745 750. * (20) R. Bonifacio, N. Piovella, M.M. Cola, L. Volpe, A. Schiavi, G.R.M. Robb, Nucl. Instr. Meth. A 593, (2008) 69-74. * (21) H.K. Avetissian and G.F Mkrtchian, Phys. Rev. E 65, (2002) 046505. * (22) Ph. Zeitoun _et al._, Nature 431, (2004) 466. * (23) G. Lambert _et al._, Nature Physics 4, (2008) 296. * (24) A. Klisnick _et al._ J. Opt. Soc. Am. B 17, (2000) 1093. * (25) Zs. Bor, S. Szatmari, and A. Müller, Appl. Phys. B 32, (1983) 101 104. * (26) J.-C. Chanteloup _et al._, J. Opt. Soc. Am. B 17, (2000) 151. * (27) S. Sebban, L. Charreyre, and Ph. Balcou, to be published. * (28) G. Pretzler, A. Kasper, and K.J. Witte, Appl. Phys. B 70, (2000) 1 9. * (29) P. L. Kapitza and P. A. M Dirac, Proc. Cambridge Philos. Soc. 29, 297 (1933). * (30) D.L. Freimund, K. Aflatooni, H. Batelaan, Nature 413, Issue 6852 (2001), 142. * (31) T.W.B. Kibble, Phys. Rev. 150, (1968) 1060. * (32) P. H. Bucksbaum, D. W. Schumacher, and M. Bashkansky , Phys. Rev. Lett. 61, (1988) 1182. * (33) T. Shiozawa, _Classical Relativistic Electrodynamics_ (Springer Verlag, Berlin 2004) 199-208. * (34) H.-P. Schlenvoigt _et al._, Nature Physics (2008). * (35) V. Malka, private communication. * (36) X. Davoine _et al._, Phys. Rev. Lett. 102, (2009) 065001. * (37) J.B. Rosenzweig _et al._, Nucl. Inst. Meth. A 557 (2006) 87 93. * (38) D. Alesini _et al._, Nucl. Inst. Meth. A 528 (2004), 586 590. * (39) M. Ferrario _et al._, Phys. Rev. Lett. 99, (2007) 234801. * (40) C. Hernandez-Gomez _et al._, J. Phys. IV France 133 (2006) 555 559.
1904.08041	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 108601, "num_imgs": 0, "llama3_tokens_count": 42026 }	[]	# The Siegel variance formula for quadratic forms Naser T. Sardari Department of Mathematics, UW-Madison, Madison, WI 53706 ntalebiz@math.wisc.edu March 2, 2024 ###### Abstract. We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of a smooth sum over the square of a functional evaluated on the $B$-th Fourier coefficients of the vector valued holomorphic Siegel modular forms which are Hecke eigenforms and obtained by the theta transfer from $O_{A_{m\times m}}$. By using the Ramanujan bound on the Fourier coefficients of the holomorphic cusp forms, we give a sharp upper bound on this variance when $n=1$. As applications, we prove a cutoff phenomenon for the probability that a unimodular lattice of dimension $m$ represents a given even number. This gives an optimal upper bound on the sphere packing density of almost all even unimodular lattices. Furthermore, we generalize the result of Bourgain, Rudnick and Sarnak [2], and also give an optimal bound on the diophantine exponent of the $p$-integral points on any positive definite $d$-dimensional quadric, where $d\geq 3$. This improves the best known bounds due to Ghosh, Gorodnik and Nevo [7] into an optimal bound. ###### Contents Contents * 1 Introduction * 2 Proof of Theorem 1.1 * 3 Proof of Theorem 1.4 * 4 The Siegel variance formula * 5 Harmonic polynomials * 6 The oscillator representations and Weyl’s sums * 7 Proof of Theorem 1.8 ## 1. Introduction ### Statement of results In this section, we discuss two applications of the Siegel variance formula (Theorem 1.8 for $n=1$). #### 1.1.1. The cutoff phenomenon in large dimension Suppose that $A_{m\times m}$ is a positive definite symmetric integral matrix with determinant 1, and $C(A)$ denotes the genus of $A$ which is a finite set. It is well known that $C(A)$ has only two possibilities, namely even or odd unimodular lattices. There is a natural probability measure defined by Siegel [29] on $C(A)$: \[\mu_{s}(A_{i}):=\dfrac{\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}}{\sum_{A_{i}\in C(A)} \frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}},\] where $s\in\{0,1\}$ depending on $C(A)$ being even or odd, and $\|O_{A_{i}}(\mathbb{Z})\|$ is the size of the integral orthogonal group of $A_{i}.$ The first application is on bounding the probability that an odd integer $q$ (even number $2q$) is representable by an odd unimodular lattice (even unimodular lattice) of dimension $m$ with respect to $\mu_{s}$. Every integer (even integer) is representable over $p$-adic integers $\mathbb{Z}_{p}$ by an odd unimodular lattice (even unimodular lattice) of dimension $m\geq 4.$ This fact and an application of the Hardy-Littlewood circle method implies that every large enough integer (even integer) with respect to $m$ is representable by every odd unimodular (every even unimodular lattice). This is a version of our theorem which shows a cutoff phenomenon at point $q\sim\frac{m}{2\pi e}$ ($2q\sim\frac{m}{2\pi e}$) for the probability measure $\mu_{s}$. Theorem 1.1.: _Let $m$ be even and $q$ be an odd integer. We have_ \[\mu_{1}\big{(}\mathbf{x}^{\intercal}A\mathbf{x}=q\text{ for some }\mathbf{x} \in\mathbb{Z}^{m}\big{)}=\begin{cases}\leq 5113^{-t}&\text{ for }q\leq\frac{m} {2\pi e}+\frac{0.5}{2\pi e}\log(m)-t-1,\\ \geq 1-5133^{-t}&\text{ for }q\geq\frac{m}{2\pi e}+\frac{1.6}{\pi e}\log(m)+t, \end{cases}\] _where $0\leq t=o(m).$ Similarly for $\mu_{0},$ we have_ \[\mu_{0}\big{(}\mathbf{x}^{\intercal}A\mathbf{x}=2q\text{ for some }\mathbf{x} \in\mathbb{Z}^{m}\big{)}=\begin{cases}\leq 5113^{-t}&\text{ for }2q\leq\frac{m }{2\pi e}+\frac{0.5}{\pi e}\log(m)-t-1,\\ \geq 1-5113^{-t}&\text{ for }2q\geq\frac{m}{2\pi e}+\frac{2.6}{\pi e}\log(m)+t ,\end{cases}\] _where $0\leq t=o(m).$ Note that $5113=\lfloor e^{\pi e}\rfloor.$_ We give the proof of Theorem 1.1 in Section 2. We have the following conjecture. Conjecture 1.2.: _Let $q\geq(1+\epsilon)\frac{m}{2\pi e}$ for some fixed $\epsilon>0.$ Then $q$ is representable by every odd unimodular lattice of dimension $m\gg_{\epsilon}1.$ Similarly, every even integer $2q\geq(1+\epsilon)\frac{m}{2\pi e}$ is representable by every even unimodular lattice of dimension $m\gg_{\epsilon}1.$_ Remark 1.3.: _Let $\delta>0$ and $L$ be an even unimodular lattice of dimension $m$. Theorem 1.1 implies the sphere packing density of $L$ is less than $m^{2+\delta}2^{-m}$ with $\mu_{0}$-probability $1+O(m^{-\delta+\epsilon})$ for any $\epsilon>0.$ Moreover, if $A$ is an odd uniomodular lattices, then the sphere packing density is less than $m^{1+\delta}2^{-m}$ with $\mu_{1}$-probability $1-m^{-\delta+\epsilon}$ for any $\epsilon>0.$ The problem of studying unimodular lattices with large sphere packing densities has studied by several authors; see [3]. The best known upper bounds on the sphere packing density of unimodular lattices is $(1.424)^{m}2^{-m}$[21], while the best known lower bound for the density of lattices (not necessarily integral) is $m\log\log(m)2^{-m}$[30]. So, there is an exponential gap between the upper bound and the lower bound for the sphere packing density of unimodular lattices. We substantially improve the upper bound and show that the sphere packing density is $o(m^{2+\epsilon}2^{-m})$ for all but a tiny fraction of unimodular lattices with respect to the Siegel mass probability. Conjecture 1.2 implies the sphere packing density of even unimodular lattices is less than $(1+\epsilon)^{m}2^{-m}.$_ #### 1.1.2. Optimal equidistribution of the integral points on quadrics The second application is on the distribution of the integral points on quadrics. Suppose that $F(x_{1},\dots,x_{m})$ is a positive definite integral quadratic form in $m\geq 3$ variables with discriminant $D$. Let $N>0$ be an integer where $\gcd(N,2D)=1$, and define \[V_{N}(R):=\big{\{}(x_{1},\dots,x_{m}):F(x_{1},\dots,x_{m})=N,\text{ and }x_{i} \in R\text{ for }1\leq i\leq m\big{\}},\] where $R$ is any commutative ring. Assume that $V_{N}(\mathbb{Z}_{p})\neq\emptyset$ for every prime $p.$ Let $O_{F}$ be the orthogonal group associate to the quadratic form $F(x_{1},\dots,x_{m}).$ Note that $V_{1}(\mathbb{R})$ is a compact homogenous variety with the action of $O_{F}(\mathbb{R}).$ Let $\mu$ be the unique $O_{F}(\mathbb{R})$ invariant probability measure defined on $V_{1}(\mathbb{R}).$ Suppose that $k(x)$ is a fixed positive smooth function supported on $(-2,2),$ and $k(x)=1$ for $x\in(-1,1).$ Let \[K_{\eta}(\mathbf{x},\mathbf{y}):=C_{\eta}k\big{(}\frac{\sqrt{F(\mathbf{x}- \mathbf{y})}}{\eta}\big{)},\] where $\mathbf{x},\mathbf{y}\in V_{1}(\mathbb{R}),$$\eta\in\mathbb{R}$ and $C_{\eta}$ is a normalization factor such that \[\int_{V_{1}(\mathbb{R})}K_{\eta}(\mathbf{x},\mathbf{y})d\mu(\mathbf{y})=1.\] We note that $K_{\eta}(\mathbf{x},\mathbf{y})$ is a point-pair invariant function, which means \[K_{\eta}(g\mathbf{x},g\mathbf{y})=K_{\eta}(\mathbf{x},\mathbf{y})\] for every $g\in O_{F}(\mathbb{R}).$ Let $R_{F}(N):=\|V_{N}(\mathbb{Z})\|.$ For $m\geq 4$ and from $\gcd(N,2D)=1$ [23, Remark 1.7], it follows that (1.1) \[N^{m/2-1-\epsilon}\ll R_{F}(N)\ll N^{m/2-1+\epsilon}.\] For $m=3,$ we further assume that $N\neq t_{i}\mathbb{Z}^{2}$ for finitely many $\{t_{i}\}$ that defines the exceptional-type square classes; see [10]. Then by Siegel’s ineffective bound $L(1,\chi_{q})>q^{-\epsilon},$ we have the same bounds as in (1.1). Define (1.2) \[\text{Var}_{F}(N,\eta):=\int_{V_{1}(\mathbb{R})}\big{(}K_{\eta}(\mathbf{x},N)- R_{F}(N)\big{)}^{2}d\mu(\mathbf{x}),\] where $K_{\eta}(\mathbf{x},N):=\sum_{\mathbf{y}\in\frac{1}{\sqrt{N}}V_{N}(\mathbb{Z}) }K_{\eta}(\mathbf{x},\mathbf{y}).$ The following theorem is an application of our main results. Theorem 1.4.: _Let $F$ and $N$ be as above. For $m=3,$ suppose that $N\neq t_{i}\square$, where $\{t_{i}\}$ defines the finitely many exceptional-type square classes. Assume either of these assumptions_ * $m$ _is even,_ * $N=sl^{2}$ _for some bounded square free integer_ $s$_,_ * _Lindelöf hypothesis holds for the holomorphic modular forms._ _Then, we have_ \[\text{Var}_{F}(N,\eta)\ll\frac{N^{\epsilon}R_{F}(N)}{\eta^{m-1}},\] _where the implied constant in $\ll$ only depends on $F$ and $\epsilon>0.$_ We give the proof of Theorem 1.4 in Section 3. When $m=3$ Theorem 1.4 essentially follows from the work of Bourgain, Rudnick and Sarnak [2]. They verified, with respect to different statistical tests, that the distribution of the integral points on the 2-sphere is similar to the distribution of a Poisson process. However, for $m\geq 4$ it was observed by Wright [33, 34] as mentioned in [24] that they are big regions on $V_{1}(\mathbb{R})$ which repels the integral points. Let $C(\mathbf{x},\eta):=B(\mathbf{x},\eta)\cap V_{1}(\mathbb{R})$ be a cap of radius $\eta>0$ centered at $\mathbf{x}\in V_{1}(\mathbb{R}),$ where $B(\mathbf{x},\eta)$ is the Euclidean ball of radius $\eta>0.$ Wright [33, 34] showed that there are caps of size $\eta\gg N^{-\frac{1}{4}}$ which does not intersect $\frac{1}{\sqrt{N}}V_{N}(\mathbb{Z}).$ In [23], we proved that every cap of size $\eta\gg N^{-\frac{1}{4}+\delta}$ contains an integral point for $m\geq 5$ and every $\delta>0.$ The following corollary implies that on average the covering properties of $\frac{1}{\sqrt{N}}V_{N}(\mathbb{Z})$ for every $m\geq 4$ is optimal and is as good as the Poisson process. Corollary 1.5.: _Assume the conditions of Theorem 1.4. Let $\eta\gg R_{F}(N)^{-\frac{1}{(m-1)}+\epsilon}\sim N^{-\frac{m-2}{2(m-1)}+\epsilon}$, then all but a tiny fraction of the caps of $V_{1}(\mathbb{R})$ with the radius $\eta$ intersects $\frac{1}{\sqrt{N}}V_{N}(\mathbb{Z}).$ On the other hand if $\eta\ll R_{F}(N)^{-\frac{1}{(m-1)}-\epsilon}$ then only a tiny fraction of them intersect $\frac{1}{\sqrt{N}}V_{N}(\mathbb{Z}).$_ We give the proof of the Corollary 1.5 in Section 3. See also the work of Ellenberg, Michel and Venkatesh [5] for the non-archimedean version of the above corollary for $F=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ under the Linnik condition on $N$. When $N=p^{2k}$ for some fixed prime number $p$ and $F=x_{1}+\dots+x_{d}^{2}$ for $d=3,4$ the above corollary follows from the work of Ghosh, Gorodnik and Nevo[7]. We discuss two related applications of Theorem 1.4 in what follows. First, we recall the definition of the averaged covering exponent of the integral points on the sphere. Let $S^{m-1}(\mathbb{R})$ be the sphere of radius $1$ in $\mathbb{R}^{m}.$ Let $S^{m-1}_{N}(\mathbb{Z})$ be the set of integral points \[S^{m-1}_{N}(\mathbb{Z}):=\big{\{}(x_{1},\dots,x_{m})\in\mathbb{Z}^{m}:x_{1}^{2 }+\dots+x_{m}^{2}=N\big{\}},\] where $0\leq N\in\mathbb{Z}.$ We have $\frac{1}{\sqrt{N}}S^{m-1}_{N}(\mathbb{Z})\subset S^{m-1}(\mathbb{R}).$ Let $N_{\delta,\epsilon}$ denote the minimum integer such that all but $\epsilon^{\delta}$ fraction of caps $C(\mathbf{x},\epsilon)$ of size $\epsilon$ on $S^{m-1}(\mathbb{R})$ contain a point of $\frac{1}{\sqrt{N}}S^{m-1}_{N}(\mathbb{Z}).$ Sarnak defined [24] the averaged covering exponent of the integral points on the sphere by: \[\begin{split}\bar{K}_{m}&:=\lim_{\delta\to 0}\limsup _{\epsilon\to 0}\frac{\log\big{(}\#S^{m-1}_{N_{\delta,\epsilon}}(\mathbb{Z}) \big{)}}{\log\big{(}1/\text{vol }(C(\mathbf{x},\epsilon))\big{)}}.\end{split}\] By the Pigeonhole principle, it is easy to see that $\bar{K}_{m}\geq 1.$ Sarnak proved that $K_{4}=1$[24]. This implies the optimal covering properties of the golden quantum gates inside $SU(2)$; see [20], [25]. Sarnak’s method is based on the spectral theory of modular forms and uses the Ramanujan bound on the Fourier coefficient of the modular forms. It relies on the coincides that $S^{3}$ is isomorphic to the units of quaternions. In particular, the analogues result for $S^{m-1}$ does not follow. The following is a corollary of Theorem 1.4. Corollary 1.6.: _Assume that $m\geq 4$ is even. Then,_ \[\bar{K}_{m}=1.\] For $\mathbf{x}\in S^{m-1}(\mathbb{Q}),$ let $H(\mathbf{x}):=\prod_{q}\max_{1\leq i\leq m}(1,\|x_{i}\|_{q}),$ where $\|.\|_{q}$ is the $q$ adic valuation of $x_{i}.$ Fix a prime $p\equiv 1\mod 4.$ Then $S^{m-1}(\mathbb{Z}[1/p])$ is dense in $S^{m-1}(\mathbb{R}).$ We prove the following quantitative form of the diophantine properties of $S^{m-1}(\mathbb{Z}[1/p]).$ Corollary 1.7.: _Let $m\geq 3.$ For almost every $x\in S^{m-1}(\mathbb{R}),$$\delta>0$, and $\varepsilon\in(0,\varepsilon_{0}(x,\delta)),$ there exists $z\in S^{m-1}(\mathbb{Z}[1/p])$ such that_ \[\|x-z\|_{\infty}\leq\epsilon\text{ and }H(z)\leq\epsilon^{-\frac{m-1}{m-2}- \delta}.\] _We note this exponent is the best possible._ The above corollary answers a question of Ghosh, Gorodnik and Nevo; see [7, 8, 9]. By using the best bound on the generalized Ramanujan conjecture, they proved the above corollary [7, Page 12] for $m=3,4$ and the following exponents for $m\geq 5$ \[H(z)\leq\epsilon^{-2-\delta}\text{ for even }m,\text{ and }H(z)\leq\epsilon^{- \frac{2(m-1)}{m+2}-\delta}\text{ for odd }m.\] They raised the question of improving these bounds in [7, Page 11]. As pointed out above and in the abstract, we give a definite answer to this question. We give the proof of Corollary 1.6 and 1.7 in Section 3. ### The Siegel variance formula In this section we discuss our method. We introduce a variance sum associated to a pair of positive definite symmetric integral matrices and a smooth compactly supported function. By using the oscillator representation, we obtain a formula for this variance sum in terms of the Fourier coefficients of the homomorphic Siegel modular forms which are Hecke eigenform. We denote this formula by the Siegel variance formula; see (1.10). We apply this formula to prove Theorem 1.1 and Theorem 1.4. More generally, this can be used to study the distribution of the integral solutions of the representation of a quadratic form by another one. Let $A$ and $B$ be two positive definite symmetric integral matrices with dimensions $m$ and $n,$ respectively. Let $C(A):=\{A_{1},\dots,A_{h}\}$ be a representative set for the genus class of $A.$ Let \[V_{A_{i},B}(R):=\{\mathbf{X}\in M_{m\times n}(R):\mathbf{X}^{\intercal}A_{i} \mathbf{X}=B\},\] where $R$ is a commutative ring. We say $V_{A_{i},B}(R)$ is the set of $R$ points of the representation variety of $B$ by $A_{i}.$ Next, we associate a variance sum associated to $V_{A_{i},B}(\mathbb{Z})\subset V_{A_{i},B}(\mathbb{R})$ for each $A_{i}\in C(A).$ The variance sum only depends on a fixed smooth bump function of size $r$ defined on $\mathbb{R}^{n}$, and it is independent of the choice of the representative $A_{i}$ in its equivalence class. Note that $O_{A_{i}}(\mathbb{R})$ acts on $V_{A_{i},B}(\mathbb{R})$ by matrix multiplication, and this action is transitive for $m\geq n$. We begin by defining a point-pair $O_{A_{i}}(\mathbb{R})$ invariant function on the representation variety $V_{A_{i},B}(\mathbb{R}).$ Suppose that $k:\mathbb{R}^{n}\to\mathbb{R}$ is a fixed positive smooth function with compact support. Define $\|\mathbf{x}\|_{i}:=\sqrt{\mathbf{x}^{\intercal}A_{i}\mathbf{x}}$ for $\mathbf{x}\in\mathbb{R}^{m}$. Let (1.3) \[K_{r,B}(\mathbf{X},\mathbf{Y}):=C_{r,B}k\big{(}\frac{\|\mathbf{x}_{1}-\mathbf{y }_{1}\|_{i}}{r},\dots,\frac{\|\mathbf{x}_{n}-\mathbf{y}_{n}\|_{i}}{r}\big{)},\] where $\mathbf{X},\mathbf{Y}\in V_{A_{i},B}(\mathbb{R}),$$\mathbf{x}_{j}$ and $\mathbf{y}_{j}$ are the $j$-th column of $X$ and $Y$ respectively for $1\leq j\leq n,$ and $C_{r,B}$ is a constant where (1.4) \[\int_{V_{A_{i},B}(\mathbb{R})}K_{r,B}(\mathbf{X},\mathbf{Y})d\mu_{i}(\mathbf{X })=\int_{V_{A_{i},B}(\mathbb{R})}K_{r,B}(\mathbf{X},\mathbf{Y})d\mu_{i}( \mathbf{Y})=1,\] where $d\mu_{i}$ is invariant by the action of $O_{A_{i}}(\mathbb{R})$ and it is normalized such that $\int_{V_{A_{i},B}(\mathbb{R})}d\mu_{i}(\mathbf{Y})=1.$ Note that $K_{r,B}(\mathbf{X},\mathbf{Y})$ is a point-pair invariant function \[K_{r,B}(g\mathbf{X},g\mathbf{Y})=K_{r,B}(\mathbf{X},\mathbf{Y}),\] where $g\in O_{A_{i}}(\mathbb{R})$ and $g\mathbf{X}$ is the matrix multiplication. This implies $C_{r,B}$ is independent of $\mathbf{X}\in V_{A_{i},B}(\mathbb{R}).$ Finally, we define the following variance sum associated to $A_{i}$, $B$ and $r$ (1.5) \[\text{Var}(A_{i},B,r):=\int_{V_{A_{i},B}(\mathbb{R})}\Big{(}\big{(}\sum_{ \mathbf{Y}\in V_{A_{i},B}(\mathbb{Z})}K_{r,B}(\mathbf{X},\mathbf{Y})\big{)}-R_ {A_{i}}(B)\Big{)}^{2}d\mu_{i}(\mathbf{X}),\] where $R_{A_{i}}(B):=\|V_{A_{i},B}(\mathbb{Z})\|.$ We define the Siegel variance of representing $B$ by the genus class of $A$ at scale $r$ by: (1.6) \[\text{Var}(B,r):=\dfrac{\sum_{A_{i}}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}\Big{(} \text{Var}(A_{i},B,r)+\big{(}R_{A_{i}}(B)-R(B)\big{)}^{2}\Big{)}}{\sum_{A_{i} \in C(A)}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}},\] where (1.7) \[R(B):=\dfrac{\sum_{A_{i}\in C(A)}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}R_{A_{i}}(B) }{\sum_{A_{i}\in C(A)}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}},\] is the weighted number of the integral representation of $B$ by the genus class of $A_{i}.$ By the Siegel mass formula [28], we have (1.8) \[R(B)=\sigma_{\infty}(A,B)\prod_{p}\sigma_{p}(A,B),\] where \[\sigma_{p}(A,B):=\lim_{k\to\infty}\frac{\|\{X\in V_{A_{i},B}(\mathbb{Z}/p^{k} \mathbb{Z}\}\|}{p^{k(mn-n(n+1)/2)}},\] and \[\sigma_{\infty}(A,B):=\alpha(m,n)\|A\|^{\frac{-n}{2}}\|B\|^{\frac{m-n-1}{2}},\] where $\alpha(m,n)$ is a fixed constant which depends only on $m$, $n$; see [29]. Note that $\text{Var}(B,r)$ measures how uniform the integral points of the representation varieties of different genus classes are distributed among balls of size $r$. Before stating our main result, we introduce some notations from the theory of automorphic forms and the oscillator representation. We give the detailed descriptions of them in Section 4.2 and Section 6. Let $\mathbb{A}_{\mathbb{Q}}=\mathbb{R}\times\hat{\prod}_{p}^{\mathbb{Z}_{p}} \mathbb{Q}_{p}$ be the ring of adeles which is the restrictive direct product of $\mathbb{R}$ and $\mathbb{Q}_{p}$ with respect to $\mathbb{Z}_{p}.$ Fix $\mathbf{E}\in V_{A,I}(\mathbb{R})$ and the lattice $(\mathbb{Z}^{m},A).$ There exists $\sqrt{B}\in M_{n\times n}(\mathbb{R})$ such that $\sqrt{B}^{\intercal}\sqrt{B}=B.$ We also fix a choice of $\sqrt{B}$ for every positive definite symmetric matrix $B.$ We note that $\mathbf{E}_{B}:=\mathbf{E}\sqrt{B}\in V_{A,B}(\mathbb{R}).$ Let $O_{\mathbf{E},A}(\mathbb{R})\times O_{A}(\prod_{p}\mathbb{Z}_{p})$ be the stabilizer of $(\mathbf{E},\mathbb{Z}^{m})$ by the action of $O_{A}(\mathbb{R})\times O_{A}(\prod_{p}\mathbb{Q}_{p}),$ which is the same as the stabilizer of $(\mathbf{E}_{B},\mathbb{Z}^{m})$. This gives the following isomorphism: \[\bigcup_{A_{i}\in C(A)}O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{R}) =O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{\mathbf{E},A}( \mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p}).\] We write the following spectral decomposition (1.9) \[L^{2}\big{(}O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{ \mathbf{E},A}(\mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})\big{)}=\bigoplus_{\pi} \bigoplus_{j=1}^{d_{\pi}}\phi_{\pi,j},\] where $\bigoplus_{j=1}^{d_{\pi}}\phi_{\pi,j}$ is a finite sum over an orthonormal basis of $O_{\mathbf{E},A}(\mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})$ invariant harmonic polynomials which generate an irreducible automorphic representation isomorphic $\pi$ of $L^{2}\left(O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})\right)$ ($\pi$ may appear with multiplicity). More explicitly, the restriction of $\phi_{\pi,j}$ to $V_{A_{i},B}(\mathbb{R})$ is a harmonic polynomial with respect to $A_{i}.$ Moreover, $\phi_{\pi,j}$ generates an irreducible representation $\tau_{\pi_{\infty}}\times\pi_{\infty}=\tau(\lambda)\times\lambda$ by the action of $GL_{n}(\mathbb{C})\times O_{A}(\mathbb{C})$ on $M_{m\times n}(\mathbb{R})$, where $\tau(\lambda)\times\lambda$ is a finite dimensional irreducible representation of $GL_{n}(\mathbb{C})\times O_{A}(\mathbb{C})$ acting on $\mathcal{H}({\lambda})$ which is the $\lambda$ isotropic subspace of harmonic polynomials; see Section 5 and [19, Section 2.5.39]. By using the result of Kashiwara and Vergne [18, Section 6], one can describe the explicit parameters of $\tau(\lambda)\times\lambda$ in terms of the highest weight vectors. In Section 6, by using the oscillator representation and fixing an appropriate Siegel theta kernel, we associate a holomorphic Siegel modular form $\Theta(\phi_{\pi,j})(Z)$ with values in the vector space $\mathcal{H}(\lambda)^{}$ (dual vector space of $\mathcal{H}(\lambda)$). In Proposition 6.2, we describe explicitly the weight and the level of the associated Siegel modular form. In Proposition 6.4, we show that $\Theta(\phi_{\pi,j})(Z)$ is an eignform of the Hecke operators defined on the space of Siegel modular forms. We also express the Fourier coefficient of the associated Siegel modular forms in terms of the Weyl sums of the automorphic forms on the orthogonal group; see Theorem 6.5. Let $h_{r,B}(\pi_{\infty})$ be the spherical transformation of the point-pair invariant function $K_{r,B}$ at $\pi_{\infty}$, see equation (4.5). Let $\Theta(\phi_{\pi,j},B)\in\mathcal{H}(\lambda)^{}$ be the $B$-th Fourier coefficient of $\Theta(\phi_{\pi,j}).$ Furthermore, we define a harmonic polynomial $p_{\lambda,\mathbf{E}}\in\mathcal{H}(\lambda);$ see Section 5.2. Finally, we state our main theorem. Theorem 1.8.: _We have_ (1.10) \[\text{Var}(B,r)=\sum_{\pi}\|h_{r}(\pi_{\infty})\|^{2}\sum_{j=1}^{d_{\pi}}\left\| \langle{\tau_{\pi_{\infty}}(\sqrt{B})^{\intercal}}^{-1}\Theta(\phi_{\pi,j},B), p_{\lambda,\mathbf{E}}\rangle\right\|^{2}.\] Remark 1.9.: _Suppose that $n=1$ and $B=N\in\mathbb{Z}^{+}$. In the Siegel variance formula, the automorphic forms $\phi_{\pi,j}$ that are associated to the degree $k$ harmonic polynomials (the total dimension is $k^{m-2}$) that contribute to the variance are the one which are the theta lift from weight $k$ holomorphic modular forms defined on $SL_{2}$ (the dimension grows linearly in $k$). By comparing the dimension of them and using Howe one-to-one correspondence, it follows that $\Theta(\phi_{\pi,j},B)=0$ for all $\phi_{\pi,j}$ unless $\phi_{\pi,j}$ comes from a lift of $SL_{2}$ weight $k$ modular form. This and the Ramanujan bound $\|N^{-k/2}\Theta(\phi_{\pi,j},N)\|^{2}\ll N^{\frac{m}{2}-1}$ are the source of the equidistribution of the integral points at the optimal scale._ ### Further motivations and techniques In this section, we give the history behind the ideas in this paper. Siegel in his study of the Hasse-Minkowski theorem generalized the classical holomorphic modular forms into Siegel modular forms. He showed that the averaged representation number of a positive definite integral symmetric matrix $B_{n\times n}$ by the genus class of $A_{m\times m}$ is the $B$-th Fourier coefficient of the theta series associated to the genus class of $A$, which is a holomorphic Siegel modular form (Eisenstein series) [29]. Weil [31, 32] gave a group theoretic interpretation of Siegel’s work and introduced the oscillator representation of the metaplictic group (double cover of the symplectic group). Shintani [27] used the oscillator representation and described the Shimura correspondence [26] between the weight $k+1/2$ holomorphic modular forms and the weight $2k$ holomorphic modular forms. Moreover, Shintani showed that the average of the integral weight modular forms $f$ over over the closed geodesics with discriminant $D$ (Weyl sums) is the $D$-th Fourier coefficient of $\theta(f)$, where $\theta(f)$ is the theta transfer of $f$; see the work of Katok and Sarnak [17] for the Maass forms. In particular, the equidistibution of the CM points or closed geodesics of a given discriminant on the modular curve follows from a sub-convex bound on the Fourier coefficients of the weight $1/2$ integral modular forms which was achieved by Iwaniec [13] for holomorphic and Duke [4] for Maass forms. Our main observation is that by using the oscillator representation and the spectral theory of the metaplictic group one can prove equidistribution results for the integral points on the homogenous variety of an orthogonal group (a different group!). One aim of this paper is to generalize Shintani’s correspondence and give a correspondence from the classical automorphic forms of the orthogonal groups to the Siegel modular forms. We describe explicitly the weight (which is a finite dimensional representation of $GL_{n}(\mathbb{C})$) and the level of the associated Siegel modular form; see Proposition 6.2. We also express the Fourier coefficient of the associated Siegel modular forms in terms of the Weyl sums of the automorphic forms on the orthogonal group; see Theorem 6.5. We use this identity to prove some new optimal results for the distribution of the integral points on homogenous varieties. Ghosh, Gorodnik and Nevo [7, 8, 9] and Sarnak [24] used the spectral theory of automorphic forms for proving some optimal results on the distribution of integral points on homogenous varieties if the associate automorphic spectrum satisfies the generalized Ramanujan conjecture [22]. Our approach is different and give some optimal results which are not achievable by the previous methods. Our main idea is to generalize the work of Shintani to the dual pairs of reductive groups $(G,G^{\prime})$ in a symplectic group [11] and relate the Weyl sums on a homogenous variety $X$ of $G$ to the period integrals of the image of the theta transfer of automorphic forms from $G$ to $G^{\prime}.$ The theta transfer has a large kernel, and as a result for all but a tiny fraction of automorphic forms of $G$, the associated Weyl sum is zero! For the remaning non-zero theta transfers, we use bounds on the generalized Ramanujan conjecture for $G^{\prime}$. This strategy gives some new optimal results; see Theorem 1.1 and Theorem 1.4, only if the automorphic spectrum of $G^{\prime}$ satisfies the Ramanujan conjecture and not necessarily the automorphic spectrum of $G$! (or even the image of the automorphic spectrum of $G$ under the theta transfer which lies inside the automorphic spectrum of $G^{\prime}$ satisfies an average version of the generalized Ramanujan conjecture). This is a new feature of our work compare to the work of Ghosh, Gorodnik and Nevo [7, 8, 9] and Sarnak [24]; see Corollary 1.6 and the discussion after it. In this paper, we work with the dual pair $(G,G^{\prime})=(O_{m},Sp_{n})\subset Sp_{mn}(\mathbb{Q})$, where $O_{m}$ is compact at the archimedean place and $X=M_{m\times n}$ is the $m\times n$ matrices. More concretely, we use the oscillator representation in order to relate the distribution of the integral points on the representation variety of pairs of positive definite symmetric integral matrices, to bound the Fourier coefficient of the Siegel modular forms. Bounding the Fourier coefficients of the classical modular forms has been extensively studied after Ramanujan’s conjecture. The natural generalization of the weight $k$ holomorphic modular forms are the vector valued Siegle modular forms with a weight $\rho:GL_{n}(\mathbb{C})\to V_{\rho}$, where $\rho$ is a finite dimensional complex representation. Unfortunately, there are very few results on bounding the Fourier coefficients of the vector valued Siegel modular forms with respect to a norm or a functional on $V_{\rho}$. Kitaoka [15, 14] generalized the Kloosterman’s method and proved the analogue of the Kloosterman’s bound when $n=2$ and $\rho$ is one dimensional. Böcherer and Raghavan [1] generalized the Rankin-Selberg method for general $n$ and one dimensional $\rho.$ We refer the reader to the work of Kohnen [16] for further discussions and the expected optimal bound when $\rho$ is one dimensional. It seems that the only known results are when $\rho$ is one dimensional. This is partly caused by the lack of the interesting applications. We give some classical application of this problem. In particular, we show that an average version of the Ramanujan bound on the Fourier coefficients of the vector valued Siegel modular forms implies the equdistribution of the integral points on the representation variety of pairs of quadratic forms at the optimal scale. In particular, our results are optimal for $n=1$; see Theorem 1.1 and Theorem 1.4. ### Acknowledgements I would like to thank Prof. Simon Marshall, Prof. Zeev Rudnick, and Prof. Peter Sarnak for their comments on the earlier version of this manuscript. ## 2. Proof of Theorem 1.1 In this section we give a proof of Theorem 1.1. We use Theorem 1.8 and a proposition, which we formulate next. Recall that $A_{m\times m}$ is a positive definite integral matrix and consider the lattice $(\mathbb{Z}^{m},A).$ In the following propositon, we give an upper bound on the number of root vectors of $A$. Recall that $\mathbf{v}\in\mathbb{Z}^{m}$ is a root vector, if $\mathbf{v}^{\intercal}A\mathbf{v}=2$ or 1. Proposition 2.1.: _The number of root vectors of $A_{m\times m}$ of length 1 and length $\sqrt{2}$ is less than $2m$ and $10m^{2},$ respectively._ We give a proof of this proposition at the end of this section. We assume this proposition and Theroem 1.8, and proceed to give a proof of Theorem 1.1. ### Proof of Theorem 1.1 Proof.: We assume that $A$ is an even unimodular lattice. The proof for the odd unimodular lattice is similar, and we briefly discuss it at the end. Let $R(N)$ be the representation mass of even integer $N$ by the genus class of $A$ that is defined in (1.7). By the Siegel mass formula (1.8) and the explicit formulas for the local densities; see [30, Lemma 2], we have (2.1) \[R(N)=\dfrac{mN^{m/2-1}\pi^{m/2}}{\Gamma(m/2+1)}(1+O(2^{-m/4})).\] By the Stirling’s formula $\Gamma(m/2+1)=\sqrt{\pi m}\big{(}\frac{m}{2e}\big{)}^{m/2}(1+O(1/m)).$ By choosing $N=\lfloor\frac{m}{2\pi e}+\frac{\log(m)}{2\pi e}-1\rfloor$, we have \[R(N)\leq\dfrac{2\pi e}{e^{\pi e}\sqrt{\pi}}(1+O(1/m))\leq 1.\] Let $q$ be an odd number such that $2q\leq\frac{m}{2\pi e}+\frac{\log(m)}{2\pi e}-t-1,$ where $0\leq t=o(m).$ Then (2.2) \[\frac{R(2q+2)}{R(2q)}=\big{(}1+\frac{4\pi e}{m}\big{)}^{m/2-1}(1+o(1))=e^{2\pi e }(1+o(1))\geq 5113^{2}.\] Hence, \[R(2q)\leq 5113^{-t}.\] Note that \[\mu_{0}\big{(}\mathbf{x}^{\intercal}A\mathbf{x}=2q\text{ for some }\mathbf{x} \in\mathbb{Z}^{m}\big{)}\leq\dfrac{\sum_{A_{i}\in C(A)}\frac{1}{\|O_{A_{i}}( \mathbb{Z})\|}R_{A_{i}}(2q)}{\sum_{A_{i}\in C(A)}\frac{1}{\|O_{A_{i}}(\mathbb{Z} )\|}}=R(2q)\leq 5113^{-t}.\] This implies the first part of Theorem 1.1. Next, we give a proof of the second part of Theorem 1.1. Assume that $q\geq\frac{m}{2\pi e}+2.6\frac{\log(m)}{\pi e}+t$, where $0\leq t=o(m).$ We use the trivial point-pair invariant function $K(\mathbf{x},\mathbf{y})=1$ in (1.5), and obtain Suppose that $R_{A_{i}}(q)=0$ for $\alpha$ proportion of $A_{i}$ with respect to the Siegel mass probability. Then, (2.3) \[\text{Var}(2q)\geq\alpha R(2q)^{2}.\] One the other hand, by Theorem 1.8, and the fact that $K(\mathbf{x},\mathbf{y})=1$, which implies $\pi_{\infty}=1$ and $\deg(\pi_{\infty})=0$ in the expansion of (1.10), we have \[\text{Var}(2q)=\sum_{\pi}\|\Theta(\phi_{\pi,j},2q)\|^{2},\] where the sum is over $\pi$ with $\pi_{\infty}=1.$ By Proposition 6.2 and 6.4, $\Theta(\phi_{\pi,j})$ is a holomorphic cusp form of weight $m/2$ and level dividing 8. Hence, we have the following multiplicative relation \[\Theta(\phi_{\pi,j},2q)=\Theta(\phi_{\pi,j},2)\lambda_{\Theta(\phi_{\pi,j})}(q)\] where $\lambda_{\Theta(\phi_{\pi,j})}(q)$ is the $q$-th Hecke eigenvalue of $\Theta(\phi_{\pi,j}).$ Since $m$ is even, by the Ramanujan bound on the Hecke eigenvalues of homomorphic cusp forms, we have \[\|\lambda_{\Theta(\phi_{\pi,j})}(q)\|^{2}\leq d(q)^{2}q^{m/2-1},\] where $d(q)$ is the number of divisors of $q.$ Therefore, we have (2.4) \[\begin{split}\text{Var}(2q)=\sum_{\pi}\|\Theta(\phi_{\pi,j},2q)\|^{ 2}&=\sum_{\pi}\|\Theta(\phi_{\pi,j},2)\|^{2}\|\lambda_{\Theta(\phi_{ \pi,j})}(q)\|^{2}\\ &\leq d(q)^{2}q^{m/2-1}\sum_{\pi}\|\Theta(\phi_{\pi,j},2)\|^{2}\\ &=d(q)^{2}q^{m/2-1}\text{Var}(2),\end{split}\] where we used \[\text{Var}(2)=\sum_{\pi}\|\Theta(\phi_{\pi,j},2)\|^{2}.\] By definition, we have \[\begin{split}\text{Var}(2)&\leq\dfrac{\sum_{A_{i}} \frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}\Big{(}\big{(}R_{A_{i}}(2)-R(2)\big{)}^{2} \Big{)}}{\sum_{A_{i}}\frac{1}{O_{A_{i}}}}\\ &\leq\max_{i}{R_{A_{i}}(2)}\dfrac{\sum_{A_{i}}\frac{1}{\|O_{A_{i}} (\mathbb{Z})\|}\big{\|}R_{A_{i}}(2)-R(2)\big{\|}}{\sum_{A_{i}}\frac{1}{O_{A_{i}}} }\\ &\leq 2\max_{i}{R_{A_{i}}(2)}R(2).\end{split}\] By Proposition 2.1, \[\max_{i}{R_{A_{i}}(2)}\leq 10m^{2}.\] Hence, \[\text{Var}(2)\leq 20m^{2}R(2).\] By equation (2.4), we have \[\text{Var}(2q)\leq 20m^{2}R(2)q^{m/2-1}d(q)^{2}.\] By the Siegel Mass formula in (2.1), we have \[R(2q)=R(2)q^{m/2-1}(1+O(2^{-m/4})).\] Hence, \[\text{Var}(2q)\leq 20m^{2}R(2q)d(q)^{2}.\] We compare this upper bound with the lower bound (2.3) and obtain \[\alpha R(2q)^{2}\leq 20R(2q)m^{2}d(q)^{2}.\] By the above and (2.2), we have \[\alpha\leq\frac{20m^{2}d(q)^{2}}{R(2q)}\ll\frac{m^{2+\epsilon}}{m^{2.1}(5113)^ {t}}\leq(5113)^{-t},\] where we used the asymptotic formula $2q\sim\frac{m}{2\pi e}.$ This completes the proof of our theorem for even unimodular lattices. The argument for odd unimodular lattices is similar. The improved bound in the case of the odd unimodular is due to our upper bound $2m$ on the number of root vectors of length 1 in Proposition 2.1. ∎ ### Proof of Proposition 2.1 We begin by proving some auxiliary lemmas. Lemma 2.2.: _Assume that the root vectors of norm $1$ of $A$ spans $\mathbb{Z}^{m}$. Then $A$ is isomorphic to the identity matrix $I$. Moreover, then number of root vectors of length $1$ is $2m,$ and the number of the root vectors of norm $\sqrt{2}$ is $2m(m-1)$._ Proof.: Let $B:=\{\mathbf{v}_{1},\dots,\mathbf{v}_{m}\}$ be a basis of root vectors such that $\mathbf{v}_{i}^{\intercal}A\mathbf{v}_{i}=1.$ By the Cauchy-Schwarz inequality, we have \[\|\mathbf{v}_{i}^{\intercal}A\mathbf{v}_{j}\|<\big{(}\|\mathbf{v}_{i}^{\intercal} A\mathbf{v}_{i}\|\|\mathbf{v}_{j}^{\intercal}A\mathbf{v}_{j}\|\big{)}^{1/2}=1,\] for $i\neq j.$ Since $\mathbf{v}_{i}$ and $A$ are integral for $1\leq i\leq m$, it follows that $\mathbf{v}_{i}^{\intercal}A\mathbf{v}_{j}=0.$ Hence, $B$ is an orthonormal basis, which implies $A$ is isomorphic to $I.$ It is easy to check that the root vectors of length 1 are $\{\pm\mathbf{v}_{1},\dots,\pm\mathbf{v}_{m}\}.$ By a simple counting, the number of root vectors of length $\sqrt{2}$ is $2m(m-1).$ This completes the proof of our lemma. ∎ Lemma 2.3.: _Assume that $A$ does not have any root vector of length 1, and there exists an orthogonal basis of root vectors of length $\sqrt{2}$ (for $\mathbb{R}^{m}$ not necessarily the lattice!) Then the number of the root vectors of length $\sqrt{2}$ is less than $6m^{2}-4m$._ Proof.: Let $B:=\{\mathbf{v}_{1},\dots,\mathbf{v}_{m}\}$ be an orthogonal basis of root vectors of length $\sqrt{2}.$ Let $\mathbf{u}\notin\pm B$ be a root vector of length $\sqrt{2}.$ Let $[\mathbf{u}]:=(\mathbf{u}^{\intercal}A\mathbf{v}_{j})\in\mathbb{Z}^{m}$, where $\mathbf{v}_{j}\in B.$ Let $\#\mathbf{u}$ denote the number of non-zero coordinates of $[\mathbf{u}].$ By the plancherel identity, we have \[2=\sum_{\mathbf{v}_{j}\in B}\frac{\|\mathbf{u}^{\intercal}A\mathbf{v}_{j}\|^{2}} {2}.\] Since $\mathbf{u}\notin\pm B$, it follows that $\#\mathbf{u}=4$ and $\mathbf{u}^{\intercal}A\mathbf{v}_{j}=\pm 1$ or 0. Let $S:=\{\mathbf{u}_{1},\dots,\mathbf{u}_{R}\}$ denote the set of all root vector $\mathbf{u}$ of length $\sqrt{2}$, where $\mathbf{u}\notin\pm B.$ Let $M_{R\times m}:=[\mathbf{u}_{i}^{\intercal}A\mathbf{v}_{j}]$, for $\mathbf{u}_{i}\in S$ and $\mathbf{v}_{j}\in B.$ In what follows, we given an upper bound on $R.$ Each row $[\mathbf{u_{i}}]$ contains exactly four $\pm 1$ and zero at other entries. So, the matrix $M$ contains $4R$ nonzero elements. By a pigeon-hole argument there exits a column, associated to $\mathbf{v}_{j}$ for some $j$, which contains at least $\frac{4R}{m}$ non-zero elements which are $\pm 1$. Without loss of generality, suppose that $\mathbf{u}_{i}^{\intercal}A\mathbf{v}_{1}=\pm 1$ for $1\leq i\leq\frac{4R}{m}.$ By the plancherel identity, \[\mathbf{u}_{i}^{\intercal}A\mathbf{u}_{j}=1/2+\sum_{k>1}\frac{(\mathbf{u}_{i}^ {\intercal}A\mathbf{v}_{k})(\mathbf{u}_{j}^{\intercal}A\mathbf{v}_{k})}{2}\in \mathbb{Z},\] where $1\leq i,j\leq\frac{4R}{m}.$ The integrality of the inner product implies $\mathbf{u}_{i}$ and $\mathbf{u}_{j}$ are non-zero at either 1 or 3 other columns. Without loss of generality assume that $\mathbf{u}_{1}^{\intercal}A\mathbf{v}_{j}\neq 0$ for $1\leq j\leq 4.$ Then for $1\leq i\leq\frac{4R}{m}$$\mathbf{u}_{i}^{\intercal}A\mathbf{v}_{j}\neq 0$ for some $2\leq j\leq 4.$ By a pigeon-hole argument for some $2\leq j\leq 4$ there are more than $\frac{4R}{3m}$, $\mathbf{u}_{i}$ such that $\mathbf{u}_{i}^{\intercal}A\mathbf{v}_{j}\neq 0.$ Without loss of generality assume that $1\leq i\leq\frac{4R}{3m},$ we have $\mathbf{u}_{i}^{\intercal}A\mathbf{v}_{2}\neq 0.$ Finally by the integrality of $\mathbf{u}_{i}^{\intercal}A\mathbf{u}_{j}$, it follows that \[\frac{4R}{3m}\geq 8(m-2).\] This implies $R\leq 6m(m-2)$. So the total number of root vectors of length $\sqrt{2}$ is less than $6m^{2}-4m.$ This completes the proof of our lemma. ∎ Lemma 2.4.: _Assume that $A$ does not have any root vector of length 1. Then the number of the root vectors of length $\sqrt{2}$ is less than $10m^{2}-4m$._ _Proof._: Let $T=\{\mathbf{w}_{1},\dots,\mathbf{w}_{q}\}$ be a maximal set of orthogonal root vectors of norm $\sqrt{2}.$ By Lemma 2.3, the number of the root vectors which are in the span of $T$ is less than $6q^{2}-4q$. We proceed and show that the number of root vectors which are not in the span of $T$ is less than $4q^{2}.$ Suppose that $\mathbf{u}$ is a root vector, and $\mathbf{u}\notin\text{span}(T).$ Let $[\mathbf{u}]:=(\mathbf{u}^{\intercal}A\mathbf{w}_{i})$, where $\mathbf{w}_{i}\in T.$ Let $\#\mathbf{u}$ denote the number of non-zero coordinates of $[\mathbf{u}].$ We show that $\#\mathbf{u}=1,2.$ By the plancherel inequality, we have \[2=\mathbf{u}^{\intercal}A\mathbf{u}\geq\sum_{\mathbf{w}_{i}\in T}\frac{\| \mathbf{u}^{\intercal}A\mathbf{w}_{i}\|^{2}}{2}.\] This shows that $\#\mathbf{u}\leq 4$. The maximality assumption of $T$ excludes $\#\mathbf{u}=0$, and $\mathbf{u}\notin\text{span}(T)$ excludes $\#\mathbf{u}=4$. Suppose that $\#\mathbf{u}=3$ and $\mathbf{w}_{1}$, $\mathbf{w}_{2}$ and $\mathbf{w}_{3}$ have non-zero inner product with $\mathbf{u}$. Then we define \[\mathbf{u}^{\prime}:=2\mathbf{u}-((\mathbf{u}^{\intercal}A\mathbf{w}_{1}) \mathbf{w}_{1}+(\mathbf{u}^{\intercal}A\mathbf{w}_{2})\mathbf{w}_{2}+(\mathbf{ u}^{\intercal}A\mathbf{w}_{3})\mathbf{w}_{3})\in\mathbb{Z}^{m}.\] We have ${\mathbf{u}^{\prime}}^{\intercal}A\mathbf{u}^{\prime}=2$. Note that $\mathbf{u}^{\prime}$ is a root vector and is orthogonal to all vectors in $T,$ which contradicts with the maximality of $T.$ So, the only possibilities for $\#\mathbf{u}$ are 1 or 2. Let $m(\mathbf{f})$ be the number of roots vectors $\mathbf{v}\notin\text{span}(T)$ such that $[\mathbf{v}]=\mathbf{f}.$ We claim that $m(\mathbf{f})\leq 2$. First, suppose that $\#\mathbf{f}=2,$ and without loss of generality $\mathbf{f}=(1,1,0,\dots,0).$ Assume the contrary that $m(\mathbf{f})>2.$ Then for some root vectors $\mathbf{u}_{1}$, $\mathbf{u}_{2}$ and $\mathbf{u}_{3}$ outside $\text{span}(T)$, we have $[\mathbf{u_{1}}]=[\mathbf{u_{2}}]=[\mathbf{u_{3}}]=\mathbf{f}$. We show that $\mathbf{u}_{i}$ are orthogonal to each other. Assume the contrary that $\mathbf{u}_{1}^{\intercal}A\mathbf{u}_{2}\neq 0.$ Then, $\mathbf{u}_{1}^{\intercal}A\mathbf{u}_{2}=\pm 1.$ Assume that $\mathbf{u}_{1}^{\intercal}A\mathbf{u}_{2}=-1.$ Then $\mathbf{u}_{1}+\mathbf{u}_{2}$ is a root vector. By plancherel inequality, we have \[2=(\mathbf{u}_{1}+\mathbf{u}_{2})^{\intercal}A(\mathbf{u}_{1}+\mathbf{u}_{2}) \geq\sum_{\mathbf{w}_{i}\in T}\frac{\|(\mathbf{u}_{1}+\mathbf{u}_{2})^{ \intercal}A\mathbf{w}_{i}\|^{2}}{2}=2\#\mathbf{u}\geq 4,\] which is a contradiction. So $\mathbf{u}_{1}^{\intercal}A\mathbf{u}_{2}=1$, and $\mathbf{u}_{1}-\mathbf{u}_{2}$ is a root vector which is orthogonal to $T.$ This contradicts with the maximality of $T$. Hence $\mathbf{u}_{1}$, $\mathbf{u}_{2}$ and $\mathbf{u}_{3}$ are orthogonal root vectors. Substitute $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ from $T$ with $\mathbf{u}_{1}$, $\mathbf{u}_{2}$ and $\mathbf{u}_{3}.$ Then the new set is an orthogonal set of root vectors which has 1 more element than $T.$ This contradicts with the maximality of $T.$ This shows that $m(\mathbf{f})\leq 2$ when $\#\mathbf{f}=2.$ Next, suppose that $\#\mathbf{f}=1$ and without loss of generality $\mathbf{f}=(1,0,\dots,0).$ Similarly, assume the contrary that $\mathbf{u}_{1}$, $\mathbf{u}_{2}$ and $\mathbf{u}_{3}$ are root vectors outside $\text{span}(T)$ with $[\mathbf{u_{1}}]=[\mathbf{u_{2}}]=[\mathbf{u_{3}}]=\mathbf{f}.$ We claim that $\mathbf{u}_{i}^{\intercal}A\mathbf{u}_{j}\neq 0$ for every $1\leq i,j\leq 3.$ Assume the contrary that $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ are orthogonal to each other then substitute $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ with $\mathbf{w}_{1}$. This again contradicts with the maximality of $T.$ Next, we show that $\mathbf{u}_{1}^{\intercal}A\mathbf{u}_{2}\neq 1.$ Otherwise, $\mathbf{u}_{1}-\mathbf{u}_{2}$ is a root vector which is orthogonal to all vectors in $T$ and this also contradicts with the maximality of $T.$ Hence $\mathbf{u}_{i}^{\intercal}A\mathbf{u}_{j}=-1$ for every $i\neq j.$ Then $\mathbf{u}_{1}+\mathbf{u}_{2}$ is a root vector. By plancherel inequality, we have \[2=(\mathbf{u}_{1}+\mathbf{u}_{2})^{\intercal}A(\mathbf{u}_{1}+\mathbf{u}_{2}) \geq\frac{\|(\mathbf{u}_{1}+\mathbf{u}_{2})^{\intercal}A\mathbf{w}_{1}\|^{2}}{2} =2,\] which implies $\mathbf{u}_{1}+\mathbf{u}_{2}=\mathbf{w}_{1}.$ Similarly, we have $\mathbf{u}_{1}+\mathbf{u}_{3}=\mathbf{w}_{1},$ and $\mathbf{u}_{2}+\mathbf{u}_{3}=\mathbf{w}_{1}.$ This implies $\mathbf{u}_{1}=\mathbf{u}_{1}=\mathbf{u}_{1}=\mathbf{w}_{1}/2$ which is a contradiction. Therefore $m(\mathbf{f})\leq 2$ for every $f\in\mathbb{Z}^{q}.$ Note that there are at most $2q$ vectors $\mathbf{f}\in\{0,\pm 1\}^{q}$ with $\#\mathbf{f}=1$. Since $m(\mathbf{f})\leq 2$, there are at most $4q$ root vectors $\mathbf{u}\notin\text{span }T$ such that $\#[\mathbf{u}]=1.$ Similarly, there are at most $2q(q-1)$ vectors $\mathbf{f}\in\{0,\pm 1\}^{q}$ with $\#\mathbf{f}=2,$ and that implies there are at most $4q(q-1)$ root vectors $\mathbf{u}\notin\text{span }T$ such that $\#[\mathbf{u}]=2.$ Therefore, the total number of root vectors $\mathbf{u}$, where $\mathbf{u}\notin\text{span}(T),$ is less than $4q^{2}.$ This completes the proof of our lemma. ∎ Finally, we give a proof of Proposition 2.1. Proof of Proposition 2.1.: Let $S:=\{\pm\mathbf{v}_{1},\dots,\pm\mathbf{v}_{p}\}$ be the set integral vectors such that $\mathbf{x}^{\intercal}A\mathbf{x}=1.$ By Lemma 2.2, it follows that $\{\mathbf{v}_{1},\dots,\mathbf{v}_{p}\}$ is an orthonormal set of vectors. Hence, the number of root vectors of length 1 is less than $2m.$ Let $V:=\text{span}_{\mathbb{Z}}\{\mathbf{v}_{1},\dots,\mathbf{v}_{p}\}\subset \mathbb{Z}^{m}$ be the lattice generated with the root vectors of length 1. It is easy to see that $\mathbb{Z}^{n}=V\oplus V^{\perp},$ where $V^{\perp}\subset\mathbb{Z}^{m}$ is the orthogonal complement of $V\subset\mathbb{Z}^{m}$ with respect to $A.$ By our assumption, all the root vectors of $V^{\perp}$ has length $\sqrt{2}$ (there is no root vector of length 1 in $V^{\perp}$). Moreover, if $u$ is any root vector with length $\sqrt{2}$ then either $u\in V$ or $u\in V^{\perp}.$ By Lemma 2.2, the number of root vectors of length $\sqrt{2}$ in $V$ is less than $2p(p-1).$ By Lemma 2.4, the number of root vectors of length $\sqrt{2}$ is less than $10(m-p)^{2}.$ This completes the proof of our Proposition. ∎ ## 3. Proof of Theorem 1.4 Recall the notations while formulating Theorem 1.4 and Theorem 1.8. In this section, we assume that $F(x_{1},\dots,x_{m})=\mathbf{x}^{\intercal}A\mathbf{x}$ for some positive definite symmetric matrix $A$, where $\mathbf{x}=\begin{bmatrix}x_{1}\\ \vdots\\ x_{m}\end{bmatrix}.$ We give a sharp upper bound on $\text{Var}(N,\eta)$ by assuming Theorem 1.8. ### Scaling the point-pair invariant function We prove a simple lemma which relates the point-pair invariant functions $K_{\eta}(\mathbf{x},\mathbf{y})$ (defined in Theorem 1.4) to $K_{r,N}(\mathbf{x},\mathbf{y})$ (defined in (1.3)). Recall that $K_{\eta}(\mathbf{x},\mathbf{y}):=C_{\eta}k\big{(}\frac{\sqrt{F(\mathbf{x}- \mathbf{y})}}{\eta}\big{)},$ where $\mathbf{x},\mathbf{y}\in V_{1}(\mathbb{R}),$$\eta\in\mathbb{R}$ and $C_{\eta}$ is a normalization factor such that $\int_{V_{1}(\mathbb{R})}K_{\eta}(\mathbf{x},\mathbf{y})d\mu(\mathbf{y})=1.$ Moreover, for $\mathbf{x},\mathbf{y}\in V_{N,A_{i}}(\mathbb{R})$, where $A_{i}\in C(A)$, we defined $K_{r,N}(\mathbf{x},\mathbf{y}):=C_{N,r}k\big{(}\frac{\|\mathbf{x}-\mathbf{y}\|_{ i}}{r}\big{)},$ where $\|\mathbf{x}-\mathbf{y}\|_{i}:=\sqrt{(\mathbf{x}-\mathbf{y})^{\intercal}A_{i}( \mathbf{x}-\mathbf{y})}$, and where $\mu_{i,N}$ is the Haar probability measure on $V_{A_{i},N}.$ Lemma 3.1.: _Assume that $\mathbf{x},\mathbf{y}\in V_{A,N}(\mathbb{R}).$ We have_ \[K_{r,N}(\mathbf{x},\mathbf{y})=K_{\eta}(\frac{\mathbf{x}}{\sqrt{N}},\frac{ \mathbf{y}}{\sqrt{N}}),\] _where $\eta=\frac{r}{\sqrt{N}}.$_ Proof.: Note that the probability measure $\mu_{N}$ on $V_{A,N}(\mathbb{R})$ is the pull back of the probability measure $\mu$ on $V_{A,1}(\mathbb{R})$ by the scaling map with $1/\sqrt{N}$. Hence, we have \[K_{r,N}(\mathbf{x},\mathbf{y})=C_{N,r}k\big{(}\frac{\|\mathbf{x}-\mathbf{y}\|}{r }\big{)},\] where $\|\mathbf{x}-\mathbf{y}\|=\sqrt{(\mathbf{x}-\mathbf{y})^{\intercal}A(\mathbf{x}- \mathbf{y})}=\sqrt{F(\mathbf{x}-\mathbf{y})}.$ The lemma follows from the above identity and the definition of $K_{\eta}(\mathbf{x},\mathbf{y})$. ∎ ### Proof of Theorem 1.4 Proof.: Let $\{\phi_{k,i}\}$ be an orthonormal basis of automorphic forms which are harmonic polynomials of degree $k$ in $L^{2}\big{(}O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O(\prod _{p}\mathbb{Z}_{p})\big{)}.$ We consider them as harmonic polynomials on the disjoint union of the following quadrics \[\bigcup\big{\{}(\mathbf{X},A_{i}):\mathbf{X}\in O_{A_{i}}(\mathbb{Z}) \backslash V_{A_{i},1}(\mathbb{R}),A_{i}\in C(A)\big{\}}.\] By Theorem 1.8, we have \[\text{Var}(N,r)=\sum_{k=1}^{\infty}h_{r}(k)^{2}N^{-k}\Theta(\phi_{k,i},N)^{2},\] where $\Theta(\phi_{k,i},N)$ is the $N$-Fourier coefficient of $\Theta(\phi_{k,i})$ that is the theta transfer of $\phi_{k,i}$. By Proposition 6.2 and 6.4, $\Theta(\phi_{k,i})$ is a Hecke holomorphic modular form of weight $m/2+k$ and level dividing $4\|A\|$. Recall that $\gcd(N,\|A\|)=1$. Hence, by the multiplicative property of the Fourier coefficients, we have \[\Theta(\phi_{k,i},N)=\lambda_{\Theta(\phi_{k,i})}(N)\Theta(\phi_{k,i},1),\] where $\lambda_{\Theta(\phi_{k,i})}(N)$ is the $N$-th Hecke eigenvalue of $\Theta(\phi_{k,i}).$ By the assumptions of Theorem 1.4 the Ramanujan bound holds for $\lambda_{\Theta(\phi_{k,i})}(N)$, and we have \[\lambda_{\Theta(\phi_{k,i})}(N)\ll N^{\frac{k+m/2-1}{2}+\epsilon},\] where the implied constant involved in $\ll$ only depends on $\epsilon$. Hence, (3.1) \[\begin{split}\text{Var}(N,r)&=\sum_{k=1}^{\infty}h_{ N,r}(k)^{2}N^{-k}\Theta(\phi_{k,i},N)^{2}\\ &\ll N^{m/2-1+\epsilon}\sum_{k=1}^{\infty}h_{N,r}(k)^{2}\Theta( \phi_{k,i},1)^{2}\end{split}\] It follows from the definition of the spherical transform $h_{N,r}(k)$ in (4.5) that \[h_{N,r}(k)=h_{1,\frac{r}{\sqrt{N}}}(k).\] Hence, by Theorem 1.8, we have \[\text{Var}(1,\frac{r}{\sqrt{N}})=\sum_{k=1}^{\infty}h_{N,r}(k)^{2}\Theta(\phi_ {k,i},1)^{2}.\] By substituting the above in (3.1), we obtain \[\text{Var}(N,r)\ll N^{m/2-1+\epsilon}\text{Var}(1,\frac{r}{\sqrt{N}}).\] Next, we give an upper bound on $\text{Var}(1,\frac{r}{\sqrt{N}}).$ For simplicity we write $\eta=\frac{r}{\sqrt{N}}$ and $K_{\eta}(\mathbf{x},\mathbf{y})=K_{\frac{r}{\sqrt{N}},1}(\mathbf{x},\mathbf{y})$ for $\mathbf{x},\mathbf{y}\in V_{A_{i},1}(\mathbb{R})$. By (1.6), we have \[\begin{split}\text{Var}(1,\eta)&\leq\max_{i}\Big{(} \int_{V_{A_{i},1}(\mathbb{R})}\Big{(}\big{(}\sum_{\mathbf{Y}\in V_{A_{i},1}( \mathbb{Z})}K_{\eta}(\mathbf{x},\mathbf{y})\big{)}-R(1)\Big{)}^{2}d\mu_{i}( \mathbf{x})\Big{)}\\ &\leq\sup_{i,\mathbf{x}}\Big{(}\big{(}\sum_{\mathbf{Y}\in V_{A_{i },1}(\mathbb{Z})}K_{\eta}(\mathbf{x},\mathbf{y})\big{)}-R(1)\Big{)}\int_{V_{A_ {i},1}(\mathbb{R})}\Big{\|}\big{(}\sum_{\mathbf{Y}\in V_{A_{i},1}(\mathbb{Z})}K _{\eta}(\mathbf{x},\mathbf{y})\big{)}-R(1)\Big{\|}d\mu_{i}(\mathbf{x}),\\ &\leq C_{1,\eta}(\max_{i}R_{A_{i}}(1)+R(1))\ll C_{1,\eta}\ll\eta^ {-(m-1)}.\end{split}\] Therefore, we have \[\text{Var}(N,r)\ll\frac{N^{m/2-1+\epsilon}}{\eta^{m-1}}.\] By assuming $m\geq 4,$ and bounding the local densities in Hardy-Littlewood formula [23, Remark 1.7], we have \[N^{m/2-1+\epsilon}\ll N^{\epsilon}R_{F}(N).\] Therefore, \[\text{Var}(N,r)\ll\frac{N^{\epsilon}R_{F}(N)}{\eta^{m-1}},\] where the implicit constant in $\ll$ only depends only on $A$ and $\epsilon.$ This completes the proof of Theorem 1.4. ∎ ### Proof of Corollary 1.5 Proof.: Assume that $\eta\ll R_{F}(N)^{-\frac{1}{(m-1)}-\epsilon}.$ We prove the second part of the corollary. Let $E(N,\eta):=\{\mathbf{x}\in V_{1}(\mathbb{R}):\forall\mathbf{y}\in\frac{1}{ \sqrt{N}}V_{N}(\mathbb{Z}),\sqrt{F(\mathbf{x}-\mathbf{y})}\geq\eta\}.$ Since $k$ is positive and $k(x)=1$ for $x\in(-1,1),$ we have \[\big{(}1-\mu(E(N,\eta))\big{)}C_{\eta}\leq\int_{V_{1}(\mathbb{R})}\sum_{ \mathbf{y}\in\frac{1}{\sqrt{N}}V_{N}(\mathbb{Z})}K_{\eta}(\mathbf{x},\mathbf{y })d\mu(\mathbf{x})=R_{F}(N).\] It is easy to check that $C_{\eta}\gg\eta^{-(m-1)}.$ Therefore, by (1.1) \[\big{(}1-\mu(E(N,\eta))\big{)}\ll R_{F}(N)\eta^{(m-1)}\ll R_{F}(N)^{-(m-1) \epsilon}\ll N^{-\epsilon}.\] This completes the proof of the second part of the corollary. Next, assume that $\eta\gg R_{F}(N)^{-\frac{1}{(m-1)}+\epsilon}$. By Chebyshev’s inequality and Theorem 1.4 \[\mu(E(N,\eta))R_{F}(N)^{2}\leq\text{Var}_{F}(N,\eta)\ll\frac{N^{\epsilon}R_{F} (N)}{\eta^{m-1}}.\] Therefore, by (1.1) (3.2) \[\mu(E(N,\eta))\ll\frac{N^{\epsilon}}{\eta^{m-1}R_{F}(N)}\ll N^{-\epsilon}.\] This completes the proof of the corollary. ∎ ### Proof of Corollary 1.6 Proof.: Recall that $\bar{K}_{m}:=\lim_{\delta\to 0}\limsup_{\epsilon_{0}\to 0}\frac{\log\big{(}\#S ^{m-1}_{N_{\delta,\epsilon_{0}}}(\mathbb{Z})\big{)}}{\log\big{(}1/\text{vol }( C(\mathbf{x},\epsilon_{0}))\big{)}},$ and $\bar{K}_{m}\geq 1.$ It is enough to show that $\bar{K}_{m}\leq 1+\epsilon$ for any $\epsilon>0.$ We have $\text{vol }(C(\mathbf{x},\epsilon_{0})\gg\epsilon_{0}^{m-1}.$ Since $m$ is even then Theorem 1.4 and inequality (3.2) holds unconditionally for $S^{m-1}$, and we have \[\mu(E(N,\epsilon_{0}))\ll\frac{N^{\epsilon}}{\text{vol }(C(\mathbf{x},\epsilon _{0})\#S^{m-1}_{N}(\mathbb{Z})}.\] Hence, by the definition of $N_{\delta,\epsilon_{0}},$ we have \[\#S^{m-1}_{N_{\delta,\epsilon_{0}}}(\mathbb{Z})\leq\text{vol }(C(\mathbf{x}, \epsilon_{0})^{(-1-\delta/(m-1)-\epsilon)}.\] Therefore, \[\bar{K}_{m}:=\lim_{\delta\to 0}\limsup_{\epsilon_{0}\to 0}\frac{\log\big{(}\#S ^{m-1}_{N_{\delta,\epsilon_{0}}}(\mathbb{Z})\big{)}}{\log\big{(}1/\text{vol }( C(\mathbf{x},\epsilon_{0}))\big{)}}\leq\lim_{\delta\to 0}{(1+\delta/(m-1)- \epsilon)}\leq 1+\epsilon.\] for every $\epsilon>0$. This concludes the proof of Corollary 1.6. ∎ ### Proof of Corollary 1.7 Proof.: The proof is based on a Borel-Cantelli argument. Define \[A_{k,\epsilon}:=\left\{x\in S^{m-1}(\mathbb{R}):\|x-z\|>\epsilon,\forall z\in S^ {m-1}(\mathbb{Z}[1/p])\text{ with }H(z)\leq p^{k}\right\}.\] We note that there is a one-to-one correspondence between $z\in S^{m-1}(\mathbb{Z}[1/p])\text{ with }H(z)\leq p^{k}$ and the integral points $\frac{1}{p^{k}}S^{m-1}_{p^{2k}}(\mathbb{Z}).$ Note that by using $N=p^{2k}$ in Theorem 1.4, it follows that inequality (3.2) holds unconditionally for $\frac{1}{p^{k}}S^{m-1}_{p^{2k}}(\mathbb{Z})$, and we have \[\mu(A_{k,\epsilon})\ll\frac{p^{2k\epsilon}}{\epsilon^{m-1}p^{k(m-2)}}.\] Let $B_{k,\delta}:=A_{k,p^{-k(\frac{m-2}{m-1}-\delta)}}.$ By the above inequality $\mu(B_{k,\delta})\ll p^{-k(\delta(m-1)-2\epsilon)}$ for any $\epsilon>0.$ Note that $\sum_{k}\mu(B_{k,\delta})\ll\sum_{k}p^{-k(\delta(m-1)-2\epsilon)}\ll\infty.$ By the Borel-Cantelli lemma, for almost all $x\in S^{m-1}(\mathbb{R}),$ there exists $k_{x}$ such that $x\notin B_{k}$ for every $k>k_{x}.$ In other words, for almost every $x\in S^{m-1}(\mathbb{R}),$$\delta>0$, and $\varepsilon\in(0,\varepsilon_{0}(x,\delta)),$ there exists $z\in S^{m-1}(\mathbb{Z}[1/p])$ such that $\|x-z\|_{\infty}\leq\epsilon\text{ and }H(z)\leq\epsilon^{-\frac{m-1}{m-2}- \delta}.$ This concludes the proof of Corollary 1.7.∎ ## 4. The Siegel variance formula Recall the definition of the Siegel variance sum $\text{Var}(B,r)$ in (1.6). In this section, we give an adelic integration formula for $\text{Var}(B,r).$ First, we write $\text{Var}(A_{i},B,r)$ and $\text{Var}(B,r)$ in terms of the $O_{A_{i}}(\mathbb{Z})$ orbits of $V_{A_{i},B}(\mathbb{Z}).$ We define the $O_{A_{i}}(\mathbb{Z})$ invariant function \[\tilde{K}_{r}(\mathbf{X},\mathbf{Y}):=\sum_{\gamma\in O_{A_{i}}(\mathbb{Z})}K_ {r,B}(\gamma\mathbf{X},\mathbf{Y}),\] where $\mathbf{X},\mathbf{Y}\in O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{Z }).$ Lemma 4.1.: _We have_ \[\frac{\text{Var}(A_{i},B,r)}{\|O_{A_{i}}(\mathbb{Z})\|}=\int_{O_{A_{i}}(\mathbb{ Z})\backslash V_{A_{i},B}(\mathbb{R})}\Big{(}\big{(}\sum_{\mathbf{Y}\in O_{A_{ i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{Z})}\frac{\tilde{K}_{r}(\mathbf{ X},\mathbf{Y})}{\|O_{A_{i},\mathbf{Y}(\mathbb{Z})}\|}\big{)}-R_{A_{i}}(B)\Big{)} ^{2}d\mu_{i}(\mathbf{X}),\] _where $\|O_{L_{i},\mathbf{Y}(\mathbb{Z})}\|$ is the size of the stabilizer of $\mathbf{Y}$ in $O_{A_{i}}(\mathbb{Z}).$_ Proof.: We have \[\sum_{\mathbf{Y}\in V_{A_{i},B}(\mathbb{Z})}K_{r,B}(\mathbf{X},\mathbf{Y})= \sum_{\mathbf{Y}\in O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{Z})} \frac{\tilde{K}_{r}(\mathbf{X},\mathbf{Y})}{\|O_{A_{i},\mathbf{Y}(\mathbb{Z})}\|}.\] Therefore, \[\begin{split}\text{Var}(A_{i},B,r)=\int_{V_{A_{i},B}(\mathbb{R})} \Big{(}\big{(}\sum_{\mathbf{Y}\in O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}( \mathbb{Z})}\frac{\tilde{K}_{r}(\mathbf{X},\mathbf{Y})}{\|O_{A_{i},\mathbf{Y}( \mathbb{Z})}\|}\big{)}-R_{A_{i}}(B)\Big{)}^{2}d\mu_{i}(\mathbf{X}).\end{split}\] The lemma follows from the fact that $\tilde{K}_{r}(\mathbf{X},\mathbf{Y})$ is $O_{A_{i}}(\mathbb{Z})$ invariant on the $\mathbf{X}$ variable and $O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{R})$ is a fundamental domain for this action. ∎ Lemma 4.2.: _We have_ \[\text{Var}(B,r)=\dfrac{\sum_{A_{i}}\int_{O_{A_{i}}(\mathbb{Z})\backslash V_{A_ {i},B}(\mathbb{R})}\Big{(}\big{(}\sum_{Y\in O_{A_{i}}(\mathbb{Z})\backslash V_ {A_{i},B}(\mathbb{Z})}\frac{\tilde{K}_{r}(\mathbf{X},\mathbf{Y})}{\|O_{L_{i}, \mathbf{Y}(\mathbb{Z})}\|}\big{)}-R(B)\Big{)}^{2}d\mu_{i}(\mathbf{X})}{{\sum_{A _{i}}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}}}.\] Proof.: By (1.4), we have \[\int_{V_{A_{i},B}(\mathbb{R})}\sum_{\mathbf{Y}\in V_{A_{i},B}(\mathbb{Z})}K_{r ,B}(\mathbf{X},\mathbf{Y})d\mu_{i}(\mathbf{X})=R_{A_{i}}(B).\] Hence, by lemma 4.1 \[\dfrac{\text{Var}(A_{i},B,r)+\big{(}R_{A_{i}}(B)-R(B)\big{)}^{2}} {O_{A_{i}}(\mathbb{Z})}\\ =\int_{O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{R})} \Big{(}\big{(}\sum_{Y\in O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}(\mathbb{Z })}\frac{\tilde{K}_{r}(\mathbf{X},\mathbf{Y})}{\|O_{L_{i},\mathbf{Y}(\mathbb{Z} )}\|}\big{)}-R(B)\Big{)}^{2}d\mu_{i}(\mathbf{X}).\] By summing both side of the above identity over $A_{i}$ and dividing by ${\sum_{A_{i}}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}}$, we conclude the lemma. ∎ ### The Siegel variance as an adelic integral In this section, we give a formula for $\text{Var}(B,r)$ in terms of an integral over a double quotient of the adelic orthogonal group. #### 4.1.1. Adelic point-pair invariant function We extend the point-pair invariant function $K_{r,B}(\mathbf{X},\mathbf{Y})$ (defined in (1.3)) into an automorphic point-pair invariant function on the adelic points of the orthogonal group $O_{A}$. We begin by defining the adelic points of the orthogonal group $O_{A}.$ Let $O_{A}(R)$ denote the orthogonal group of $A$ with coefficients in a commutative ring $R,$ which we consider as a subset of $GL_{m}(R):$ \[O_{A}(R):=\big{\{}X\in GL_{m}(R):X^{\intercal}AX=A\big{\}}.\] Let $\mathbb{A}_{f}=\hat{\prod}_{p}^{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ be the ring of finite adeles which is the restrictive direct product of $\mathbb{Q}_{p}$ with respect to $\mathbb{Z}_{p}.$ Let $\mathcal{L}_{A,B}$ denote the space of $(\mathbf{X},L),$ where $\mathbf{X}\in V_{A,B}(\mathbb{R})$ and $L\subset\mathbb{Q}^{m}$ is a lattice where $(L,A)$ has the same genus as $(\mathbb{Z}^{m},A):$ \[\mathcal{L}_{A,B}:=\big{\{}(\mathbf{X},L):\mathbf{X}\in V_{A,B}(\mathbb{R}),L \subset\mathbb{Q}^{m},\text{ and }(L\otimes\mathbb{Z}_{p},A)\sim(\mathbb{Z}_{p }^{d},A),\forall\text{ prime }p\big{\}},\] where $(L\otimes\mathbb{Z}_{p},A)\sim(\mathbb{Z}_{p}^{d},A)$ means there exists $g\in O_{A}(\mathbb{Q}_{p})$ such that $g\mathbb{Z}_{p}^{d}=L\otimes\mathbb{Z}_{p}.$ Note that $O_{A}(\mathbb{A}_{\mathbb{Q}})$ acts transitively on $\mathcal{L}_{A,B}$ by: where $(g_{\infty},\prod_{p}g_{p})\in O_{A}(\mathbb{A}_{\mathbb{Q}}).$ It is well-known that $C(A)$, the genus class of $A$, is isomorphic to $O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{f})/O_{A}(\prod_{p}\mathbb{Z}_{p }).$ Suppose that $\{L_{i}\subset\mathbb{Q}^{n}:1\leq i\leq h\}$ is a representative set for the genus class of the lattice $(\mathbb{Z}^{n},A)$ such that $(L_{i},A)$ is isomorphic to $(\mathbb{Z}^{m},A_{i}),$ which means $L_{i}^{\intercal}AL_{i}=A_{i}.$ We extend $K_{r,B}(\mathbf{X},\mathbf{Y})$ into a function on $\mathcal{L}_{A,B}$ and denote the extension by $K_{r,B}\big{(}(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}$ again. Define \[K_{r,B}\big{(}(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}:=\begin{cases}K_{r, B}(\mathbf{X},\mathbf{Y})&\text{ if }L_{1}=L_{2},\\ 0&\text{ otherwise.}\end{cases}\] We sum $K_{r,B}(\mathbf{X},\mathbf{Y})$ over the orbit of $O_{A}(\mathbb{Q})$, and obtain \[\mathcal{K}_{r,B}\big{(}(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}:=\sum_{ \gamma\in O_{A}(\mathbb{Q})}K_{r,B}\big{(}\gamma(\mathbf{X},L_{1}),(\mathbf{X} ,L_{2})\big{)}.\] Note that $\mathcal{K}_{r,B}$ is invariant by the action of $O_{A}(\mathbb{Q})$ on the left: \[\mathcal{K}_{r,B}\big{(}(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}=\mathcal{ K}_{r,B}\big{(}\gamma(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}=\mathcal{K}_ {r,B}\big{(}(\mathbf{X},L_{1}),\gamma(\mathbf{Y},L_{2})\big{)},\] for every $\gamma\in O_{A}(\mathbb{Q}).$ Hence, $\mathcal{K}_{r,B}\big{(}(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}$ defines an automorphic kernel on $O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}\times O_{A}(\mathbb{Q})\backslash \mathcal{L}_{A,B}.$ Given $(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\in\mathcal{L}_{A,B},$ it follows that (4.1) \[\mathcal{K}_{r,B}\big{(}(\mathbf{X},L_{1}),(\mathbf{Y},L_{2})\big{)}=\delta(L_ {1},L_{2})\sum_{\gamma\in O_{L_{2}}}K_{r,B}(\gamma\gamma_{(1,2)}\mathbf{X}, \mathbf{Y}),\] where \[\delta(L_{1},L_{2})=\begin{cases}1\text{ if $L_{1}$ and $L_{2}$ are in the same genuss class of lattices, }\\ 0\text{ otherwise,}\end{cases}\] and $O_{L_{2}}$ is the stabilizer of $L_{2}$ in the orthogonal group $O_{A}(Q),$ and if $\delta(L_{1},L_{2})=1,$ then there exists $\gamma_{(1,2)}\in O_{A}(\mathbb{Q})$ which maps $L_{1}$ to $L_{2}$. Recall that $\mathbf{E}_{B}=\mathbf{E}\sqrt{B}.$ Fix $\mathbf{\mathcal{E}_{0}}:=(\mathbf{E}_{B},\mathbb{Z}^{m})\in\mathcal{L}_{A,B}.$ Let $O_{A,\mathbf{E}}(\mathbb{R})$ denote the stabilizer of $\mathbf{E}_{B},$ then $O_{A,\mathbf{E}}(\mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})$ is the stabilizer of $\mathcal{E}_{0}\in\mathcal{L}_{A,B},$ and we have the following isomorphism \[\mathcal{L}_{A,B}=O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{\mathcal{E}_{0}} }(\mathbb{A}_{\mathbb{Q}}).\] Therefore, we can view $\mathcal{K}_{r,B}$ as an automorphic point-pair invariant function on \[O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{ \mathcal{E}_{0}}}(\mathbb{A}_{\mathbb{Q}})\times O_{A}(\mathbb{Q})\backslash O _{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{\mathcal{E}_{0}}}(\mathbb{A}_{ \mathbb{Q}}).\] #### 4.1.2. Integration formula for $\text{Var}(B,r)$ Let \[\mathcal{S}_{B}:=\{(\mathbf{X},L)\in\mathcal{L}_{A,B}:\mathbf{x}_{j}\in L, \text{ where }\mathbf{x}_{j}\text{ is the $j$-th column of }\mathbf{X}\text{ for }1\leq j\leq n\}.\] Note that $\mathcal{S}_{B}=\cup_{i=1}^{h}\mathcal{S}_{L_{i},B},$ where (4.2) \[\mathcal{S}_{L_{i},B}:=\{(\mathbf{X},L)\in\mathcal{S}_{B}:(L,A)\text{ is equivalent to }(L_{i},A)\}.\] Note that $\mathcal{S}_{L_{i},B}$ is invariant by the action of $O_{A}(\mathbb{Q}).$ Finally, we define the adelic variance: \[\mathcal{VAR}(B,r):=\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}\Big{(} \sum_{\mathcal{Q}\in O_{A}(\mathbb{Q})\backslash\mathcal{S}_{B}}\dfrac{ \mathcal{K}_{r,B}(\mathcal{X},\mathcal{Q})}{\|O_{\mathcal{Q}}(\mathbb{Q})\|}- \mathcal{R}(B)\Big{)}^{2}d\tilde{\mu}(\mathcal{X}),\] where $\|O_{\mathcal{Q}}(\mathbb{Q})\|$ is the size of the stabilizer of a representative of $\mathcal{Q}\in O_{A}(\mathbb{Q})\backslash\mathcal{S}_{B},$ and $d\tilde{\mu}(\mathcal{X})$ is a normalized $O(\mathbb{A}_{Q})$ invariant measure such that \[\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}d\tilde{\mu}(\mathcal{X})=1,\] and (4.3) \[\mathcal{R}(B)=\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}\sum_{ \mathcal{Q}\in O_{A}(\mathbb{Q})\backslash\mathcal{S}_{B}}\dfrac{\mathcal{K}_{ r,B}(\mathcal{X},\mathcal{Q})}{\|O_{\mathcal{Q}}(\mathbb{Q})\|}d\tilde{\mu}( \mathcal{X}).\] Proposition 4.3.: _We have_ (4.4) \[\begin{split}\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}} \mathcal{K}_{r,B}(\mathcal{X},\mathcal{Y})d\tilde{\mu}(\mathcal{X})& =\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}\mathcal{K}_{ r,B}(\mathcal{X},\mathcal{Y})d\tilde{\mu}(\mathcal{Y})=1,\\ \mathcal{R}(B)&=R(B),\\ \mathcal{VAR}(B,r)&=\text{Var}(B,r).\end{split}\] Proof.: We have $O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}=\cup_{i=1}^{h}\{(L_{i},\mathbf{X} ):\mathbf{X}\in O_{L_{i}}\backslash V_{A,B}(\mathbb{R})\},$ where $O_{L_{i}}$ is the stabilizer of $L_{i}$ by the action of $O_{A}(\mathbb{Q}).$ Since the action of $O_{A}(\mathbb{A}_{\mathbb{Q}})$ is transitive on $\mathcal{L}_{A,B}$ and $d\mu$ is a Haar measure, it follows that for every $1\leq i,j\leq h.$ Since $\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}d\tilde{\mu}(\mathcal{X})=1$, we have \[\tilde{\mu}\Big{(}\{(L_{i},\mathbf{X}):\mathbf{X}\in O_{L_{i}}\backslash V_{A, B}(\mathbb{R})\}\Big{)}=\frac{\frac{1}{\|O_{L_{i}}\|}}{\Big{(}\sum_{L_{i}}\frac{ 1}{\|O_{L_{i}}\|}\Big{)}}.\] Recall that $\int_{V_{A_{i},B}(\mathbb{R})}d\mu_{i}(\mathbf{Y})=1$ and $L_{i}^{\intercal}AL_{i}=A_{i}.$ This implies $\Big{(}\sum_{L_{i}}\frac{1}{O_{L_{i}}}\Big{)}d\tilde{\mu}$ restricted to $\{(L_{i},\mathbf{X}):\mathbf{X}\in O_{L_{i}}\backslash V_{A,B}(\mathbb{R})\}$ is equal to $d\mu$ on $O_{L_{i}}\backslash V_{A,B}(\mathbb{R}).$ Let $\mathcal{Y}=(L,Y).$ By (4.1), we have \[\begin{split}\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}} \mathcal{K}_{r,B}(\mathcal{X},\mathcal{Y})d\tilde{\mu}(\mathcal{X})& =\int_{O_{L}\backslash V_{A,B}(\mathbb{R})}\sum_{\gamma\in O_{L}} K_{r,B}(\gamma\mathbf{X},\mathbf{Y})d\mu(\mathbf{X})\\ &=\int_{V_{A,B}(\mathbb{R})}K_{r,B}(\mathbf{X},\mathbf{Y})d\mu( \mathbf{X})=1.\end{split}\] This completes the proof of the first identity. For the second identity, we have \[\begin{split}\Big{(}\sum_{A_{i}}\frac{1}{O_{L_{i}}}\Big{)} \mathcal{R}(B)=\sum_{i=1}^{h}\int_{O_{L_{i}}\backslash V_{A,B}(\mathbb{R})} \sum_{\mathcal{Q}\in O_{A}(\mathbb{Q})\backslash\mathcal{S}_{L_{i},B}}\dfrac{ \mathcal{K}_{r,B}((L_{i},\mathbf{X}),\mathcal{Q})}{\|O_{\mathcal{Q}}(\mathbb{Q} )\|}d\mu(\mathbf{X}).\end{split}\] By unfolding $\sum_{\mathcal{Q}\in O_{A}(\mathbb{Q})\backslash\mathcal{S}_{L_{i},B}}\dfrac{ \mathcal{K}_{r,B}((L_{i},\mathbf{X}),\mathcal{Q})}{\|O_{\mathcal{Q}}(\mathbb{Q} )\|}$, we have \[\begin{split}\Big{(}\sum_{A_{i}}\frac{1}{O_{L_{i}}}\Big{)} \mathcal{R}(B)&=\sum_{i=1}^{h}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|} \int_{V_{A_{i},B}(\mathbb{R})}\sum_{Q\in V_{A_{i},B}(\mathbb{Z})}{K_{r,B}( \mathbf{X},Q)}d\mu_{i}(\mathbf{X})\\ &=\sum_{i=1}^{h}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}R_{A_{i}}(B)= \Big{(}\sum_{A_{i}}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}\Big{)}R(B).\end{split}\] Finally, by Lemma 4.2, we have \[\begin{split}\Big{(}\sum_{A_{i}}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|} \Big{)}\mathcal{VAR}(B,r)&=\sum_{i=1}^{h}\int_{O_{L_{i}} \backslash V_{A,B}(\mathbb{R})}\Big{(}\big{(}\sum_{\mathcal{Q}\in O_{A}( \mathbb{Q})\backslash\mathcal{S}_{L_{i},B}}\dfrac{\mathcal{K}_{r,B}(\mathcal{X },\mathcal{Q})}{\|O_{\mathcal{Q}}(\mathbb{Q})\|}\big{)}-R(B)\Big{)}^{2}d\tilde{ \mu}(\mathcal{X})\\ &=\sum_{A_{i}}\int_{O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i},B}( \mathbb{R})}\Big{(}\big{(}\sum_{Y\in O_{A_{i}}(\mathbb{Z})\backslash V_{A_{i}, B}(\mathbb{Z})}\frac{\tilde{K}_{r}(\mathbf{X},\mathbf{Y})}{\|O_{L_{i},\mathbf{Y }(\mathbb{Z})}\|}\big{)}-R(B)\Big{)}^{2}d\mu(\mathbf{X})\\ &=\Big{(}\sum_{A_{i}}\frac{1}{\|O_{A_{i}}(\mathbb{Z})\|}\Big{)} \text{Var}(B,r).\end{split}\] This completes the proof of the lemma. ∎ ### Siegel variance in terms of the Weyl sums In this section, we write the spectral decomposition of $\mathcal{K}_{r,B}.$ Let $\{\phi_{\pi,j}(\alpha)\}$ be an orthonormal basis of $L^{2}\Big{(}O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A, \mathbf{E}}(\mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})\Big{)},$ where $\pi$ is an automorphic representation and $\phi_{\pi,j}$ is an $O_{A,\mathbf{E}}(\mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})$ invariant vectors in $\pi.$ We write $\pi=\pi_{\infty}\prod_{p}\pi_{p},$ where $\pi_{p}$ and $\pi_{\infty}$ are the local components of the automorphic representation $\pi.$ We identify $O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{E}}( \mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})=O_{A}(\mathbb{Q})\backslash\mathcal{ L}_{A,B}.$ By Lemma 4.3, we have \[\mathcal{K}_{r,B}(\alpha,\beta)=1+\sum_{\pi}\sum_{j=1}^{d_{\pi}}h_{r}(\pi_{ \infty})\phi_{\pi,j}(\alpha)\bar{\phi}_{\pi,i}(\beta),\] where the sum is over $\phi_{\pi,j}$ such that \[\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}\phi_{\pi,j}(\alpha)d\tilde {\mu}(\alpha)=0,\] and $h_{r}(\pi_{\infty})$ is the spherical transformation of the point-pair invariant kernel $K_{r,B},$ which is defined by: (4.5) \[h_{r}(\pi_{\infty})\phi_{\pi,j}(\alpha)=\int_{O_{A}(\mathbb{Q})\backslash \mathcal{L}_{A,B}}\mathcal{K}_{r,B}(\alpha,\beta)\phi_{\pi,j}(\beta)d\tilde{ \mu}(\beta).\] By Lemma 4.3, we obtain \[\mathcal{VAR}(B,r)=\int_{O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}}\Big{(} \sum_{\pi}\sum_{j=1}^{d_{\pi}}h_{r}(\pi_{\infty})\phi_{\pi,j}(\alpha)W(\phi_{ \pi,j},B)\Big{)}^{2}d\tilde{\mu}(\alpha),\] where (4.6) \[W(\phi_{\pi,j},B):=\sum_{\mathcal{Q}\in O_{A}(\mathbb{Q})\backslash\mathcal{S} _{B}}\dfrac{\phi_{\pi,j}(\mathcal{Q})}{\|O_{\mathcal{Q}}(\mathbb{Q})\|},\] which is a generalization of Weyl’s sum associated to $\phi_{\pi,j}.$ By using the orthogonality of $\phi_{\pi,j}$, only the diagonal terms contribute to $\mathcal{VAR}(B,r)$, and we have the following proposition. Proposition 4.4.: _We have_ (4.7) \[\mathcal{VAR}(B,r)=\sum_{\pi}\sum_{j=1}^{d_{\pi}}\|h_{r}(\pi_{\infty})\|^{2}\|W( \phi_{\pi,j},B)\|^{2}.\] ## 5. Harmonic polynomials ### Harmonic polynomials for n=1 In this section, we restrict ourself to the case $n=1$ and cite some standard results on the spherical harmonic polynomials. Let $F(\mathbf{x}):=\mathbf{x}^{\intercal}A\mathbf{x},$ where $\mathbf{x}=\begin{bmatrix}x_{1}\\ \vdots\\ x_{m}\end{bmatrix}.$ Let $A^{-1}=[a^{ij}]$ denote the inverse of $A,$ and $\Delta_{A}:=\sum_{i,j}a^{ij}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}$ be the Laplacian operator associated to $A.$ Let $H_{k}$ be the space of harmonic polynomials of degree $k$ with respect to the symmetric matrix $A$, which is \[H_{k}:=\left\{p(\mathbf{x}):\Delta_{A}p(\mathbf{x})=0,\text{ and }\deg{p}=k \right\}.\] Let $\mathbf{r}\in\mathbb{C}^{m}$ and $\mathbf{r}^{\intercal}A\mathbf{r}=0.$ It is easy to check that $\Delta_{A}\langle\mathbf{x},\mathbf{r}\rangle^{k}=0,$ for $k=1$ the condition $\mathbf{r}^{\intercal}A\mathbf{r}=0$ is not necessary. It is well-known that $H_{k}$ is the span of polynomials of the form $\langle\mathbf{x},\mathbf{r}\rangle^{k}.$ Moreover $H_{k}$ is invariant under the action of $O_{A}(\mathbb{C})$ and form an irreducible representation of this group. Let $V_{N}(\mathbb{R})=\{\mathbf{x}\in\mathbb{R}^{m}:F(\mathbf{x})=N\}$ and $f\in H_{k}$. Then the restriction of $f$ to $V_{N}(\mathbb{R})\subset\mathbb{R}^{m}$ defines an embedding of $H_{k}$ into $L^{2}(V_{N}(\mathbb{R})).$ Next we give the spectral decomposition of $L^{2}(V_{N}(\mathbb{R}))$ in terms of the harmonic polynomials. Proposition 5.1.: _We have_ \[L^{2}(V_{N}(\mathbb{R}))=\oplus_{k}H_{k}\] Proof.: This proposition is standard; see [19, Section 2.5.12] for the proof. ∎ Fix $\mathbf{e}\in V_{N}(\mathbb{R})$ and let $O_{A,\mathbf{e}}\subset O_{A}(\mathbb{R})$ be the centralizer of $\mathbf{e}$. It follows that there exists a unique $p_{k,\mathbf{e}}(\mathbf{x})\in H_{k},$ such that (5.1) \[\begin{cases}p_{k,\mathbf{e}}(\mathbf{x})=p_{k,\mathbf{e}}(g\mathbf{x}),\text{ for every }g\in O_{A,\mathbf{e}}\text{ and }\mathbf{x}\in V_{N}(\mathbb{R}), \\ p_{k,\mathbf{e}}(\mathbf{e})=1.\end{cases}\] The following mean value theorem is standard for the harmonic polynomials. Lemma 5.2.: _Let $p_{k,\mathbf{e}}(\mathbf{x})$ be as above and $q(\mathbf{x})\in H_{k}.$ We have_ \[\begin{split}\int_{O_{A,\mathbf{E}}}q(g\mathbf{x})d\mu(g)& =q(\mathbf{e})p_{k,\mathbf{e}}(\mathbf{x}),\\ p_{k,\mathbf{e}}(u\mathbf{e})&=p_{k,\mathbf{e}}(u^{ -1}\mathbf{e}),\text{ where }u\in O_{F}(\mathbb{R}),\\ \int_{V_{N}(\mathbb{R})}q(\mathbf{x})\overline{p_{k,\mathbf{e}}}( \mathbf{x})d\mu(\mathbf{x})&=q(\mathbf{e})\|p_{k,\mathbf{e}}\|^{2}, \text{ where }\|p_{k,\mathbf{e}}\|^{2}=\int_{V_{N}(\mathbb{R})}\|p_{k,\mathbf{e}} (\mathbf{x})\|^{2}d\mu(\mathbf{x}).\end{split}\] Proof.: The proof is an easy consequence of the uniqueness of $p_{k,\mathbf{e}}(\mathbf{x}).$ ∎ ### Harmonic polynomials for general $n$ In this section, we record a generalization of the results of the previous section from the work of Kashiwara and Vergne [18]. We give an orthonormal basis consisting of the generalized harmonic polynomials for $L^{2}(V_{A,B}(\mathbb{R})).$ We will use the result of this section later in Section 6.2 and 6.5 and describe the weight of the Siegel modular forms which appears in formula (1.10). We begin by defining some notations. Let $W:=\mathbb{R}^{n}$ and $W^{}$ be its dual vector space. We take the symplectic space $V:=W+W^{}$ with the symplectic form $B(x_{1}+f_{1},x_{2}+f_{2})=f_{2}(x_{1})-f_{1}(x_{2}).$ Then $W$ and $W^{}$ are complementary Lagrangian subspaces in $(V,B).$ Let $E:=(\mathbb{R}^{m},A)$ be the inner product space with respect to the symmetric form $A.$ Let $\mathcal{P}$ be the vector space of all complex valued polynomials on $Hom(W,E)=M_{m\times n}[\mathbb{R}],$ which is isomorphic to the space of complex valued polynomials on $Hom(W^{\mathbb{C}},E^{\mathbb{C}})=M_{m\times n}[\mathbb{C}].$ We denote by $O_{A}(\mathbb{C})$ the orthogonal group of $A$ with complex coefficients. The group $GL(n,\mathbb{C})\times O_{A}(\mathbb{C})$ acts on $\mathcal{P}$ via $(H,\sigma)P=P(\sigma^{-1}XH)$, where $H\in GL_{n}(\mathbb{C})$ and $\sigma\in O_{A}(\mathbb{C}).$ For $X\in Hom(W^{\mathbb{C}},E^{\mathbb{C}})$ consider the symmetric matrix $X^{\intercal}AX.$ The coefficients $(X^{\intercal}AX)_{i,j}$ for $1\leq i,j\leq n$ generate the algebra of all $O_{A}(\mathbb{C})$ invariant polynomials on $Hom(W,E).$ Thus we can describe the algebra $\mathcal{D}_{A}$ of all $O_{A}(\mathbb{C})$-invariant constant coefficient differential operators on $Hom(W,E)$ as follows. We fix a basis of $W^{\mathbb{C}}$ and an orthogonal basis of $E^{\mathbb{C}}.$ Writing $X$ in $Hom(W^{\mathbb{C}},E^{\mathbb{C}})$ as $X=[x_{i,j}]_{m\times n}.$ The algebra $\mathcal{D}_{A}$ is generated by the operators: \[\Delta_{i,j}=\sum_{l=1}^{m}\frac{\partial}{\partial x_{li}}\frac{\partial}{ \partial x_{lj}},\] for any $1\leq i,j\leq n.$ We define the space of the harmonic polynomials by \[\mathcal{H}:=\big{\{}P\in\mathcal{P}:\text{ such that }\Delta_{i,j}P=0\text{ for all }1\leq i,j\leq n\big{\}}.\] $\mathcal{H}$ is stable under the action of $GL(n,\mathbb{C})\times O_{A}(\mathbb{C}).$ We write $\mathcal{H}=\oplus\mathcal{H}(\lambda)$ for the decomposition of $\mathcal{H}$ in isotopic components under $O_{A}(\mathbb{C}).$ We cite the following theorem from [19, Theorem 2.5.41] Theorem 5.3.: _The isotopic component $\mathcal{H}(\lambda)$ of $\mathcal{H}$ of type $\lambda$ under $O_{A}(\mathbb{C})$ is irreducible under $GL(n,\mathbb{C})\times O_{A}(\mathbb{C})$ and it is isomorphic to $\tau\otimes\lambda$ for some finite dimensional irreducible representation of $GL(n,\mathbb{C}).$ Moreover, the isotopic component of $\mathcal{H}(\tau)$ of $\mathcal{H}$ of type $\tau$ under $GL_{n}(\mathbb{C})$ is irreducible under $GL(n,\mathbb{C})\times O_{A}(\mathbb{C}).$ In other words the correspondence $\lambda\to\tau$ is injective._ Let $f\in\mathcal{H}(\lambda)$. Then the restriction of $f$ to $V_{A,B}(\mathbb{R})\subset Hom(W,E)=M_{m\times n}[\mathbb{R}]$ defines an embedding of $\mathcal{H}(\lambda)$ into $L^{2}(V_{N}(\mathbb{R})).$ We have the following generalization of Proposition 5.1. Theorem 5.4.: _We have_ \[L^{2}(V_{A,B}(\mathbb{R}))=\oplus\mathcal{H}(\lambda)\] Proof.: The space of all polynomial is dense in $L^{2}(V_{A,B}(\mathbb{R}))$. Let $Inv$ be the sub-algebra of the $O_{A}(\mathbb{C})$ invariant polynomials. The space of all polynomials is the direct sum of $\mathcal{P}=\mathcal{H}+\mathcal{H}Inv$; see [19, Section 2.5.11]. Since the restriction of $Inv$ is constant on $V_{A,B}(\mathbb{R})$. Hence, $L^{2}(V_{A,B}(\mathbb{R}))=\mathcal{H}=\oplus\mathcal{H}(\lambda).$ ∎ Let $\mathcal{V}_{\mathbf{E}_{B}}:\mathcal{P}\to\mathbb{C}$ be the evaluation of the polynomials at $\mathbf{E}_{B}.$ There exists a unique $p_{\lambda,\mathbf{E}_{B}}(\mathbf{X})\in\mathcal{H}(\lambda)$ that represent the restriction of $\mathcal{V}_{\mathbf{E}_{B}}$ to $\mathcal{H}(\lambda)$, which means for every $q(\mathbf{X})\in\mathcal{H}(\lambda)$, we have (5.2) \[\int_{V_{A,B}(\mathbb{R})}q(\mathbf{X})\overline{p_{\lambda,\mathbf{E}_{B}}}( \mathbf{X})d\mu(\mathbf{X})=q(\mathbf{E}_{B}).\] Lemma 5.5.: _We have_ \[p_{\lambda,\mathbf{E}_{B}}(\alpha\mathbf{X})=p_{\lambda,\mathbf{E}_{B}}( \mathbf{X})\] _for every $\alpha\in O_{A,\mathbf{E}}(\mathbb{R}).$ Moreover, we have_ \[p_{\lambda,\mathbf{E}_{B}}(g^{-1}\mathbf{E}_{B})=p_{\lambda,\mathbf{E}_{B}}(g \mathbf{E}_{B})\] _for every $g\in O_{A}(\mathbb{R}).$ Finally_ \[p_{\lambda,\mathbf{E}_{B}}=\tau(\sqrt{B})^{-1}p_{\lambda,\mathbf{E}}.\] Proof.: Note that the functional $\mathcal{V}_{\mathbf{E}_{B}}$ is invariant by $O_{A,\mathbf{E}}(\mathbb{R}),$ which means \[\mathcal{V}_{\mathbf{E}_{B}}(q(\mathbf{X}))=\mathcal{V}_{\mathbf{E}_{B}}(q( \alpha\mathbf{X}))\] for every $\alpha\in O_{A,\mathbf{E}}(\mathbb{R}).$ This concludes the first part of the lemma. Let $\mathcal{P}_{O_{A,\mathbf{E}}(\mathbb{R})}$ be the set of harmonic polynomials which are invariant by $O_{A,\mathbf{E}}(\mathbb{R}).$ There is an involution $\sigma$ defined on $\mathcal{P}_{O_{A,\mathbf{E}}(\mathbb{R})}$ as follows. For $q\in\mathcal{P}_{O_{A,\mathbf{E}}(\mathbb{R})}$ and $\mathbf{X}=g\mathbf{E}_{B}$ define \[\sigma(q)(\mathbf{X}):=q(g^{-1}\mathbf{E}_{B}).\] It is easy to see that $\sigma(q)\in\mathcal{P}_{O_{A,\mathbf{E}}(\mathbb{R})}$ and $\mathcal{V}_{\mathbf{E}_{B}}(q)=\mathcal{V}_{\mathbf{E}_{B}}(\sigma(q))$. This implies $\sigma(p_{\lambda,\mathbf{E}_{B}})=p_{\lambda,\mathbf{E}_{B}}$, which concludes the second part of the lemma. Finally, we have (5.3) \[\begin{split}\int_{V_{A,B}(\mathbb{R})}q(\mathbf{X})\overline{ \tau(\sqrt{B})^{-1}p_{\lambda,\mathbf{E}}}(\mathbf{X})d\mu(\mathbf{X})& =\int_{V_{A,B}(\mathbb{R})}q(\mathbf{X})\overline{p_{\lambda, \mathbf{E}}}(\mathbf{X}\sqrt{B}^{-1})d\mu(\mathbf{X})\\ &=\int_{V_{A,I}(\mathbb{R})}q(\mathbf{Y}\sqrt{B})\overline{p_{ \lambda,\mathbf{E}}}(\mathbf{Y})d\mu(\mathbf{Y})\\ &=q(\mathbf{E}\sqrt{B})=q(\mathbf{E}_{B}).\end{split}\] This concludes the proof of the lemma. ∎ ### The weight space with a functional Let $\mathcal{H}(\lambda)^{}$ be the dual vector space of $\mathcal{H}(\lambda)$. $GL_{n}(\mathbb{C})$ acts on $\mathcal{H}(\lambda)^{}$ by ${\tau^{\intercal}}^{-1}.$ Every $f\in\mathcal{H}(\lambda)$ defines a functional $\langle f,\mathcal{H}(\lambda)^{}\rangle\to\mathbb{C}.$ ## 6. The oscillator representations and Weyl’s sums In this section, we describe the Schrödinger Model of the oscillator representation. We use this model to construct an explicit automorphic Siegel’s theta kernel. Next, we define the theta transfer $\Theta(\pi)$ of an automorphic representation $\pi$ of $O_{A}$. We show that $\Theta(\phi_{\pi})$ is a holomorphic Siegel modular form with values in the dual space of vectors of $\pi_{\infty}$ and describe explicitly its weight and its level in terms of $\pi_{\infty}$ and $A_{m\times m}$. We also show that $\Theta(\pi)$ is an eigenfunction of the Hecke operators at the unramified places. Finally, we relate the Weyl sums $W(\phi_{\pi},B)$ to $\langle\Theta(\pi,B),\phi_{\pi}\rangle,$ where $\Theta(\pi,B)$ is the $B$-th Fourier coefficient of $\Theta(\pi).$ This generalizes the result of Shintani [27]. ### The Schrödinger Model of the oscillator representation We begin by describing the oscillator representation. Let $W:=\mathbb{Q}^{n}$ and $W^{}$ be its dual vector space. Consider the $2n$ dimensional symplectic vector space $W\oplus W^{}$with the symplectic form: \[\langle(x_{1},y_{1}),(x_{2},y_{2})\rangle:=y_{2}(x_{1})-y_{1}(x_{2}).\] We fix the lattices $L_{W}:=\mathbb{Z}^{n}\subset W$ and $L_{W^{}}:=\mathbb{Z}^{n}\subset W^{}.$ Let $E=\mathbb{Q}^{m}$ be an orthogonal vector space with the positive definite symmetric form \[(\mathbf{x},\mathbf{y})=\mathbf{x}^{\intercal}A\mathbf{y}.\] We fix the lattice $L_{E}:=\mathbb{Z}^{m}\subset E$ and denote its dual lattice by $L_{E}^{}:=A^{-1}\mathbb{Z}^{m}\subset E.$ Consider the $2mn$ dimensional symplectic vector space $(W\oplus W^{})\otimes E$ with the symplectic form \[\langle w_{1}\otimes v_{1},w_{2}\otimes v_{2}\rangle=\langle w_{1},w_{2} \rangle(v_{1},v_{2}),\] where $v_{1},v_{2}\in E$ and $w_{1},w_{2}\in W\oplus W^{}.$ Note that $L:=L_{W}\otimes L_{E}^{}\oplus L_{W^{}}\otimes L_{E}$ is a self dual lattice inside $(W\oplus W^{})\otimes E.$ We write a complete polarization as $(W\oplus W^{})\otimes E=W\otimes E\oplus W^{}\otimes E,$ which means $W\otimes E$ and $W^{}\otimes E$ are the isotropic subspace of the symplectic vector space $(W\oplus W^{})\otimes E.$ We consider the adelic points of $(W\oplus W^{})\otimes E$ with respect to the self dual lattice $L.$ We identify $W^{}\otimes E$ with $Hom(W,E).$ Let $\mathcal{S}\left(Hom(W,E)\otimes\mathbb{A}_{\mathbb{Q}}\right)$ be the Schwartz-Bruhat functions defined on the adelic space $Hom(W,E)\otimes\mathbb{A}_{\mathbb{Q}}.$ Fix $\psi$ to be the continuous additive character on $\mathbb{Q}\backslash A_{\mathbb{Q}}/\prod_{p}\mathbb{Z}_{p}$ which is defined as follows on a complete representative set: \[\psi\big{(}(a_{\infty},0,0,\dots)\big{)}:=\exp(2\pi ia_{\infty}).\] By using standard the standard basis in the lattices $L_{W\oplus W^{}}$ and $L$, we identify the symplectic group $SP_{W\oplus W^{}}(\mathbb{A}_{\mathbb{Q}})$ with $SP_{2n}(\mathbb{A}_{\mathbb{Q}})$ and $SP_{(W\oplus W^{})\otimes E}(\mathbb{A}_{\mathbb{Q}})$ with $SP_{2mn}(\mathbb{A}_{\mathbb{Q}}).$ We note that under these coordinates the matrix representation of $s\otimes I_{m\times m}\in SP_{2mn}$ for $s=\begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}\in SP_{2n}(\mathbb{A}_{\mathbb{Q}})$ is (6.1) \[\begin{bmatrix}g_{11}\otimes I_{m\times m}&g_{12}\otimes A\\ g_{21}\otimes A^{-1}&g_{22}\otimes I_{m\times m}\end{bmatrix}.\] Weil defined the Metaplictic group $\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}]$ (double cover of the symplectic group $SP_{2mn}[\mathbb{A}_{\mathbb{Q}}]$) and constructed the unitary oscillator representation $\omega_{\psi}.$ In what follows, we record some properties of $\omega_{\psi}$ from [12, Section 2]; we refer the reader to [12, Section 2] and [6] for the definition and further properties of $\omega_{\psi}$. In the Schrödinger Model of the oscillator representation, $\omega_{\psi}$ acts on $L^{2}\left(Hom(W,E)\otimes\mathbb{A}_{\mathbb{Q}}\right).$ It is not convenient and necessary for our purpose to give the action of $\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}].$ We only need the action of a parabolic subgroup of $\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}],$ which we describe next. Let $P\subset SP_{2mn}$ be the stabilizer of $W\otimes E.$ Let $M\subset P$ (maximal levi subgroup) be the stabilizer of $W\otimes E$ and $W^{}\otimes E$ and $N\subset P$ (maximal unipotent subgroup) be the subgroup which acts as identity on $W\otimes E.$ We have a factorization $P=MN.$ More concretely, (6.2) \[\begin{split} P&=\left\{\begin{bmatrix}(g^{t})^{-1}& \\ 0&g\end{bmatrix}:g\in GL(W^{}\otimes E)\right\},\\ M&=\left\{\begin{bmatrix}(g^{t})^{-1}&0\\ 0&g\end{bmatrix}:g\in GL(W^{}\otimes E)\right\},\\ N&=\left\{\begin{bmatrix}I_{m}&n\\ 0&I_{m}\end{bmatrix}:n:W^{}\otimes E\to W\otimes E\ \text{ and }n=n^{ \intercal}\right\}.\end{split}\] We denote the inverse image of $P$ and $M$ in $\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}]$ by $\widetilde{P}$ and $\widetilde{M}.$ It follows that $N$ has a unique lift in $\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}]$, so we may regard $N\subset\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}].$ Given $\tilde{g}\in\widetilde{M}$, its image in $M$ will be denoted by of $g.$ The oscillator representation acts as follows in the Schrödinger Model; see [12, 6]. For $\Phi\in L^{2}(W^{}\otimes E)$ we have (6.3) \[\begin{split}\omega_{\psi}\left(\begin{bmatrix}I_{m}&n\\ 0&I_{m}\end{bmatrix}\right)\Phi(X)=\psi\left(\frac{1}{2}\langle X,n(X)\rangle \right)\Phi(X),\\ \omega_{\psi}\left(\tilde{g}\right)\Phi(X)=\gamma(\tilde{g})\|\det (g)\|^{-1/2}\Phi(g^{-1}(X)),\end{split}\] where $X\in W^{}\otimes E,$$g\in GL(W^{}\otimes E)$ and $\gamma(\tilde{g})$ is a certain root of unity, and det is the usual determinant function on $GL(W^{}\otimes E)$, and $\|.\|$ denotes the standard absolute value on $A_{\mathbb{Q}}$. In particular, for $(\alpha,\tilde{s})\in O_{A}\times\widetilde{GL}(W^{})\subset GL(W^{}\otimes E),$ we have (6.4) \[\omega_{\psi}((\tilde{\alpha},\tilde{s}))\Phi(X)=\gamma(\tilde{s})\|\det(s)\|^{- m/2}\Phi(\alpha^{-1}\circ X\circ{s^{\intercal}}^{-1}),\] where $s^{\intercal}\in GL(W)$ is the transpose of $s$ and $\alpha^{-1}\circ X\circ{s^{\intercal}}^{-1}\in Hom(W,E)$ is the composition of the linear maps. Here we have for convenience replaced $\widetilde{O_{A}}$ with $O_{A}$ itself and identified $\widetilde{SP}_{2mn}[\mathbb{A}_{\mathbb{Q}}]$ with the image of $\omega_{\psi}$; see [12, Section 4] for further discussion. ### Construction of the Siegel theta kernel In this section, we construct Siegel’s theta kernel. We begin by defining the Siegel upper half place associated to the symplectic space $(W\oplus W^{},\langle,\rangle).$ Let \[\mathbb{D}:=\left\{Z\in W^{}\otimes{\mathbb{C}}\to W\otimes{\mathbb{C}}\text{ such that }Z^{\intercal}=Z,\text{ and }\Im(Z)>0\right\},\] where $\Im(Z)$ is obtained by taking the imaginary part of every matrix entry of $Z$ and $\Im(Z)>0$ means $\Im(Z)$ is positive definite. Let $s:=\begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}\in SP_{2n}(\mathbb{Q}),$ where $g_{11}\in Hom(W,W),$$g_{12}\in Hom(W^{},W),$$g_{21}\in Hom(W,W^{})$ and $g_{22}\in Hom(W^{},W^{}).$ Then $s$ acts on $\mathbb{D}$ as follows: (6.5) \[s:Z\to(g_{11}\circ Z+g_{12})\circ(g_{21}\circ Z+g_{22})^{-1}\] where $\circ$ is the composition of linear maps in $Hom$. For $Z\in\mathbb{D}$ and $f\in\mathcal{H}(\lambda)=\lambda\otimes\tau,$ where $\mathcal{H}(\lambda)$ is the irreducible representation of $GL(n,\mathbb{C})\times O_{A}(\mathbb{C})$ defined in Theorem 5.3, we define $\varphi_{f,Z}\in\mathcal{S}\left(Hom(W,E)(\mathbb{A}_{\mathbb{Q}})\right)$, as follows: \[\varphi_{f,Z}(\mathbf{X}_{\infty},\prod_{p}{\mathbf{X}_{p}}):=\exp\left(i\pi tr (Z\mathbf{X}_{\infty}^{\intercal}A\mathbf{X}_{\infty})\right)f(\mathbf{X}_{ \infty})\prod_{p}1_{\mathbb{Z}_{p}}(\mathbf{X}_{p}),\] where $1_{\mathbb{Z}_{p}}(\mathbf{X}_{p})=1$ if $\mathbf{X}_{p}\in\mathbb{Z}_{p}^{d}$ and $1_{\mathbb{Z}_{p}}(\mathbf{X}_{p})=0$ otherwise. Let $\mathcal{H}(\lambda)^{}$ be the dual vector space of $\mathcal{H}(\lambda)$ that is defined in Section 5.3. We define to be the unique function that satisfies: $\langle\phi_{\lambda,Z},g\rangle=\varphi_{g,Z}$ for every $g\in\mathcal{H}(\lambda).$ Next, we describe the automorphic properties of $\varphi_{f,Z}$ and $\phi_{\lambda,Z}$ as a function of $Z$ on the Siegel half plane. Recall that by Theorem 5.3, \[\tau\left(((g_{21}Z+g_{22})^{\intercal})^{-1}\right)f(\mathbf{X_{\infty}})=f \left(\mathbf{X_{\infty}}\circ((g_{21}Z+g_{22})^{\intercal})^{-1}\right).\] Lemma 6.1.: _Let $f\in\mathcal{H}(\lambda)$ and $\tilde{s}\in\widetilde{SP}_{2n}(\mathbb{R})$ where $s=\begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}\in SP_{2n}(\mathbb{R}).$ We have_ (6.6) \[\omega_{\psi}(\tilde{s})\varphi_{f,Z}=\gamma(\tilde{s})\det(g_{21}Z+g_{22})^{- m/2}\varphi_{\left(\tau(g_{21}Z+g_{22})^{\intercal}\right)^{-1}f,s(Z)},\] _and equivalently,_ (6.7) \[\omega_{\psi}(\tilde{s})\phi_{\lambda,Z}=\gamma(\tilde{s})\det(g_{21}Z+g_{22}) ^{-m/2}\left(\tau(g_{21}Z+g_{22})^{\intercal}\right)^{-1}\phi_{\lambda,s(Z)},\] Proof.: We refer the reader to [19, Section 2.5.42]. ∎ Let $\Theta$ be the following distribution on $\mathcal{S}\left(Hom(W,E)\right)$ which sends a function to the sum of its values on the rational points of $Hom(W,E)(\mathbb{Q})$: \[\Theta(f):=\sum_{a\in Hom(W,E)(\mathbb{Q})}f(a).\] Let $\vartheta(g,f):=\Theta(\omega_{\psi}(g)f).$ It is well-known that $SP_{2mn}(\mathbb{Q})$ splits in $\widetilde{SP}_{2mn}(\mathbb{A}_{\mathbb{Q}})$ and we consider $SP_{2mn}(\mathbb{Q})\subset\widetilde{SP}_{2mn}(\mathbb{A}_{\mathbb{Q}}).$ It follows (by a generalized poisson formula) that $\vartheta(g,f)$ is invariant by the action of $SP_{2mn}(\mathbb{Q})$ on the left and it defines an automorphic function on $L^{2}\left(SP_{2mn}(\mathbb{Q}))\backslash\widetilde{SP}_{2mn}(\mathbb{A}_{ \mathbb{Q}})\right).$ For $\alpha\in O_{A}(\mathbb{A}_{\mathbb{Q}})$, $s\in\widetilde{SP}_{2n}(\mathbb{\mathbb{A}_{\mathbb{Q}}})$ and $f\in\mathcal{H}(\lambda)$ for some $\lambda$, we define the Siegel theta kernel $\vartheta(\alpha,\tilde{s},f,Z)$ to be the following: (6.8) \[\begin{split}\vartheta(\alpha,\tilde{s},f,Z):=\Theta(\omega_{\psi }((\alpha,\tilde{s}))\varphi_{f,Z}).\end{split}\] Note that $\vartheta(\alpha,\tilde{s},f,Z)$ is $O_{A}(\mathbb{Q})\times SP_{2n}(\mathbb{Q})$ invariant, and it defines a kernel which transfers the space cusp forms $\mathcal{A}^{0}\big{(}O_{A}(\mathbb{Q})\backslash O_{F}(\mathbb{A}_{\mathbb{Q} })\big{)}$ to the automorphic forms of $L^{2}\big{(}SP_{2n}(\mathbb{Q})\backslash\widetilde{SP}_{2n}(\mathbb{A}_{ \mathbb{Q}})\big{)}$ (possibly zero) and vice versa. Similarly, we define $\theta(\alpha,\tilde{s},\lambda,Z)$ with values in $\mathcal{H}(\lambda)^{}$ to be the unique function which satisfies \[\langle\theta(\alpha,\tilde{s},\lambda,Z),g\rangle=\vartheta(\alpha,\tilde{s}, g,Z)\] for every $g\in\mathcal{H}(\lambda).$ ### The weight and the level of the theta lift For $\alpha\in O_{A}(\mathbb{Q})\backslash O_{F}(\mathbb{A}_{\mathbb{Q}})$ we write $\theta(\alpha,\lambda,Z):=\theta(\alpha,\tilde{I}_{n\times n},\lambda,Z),$ where $\tilde{I}_{n\times n}$ is the identity element of $\widetilde{SP}_{2n}(\mathbb{A}_{\mathbb{Q}})$. In this section, we show that $\theta(\alpha,\lambda,Z)$ is a holomorphic Siegel modular form of $Z$ with values in the vector space $\mathcal{H}(\lambda)^{}$. Moreover, we show that its weight is given by the irreducible representation $\gamma\det^{m/2}(\tau^{\intercal})^{-1}$ and its level by the level of $A$. We begin by defining the associated congruence subgroup of $SP_{2n}(\mathbb{Z}).$ Let $D$ be the level of $A$ which is the smallest integer such that $DA^{-1}$ is integral and has even entries on its diagonal. We define the congruence subgroup $\Gamma_{0}^{n}(D)\subset SP_{n}(\mathbb{Z}):$ (6.9) \[\Gamma_{0}^{n}(D):=\Big{\{}\begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}\in SP_{n}(\mathbb{Z}):g_{21}\in DM_{n\times n}( \mathbb{Z}),\text{ and }g_{12}\in 2M_{n\times n}(\mathbb{Z})\Big{\}}.\] Proposition 6.2.: _Let $s_{0}=\begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}\in\Gamma_{0}^{n}(D)$. We have_ (6.10) \[\theta(\alpha,\lambda,Z)=\gamma(s_{0})\det(g_{21}Z+g_{22})^{-m/2}({\tau(g_{21} Z+g_{22})^{\intercal}})^{-1}\theta\left(\alpha,\lambda,s_{0}Z\right).\] Proof.: It is enough to show that for every $g\in\mathcal{H}(\lambda),$ we have \[\vartheta(\alpha,\tilde{I}_{n\times n},g,Z)=\gamma(\tilde{s})\det(g_{21}Z+g_{2 2})^{-m/2}\vartheta\left(\alpha,\tilde{I}_{n\times n},({\tau(g_{21}Z+g_{22})^{ \intercal}})^{-1}g,s_{0}Z\right).\] Since $SP_{2n}$ and $O_{A}$ commute in $SP_{2mn}$, $s_{0}\in\Gamma_{0}^{n}(D)\subset SP_{2n}(\mathbb{Q})$ and $\Theta$ is invariant by $\widetilde{SP}_{2mn}(\mathbb{A}_{\mathbb{Q}}),$ we have \[\vartheta(\alpha,\tilde{I}_{n\times n},g,Z)=\Theta(\omega_{\psi}(s_{0})\circ \omega_{\psi}(\alpha)\varphi_{g,Z})=\Theta(\omega_{\psi}(\alpha)\circ\omega_{ \psi}(s_{0})\varphi_{g,Z}).\] By (6.1), the image of $s_{0}$ inside $SP_{2mn}$ is \[\begin{bmatrix}g_{11}\otimes I_{m\times m}&g_{12}\otimes A\\ g_{21}\otimes A^{-1}&g_{22}\otimes I_{m\times m}\end{bmatrix}.\] We write $s_{0}=s_{0}^{\infty}s_{0,\infty},$ where $s_{0,\infty}\in\widetilde{SP}_{2n}\left(\prod_{p}SP_{2mn}(\mathbb{Q}_{p})\right)$ and $s_{0,\infty}\in\widetilde{SP}_{2n}(\mathbb{R}).$ By the definition of $D$, $\varphi_{g,Z}$ and (6.3), it follows that $\omega_{\psi}(s_{0}^{\infty})\varphi_{g,Z}=\varphi_{g,Z}.$ Finally by Lemma 6.1, we have \[\omega_{\psi}(s_{0,\infty})\varphi_{g,Z}=\gamma(\tilde{s}_{0})\det(g_{21}Z+g_{ 22})^{-m/2}\varphi_{\left(\tau(g_{21}Z+g_{22})^{\intercal}\right)^{-1}g,s(Z)}.\] This concludes the proof of our Proposition. ∎ Proposition 6.2 implies that $\theta(\alpha,\lambda,Z)$ has weight $(\tau_{\pi_{\infty}}^{\intercal})^{-1}$ and level $\Gamma_{0}^{n}(D)\subset SP_{n}(\mathbb{Z}),$ where $D$ is the discriminant of $A.$ ### Hecke operators and the theta lift In this section, we briefly explain the Hecke algebra of the orthogonal group $O_{A}$ and its dual pair $\widetilde{SP_{2n}}$ at the unramified primes. We cite a result of Howe [11, Theorem 7.1], that implies the theta transfer sends the eigenfunction of the Hecke operators of $O_{A}$ to the eigenfunction of the Hecke operators of $SP_{2n}$. Let $p$ be a prime number where $\gcd(p,D)=1.$ Let $\tilde{J}_{p}$ be the maximal compact subgroup of $\widetilde{SP}_{2mn}(\mathbb{Q}_{p}).$ It follows that $\tilde{J}_{p}$ splits and is isomorphic to $\tilde{J}_{p}=SP_{2mn}(\mathbb{Z}_{p})\times\{\pm 1\}$; see [11, Section 3]. Let $K_{p}$ and $K^{\prime}_{p}$ be the maximal compact subgroups of $O_{A}(\mathbb{Q}_{p})$ and $\widetilde{SP}_{2n}(\mathbb{Q}_{p}).$ Up to conjugation, we can assume that $K_{p}$ and $K^{\prime}_{p}$ contained in $J_{p}.$ Let $C^{\infty}_{c}(O_{A}(\mathbb{Q}_{p})//K_{p})$ be the (Hecke) algebra of $K$-bi-invariant functions on $O_{A}(\mathbb{Q}_{p})$. Define $C^{\infty}_{c}(SP_{2n}(\mathbb{Q}_{p})//K^{\prime}_{p})$ similarly. Let $I(K_{p},K^{\prime}_{p})$ be the vectors fixed by $\omega_{\psi}(K_{p})$ and by $\omega_{\psi}(K^{\prime}_{p}).$ Then $\omega_{\psi}\left(C^{\infty}_{c}(O_{A}(\mathbb{Q}_{p})//K_{p})\right)$ and $\omega_{\psi}\left(C^{\infty}_{c}(SP_{2n}(\mathbb{Q}_{p})//K^{\prime}_{p})\right)$ leaves $I(K_{p},K^{\prime}_{p})$ invariant. We consider the restrictions $\omega_{\psi}\left(C^{\infty}_{c}(O_{A}(\mathbb{Q}_{p})//K_{p})\right)\|I(K_{p} ,K^{\prime}_{p}).$ We cite the following result of Howe [11, Theorem 7.1] Theorem 6.3* (Howe).: _The restrictions $\omega_{\psi}\left(C^{\infty}_{c}(O_{A}(\mathbb{Q}_{p})//K_{p})\right)\|I(K_{p} ,K^{\prime}_{p})$ and $\omega_{\psi}\left(C^{\infty}_{c}(SP_{2n}(\mathbb{Q}_{p})//K^{\prime}_{p}) \right)\|I(K_{p},K^{\prime}_{p})$ are the same algebra of operators._ Suppose that $\phi_{\pi}$ is a smooth function which belongs to the automorphic irreducible representation of $\pi$ of $L^{2}\big{(}O_{A}(\mathbb{Q})\backslash O_{F}(\mathbb{A}_{\mathbb{Q}})\big{)}$ and is invariant by $K_{p}$ and $\pi_{\infty}=\lambda.$ We define (6.11) Proposition 6.4.: $\Theta(\phi_{\pi})(Z,f)$ _is an eigenfunction of $C^{\infty}_{c}(SP_{2n}(\mathbb{Q}_{p})//K^{\prime}_{p}).$_ Proof.: This is a consequence of of Theorem 6.4. See also Howe [12, Proposition 2.3] for more details. ∎ ### Weyl’s sums and the Fourier coefficient of the theta lift Let $B\in Hom(W,W^{})(\mathbb{Z})$ be a positive symmetric definite matrix $B^{\intercal}=B.$ Recall \[N=\left\{\begin{bmatrix}I_{m}&n\\ 0&I_{m}\end{bmatrix}\Big{\|}n:W^{}\otimes E\to W\otimes E\ \text{ and }n=n^{ \intercal}\right\}\subset SP_{2n}.\] Note that by definition 6.9, $\Theta(\phi_{\pi,j})(\lambda,Z)$ is invariant by sending $Z$ to $Z+2n$ where $n\in N(\mathbb{Z}).$ We define the $B$-th Fourier coefficient of $\Theta(\phi_{\pi})(Z,f)$ which is an element of $\mathcal{H}(\lambda)^{}$ as follows: (6.12) \[\Theta(\phi_{\pi,j})(\lambda,B):=\exp\left(-i\pi tr(ZB)\right)\int_{N(\mathbb{ Q})\backslash N(A_{\mathbb{Q}})}\Theta(\phi_{\pi})(\lambda,Z+2n)\psi\left(-tr( nB)\right)dn.\] Recall the Weyl sums $W(\phi_{\pi,j},B)$ and $p_{\lambda,\mathbf{E}_{B}}\in\mathcal{H}(\lambda)$ defined in (4.6) and (5.2) respectively. Theorem 6.5.: _We have_ \[\langle\Theta(\phi_{\pi,j})(\lambda,B),p_{\lambda,\mathbf{E}_{B}}\rangle=W( \phi_{\pi,j},B).\] Proof.: By Lemma 6.1, we have $\varphi_{f,Z+2n}=\omega_{\psi}(2n)\varphi_{f,Z}.$ By (6.3), we have \[\varphi_{f,Z+2n}=\omega_{\psi}(2n)\varphi_{f,Z}=\psi\left(\langle\alpha^{-1} \mathbf{H},n(\alpha^{-1}\mathbf{H})\rangle\right)\varphi_{f,Z}\] Therefore, where \[\varphi_{p_{\lambda,\mathbf{E}_{B}},Z}(\mathbf{X}_{\infty},\prod_{p}{\mathbf{X }_{p}}):=\exp\left(i\pi tr(Z\mathbf{X}_{\infty}^{\intercal}A\mathbf{X}_{\infty })\right)p_{\lambda,\mathbf{E}_{B}}(\mathbf{X}_{\infty})\prod_{p}1_{\mathbb{Z} _{p}}(\mathbf{X}_{p}).\] Note that $\langle\alpha^{-1}\mathbf{H},n(\alpha^{-1}\mathbf{H})\rangle=tr(n\mathbf{H}^{ \intercal}({\alpha^{-1}})^{\intercal}A\alpha^{-1}\mathbf{H})=tr(n\mathbf{H}^{ \intercal}A\mathbf{H})$ is independent of $\alpha.$ Moreover, $\varphi_{p_{\lambda,\mathbf{E}_{B}},Z}(\alpha^{-1}\mathbf{H})=0$ unless $\alpha^{-1}\mathbf{H}\in Hom(W,E)(\prod_{p}\mathbb{Z}_{p})$, which implies $\mathbf{H}^{\intercal}A\mathbf{H}\in Hom(W,E)(\mathbb{Z}).$ By the orthogonality of the additive character $\psi,$ for $\varphi_{p_{\lambda,\mathbf{E}_{B}},Z}(\alpha^{-1}\mathbf{H})\neq 0$ we have \[\int_{N(\mathbb{Q})\backslash N(A_{\mathbb{Q}})}\psi\left(\langle\alpha^{-1} \mathbf{H},n(\alpha^{-1}\mathbf{H})\rangle-tr(nB)\right)dn=\delta(\mathbf{H}^{ \intercal}A\mathbf{H}=B).\] where \[\delta(X,Y)=\begin{cases}1&\text{ if }X=Y\\ 0&\text{ otherwise.}\end{cases}\] Therefore, \[\begin{split}\langle\Theta(\phi_{\pi,j})(\lambda,B), p_{\lambda,\mathbf{E}_{B}}\rangle=\int_{O_{A}(\mathbb{Q})\backslash O_{A}( \mathbb{A}_{\mathbb{Q}})}\sum_{\mathbf{H}\in V_{A,B}(\mathbb{Q})}\varphi_{p_{ \lambda,\mathbf{E}_{B}},Z}(\alpha^{-1}\mathbf{H})\bar{\phi}_{\pi,i}(\alpha) \prod_{p}1_{\mathbb{Z}_{p}}(\alpha^{-1}\mathbf{H}_{p})d\alpha.\end{split}\] Recall that $\phi_{\pi,i}$ is defined on $O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{E}}( \mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p}).$ By Lemma 5.5, $\varphi_{p_{\lambda,\mathbf{E}_{B}},Z}(\alpha^{-1}\mathbf{H}),$ as a function of $\alpha$, is also defined on $O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{E}}(\mathbb{R})O_{A}(\prod_{p} \mathbb{Z}_{p}).$ We identify $O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_{\mathbb{Q}})/O_{A,\mathbf{E}}( \mathbb{R})O_{A}(\prod_{p}\mathbb{Z}_{p})$ with $O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}.$ Recall from 4.1$O_{A}(\mathbb{Q})\backslash\mathcal{L}_{A,B}=\cup_{i=1}^{h}\{(L_{i},\mathbf{X} ):\mathbf{X}\in O_{L_{i}}\backslash V_{A,B}(\mathbb{R})\}.$ Hence, \[\begin{split}&\int_{O_{A}(\mathbb{Q})\backslash O_{A}(\mathbb{A}_ {\mathbb{Q}})}\sum_{\mathbf{H}\in V_{A,B}(\mathbb{Q})}\varphi_{p_{\lambda, \mathbf{E}_{B}},Z}(\alpha^{-1}\mathbf{H})\bar{\phi}_{\pi,i}(\alpha)\prod_{p}1_ {\mathbb{Z}_{p}}(\alpha^{-1}\mathbf{H}_{p})d\alpha\\ &=\sum_{L_{i}}\frac{1}{\|O_{L_{i}}\|}\sum_{\mathbf{H}\in V_{A,B}( \mathbb{Q})}\int_{O_{A}(\mathbb{R})/O_{A,\mathbf{E}}}\phi_{\pi,j}((L_{i},g \mathbf{E}))p_{\lambda,\mathbf{E}_{B}}(g^{-1}\mathbf{H})\prod_{p}1_{\mathbb{Z} _{p}}(\alpha_{i}^{-1}\mathbf{H}_{p})dg,\end{split}\] where we have; see (4.2) \[\prod_{p}1_{\mathbb{Z}_{p}}(\alpha_{i}^{-1}\mathbf{H}_{p})\begin{cases}1&\text { if }H\in\mathcal{S}_{L_{i},B}\\ 0&\text{ otherwise.}\end{cases}\] Therefore, \[\begin{split}\langle\Theta(\phi_{\pi,j})(\lambda,B),p_{\lambda, \mathbf{E}_{B}}\rangle=\sum_{L_{i}}\frac{1}{\|O_{L_{i}}\|}\sum_{\mathbf{H}\in \mathcal{S}_{L_{i},B}}\int_{O_{A}(\mathbb{R})/O_{A,\mathbf{E}}}\phi_{\pi,j}((L _{i},g\mathbf{E}))p_{\lambda,\mathbf{E}_{B}}(g^{-1}\mathbf{H})dg.\end{split}\] Since $\mathbf{H}\in V_{A,B}(\mathbb{Q}),$ there exists $\alpha_{\mathbf{H}}\in O_{A}(\mathbb{R})/O_{A,\mathbf{E}}$ such that $\alpha_{\mathbf{H}}\mathbf{E}=\mathbf{H}.$ By the symmetry of $p_{\lambda,\mathbf{E}_{B}}$ proved in Lemma 5.5, we have \[\begin{split}\int_{O_{A}(\mathbb{R})/O_{A,\mathbf{E}}}\phi_{\pi,j }((L_{i},g\mathbf{E}))p_{\lambda,\mathbf{E}_{B}}(g^{-1}\mathbf{H})dg& =\int_{O_{A}(\mathbb{R})/O_{A,\mathbf{E}}}\phi_{\pi,j}((L_{i},g \mathbf{E}))p_{\lambda,\mathbf{E}_{B}}(g^{-1}\alpha_{\mathbf{H}}\mathbf{E})dg \\ &=\int_{O_{A}(\mathbb{R})/O_{A,\mathbf{E}}}\phi_{\pi,j}((L_{i}, \alpha_{\mathbf{H}}g\mathbf{E}))p_{\lambda,\mathbf{E}_{B}}(g^{-1}\mathbf{E})dg \\ &=\int_{O_{A}(\mathbb{R})/O_{A,\mathbf{E}}}\phi_{\pi,j}((L_{i}, \alpha_{\mathbf{H}}g\mathbf{E}))p_{\lambda,\mathbf{E}_{B}}(g\mathbf{E})dg\\ &=\phi_{\pi,j}((L_{i},\alpha_{\mathbf{H}}\mathbf{E}))=\phi_{\pi,j }((L_{i},\mathbf{H})).\end{split}\] Therefore, \[\begin{split}\langle\Theta(\phi_{\pi,j})(\lambda,B),p_{\lambda, \mathbf{E}_{B}}\rangle&=\sum_{L_{i}}\frac{1}{\|O_{L_{i}}\|}\sum_{ \mathbf{H}\in\mathcal{S}_{L_{i},B}}\phi_{\pi,j}((L_{i},\mathbf{H}))=W(\phi_{ \pi,j},B).\end{split}\] This concludes the proof of our theorem. ∎ ## 7. Proof of Theorem 1.8 Proof.: By Propositions 4.3 and 4.4, we have \[\text{Var}(B,r)=\sum_{\pi}\sum_{j=1}^{d_{\pi}}\|h_{r}(\pi_{\infty})\|^{2}\|W(\phi _{\pi,j},B)\|^{2}.\] By Theorem 6.5 and Lemma 5.5, we have (7.1) \[\begin{split} W(\phi_{\pi,j},B)=\langle\Theta(\phi_{\pi,j})( \lambda,B),p_{\lambda,\mathbf{E}_{B}}\rangle&=\langle\Theta(\phi_ {\pi,j})(\lambda,B),\tau(\sqrt{B})^{-1}p_{\lambda,\mathbf{E}}\rangle\\ &=\langle{\tau(\sqrt{B})^{\intercal}}^{-1}\Theta(\phi_{\pi,j})( \lambda,B),p_{\lambda,\mathbf{E}}\rangle.\end{split}\] This concludes the proof of Theorem 1.8. ∎ ## References [BR88] S. Böcherer and S. Raghavan. On Fourier coefficients of Siegel modular forms. _J. Reine Angew. Math._, 384:80–101, 1988. * [BRS17] J. Bourgain, Z. Rudnick, and P. Sarnak. Spatial statistics for lattice points on the sphere I: Individual results. _Bull. Iranian Math. Soc._, 43(4):361–386, 2017. * [CS99] J. H. Conway and N. J. A. Sloane. _Sphere packings, lattices and groups_, volume 290 of _Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_. Springer-Verlag, New York, third edition, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. * [Duk88] W. Duke. Hyperbolic distribution problems and half-integral weight Maass forms. _Invent. Math._, 92(1):73–90, 1988. * [EMV13] Jordan S. Ellenberg, Philippe Michel, and Akshay Venkatesh. Linnik’s ergodic method and the distribution of integer points on spheres. In _Automorphic representations and $L$-functions_, volume 22 of _Tata Inst. Fundam. Res. Stud. Math._, pages 119–185. Tata Inst. Fund. Res., Mumbai, 2013. * [Gel79] Stephen Gelbart. Examples of dual reductive pairs. In _Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1_, Proc. Sympos. Pure Math., XXXIII, pages 287–296. Amer. Math. Soc., Providence, R.I., 1979. * [GGN13] Anish Ghosh, Alexander Gorodnik, and Amos Nevo. Diophantine approximation and automorphic spectrum. _Int. Math. Res. Not. IMRN_, (21):5002–5058, 2013. * [GGN15] Anish Ghosh, Alexander Gorodnik, and Amos Nevo. Diophantine approximation exponents on homogeneous varieties. In _Recent trends in ergodic theory and dynamical systems_, volume 631 of _Contemp. Math._, pages 181–200. Amer. Math. Soc., Providence, RI, 2015. * [GGN16] Anish Ghosh, Alexander Gorodnik, and Amos Nevo. Best possible rates of distribution of dense lattice orbits in homogeneous spaces. _J. reine angew. Math._, 2016. * [Han04] Jonathan Hanke. Some recent results about (ternary) quadratic forms. In _Number theory_, volume 36 of _CRM Proc. Lecture Notes_, pages 147–164. Amer. Math. Soc., Providence, RI, 2004. * [How79] R. Howe. $\theta$-series and invariant theory. In _Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1_, Proc. Sympos. Pure Math., XXXIII, pages 275–285. Amer. Math. Soc., Providence, R.I., 1979. * [HPS83] Roger Howe and I. I. Piatetski-Shapiro. Some examples of automorphic forms on ${\rm Sp}_{4}$. _Duke Math. J._, 50(1):55–106, 1983. * [Iwa87] Henryk Iwaniec. Fourier coefficients of modular forms of half-integral weight. _Invent. Math._, 87(2):385–401, 1987. * [Kit84] Yoshiyuki Kitaoka. Fourier coefficients of Siegel cusp forms of degree two. _Nagoya Math. J._, 93:149–171, 1984. * [Kit86] Y. Kitaoka. _Lectures on Siegel modular forms and representation by quadratic forms_, volume 77 of _Tata Institute of Fundamental Research Lectures on Mathematics and Physics_. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1986. * [Koh04] Winfried Kohnen. On the growth of Fourier coefficients of certain special Siegel cusp forms. _Math. Z._, 248(2):345–350, 2004. * [KS93] Svetlana Katok and Peter Sarnak. Heegner points, cycles and Maass forms. _Israel J. Math._, 84(1-2):193–227, 1993. * [KV78] M. Kashiwara and M. Vergne. On the Segal-Shale-Weil representations and harmonic polynomials. _Invent. Math._, 44(1):1–47, 1978. * [LV80] Gérard Lion and Michèle Vergne. _The Weil representation, Maslov index and theta series_, volume 6 of _Progress in Mathematics_. Birkhäuser, Boston, Mass., 1980. * [PS18] Ori Parzanchevski and Peter Sarnak. Super-golden-gates for $PU(2)$. _Adv. Math._, 327:869–901, 2018. * [RS98] E. M. Rains and N. J. A. Sloane. The shadow theory of modular and unimodular lattices. _J. Number Theory_, 73(2):359–389, 1998. * [Sar05] Peter Sarnak. Notes on the generalized Ramanujan conjectures. In _Harmonic analysis, the trace formula, and Shimura varieties_, volume 4 of _Clay Math. Proc._, pages 659–685. Amer. Math. Soc., Providence, RI, 2005. * [Sar15a] N. T Sardari. Optimal strong approximation for quadratic forms. _Accepted for publication at Duke Math Journal_, October 2015. * [Sar15b] Peter Sarnak. _Letter to Scott Aaronson and Andy Pollington on the Solovay-Kitaev Theorem_, February 2015. * [Sar17] N. T Sardari. Complexity of strong approximation on the sphere. _ArXiv e-prints_, March 2017. * [Shi73] Goro Shimura. On modular forms of half integral weight. _Ann. of Math. (2)_, 97:440–481, 1973. * [Shi75] Takuro Shintani. On construction of holomorphic cusp forms of half integral weight. _Nagoya Math. J._, 58:83–126, 1975. * [Sie44] Carl Ludwig Siegel. On the theory of indefinite quadratic forms. _Ann. of Math. (2)_, 45:577–622, 1944. * [Sie63] Carl Ludwig Siegel. _Lectures on the analytical theory of quadratic forms_. Notes by Morgan Ward. Third revised edition. Buchhandlung Robert Peppmüller, Göttingen, 1963. * [Ven13] Akshay Venkatesh. A note on sphere packings in high dimension. _Int. Math. Res. Not. IMRN_, (7):1628–1642, 2013. * [Wei64] André Weil. Sur certains groupes d’opérateurs unitaires. _Acta Math._, 111:143–211, 1964. * [Wei65] André Weil. Sur la formule de Siegel dans la théorie des groupes classiques. _Acta Math._, 113:1–87, 1965. * [Wri33] E. M. Wright. The representation of a number as a sum of five or more squares (ii). _Quart. J. Math. Oxford_, 4:37–51 ; 228–232., 1933. * [Wri37] E. M. Wright. The representation of a number as a sum of four almost equal squares. _Quart. J. Math. Oxford_, 8:278–279., 1937.
1008.3167	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 95647, "num_imgs": 20, "llama3_tokens_count": 28883 }	[ "content_image/1008.3167/x1.png", "content_image/1008.3167/x2.png", "content_image/1008.3167/x3.png", "content_image/1008.3167/x4.png", "content_image/1008.3167/x5.png", "content_image/1008.3167/x6.png", "content_image/1008.3167/x18.png", "content_image/1008.3167/x22.png", "content_image/1008.3167/x23.png", "content_image/1008.3167/x24.png", "content_image/1008.3167/x25.png", "content_image/1008.3167/x26.png", "content_image/1008.3167/x27.png", "content_image/1008.3167/x28.png", "content_image/1008.3167/x29.png", "content_image/1008.3167/x30.png", "content_image/1008.3167/x31.png", "content_image/1008.3167/x32.png", "content_image/1008.3167/x33.png", "content_image/1008.3167/x34.png" ]	# The SL2S Galaxy-scale Lens Sample. II. Cosmic evolution of dark and luminous mass in early-type galaxies Andrea J. Ruff¹²${}^{}$ Raphaël Gavazzi³ Philip J. Marshall¹⁴ Tommaso Treu¹${}^{{\dagger}}$ Matthew W. Auger¹ Florence Brault³ [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] ###### Abstract We present a joint gravitational lensing and stellar-dynamical analysis of 11 early-type galaxies (median deflector redshift $z_{\rm d}=0.5$) from Strong Lenses in the Legacy Survey (SL2S). Using newly measured redshifts and stellar velocity dispersions from Keck spectroscopy with lens models from Paper I, we derive the total mass density slope inside the Einstein radius for each of the 11 lenses. The average total density slope is found to be $\langle\gamma^{\prime}\rangle$$\,=\,$$2.16^{+0.09}_{-0.09}$ ($\rho_{\rm tot}\propto r^{-\gamma^{\prime}}$), with an intrinsic scatter of $0.25^{+0.10}_{-0.07}$. We also determine the dark matter fraction for each lens within half the effective radius, $R_{\rm eff}/2$ and find the average projected dark matter mass fraction to be $0.42^{+0.08}_{-0.08}$ with a scatter of $0.20^{+0.09}_{-0.07}$ for a [66] IMF. By combining the SL2S results with those from the Sloan Lens ACS Survey (median $z_{\rm d}=0.2$) and the Lenses Structure and Dynamics survey (median $z_{\rm d}=0.8$), we investigate cosmic evolution of $\gamma^{\prime}$ and find a mild trend $\partial\langle\gamma^{\prime}\rangle/\partial z_{\rm d}$$\,=\,$$-0.25^{+0.10}_{-0.12}$. This suggests that the total density profile of massive galaxies has become slightly steeper over cosmic time. If this result is confirmed by larger samples, it would indicate that dissipative processes played some role in the growth of massive galaxies since $z\sim 1$. Subject headings:galaxies: fundamental parameters — gravitational lensing — [FOOTNOTE:][ENDFOOTNOTE] ${\dagger}$ [FOOTNOTE:${\dagger}$][ENDFOOTNOTE] ## 1. Introduction Early-type (_i.e._ elliptical and lenticular) galaxies in the local universe are considered to be simple objects (_e.g._ Bertin & Stiavelli 1993; Merritt 1999; Ciotti 2009). In the central few kpc most of their mass is dominated by stars, while at larger radii there is convincing evidence – at least for the most massive systems – that dark matter halos are dominant. Their stellar populations are relatively simple, dominated by old stars with little or negligible star formation (_e.g._ Renzini 2006). Remarkably, many global properties, ranging from the chemical composition of their stars to their size, luminosity and mass of the central black hole, correlate tightly with their stellar velocity dispersion, $\sigma$(_e.g._ Bernardi et al. 2005; Graves et al. 2009). In spite of this suspected simplicity, their formation and evolution are still poorly understood, and therefore they are the subject of many observational and theoretical investigations. A number of observational facts have proved difficult to explain by theoretical models. These include: i) the tightness of the empirical correlations with $\sigma$(_e.g._ , Bernardi et al. 2005; Graves et al. 2009; Nipoti et al. 60); ii) the so-called downsizing trend of their stellar populations, _i.e._ the correlation between mean stellar age and present day stellar mass (_e.g._ , Thomas et al. 2005; Treu et al. 2005; van der Wel et al. 2005; Juneau et al. 2005; di Serego Alighieri et al. 2005); iii) the evolution of the upper end of their mass function since $z\sim 1$(_e.g._ , Bundy et al. 2005, 2007; Cimatti et al. 2006; van der Wel et al. 2009; Hopkins et al. 2010); iv) the unusually compact size of high redshift massive red galaxies (_e.g._ , Treu et al. 1998; Daddi et al. 2005; van Dokkum et al. 2008; Saracco et al. 2009; Cassata et al. 2010; Mancini et al. 2010; Newman et al. 2010). From a theoretical standpoint, it is clear that understanding the interplay between baryons, black holes, and dark matter is essential to develop a scenario that can quantitatively reproduce all observations. The physical processes that need to be accurately modeled in a successful theory appear to include black hole accretion and the related energy and momentum feedback (_e.g._ , Croton et al. 2006; Ciotti et al. 2009), dry and wet major mergers (_e.g._ , Khochfar & Silk 2006; Ciotti et al. 2007; Robertson et al. 2006), as well as minor mergers and dry accretion of minor satellites (_e.g._ , Naab et al. 2009; Hopkins et al. 2010). Most of the observational studies, including those listed in the previous paragraph, are concerned with global parameters of early-type galaxies. An entirely new line of investigation can be opened up if we are able to dissect early-type galaxies and map their internal dynamical structure as a function of cosmic time. By decomposing the internal mass distribution of early-type galaxies into luminous and dark components we can start addressing the following questions, the answers to which would provide essential clues as to their formation and evolution. How and when is mass assembled to form early-type galaxies? How are baryons converted into stars and accumulated inside dark matter halos? Is the mass density profile of early-type galaxies comparable to that observed in numerical simulations? Do isolated early-type galaxies undergo internal structural and dynamical evolution? As numerical simulations of early-type galaxies become more and more realistic, detailed knowledge of their internal structure (_e.g._ their distribution functions) will provide more and more stringent tests of the current paradigm of structure formation (_e.g._ Meza et al. 2003; Naab et al. 2007; Oñorbe et al. 2007; Lackner & Ostriker 2010). Great progress in answering these questions has been achieved in the past few years with the systematic study of early-type galaxies acting as strong gravitational lenses. For these systems, strong lensing provides an absolutely calibrated measurement of mass at a fiducial radius (the Einstein radius), which is typically comparable in size to the effective radius. By combining this mass tracer with traditional diagnostics such as stellar velocity dispersion (Miralda-Escude 1995; Natarajan & Kneib 1996; Treu & Koopmans 2002), and stellar mass maps from multicolor imaging and/or spectroscopy, one can break many of the degeneracies inherent to each method alone, including the mass-anisotropy degeneracy, bulge-halo degeneracy, and the stellar mass/initial mass function (IMF) degeneracy (_e.g._ Koopmans & Treu 2003; Treu & Koopmans 2004; Treu et al. 2010). Additional information can be gathered with the addition of weak-lensing (Gavazzi et al. 2007; Jiang & Kochanek 2007; Lagattuta et al. 2010), although at the moment this is not possible for individual galaxies. The SLACS team applied this methodology to a sample of more than 80 early-type galaxies (Bolton et al. 2006, 7; Auger et al. 2009), which have been shown to be indistinguishable from equally massive non-lensing early-type galaxies in terms of their internal properties and environment (Bolton et al. 2006; Treu et al. 2006, 2009). Among the most relevant findings of SLACS is that the total mass-density profile $\rho_{\rm tot}\propto r^{-\gamma^{\prime}}$ of early-type galaxies is close to isothermal with $\gamma^{\prime}=2.085^{+0.025}_{-0.018}$ with intrinsic scatter less than 0.1 (Koopmans et al. 2009; Barnabè et al. 2009; Auger et al. 2010), even though neither the stars nor the dark matter obey a simple power law profile (see also Wucknitz et al. 2004). This “bulge-halo conspiracy”, similar to the disk-halo conspiracy found for spiral galaxies (_e.g._ van Albada & Sancisi 1986), has implications both for lensing studies but also for galaxy formation studies. Since the total mass density profile is preserved by dry-mergers Dehnen (2005); Kazantzidis et al. (2006); Nipoti et al. (61), and dark matter-only profiles are not isothermal (_e.g._ Navarro et al. 2010), the isothermal nature has to be established through dissipational processes (_e.g._ , Koopmans et al. 2006). The other main finding of the SLACS survey is that (assuming a constant stellar IMF) the fraction of dark matter $f_{\rm DM}$ within a fixed fraction of the effective radius increases with galaxy mass or stellar velocity dispersion (see also Jiang & Kochanek 2007; Grillo et al. 2008; Cardone et al. 2009; Cardone & Tortora 2010). Possible explanations for this trend are: varying efficiency in converting baryons into stars as a function of halo mass, varying inner slope of the dark matter halo with mass, and evolutionary processes. For example, dry mergers can increase the fraction of dark matter within the effective radius (_e.g._ , Nipoti et al. 61). The SLACS, however, sample is limited to low redshift by the selection function of the parent sample of SDSS luminous galaxies. Therefore, evolutionary studies with the SLACS sample are limited to a short baseline. Suitable samples of strong lenses are smaller at high redshift. In fact, most of the strong lensing galaxies known to date at $z>0.4$ are too faint and/or dominated by strongly lensed bright quasars to allow for detailed kinematic studies of the deflector. Of the handful of exceptions (_e.g._ , Faure et al. 2008; Lagattuta et al. 2010), only a few of them have published lensing and dynamical analysis (Treu & Koopmans 2002; Ohyama et al. 2002; Treu & Koopmans 2004; Suyu et al. 2010). These early studies found results similar to SLACS, _i.e._ the internal slope $\gamma^{\prime}$ is close to isothermal, and the central dark matter fraction appears to increase with mass as found by the LSD Survey (Treu & Koopmans 2004). However, the results also hint that things may have been different at $z\sim 1$ when the universe was less than half its present age: the scatter in $\gamma^{\prime}$ might have been larger (Treu & Koopmans 2004) and perhaps even the average might have been different (Koopmans et al. 2006). Unfortunately, current samples beyond $z\sim 0.4$ with lensing and dynamical data are too small to probe the evolution of the internal structure of early-type galaxies, its scatter and trends with mass at the same time. In this paper we present the first detailed study of a sample of early-type lens galaxies identified by the Strong Lenses in the Legacy survey (SL2S). We use new spectroscopic data obtained at the W.M. Keck telescopes, in combination with multicolor photometry from the CFHT Legacy Survey to infer dynamical and stellar mass for the deflector galaxies. This imaging data-set is complemented by _HST_ imaging that allowed to confirm the lensing nature of the SL2S candidates and detailed lens modeling. The newly measured source and deflector redshifts are combined with gravitational lens models from Paper I (Gavazzi et al., in preparation) to infer lensing masses. We determine the total mass density profile of the early-type lens galaxies (quantified by $\gamma^{\prime}$) and their central dark matter fraction ($f_{\rm DM}$). The median redshift of the sample is $z=0.494$, providing an ideal complement to the earlier SLACS and LSD samples. By combining the three samples we investigate evolutionary trends in these quantities. This paper is organized as follows. In Section 2 we describe the SL2S survey, and how the galaxy-scale lens candidates were selected. We then present our spectroscopic measurements in Section 3 before combining them with lens model parameters to investigate the mass structure of massive galaxies since redshift 0.9 in Section 7. In Section 4 we incorporate stellar mass estimates in order to separate the dark and luminous components of the lens galaxies. The cosmic evolution of the total mass density slope and dark matter fraction are discussed in Section 8. After a brief discussion of our results in Section 9 we conclude in Section 10. Throughout this paper magnitudes are given in the AB system. We assume a concordance cosmology with matter and dark energy density $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and Hubble constant H${}_{0}$=70 km s${}^{-1}$Mpc${}^{-1}$. ## 2. Strong Lenses in the Legacy Survey: SL2S In this section we describe our feeder survey, the CFHT Legacy Survey, and how we identified our new sample of galaxy scale lenses at intermediate redshift. A more detailed description is given in Paper I. Strong lensing candidates were selected from the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS)¹. In brief, the survey consists of two main components of sufficient depth and image quality to be interesting for lens searching². Both are imaged in the $u^{}$, $g$, $r$, $i$ and $z$ bands with the 1 deg${}^{2}$ field-of-view Megacam Camera. The multi epoch Deep survey covers 4 pointing of 1 deg${}^{2}$ each. Two different image stacks were produced: D-85 contains the 85% best seeing images whereas the D-25 only includes the 25% best seeing images. For finding lenses we only considered these latter, better resolution stacks. They reach a typical depth of $u^{}\simeq 26.18$, $g=25.96\simeq 25.47$, $r\simeq 25.43$, $i\simeq 25.08$ and $z\simeq 24.57$ (80% completeness for point sources) with typical FWHM point spread functions of $0\farcs 75$, $0\farcs 69$, $0\farcs 64$, $0\farcs 62$ and $0\farcs 61$, respectively. The Wide survey is a single epoch imaging survey, covering some 171 deg${}^{2}$ in 4 patches of the sky. It reaches a typical depth of $u^{}\simeq 25.35$, $g\simeq 25.47$, $r\simeq 24.83$, $i\simeq 24.48$ and $z\simeq 23.60$ (AB mag of 80% completeness limit for point sources) with typical FWHM point spread functions of $0\farcs 85$, $0\farcs 79$, $0\farcs 71$, $0\farcs 64$ and $0\farcs 68$, respectively. Because of the greater area, the Wide component is our main provider of lens candidates. [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] Images from both the Deep and Wide survey modes were analyzed to find strong lens candidates using several algorithms, as described by Cabanac et al. (2007) at the group and cluster mass scales, and by Gavazzi et al. (in preparation) at the galaxy scale with the RingFinder algorithm. The ring-detecting algorithm is aimed at detecting compact rings around centers of isolated galaxies ($<10^{13}h^{-1}M_{\odot}$), and works by focusing on the achromatic image excesses around early-type lens galaxies that are indicative of the presence of lensed arcs. For each of a sample of pre-selected bright ($i_{\rm AB}\leq 22.5$) red galaxies, a scaled, PSF-matched version of the $i$-band cutout image was subtracted from the $g$-band image of the same system. The rescaling in this operation is performed such that the early-type galaxy light is efficiently removed, leaving only objects with an SED different from that of the target galaxy. These (typically) blue residuals are then characterized with an object detector, and analyzed for their position, ellipticity, and orientation, and those showing characteristic properties of lensed arcs are kept as lens candidates. A sample of several hundred good candidates were visually inspected, and ranked for follow-up with _HST_. Currently, 65 CFHT galaxy-scale lens candidates have been observed firstly with ACS, then WFPC2, and finally with WFC3 as snapshot programs over cycles 15, 16 and the ongoing cycle 17. Details of these observations are given in Paper I. Approximately 50% of the lens candidates were confirmed as lenses in this way. The sources are all faint blue galaxies, with very few showing signs of an active nucleus. Those with the most convincing lens models (see Paper I) were selected for spectroscopic follow up to obtain high precision redshifts for lens and source galaxies, and lens galaxy velocity dispersions. These observations are described in the next section. ## 3. Spectroscopic observations <figure><img src="content_image/1008.3167/x1.png"><figcaption>Figure 1.— Keck/LRIS spectra for 17 SL2S lenses. An elliptical galaxytemplate is shown in a thin black line beneath each spectrum. The dotted blacklines mark features of the lens galaxy, while source emission lines arehighlighted in red. The shaded regions indicate A and B band telluricabsorption features.</figcaption></figure> <figure><img src="content_image/1008.3167/x2.png"><figcaption>Figure 1.— continued.</figcaption></figure> <figure><img src="content_image/1008.3167/x3.png"><figcaption>Figure 1.— continued.</figcaption></figure> <figure><img src="content_image/1008.3167/x4.png"><figcaption>Figure 1.— continued.</figcaption></figure> <figure><img src="content_image/1008.3167/x5.png"><figcaption>Figure 1.— continued.</figcaption></figure> Follow up spectroscopy of 17 selected lens candidates was obtained using the Low Resolution Imaging Spectrograph (LRIS) on the Keck I telescope over six nights from 2006 to 2010, with the aim of measuring the deflector and source redshifts and the velocity dispersion of the deflector. Because of incompleteness in the execution of the _HST_ snapshot observations mentioned above, we have spectra of 6 systems where _HST_ data is not yet available. These lens candidates, however, were very promising on the basis of their CFHT images alone. In this section we describe the observations and their analysis in some detail, outlining our methodology for measuring the lens redshifts, source redshifts, and lens velocity dispersions. ### Details of the observations and data reduction Observations were made in long slit mode, with the slit centered on the deflector. The slit orientation was chosen to maximize both the source flux and the spatial separation of the deflector and arc. The total flux is dominated by the deflector, so maximizing the source flux maximizes the likelihood of measuring the source redshift, $z_{\rm s}$. Spatial separation within the slit also separates the deflector and arc traces in the 2D spectra, allowing us to look for source emission lines directly, as discussed in Section 3.4. The seeing ranged between 0$\farcs$55 and 1$\farcs$3 and total exposure times varied from 20 to 45 min, typically with two consecutive 15 min intervals. Between each exposure, we dithered along the slit by 10${}^{\prime\prime}$ to improve defect removal and sky subtraction. A summary of each observing run is given in Table 1. run \| obsdate \| graname \| grisname \| slit \| time \| seeing ---\|---\|---\|---\|---\|---\|--- 1 \| 20 Jul 2006 \| 831/8200 \| 600/4000 \| 1\farcs5 \| 20 \| 0\farcs84 2a \| 23 Dec 2006 \| 831/8200 \| 300/5000 \| 1\farcs5 \| 60 \| 0\farcs60 2b \| 23 Dec 2006 \| 831/8200 \| 300/5000 \| 1\farcs5 \| 40 \| 0\farcs80 3a \| 13 Sep 2007 \| 400/8500 \| 400/3400 \| 1\farcs0 \| 45 \| 0\farcs73 3b \| 13 Sep 2007 \| 400/8500 \| 400/3400 \| 1\farcs0 \| 30 \| 0\farcs75 4a \| 14 Sep 2007 \| 400/8500 \| 400/3400 \| 1\farcs0 \| 30 \| 0\farcs55 4b \| 14 Sep 2007 \| 600/7500 \| 600/4000 \| 0\farcs7 \| 30 \| 0\farcs55 4c \| 14 Sep 2007 \| 831/8200 \| 600/4000 \| 1\farcs0 \| 90 \| 0\farcs55 5 \| 9 Sep 2009 \| 600/7500 \| 300/5000 \| 1\farcs0 \| 30 \| 1\farcs0 6 \| 14 Jan 2010 \| 600/7500 \| 300/5000 \| 1\farcs0 \| 45 \| 1\farcs3 Note. – The exposure time is given in minutes. The plate scale for both the red and blue chips is 0\farcs135/pixel for observations from 2009 and after. For observing runs before 2009, the red chip had plate scale of 0\farcs211/pixel. Table 1 Observing logs The later setups took greater advantage of the blue sensitivity of LRIS. We found that the most effective method to measure all three quantities was using a 680 dichroic, so that the deflector redshift and velocity dispersion could be measured in the blue. The 680 dichroic allowed us to choose a large wavelength coverage for the red, which was useful in detecting [O ii] out to $z>1.1$. The data was reduced using a Python pipeline (developed by MWA). On the red side, the night sky lines were used to determine the wavelength solution, while standard arclamps were used on the blue side. Two different extraction windows were chosen: one wide extraction window to ensure that light from the source was also included in the final spectrum (half widths between 5 and 12 pixels from the central trace) and another narrower extraction window to increase the signal to noise ratio, which is important for measuring velocity dispersions. The reduced spectra are shown in Figure 1. ### Measuring deflector redshifts Deflector redshifts were measured using the centroids and known rest frame wavelengths of prominent absorption features. In the majority of cases, a minimum of the Ca ii H and K lines were used, however other absorption features were also used when available. The additional absorption features used were: H$\eta$ (3835Å), G band (4305Å), H$\gamma$ (4341Å), H$\beta$ (4861Å), Mg ii ($5175$Å), Na ii ($5892$Å) and H$\alpha$ (6563Å). Typically, between three and five absorption features were centroided to measure the redshift. The measured redshifts are listed in Table 2. ### Measuring deflector velocity dispersions Velocity dispersions were measured by fitting combinations of stellar spectra over regions with prominent absorption features and high signal to noise. Linear combinations of stellar spectra were used to fit a model to the data and calculate a velocity dispersion. A Python based implementation of the van der Marel (1994) velocity dispersion code, developed by MWA and described by Suyu et al. (2009), was used. We use a set of templates from the INDO-US stellar library containing spectra for a set of seven K and G giants with a variety of temperatures and spectra. K and G giants were used because they provide a good description of the spectra of our deflectors, as expected for massive ellipticals. The value of the velocity dispersion, $\sigma$ for each deflector was determined by finding a consistent value over several spectral regions and features. If the mean signal to noise (S/N) ratio per rest-frame angstrom was $\;\buildrel>\over{\sim}\;$$10$ in the 4000–5000Å range, then consistent and reliable $\sigma$ values could be measured. Of the 17 SL2S lenses, 12 spectra had sufficient S/N to measure $\sigma$. In general, the rest frame 4000–5000Å range was used, as the G-band absorption feature is often uncontaminated by atmospheric absorption, and contains no sharp changes in the continuum and the CCD efficiency is also good over this range. The average signal to noise ratio per angstrom in the 4000–5000Å range for the 12 objects with measured velocity dispersions is 24.8. Other regions that were used to fit the stellar templates were: the 5000–6000Å range (matching the Mg ii and Na ii absorption features) and the 3500–4000Å range was also used, but only where the depth and continuum fit to the spectra were good. Generally, three regions of the spectrum that produced good fits and consistent $\sigma$ values were used to calculate the final $\sigma$ and its associated uncertainty. For each lens, one of the models generated to measure $\sigma$ is shown in Figure 6. Regions where atmospheric absorption was a problem, or the templates did not produce a good fit were masked out, as shown in the grey regions of Figure 6. The results of the velocity dispersion measurements and the mean S/N per angstrom in the rest frame 4000–5000Å range for each object are listed in Table 2. The mean measured $\sigma$ for the SL2S sample is 250 km s${}^{-1}$. The measured velocity dispersions were then corrected to a uniform physical aperture using the slit width, the size of the extraction window and the empirical power-law relation of Jørgensen et al. (1995). The corrected velocity dispersion, $\sigma_{e2}$, measures the velocity dispersion at $R_{\rm eff}/2$, as used by Bolton et al. (9). Inside the effective radius, the relation is well described by a power law: \[\sigma_{\rm e2}=\sigma_{\rm ap}\left(\frac{R_{\rm eff}}{2\,r_{\rm ap}}\right)^ {-0.04},\] (1) where $2r_{\rm ap}\approx 2(xy/\pi)^{1/2}$ and where $x$ and $y$ are the width and length of the rectangular aperture. <figure><img src="content_image/1008.3167/x6.png"><figcaption>Figure 6.— Measuring the stellar velocity dispersions of the lens galaxies,by accurate absorption line fitting. For each object, we show the relevantportion of the LRIS spectrum (black line), compared to a model generated fromall 9 INDO-US templates and a sixth order polynomial continuum (in red, withthe continuum alone shown in green). The grey shaded wavelength ranges werenot included in the fits. The lower sub-panels show the fit residuals in eachcase.</figcaption></figure> ### Measuring source redshifts Source redshifts were measured for five objects. Two objects, SL2SJ021737-051329 and SL2SJ141137+565119 ($z_{s}$$\,=\,$1.847 and 1.420, respectively) have multiple source emission lines that were centroided to measure the redshift. As can be seen in Figure 1, SL2SJ021737-051329 has narrow emission lines typical of Type II AGN (C iv $\lambda$1549, He ii $\lambda$1640, O iii] $\lambda$1666 and C iii] $\lambda$1909). SL2SJ141137+565119 shows a clear splitting of the [O ii] doublet as well as strong C iii] and O iii] emission. The three remaining source redshifts were identified using [O ii] $\lambda\lambda$3726.1, 3728.8 only. SL2SJ022610-042011 and SL2SJ022511-045433 ($z_{s}$$\,=\,$1.232 and 1.1988, respectively) show a clear splitting of the doublet in the 2D spectra. For the objects where there was no clear splitting of the [O ii] doublet, the redshift measurement was more difficult. We used two methods to search for source emission lines. Firstly, we looked for emission lines in the 2D spectra at the expected position of the source trace. Secondly, we looked at the residuals from both a simple fit to a standard elliptical galaxy template and also the residuals from the velocity dispersion measurement, discussed in Section 3.3. Despite that three source redshifts were identified using [O ii] alone, no other strong emission features are observed blue-ward of this feature, corroborating the identification, as we would expect to see other lines if the detected feature were redder (_e.g._ O iii], H$\alpha$ or H$\beta$). <figure><img src="content_image/1008.3167/x18.png"><figcaption>Figure 7.— The [O ii] doublet is shown for each object with a measured sourceredshift and observed [O ii] doublet.</figcaption></figure> The 2D spectrum of SL2SJ022511-045433 shows a slight offset between the [O ii] emission of the arc and the arclet. We estimate the difference in redshifts to be $\Delta z=0.0005$, corresponding to a relative motion of $\sim$150 km/s. This is most likely due to kinematic structure of the source. Consistent with this interpretation, the lens model indicates the presence of two separate peaks in the surface brightness distribution of the lens, _i.e._ possibly a pair of source galaxies (see Paper I for details). ### Systems with no source emission lines For the remaining 11 sources, for which no source emission lines could be detected, we must estimate the source redshift from whatever information we have. We first consider the the _HST_ and CFHTLS photometry. We did not attempt to infer photometric redshifts, since the disentanglement of lens and source colors in the low-resolution, multi-filter CFHTLS data was deemed likely to lead to significant systematic uncertainty. Instead, we conservatively used the redshift distribution of the faint galaxies in the COSMOS survey to provide a broad probability density function (PDF) for $z_{\rm s}$. Leauthaud et al. (2007) model this distribution using the following functional form: \[{\rm Pr}(z_{\rm s}\|m) \propto z_{\rm s}^{2}\,\exp{\left[-\left(\frac{z}{z_{0}(m)}\right)^{1.5} \right]}\] (2) \[z_{0}(m) = \frac{(0.18\,m-3.3)}{1.412}.\] (3) Here $m$ is the AB magnitude of the source in the F814W filter. In most cases we have either a F606W magnitude from the _HST_ image modeling, or a $g$-band magnitude from the CFHTLS image modeling: we assume that the sources, as faint blue galaxies, have spectral energy distributions consistent with flat in the AB magnitude system, and hence substitute our blue unlensed source magnitudes (output from the lens inversions) directly. This approximate transformation introduces a small additional uncertainty which we neglect relative to the intrinsic PDF width. This model source redshift distribution was used by, among others, Gavazzi et al. (2007) and Lagattuta et al. (2010) when estimating the redshift distribution of background sources. We note that this approach is qualitatively different from that taken by Gavazzi et al. (2007) and Lagattuta et al. (2010): in their weak lensing studies they did not have a single well-defined source magnitude, but instead integrated over the number counts down to the magnitude limit. Whereas we are able to use the small amount of information that the source brightness provides. We neglect the small uncertainty in $z_{0}(m)$ and the photometric uncertainty in $m$, and truncate the density to zero at $z_{\rm s}\leq z_{\rm d}$. We then draw samples from ${\rm Pr}(z_{\rm s}\|m)$ that we can then transform into distances, physical masses and so on. This gives the broader PDF, ${\rm Pr}(z_{\rm s})$. While the source magnitude provides a very rough photometric redshift, we can also use the lens geometry to give a similarly rough _geometric_ redshift. This involves multiplying the COSMOS prior PDF ${\rm Pr}(z_{\rm s}\|m)$ by an additional distribution describing our prior knowledge of the lens strength, as follows. In practice we do this by importance sampling the COSMOS prior (see _e.g._ Lewis & Bridle 2002; Suyu et al. 2010, for descriptions of how this process works). Given that the velocity dispersion of the dark matter is approximately equal to the central stellar velocity dispersion, and the apparently universal (approximate) isothermal profile of lens galaxies (see _e.g._ Kochanek 1994; Koopmans et al. 2006), we can down weight some predicted $z_{\rm s,pdf}$ values based on the velocity dispersions they predict. The total deflector mass is modeled as a Singular Isothermal Ellipsoid (SIE), with a velocity dispersion \[\sigma_{\mathrm{SIE}}=186.2\,{\rm km}\,{\rm s}^{-1}\,\times\sqrt{\frac{R_{\rm Ein }}{\rm arcsec}\frac{D_{\rm ds}}{D_{\rm s}}}\] (4) where $D_{\rm ds}$ and $D_{\rm s}$ are the angular diameter distances between the deflector and source and observer and source, respectively. The calculated $\sigma_{\mathrm{SIE}}$ for each $z_{\rm s,pdf}$ was used to give a ${\rm Pr}(\sigma_{\mathrm{SIE}})$. A joint prior, ${\rm Pr}(z_{\rm s}\|m){\rm Pr}(\sigma_{\mathrm{SIE}})$, using the additional information from $\sigma_{\mathrm{SIE}}$ was then calculated and used to tighten the constraint on $z_{\rm s}$. <figure><img src="content_image/1008.3167/x22.png"><figcaption>Figure 8.— The distribution of measured velocity dispersions is shown withthe Sheth et al. fitting function overlayed. The solid and dashed curves showthe fitting function multiplied by σ8 (lensing and luminosity selectionfunction) and σ4 (only lensing), respectively. The fitting functions werenormalized to the unit area of the histogram. Note that the mean of the SL2S σdistribution is in good agreement with the peak of the solid curve.</figcaption></figure> <figure><img src="content_image/1008.3167/x23.png"><figcaption>Figure 9.— The zs,pdf distributions are shown for the final sample of lenses.The dashed histogram shows the source redshift inferred from the COSMOSdistribution. The thick histogram shows the final zs,pdf, where the inferredsource redshifts were weighted by the Sheth et al. fitting function(multiplied by σ8), as discussed in the text. For lenses with a measured zs,the grey vertical line shows the measured source redshift. Note that the PDFshave been normalized to unit area and that y-axis on the bottom row is on adifferent scale.</figcaption></figure> To calculate ${\rm Pr}(\sigma_{\mathrm{SIE}})$, we assumed that the SL2S lens galaxies are at the high mass end of the velocity function of bright galaxies – high mass, because we know they are acting as gravitational lenses. To describe the velocity function of the SL2S lenses, the velocity dispersion fitting function determined by Sheth et al. (2003) was used. The parameters in the fitting function were determined using measurements of a large sample of early-type galaxies drawn from the SDSS database. The fitting function was multiplied by $\sigma^{4}$ to mimic the lensing selection function (see _e.g._ Auger et al. 2010). However, the lenses, particularly those with measured $\sigma$, are also luminosity selected because SDSS is a flux-selected sample and more luminous objects are drawn from a large volume (Hyde & Bernardi 2009; Auger et al. 2010). To account for the luminosity selection, the [69] fitting function was heuristically multiplied by $\sigma^{4}$. Figure 8 shows the distribution of the measured SL2S velocity dispersions with the [69] fitting function, multiplied by two different selection functions, overlayed. The dashed curve shows the fitting function multiplied by $\sigma^{4}$ (mimicking the lensing selection function), while the solid curve shows the fitting function multiplied by both the lensing and luminosity selection functions. Despite the small number of objects, the mean $\sigma$ of the SL2S deflectors ($250\pm 39$ km s${}^{-1}$) is in very good agreement with the peak of the solid curve ($243\pm 48$ km s${}^{-1}$). The prior adopted is then the [69] fitting function multiplied by $\sigma^{8}$; the resulting distribution was then normalized and used to compute importances to weight the $z_{\rm s,pdf}$ values drawn from the COSMOS prior. We note that final result is relatively insensitive to the precise choice of the exponent, since the dominant effect is the exponential cutoff, which eliminates unrealistically high stellar velocity dispersions. Our final prior PDF for the velocity dispersions of the SL2S galaxies is then given by: \[{\rm Pr}(\sigma_{\mathrm{SIE}})\propto\sigma^{\alpha+7}\,e^{-\left(\frac{ \sigma}{\sigma_{}}\right)^{\beta}},\] (5) where the best fit parameters determined by Sheth et al. (2003) for early-type galaxies are: $\sigma_{}$=88.8$\pm$17.7 km s${}^{-1}$, $\alpha$=6.5$\pm$1.0 and $\beta$=1.93$\pm$0.22. To account for the scatter in $f_{\rm SIE}=\sigma_{\mathrm{SIE}}/\sigma_{}$, the function was conservatively smeared by $20\%$. This distribution is quite broad: the assumption of an approximately isothermal mass profile does not significantly bias our results when inferring the profile slopes of individual lenses, since we include plausible scatter on $\sigma$. Distributions of $z_{\rm s,pdf}$ for the final sample of lenses are shown in Figure 9. The dashed line shows $z_{\rm s,pdf}$ distributions from the COSMOS prior without weightings and the solid line indicates the final $z_{\rm s,pdf}$ for each of the lens. The primary effect of ${\rm Pr}(\sigma_{\mathrm{SIE}})$ is to disfavor lower redshift solutions to the lens equation, that are offered by the COSMOS redshift distribution with its relatively low $z_{\rm s}$ peak, which would indicate unrealistically large stellar velocity dispersions, well above 400 km s${}^{-1}$. For each non-spectroscopic case, we give the median of each lens system’s source redshift PDF in Table 2, along with the 16${}^{\rm th}$ and 84${}^{\rm th}$ percentiles as indicators of the uncertainty. The method was verified by comparing the 5 spectroscopically measured redshifts to those inferred via this procedure, using the same 5 systems. The grey vertical lines in Figure 9 show the measured source redshifts, which are in good agreement with $z_{\rm s,pdf}$ after weighting by the Sheth et al. (2003) fitting function and the selection function. ## 4. Photometric measurements We now turn to our photometric data, in order to measure sizes and stellar masses of our lens galaxies. The latter are required in order for us to probe the dark matter fractions in the core regions of these objects, and are obtained by stellar population analysis of the CFHTLS photometry. ### Model size and magnitude estimation We fit an elliptically symmetric de Vaucouleurs profile (de Vaucouleurs 1948) surface brightness distributions to each of the lens galaxies in our sample, to measure the total magnitudes and effective radii. For each lens, we used the galfit software package to fit for the position, effective (half-light) radius, ellipticity, orientation angle and total magnitude. When available, we also used the _HST_ data that generally provide a better estimate of the deflector effective radius. In particular, the light profile in the reddest available filter should be a better tracer of the underlying stellar mass distribution profile. Otherwise we used the average effective radius of the CFHT r, i and z bands and the internal filter-to-filter r.m.s. scatter to get a precise estimate of the effective radius and its associated measurement error. Under the assumption of a spheroidal distribution of stars, with a single stellar population and no color gradients, the scatter across the filter set in each morphological parameter gives a simple estimate of the uncertainty of each parameter, and in particular on $R_{\rm eff}$. In most cases, the typical error on $R_{\rm eff}$ is of order 5%. Note also that $R_{\rm eff}$ is expressed along the geometric mean axis, that is the radius of the circle enclosing the same area as the elliptical isophote enclosing half the light. To build up a picture of each lens’ spectral energy distribution (SED), with the deflector photometry devoid of light coming from the lensed background source, we follow a two-step iterative process to mask out the lensed images: first we fit models to the images with no masking, and then use the residual image to manually mask out the lensed features, before refitting the model. Apparent magnitudes are then corrected for Galactic dust extinction (Schlegel et al. 1998). Errors are dominated by systematic photometric zero point uncertainties, dust correction errors and variations of galfit results depending on the masking strategy. All together these amount to systematic errors of $\sim$0.05 magnitudes in each CFHT bands. The resulting effective radii and magnitudes are given in Table 2. ### SED fitting methodology Estimates of the stellar masses were calculated using the CFHT galfit model magnitudes and composite stellar population models using a code developed by Auger et al. (2009). The code employs a Bayesian exploration of the stellar populations of galaxies using composite stellar population models produced by Bruzual & Charlot (2003). The code takes a set of photometric data for each object (g, r, i, z) magnitudes and their uncertainties from the CFHT filters and the measured redshift. The free parameters of the star formation history are: the time when star formation began, the time scale of the exponential decay of the star formation rate, internal reddening due to dust extinction, and metallicity. The parameter space was then explored using a Markov Chain Monte Carlo (MCMC) routine, allowing a full determination of the posterior probability distribution function for each parameter; see Auger et al. (2009) for a comprehensive description of the method. We employ uniform priors for all of the model parameters (including the metallicity) due to the absence of any well-calibrated priors for the stellar population model parameters of massive galaxies at the redshifts of these lenses. The code was run for both Salpeter (1955) and Chabrier (2003) initial mass functions (IMFs). The masses derived from the [66] IMF were a factor of $\sim$1.7 greater than the masses derived using the [19] IMF. In a recent study of 56 gravitational lenses identified by the SLACS survey, Treu et al. (2010) found that for massive early-type galaxies, the [66] IMF provides stellar masses in approximate agreement with those inferred by lensing and dynamical models, whereas the Chabrier IMF gives (on average) an underestimate. However, the IMF normalization may be mass dependent being more similar to a [19] IMF for the lower mass systems in the SLACS sample, with $\sigma$$\,\sim\,$200 km s${}^{-1}$ (Barnabe et al. 2010; Auger et al. 2010; see also Cappellari et al. 2006, 2009). For simplicity, we adopt the [66] IMF normalization, although they can easily be converted to other normalizations, by multiplying by an appropriate factor (_e.g._ Auger et al. 2009). Total stellar masses for a [66] IMF are given in Table 3. LSD stellar masses were also calculated using the method described above with photometry from Koopmans & Treu (2002, 2003) and Treu & Koopmans (2004). This method could only be applied to 4 of the 5 LSD lenses, as photometry in multiple bands is required. Using this method we inferred total stellar masses (in units of 10${}^{11}$$M_{\odot}$) of: 5.01$\pm$2.06, 2.24$\pm$0.58, 4.26$\pm$0.95 and 2.49$\pm$0.82 for H1417+526, H1543+535, MG2016+112 and 0047-281, respectively. ## 5. Lens modeling The modeling of our high resolution _HST_ images is described in detail in Paper I. Here, we briefly outline the procedure followed and describe the relevant output parameters that we use in our joint analysis. Following standard image reduction, the bright lens galaxies were subtracted from postage stamp images of the lenses, iteratively masking the lensed arcs. The remaining residual image was fitted using a flexibly parametrized source (an elliptical exponential profile with free position, ellipticity, orientation, size and flux) traced through a simple elliptically symmetric lens potential. We model each lens mass distribution using a Singular Isothermal Ellipsoid (SIE) model (Kormann et al. 1994), with centroid fixed at the centroid of the galfit model lens light distribution. The lens has three free parameters: Einstein radius $R_{\rm Ein}$, orientation $\phi$ and axis ratio $q$. In all cases the lens redshift is known spectroscopically, allowing us to work with Einstein radius in kpc. The parameters of each lens were inferred by exploring their posterior PDF using a Markov Chain Monte Carlo (MCMC) sampler. While the samples do characterize the posterior PDF as required, the simplicity of the lens model and the idealized source light distribution mean that, in practice, the posterior width does not provide a good estimate of the uncertainty on, for example, the Einstein radius. Instead, we use independent reconstructions in several bands (which we have for a subset of the lenses) to estimate the uncertainties (see _e.g._ Marshall et al. 2007), which are found to be about 5% on $R_{\rm Ein}$. The properties of the lens models are discussed in detail in Paper I; here we are concerned only with the combination of the lensing mass with the dynamical mass from our spectroscopic measurements. With this in mind, the definition of elliptical radius in the lens modeling was chosen to preserve the enclosed mass with circular apertures, adopting the same definition as the SLACS survey (Koopmans et al. 2006; Bolton et al. 9; Auger et al. 2009). The mass enclosed within the Einstein radius, $M_{\rm Ein}$, is essentially independent of the profile of the lens density profile: although we used an SIE model to infer $M_{\rm Ein}$, we can treat it as a profile-independent, circularly symmetric enclosed mass and then combine it with a dynamical spherical mass estimate in order to infer the total density profile slope (see Section 7.4 below). For more discussion of the accuracy of this assumption see Auger et al. (2009). Three systems J140614+520252, J220629+005728 and J221606-175131, were observed with LRIS before attempting to perform a lens model. But subsequently we were not able to find a satisfactory lens model and we failed in measuring their Einstein radius essentially because _HST_ imaging is not available for them yet. Deflector photometry along with effective radius and redshift are nevertheless reported in Table 2. A final conclusion on their lensing nature is left for future work (see also Paper I for further discussion). ## 6. Characterization of the SL2S sample In this section we explore the basic properties of the SL2S lens galaxies, and compare them with two previously studied samples of lenses. The SLACS sample is a lower redshift sample, $0.08<z_{\rm d}<0.51$, of 85 lenses from the Sloan Digital Sky Survey (Bolton et al. 2006, 7). The LSD survey measured the internal kinematics of 5 early-type lens galaxies over a large range of redshifts, $0.48<z_{\rm d}<1.00$, and masses (Treu & Koopmans 2004). ### The redshift distributions of SL2S lenses <figure><img src="content_image/1008.3167/x24.png"><figcaption>Figure 10.— Redshift distributions of the SL2S, SLACS and LSD surveys. Allthree distributions have been normalized to one.</figcaption></figure> It is instructive to compare the distribution of lens redshifts in the SL2S sample, described in the preceding section, with those of the lower and higher redshift reference samples, SLACS and LSD, respectively. In Figure 10, we show histograms of deflector redshifts for the SL2S (red), SLACS (grey) and LSD (black) samples. We see that, as anticipated, the SL2S lenses lie at higher redshift than the SLACS sample. The median and 68% confidence intervals are $z_{\rm d}$=0.494${}^{+0.16}_{-0.16}$, $z_{\rm s}$=1.199${}^{+0.22}_{-0.18}$ for the SL2S sample, $z_{\rm d}$=0.189${}^{+0.096}_{-0.066}$$z_{\rm s}$=0.606${}^{+0.201}_{-0.157}$ for SLACS and $z_{\rm d}$=0.810${}^{+0.096}_{-0.066}$$z_{\rm s}$=3.263${}^{+0.61}_{-0.61}$ for LSD. ### Comparison to previous lens samples The measured stellar velocity dispersions and effective radii provide us with a means to compare the SL2S, SLACS and LSD samples, to assess whether we are studying the same _types_ of galaxies. In Figure 11, we plot our spectroscopic velocity dispersions, $\sigma$ against the effective radii, $R_{\rm eff}$ inferred from the CFHTLS data above, in order to compare the types of galaxies selected in the SL2S, SLACS and LSD samples. The stellar velocity dispersions are consistent between the samples, while the effective radii of the SLACS galaxies are slightly higher. Note that the stellar velocity dispersions have been normalized to a standard aperture, $\sigma_{\rm e2}$ for all samples. <figure><img src="content_image/1008.3167/x25.png"><figcaption>Figure 11.— Stellar velocity dispersions are plotted against the effectiveradius in the V band. The SL2S, SLACS (Auger et al. 2009) and LSD (Treu &Koopmans 2004) data points are shown in red, grey and black, respectively.Note that Reff is measured at the intermediate axis and the stellar velocitydispersions have been corrected to σe2.</figcaption></figure> To compare the samples quantitatively, we applied the 2D Kolmogorov-Smirnov test in the $\sigma-R_{\rm eff}$ plane. We found the KS statistic to be 0.28, and the significance level to be 91%. Given this high significance, we conclude that we are drawing from the same population of galaxies, despite the marginal tendency towards larger sizes in the SLACS sample. <figure><img src="content_image/1008.3167/x26.png"><figcaption>Figure 12.— The effective radius is plotted against the Einstein radius. TheSL2S, SLACS and LSD points are shown in red, grey and black, respectively.Typical uncertainties on REin and Reff are 5% and 10%, respectively.</figcaption></figure> Figure 12 shows the distribution of $R_{\rm eff}$ and $R_{\rm Ein}$ for the SL2S, SLACS and LSD samples. The higher redshift samples (SL2S and LSD), have much higher physical $R_{\rm Ein}$ values while the effective radii are slightly smaller, as discussed above. The difference in $R_{\rm Ein}$ is partly due to the method used to select the lens candidates: resolvable SL2S ring radii of 0.5-2 arcsec correspond to larger physical radii than those of the SLACS sample. SLACS lenses were pre-selected to be bright SDSS target galaxies. This selection method ensures that they lie at relatively low redshift, which in turn means that the characterizing angular size of Einstein rings translates to a smaller physical Einstein radii for the SLACS lenses than the SL2S lenses. Furthermore, the source redshifts are on average significantly higher for the LSD and SL2S samples, implying larger Einstein Radii for the same mass distribution. We conclude that we do seem to be measuring similar types of galaxies, with similar sizes, in a similar mass range: the SLACS and SL2S lenses are consistent with having been drawn from the same population. However, the different lens geometry (due to the difference in redshift between the samples) means that we are sampling the mass density profile at a larger radius, which is approximately a factor of 2 times greater compared to the SLACS lenses. ## 7. The internal mass structure of SL2S lensing galaxies In this section we use our new spectroscopic measurements to carry out a joint lensing and dynamics analysis of the SL2S sample, estimating the total mass density profile slope and dark matter fraction of each individual lens galaxy. We first describe the simple model within which we work, and then present our inferences of these two key parameters. ### Power-law density profiles We choose to work in the context of a spherically-symmetric power law total mass density profile model (_e.g._ Treu & Koopmans 2002; Koopmans et al. 2006; Suyu et al. 2010): \[\rho_{tot}(r) \propto r^{-\gamma^{\prime}}\] (6) \[\Sigma(R) \propto R^{1-\gamma^{\prime}}.\] (7) As in the lens modeling, we use capitalized $R$ to denote projected radius. We can normalize these power law profiles in terms of $M_{\rm Ein}$, the robustly-estimated, profile-independent Einstein Mass, from Section 5 above: our approach is to use the well-measured Einstein radii and Einstein masses from the lens models of Paper I, and re-interpret them within the context of this spherical power-law model. The spherical power law model allows us to predict the observed spectroscopic velocity dispersion of the tracer stellar population, via the Jeans equation (see, _e.g._ Treu & Koopmans 2004; Koopmans et al. 2006; Auger et al. 2010; Suyu et al. 2010). Normalizing the surface density and integrating to the (observable) Einstein radius $R_{\rm Ein}$, we find that \[\Sigma(R) = \frac{(3-\gamma^{\prime})}{2}\,\Sigma_{\rm crit}\left(\frac{R}{R_ {\rm Ein}}\right)^{1-\gamma^{\prime}}\] (8) \[\rho_{tot}(r) = \frac{(3-\gamma^{\prime})}{2\pi^{1/2}}\frac{\Gamma(\frac{\gamma^{ \prime}}{2})}{\Gamma(\frac{\gamma^{\prime}-1}{2})}\frac{\Sigma_{\rm crit}}{R_{ \rm Ein}}\left(\frac{r}{R_{\rm Ein}}\right)^{-\gamma^{\prime}}.\] (9) Since $M_{\rm Ein}=\pi R_{\rm Ein}^{2}\Sigma_{\rm crit}$, we can choose the two parameters of our model to be $M_{\rm Ein}$ and $\gamma^{\prime}$. In the next subsection we outline how the stellar velocity dispersion is predicted from these parameters. ### Stellar dynamics analysis As described by Suyu et al. (2010) and Auger et al. (2010), we predict the stellar velocity dispersion at the appropriate aperture radius using the spherical Jeans equation. Qualitatively we use the power law profile to compute the total spherical mass enclosed within radius $r$, and assume the tracer stars follow a Hernquist distribution with scale radius related to the measured effective radius, and then integrate to predict the stellar velocity dispersion, $\sigma(M_{\rm Ein},\gamma^{\prime})$. We direct the reader to Section 2.3 of Suyu et al. (2010) for all relevant equations. We assume isotropic orbits for all of the systems, which is approximately found to be the case from a more detailed analysis of the resolved kinematics of several SLACS lenses (Barnabè et al. 2009). ### Joint analysis We propagate the uncertainties in the Einstein mass (which can be significant, for systems with no spectroscopic source redshift), and the spectroscopic velocity dispersion, in the following way. The output of the lens modeling procedure for each lens is the posterior probability distribution for the Einstein mass given the _HST_ image data $\mathbf{d}$, ${\rm Pr}(M_{\rm Ein}\|\mathbf{d},z_{\rm d},z_{\rm s})$, characterized as a set of sample $M_{\rm Ein}$ values – this can be viewed as the _prior_ PD for $M_{\rm Ein}$ for the joint analysis. The analysis of that lens’ spectrum yields the likelihood, ${\rm Pr}(\sigma^{\rm obs}\|M_{\rm Ein},\gamma^{\prime})$, which we assume to be Gaussian in $\sigma^{\rm obs}$, with mean equal to the Jeans prediction $\sigma(M_{\rm Ein},\gamma^{\prime})$ from the previous subsection. The posterior PDF that we seek for each lens is then \[{\rm Pr}(M_{\rm Ein},\gamma^{\prime}\|\sigma^{\rm obs},\mathbf{d}, z_{\rm d},z_{\rm s}) \propto {\rm Pr}(\sigma^{\rm obs}\|M_{\rm Ein},\gamma^{\prime})\] (10) \[\cdot{\rm Pr}(M_{\rm Ein}\|\mathbf{d},z_{\rm d},z_{\rm s})\] \[\cdot{\rm Pr}(\gamma^{\prime}).\] Since we are only working in the context of a single model, we do not compute the evidence to normalize the right-hand side of this equation; all we need are samples drawn from the posterior PDF ${\rm Pr}(M_{\rm Ein},\gamma^{\prime}\|\sigma^{\rm obs})$. For the prior on the slope $\gamma^{\prime}$ we assume a uniform distribution with limits of $-1.2$ and $-2.8$, which we find to be broad enough to enclose all the likelihood (and are in fact close to the mathematical limits required for the normalizability of the profile). We draw an equal number of sample $\gamma^{\prime}$ values from this uniform distribution to match the $M_{\rm Ein}$ samples, and then use the likelihood evaluated at each 2-dimensional sample position as weights: for each sample $\{M_{\rm Ein},\gamma^{\prime}\}$ we solve the Jeans equation to calculate $\sigma(M_{\rm Ein},\gamma^{\prime})$, and then evaluate the Gaussian function ${\rm Pr}(\sigma^{\rm obs}\|M_{\rm Ein},\gamma^{\prime})$. (For more details on importance sampling, see the appendix of [70] [2010].) This set of weighted samples can then be used to compute integrals over the posterior, confidence limits, histograms to represent the marginalized distributions and so on. We present these inferences in the following two sections, and summarize our numerical results in Table 3. ### The density profile slope $\gamma^{\prime}$ from lensing and stellar dynamics In total we have 11 SL2S lenses with both lensing and dynamical mass estimates. In Figure 13 we show the posterior probability distributions for the logarithmic density profile slope $\gamma^{\prime}$, resulting from the joint lensing and dynamics analysis. We see that the mean of the population lies at $\langle\gamma^{\prime}\rangle$$\,=\,$$2.16^{+0.09}_{-0.09}$, and is shown by a shaded region in Figure 13. The intrinsic (Gaussian) scatter of the sample, inferred assuming Gaussian errors on the individual $\gamma^{\prime}$ values, is $S_{\gamma^{\prime}}$$\,=\,$$0.25^{+0.10}_{-0.07}$. For each of the 11 lenses in the final sample, the median $\gamma^{\prime}$ (with 16${}^{\rm th}$ and 84${}^{\rm th}$ percentiles) is given in Table 3 <figure><img src="content_image/1008.3167/x27.png"><figcaption>Figure 13.— Posterior probability distributions for γ′ using a uniform prior.The solid and dashed curves show γ′ distributions for individual lenses withand without measured source redshifts, respectively. The shaded regionindicates the posterior PDF for the mean of the Gaussian distribution fromwhich the sample was inferred to have been drawn: ⟨γ′⟩=2.16+0.09−0.09. Notethat the distributions have been normalized to unit area.</figcaption></figure> <figure><img src="content_image/1008.3167/x28.png"><figcaption>Figure 14.— The slope of the density profile is plotted against Einsteinradius, projected mass within the Einstein radius, stellar ellipticity andeffective radius. The error bars show the 16th and 84th percentiles. The solidline shows the linear best fit to the data and the dashed lines indicate thescatter.</figcaption></figure> We explore how $\gamma^{\prime}$ varies with: $R_{\rm Ein}/R_{\rm eff}$, total mass within $R_{\rm Ein}$, axis ratio and effective radius of the lenses in Figure 14. In each panel, the solid line shows the linear best fit to the data and the dashed lines show the scatter. The gradients of the fits, including 16${}^{\rm th}$ and 84${}^{\rm th}$ percentiles are: \[d\gamma^{\prime}/d(R_{\rm Ein}/R_{\rm eff}) = -0.01^{+0.21}_{-0.11}\] \[d\gamma^{\prime}/dM_{R_{\rm eff}} = -0.03^{+0.17}_{-0.05}\] \[d\gamma^{\prime}/dq_{} = 0.22^{+0.29}_{-0.17}\] \[d\gamma^{\prime}/dR_{\rm eff} = 0.12^{+0.14}_{-0.07},\] with $M_{R_{\rm eff}}$ in units of $10^{11}M_{\odot}$, and $R_{\rm eff}$ in kpc. The large uncertainties on the gradients indicate negligible trends between $\gamma^{\prime}$ and these variables. As discussed in Section 9, cosmic evolution of $\gamma^{\prime}$ could be mimicked by a dependence on $R_{\rm Ein}/R_{\rm eff}$. However, a negligible correlation between $\gamma^{\prime}$ and $R_{\rm Ein}/R_{\rm eff}$ suggests that this is unlikely. We find no correlation between $\gamma^{\prime}$ and either $R_{\rm eff}$ or the projected mass inside $R_{\rm eff}$, indicating that $\gamma^{\prime}$ is independent of galaxy size and mass. ### The dark matter fraction in the SL2S lenses We now turn to our second key parameter: the dark matter mass fraction, $f_{\rm DM}$ in the cores of lens galaxies. Following the SLACS analyses of Koopmans et al. (2006) and Auger et al. (2009), we use $R_{\rm eff}/2$ as the fiducial aperture radius for dark matter fraction estimation. The stellar masses calculated in Section 4.2 are total stellar masses. These must be corrected to the mass within our fiducial aperture by integrating (under the assumption that stellar surface density follows surface brightness) the de Vaucouleurs profile, which is given by: \[I(r)=I_{\rm eff}\,{\rm e}^{-7.67\left[\left(\frac{r}{R_{\rm eff}}\right)^{1/4} -1\right]}.\] (11) The fraction of mass enclosed within $R_{\rm eff}$/2 is therefore given by: \[\frac{M_{}(R_{\rm eff}/2)}{M_{,\rm total}}=\frac{\int_{0}^{R_{\rm eff}/2}I(r )\,r\,dr}{\int_{0}^{\infty}I(r)\,r\,dr}=0.320.\] (12) We obtained total masses within the same radius by integrating the power law total surface density profile: \[\frac{M(R_{\rm eff}/2)}{M_{\rm Ein}}=\left(\frac{R_{\rm eff}/2}{R_{\rm Ein}} \right)^{3-\gamma^{\prime}}.\] (13) Total masses within the Einstein radius and total stellar masses are given in Table 3. The dark matter fraction is then defined by: \[f_{\rm DM}=1-\frac{M_{}(R_{\rm eff}/2)}{M(R_{\rm eff}/2)}.\] (14) We do not just have a single number for each of $M_{}(R_{\rm eff}/2)$ and $M(R_{\rm eff}/2)$; rather, we have probability distributions for each. We combine these to form the posterior PDF for $f_{\rm DM}$, by applying the formulae in Equations 12, 13 and 14 to each sample $\{M_{},M\}$ drawn from the product of ${\rm Pr}(M_{})$ (from the SED modeling) and ${\rm Pr}(M)$ (from the lensing plus dynamics joint analysis) – since these PDFs are independent, the members of each $\{M_{},M\}$ pair can be drawn randomly from each individual ensemble. The resulting PDF for $f_{\rm DM}$ (visualized as a histogram of $f_{\rm DM}$ samples) can have significant probability at $f_{\rm DM}<0$, due to the uncertainty in each mass estimate, and the fact that our model only enforces positivity for the total and stellar mass distributions, not their difference. For consistency with the SLACS analysis, we allow negative values of $f_{\rm DM}$ (equivalent to $M_{}>M$). We found that truncating the PDFs at $f_{\rm DM}\geq 0$ for the SL2S lenses did not significantly affect the numerical results. <figure><img src="content_image/1008.3167/x29.png"><figcaption>Figure 15.— Posterior probability distributions for the dark matter fractionin the SL2S lenses. The dark matter distributions of lenses with measuredsource redshifts are shown in a solid line. The dashed lines show thedistributions for lenses with no measured source redshift. Note that thedistributions have been normalized to unit area. The shaded region indicatesthe posterior PDF for the mean of the Gaussian distribution from which thesample was inferred to have been drawn: ⟨fDM⟩=0.42+0.08−0.08.</figcaption></figure> In Figure 15 we show the resulting dark matter fraction posterior probability distribution for each SL2S lens, assuming a [66] IMF. The median and 16${}^{\rm th}$ and 84${}^{\rm th}$ percentiles of each dark matter fraction distribution are given in Table 3. We see that the mean of the population lies at $0.42^{+0.08}_{+0.08}$, as shown by the shaded region. This distribution has an intrinsic width $0.20^{+0.09}_{-0.07}$. Our results are sensitive to the choice of a universal [66] IMF in the stellar populations analysis. If we instead assert a universal [19] IMF, we find that the mean dark matter fraction is $0.68^{+0.04}_{+0.06}$, while the scatter is essentially unchanged ($0.11^{+0.06}_{-0.04}$). As expected, a [19] IMF predicts a higher dark matter fraction, as shown in Table 3. <figure><img src="content_image/1008.3167/x30.png"><figcaption>Figure 16.— Dark matter fraction plotted against REin/Reff, the total masswithin Reff, axis ratio, total mass density profile slope, and Reff. The errorbars show the 16th and 84th percentiles. The solid line shows the linear bestfit to the data and the dashed lines indicate the scatter.</figcaption></figure> In Figure 16 we show how the SL2S lens dark matter fractions vary with: $R_{\rm Ein}/R_{\rm eff}$, the total mass within $R_{\rm eff}$, axis ratio, density profile slope, and $R_{\rm eff}$. In each panel, the solid line shows the linear best fit to the data and the dashed lines show the scatter. The gradients of the fits, including 16${}^{\rm th}$ and 84${}^{\rm th}$ percentiles are: \[df_{\rm DM}/d(R_{\rm Ein}/R_{\rm eff}) = -0.02^{+0.14}_{-0.12}\] \[df_{\rm DM}/dM_{R_{\rm eff}} = 0.034^{+0.048}_{-0.045}\] \[df_{\rm DM}/dq_{} = 0.23^{+0.29}_{-0.32}\] \[df_{\rm DM}/dR_{\rm eff} = 0.08^{+0.10}_{-0.08}\] \[df_{\rm DM}/d\gamma^{\prime} = 0.40^{+0.31}_{-0.12},\] with $M_{R_{\rm eff}}$ in units of $10^{11}M_{\odot}$, $R_{\rm eff}$ in kpc and $f_{\rm DM}$ is the dark matter within $R_{\rm eff}/2$. There is evidence for a correlation between $f_{\rm DM}$ and $\gamma^{\prime}$. For all other variables, the large uncertainties on the gradients indicate that there is only a negligible trend with $f_{\rm DM}$. ## 8. Cosmic evolution We now investigate evolution in the properties of massive galaxies using our new lens sample. We focus on the two key quantities studied in Section 7: the density profile slope $\gamma^{\prime}$, and the dark matter fraction $f_{\rm DM}$. The low redshift reference measurements of $\gamma^{\prime}$ and $f_{\rm DM}$ come from the SLACS analysis, which found density profiles very close to isothermal inside one effective radius in a sample of 63 SLACS strong-lens early-type galaxies: $\langle\gamma^{\prime}_{\rm SLACS}\rangle=2.078\pm 0.027$, with a scatter of $0.16$(Auger et al. 2010). Likewise Auger et al. (2009) inferred a mean $f_{\rm DM}$ within $R_{\rm eff}/2$ for 85 SLACS lenses of 0.3 with a scatter of 0.2, using a [66] IMF. These results are broadly consistent with our findings for SL2S systems. We can therefore anticipate only a mild cosmic evolution of these quantities. In Figure 17, we show how the density profile slope $\gamma^{\prime}$ varies with redshift, using the SL2S, SLACS and LSD samples together to cover the redshift range 0.05 to 1. We quantify this statement by fitting the $\gamma^{\prime}(z_{\rm d})$ data with a linear relation in the mean slope, still including Gaussian scatter about that relation: \[\langle\gamma^{\prime}\rangle(z_{\rm d})=\langle\gamma^{\prime}_{0}\rangle+ \frac{\partial\langle\gamma^{\prime}\rangle}{\partial z_{\rm d}}\,z_{\rm d}\pm S _{\gamma^{\prime}}.\] (15) For the SL2S data alone, we find $\langle\gamma^{\prime}_{0}\rangle$$\,=\,$$2.22^{+0.17}_{-0.21}$, $\partial\langle\gamma^{\prime}\rangle/\partial z_{\rm d}$$\,=\,$$-0.16^{+0.48}_{-0.51}$ for the gradient and, in this evolving $\gamma^{\prime}$ case, the scatter is $S_{\gamma^{\prime}}$$\,=\,$$0.23^{+0.09}_{-0.06}$. When we include the SLACS and LSD data points, we find $\langle\gamma^{\prime}_{0}\rangle$$\,=\,$$2.12^{+0.03}_{-0.04}$, $\partial\langle\gamma^{\prime}\rangle/\partial z_{\rm d}$$\,=\,$$-0.25^{+0.10}_{-0.12}$, and $S_{\gamma^{\prime}}$$\,=\,$$0.17^{+0.02}_{-0.02}$. These results are inconsistent with no evolution in the total density profile slope of massive lens galaxies since $z\simeq 1$: the probability of the linear gradient in $\langle\gamma^{\prime}\rangle$ being positive is just 2%. The lens data suggest (at approximately the 2-$\sigma$ level) that the mean total density profile of massive galaxies has become slightly steeper over cosmic time. <figure><img src="content_image/1008.3167/x31.png"><figcaption>Figure 17.— Cosmic evolution of total mass density slope, γ′. The SLACS andLSD values were taken from: (Auger et al. 2010) and (Treu & Koopmans 2002;Koopmans & Treu 2003; Treu & Koopmans 2004), respectively. The error bars showthe 16th and 84th percentiles. The best fit to the data is shown by the solidline and the scatter is shown by the dashed lines.</figcaption></figure> <figure><img src="content_image/1008.3167/x32.png"><figcaption>Figure 18.— The dark matter fraction within Reff/2 as a function of measuredvelocity dispersion. The error bars show the 16th and 84th percentiles. Forall samples, the velocity dispersion was normalized to a standard aperture,Reff/2.</figcaption></figure> <figure><img src="content_image/1008.3167/x33.png"><figcaption>Figure 19.— Evolution of dark matter fraction with redshift for a SalpeterIMF. The error bars show the 16th and 84th percentiles. The solid and dashedlines show the best fit to the data and the scatter, respectively.</figcaption></figure> Figure 18 shows the dark matter fraction as a function of the measured stellar velocity dispersion, $\sigma_{\rm e2}$ for the SL2S, SLACS and LSD samples. Together, these samples cover a large rage in stellar velocity dispersion, however, no significant trend with central dark matter fraction was found. In Figure 19, we show how the dark matter fraction varies with redshift, using the SL2S, SLACS and LSD samples together to cover the redshift range 0.05 to 1.0, and focusing on the [66] IMF. (The Chabrier IMF assignment only affects the overall normalization of the stellar masses and not their evolution.) Only 4 of the 5 LSD lenses were included, since photometry in multiple bands was required to infer stellar masses (given in Section 4.2). We find that the mean dark matter fraction has not evolved strongly with cosmic time, however, there is marginal evidence for some change in the population. Again, we quantify this statement by fitting the $f_{\rm DM}(z_{\rm d})$ data with a linear relation in the mean slope, still including Gaussian scatter about that relation: \[\langle f_{\rm DM}\rangle(z_{\rm d})=\langle f_{{\rm DM,}0}\rangle+\frac{ \partial\langle f_{\rm DM}\rangle}{\partial z_{\rm d}}\,z_{\rm d}\pm S_{f_{\rm DM }}.\] (16) For the SL2S data alone, we find $\langle f_{{\rm DM,}0}\rangle$$\,=\,$$0.38^{+0.12}_{-0.50}$, $\partial\langle f_{\rm DM}\rangle/\partial z_{\rm d}$$\,=\,$$0.07^{+0.36}_{-0.36}$ for the gradient and $S_{f_{\rm DM}}$$\,=\,$$0.18^{+0.08}_{-0.07}$ for the scatter. Including the SLACS and LSD data points, we find $\langle f_{{\rm DM,}0}\rangle$$\,=\,$$0.27^{+0.06}_{-0.06}$, $\partial\langle f_{\rm DM}\rangle/\partial z_{\rm d}$$\,=\,$$0.36^{+0.18}_{-0.24}$, and $S_{f_{\rm DM}}$$\,=\,$$0.13^{+0.02}_{-0.02}$. The probability of the linear gradient in $\langle f_{\rm DM}\rangle$ being positive is 98%: the lens data suggest (again at approximately the 2-$\sigma$ level) that the mean projected dark matter fraction in massive galaxies, within half their effective radius, has decreased slightly over cosmic time. ## 9. Discussion By combining the inferred total mass density slopes of the SL2S, SLACS and LSD samples, we found a tantalizing suggestion that $\gamma^{\prime}$ has become slightly steeper over cosmic time. <figure><img src="content_image/1008.3167/x34.png"><figcaption>Figure 20.— Evolution of REin/Reff with redshift. The ratio of REin/Refftends to increase with redshift.</figcaption></figure> Before interpreting this result, and its implications for our understanding of the formation and evolution of early-type galaxies, it is important to address a potential source of systematic error. As shown in Figure 20, the ratio of $R_{\rm Ein}/R_{\rm eff}$ increases with redshift, mainly because the physical size of $R_{\rm Ein}$ increases with redshift (as discussed in Section 6.2), but also because of the increasing source redshift. Since the slope of the total mass density is determined by measuring the total mass at two different radii, one of which is the Einstein radius, SL2S lenses sample $\gamma^{\prime}$ at larger radii than SLACS lenses. Therefore, a trend in $R_{\rm Ein}/R_{\rm eff}$ with redshift could mimic the inferred evolution of $\gamma^{\prime}$. However, for this effect to mimic the evolution of $\gamma^{\prime}$, the density profile would have to become shallower and then steeper with increasing radius, which is unlikely given the total mass density profile of local early-type galaxies. We also note that $\gamma^{\prime}$ and $f_{\rm DM}$ vary differently with cosmic time, which is unexpected if the evolution being mimicked by a trend in $R_{\rm Ein}/R_{\rm eff}$ with redshift. As discussed in Sections 7.4 and 7.5, Figures 14 and 16 show that both $\gamma^{\prime}$ and $f_{\rm DM}$ show a negligible trend with $R_{\rm Ein}/R_{\rm eff}$, suggesting that the evolution of $\gamma^{\prime}$ is not an artifact due to a dependence on $R_{\rm Ein}/R_{\rm eff}$. For these reasons we conclude that true evolution of the average mass density profile is the most likely interpretation of our findings, although additional measurements of $\gamma^{\prime}$ at different radii are needed to conclusively rule out alternate interpretations. It is important to stress that both the mean value and evolution of $\gamma^{\prime}$ are completely consistent with previous results, $\langle\gamma^{\prime}_{0,\rm SLACS+LSD}\rangle$$\,=$$\,$$2.10\pm 0.07$ and $\partial\langle\gamma^{\prime}\rangle/\partial z_{\rm d}$$\,=\,$$-0.23\pm 0.16$(Koopmans et al. 2006). It is only by virtue of our larger sample that we have been able to reduce the error bars and find marginal evidence for evolution. If this evidence for evolution is confirmed by larger samples, it would indicate that growth of massive galaxies since $z\sim 1$ has not occurred through dry (dissipationless) mergers alone, since they preserve $\gamma^{\prime}$(see _e.g._ Nipoti et al. 61). In dissipative merging events, $\gamma^{\prime}$ increases as a result of baryons cooling and sinking towards the center. Therefore, dissipative processes would be required to contribute, at least partially, to the evolution of early-type galaxies since $z\sim 1$. Although they cannot be the dominant process, because of tight limits on recent star formation in massive early-type galaxies since $z\sim 1$(e.g., Treu et al. 2005), our result seems to suggest that they cannot be completely neglected either. This is consistent with evidence for a “frosting” of recent stars found in detailed studies of the stellar populations of massive galaxies (e.g., Trager et al. 2000). Another notable feature of the SL2S $\gamma^{\prime}$ distribution is its large intrinsic scatter. The intrinsic scatter in the SLACS sample is just 0.16$\pm$0.02 (Auger et al. 2010), compared to $0.25^{+0.10}_{-0.07}$ for SL2S. From the LSD lenses, Treu & Koopmans (2004) also found that the intrinsic scatter in $\gamma^{\prime}$ was larger at $z=1$. This may indicate an overall trend toward more complete dynamical relaxation over Gyr timescales, and perhaps a reduced contribution of external convergence, due to line of sight structure in nearby systems, where the Einstein radii are smaller. Again, a larger sample size is required to confirm this interpretation. Finally, we find that the dark matter fraction within half of the effective radius has decreased slightly with cosmic time. This is again consistent with some contribution from dissipational processes in early-type galaxy formation and evolution, where baryons move to the central region as they cool. However, it is important to keep in mind residual uncertainties in estimating the stellar mass, which may be redshift dependent, since the average age of the stellar populations is a function of cosmic time. Cosmic evolution of the IMF would be another source of ambiguity in interpreting these results (van Dokkum 2008; Treu et al. 2010). To conclude, we have developed a new method to estimate source redshifts, in the absence of spectroscopic measurements. The method is completely general, does not require accurate multiband photometry of the sources, and allows us to infer $\gamma^{\prime}$ and $f_{\rm DM}$ with errors that are just 2-3 times as large as if we had spectroscopic redshifts. Although spectroscopic redshifts are in general preferable, this may be a good strategy for exploiting future surveys of thousands of strong lenses, where wholesale spectroscopy of the complete sample may not be practical. ## 10. Summary and Conclusions New spectroscopic measurements with deflector redshifts and velocity dispersions were presented for 11 lenses. These spectroscopic measurements were combined with lens models and photometry, described in Paper I, to infer the total density slope and dark matter fraction of each of the 11 SL2S galaxies in the final sample. The main results are summarized below: 1. The SL2S sample has a median deflector redshift, $z_{\rm d}$$\,=\,$$0.494$, source redshift, $z_{\rm s}$$\,=\,$$1.199$ and velocity dispersion, $\sigma_{\rm e2}$$\,=\,$$273$ km s${}^{-1}$. 2. The SL2S, SLACS and LSD lenses are the same types of galaxies, however, the physical size of $R_{\rm Ein}$ is generally larger for higher redshift deflectors. 3. We developed a new method to estimate the source redshift probability distribution function for lenses with no spectroscopic $z_{\rm s}$. This lack of accurate source redshift produces uncertainties on $\gamma^{\prime}$ and $f_{\rm DM}$ that are only a factor 2-3 greater. Uncertainties would eventually further decrease with better multiband source photometry, and this would allow large samples of high redshift lenses to be analyzed in the next generation of cosmological surveys, where spectroscopy of all the systems may not be affordable. 4. The average total density slope measured from the SL2S sample alone is: $\langle\gamma^{\prime}_{0}\rangle$$\,=\,$$2.16^{+0.09}_{-0.09}$, with a scatter of $0.25^{+0.10}_{-0.07}$. 5. Combining the SL2S $\gamma^{\prime}$ measurements with previous analyses from SLACS and LSD, we find, $\langle\gamma^{\prime}_{0}\rangle$$\,=\,$$2.12^{+0.03}_{-0.04}$, $\partial\langle\gamma^{\prime}\rangle/\partial z_{\rm d}$$\,=\,$$-0.25^{+0.10}_{-0.12}$, and $S_{\gamma^{\prime}}$$\,=\,$$0.17^{+0.02}_{-0.02}$. This suggests (at approximately the 2-$\sigma$ level) that the mean total density profile of massive galaxies has become slightly steeper over cosmic time. 6. Stellar masses were estimated using CFHT photometry, and enabled us to disentangle the total and stellar masses. 7. From this we inferred the dark matter fraction within half the effective radius: $\langle f_{{\rm DM,}0}\rangle$$\,=\,$$0.38^{+0.12}_{-0.50}$, with a scatter $S_{f_{\rm DM}}$$\,=\,$$0.18^{+0.08}_{-0.07}$ for the SL2S sample, using a [66] IMF. 8. The combined dark matter fractions from all three samples suggest that the dark matter fraction within $R_{\rm eff}/2$ has decreased slightly since $z\sim 1$. We find the mean dark matter fraction within $R_{\rm eff}/2$, gradient of evolution over cosmic time and scatter to be: $\langle f_{{\rm DM,}0}\rangle$$\,=\,$$0.27^{+0.06}_{-0.06}$, $\partial\langle f_{\rm DM}\rangle/\partial z_{\rm d}$$\,=\,$$0.36^{+0.18}_{-0.24}$, and $S_{f_{\rm DM}}$$\,=\,$$0.13^{+0.02}_{-0.02}$, respectively. We thank our friends of the SLACS and SL2S collaborations for many useful and insightful discussions over the course of the past years. AJR acknowledges the support of an Australian Postgraduate Award. RG and FB acknowledge support from the Centre National des Etudes Spatiales (CNES). PJM was given support by the TABASGO and Kavli foundations in the form of two research fellowships. TT acknowledges support from the NSF through CAREER award NSF-0642621, and from the Packard Foundation through a Packard Research Fellowship. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. This research is supported by NASA through Hubble Space Telescope programs GO-10876, GO-11289, GO-11588 and in part by the National Science Foundation under Grant No. PHY99-07949, and is based on observations made with the NASA/ESA Hubble Space Telescope and obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555, and at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. ## References Auger et al. (2009) Auger, M. W., Treu, T., Bolton, A. S., Gavazzi, R., Koopmans, L. V. E., Marshall, P. J., Bundy, K., & Moustakas, L. A. 2009, ApJ, 705, 1099 * Auger et al. (2010) Auger, M. W., Treu, T., Bolton, A. S., Gavazzi, R., Koopmans, L. V. E., Marshall, P. J., & Burles, S. 2010, ApJ * Barnabe et al. (2010) Barnabe, M., Auger, M. W., Treu, T., Koopmans, L., Bolton, A. S., Czoske, O., & Gavazzi, R. 2010, MNRAS, 955 * Barnabè et al. (2009) Barnabè, M., Czoske, O., Koopmans, L. V. E., Treu, T., Bolton, A. S., & Gavazzi, R. 2009, MNRAS, 399, 21 * Bernardi et al. (2005) Bernardi, M., Sheth, R. K., Nichol, R. C., Schneider, D. P., & Brinkmann, J. 2005, AJ, 129, 61 * Bertin & Stiavelli (1993) Bertin, G., & Stiavelli, M. 1993, Reports on Progress in Physics, 56, 493 * Bolton et al. (2008a) Bolton, A. S., Burles, S., Koopmans, L. V. E., Treu, T., Gavazzi, R., Moustakas, L. A., Wayth, R., & Schlegel, D. J. 2008a, ApJ, 682, 964 * Bolton et al. (2006) Bolton, A. S., Burles, S., Koopmans, L. V. E., Treu, T., & Moustakas, L. A. 2006, ApJ, 638, 703 * Bolton et al. (2008b) Bolton, A. S., Treu, T., Koopmans, L. V. E., Gavazzi, R., Moustakas, L. A., Burles, S., Schlegel, D. J., & Wayth, R. 2008b, ApJ, 684, 248 * Bruzual & Charlot (2003) Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 * Bundy et al. (2005) Bundy, K., Ellis, R. S., & Conselice, C. J. 2005, ApJ, 625, 621 * Bundy et al. (2007) Bundy, K., Treu, T., & Ellis, R. S. 2007, ApJ, 665, L5 * Cabanac et al. (2007) Cabanac, R. A., Alard, C., Dantel-Fort, M., Fort, B., Gavazzi, R., Gomez, P., Kneib, J. P., Le Fèvre, O., Mellier, Y., Pello, R., Soucail, G., Sygnet, J. F., & Valls-Gabaud, D. 2007, A&A, 461, 813 * Cappellari et al. (2006) Cappellari, M., Bacon, R., Bureau, M., Damen, M. C., Davies, R. L., de Zeeuw, P. T., Emsellem, E., Falcón-Barroso, J., Krajnović, D., Kuntschner, H., McDermid, R. M., Peletier, R. F., Sarzi, M., van den Bosch, R. C. E., & van de Ven, G. 2006, MNRAS, 366, 1126 * Cappellari et al. (2009) Cappellari, M., di Serego Alighieri, S., Cimatti, A., Daddi, E., Renzini, A., Kurk, J. D., Cassata, P., Dickinson, M., Franceschini, A., Mignoli, M., Pozzetti, L., Rodighiero, G., Rosati, P., & Zamorani, G. 2009, ApJ, 704, L34 * Cardone & Tortora (2010) Cardone, V. F., & Tortora, C. 2010, ArXiv e-prints * Cardone et al. (2009) Cardone, V. F., Tortora, C., Molinaro, R., & Salzano, V. 2009, A&A, 504, 769 * Cassata et al. (2010) Cassata, P., Giavalisco, M., Guo, Y., Ferguson, H., Koekemoer, A. M., Renzini, A., Fontana, A., Salimbeni, S., Dickinson, M., Casertano, S., Conselice, C. J., Grogin, N., Lotz, J. M., Papovich, C., Lucas, R. A., Straughn, A., Gardner, J. P., & Moustakas, L. 2010, ApJ, 714, L79 * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763 * Cimatti et al. (2006) Cimatti, A., Daddi, E., & Renzini, A. 2006, A&A, 453, L29 * Ciotti (2009) Ciotti, L. 2009, Nature, 460, 333 * Ciotti et al. (2007) Ciotti, L., Lanzoni, B., & Volonteri, M. 2007, ApJ, 658, 65 * Ciotti et al. (2009) Ciotti, L., Ostriker, J. P., & Proga, D. 2009, ApJ, 699, 89 * Croton et al. (2006) Croton, D. J., Springel, V., White, S. D. M., De Lucia, G., Frenk, C. S., Gao, L., Jenkins, A., Kauffmann, G., Navarro, J. F., & Yoshida, N. 2006, MNRAS, 365, 11 * Daddi et al. (2005) Daddi, E., Renzini, A., Pirzkal, N., Cimatti, A., Malhotra, S., Stiavelli, M., Xu, C., Pasquali, A., Rhoads, J. E., Brusa, M., di Serego Alighieri, S., Ferguson, H. C., Koekemoer, A. M., Moustakas, L. A., Panagia, N., & Windhorst, R. A. 2005, ApJ, 626, 680 * de Vaucouleurs (1948) de Vaucouleurs, G. 1948, Annales d’Astrophysique, 11, 247 * Dehnen (2005) Dehnen, W. 2005, MNRAS, 360, 892 * di Serego Alighieri et al. (2005) di Serego Alighieri, S., Vernet, J., Cimatti, A., Lanzoni, B., Cassata, P., Ciotti, L., Daddi, E., Mignoli, M., Pignatelli, E., Pozzetti, L., Renzini, A., Rettura, A., & Zamorani, G. 2005, A&A, 442, 125 * Faure et al. (2008) Faure, C., Kneib, J.-P., Covone, G., Tasca, L., Leauthaud, A., Capak, P., Jahnke, K., Smolcic, V., de la Torre, S., Ellis, R., Finoguenov, A., Koekemoer, A., Le Fevre, O., Massey, R., Mellier, Y., Refregier, A., Rhodes, J., Scoville, N., Schinnerer, E., Taylor, J., Van Waerbeke, L., & Walcher, J. 2008, ApJS, 176, 19 * Gavazzi et al. (2007) Gavazzi, R., Treu, T., Rhodes, J. D., Koopmans, L. V. E., Bolton, A. S., Burles, S., Massey, R. J., & Moustakas, L. A. 2007, ApJ, 667, 176 * Graves et al. (2009) Graves, G. J., Faber, S. M., & Schiavon, R. P. 2009, ApJ, 693, 486 * Grillo et al. (2008) Grillo, C., Gobat, R., Rosati, P., & Lombardi, M. 2008, A&A, 477, L25 * Hopkins et al. (2010) Hopkins, P. F., Bundy, K., Hernquist, L., Wuyts, S., & Cox, T. J. 2010, MNRAS, 401, 1099 * Hyde & Bernardi (2009) Hyde, J. B., & Bernardi, M. 2009, MNRAS, 396, 1171 * Jiang & Kochanek (2007) Jiang, G., & Kochanek, C. S. 2007, ApJ, 671, 1568 * Jørgensen et al. (1995) Jørgensen, I., Franx, M., & Kjaergaard, P. 1995, MNRAS, 276, 1341 * Juneau et al. (2005) Juneau, S., Glazebrook, K., Crampton, D., McCarthy, P. J., Savaglio, S., Abraham, R., Carlberg, R. G., Chen, H., Le Borgne, D., Marzke, R. O., Roth, K., Jørgensen, I., Hook, I., & Murowinski, R. 2005, ApJ, 619, L135 * Kazantzidis et al. (2006) Kazantzidis, S., Zentner, A. R., & Kravtsov, A. V. 2006, ApJ, 641, 647 * Khochfar & Silk (2006) Khochfar, S., & Silk, J. 2006, ApJ, 648, L21 * Kochanek (1994) Kochanek, C. S. 1994, ApJ, 436, 56 * Koopmans et al. (2009) Koopmans, L. V. E., Bolton, A., Treu, T., Czoske, O., Auger, M. W., Barnabè, M., Vegetti, S., Gavazzi, R., Moustakas, L. A., & Burles, S. 2009, ApJ, 703, L51 * Koopmans & Treu (2002) Koopmans, L. V. E., & Treu, T. 2002, ApJ, 568, L5 * Koopmans & Treu (2003) —. 2003, ApJ, 583, 606 * Koopmans et al. (2006) Koopmans, L. V. E., Treu, T., Bolton, A. S., Burles, S., & Moustakas, L. A. 2006, ApJ, 649, 599 * Kormann et al. (1994) Kormann, R., Schneider, P., & Bartelmann, M. 1994, A&A, 284, 285 * Lackner & Ostriker (2010) Lackner, C. N., & Ostriker, J. P. 2010, ApJ, 712, 88 * Lagattuta et al. (2010) Lagattuta, D. J., Fassnacht, C. D., Auger, M. W., Marshall, P. J., Bradač, M., Treu, T., Gavazzi, R., Schrabback, T., Faure, C., & Anguita, T. 2010, ApJ, 716, 1579 * Leauthaud et al. (2007) Leauthaud, A., et al. 2007, ApJS, 172, 219 * Lewis & Bridle (2002) Lewis, A., & Bridle, S. 2002, Phys. Rev. D, 66, 103511 * Mancini et al. (2010) Mancini, C., Daddi, E., Renzini, A., Salmi, F., McCracken, H. J., Cimatti, A., Onodera, M., Salvato, M., Koekemoer, A. M., Aussel, H., Floc’h, E. L., Willott, C., & Capak, P. 2010, MNRAS, 401, 933 * Marshall et al. (2007) Marshall, P. J., Treu, T., Melbourne, J., Gavazzi, R., Bundy, K., Ammons, S. M., Bolton, A. S., Burles, S., Larkin, J. E., Le Mignant, D., Koo, D. C., Koopmans, L. V. E., Max, C. E., Moustakas, L. A., Steinbring, E., & Wright, S. A. 2007, ApJ, 671, 1196 * Merritt (1999) Merritt, D. 1999, PASP, 111, 129 * Meza et al. (2003) Meza, A., Navarro, J. F., Steinmetz, M., & Eke, V. R. 2003, ApJ, 590, 619 * Miralda-Escude (1995) Miralda-Escude, J. 1995, ApJ, 438, 514 * Naab et al. (2009) Naab, T., Johansson, P. H., & Ostriker, J. P. 2009, ApJ, 699, L178 * Naab et al. (2007) Naab, T., Johansson, P. H., Ostriker, J. P., & Efstathiou, G. 2007, ApJ, 658, 710 * Natarajan & Kneib (1996) Natarajan, P., & Kneib, J.-P. 1996, MNRAS, 283, 1031 * Navarro et al. (2010) Navarro, J. F., Ludlow, A., Springel, V., Wang, J., Vogelsberger, M., White, S. D. M., Jenkins, A., Frenk, C. S., & Helmi, A. 2010, MNRAS, 402, 21 * Newman et al. (2010) Newman, A. B., Ellis, R. S., Treu, T., & Bundy, K. 2010, ApJ, 717, L103 * Nipoti et al. (2009a) Nipoti, C., Treu, T., Auger, M. W., & Bolton, A. S. 2009a, ApJ, 706, L86 * Nipoti et al. (2009b) Nipoti, C., Treu, T., & Bolton, A. S. 2009b, ApJ, 703, 1531 * Oñorbe et al. (2007) Oñorbe, J., Domínguez-Tenreiro, R., Sáiz, A., & Serna, A. 2007, MNRAS, 376, 39 * Ohyama et al. (2002) Ohyama, Y., Hamana, T., Kashikawa, N., Chiba, M., Futamase, T., Iye, M., Kawabata, K. S., Aoki, K., Sasaki, T., Kosugi, G., & Takata, T. 2002, AJ, 123, 2903 * Renzini (2006) Renzini, A. 2006, ARA&A, 44, 141 * Robertson et al. (2006) Robertson, B., Cox, T. J., Hernquist, L., Franx, M., Hopkins, P. F., Martini, P., & Springel, V. 2006, ApJ, 641, 21 * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161 * Saracco et al. (2009) Saracco, P., Longhetti, M., & Andreon, S. 2009, MNRAS, 392, 718 * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 * Sheth et al. (2003) Sheth, R. K., Bernardi, M., Schechter, P. L., Burles, S., Eisenstein, D. J., Finkbeiner, D. P., Frieman, J., Lupton, R. H., Schlegel, D. J., Subbarao, M., Shimasaku, K., Bahcall, N. A., Brinkmann, J., & Ivezić, Ž. 2003, ApJ, 594, 225 * Suyu et al. (2010) Suyu, S. H., Marshall, P. J., Auger, M. W., Hilbert, S., Blandford, R. D., Koopmans, L. V. E., Fassnacht, C. D., & Treu, T. 2010, ApJ, 711, 201 * Suyu et al. (2009) Suyu, S. H., Marshall, P. J., Blandford, R. D., Fassnacht, C. D., Koopmans, L. V. E., McKean, J. P., & Treu, T. 2009, ApJ, 691, 277 * Thomas et al. (2005) Thomas, D., Maraston, C., Bender, R., & Mendes de Oliveira, C. 2005, ApJ, 621, 673 * Trager et al. (2000) Trager, S. C., Faber, S. M., Worthey, G., & González, J. J. 2000, AJ, 120, 165 * Treu et al. (2010) Treu, T., Auger, M. W., Koopmans, L. V. E., Gavazzi, R., Marshall, P. J., & Bolton, A. S. 2010, ApJ, 709, 1195 * Treu et al. (2005) Treu, T., Ellis, R. S., Liao, T. X., van Dokkum, P. G., Tozzi, P., Coil, A., Newman, J., Cooper, M. C., & Davis, M. 2005, ApJ, 633, 174 * Treu et al. (2009) Treu, T., Gavazzi, R., Gorecki, A., Marshall, P. J., Koopmans, L. V. E., Bolton, A. S., Moustakas, L. A., & Burles, S. 2009, ApJ, 690, 670 * Treu et al. (2006) Treu, T., Koopmans, L. V., Bolton, A. S., Burles, S., & Moustakas, L. A. 2006, ApJ, 640, 662 * Treu & Koopmans (2002) Treu, T., & Koopmans, L. V. E. 2002, ApJ, 575, 87 * Treu & Koopmans (2004) —. 2004, ApJ, 611, 739 * Treu et al. (1998) Treu, T., Stiavelli, M., Walker, A. R., Williams, R. E., Baum, S. A., Bernstein, G., Blacker, B. S., Carollo, C. M., Casertano, S., Dickinson, M. E., De Mello, D. F., Ferguson, H. C., Fruchter, A. S., Lucas, R. A., MacKenty, J., Madau, P., & Postman, M. 1998, A&A, 340, L10 * van Albada & Sancisi (1986) van Albada, T. S., & Sancisi, R. 1986, Royal Society of London Philosophical Transactions Series A, 320, 447 * van der Marel (1994) van der Marel, R. P. 1994, MNRAS, 270, 271 * van der Wel et al. (2009) van der Wel, A., Bell, E. F., van den Bosch, F. C., Gallazzi, A., & Rix, H.-W. 2009, ApJ, 698, 1232 * van der Wel et al. (2005) van der Wel, A., Franx, M., van Dokkum, P. G., Rix, H.-W., Illingworth, G. D., & Rosati, P. 2005, ApJ, 631, 145 * van Dokkum (2008) van Dokkum, P. G. 2008, ApJ, 674, 29 * van Dokkum et al. (2008) van Dokkum, P. G., Franx, M., Kriek, M., Holden, B., Illingworth, G. D., Magee, D., Bouwens, R., Marchesini, D., Quadri, R., Rudnick, G., Taylor, E. N., & Toft, S. 2008, ApJ, 677, L5 * Wucknitz et al. (2004) Wucknitz, O., Biggs, A. D., & Browne, I. W. A. 2004, MNRAS, 349, 14 Name \| zd \| zs \| zs,pdf \| σ \| S/N \| REin \| qmass \| Reff \| q∗ \| mu \| mg \| mr \| mi \| mz \| Flag \| Run ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| \| (km s−1) \| (Å−1) \| (arcsec) \| \| (arcsec) \| \| \| \| \| \| \| \| J021411−040502 \| 0.6080 \| - \| 1.57+0.50−0.35 \| − \| 6.1 \| 0.918 \| 0.33 \| 0.94 \| 0.89 \| 22.46 \| 21.08 \| 19.57 \| 18.78 \| 19.41 \| HST \| 2b J021737−051329 \| 0.6458 \| 1.847 \| 1.74+0.53−0.37 \| 257±26 \| 14.5 \| 1.268 \| 0.91 \| 0.77 \| 0.89 \| 21.57 \| 20.45 \| 19.51 \| 19.47 \| 19.16 \| HST \| 2b,4c J021902−082934 \| 0.3898 \| - \| 1.30+0.65−0.43 \| 305±25 \| 19.7 \| 0.918 \| 0.53 \| 0.90 \| 0.72 \| 22.50 \| 20.77 \| 20.14 \| 19.33 \| 18.70 \| CFHT \| 3a J022056−063934 \| 0.3297 \| - \| 1.47+0.68−0.49 \| 242±28 \| 25.7 \| 1.250 \| 0.73 \| 1.47 \| 0.57 \| 21.09 \| 20.12 \| 18.30 \| 18.35 \| 17.62 \| CFHT \| 3b J022511−045433 \| 0.2380 \| 1.1988 \| 1.25+0.55−0.39 \| 241±12 \| 53.0 \| 1.770 \| 0.58 \| 1.90 \| 0.75 \| 19.37 \| 17.91 \| 17.39 \| 16.84 \| 16.74 \| HST \| 5 J022610−042011 \| 0.4943 \| 1.232 \| 1.54+0.60−0.40 \| 266±21 \| 16.4 \| 1.153 \| 0.92 \| 0.56 \| 0.94 \| 22.04 \| 20.47 \| 19.66 \| 18.78 \| 18.58 \| HST \| 4a J022648−040610 \| 0.7663 \| - \| 2.18+0.67−0.47 \| − \| 9.4 \| 1.306 \| 0.82 \| 1.20 \| 0.37 \| 23.34 \| 22.15 \| 21.40 \| 20.00 \| 19.60 \| HST \| 2a J022648−090421 \| 0.4563 \| - \| 1.98+0.78−0.57 \| 301±18 \| 30.8 \| 1.582 \| 0.83 \| 1.30 \| 0.80 \| 22.47 \| 20.06 \| 18.14 \| 18.32 \| 17.73 \| CFHT \| 4a J023251−040823 \| 0.3516 \| - \| 1.52+0.74−0.53 \| 264±17 \| 22.6 \| 1.102 \| 0.80 \| 0.81 \| 0.67 \| 21.87 \| 20.13 \| 18.41 \| 18.58 \| 18.52 \| HST \| 3a J140123+555705 \| 0.5263 \| - \| 1.62+0.58−0.42 \| − \| 10.0 \| 1.186 \| 0.49 \| 0.76 \| 0.73 \| 22.77 \| 21.35 \| 20.15 \| 19.14 \| 18.58 \| HST \| 1 J140614+520252 \| 0.4797 \| - \| − \| − \| 8.5 \| - \| - \| 2.15 \| 0.49 \| 22.05 \| 20.06 \| 18.47 \| 18.29 \| 17.93 \| CFHT \| 1 J141137+565119 \| 0.3218 \| 1.420 \| 1.37+0.77−0.52 \| 228±20 \| 34.0 \| 0.924 \| 0.90 \| 0.76 \| 0.74 \| 21.08 \| 19.75 \| 18.59 \| 18.49 \| 18.32 \| HST \| 6 J220629+005728 \| 0.7044 \| - \| − \| − \| 10.4 \| - \| - \| 2.25 \| 0.99 \| 23.11 \| 21.47 \| 20.64 \| 20.59 \| 19.04 \| CFHT \| 3a J221326−000946 \| 0.3378 \| - \| 1.30+0.63−0.44 \| 183±34 \| 18.9 \| 1.076 \| 0.19 \| 0.41 \| 0.30 \| 23.93 \| 22.31 \| 20.10 \| 20.00 \| 19.49 \| HST \| 5 J221407−180712 \| 0.6505 \| - \| 1.12+0.46−0.22 \| 167±43 \| 8.3 \| 0.411 \| 0.69 \| 0.57 \| 0.65 \| 23.91 \| 21.96 \| 21.04 \| 20.43 \| 19.79 \| HST \| 3b J221606−175131 \| 0.8602 \| - \| − \| 282±44 \| 13.5 \| - \| - \| 0.93 \| 0.87 \| 23.35 \| 22.39 \| 21.36 \| 20.68 \| 19.72 \| CFHT \| 3a J221929−001743 \| 0.2888 \| 1.0232 \| 0.91+0.54−0.32 \| 263±26 \| 44.4 \| 0.736 \| 0.75 \| 1.00 \| 0.75 \| 20.39 \| 18.38 \| 16.78 \| 17.78 \| 17.61 \| CFHT \| 4a,4b Note. – All spectroscopically-observed lens systems are shown. Reff values are measured at the intermediate axis. The signal to noise ratio per pixel was calculated over the rest wavelength range 4000–5000Å. Typical uncertainties on REin and Reff are 5% and 10%, respectively. The column, Flag, indicates whether HST or CFHT data was used to measure REin and Reff. Table 2 Measured SL2S galaxy-scale lens properties Name \| REin \| σe2 \| σSIE \| M∗ \| MEin \| γ′ \| fSalpDM,Reff2 \| fChabDM,Reff2 ---\|---\|---\|---\|---\|---\|---\|---\|--- \| (kpc) \| (km s−1) \| (km s−1) \| (1011M⊙) \| (1011M⊙) \| \| \| J021737−051329 \| 8.76 \| 273 ± 27 \| 289 \| 1.77 +0.53−0.37 \| 5.35 \| 1.99 +0.11−0.12 \| 0.65 +0.11−0.08 \| 0.80 +0.11−0.04 J021902−082934 \| 4.85 \| 324 ± 26 \| 228+37−13 \| 1.25 +0.29−0.19 \| 1.85 +0.65−0.21 \| 2.55 +0.13−0.17 \| 0.71 +0.07−0.06 \| 0.83 +0.13−0.03 J022056−063934 \| 5.94 \| 252 ± 29 \| 256+37−12 \| 3.48 +0.89−0.68 \| 2.81 +0.72−0.25 \| 2.04 +0.19−0.19 \| 0.34 +0.18−0.14 \| 0.63 +0.19−0.08 J022511−045433 \| 6.67 \| 251 ± 12 \| 287 \| 2.82 +0.59−0.50 \| 4.02 \| 1.88 +0.07−0.07 \| 0.55 +0.10−0.08 \| 0.74 +0.07−0.04 J022610−042011 \| 6.99 \| 288 ± 22 \| 279 \| 2.74 +0.55−0.49 \| 3.99 \| 2.09 +0.08−0.09 \| 0.21 +0.19−0.15 \| 0.56 +0.08−0.08 J022648−090421 \| 9.18 \| 315 ± 18 \| 315+72−23 \| 6.38 +1.17−1.27 \| 6.52 +2.41−0.86 \| 2.16 +0.08−0.09 \| 0.24 +0.16−0.15 \| 0.57 +0.08−0.09 J023251−040823 \| 5.46 \| 282 ± 18 \| 239+31−10 \| 1.64 +0.31−0.27 \| 2.29 +0.74−0.21 \| 2.27 +0.10−0.12 \| 0.53 +0.10−0.08 \| 0.73 +0.10−0.05 J141137+565119 \| 4.32 \| 246 ± 21 \| 214 \| 1.33 +0.29−0.24 \| 1.45 \| 2.30 +0.12−0.14 \| 0.45 +0.13−0.10 \| 0.69 +0.12−0.06 J221326−000946 \| 5.19 \| 203 ± 37 \| 241+38−13 \| 0.92 +0.20−0.18 \| 2.17 +0.59−0.21 \| 1.84 +0.19−0.20 \| 0.12 +0.33−0.23 \| 0.52 +0.19−0.14 J221407−180712 \| 2.85 \| 181 ± 46 \| 171+29−9 \| 1.68 +0.32−0.23 \| 0.57 +0.16−0.06 \| 1.64 +0.53−0.29 \| 0.03 +0.35−0.26 \| 0.39 +0.53−0.18 J221929−001743 \| 3.19 \| 278 ± 27 \| 197 \| 2.44 +0.50−0.40 \| 0.91 \| 2.66 +0.13−0.16 \| 0.02 +0.21−0.16 \| 0.40 +0.13−0.11 Note. – Only the 11 modeled lenses with measured velocity dispersions are shown. σe2 is the measured stellar velocity dispersion corrected to a standard aperture. σSIE values calculated from measured source redshifts are given without uncertainty and σSIE values calculated from zs,pdf are given with the 16th and 84th percentiles (and were also weighted by the Sheth et al. fitting function and selection function). M∗ is the total stellar mass. MEin is the total mass enclosed at the Einstein radius. Dark matter fractions are given at Reff/2 for both Salpeter and Chabrier IMFs. Table 3 Inferred SL2S galaxy-scale lens properties
1706.04791	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 47677, "num_imgs": 10, "llama3_tokens_count": 12082 }	[ "content_image/1706.04791/x1.png", "content_image/1706.04791/x3.png", "content_image/1706.04791/x4.png", "content_image/1706.04791/x5.png", "content_image/1706.04791/x6.png", "content_image/1706.04791/x7.png", "content_image/1706.04791/x8.png", "content_image/1706.04791/x9.png", "content_image/1706.04791/x10.png", "content_image/1706.04791/x11.png" ]	# Obstacle-shape effect in a two-dimensional granular silo flow field K. Endo${}^{1}$, K. Anki Reddy${}^{2}$, and H. Katsuragi${}^{1}$ ${}^{1}$Department of Earth and Environmental Sciences, Nagoya University, Nagoya 464-8601, Japan ${}^{2}$Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India February 21, 2024 ###### Abstract We conducted simple experiment and numerical simulation of two-dimensional granular discharge flow driven by gravity under the influence of an obstacle. According to the previous work (Zuriguel _et al., Phys. Rev. Lett._ 107: 278001, 2011), the clogging of granular discharge flow can be suppressed by putting a circular obstacle at a proper position. In order to investigate the details of obstacle effect in granular flow, we focused on particle dynamics in this study. From the experimental and numerical data, we found that the obstacle remarkably affects the horizontal-velocity distribution and packing fraction at the vicinity of the exit. In addition to the circular obstacle, we utilized triangular, inverted-triangular, and horizontal-bar obstacles to discuss the obstacle-shape effect in granular discharge flow. Based on the investigation of dynamical quantities such as velocity distributions, granular temperature, and volume fraction, we found that the triangular obstacle or horizontal bar could be very effective to prevent the clogging. From the obtained result, we consider that the detouring of particles around the obstacle and resultant low packing fraction at the exit region effectively prevent the clogging in a certain class of granular discharge flow. pacs: 45.70.-n,47.57.Gc ## I Introduction Gravity-driven granular silo flow shows some intriguing phenomena. Its behavior is quite different from usual fluid behavior. For instance, discharge flow rate of gravity-driven granular silo flow is independent of its height (thickness) in the silo. Such a constant flow cannot be achieved when usual fluid is used. In usual fluid, the discharge flow rate varies depending on the layer’s height (Torricelli’s theorem Faber (1995)). The pressure structure within a granular silo tends to saturate in deep part due to Janssen effect Janssen (1895). This saturated (constant) pressure could be a reason for the constant flow rate. However, according to the recent study Aguirre _et al._ (2010), the constant granular flow might not relate to Janssen effect. Namely, the origin of steadiness in granular silo flow has not yet been fully understood. Beverloo _et al._ experimentally obtained the scaling for the flow rate in granular silo flow which depends on the sizes of particles and exit Beverloo _et al._ (1961). In particular, when the exit width $W$ decreases to a certain limit (e.g. $W\simeq 6D$, where $D$ is the particle diameter), an arch structure of the granular particles could easily be formed around the exit Helbing _et al._ (2006). Due to this arch formation, particles suddenly clog to arrest the flow To _et al._ (2001); To (2005); Zuriguel _et al._ (2005); Janda _et al._ (2008). The clogging phenomenon is often a severe problem in transporting granular materials in industry. An obstacle placed within the granular silo flow could affect the flow property such as clogging condition. In fact, the clogging probability is decreased by inserting an obstacle into the granular silo at a proper position Zuriguel _et al._ (2011); Lozano _et al._ (2012); Zuriguel _et al._ (2014). The clogging phenomena have also been observed in crowd and animal flows, and obstacles have been used to control these flows as well D. Helbing and Werner (2005); Frank and Dorso (2011); Garcimartín _et al._ (2015). By the experimental simulation of evacuating exit flow, it has been revealed that the obstacle set in front of the exit can decrease the outflow period Kawaguchi and Shimizu (2012); Zuriguel _et al._ (2014). Although the vibration or concentrated air flow can also be utilized to avoid the granular clogging Mankoc _et al._ (2009); Zuriguel _et al._ (2005), obstacle usage is much easier than vibration and air flow in various situations. Arch formation by particles at the exit region is the fundamental process to proceed to the clogging. The smaller the exit size, the larger the clogging probability becomes. Recently, non-zero clogging probability even for very large exit has been proposed on the basis of experimental result Thomas and Durian (2015). The self-similar density and velocity profiles around the exit have also been observed Janda _et al._ (2012). In addition, velocity field in granular silo flow without an obstacle has been studied in various works Choi _et al._ (2004); Moka and Nott (2005); Orpe and Kudrolli (2007); Thomas and Durian (2016). However, the characteristics of granular-flow field under the influence of obstacle has not yet been understood well. Moreover, details of flow-field conditions must be influenced by the shape of obstacle and/or particles as well. Although the particle-shape dependence of granular silo flow has recently been studied Ashour _et al._ (2017), the effect of shape of obstacle has not yet been studied well so far. Therefore, we conducted simple experiment and numerical simulation of two-dimensional granular silo flow driven by gravity under the influence of an obstacle. In the experiment, we traced individual particles from the flow images acquired by a high-speed camera. Numerical simulation of granular silo flow has also been performed to confirm the major effect of obstacle to prevent the clogging. Particularly, we mainly compare the various obstacle-shape results in order to discuss the physical origin of clogging reduction by obstacle. ## II Methods ### Experiment An experimental system we built and pictures of obstacles used are shown in Fig. 1. In this experiment, we prepare a two-dimensional cell consisting of two acrylic transparent plates and aluminum rectangles for the side and bottom walls. The inner dimension of the cell is 6.50 $\times$ 210 $\times$ 300 mm (thickness $\times$ width $\times$ height). The obstacles are made of stainless steel and have 6.0 mm thickness. We use three types of obstacles : a circle of 50 mm diameter, an equilateral triangle of one side 50 mm, and an inverted equilateral triangle of the identical size. After inserting an obstacle and filling the cell with stainless-steel particles having 6.35 $\pm$ 0.05 mm diameter, a discharge flow is triggered by opening a small exit at the center of the cell’s bottom. We do not refill the container by particles. Namely, the measurement lasts until all the particles are discharged or the clogging occurrence. Discharged particles are caught by a container placed below the cell. Load cell sensors (LMB-A, KYOWA) are used to measure the mass of discharged particles. The obstacle is fixed to a stainless-steel pole of 6.0 mm diameter which is connected to an universal testing machine (AG-X, SHIMADZU) to control the position of the obstacle. Drag force exerting on the obstacle can also be measured using the testing machine. However, here we focus on the clogging problem in this paper. The drag force characterization will be presented elsewhere Katsuragi _et al._ (2017). Flow rate data are taken at 100 Samples/s sampling rate. We also measure the microscopic granular flow field by using a high-speed camera (FASTCAM SA5, Photron). The acquired images are analyzed by means of particle tracking velocimetry (PTV) method implemented by LabVIEW software (vision toolkit). The frame rate is fixed at 500 fps. The spatial resolution of the images is 0.26 mm/pixel, and the images consist of $832\times 880$ pixels corresponding to the field of view $220\times 233$ mm${}^{2}$. The main parameters in this experiment are the width of the exit $W$ and the vertical distance between the exit and obstacle $L$. Specifically, $W$ is varied as 25, 30, 40 and 60 mm. $L$ is changed by 5 mm from $L=0$ mm to $L=50$ mm and by 10 mm from $L=50$ mm to $L=100$ mm. The case without any obstacle is denoted by $L=\infty$. <figure><img src="content_image/1706.04791/x1.png"><figcaption>Figure 1: Experimental apparatus and obstacles. A schematic diagram of theexperimental setup is shown above. Width of the exit W and distance betweenthe exit and obstacle L are principal parameters in this system. Below imagesshow obstacles used in this experiment: a circle of 50 mm in diameter, anequilateral triangle of one side 50 mm, and an inverted equilateral triangleof one side 50 mm. Holes in the obstacles were opened for making attachingmechanism.</figcaption></figure> ### Simulation methodology We used the contact force models based on Brilliantov _et al._ (1996) and explained further in Silbert _et al._ (2001) to simulate the gravity driven flow of granular media. Suppose if particle _i_ is in contact with particle _j_, the contact force on particle _i_ resolved into two components (along the line joining centers and perpendicular to the line joining centers) can be expressed as $\textbf{F}_{ij}=\textbf{F}_{ij}^{n}+\textbf{F}_{ij}^{t}$. For the present model, the normal and tangential force components are given by $\textbf{F}_{ij}^{n}=\sqrt{\delta_{ij}}\sqrt{\frac{R_{i}R_{j}}{R_{i}+R_{j}}}(k_ {n}\delta_{ij}\textbf{n}_{ij}-m_{\textup{eff}}\gamma_{n}\textbf{v}_{ij}^{n})$ and $\textbf{F}_{ij}^{t}=\sqrt{\delta_{ij}}\sqrt{\frac{R_{i}R_{j}}{R_{i}+R_{j}}}(k_ {t}\textbf{u}_{ij}^{t}-m_{\textup{eff}}\gamma_{t}\textbf{v}_{ij}^{t})$, respectively. Here $R_{i}$ and $R_{j}$ are the radii of particles $i$ and $j$. If particle _i_ and particle _j_ respectively have positions $\textbf{r}_{i}$ and $\textbf{r}_{j}$, masses $m_{i}$ and $m_{j}$, linear velocities $\textbf{v}_{i}$ and $\textbf{v}_{j}$, angular velocities $\textbf{w}_{i}$ and $\textbf{w}_{j}$, quantities used in the above force equation are defined as $\delta_{ij}=d-\|\textbf{r}_{i}-\textbf{r}_{j}\|$, $\textbf{v}_{ij}^{n}=(\textbf{v}_{ij}\cdot\textbf{n}_{ij})\textbf{n}_{ij}$, $\textbf{v}_{ij}=\textbf{v}_{i}-\textbf{v}_{j}$, $\textbf{v}_{ij}^{t}=\textbf{v}_{ij}-\textbf{v}_{ij}^{n}-\frac{1}{2}\textbf{r}_ {ij}\times(\textbf{w}_{i}+\textbf{w}_{j})$, $\textbf{n}_{ij}=\textbf{r}_{ij}/\|\textbf{r}_{ij}\|$ and $m_{\textup{eff}}=m_{i}m_{j}/(m_{i}+m_{j})$. Once the force acting on particle _i_ is known, classical equations of motion can be integrated with the suitable numerical technique to update its position and velocities. In the simulation, elastic constant for normal contact ($k_{n}$) is $2\times 10^{8}mg/d^{2}$, whereas the the elastic constant in tangential direction ($k_{t}$) is $\frac{2}{7}k_{n}$ Silbert _et al._ (2001). Here, $m$ is the mass and $d$ is the diameter of the particle and $g$ is the acceleration due to gravity. The viscoelastic damping constant for normal contact ($\gamma_{n}$) is $1850\sqrt{g}/d^{1.5}$ and for tangential contact $\gamma_{t}$ is half of the $\gamma_{n}$. The chosen constants represent the properties of material used in the experimental study. It is important to mention that we included memory effect in the modeling of tangential forces. Tangential displacement between the two contact particles $(\textbf{u}_{ij}^{t})$ will be computed from the initiation of the contact (it will be zero at the initiation of contact) and it is adjusted so that the local yield criterion $\|\textbf{F}_{ij}^{t}\|<\|\mu\textbf{F}_{ij}^{n}\|$ is satisfied. Rate of change of this elastic tangential displacement $(\textbf{u}_{ij}^{t})$ is given by ${d\textbf{u}_{ij}^{t}}/{dt}=\textbf{v}_{ij}^{t}-(\textbf{u}_{ij}^{t}\cdot \textbf{v}_{ij})\textbf{r}_{ij}/\|\textbf{r}_{ij}\|^{2}$. In the present simulations, the friction coefficient $\mu$ has been set equal to 0.36 for particle-particle interactions and 0.5 for particle-wall interactions. The timestep in the present simulation is 0.000002. We arrived at this value of time step by computing the collision time with the help of equations given in Schwager and Pöschel (1998); Brilliantov _et al._ (1996) and this collision time was found to be $O(10^{-4})$. All the lengths are scaled by particle diameter ($d$), which is $1.0$ in the present case. Time, velocities and forces are measured in units of $\sqrt{d/g}$, $\sqrt{gd}$, and $mg$, respectively. Elastic constants and viscoelastic damping constants are measured in units of $(mg/d^{2})$ and $\frac{1}{(\mbox{time}\mbox{distance})}$. Density of the particles is also set as $1.0$. Obstacle shape in the present simulation is made using spherical particles of diameter 1.0. For example, to create a circular obstacle of diameter $8d$, we need approximately 22 particles to make the outer layer of circle. Similarly, to make a triangular obstacle which has length of side as $8d$, we need 36 particles. Initial configuration is prepared by pouring the particles into a container under the gravity. There are 1900 particles in the simulation which is in correspondence with the number of particles used in experimental study discussed in this paper. This dimension of a simulation box is $34d\times 48d\times 1d$. The simulation system is as shown in Fig. 2. Flat frictional walls are present in the $x$-direction and periodic boundary conditions are applied in the $z$-direction. Bottom wall in the $y$-direction is made up of spherical particles of diameter 1.0 $d$ and to simulate the discharge of particles through orifice, certain number of spheres will be removed from the bottom $y$-wall (For $W=25$ mm, we need to remove 4 spheres from the bottom wall). All the simulations are carried out using the Large Atomic Molecular Massively Parallel Simulator (LAMMPS) Plimpton (1995). In the experiment, the mean grain diameter is 6.35 mm. To make a comparison between the numerical and experimental data, $L$ and $W$ values in the simulation are transformed to the real length scale by multiplying the grain diameter with 6.35 mm in the following data plots. Furthermore, the elastic constant for normal contact used in the simulation results in Young’s modulus $(Y)$ of $108.5$ GPa for a stainless steel spherical particle of diameter 6.35 mm if we use the equation $k_{n}=\frac{2Y}{3(1-\nu^{2})}$. Here density and Poisson’s ratio ($\nu$) for stainless steel are taken as $8000$ kg/m${}^{3}$ and $0.275$. These values are more or less reasonable for the steel. <figure><img src="content_image/1706.04791/x3.png"><figcaption>Figure 2: (a) Simulation system with a horizontal-bar obstacle consisting ofgreen-color spheres. Typical clogging configurations observed in simulationsfor (b) the flow past horizontal-bar obstacle, L=45 and W=25 mm and for (c)the flow past inverted-triangle obstacle, L=60 and W=25 mm.</figcaption></figure> ## III Experimental results and analyses ### Clogging diagram First, the clogging-occurrence conditions depending on $W$ and $L$ are examined. In Fig. 3, obtained clogging diagrams for (a) circular, (b) triangular, and (c) inverted-triangular cases are shown. Three or more experimental runs are performed for each experimental condition. If the flow clogs during discharge at least once, that condition is regarded as a clogging case. On the other hand, if all the particles are discharged without clogging for all experimental runs, that condition is assigned to the no-clogging case. In Fig. 3, no clogging is expressed by circular marks while triangle, square, and cross symbols indicate the clogging conditions. In this study, the number of experimental runs might not be sufficient to precisely discuss the clogging probability. Moreover, we did not compute the avalanche size distribution. Thus, the clogging diagrams shown in Fig. 3 are more or less qualitative ones. Therefore, the quantitative analysis for the clogging probability would not be discussed in this study. However, some characteristic features can be confirmed in these diagrams as discussed below. <figure><img src="content_image/1706.04791/x4.png"><figcaption>Figure 3: Clogging diagrams for (a) circular, (b) triangular, and (c)inverted-triangular obstacle cases. Experiments were conducted three or moretimes (in W≤ 40 mm) to make this diagram. L=∞ means the case without anobstacle. Circular symbols indicate that the clogging did not occur.Triangular symbols indicate that the clogging by arching at the exit (Fig.4(a)) occurred. Square symbols indicate that the clogging occurred between theobstacle and the bottom wall (Fig. 4(b)), and × indicates that granular flowdid not occur at all. Green broken line corresponds to W=6D.</figcaption></figure> As shown in Figs. 2(c) and 4(a), the clogging is basically induced by an arch structure formed at the exit region. If an obstacle is too close to the exit (i.e., $L$ is too small), arch formation occurs between the obstacle and bottom wall (Fig. 4(b)). This type of arch formation is different from the ordinary one which is formed around the exit (Fig.4(a)). In the clogging diagram (Fig. 3), the arch formation around the exit is denoted by triangle symbols, and the arch formation between the obstacle and bottom wall is presented by square symbols. If the flow is not induced from the very initial state, cross symbols are assigned. In the numerical simulation, a slightly different clogging mode (Fig. 2(b)) is also observed. In this clogging mode, the arch is formed around the obstacle edge and the stable triangular lattice slope. Although this could be regarded as a type of obstacle-wall clogging, this mode cannot be observed in the experiment since it is difficult to make the stable triangular lattice slope in the actual experiment. When $W$ is large enough, the flow is relatively smooth and difficult to clog. By contrast, when $W=25$ mm, clogging frequently occurs. In Fig. 3(a) (circle case), $L$ dependence of the clogging occurence is not clear. In Fig. 3(c) (inverted-triangle case), although some no-clogging states can be found in $W=25$ mm, any clear trend of $L$-dependent clogging occurence cannot be confirmed. Contrastively, in Fig. 3(b) (triangle case), clear reduction of clogging occurence can be confirmed in the relatively small $L$ regime ($30$ mm $\leq L\leq 40$ mm). This anti-clogging tendency due to the obstacle is qualitatively consistent with the previous work using a circular obstacle Zuriguel _et al._ (2011). In this study, the anti-clogging by putting an obstacle can clearly be confirmed particularly in the triangle case. <figure><img src="content_image/1706.04791/x5.png"><figcaption>Figure 4: (a) Clogging by arch formation around the exit. (b) Clogging by archformation between the obstacle and bottom wall.</figcaption></figure> From the above-mentioned observations, we consider that the flow situation could significantly depend on the shape of the obstacle. At $W=30$ mm, clogging due to the arch formation at the exit region occurs only at $L=\infty$ (no-obstacle case). The case of $W=25$ mm is more interesting because the clogging occurence variation depending on $L$ can be observed. The condition to prevent the clogging might be revealed by comparing these cases (no-obstacle, circle, triangle, and inverted triangle) at $W=25$ mm. Therefore, we focus on the case of $W=25$ mm in the following. ### Granular flow field In this study, we would like to relate macroscopic behaviors of granular discharge flow to the statistics of motion of individual particles. Thus, we track particle motions using PTV method. The examples of velocity fields computed from the series of images of flowing particles acquired by a high-speed camera are shown in Fig. 5. Figures 5(a,b), (c,d), (e,f), and (g,h) correspond to the cases with no-obstacle, a circle, a triangle, and an inverted triangle, respectively. Experimental conditions in these data are $W=25$ mm and $L=30$ mm, at which the flow is in the marginal state between smooth flow and clogging. Each case has a snapshot of the raw particle image on the left side (Figs. 5(a), (c), (e), and (g)) and a corresponding velocity field in vector representation on the right side (Figs. 5(b), (d), (f), and (h)). We first discuss on the no-obstacle case. In Fig. 5(b), we can confirm that the vertical component dominates the particle velocities. The relatively large-speed particles distribute at the central zone of the cell (right above the exit). The meandering of flow in this state might relate to the local crystallization due to the monodispersity of particles. Although this velocity field results in the large flow rate, dense flow at the exit region might cause the clogging due to the arch formation. Indeed, the largest flow rate in no-obstacle case has been confirmed (see the legend in Fig. 7(a)). It should be noted that all the experiments show steady flow Katsuragi _et al._ (2017); Endo and Katsuragi (2017) even right before the clogging. Although some numerical simulations have reported the increase of the flow rate by the effect of obstacle Alonso-Marroquin _et al._ (2012); Murray and Alonso-Marroquin (2016), these studies have simulated the inclined-bottom-wall silo flow. Furthermore, Murray and Alonso-Marroquin (2016) has used the fluctuation of grains and vibration of walls to prevent the clogging. This could significantly affect the flow rate as well. Experiments with horizontal-bottom-wall silo has also shown similar increase of flow rate by the effect of obstacle Lozano _et al._ (2012). However, its increase trend of flow rate was not very significant. Next, let us focus on the circular-obstacle case. In Fig. 5(c,d), the particle configuration and corresponding velocity field of granular flow with a circular obstacle are presented. As can be seen in Fig. 5(c), more structural defects in the particles configuration are introduced (compared to no-obstacle case) due to the presence of a large circular obstacle. In addition, the velocity field in this case becomes asymmetric as shown in Fig. 5(d). At this moment, particles in the left region are much more active than the other side. Actually, this asymmetry results in the temporal oscillation of the active zone, i.e., the alternate flow is developed. Namely, the obstacle triggers the spatiotemporal inhomogeneity in the granular discharge flow. This inhomogeneity could be a possible reason for avoiding the clogging. However, as shown in the clogging diagram (Fig. 3(a)), the relation between the clogging and the circular obstacle is not so clear. Moreover, we have confirmed that discharge flow rate is almost always steady even in the alternate flow regime Endo and Katsuragi (2017). The effect of obstacle is exaggerated by using a triangular obstacle. In Fig. 5(e,f), the particle configuration and corresponding velocity field with a triangular obstacle are shown. Qualitative characteristics observed in Fig. 5(e,f) are more or less similar to those seen in Fig. 5(c,d). Spatiotemporal inhomogeneity including alternate flow can be induced in this case as well. Moreover, particles-number density (packing fraction) beneath the obstacle (above the exit) significantly decreases in Fig. 5(e,f). Since the clogging prevention by the obstacle is most significant in this triangle case, an essential key effect to prevent the clogging must present in this triangular-obstacle case. This point will be discussed later from the viewpoint of velocity distribution and packing fraction. When we use a triangular obstacle fixed upside down (inverted triangle), different flow field is observed as shown in Fig. 5(g,h). Since the angle of triangular shape is commensurate with the triangular lattice structure, particle configuration shows the crystalline structure even in the region between the obstacle and bottom wall. And the converging flow at the exit region can be observed. The flow field is relatively symmetric compared to circle and triangle cases while it still shows a slight asymmetry. Furthermore, the packing fraction right above the exit is not very much reduced in this case. <figure><img src="content_image/1706.04791/x6.png"><figcaption>Figure 5: Velocity fields of granular discharge flow acquired by a high-speedcamera and computed by PTV method. Panel (a) indicates a snapshot of particleconfiguration without an obstacle. Panel (b) is its velocity field computed byeach particle’s motion in vector representation. Panel (c) indicates asnapshot of particle configuration with a circular obstacle. Panel (d) is itscorresponding velocity field． Panel (e) indicates a snapshot of particleconfiguration with a triangular obstacle. Panel (f) is its correspondingvelocity field． Panel (g) indicates a snapshot of particle configuration withan inverted-triangular obstacle. Panel (h) is its corresponding velocityfield. Experimental conditions are fixed at W=25 mm and L=30 mm．</figcaption></figure> ### Analyses of particle motions In order to discuss the principal effect for avoiding the clog, here we analyze the region above the exit A (the yellow square region shown in Fig. 6). All data shown in this subsection are based on the analyses in the region A. The width and height of region A are basically fixed to 50 mm and 30 mm, respectively. When $L<30$ mm, however, height of the region A is adjusted to $L$. With respect to the definition of the spatial axis, top left corner in the acquired image corresponds to the origin of vertical ($y$) and horizontal ($x$) axis. And the positive direction of $y$ axis is taken to be the gravitational downward direction. <figure><img src="content_image/1706.04791/x7.png"><figcaption>Figure 6: Definition of the focussed region A. The yellow square region(region A) is fixed at the central part above the exit with 50 mm wide and 30mm high. When L is smaller than 30 mm, the height is adjusted to the value ofL.</figcaption></figure> #### iii.3.1 Vertical velocity To characterize the velocity field, its probability density function (PDF) is computed from the PTV data. First, the vertical component $V_{y}$ is focused. Measured PDF of $V_{y}$ is shown in Fig. 7(a). In Fig. 7(a), different colors (and line codes) represent the different obstacle shapes. To reveal the effect of obstacle shape, experimental conditions are fixed at $W=25$ mm and $L=30$ mm. The qualitative form of PDF is basically identical among all the data shown in Fig. 7(a). In large $V_{y}$ regime, exponential-like tail can be observed. This part must principally determine the discharge flow rate. Obviously, the velocity level is reduced by the effect of obstacle. In other words, probability of large $V_{y}$ in the no-obstacle case is larger than other obstacle cases. The corresponding flow rates are shown in the legend of Fig. 7(a). This tendency is consistent with the flow rate reduction due to the obstacle effect Katsuragi _et al._ (2017); Endo and Katsuragi (2017). Note that, the discharge flow rate is always steady independently of obstacle shapes Katsuragi _et al._ (2017); Endo and Katsuragi (2017). In Fig. 7(a), relatively small amount of particles have negative $V_{y}$. This negative $V_{y}$ stems from the effective back scattering due to the particle-particle collisions. <figure><img src="content_image/1706.04791/x8.png"><figcaption>Figure 7: (a) Vertical velocity distribution in the region A. PDF of Vy atW=25 mm and L=30 mm is shown. The positive direction of Vy corresponds to thegravitational (downward) direction. (b) Horizontal velocity distribution inthe region A (PDF of Vx at W=25 mm and L=30 mm).</figcaption></figure> #### iii.3.2 Horizontal velocity and granular temperature Next, we focus on the behavior of horizontal velocity $V_{x}$. Figure 7(b) shows PDF of $V_{x}$ at $W=25$ mm and $L=30$ mm. These PDFs have the symmetric form because the shape of obstacle is symmetric. The effect of alternate flow is wiped out by the spatiotemporal average to make PDF. The width of PDF depends on the shape of obstacle. In the excited (dilute) granular gas, stretched-exponential-type PDF has been observed in many experiments Losert _et al._ (1999); Rouyer and Menon (2000); van Zon and MacKintosh (2004). However, here we are not going to examine the detail structure of PDF because the statistics to discuss the detail shape is limited in this experiment. Instead, we simply discuss the global-shape characterization of the PDF. Specifically, the width of PDF becomes the largest in the triangular-obstacle case while others are somewhat similar in large $\|V_{x}\|$ regime. Also, in the small $\|V_{x}\|$ regime ($\|V_{x}\|\leq 100$ mm/s), the obstacle dependence in PDF structure can be observed. No-obstacle and inverted-triangular obstacle result in qualitatively similar PDF forms in which the population of smaller $\|V_{x}\|$ is always larger than that of larger $\|V_{x}\|$. On the other hand, the population dips at relatively small $\|V_{x}\|$ regime can be confirmed in the cases of circular and triangular obstacles. Sharp peaks confirmed at $V_{x}\simeq 0$ come from the almost stopping particles in the silo. Similar peaks can also be found in $V_{y}$ PDF (Fig.7(a)). To characterize the statistical property of $V_{x}$, here we introduce the granular temperature in horizontal direction, $T_{gx}\sim\langle\delta V_{x}^{2}\rangle=\langle(V_{x}-\langle V_{x}\rangle)^{ 2}\rangle$, where $\langle\cdot\rangle$ indicates the spatiotemporal average. Since this granular temperature corresponds to the variance of PDF, it can be used to characterize the width of the distribution. The measured $T_{gx}$ as a function of $L$ at $W=25$ mm is shown in Fig. 8(a). As expected, $T_{gx}$ with the triangular-obstacle case is larger than the other obstacle cases in small $L$ regime. The sudden increase of $T_{gx}$ at small $L$ comes from the detouring of particles around the obstacle. The detouring followed by collisions below the obstacle results in the large horizontal velocity component. Note that, however, $T_{gx}$ in the case of circular obstacle also significantly increases in the very small $L$ regime. Since the clear and significant decrease of clogging occurence can be detected only for triangular obstacle at small $W$ and $L$ regime (Fig. 3), it is difficult to explain the anti-clogging mechanism solely by large $T_{gx}$. <figure><img src="content_image/1706.04791/x9.png"><figcaption>Figure 8: L dependences of (a) horizontal granular temperature Tgx and (b)packing fraction ϕ at W=25 mm. As the obstacle approaches the exit (L becomessmall), (a) Tgx increases, and (b) ϕ decreases. Horizontal dashed linesindicate the levels of no-obstacle case.</figcaption></figure> #### iii.3.3 Packing fraction Another possible reason for reducing the clogging risk is the low packing fraction $\phi$. Here, $\phi$ is defined by the ratio of the particles area relative to the total area in the region A. The measured $\phi$ as a function of $L$ at $W=25$ mm is shown in Fig. 8(b). As expected, $\phi$ decreases as $L$ decreases. Furthermore, the decreasing rate of $\phi$ is much more significant in the triangular-obstacle case than other two cases. This result is consistent with the fact that the triangular-obstacle case shows the clear reduction of clogging occurence in small $L$ and $W$ regime (Fig. 3). Because the clogging reduction is clearly confirmed in $L<40$ mm (at $W=25$ mm) with the triangular obstacle (Fig. 3(b)), we can consider that the sufficiently small $\phi(\leq 0.6)$ can safely prevent the clogging in two-dimensional granular flow through a narrow exit. This result is qualitatively consistent with Roussel _et al._ (2007). Obviously, the very small packing fraction is better to prevent clogging. In the very small $L$($\leq 25$ mm) regime, clogging by the arch formation between the obstacle and bottom wall (Fig. 4(b)) is observed. In this regime, $\phi$ in the region A is no longer relevant to discuss te clogging. Rather, the packing fraction in the lateral side regions would be important in this regime. Namely, the important quantity is the packing fraction at the clogging (arch formation) region. In the case of inverted triangle, $\phi$ remains large $(\geq 0.65)$. Nevertheless, the inverted triangle can make smooth flow in some (seemingly random) regimes in the clogging diagram (Fig. 3(c)). This point will be discussed later. Finally, we directly compare $\phi$ with $T_{gx}$ at $W=25$ mm. We can see the negative correlation between $\phi$ and $T_{gx}$ in large $T_{gx}$ regime (Fig. 9). Decreasing rate of $\phi$ becomes the maximum in the triangle case. This means that the triangular shape is the most efficient one to reduce $\phi$ by the identical $T_{gx}$. The flat bottom of the obstacle could play a crucial role to achieve this efficiency. To investigate the details of this shape dependence, force visualization by photoelastic material Tang and Behringer (2011); Vivanco _et al._ (2012); Iikawa _et al._ (2015, 2016) and/or numerical simulation Hidalgo _et al._ (2013) might be helpful. In this study, we perform the numerical simulation as presented below. <figure><img src="content_image/1706.04791/x10.png"><figcaption>Figure 9: Correlation between packing fraction ϕ and horizontal granulartemperature Tgx at W=25 mm.</figcaption></figure> ## IV Numerical results and analyses Next, we compare the numerical result with experiment. In Fig. 10, numerically obtained PDF of $V_{y}$ and $V_{x}$, horizontal granular temperature $T_{gx}$, and packing fraction $\phi$ are shown. The parameter values of $W$ and $L$ used here are same as those used in the experiment. From the experimental observation, we guess that the particles detouring due to the obstacle and the horizontal bottom shape are the principal factors to make small $\phi$. Thus, in the numerical simulation, we employ a horizontal-bar obstacle instead of the triangle. We also examine the cases with circular and inverted-triangular obstacles. As can be seen in Fig. 10, almost all qualitative characteristics observed in the experiment are reproduced by the numerical simulation. Then, the horizontal bar in numerical simulation should correspond to the triangle in the experiment. Although the PDF shapes of $V_{y}$ and $V_{x}$ are slightly different from experimentally obtained ones, obstacle-shape dependence of the PDF shape and $L$ dependences of $T_{gx}$ and $\phi$ are basically captured by the numerical result. This result supports our speculation: particles detouring around the obstacle and horizontal bottom shape of the obstacle significantly reduce $\phi$ below the obstacle (at the exit region). And the reduced $\phi$ decreases the clogging risk (and flow rate). <figure><img src="content_image/1706.04791/x11.png"><figcaption>Figure 10: Numerically obtained data of (a) vertical velocity PDF, (b)horizontal velocity PDF, (c) horizontal granular temperature, and (d) packingfraction. For (a) and (b), W=25 mm and L=50 mm. For (c) and (d), W=25 mm.</figcaption></figure> ## V Discussion From the above observations, we suppose that the principal effect induced by the obstacle in granular discharge flow field is as follows. By the presence of an obstacle at the vicinity of exit, particles have to detour to approach the exit. Thereby, particles-supply rate to the exit region is decreased by the detouring effect. This effect directly reduces local packing fraction at the exit region. The most significant detouring can be made by triangular obstacle (or horizontal bar). In addition, at the central part beneath the obstacle, strong collisions occur among the particles coming from the left and right sides. Then, the random motion of particles, which corresponds to the “temperature” in this system (granular temperature), is increased. Finally, the packing fraction right above the exit is further reduced by the excluded volume effect of active (high temperature) particles. As a consequence, the resultant low packing fraction prevents the arch formation at the exit. Thus, these obstacle effects can avoid the clogging occurrence. In this sense, the obstacle is useful to prevent the clogging by making small $\phi$. This simple scenario to prevent the clogging cannot explain all the situations. For instance, when the inverted-triangular obstacle is used, the clogging can be observed rather randomly at $W=25$ mm (Fig. 3(c)) in spite of large $\phi$. This result implies that the clogging prevention can be achieved even in large $\phi$ regime. We speculate that the crystal-like ordered structure made at the exit region could be a key to keep large $\phi$ by the inverted triangle (Fig. 5(g)). Although such an ordered structure yields the large $\phi$, its structure could flow systematically forming a cluster without developing an arch, as long as it does not create the obstacle-wall arch like Fig. 2(b). Then, the competition between high density and clustering might result in a very stochastic (not systematic) clogging as observed in the inverted-triangular case at $W=25$ mm. Moreover, the pressure decrease above the obstacle will also be effective to prevent the clogging Zuriguel _et al._ (2011). Namely, there might be plural mechanisms to prevent the clogging by the obstacle. Detail classification and characterization of various obstacle effects are still important future problems. There are some interesting issues we have not discussed in this paper. For example, spatiotemporal analyses of alternate flow typically observed in circular- and triangular-obstacle cases have not been performed. Actually, the experimental conditions to reproduce the alternate flow could not be precisely determined from our experimental result. It occurred rather in a random fashion. Perhaps, its classification might be more complex than making the clogging diagram. And the preliminary-measured switching period of the alternate flow also seemed to be neither universal nor systematic. Its characterization is one of the challenging issues opened to future study. In addition, the forms of PDF (Fig. 7) have not been discussed in detail. Only $T_{gx}$ has been used in this study. Its detail characterization is also an interesting future problem. ## VI Conclusion We conducted simple experiment and numerical simulation of two-dimensional granular flow driven by gravity under the influence of an obstacle. From the images of the granular exit flow acquired by a high-speed camera, we tracked the motion of individual particles at the exit region by means of PTV method. By the triangular obstacle, the clogging occurence can be reduced at a certain distance range from the exit. The principal reason for this clogging prevention is the low packing fraction at the vicinity of exit. The free space among the particles seems to be effectively created when the triangular obstacle approaches the exit. At the same time, the triangular obstacle also results in the large granular temperature in horizontal direction. The detouring by the triangle can decrease the supply of the particles to the exit region, and also activate the particles by collisions to have large velocity component in a horizontal direction. These behaviors were reproduced by the numerical simulation as well. And, the numerical simulation also revealed that the horizontal bar is enough to make the above-mentioned situations: small $\phi$ below the obstacle (above the exit region). ## Acknowledgement We would like to acknowledge S. Watanabe, H. Kumagai, S. Sirono, and T. Morota for helpful discussion. This work has been supported by JSPS KAKENHI No. 15H03707. K. Anki Reddy would like to thank IITG start-up research grant. ## References Faber (1995)T. E. Faber, _FLUID DYNAMICS FOR PHYSICISTS_ (Cambridge University Press, Cambridge, 1995). * Janssen (1895)H. A. Janssen, “Versuche über getreidedruck in silozellen,” Zeitschr. d. Vereines deutscher Ingenieure 39, 1045 (1895). * Aguirre _et al._ (2010)M A Aguirre, J G Grande, A Calvo, L A Pugnaloni, and J C Géminard, “Pressure Independence of Granular Flow through an Aperture,” Phys. Rev. Lett. 104, 238002–4 (2010). * Beverloo _et al._ (1961)W.A. Beverloo, H.A. Leniger, and J. van de Velde, “The flow of granular solids through orifices,” Chemical Engineering Science 15, 260 – 269 (1961). * Helbing _et al._ (2006)Dirk Helbing, Anders Johansson, Joachim Mathiesen, Mogens H. Jensen, and Alex Hansen, “Analytical approach to continuous and intermittent bottleneck flows,” Phys. Rev. Lett. 97, 168001 (2006). * To _et al._ (2001)Kiwing To, Pik-Yin Lai, and H. K. Pak, “Jamming of granular flow in a two-dimensional hopper,” Phys. Rev. Lett. 86, 71–74 (2001). * To (2005)Kiwing To, “Jamming transition in two-dimensional hoppers and silos,” Phys. Rev. E 71, 060301 (2005). * Zuriguel _et al._ (2005)Iker Zuriguel, Angel Garcimartín, Diego Maza, Luis A. Pugnaloni, and J. M. Pastor, “Jamming during the discharge of granular matter from a silo,” Phys. Rev. E 71, 051303 (2005). * Janda _et al._ (2008)A. Janda, I. Zuriguel, A. Garcimartín, L. A. Pugnaloni, and D. Maza, ‘‘Jamming and critical outlet size in the discharge of a two-dimensional silo,” EPL (Europhysics Letters) 84, 44002 (2008). * Zuriguel _et al._ (2011)Iker Zuriguel, Alvaro Janda, Angel Garcimartín, Celia Lozano, Roberto Arévalo, and Diego Maza, “Silo clogging reduction by the presence of an obstacle,” Phys. Rev. Lett. 107, 278001 (2011). * Lozano _et al._ (2012)Celia Lozano, Alvaro Janda, Angel Garcimartín, Diego Maza, and Iker Zuriguel, “Flow and clogging in a silo with an obstacle above the orifice,” Phys. Rev. E 86, 031306 (2012). * Zuriguel _et al._ (2014)Iker Zuriguel, Daniel R. Parisi, Raúl C. Hidalgo, Celia Lozano, Alvaro Janda, Paula A. Gago, Juan P. Peralta, Luis M. Ferrer, Luis A. Pugnaloni, Eric Clément, Diego Maza, Ignacio Pagonabarraga, and Angel Garcimartín, “Clogging transition of many-particle systems flowing through bottlenecks,” Scientific Reports 4, 7324+ (2014). * D. Helbing and Werner (2005)A. Johansson D. Helbing, L. Buzna and T. Werner, “Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions,” Transportation Science 39, 1–24 (2005). * Frank and Dorso (2011)G.A. Frank and C.O. Dorso, “Room evacuation in the presence of an obstacle,” Physica A: Statistical Mechanics and its Applications 390, 2135 – 2145 (2011). * Garcimartín _et al._ (2015)A. Garcimartín, J. M. Pastor, L. M. Ferrer, J. J. Ramos, C. Martín-Gómez, and I. Zuriguel, “Flow and clogging of a sheep herd passing through a bottleneck,” Phys. Rev. E 91, 022808 (2015). * Kawaguchi and Shimizu (2012)T. Kawaguchi and T. Shimizu, “Particle simulation of crowd evacuation,” Safety science review , 75–84 (2012). * Mankoc _et al._ (2009)Cristian Mankoc, Angel Garcimartín, Iker Zuriguel, Diego Maza, and Luis A. Pugnaloni, “Role of vibrations in the jamming and unjamming of grains discharging from a silo,” Phys. Rev. E 80, 011309 (2009). * Thomas and Durian (2015)C C Thomas and D J Durian, “Fraction of Clogging Configurations Sampled by Granular Hopper Flow,” Phys. Rev. Lett. 114, 178001–5 (2015). * Janda _et al._ (2012)Alvaro Janda, Iker Zuriguel, and Diego Maza, “Flow rate of particles through apertures obtained from self-similar density and velocity profiles,” Phys. Rev. Lett. 108, 248001 (2012). * Choi _et al._ (2004)Jaehyuk Choi, Arshad Kudrolli, Rodolfo R. Rosales, and Martin Z. Bazant, “Diffusion and mixing in gravity-driven dense granular flows,” Phys. Rev. Lett. 92, 174301 (2004). * Moka and Nott (2005)Sudheshna Moka and Prabhu R Nott, “Statistics of Particle Velocities in Dense Granular Flows,” Phys. Rev. Lett. 95, 068003–4 (2005). * Orpe and Kudrolli (2007)Ashish V. Orpe and Arshad Kudrolli, “Velocity correlations in dense granular flows observed with internal imaging,” Phys. Rev. Lett. 98, 238001 (2007). * Thomas and Durian (2016)C C Thomas and D J Durian, “Intermittency and velocity fluctuations in hopper flows prone to clogging,” Phys. Rev. E 94, 022901 (2016). * Ashour _et al._ (2017)A. Ashour, S. Wegner, T. Trittel, T. Borzsonyi, and R. Stannarius, “Outflow and clogging of shape-anisotropic grains in hoppers with small apertures,” Soft Matter 13, 402–414 (2017). * Katsuragi _et al._ (2017)H Katsuragi, K Anki-Reddy, and K Endo, “Shape-dependence of quasi-static drag coefficient in a gravity-driven granular flow,” (2017), in preparation. * Brilliantov _et al._ (1996)Nikolai V. Brilliantov, Frank Spahn, Jan-Martin Hertzsch, and Thorsten Pöschel, “Model for collisions in granular gases,” Phys. Rev. E 53, 5382–5392 (1996). * Silbert _et al._ (2001)L. E. Silbert, D. Ertas, G. S. Grest, T. C. Halsey, D. Levine, and S. J. Plimpton, “Granular flow down an inclined plane: Bagnold scaling and rheology,” Phys. Rev. E 64, 051302 (2001). * Schwager and Pöschel (1998)Thomas Schwager and Thorsten Pöschel, “Coefficient of normal restitution of viscous particles and cooling rate of granular gases,” Phys. Rev. E 57, 650–654 (1998). * Plimpton (1995)Steve Plimpton, “Fast Parallel Algorithms for Short-Range Molecular Dynamics ,” Journal of Computational Physics 117, 1 –19 (1995). * Endo and Katsuragi (2017)K. Endo and H Katsuragi, “Statistical properties of gravity-driven granular discharge flow under the influence of an obstacle,” EPJ Web of Conferences (2017), in press, arXiv:1611.03927. * Alonso-Marroquin _et al._ (2012)F Alonso-Marroquin, S I Azeezullah, S A Galindo-Torres, and L M Olsen-Kettle, “Bottlenecks in granular flow: When does an obstacle increase the flow rate in an hourglass?” Phys. Rev. E 85, 020301 (2012). * Murray and Alonso-Marroquin (2016)Alan Murray and Fernando Alonso-Marroquin, “Increasing granular flow rate with obstructions,” Pap. Phys. 8, 1–7 (2016). * Losert _et al._ (1999)W Losert, D G W Cooper, J Delour, A Kudrolli, and J P Gollub, “Velocity statistics in excited granular media,” Chaos 9, 682–690 (1999). * Rouyer and Menon (2000)Florence Rouyer and Narayanan Menon, “Velocity Fluctuations in a Homogeneous 2D Granular Gas in Steady State,” Phys. Rev. Lett. 85, 3676–3679 (2000). * van Zon and MacKintosh (2004)J S van Zon and F C MacKintosh, “Velocity Distributions in Dissipative Granular Gases,” Phys. Rev. Lett. 93, 038001 (2004). * Roussel _et al._ (2007)N. Roussel, Thi Lien Huong Nguyen, and P. Coussot, “General probabilistic approach to the filtration process,” Phys. Rev. Lett. 98, 114502 (2007). * Tang and Behringer (2011)J. Tang and R. P. Behringer, “How granular materials jam in a hopper,” Chaos 21, 041107 (2011), http://dx.doi.org/10.1063/1.3669495. * Vivanco _et al._ (2012)Francisco Vivanco, Sergio Rica, and Francisco Melo, “Dynamical arching in a two dimensional granular flow,” Granular Matter 14, 563–576 (2012). * Iikawa _et al._ (2015)Naoki Iikawa, Mahesh M Bandi, and Hiroaki Katsuragi, “Structural Evolution of a Granular Pack under Manual Tapping,” J. Phys. Soc. Jpn. 84, 094401–7 (2015). * Iikawa _et al._ (2016)N Iikawa, M M Bandi, and H Katsuragi, “Sensitivity of Granular Force Chain Orientation to Disorder-Induced Metastable Relaxation,” Phys. Rev. Lett. 116, 128001–5 (2016). * Hidalgo _et al._ (2013)R C Hidalgo, C Lozano, I Zuriguel, and A Garcimartín, “Force analysis of clogging arches in a silo,” Granular Matter 15, 841–848 (2013).
1309.7258	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 26800, "num_imgs": 1, "llama3_tokens_count": 7903 }	[ "content_image/1309.7258/x1.png" ]	# Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3 Yogesh Anbalagan 1 School of Computer Science, McGill University. yogesh.anbalagan@mail.mcgill.ca1 Sergey Norin 2Department of Mathematics and Statistics, McGill University. snorin@math.mcgill.ca2 Rahul Savani 3Department of Computer Science, University of Liverpool. rahul.savani@liverpool.ac.uk3 Adrian Vetta 4Department of Mathematics and Statistics, and School of Computer Science, McGill University. vetta@math.mcgill.ca4 ###### Abstract In an $\epsilon$-approximate Nash equilibrium, a player can gain at most $\epsilon$_in expectation_ by unilateral deviation. An $\epsilon$-_well-supported_ approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within $\epsilon$ of the best response payoff. Daskalakis, Mehta and Papadimitriou [8] conjectured that every win-lose bimatrix game has a $\frac{2}{3}$-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist _subject to_ the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a $\frac{2}{3}$ payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least $\Omega(\sqrt[3]{\log n})$ in any $\epsilon$-well supported equilibrium with $\epsilon<\frac{2}{3}$. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Häggkvist conjecture [4]. A probabilistic argument [13] shows that there exist $\epsilon$-well-supported equilibria with supports of cardinality $O(\frac{1}{\epsilon^{2}}\cdot\log n)$, for any $\epsilon>0$; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any $\delta>0$, there exist win-lose games for which no pair of strategies with support sizes at most two is a $(1-\delta)$-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in $[0,1]$ has a $\frac{1}{2}$-approximate Nash equilibrium where the supports of the players have cardinality at most two [8]. ## 1 Introduction A Nash equilibrium of a bimatrix game is a pair of strategies in which the supports of both players consist only of best responses. The apparent hardness of computing an exact Nash equilibrium [6, 5] even in a bimatrix game has led to work on computing approximate Nash equilibria, and two notions of approximate Nash equilibria have been developed. The first and more widely studied notion is of an _$\epsilon$-approximate Nash equilibrium_ ($\epsilon$-Nash). Here, no restriction is placed upon the supports; any strategy can be in the supports provided each player achieves an expected payoff that is within $\epsilon$ of a best response. Therefore, $\epsilon$-Nash equilibria have a practical drawback: a player might place probability on a strategy that is arbitrarily far from being a best response. The second notion, defined to rectify this problem, is called an _$\epsilon$-well supported approximate Nash equilibrium_ ($\epsilon$-WSNE). Here, the content of the supports are restricted, but less stringently than in an exact Nash equilibrium. Specifically, both players can only place positive probability on strategies that have payoff within $\epsilon$ of a pure best response. Observe that the latter notion is a stronger equilibrium concept: every $\epsilon$-WSNE is an $\epsilon$-Nash, but the converse is not true. Approximate well-supported equilibria recently played an important role in understanding the hardness of computing Nash equilibria. They are more useful in these contexts than $\epsilon$-Nash equilibria because their definition is more combinatorial and more closely resembles the _best response condition_ that characterizes exact Nash equilibria. Indeed, approximate well-supported equilibria were introduced in [12, 6] in the context of PPAD reductions that show the hardness of computing (approximate) Nash equilibria. They were subsequently used as the notion of approximate equilibrium by Chen et al. [5] that showed the PPAD-hardness of computing an exact Nash equilibrium even for bimatrix games. Another active area of research is to investigate the best (smallest) $\epsilon$ that can be guaranteed in polynomial time. For $\epsilon$-Nash, the current best algorithm, due to Tsaknakis and Spirakis [18], achieves a $0.3393$-Nash equilibrium; see [8, 7, 3] for other algorithms. For the important class of win-lose games – games with payoffs in $\{0,1\}$ – [18] gives a $\frac{1}{4}$-Nash equilibrium. For $\epsilon$-WSNE, the current best result was given by Fearnley et al. [10] and finds a $(\frac{2}{3}-\zeta)$-WSNE, where $\zeta=0.00473$. It builds on an approach of Kontogiannis and Spirakis [13], which finds a $\frac{2}{3}$-WSNE in polynomial time using linear programming. The algorithm of Kontogiannis and Spirakis produces a $\frac{1}{2}$-WSNE of win-lose games in polynomial time, which is best-known (the modifications of Fearnley et al. do not lead to an improved approximation guarantee for win-lose games). It is known that this line of work cannot extend to a fully-polynomial-time approximation scheme (FPTAS). More precisely, there does not exist an FPTAS for computing approximate Nash equilibria unless PPAD is in P [5]. Recall, an FPTAS requires a running time that is polynomial in both the size of the game input and in $\frac{1}{\epsilon}$. A polynomial-time approximation scheme (PTAS), however, need not run in time polynomial in $\frac{1}{\epsilon}$. It is not known whether there exists a PTAS for computing an approximate Nash equilibrium and, arguably, this is the biggest open question in equilibrium computation today. While the best-known approximation guarantee for $\epsilon$-Nash that is achievable in polynomial time is much better than that for $\epsilon$-WSNE, the two notions are polynomially related: there is a PTAS for $\epsilon$-WSNE if and only if there is a PTAS for $\epsilon$-Nash [5, 6]. ### Our Results The focus of this paper is on the combinatorial structure of equilibrium. Our first result shows that well-supported Nash equilibria differ structurally from approximate Nash equilibria in a significant way. It is known that there are $\frac{1}{2}$-Nash equilibria with supports of cardinality at most two [12], and that this result is tight [11]. In contrast, we show in Theorem 2.2 that for any $\delta>0$, there exist win-lose games for which no pair of strategies with support sizes at most two is a $(1-\delta)$-well-supported Nash equilibrium.¹ [FOOTNOTE:1][ENDFOOTNOTE] With supports of cardinality three, Daskalakis et al. conjectured, in the first paper that studied algorithms for finding $\epsilon$-WSNE [8], that $\frac{2}{3}$-WSNE are obtainable in every win-lose bimatrix game. Specifically, this would be a consequence of the following graph-theoretic conjecture. Conjecture 1 ([8]): Every digraph either has a cycle of length at most 3 or an undominated set² of 3 vertices. [FOOTNOTE:2][ENDFOOTNOTE] The main result in this paper, Theorem 2.1, shows that one cannot do better with constant size supports. We prove that there exist win-lose games that require supports of cardinality at least $\Omega(\sqrt[3]{\log n})$ in any $\epsilon$-WSNE with $\epsilon<\frac{2}{3}$. We prove this existence result probabilistically. The key tool in showing correctness is a proof of a bipartite digraph variant of the well-known Caccetta-Häggkvist conjecture [4]. A polylogarithmic cardinality bound, as presented here, is the best we can hope for – a probabilistic argument [13] shows that there exist $\epsilon$-WSNE with supports of cardinality $O(\frac{1}{\epsilon^{2}}\cdot\log n)$, for any $\epsilon>0$.³ [FOOTNOTE:3][ENDFOOTNOTE] ## 2 A Lower Bound on the Support Size of Well Supported Nash Equilibria We begin by formally defining bimatrix win-lose games and well-supported Nash equilibria. A _bimatrix game_ is a $2$-player game with $m\times n$ payoff matrices $A$ and $B$; we may assume that $m\leq n$. The game is called _win-lose_ if each matrix entry is in $\{0,1\}$. A pair of mixed strategies $\{{\bf p},{\bf q}\}$ is a _Nash equilibrium_ if every pure row strategy in the support of p is a best response to q and every pure column strategy in the support of q is a best response to p. A relaxation of this concept is the following. A pair of mixed strategies $\{{\bf p},{\bf q}\}$ is an $\epsilon$_-well supported Nash equilibrium_ if every pure strategy in the support of p (resp. q) is an $\epsilon$-approximate best response to q (resp. p). That is, for any row $r_{i}$ in the support of ${\bf p}$ we have \[{\bf e_{i}}^{T}A{\bf q}\geq\max_{\ell}{\bf e_{\ell}}^{T}A{\bf q}-\epsilon\] and, for any column $c_{j}$ in the support of ${\bf q}$ we have \[{\bf p}^{T}B{\bf e_{j}}\geq\max_{\ell}{\bf p}^{T}B{\bf e_{\ell}}-\epsilon\ .\] In this section we prove our main result. Theorem 2.1: _For any $\epsilon<\frac{2}{3}$, there exist win-lose games for which every $\epsilon$-well-supported Nash equilibrium has supports of cardinality $\Omega(\sqrt[3]{\log n})$._ To prove this result, we first formulate our win-lose games graphically. This can be done in a straight-forward manner. Simply observe that we may represent a $2$-player win-lose game by a directed bipartite graph $G=(R\cup C,E)$. There is a vertex for each row and a vertex for each column. There is an arc $(r_{i},c_{j})\in E$ if and only if $(B)_{ij}=b_{ij}=1$; similarly there is an arc $(c_{j},r_{i})\in E$ if and only if $(A)_{ij}=a_{ij}=1$. Consequently, we are searching for a graph whose corresponding game has no high quality well-supported Nash equilibrium with small supports. We show the existence of such a graph probabilistically. The Construction. Let $T=(V,E)$ be a random tournament on $N$ nodes. Now create from $T$ an auxiliary bipartite graph $G(T)=(R\cup C,A)$ corresponding to a $2$-player win-lose game as follows. The auxiliary graph has a vertex-bipartition $R\cup C$ where there is a vertex of $R$ for each node of $T$ and there is a vertex of $C$ for each set of $k$ distinct nodes of $T$. (Observe that, for clarity we will refer to nodes in the tournament $T$ and vertices in the bipartite graph $G$.) There are two types of arc in $G(T)$: those oriented from $R$ to $C$ and those oriented from $C$ to $R$. For arcs of the former type, each vertex $X\in C$ will have in-degree exactly $k$. Specifically, let $X$ correspond to the $k$-tuple $\{v_{1},\dots,v_{k}\}$ where $v_{i}\in V(T)$, for all $1\leq i\leq k$. Then there are arcs $(v_{i},X)$ in $G$ for all $1\leq i\leq k$. Next consider the latter type of arc in $G$. For each node $u\in R$ there is an arc $(X,u)$ in $G$ if and only if $u$ dominates $X=\{v_{1},\dots,v_{k}\}$ in the tournament $T$, that is if $(u,v_{i})$ are arcs in $T$ for all $1\leq i\leq k$. This completes the construction of the auxiliary graph (game) $G$. We say that a set of vertices $W=\{w_{1},\dots,w_{t}\}$ is _covered_ if there exists a vertex $y$ such that $(w_{j},y)\in A$, for all $1\leq j\leq t$. Furthermore, a bipartite graph is $k$_-covered_ if every collection of $k$ vertices that lie on the same side of the bipartition is covered. Now with positive probability the auxiliary graph $G(T)$ is $k$-covered. Lemma 1: _For all sufficiently large $n$ and $k\leq\sqrt[3]{\log n}$, there exists a tournament $T$ whose auxiliary bipartite graph $G(T)$ is $k$-covered._ Proof: Observe that the payoff matrices that correspond to $G(T)$ have $m=N$ rows and $n={N\choose k}$ columns. Furthermore, by construction, any set of $k$ vertices in $R$ is covered. Thus, first we must verify that any set of $k$ vertices in $C$ is also covered. So consider a collection $\mathcal{X}=\{X_{1},\dots,X_{k}\}$ of $k$ vertices in $C$. Since each $X_{i}\in C$ corresponds to a $k$-tuple of nodes of $T$, we see that $\mathcal{X}$ corresponds to a collection of at most $k^{2}$ nodes in $T$. Thus, for any node $u\notin\cup_{i}X_{i}$, we have that $u$ has an arc in $T$ to every node in $\cup_{i}X_{i}$ with probability at least $2^{-k^{2}}$. Thus with probability at most $(1-\frac{1}{2^{k^{2}}})^{N-k^{2}}$ the subset $\mathcal{X}$ of $C$ not covered in $G(T)$. Applying the union bound we have that there exists the desired tournament if \[{n\choose k}\cdot\left(1-\frac{1}{2^{k^{2}}}\right)^{N-k^{2}}<1\] (1) Now set $k={\log^{\frac{1}{3}}n}$. Therefore $\log n^{\frac{1}{k}}=\log^{\frac{2}{3}}n=k^{2}$. In addition, because $n={N\choose k}$, we have that $N\geq\frac{k}{e}\cdot n^{\frac{1}{k}}$. Hence, $N-k^{2}>n^{\frac{1}{k}}$. (Note that, since $N\geq k$ this implies that $G(T)$ is defined.) Consequently, \[{n\choose k}\cdot\left(1-\frac{1}{2^{k^{2}}}\right)^{N-k^{2}} \leq n^{k}\cdot\left(1-\frac{1}{2^{k^{2}}}\right)^{n^{\frac{1}{k}}}\] \[\leq n^{k}\cdot e^{-\frac{1}{2^{k^{2}}}\cdot{n^{\frac{1}{k}}}}\] \[\leq n^{k}\cdot e^{-\frac{1}{e^{k^{2}\cdot\log 2}}\cdot{n^{\frac{1}{k }}}}\] Thus, taking logarithms, we see that Inequality (1) holds if \[e^{k^{2}\cdot\log 2}\cdot k\cdot\log n<n^{\frac{1}{k}}\] (2) But $n^{\frac{1}{k}}=e^{k^{2}}$, so Inequality (2) clearly holds for large $n$. The result follows. A property of the auxiliary graph $G(T)$ that will be very useful to us is that it contains no cycles with less than six vertices. Lemma 2: _The auxiliary graph $G(T)$ contains no digons and no $4$-cycles._ Proof: Suppose $G(T)$ contains a digon $\{w,X\}$. The arc $(w,X)$ implies that $X=\{x_{1},\dots,x_{k-1},w\}$. On the other-hand, the arc $(X,w)$ implies that $w$ dominates $X$ in $T$ and, thus, $w\notin X$. Suppose $G(T)$ contains a $4$-cycle $\{w,X,z,Y\}$ where $w$ and $z$ are in $R$ and where $X=\{x_{1},\dots,x_{k-1},w\}$ and $Y=\{y_{1},\dots,y_{k-1},z\}$ are in $C$. Then $z$ must dominate $X$ in $T$ and $w$ must dominate $Y$ in $T$. But then we have a digon in $T$ as $(w,z)$ and $(z,w)$ must be arcs in $T$. This contradicts the fact that $T$ is a tournament. Lemmas 1 and 2 are already sufficient to prove a major distinction between approximate-Nash equilibria and well-supported Nash equilibria. Recall that there always exist $\frac{1}{2}$-Nash equilibria with supports of cardinality at most two [8]. In sharp contrast, for supports of cardinality at most two, no constant approximation guarantee can be obtained for $\epsilon$-well-supported Nash equilibria. Theorem 2.2: _For any $\delta>0$, there exist win-lose games for which no pair of strategy vectors with support sizes at most two is a $(1-\delta)$-well-supported Nash equilibrium._ Proof: Take the auxiliary win-lose game $G(T)$ from Lemma 1 for the case $k=2$. Now consider any pair of strategy vectors ${\bf p_{1}}$ and ${\bf p_{2}}$ with supports of cardinality $2$ or less. Since $G(T)$ is $2$-covered, the best responses to ${\bf p_{1}}$ and ${\bf p_{2}}$ both generate payoffs of exactly $1$. Thus $\{{\bf p_{1}},{\bf p_{2}}\}$ can be a $(1-\delta)$-well-supported Nash equilibrium only if each strategy in the support of ${\bf p_{1}}$ is a best response to at least one of the pure strategies in the support of ${\bf p_{2}}$ and vice versa. Therefore, in the subgraph $H$ of $G(T)$ induced by the supports of ${\bf p_{1}}$ and ${\bf p_{2}}$, each vertex has in-degree at least one. Thus, $H$ contains a directed cycle. But $G(T)$ has no digons or $4$-cycles, by Lemma 2. Hence, we obtain a contradiction as $H$ contains at most four vertices. In light of Lemma 2, we will be interested in the minimum in-degree required to ensure that a bipartite graph contains a $4$-cycle. The following theorem may be of interest on its own right, as it resolves a variant of the well-known Caccetta-Häggkvist conjecture [4] for bipartite digraphs. For Eulerian graphs, a related but different result is due to Shen and Yuster [17]. Theorem 2.3: _Let $H=(L\cup R,A)$ be a directed $k\times k$ bipartite graph. If $H$ has minimum in-degree $\lambda\cdot k$ then it contains a $4$-cycle, whenever $\lambda>\frac{1}{3}$._ Proof: To begin, by removing arcs we may assume that every vertex has in-degree exactly $\lambda\cdot k$. Now take a vertex $v$ with the maximum out-degree in $H$, where without loss of generality $v\in L$. Let $A_{1}$ be the set of out-neighbours of $v$, and set $\alpha_{1}\cdot k=\|A_{1}\|$. Similarly, let $B_{t}$ be the set of vertices with paths to $v$ that contain exactly $t$ arcs, for $t\in\{1,2\}$, and set set $\beta_{t}\cdot k=\|B_{t}\|$. Finally, let $C_{1}$ be the vertices in $R$ that are not adjacent to $v$, namely $C_{1}=L-(A_{1}\cup B_{1})$. Set $\gamma_{1}\cdot k=\|C_{1}\|$. These definitions are illustrated in Figure 1. <figure><img src="content_image/1309.7258/x1.png"><figcaption>Figure 1:</figcaption></figure> Observe that we have the following constraints on $\alpha_{1},\beta_{1}$ and $\gamma_{1}$. By assumption, $\beta_{1}=\lambda$. Thus, we have $\gamma_{1}=1-\alpha_{1}-\lambda$. Moreover, by the choice of $v$, we have $\alpha_{1}\geq\lambda$, since the maximum out-degree must be at least the average in-degree. Note that if there is an arc from $A_{1}$ to $B_{2}$ then $H$ contains a $4$-cycle. So, let’s examine the in-neighbours of $B_{2}$. We know $B_{2}$ has exactly $\lambda\cdot k\cdot\|B_{2}\|$ incoming arcs. We may assume all these arcs emanate from $B_{1}\cup C_{1}$. On the other-hand, there are exactly $\lambda\cdot k\cdot\|B_{1}\|$ arcs from $B_{2}$ to $B_{1}$. Thus, there are at most $\|B_{1}\|\cdot\left(\|B_{2}\|-\lambda\cdot k\right)$ arcs from $B_{1}$ to $B_{2}$. So the number of arcs from $C_{1}$ to $B_{2}$ is at least \[\lambda\cdot k\cdot\|B_{2}\|-\|B_{1}\|\cdot\left(\|B_{2}\|-\lambda\cdot k\right) = \lambda\cdot k\cdot\beta_{2}\cdot k-\beta_{1}\cdot k\cdot\left( \beta_{2}\cdot k+\lambda\cdot k\right)\] \[= \lambda\cdot k\cdot\beta_{2}\cdot k-\lambda\cdot k\cdot\left( \beta_{2}\cdot k+\lambda\cdot k\right)\] \[= \lambda^{2}\cdot k^{2}\] Since the maximum degree is $\alpha_{1}\cdot k$, the number of arcs emanating from $C_{1}$ is at most $\gamma_{1}\cdot\alpha_{1}\cdot k^{2}$. Thus $\gamma_{1}\cdot\alpha_{1}\cdot(1-\alpha_{1}-\lambda)\geq\lambda^{2}$. Rearranging we obtain the quadratic inequality \[\alpha_{1}^{2}-\alpha_{1}(1-\lambda)+\lambda^{2}\leq 0\] The discriminant of this quadratic is $1-2\lambda-3\lambda^{2}$. But $1-2\lambda-3\lambda^{2}=(1-3\lambda)(1+\lambda)$ and this is non-negative if and only if $\lambda\leq\frac{1}{3}$. This completes the proof. We may now prove our main result: no approximation guarantee better than $\frac{2}{3}$ can be achieved unless the well-supported equilibria has supports with cardinality $\Omega(\sqrt[3]{\log n})$. Proof of Theorem 2.1. Take a tournament $T$ whose auxiliary bipartite graph is $k$-covered. By Lemma 1, such a tournament exists. Consider the win-lose game corresponding to the auxiliary graph $G(T)$, and take strategy vectors ${\bf p_{1}}$ and ${\bf p_{2}}$ with supports of cardinality $k$ or less. Without loss of generality, we may assume that $\bf{p_{1}}$ and $\bf{p_{2}}$ are rational. Denote these supports as $S_{1}\subseteq R$ and $S_{2}\subseteq C$, respectively. As $G(T)$ is $k$-covered, there is a pure strategy $c^{}\in C$ that covers $S_{1}$ and a pure strategy $r^{}\in R$ that covers $S_{2}$. Thus, in the win-lose game, $c^{}\in C$ has an expected payoff of $1$ against ${\bf p_{1}}$ and $r^{}\in R$ has an expected payoff of $1$ against ${\bf p_{2}}$. Suppose ${\bf p_{1}}$ and ${\bf p_{2}}$ form a $\epsilon$-well-supported equilibrium for some $\epsilon<\frac{2}{3}$. Then it must be the case that each $r_{i}\in S_{1}$ has expected payoff at least $1-\epsilon>\frac{1}{3}$ against ${\bf p_{2}}$. Similarly, each $c_{j}\in S_{2}$ has expected payoff at least $1-\epsilon>\frac{1}{3}$ against ${\bf p_{1}}$. But this cannot happen. Consider the subgraph of $G(T)$ induced by $S_{1}\cup S_{2}$ where each $r_{i}\in S_{1}$ has weight $w_{i}=p_{1}(r_{i})$ and each $c_{j}\in S_{2}$ has weight $w_{j}=p_{2}(c_{j})$. We convert this into an unweighted graph $H$ by making $L\cdot w_{v}$ copies of each vertex $v$, for some large integer $L$. Now $H$ is an $L\times L$ bipartite graph with minimum in-degree $(1-\epsilon)\cdot L>\frac{1}{3}\cdot L$. Thus, by Theorem 2.3, $H$ contains a $4$-cycle. This is a contradiction, by Lemma 2. ∎ We remark that the $\frac{2}{3}$ in Theorem 2.1 cannot be improved using this proof technique. Specifically the minimum in-degree requirement of $\frac{1}{3}\cdot k$ in Theorem 2.3 is tight. To see this, take a directed $6$-cycle $C$ and replace each vertex in $C$ by $\frac{1}{3}\cdot k$ copies. Thus each arc in $C$ now corresponds to a complete $\frac{k}{3}\times\frac{k}{3}$ bipartite graph with all arc orientations in the same direction. The graph $H$ created in this fashion is bipartite with all in-degrees (and all out-degrees) equal to $\frac{1}{3}\cdot k$. Clearly the minimum length of a directed cycle in $H$ is six. ## 3 Conclusion An outstanding open problem is whether any constant approximation guarantee better than 1 is achievable with constant cardinality supports. We have shown that supports of cardinality two cannot achieve this; a positive resolution of Conjecture 1 would suffice to show that supports of cardinality three can. However, Conjecture 1 seems a hard graph problem and it is certainly conceivable that it is false.⁴ If so, that would lead to the intriguing possibility of a very major structural difference between $\epsilon$-Nash and $\epsilon$-WSNE; namely, that for any $\delta>0$, there exist win-lose games for which no pair of strategies with constant cardinality supports is a $(1-\delta)$-well-supported Nash equilibrium. [FOOTNOTE:4][ENDFOOTNOTE] The existence of small support $\epsilon$-WSNE clearly implies the existence of of polynomial time approximation algorithms to find such equilibria. Obtaining better approximation guarantees using more complex algorithms is also an interesting question. As discussed, the best known polynomial-time approximation algorithm for well-supported equilibria in win-lose games finds a $\frac{1}{2}$-well supported equilibrium [13] by solving a linear program (LP). For games with payoffs in $[0,1]$ that algorithm finds a $\frac{2}{3}$-well-supported equilibrium. The algorithm has been modified in [10] to achieve a slightly better approximation of about $\frac{2}{3}-\zeta$ where $\zeta=0.00473$. That modification solves an almost identical LP as [13] and then either transfers probability mass within the supports of a solution to the LP or returns a small support strategy profile that uses at most two pure strategies for each player. The results of this paper show that both parts of that approach are needed, and any improvement to the approximation guarantee must allow for super-constant support sizes. Acknowledgements. We thank John Fearnley and Troels Sørensen for useful discussions. The first author is supported by a fellowship from MITACS and an NSERC grant. The second author is supported by an NSERC grant 418520. The third author is supported by an EPSRC grant EP/L011018/1. The fourth author is supported by NSERC grants 288334 and 429598. ## References * [1] I. Althöfer, “On sparse approximations to randomized strategies and convex combinations”, _Linear Algebra and its Applications_, 199, pp339-355, 1994. * [2] I. Bárány, S. Vempala, and A. Vetta, “Nash equilibria in random games”, _Random Struct. Algorithms_, 31(4), pp391-405, 2007. * [3] H. Bosse, J. Byrka, and E. Markakis, “New algorithms for approximate Nash equilibria in bimatrix games”, _Theoretical Computer Science_, 411(1), pp164-173, 2010. * [4] L. Caccetta and R. Häggkvist, “On minimal digraphs with given girth”, _Congressus Numerantium_, 21, pp181-187, 1978. * [5] X. Chen, X. Deng, and S. Teng, “Settling the complexity of computing two-player Nash equilibria”, _Journal of the ACM_, 56(3), pp1-57, 2009. * [6] C. Daskalakis, P. Goldberg, and C. Papadimitriou, “The complexity of computing a Nash equilibrium”, _SIAM Journal on Computing_, 39(1), pp195-259, 2009. * [7] C. Daskalakis, A. Mehta, and C. Papadimitriou “Progress in approximate Nash equilibria”, _ACM Conference on Electronic Commerce (EC)_, pp355-358, 2007. * [8] C. Daskalakis, A. Mehta, and C. Papadimitriou, “A note on approximate Nash equilibria”, _Theoretical Computer Science_, 410(17), pp1581-1588, 2009. * [9] C. Even-Zohar and N. Linial, “Triply existentially complete triangle-free graphs”, _arxiv.org/1306.5637_, 2013. * [10] J. Fearnley, P. Goldberg, R. Savani, and T. Sørensen, “Approximate well-supported Nash equilibria below two-thirds”, _International Symposium on Algorithmic Game Theory (SAGT)_, pp108-119, 2012. * [11] T. Feder, H. Nazerzadeh, and A. Saberi, “Approximating Nash equilibria using small-support strategies”, _ACM Conference on Electronic Commerce (EC)_, 2007. * [12] P. Goldberg and C. Papadimitriou, “Reducability among equilibrium problems”, _STOC_, 2006. * [13] S. Kontogiannis and P. Spirakis, “Well supported approximate equilibria in bimatrix games”, _Algorithmica_, 57, pp653-667, 2010. * [14] S. Kontogiannis and P. Spirakis, “Efficient algorithms for constant well supported approximate equilibria in bimatrix games”, _ICALP_, pp595-606, 2007. * [15] R. Lipton and N. Young, “Simple strategies for zero-sum games with applications to complexity theory”, _STOC_, pp734-740, 1994. * [16] R. Lipton, E. Markakis, and A. Mehta, “Playing large games using simple startegies’”, _ACM Conference on Electronic Commerce (EC)_, pp36-41, 2003. * [17] J. Shen and R. Yuster, “A note on the number of edges guaranteeing a $C_{4}$ in Eulerian bipartite digraphs”, _Electronic Journal of Combinatorics_, 9(1), Note 6, 2002. * [18] H. Tsaknakis and P. Spirakis, “An optimization approach for approximate Nash equilibria”, _Internet Mathematics_, 5(4), pp365-382, 2008.
1809.03891	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 89151, "num_imgs": 6, "llama3_tokens_count": 21844 }	[ "content_image/1809.03891/itifulltextcountonly.png", "content_image/1809.03891/dendrogram_sent_100.png", "content_image/1809.03891/dendrogram_sent_100_noreuse.png", "content_image/1809.03891/lifereuselemAfinal.png", "content_image/1809.03891/newlemmasallA.png", "content_image/1809.03891/Wordfreqs.png" ]	# Studying the History of the Arabic Language † [FOOTNOTE:†][ENDFOOTNOTE] Language Technology and a Large-Scale Historical Corpus Yonatan Belinkov* Y. Belinkov MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA 02139, USA ¹A. Magidow Department of Modern and Classical Languages and Literatures, University of Rhode Island, USA ²A. Barrón-Cedeño Qatar Computing Research Institute, HBKU, Doha, Qatar ³A. Shmidman Department of Hebrew Literature, Bar-Ilan University, Israel Dicta: The Israel Center for Text Analysis ⁴M. Romanov Department of History, University of Vienna, Vienna, Austria ⁵ Alexander Magidow* Y. Belinkov MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA 02139, USA ¹A. Magidow Department of Modern and Classical Languages and Literatures, University of Rhode Island, USA ²A. Barrón-Cedeño Qatar Computing Research Institute, HBKU, Doha, Qatar ³A. Shmidman Department of Hebrew Literature, Bar-Ilan University, Israel Dicta: The Israel Center for Text Analysis ⁴M. Romanov Department of History, University of Vienna, Vienna, Austria ⁵ Alberto Barrón-Cedeño Y. Belinkov MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA 02139, USA ¹A. Magidow Department of Modern and Classical Languages and Literatures, University of Rhode Island, USA ²A. Barrón-Cedeño Qatar Computing Research Institute, HBKU, Doha, Qatar ³A. Shmidman Department of Hebrew Literature, Bar-Ilan University, Israel Dicta: The Israel Center for Text Analysis ⁴M. Romanov Department of History, University of Vienna, Vienna, Austria ⁵ Avi Shmidman Y. Belinkov MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA 02139, USA ¹A. Magidow Department of Modern and Classical Languages and Literatures, University of Rhode Island, USA ²A. Barrón-Cedeño Qatar Computing Research Institute, HBKU, Doha, Qatar ³A. Shmidman Department of Hebrew Literature, Bar-Ilan University, Israel Dicta: The Israel Center for Text Analysis ⁴M. Romanov Department of History, University of Vienna, Vienna, Austria ⁵ Maxim Romanov Y. Belinkov MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA 02139, USA ¹A. Magidow Department of Modern and Classical Languages and Literatures, University of Rhode Island, USA ²A. Barrón-Cedeño Qatar Computing Research Institute, HBKU, Doha, Qatar ³A. Shmidman Department of Hebrew Literature, Bar-Ilan University, Israel Dicta: The Israel Center for Text Analysis ⁴M. Romanov Department of History, University of Vienna, Vienna, Austria ⁵ [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:6][ENDFOOTNOTE] [FOOTNOTE:8][ENDFOOTNOTE] [FOOTNOTE:10][ENDFOOTNOTE] Received: date / Accepted: date ###### Abstract Arabic is a widely-spoken language with a long and rich history, but existing corpora and language technology focus mostly on modern Arabic and its varieties. Therefore, studying the history of the language has so far been mostly limited to manual analyses on a small scale. In this work, we present a large-scale historical corpus of the written Arabic language, spanning 1400 years. We describe our efforts to clean and process this corpus using Arabic NLP tools, including the identification of reused text. We study the history of the Arabic language using a novel automatic periodization algorithm, as well as other techniques. Our findings confirm the established division of written Arabic into Modern Standard and Classical Arabic, and confirm other established periodizations, while suggesting that written Arabic may be divisible into still further periods of development. Keywords:Arabic Corpus Periodization Text Reuse Historical Linguistics ∎ ## 1 Introduction Language is complex and dynamic. It changes across space and time, from generation to generation. New words are introduced and old words go out of use; words acquire new meanings and change their old meanings; and grammatical norms that existed in the past may become obsolete in the future. Language use is partly documented by texts which preserve traces of that change including variations in spelling, prefixes or suffixes that appear or disappear across eras and changes in word meanings. The Arabic language is no exception to this, but provides a challenge for the historical linguist. Unlike many languages where the standardization of writing was a long and contended process, leaving a trace of that process in the written record, Arabic writing was standardized quite early. This created a divergence between the spoken and written languages and while spoken Arabic has continued to evolve, written Arabic was essentially _fossilized_ in the 8th century if not earlier. Between that period and today, spellings vary little, the grammar of formal written Arabic shows few changes from the earliest texts, and aside from a period of modernization in the 19th century, there is little apparent change in the lexicon. While many studies have been conducted on non-standard texts that do diverge from this norm, or on the history of spoken Arabic, comparatively little attention has been paid to standard written Arabic. In this paper we seek to develop tools that enable us to determine whether formal written Arabic is as unchanging and homogeneous as it first appears, or whether we can divide it into separate periods, analogous to those in other languages (e.g., “Old English”,“Early Modern English”). With 1400 years of written Arabic texts across many different genres and with very subtle differences between eras, this is a task that is incredibly difficult to undertake using traditional philological methods, which may explain the small number of previous studies. For that reason, we seek to develop computational resources and methods to investigate the periodization of standard written Arabic. Specifically, we seek to answer the following: 1. What computational tools and resources are needed to investigate the periodization of Arabic? 2. Can formal written Arabic be divided into temporally-distinct periods based on linguistic evidence? 3. Are previously proposed periodizations of written Arabic accurate? After a brief overview of Arabic, and a review of previous studies in this area (Section 3), the rest of the paper is divided in two parts. In the first part (Section 4), we describe our efforts to process OpenITI— a large-scale diachronic corpus of Arabic with approximately $1.5~{}G$ words. OpenITI is the largest publicly-available historical corpus of Arabic that we are aware of. We make the processed corpus available for the community in order to facilitate data-driven research of the history of the language. In the second part of the paper, we develop computational methods for investigating the history of the Arabic language. In Section 5 we adapt a text reuse algorithm that was previously applied to a relatively small Hebrew/Aramaic corpus in order to identify exact and approximate matches in the large OpenITI corpus. Identifying matches is especially important because many of the documents in the corpus quote and paraphrase large quantities of texts from earlier works, sometimes many centuries earlier. Texts that contain language from very different eras make all forms of historical linguistic analysis more difficult. In Section 6 we develop a novel data-driven periodization algorithm that is based on word embeddings. Contrary to previous methods, our algorithm captures language use on the level of full corpora or subsets of corpora, rather than being limited to a handful of linguistic features. We apply this algorithm to the OpenITI historical corpus and find well-known as well as new periodizations. In Section 7 we utilize the corpus for an expert study of Arabic periodization. First, we verify that written Arabic does indeed change remarkably slowly by tracking the lifespan of Arabic words in contrast to English words. We also study several important linguistic phenomena that have so far only been anecdotally analyzed in the literature. The main contributions of our work are: * Preprocessing OpenITI, a diverse large-scale historical corpus of Arabic, including morphological segmentation, part-of-speech tags, lemmatization, and syntactic parse trees. * A complete identification of parallel matches in the corpus based on a novel adaptation of a text reuse algorithm. The algorithm is adapted for Arabic and runs efficiently on the large-scale OpenITI corpus. We are able to identify and remove $292~{}M$ words of reused text, nearly $20\%$ of the total corpus. * A novel periodization algorithm relying on word embeddings, which can be applied to any large-scale historical corpus. We demonstrate its applicability to the Arabic case on OpenITI. * New insights regarding the history of the Arabic language, that illuminate its development from early times to the modern days. In particular, our computational methods affirm the established periodization of Standard Arabic into Classical and Modern Standard Arabic, and point to new periodizations for Classical Arabic. ## 2 Linguistic Background Arabic is a _diglossic_ language, meaning there is a single formal language, Standard Arabic (SA),¹ which is used as a language of writing and formal communication. Everyday life is conducted in a divergent set of spoken languages, referred to collectively as “colloquial Arabic”. Written Arabic in various forms is attested for some time prior to Islam, but the coming of Islam marks the beginning of a vast written tradition. Even prior to Islam, SA was a relatively homogeneous register used for oral literature [24]. However the coming of Islam, dated to 622 CE, when the early Muslim community moved from Mecca to the city of Medina, brought a huge increase in written production. SA was largely standardized even before the 8th century CE, but that is the time period most strongly associated with the establishment of explicit linguistic standards. Oral texts from prior to that time were committed to writing, but were almost certainly edited later to conform more closely to the standard [42]. Such early standardization means that the SA of the 8th century CE is still basically accessible to a reader today — for example, the collection of stories _Kalila wa Dimna_ from ca. 750 CE is considered appropriate reading material at the middle school level today, with archaic terms or structures elucidated by footnotes. Though the literary style of SA varies between genres and eras, the basic orthography, morphology, syntax, and even vocabulary appear to have remained largely the same since that time. Impressionistically, there is very little variation between a modern formal text in Arabic and a text from nearly a thousand years ago, if they cover similar topics. [FOOTNOTE:1][ENDFOOTNOTE] This is not to say that all Arabic writing is homogeneous. There are variants of written Arabic which diverge significantly from SA, even in the pre-modern era: the language of the Ancient North Arabian inscriptions predates the standardization of Arabic, and probably the coming of Islam, and so differs in alphabet, spelling, and lexicon [2]. “Middle Arabic”, a term which is not chronological, refers to writings which do not match the norms of SA, whether they are early Islamic era papyri or Judeo–Arabic letters from the 14th century [71]. These texts are of great use for historical linguists, but a relatively small quantity have been digitized. In the case of Ancient North Arabian texts, they are far too different in genre and vocabulary to be easily comparable to SA texts. By far the largest body of writing is in SA, as this is the language of writing and publishing. Even texts which may have been closer to Middle Arabic at one point were likely corrected by later editors to better conform to the SA style [42]. Therefore, a major question, is whether the apparently homogeneous SA can be divided into eras in the same manner as many European languages, which generally did not experience standardization until late in their literary history. The autochthonous linguistic tradition typically treats SA as a single, undifferentiated language, while western Arabists divide SA into Classical Arabic (CA), the language used in pre-modern texts, and Modern Standard Arabic (MSA). MSA is said to have arisen due to increased contact with the West, starting in the late 18th and early 19th centuries. The largest changes were in lexis, as Western (French, English, German, Italian) terms were adopted or translated, though there may have also been some changes in syntax under the influence of European languages, or even of the spoken colloquial Arabic dialects [50]. Rarely is CA itself divided into other eras, with most variation in CA attributed to register or geography, rather than temporal variation (33, e.g., pp. 38–41). A small quantity of research supports dividing CA further. There are two slightly different accounts in the literature. Fischer suggests a tri-partite division of pre-Standardized CA, Standardized CA, and Post-CA[25]. Pre-Standardized Classical Arabic (PSCA) is pre-Islamic and early Islamic, primarily attested in quotations of older texts, and represents a very limited corpus. Standardized Classical Arabic (SCA) fully develops by the 8th century CE, less than two centuries after the coming of Islam, and there are only a small number of minor changes that are claimed to separate it from the Pre-Standardized Era. The period of Post-Classical Arabic (PCA) is dated to the fall of the Abbasid Empire at the hands of the Mongols (1258 CE), when the power of Arabic-using regimes in Iraq became decentralized, with power shifting elsewhere [58]. Ali suggests a slightly different timeline in which Arabic undergoes two ‘renaissances’ [5]. The first begins under the Abbasid Empire in Baghdad (750–1258), witnessing a huge growth in text production and translation of texts from Syriac and Greek. Following the end of the Abbasid Empire empire, Arabic experiences a period of decline until the 19th century renaissance that produces MSA. These models make slightly different predictions. Fischer’s model has SCA coming early and being distinguished primarily from pre-Islamic Arabic and very early texts. Ali’s model predicts significant changes during the Abbasid period, particularly an increase in vocabulary during this time. Fischer recognizes PCA as a development of the language, whereas Ali sees it as the end of a renaissance, and hence would not predict rapid change or growth in PCA. Both models treat the fall of the Abbasids in 1258 as a pivotal event in the history of Arabic, but it should be emphasized that this is a political change and only limited linguistic evidence has been shown to separate the two eras [44]. Within this work, we find strong support for the existence of MSA as separate from CA, but within CA the results are less definitive. There is very limited evidence for linguistic changes that occur shortly before the fall of the Abbasids. Other evidence supports the Abbasid renaissance model, with a break between the language of the earliest texts and those of the 4th century AH onward. ## 3 Related Work In this section we give an overview of existing Arabic corpora as well as some existing methods for text reuse identification, periodization, and dating. ### Arabic Corpora Though there has been increasing interest in compiling Arabic corpora in the past decade, very little work has been done on compiling historical corpora reflecting the long history of the Arabic language. Most of the existing corpora focus on modern written Arabic texts, particularly online news media, although there are a growing number of corpora which feature written and to a lesser degree spoken material from Arabic dialects. We mention here several relevant corpora and refer to other surveys for more details [3; 4; 66; 72]. To date, only a small number of diachronically oriented corpora of Arabic have been produced and made available. The King Saud University Corpus of Classical Arabic (KSUCCA) [6]² consists of approximately $50.6~{}M$ words from the first 4 Islamic centuries. It has been morphologically analyzed with the MADA tool [28; 29]. Almost all of the texts are derived from the Shamela website which is a major source of texts for OpenITI.³ Text metadata is given by century, so more granular buckets are not possible in the current state of this corpus. The Historical Arabic Corpus (HAC) [32] has about $45~{}M$ words from diverse time periods, with text data given by century, as well as automatic part-of-speech tagging information. Other Classical Arabic corpora that are worth mentioning include a $5~{}M$ word corpus [23], which does not seem to be publicly available, another $2.5~{}M$ word corpus [56],⁴ and Tashkeela, a $76~{}M$ word corpus of texts from the Shamela website.⁵ These corpora are either small or lack high-quality temporal metadata. [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:5][ENDFOOTNOTE] Finally, a few large corpora are available only via online search interfaces: KACST Arabic Corpus [4] has more than $700~{}M$ words, including around $16~{}M$ words from the beginning of the Islamic era. The Leeds Arabic Internet Corpus⁶ and the International Corpus of Arabic⁷ contain $300~{}M$ and $100~{}M$ words, respectively, but they include mostly modern texts. The well-known ArabiCorpus⁸ has more than $170~{}M$ words from diverse periods of time, and arTenTen [7] is a $5.8~{}G$ word Web corpus, with a sub-corpus of $115~{}M$ words available through Sketch Engine [38]. There is also CLAUDia [73], another Shamela-based corpus, but with added genre metadata; however, only a subset appears to be accessible via a Web interface.⁹ While these corpora are very large and may contain texts from different periods, they are not directly accessible and lack sufficient diachronic information. [FOOTNOTE:6][ENDFOOTNOTE] [FOOTNOTE:7][ENDFOOTNOTE] [FOOTNOTE:8][ENDFOOTNOTE] [FOOTNOTE:9][ENDFOOTNOTE] In contrast to previous resources, our corpus has fine-grained time information, it covers most of the history of the written Arabic language, and it is available for developing NLP applications or supporting digital humanities projects (cf. Section 4). ### Identification of Text Reuse A popular approach to text reuse addressed the problem in the context of domains such as newspaper texts and law bills [16; 67; 70]. A standard approach to approximate-matching tasks is the use of edit-distance measures such as Levenshtein Distance (e.g., [62]); however, such an approach is not efficient, particularly given a corpus of this size. More successful and efficient models rely on either word- or character-level $n$-grams to align similar chunks of text [37; 49]. Documents are broken down into overlapping sequences of relatively short $n$-grams ($\sim 4$ for words; $\sim 16$ for characters) and the resulting hashes are either indexed for search or compared pairwise in order to find collisions. Some text reuse detection models have been open-sourced, such as passim.¹⁰ A recent international challenge focused on text reuse detection in Arabic [10], but the reuse cases were artificially generated in order to allow for an objective evaluation. The approaches that addressed the task were mostly based on $n$-grams comparison as well, with some Arabic-specific preprocessing. An especially interesting study of real text reuse is [74], which detected quotations in the CLAUDia corpus (cf. Section 3.1) and built a network of documents based on metadata and quoted texts. However, their method focuses on long verbatim quotations, whereas we are interested in approximately-matching parallel passages with possible variations. Instead, we follow a recent approach for finding parallel passages across a large Hebrew/Aramaic corpus [65], adapt it to the Arabic language, and scale it up to handle the large corpus. [FOOTNOTE:10][ENDFOOTNOTE] We refer the reader to [15; 43; 67] for a broader overview of text reuse detection models and their applications. ### Periodization, Text Dating, and Language Change Most approaches to periodization, qualitative and quantitative, have either assumed standard periodizations proposed by traditional linguistics, or worked with pre-determined temporal bins (e.g. decades or centuries). Some recent works use clustering algorithms to determine more natural periodizations, but these approaches require pre-selected variables for classification. For example, in a study on automatic clustering of English, frequencies of get-passives and verb conjugation suffixes _-(e)th_ and _-(e)s_ were used as input for the clustering algorithm [26]. In a study on Chinese, the variables were already selected with an awareness of the history of Chinese morphosyntax, and the variables themselves were encoded into the annotated corpus [34]. Although these methods are promising, they require an existing sense of meaningful variables which vary diachronically, whereas we seek to periodize language without subscribing to pre-defined variables. There is a fairly large body of work on text dating, especially using clues like time expressions, but also various other features [13; 17; 51; 54]. Previous research operated at different granularity levels and algorithmic methods, including pairwise learning-to-rank, multi-class support vector machines, and language models [51; 54; 36]. These methods aim to learn to assign dates to undated documents. Our main motivation in this paper is different: given a corpus of dated texts, we seek to find a division into historical time periods. Another line of work has applied computational methods for studying diachronic language change. These methods learn the context in which words appear in large corpora, and define semantic change as a change to that context. Earlier attempts used this principle to detect meaning change in 19th century British English [61], or in a more modern corpus [69]. However, they tested semantic change for a very small set of words. In contrast, recent work has extended the scope of such analysis, and is now able to automatically detect words with significant semantic change for the entire vocabulary, building on word embeddings learned from large raw texts [22; 31; 39; 40]. We develop a periodization algorithm based on word embeddings to automatically cluster periods of similar language use. ## 4 Corpus Construction In this section we describe the OpenITI corpus as well as the preprocessing we carried out on it in order to perform our computational linguistic analyses. ### Text Collection The corpus we present in this work is the Arabic portion of the works collected by the Open Islamicate Texts Initiative (OpenITI), an on-going effort to collect texts in Arabic, Persian, and other languages.¹¹ The documents are collected from freely available editions of primarily religious and literary texts from different time periods. The main sources of these texts include Al-Maktaba Al-Shamela,¹² referred to here as Shamela, the Shia online library,¹³ and Al-Jami’ Al-Kabir (JK).¹⁴ The documents were converted into a unified format and organized into a machine-readable corpus, which is now openly available.¹⁵ Accompanying metadata information includes standardized author names, text title and author date of death.¹⁶ Though Shamela includes text genre data, it is not very reliable and so this was not retained. We use the author date of death as the document date throughout this work. Author lifespans average around 70 years, so there is a lag surrounding all of the data that needs to be taken into consideration during analysis (59, p. 239). All dates in the corpus are based on the Islamic calendar, which begins in 622 and which uses lunar years. Where reasonable, we have converted these dates to Gregorian years, but where they would produce unrounded bins and boundaries, we have retained the Islamic century numbering. [FOOTNOTE:11][ENDFOOTNOTE] [FOOTNOTE:12][ENDFOOTNOTE] [FOOTNOTE:13][ENDFOOTNOTE] [FOOTNOTE:14][ENDFOOTNOTE] [FOOTNOTE:15][ENDFOOTNOTE] [FOOTNOTE:16][ENDFOOTNOTE] \| \| Words \| Documents ---\|---\|--- OpenITI full \| 1.5 G \| 7,144 OpenITI core \| 725 M \| 4,322 (a) The current OpenITI corpus actually retains all duplicate texts. We refer to this as OpenITI full. De-duplication has systematically chosen an instantiated text for each unique work from the source corpora. We refer to this as OpenITI core. Table 3 shows some statistics of the two versions. After de-duplication, OpenITI core has 1106 texts (26%) from JK, 2375 (55%) from Shamela, and 823 (19%) from the Shia library; 8 texts are from elsewhere. The de-duplication was conducted in a semi-automatic manner.¹⁷ [FOOTNOTE:17][ENDFOOTNOTE] Table 4 shows detailed statistics on the OpenITI full corpus of texts, including the distribution of texts, sentences,¹⁸ and words over time as reflected in the corpus. As illustrated in Figure 5, we observe three major jumps in the number of documents: one in the early period, ca, 800 CE, one in the middle period, ca. 1400 CE, and one in the modern period, after 1800 CE. These mostly correspond to increases in the number of words as well. [FOOTNOTE:18][ENDFOOTNOTE] As is evident from the statistics, some time periods are more represented in the corpus than others. As in most historical corpora, guaranteeing representativeness is often not possible [14]. Therefore, we decided to keep all available texts rather than artificially selecting a balanced sub-set of them, in an effort to be as comprehensive as possible. Period (AH) \| Texts \| Sentences \| Words \| Period \| Texts \| Sentences \| Words ---\|---\|---\|---\|---\|---\|---\|--- 1–50 \| 43 \| 33K \| 462K \| 751–800 \| 407 \| 3,980K \| 112M 51–100 \| 17 \| 33K \| 546K \| 801–850 \| 226 \| 2,105K \| 47M 101–150 \| 48 \| 122K \| 3M \| 851–900 \| 240 \| 3,396K \| 98M 151–200 \| 120 \| 549K \| 11M \| 901–950 \| 288 \| 2,248K \| 55M 201–250 \| 325 \| 1,487K \| 39M \| 951–1000 \| 122 \| 1,536K \| 45M 251–300 \| 504 \| 1,779K \| 39M \| 1001–1050 \| 112 \| 858K \| 29M 301–350 \| 495 \| 2,539K \| 65M \| 1051–1100 \| 103 \| 1,533K \| 37M 351–400 \| 568 \| 2,850K \| 59M \| 1101–1150 \| 82 \| 1,713K \| 47M 401–450 \| 458 \| 2,075K \| 46M \| 1151–1200 \| 81 \| 482K \| 14M 451–500 \| 481 \| 3,350K \| 102M \| 1201–1250 \| 211 \| 2,666K \| 66M 501–550 \| 266 \| 1,919K \| 41M \| 1251–1300 \| 100 \| 1,004K \| 37M 551–600 \| 443 \| 3,392K \| 101M \| 1301–1350 \| 201 \| 1,713K \| 36M 601–650 \| 291 \| 2,361K \| 67M \| 1351–1400 \| 53 \| 1,537K \| 34M 651–700 \| 313 \| 2,216K \| 63M \| 1401–1450 \| 90 \| 2,714K \| 56M 701–750 \| 456 \| 9,597K \| 186M \| Total \| 7144 \| 62M \| 1,537M Table 4: Number of texts, sentences, and words in each 50-year time period in the OpenITI full corpus. AH refers to the Islamic calendar period. <figure><img src="content_image/1809.03891/itifulltextcountonly.png"><figcaption>(a) OpenITI full document count.</figcaption></figure> Comparison with ShamelaInitial findings of this work have been reported in the 2016 Workshop on Language Technology Resources and Tools for Digital Humanities (LT4DH) [9]. That work made use of texts from the Shamela text collection. This text collection has good coverage of pre-modern texts, and an excellent section of modern texts. However, many foundational texts, particularly texts which are more literary than religious, were absent from that collection. The current work is based on the Open Islamicate Texts Initiative, which integrates texts from a variety of text collections — including semi-automated selection of the ideal copy of each text where multiple copies are present. This has the advantage of gathering a greater coverage of the pre-modern era, such that OpenITI has $500$ more texts from the period before 1800 ($15~{}M$ more words). However, less effort has been made thus far to reintegrate modern texts into OpenITI, so Shamela contains $860$ more modern texts than OpenITI, totaling $77~{}M$ words. Ongoing initiatives exist to reintegrate the Shamela texts into OpenITI and to increase the number of texts, specifically texts from the early modern era (1800-1900) as that period is when MSA is said to have started developing. ### Preprocessing We used a combination of OpenNLP¹⁹ and the _Farasa_ toolkit [1]²⁰ to pre-process the corpus. Farasa is a popular toolkit for Arabic NLP which was found by multiple studies [19] to perform comparably to MADAMIRA [1; 53; 20], another popular toolkit. The two toolkits were developed mainly for MSA, but there is no known comprehensive evaluation of them on historical Arabic.²¹ We provide a small manual evaluation of Farasa on our corpus below, as well as compare the preprocessing results of Farasa and MADAMIRA quantitatively. Our evaluation demonstrates that there is little difference in their performance, and that both are viable choices for preprocessing historical Arabic. [FOOTNOTE:19][ENDFOOTNOTE] [FOOTNOTE:20][ENDFOOTNOTE] [FOOTNOTE:21][ENDFOOTNOTE] Next we describe our preprocessing pipeline.²² [FOOTNOTE:22][ENDFOOTNOTE] Sentence splitting.: This is the only component not available in Farasa. Punctuation is inconsistently used in Arabic and a stretch of text between periods may contain many sentences. We used OpenNLP and trained the sentence splitting model on $5~{}K$ sentences from the AQMAR Arabic Wikipedia Supersense corpus [63] and NIST’s MT06 corpus.²³ [FOOTNOTE:23][ENDFOOTNOTE] Morphological segmentation.: The segmenter breaks words into their underlying morphemes. Farasa’s segmenter uses SVM${}^{rank}$ [35] with a linear kernel to determine the best segmentation for each word. Lemmatization.: The lemmatiser is a rule-based system. First, a word is looked up in a word–{diacritization, lemma} dictionary and the lemma of the most frequent diacritized word is returned if located. Otherwise, the Farasa segmenter is applied to extract the word’s morphemes and the first non-prefix morpheme is looked up again in the dictionary. If found, the mapped lemma is returned; otherwise, the morpheme is returned. Part-of-speech tagging.: The POS tagger uses the simplified PATB tagset proposed by [18]. It attempts to find the optimal tag for each morpheme produced by the segmenter, as well as determining the gender (masculine or feminine) and number for nouns and adjectives (singular, dual, or plural). The POS tagger uses SVM${}^{Rank}$ to find the best tag for each morpheme as well. Constituency parsing.: This is a re-implementation of the Epic parser [30], which performed best at SPMRL 2013 [11]. It uses a conditional random fields model trained on features derived from the POS tagger. We utilize the different output formats produced by the preprocessing throughout this work, except for the parse trees. We still make the trees available to facilitate future syntactic work on the history of Arabic. Qualitative evaluationFarasa is tuned for the news domain and for MSA; still, “it can handle other genres along with classical and dialectal Arabic” [60]. We performed a qualitative analysis of the Farasa results to ensure that it still performed well with Classical Arabic texts. We analyzed the results of the segmentation, POS-tagging, and lemmatization in a set of texts chosen based on genre and date.²⁴ In all of the texts, the quality of the Farasa results was high, with relatively few errors in the segmentation and lemmatization, even for words that are obsolete or Classical. In a small sample chosen for manual quantitative analysis, comprising a total of 507 words in 4 texts, there were 41 mistakes of all kinds, with Farasa correctly segmenting 98.82%, lemmatizing 98.62% and POS-tagging 94.28% of words in this small sample. Classical coverage appears adequate --- for example, in a pre-Islamic poem,²⁵ the lemmatizer correctly derived the lemma _A$Abp_ ‘mixed group of people’ from the plural form _A$A}b_, though this word is almost never used in modern language. This suggests that Farasa’s lexicon has an adequate coverage of CA vocabulary in spite of being designed for MSA. On the other hand, the POS tagging occasionally produced incorrect gender information, and mislabeled verbs as nouns, though the lemmatization produced the correct form in most cases, and this is an ongoing issue even in MSA texts. One instance of a specifically CA structure that posed a difficulty for Farasa was the energetic form _l-yltms-n_ ‘he will seek out’, which Farasa treated as a single word in all analyses. The energetic is largely absent from MSA, and marginal in its use in CA. However, Farasa had some difficulty segmenting verbs, especially present-tense verbs, from conjunctions and other prefixes, so this may be a related issue. The output of Farasa is generally useful for a variety of research types, but is specifically useful for concordancing and word frequency counts, since removal of clitics and identification of lemmas poses a significant problem in Arabic computational linguistics. [FOOTNOTE:24][ENDFOOTNOTE] [FOOTNOTE:25][ENDFOOTNOTE] Farasa vs. MADAMIRABoth Farasa and MADAMIRA are high-quality toolkits for Arabic NLP. For periodization, we do not expect to see a large impact of the choice of tool, as we are interested in high-level trends, rather than obtaining a few more points in performance. Nevertheless, we provide here quantitative and qualitative results showing that the differences between the tools are minor and insignificant for our purposes. To test this, we ran the two toolkits on 58 texts from our corpus, one from each 25 year time period, and compared their results by computing the character edit distance for each preprocessed sentence. When comparing morphological segmentation, the average edit distance was about 20.1 edit operations, corresponding to 11.2% of the characters (when dividing by the total sentence length). After accounting for different normalization schemes, the average edit distance was about 8.9 operations, corresponding to just 5.9% of the characters. We also ran a similar evaluation for lemmatization and found even smaller differences between Farasa and MADAMIRA. On lemmatized texts, the average edit distance was 6.8 edit operations, which correspond to 5.4% of the characters. These results indicate that the choice of toolkits for preprocessing would not have a major effect on our analysis. More importantly, such small differences do not have a high impact in periodization, where we are concerned with global, overall trends, rather than with improving the state-of-the-art by some small fraction. ## 5 Text Reuse In order to identify instances of text reuse within the corpus, we adapt SKP [65], a recent approach designed to efficiently find approximately-parallel passages across a large Hebrew/Aramaic corpus. This method is appealing to use in our case due to the similarity between Arabic, Aramaic and Hebrew: all are languages of Semitic origin with a high morpheme-per-word ratio. First we detail two modifications that we had to apply before running the algorithm in order to handle the nature of the OpenITI corpus. We refer to this enhanced algorithm as SKP-Ar. The whole procedure is summarized in Algorithm 1. ### Identification of Boilerplate Passages Before running the processor-intensive algorithm to identify approximate matches within the corpus, we isolate “boilerplate” passages that recur verbatim dozens, hundreds, or even thousands of times within the corpus (e.g., Quranic verses are quoted extremely frequently). A common genre in the corpus is Hadith collections, Prophetic reports, which must be preceded with a chain of transmission from the person who heard it directly from the Prophet through the main person who transmitted it to the latter transmitter. Thus, both the chain of transmission and the quotation itself tend to be widely repeated. Others are simply frequently mentioned anecdotes or sayings. To identify boilerplate passages, overlapping phrases of length 20 are extracted and counted from the whole corpus (line 1) and those phrases which have appeared 25 or more times verbatim are marked (line 1). Phrases are clustered and conflated if they appear overlapped or juxtaposed (allowing for a gap of up to $10$ words between the $20$-gram segments) and the resulting text fragments are marked as boilerplate passages. These text fragments are ignored by the subsequent stages of the approximate-matching algorithm, allowing them to focus on the more meaningful parts of the text, without getting bogged down in these commonly-recurring exact matches. Appendix B shows examples of boilerplate passages identified in this step. ``` input : $Texts$: a list of $T$ documents, ordered chronologically output : $BoilerPlates$: the boiler plate fragments $ReusedFragments$: the reused fragments 1Function FindBoilerPlates($Texts$) 2 Initialize dictionary $Phrases(phrase,counter)$for $i\gets 1$to$T$ do 3 for $j\gets 1$to$J_{i}\mid\|j\|=20,step=1$ do 4 $Phrases\gets j$ 5 end for 6 7 end for 8 $Phrases\leftarrow[\forall phrase\in Phrases\mid freq(phrase)\geq 25]$$BoilerPlates\leftarrow$Conflate($Phrases$)returnBoilerplates 9 10Function FindFrequentPhrases($Texts$) 11 Initialize dictionary $Phrases(phrase,counter)$for $i\gets 1$to$T$ do 12 for $j\gets 1$to$J_{i}\mid\|j\|=4,step=1$ do 13 $Phrases\gets j$ 14 end for 15 16 end for 17 $Phrases\leftarrow$SortByFrequency($Phrases$)$ExtremelyFrequentPhrases\leftarrow$Top($Phrases,35000$)returnExtremelyFrequentPhrases 18 19Function IdentifyReuse($Texts$) 20 $BoilerPlates\leftarrow$FindBoilerPlates($Texts$)$Texts^{\prime}\leftarrow[Texts\setminus BoilerPlates]$$XtremFreqPhrases\leftarrow$FindFrequentPhrases($Texts^{\prime}$)$Text^{\prime\prime}\leftarrow$ConflateFreqPhrasesIntoSingleWords($Texts^{\prime},XtremFreqPhrases$)$SkipGrams\leftarrow$ComputeSkipGrams($Texts^{\prime\prime}$)$SkipGramsHashes\leftarrow$HashSkipGrams($SkipGrams$) Initialize list $ReusedFragments$for $i\gets 1$to$T$ do 21 for $j\gets 1$to$T$ do 22 $BaseMatches\leftarrow$GetMatchingHashes($T_{i},T_{j},SkipGramHashes$)$FullMatches\leftarrow$ExtendMatches($BaseMatches$)Append($ReusedFragments,FullMatches$) 23 end for 24 25 end for 26 returnBoilerPlates, ReusedFragments 27 ``` Algorithm 1The SKP-Ar algorithm for text reuse identification. ### Identification of Extremely Frequent Short Phrases The second step involves the identification of frequently recurring $4$-grams. The core of the SKP algorithm involves hashing and indexing $4$-grams throughout the whole corpus. This works well for Hebrew and Aramaic, since frequently-recurring Hebrew and Aramaic formulaic phrases tend to be limited to two (sometimes three) words, and thus $4$-grams prove to be effective units in processing the text. However, within the OpenITI corpus, we find many $4$-grams that recur repeatedly. These are often blessings upon the Prophet Muhammad, which are extremely frequent (e.g., SlY Allh Elyh wslm, “peace be upon him”, recurs over $5~{}M$ times). The prevalence of such phrases could render SKP [65] highly ineffective. In order to solve this problem, we identify the top $35~{}K$ frequently-recurring $4$-grams in the corpus (function FindFrequentPhrases). We assign each of these phrases a unique 16-bit hash, and henceforth treat each of those phrases as a single word unit, each one with its own unique hash. When launched on the OpenITI corpus, the selected phrases appeared between 515 and $5~{}M$ times each. ### Identification of Approximate Matches The function IdentifyReuse details the main steps in the SKP-Ar algorithm. As an intermediate step towards finding long parallel passages, we first want to identify all cases of approximately-matching short phrases between the documents. For our purposes, we regard an approximately-matched short phrase as cases of phrases of length five in which four out of the five words are nearly identical. We define a skipgram as any four-word subset of a five-word string. For every 4-out-of-5 skipgram within the text (excluding the boilerplate material, as described above), we assign a 64-bit hash, comprised of the two least-frequent letters of each of the four words (line 1 in Algorithm 1). An initial pass of the program reviews the entire corpus and builds a character frequency table of all characters classified as Arabic characters by the Unicode specification. The corpus includes 131 such characters; thus any two-letter combination fits easily into 16-bits, and also allows space for the $35~{}K$ unique hashes for the frequently-recurring $4$-gram phrases detailed in the previous step. The determination of the two least-frequent letters in a given word is also based upon this initial review of the character inventory within the corpus. This method conveniently facilitates approximate matches. The use of 4-out-of-5 skipgrams allows passages to match up even though a given word may be subtracted, added, or replaced within the unit. Similarly, the two-letter word hashing allows words to be considered equal despite differences in prefixes, suffixes, or matres lectionis. Now, for any given document (the “base document”), we tabulate all cases in which one of its skipgram hashes matches a skipgram hash from another document (the “target document”) (line 1). We wish to identify cases in which multiple skipgram matches are in close proximity with one another to form a passage of substantial length. To do so, we generate a two-dimensional graph, wherein each skipgram match is plotted on one axis according to the starting word position in the base text, and on the other axis according to the starting word position in the target text, similar to a dotplot [8; 27]. We are interested in the cases in which multiple skipgram matches cluster on a more-or-less diagonal line on the graph. To efficiently find such cases, we bin the skipgrams based upon the difference between their two coordinates. We review the bins which contain multiple skipgrams and consider whether those skipgrams can cluster together to form a match containing 16 or more identical words, allowing up to 3 non-matching word positions in between any two matching skipgrams (line 1). The use of 4-out-of-5 64-bit hashes casts a rather wide net from the start, wherein many identical skipgrams actually point to very different phrases. However, this is compensated by the requirement to have a series of adjacent matching skipgrams. As the number of adjacent skipgrams cluster together, the number of false positives drops progressively lower. In our case, where we require a series of matching skipgrams which match up at a minimum of 16 word positions, we find that the resulting passages are virtually always legitimate cases of text reuse. ### Results We ran the SKP-Ar algorithm on the entire $1.5~{}G$ word OpenITI corpus. The algorithm first isolated 230,530 unique (though possibly overlapping) frequently-occurring $20$-word phrases (boilerplate strings). Each phrase occurs at least $25$ times within the corpus, with the most frequent phrase occuring 4,495 times. In total, we mark 28,491,859 words out of the total $1.5~{}G$ words as boilerplate text. After eliminating the boilerplate text, the approximate-matching algorithm returned $76~{}M$ pairwise matches, with an average length of $46.7\pm 249$ words per match. The process took 48 hours, running in parallel on 64 CPUs. Given that the texts are dated by author’s date of death, we allowed some relaxation in finding reused text chunks. We counted only cases in which at least $50$ years elapsed between the dates of the earlier and and latter document. After this filter, we are left with $57~{}M$ pairwise matches, with an average length of $31.58\pm 35.12$. Note that this filter also eliminates duplicate texts that have the same date. Finally, we calculated the total number of reused words within the corpus, leveraging both the set of boilerplate phrases as well as the set of approximately-matching passages. After applying the 50-year filter, we were left with a total of $292~{}M$ reused words within the corpus. Efforts have been carried out to standardize the evaluation of text reuse models [55], but they rely on artificially-generated cases of reuse. Evaluating the performance of a model when applied to a real-life corpus, such as OpenITI, remains a difficult task. Most challenging is evaluating the exhaustiveness of matching. We expect to find extremely important texts to be quoted the most, so the number of matches by text are indicative of whether matches are being correctly identified. Indeed, the largest numbers of boilerplate quotations, i.e. quotations repeated 25 times or more, come from the Quran and major works of religious exegesis and historical works, all of which we expect to see widely quoted. Manual examination of the approximate pairwise matches is also positive — though not quantifiable, rapid scrolling through the lists of results provides easy visual identification of the similarity of the matches, and closer evaluation reveals slight variations in the quotations, often slight reformulation of the phrasing. Visually checking several lists of matches by file did not reveal any obviously flawed matches. ## 6 Automatic Periodization With access to a diachronic corpus, we are able to investigate the linguistic developments which have been claimed to characterize the different stages of CA. We developed an automatic algorithm for dividing a historical text corpus into time periods. We then applied it to the OpenITI corpus and analyzed the obtained results.²⁶ [FOOTNOTE:26][ENDFOOTNOTE] ### Word-Embedding-based Neighbor Clustering Given the nature of language change, we note that the language in two consecutive time periods should in principle be more similar than the language in two remote time periods. Therefore, we apply a chronologically-constrained hierarchical clustering algorithm that is only allowed to merge consecutive time periods. The core of the algorithm is based on a word-embedding function and a distance function. The word-embedding function takes a text document and generates a word-embedding matrix. The distance function takes two word-embedding matrices, corresponding to two time periods, and computes the distance between them. Below we discuss specific instantiations of these functions. Our algorithm can be seen as a word-embedding-based variant of the Variability-based Neighbor Clustering (VNC) algorithm for periodization [26], where we replace the measure of variability by a distance measure based on word-embedding matrices. Word embeddings are attractive to use for this purpose because they provide a soft notion of language use, with similar words having similar vectors in the word embedding space [47]. We name our periodization algorithm Word-Embedding-based Neighbor Clustering (WENC). We assume a collection of texts, $\mathcal{T}$, with known dates. We first bin the texts into initial time periods $\mathcal{P}=\{P_{1},...,P_{\|\mathcal{P}\|}\}$, that are ordered chronologically (e.g., centuries). That is, for each $i<j$, all the texts in $P_{i}$ are dated earlier than all the texts in $P_{j}$. The texts in each time period are concatenated into documents that are input to the periodization algorithm (see Algorithm 2). We start by training initial word embedding models (line 2). Then, we iteratively look for the next best possible merge of time periods until there are no more time periods to merge (line 2). At each iteration we record the best merges and distances (lines 2-2). The algorithm utilizes a function FindBestMerge that takes a collection of documents and their corresponding word embedding models and computes the distances between each consecutive pair of documents (line 2). It then finds the best pair (line 2), concatenates the two documents (line 2), trains a word embedding model on the new concatenated document (line 2), and updates the list of texts and models (lines 2-2). ``` input : $Texts$: a list of $T$ documents, ordered chronologically output : $MergedPairs$: the merged clusters $MergedDistances$: the corresponding distances 1Function Periodize($Texts$) 2 Initialize list $Models$for $i\gets 1$to$T$ do 3 $Models[i]$$\leftarrow$TrainWordEmbModel($Texts[i]$) 4 end for 5 Initialize lists $MergedPairs$, $MergedDistances$while $\|Texts\|>1$ do 6 $BestPair$, $BestDist$, $Texts$, $Models$$\leftarrow$FindBestMerge($Texts$, $Models$)Append($MergedPairs$, $BestPair$)Append($MergedDistances$, $BestDistance$) 7 end while 8 return$MergedPairs$, $MergedDistances$ 9 10Function FindBestMerge($Texts$, $Models$) 11 Initialize list $Distances$for $i\gets 1$to$\|Texts\|-1$ do 12 $Distances[i]\leftarrow$ComputeDistance($Texts[i]$, $Texts[i+1]$) 13 end for 14 $BestPair$, $BestDistance$$\leftarrow$ArgMin($Distances$)$MergedText\leftarrow$Concat($Texts[BestPair]$)$MergedModel\leftarrow$TrainWordEmbModel($MergedText$)$Texts\leftarrow$UpdateList($Texts$, $MergedText$)$Models\leftarrow$UpdateList($Models$, $MergedModel$)return$BestPair$, $BestDistance$, $Texts$, $Models$ 15 ``` Algorithm 2WENC: word-embedding-based neighbor clustering for automatic periodization. #### 6.1.1 Word Embeddings and a Distance Measure The periodization algorithm relies on training word embeddings on texts in each time period (line 2 in Algorithm 2). We obtain the word embeddings using Word2Vec [45; 46; 48] as implemented in gensim[57]. Specifically, we train the CBOW algorithm with negative sampling and the following default settings defined in gensim: word embedding dimensionality of 100, 5 negative samples, and a window size of 5 words. Given two word embedding matrices, $W_{1}$ and $W_{2}$, trained on different corpora, we need to define a distance measure between them. One option could be to directly calculate the distance with respect to some norm: \[\texttt{ComputeDistance}(W_{1},W_{2})=\|\|W_{1}-W_{2}\|\|\] (1) However, since the two word embedding matrices are trained independently, we have no guarantee that this distance measure would yield meaningful results. Moreover, the stochastic nature of the word embedding training algorithm precludes a direct comparison of words from different embedding models. To avoid this problem, we follow [31] and align the two matrices using orthogonal Procrustes: \[\texttt{ComputeDistance}(W_{1},W_{2})=\min_{Q:Q^{T}Q=I}\|\|QW_{1}-W_{2}\|\|_{F}\] (2) where $\|\|\cdot\|\|_{F}$ is the Frobenius norm of matrix $\cdot$ and $Q$ is an orthogonal matrix that rotates the word embedding matrix $W_{1}$ towards $W_{2}$. The solution to this minimization problem is given by the best rotation matrix and can be found with singular value decomposition (SVD) [64]: \[R=\operatorname{arg\,min}_{Q:Q^{T}Q=I}\|\|QW_{1}-W_{2}\|\|_{F}=UV^{T}\] (3) where $W_{2}W_{1}^{T}=U\Sigma V^{T}$ is the SVD decomposition. ### Experiments For the periodization experiments, we consider two possible initial time divisions: 50 and 100 year bins.²⁷ Table 4 shows the number of texts, sentences, and words, for each 50-year bin. The 100-year bins are simply a concatenation of each two consecutive 50-year bins. We find that the very early time periods contain much less text, so we merge the bins for the first 200 years in all of the experiments (2 and 4 bins are merged in the case of the 100-year and 50-year bins, respectively). In the following, we investigate the effect of preprocessing on periodization, by considering two input formats for the periodization algorithm: plain text and lemmas. Working with lemmas allows the periodization algorithm to focus more on lexical properties rather than surface forms. However, the morphological lemmatization performed by Farasa (Section 4.2) is an automatic process that may produce errors, so we also run the periodization algorithm on plain text. [FOOTNOTE:27][ENDFOOTNOTE] We also investigate the effect of text reuse by comparing the periodization results on the full corpus to running the same algorithm on a version of the corpus where reused text chunks were removed. In all cases, we run the WENC periodization algorithm (Algorithm 2), record the hierarchical merges and their distances, and plot the results in dendrograms. <figure><img src="content_image/1809.03891/dendrogram_sent_100.png"><figcaption>(a) 100-year bins, plain.</figcaption></figure> Figures (a)a and (b)b show the results of running WENC on 100-year time periods, using plain and lemmatized texts, respectively. In both cases, we see a split into three main periods: early period until 200/300 AH, middle period from 200/300 AH to around 1300 AH, and a late period from that time to modern days. In the case of 50-year bins (Figures (c)c and (d)d), we can observe a more fine-grained periodization. The two figures are very different: the periodization based on plain texts leads again to three large time periods, more or less corresponding to the results with 100-year bins. The periodization based on lemmatized texts exhibits a very large middle period, with comparatively shorter early and late periods. The reason may be that some of the differences between time periods are not as strong when abstracting over the word forms and working with lemmas. Interestingly, the algorithm does not always show the split between texts before and after 1300 AH (1882 CE) that should represent the CA and MSA boundary as it is normally portrayed in the literature. However, all but Figure (c)c seem to point towards a differentiation between a modern and pre-modern period. The algorithm should in principle abstract over genre effects by comparing only shared word embeddings between time periods. However, word embeddings are trained based on their contexts which in turn are influenced by genre. Therefore it should be noted that a consistent grouping which splits the earliest bin from later bins could be due in part to genre effects. The 200 year bin includes all texts from the 1st century, which is dominated by poetic collections. Texts which have the title “diwan”, meaning a poetic collection (some poetic collections may have other titles), are primarily found in the first several centuries. Of the 148 distinct works in the first two centuries, 67 or 44% are diwans, but by the year 300, only 14% of distinct works are diwans and only 3.4% of all books in the corpus are diwans. Most of the corpus is prose and may differ significantly from poetry in style and use of words in specific contexts. In interpreting the results of the periodization, we should be cautious due to the interaction between genre and text date in the corpus. #### 6.2.1 Text Reuse and Periodization The detection of periods of language change can be affected by quotation and general reuse of text chunks from early periods. The text reuse algorithm detects such texts and so we use it here to remove all cases of reuse. The “un-reused” or “hollowed” corpus is obtained by removing, for each detected match, all later cases of reuse, while keeping the earliest instance. Figure 13 shows the resuts of running WENC on the hollowed corpus in 50-year and 100-year bins (showing the plain text version). The results are fairly similar to the periodization results on the full corpus (Figure 10), but there appears to be a clearer separation around 800-900AH, especially in the 100-year bins case. Thus, after removing instances of text reuse, we find a clearer division into different pre-modern time periods. <figure><img src="content_image/1809.03891/dendrogram_sent_100_noreuse.png"><figcaption>(a) 100-year bins, “hollowed”</figcaption></figure> The separation of the modern period from the pre-modern period becomes more evident once textual quotations are removed, with the 1400s CE (1979 CE to present) clearly separated, and in Figure (b)b, the more expected differentiation of 1250 AH (1770 CE) separating from the previous period. This is right at the beginning of the period in which MSA developed. With 100-year bins, 1400 AH would represent authors who died between 1300 AH (1882 CE) and 1979 CE, i.e. a group born around the 1820s to the early 20th century. These represent the second or third generation of modernizing authors. That this would become somewhat more evident in the “hollowed” corpus is logical, as the texts in the corpus are inherently conservative and religious in nature, quoting heavily from earlier works. Removing those quotations makes the modern sections of these texts more salient to the algorithm. Overall the automatic periodization separates out the earliest and latest texts (200-300 and 1400-1450, sometimes including 1350) from a core bin that occupies the periods roughly between 400 AH (1009 CE) and 1300 AH (1882 CE). The first bin appears to correspond to Fischer’s PSCA era, containing a large quantity of pre-Islamic and early Islamic poetic texts, while the core bin is CA proper. MSA comes surprisingly late, even assuming a 70-year author lifespan. There is some support in the periodizations for the claimed change in the language due to the end of the Abbasid empire in 1258 CE (656 AH), with a cluster break clearest in the “hollowed” periodizations. A surprising result which requires further research is the consistent break around 900 AH (1494 CE). Though the literature has not considered this as a transition period, it does corresponds to the end of the Islamic state in the Iberian peninsula (1492 CE), and the rise of the Ottoman empire (conquest of Syria and Egypt during 1510s CE). ## 7 Expert Periodization of Arabic The corpus also supports less automated approaches to periodization that still rely on computational tools. These approaches are particularly useful for assessing the validity of periodizations based on impressionistic analyses of Arabic from earlier publications. ### The Lifespan of Arabic Words It is possible that the impression of SA as unchanging is actually a myth rather than reality, that SA actually changes just as quickly as other written languages. In order to investigate this question we use the corpus to check whether there is a quantitative difference in the development of Arabic writing and other languages for which historical corpora are available. To do this, we track the “lifespan” of Arabic and English words in two corpora: OpenITI core and the Corpus of Historical American English (COHA) [21]. We used lemmas rather than words in both cases. For OpenITI, we use the lemmatization provided by Farasa, but since Farasa does not reject non-words, we use MADAMIRA [53] to discard lemmas that are not actually Arabic words (incorrectly spelled words, etc.). For every lemma in the corpus, we find its first and last chronological usages. We discard words which occur only once or in a single year, which may be misspellings (often the case in COHA), or which have apparently short lifespans since they occur in the very last year/decade (in COHA) in the corpus. <figure><img src="content_image/1809.03891/lifereuselemAfinal.png"><figcaption>(a) OpenITI core.</figcaption></figure> Figure 18 shows the difference between Arabic and English word lifespans. The comparison reveals that Arabic words tend to have a very long life span in comparison with English words. In OpenITI the average Arabic word lifespan is 1,190 years (see figures for further descriptive statistics), approximately $83\%$ of the time span of the entire corpus. In COHA, the average English word lifespan is 88 years, about $45\%$ of the overall time span of the 190 year corpus. Of course, COHA is a much narrower corpus, only covering 200 years, from 1810-2000s. To control for the 200 year span of COHA versus the much longer span of our corpus, we also ran the same analysis on the set of all thirteen 200 year spans of the corpus (in 100-year increments, with one 225 years span for the final bucket). All lifespan analyses exhibited a similar trend, with long left-tailed distributions; Figure (c)c shows an example for the last time span. Morover, under this analysis, the mean lifespan of words was 161 years, nearly double COHA’s average word lifespan. These results confirm that Arabic words tend to have longer lifespans than English ones. Arabic words do not disappear as quickly from the lexicon as English words do, suggesting that indeed Arabic does change less quickly than English. This provides quantitative confirmation of an earlier qualitative observation: while Arabic-speaking schoolchildren can read a text from 750 CE, an English-speaking middle school student would have a much more difficult time reading Beowulf, an Old English poem from ca. 1000 CE. It also means that any measures of change in Arabic will need to be more sensitive than measures used for English in order to produce a meaningful result. It is possible that the apparently long lifespan of these words is due to extensive quotation of texts including archaic words. In a sense, this does not really matter from the perspective of a language user, since they still must be able to understand the quotation regardless (footnotes may be provided to help with this in some texts, but not all.) To determine the extent that quotation influences word lifespan in Arabic, we ran the same lifespan measurements on the “hollowed” version of the lemmatized corpus with reuse removed. The results are shown in Figure (b)b, and differ very little from the results on the normal lemmatized corpus. This shows that the long lifespans of Arabic words are clearly not the result of quotation alone. ### New Words Over Time Another claim in the literature is that the number of new words increased significantly during the development of MSA, as new terms were needed to refer to European technology and ideas suddenly becoming available in the Arab world due to colonization and modernization efforts [50]. We can use the corpus to confirm that this is indeed accurate, and to investigate whether there are other periods when an increase occurred in the number of new vocabulary items. <figure><img src="content_image/1809.03891/newlemmasallA.png"><figcaption>(a) OpenITI.</figcaption></figure> Figure (a)a shows that, as expected, new lemmas in Arabic developed very rapidly, a corollary of the long lifespan of Arabic words. Compare this to the English results which show a steadier rise (Figure (b)b). Zooming into the modern era in Figure (c)c, we see that fully $1\%$ of Arabic words were added just in the 20th century (really earlier, since these are date-of-death measures), a much more rapid increase than in the previous two centuries, which added slightly more than $0.5\%$ of lemmas to the lexicon. There appears to be another period of rapid growth at the beginning of the second millennium, with a large jump in vocabulary between 1000 and 1100, and again a jump starting in 1150, but they are not as clear as the increase in the modern period (Figure (d)d). Some caution is needed as a single author or work could easily cause these jumps in the vocabulary, but this is an interesting result: the account of Arabic history that claims a decline in the language following the end of the Abbasid Empire empire would predict that vocabulary addition would level at this point, rather than increasing. On the other hand, the breakdown of central authority might decrease standardization of vocabulary and increase diversity in specialized terminology. ### Verifying Previous Periodizations Pre-Standardized Classical Arabic (PSCA) does have distinguishing linguistic features, though these are largely found in the Quran or in rare poetic attestations. Moreover, most of these are slight differences in assimilation or vocalization, and would not show up in the written text in an easily distinguishable way. One of the few testable claims is that prior to the 8th century CE, the formation of abstract conceptual nouns was done via a phrase, _ElY jhp Al-_ ‘from the perspective of’, but was replaced with a suffix _-yp_ ‘-ity, -ness’ in the period of Standardized Classical Arabic (SCA) [5]. Two suggested phrases based on the _ElY jhp Al-_ ‘from the perspective of’ structure are _ElY jhp Al-xyr_ ‘charity, goodness’ and _ElY jhp Al-Edl_ ‘justice, fairness’. A concordance search on the lemmatized corpus finds these structures are basically unattested in the data with a total of less than 30 attestations, all of which post-date the 8th century CE. Nor is this structure attested in Arabic papyri via the Arabic Papyrology Database.²⁸ It is therefore unclear where this claim originates since it is so poorly supported by the data.²⁹ [FOOTNOTE:28][ENDFOOTNOTE] [FOOTNOTE:29][ENDFOOTNOTE] More testable features are claimed for the break between SCA and PCA. These include the use of the adverb _AyDA_ ‘also’³⁰, and the development of adjectives that show a suffix _-Any_ such as _jsmAny_ ‘bodily’ and _rwHAny_ ‘spiritual’ [25]. These are relatively insignificant changes, but at least they can now be verified or disproved using OpenITI. [FOOTNOTE:30][ENDFOOTNOTE] <figure><img src="content_image/1809.03891/Wordfreqs.png"><figcaption>Figure 24: Relative frequencies of words and suffixes, with actual frequenciesand LOESS smoothed lines.</figcaption></figure> Using the segmentation produced by Farasa to remove extraneous clitics (Section 4.2), we are able to investigate whether these claims are accurate. Figure 24 shows the relative frequencies for the word _AyDA_, and the combined frequencies for the words _jsmAny_ and _rwHAny_, as well as several other words suggested in the literature, which use an abstracting suffix _-yp_.³¹ For the word _AyDA_, there is no relationship to different eras, with this word functioning as an adverb even in very early texts. In OpenITI our earliest clear attestation is in a text with author DOD of 68 HA (688 CE)³² It increases in usage over time, but shows no periods of significantly greater growth. It is worth noting that its use was judged negatively, with Ibn al-Sikkit’s (d. 858 CE) dictionary providing precise rejoinders to mock anyone who uses the word in its adverbial meaning as ‘also’ [41]. This establishes that the ‘also’ meaning is attested at least by the 9th century, but also that this term was undergoing some form of change for it to be subject to meta-linguistic judgment. However, this significantly predates the claimed PCA development date, nor is there a discernible increase in its frequency at that time. [FOOTNOTE:31][ENDFOOTNOTE] [FOOTNOTE:32][ENDFOOTNOTE] The other two variables do correspond roughly to the claimed PCA era, but increases in their usage actually seem to precede the Mongol conquests in 1258 CE — since all figures here are dates of death, clearly use of these suffixes was on the rise well before that time, and recent research suggests that the Mongols simply finished off a process of decline already underway in Iraq and Iran during that era [58]. The two variables do show remarkable similarity in their frequencies over time, suggesting that the increase in their usage does reflect a change in linguistic eras, even if it begins prior to 1258 CE. ## 8 Conclusions The Arabic language has a long and diverse history spanning more than 1400 years. The written Arabic language, Standard Arabic (SA), is often thought to be a more or less monolithic language with little change before the modern period. In this work, we process OpenITI, a large-scale diachronic corpus of Arabic, and investigate the question of periodization in different ways. We identify instances of text reuse in the corpus, develop an automatic periodization algorithm, and investigate existing claims about the periodization of Arabic. We find that although words do persist relatively longer in Arabic than in English, there is evidence for several distinct periods in the language’s development. OpenITI represents the largest publicly available diachronic corpus of Arabic to date. We preprocessed the corpus to make it more amenable to natural language processing, and the results of this are also available for use. The nature of the corpus is such that texts frequently quote from one another, and using an efficient algorithm we were able to identify $292~{}M$ words of reused text, nearly $20\%$ of the total corpus. We were able to produce a “hollowed” version of the corpus with this data removed, both for the plain-text and preprocessed versions of the corpus. This corpus and the tools we develop allow us to answer open questions about the history of Arabic. The corpus allows us to establish that Arabic vocabulary does indeed change more slowly over time than in English. The automatic periodization algorithm we develop confirms established periodizations of Arabic, while suggesting new ones. It shows that the oldest periods of Arabic and the most modern ones are both separate from a core period stretching from approximately 400 CE (1009 CE) until 1300 AH (1882 CE), reflecting the prototypical Classical Arabic (CA). The data from the automatic periodization and from the evidence of new words in Arabic both strongly support established periodizations that divide Modern Standard Arabic (MSA) from CA. Both automatic periodization and computational evaluation of established periodizations provide some support for a break between an early and later Classical Arabic around the 1000s CE, which is somewhat earlier than typically believed, and suggests an additional previously unexplored division in the late 15th century CE. The processed corpus and the code associated with this work are available to the research community. We hope that future work will illuminate other aspects of the history of the Arabic language, as well as utilize the methods proposed in this work for studying other languages. ## References (1) Abdelali, A., Darwish, K., Durrani, N., Mubarak, H.: Farasa: A Fast and Furious Segmenter for Arabic. In: Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations, pp. 11–16. Association for Computational Linguistics (2016). DOI 10.18653/v1/N16-3003. URL http://aclanthology.coli.uni-saarland.de/pdf/N/N16/N16-3003.pdf * (2) Al-Jallad, A.: An outline of the grammar of the Safaitic inscriptions. No. 80 in Studies in Semitic languages and linguistics. Brill (2015) * (3) Al-Sulaiti, L.: Designing and Developing a Corpus of Contemporary Arabic. Master’s thesis, The University of Leeds, Leeds, UK (2004) * (4) Al-Thubaity, A.O.: A 700M+ Arabic Corpus: KACST Arabic Corpus Design and Construction. Lang. Resour. Eval. 49(3), 721–751 (2015) * (5) Ali, A.S.M.: A linguistic study of the development of scientific vocabulary in Standard Arabic. Keegan Paul International (1987) * (6) Alrabiah, M., Al-Salman, A., Atwell, E.: The design and construction of the 50 million words KSUCCA. In: Proceedings of WACL’2 Second Workshop on Arabic Corpus Linguistics, pp. 5–8 (2013) * (7) Arts, T., Belinkov, Y., Habash, N., Kilgarriff, A., Suchomel, V.: arTenTen: Arabic Corpus and Word Sketches. Journal of King Saud University - Computer and Information Sciences 26(4), 357 – 371 (2014). Special Issue on Arabic NLP * (8) Basile, C., Benedetto, D., Caglioti, G., Degli Esposti, M.: A Plagiarism Detection Procedure in Three Steps: Selection, Matches and Squares. In: Stein et al. [68], pp. 19–23. Http://ceur-ws.org/Vol-502 * (9) Belinkov, Y., Magidow, A., Romanov, M., Shmidman, A., Koppel, M.: Shamela: A large-scale historical arabic corpus. In: Proceedings of the Workshop on Language Technology Resources and Tools for Digital Humanities (LT4DH at Coling), pp. 45–53. The COLING 2016 Organizing Committee, Osaka, Japan (2016) * (10) Bensalem, I., Boukhalfa, I., Rosso, P., Lahsen, A., Darwish, K., Chikhi, S.: Overview of the AraPlagDet PAN@ FIRE2015 Shared Task on Arabic Plagiarism Detection. In: Notebook Papers of FIRE 2015 (CEUR-WS vol. 1587), pp. 111–122. Gandhinagar, India (2015) * (11) Björkelund, A., Çetinoğlu, Ö., Farkas, R., Mueller, T., Seeker, W.: (Re) Ranking Meets Morphosyntax: State-of-the-Art Results from the SPMRL 2013 Shared Task pp. 135––145 (2013) * (12) Braschler, M., Harman, D. (eds.): Notebook Papers of CLEF 2010 LABs and Workshops. Padua, Italy (2010) * (13) Chambers, N.: Labeling Documents with Timestamps: Learning from their Time Expressions. In: Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 98–106. Jeju Island, Korea (2012) * (14) Claridge, C.: Historical corpora. Corpus linguistics: An international handbook 1, 242–259 (2008) * (15) Clough, P., Gaizauskas, R.: Corpora and Text Re-Use. In: A. Lüdeling, M. Kytö, T. McEnery (eds.) Handbook of Corpus Linguistics, Handbooks of Linguistics and Communication Science, pp. 1249—1271. Mouton de Gruyter (2009) * (16) Clough, P., Gaizauskas, R., Piao, S., Wilks, Y.: Measuring Text Reuse. In: Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics (ACL 2002), pp. 152–159. Association for Computational Linguistics, Philadelphia, PA (2002) * (17) Dalli, A., Wilks, Y.: Automatic Dating of Documents and Temporal Text Classification. In: Proceedings of the Workshop on Annotating and Reasoning about Time and Events, pp. 17–22. Sydney, Australia (2006) * (18) Darwish, K., Abdelali, A., Mubarak, H.: Using Stem-Templates to Improve Arabic POS and Gender/Number Tagging. In: Proceedings of the Ninth International Conference on Language Resources and Evaluation (LREC’14), LREC 2014, pp. 2926–2931. European Language Resources Association (ELRA) (2014). URL http://www.lrec-conf.org/proceedings/lrec2014/pdf/335_Paper.pdf * (19) Darwish, K., Mubarak, H.: Farasa: A New Fast and Accurate Arabic Word Segmenter. In: LREC (2016) * (20) Darwish, K., Mubarak, H., Abdelali, A., Eldesouki, M.: Arabic POS Tagging: Don’t Abandon Feature Engineering Just Yet. In: Proceedings of the Third Arabic Natural Language Processing Workshop, pp. 130–137. Association for Computational Linguistics (2017). URL http://aclweb.org/anthology/W17-1316 * (21) Davies, M.: The Corpus of Historical American English: 400 million words, 1810-2009. http://corpus.byu.edu/coha (2010) * (22) Dubossarsky, H., Tsvetkov, Y., Dyer, C., Grossman, E.: A bottom up approach to category mapping and meaning change. In: V. Pirrelli, C. Marzi, M. Ferro (eds.) Word Structure and Word Usage. Proceedings of the NetWordS Final Conference (2015) * (23) Elewa, A.H.: Collocation and Synonymy in Classical Arabic: A Corpus-based Approach. Ph.D. thesis, The University of Manchester, Manchester, UK (2004) * (24) Ferrando, I.: History of Arabic. In: K. Versteegh (ed.) Encyclopedia of Arabic Language and Linguistics, vol. 2, pp. 604–611. Brill, Leiden (2007) * (25) Fischer, W.: Classical Arabic. In: K. Versteegh (ed.) Encyclopedia of Arabic Language and Lingusitics, vol. 1, pp. 397–405. Brill, Leiden (2006) * (26) Gries, S.T., Hilpert, M.: Variability-based Neighbor Clustering: A bottom-up approach to periodization in historical linguistics. In: T. Nevalainen, E.C. Traugott (eds.) The Oxford Handbook of the History of English, pp. 134–144. Oxford University Press, Oxford (2012) * (27) Grozea, C., Popescu, M.: The Encoplot Similarity Measure for Automatic Detection of Plagiarism - Notebook for PAN at CLEF 2011. In: V. Petras, P. Forner, P. Clough (eds.) Notebook Papers of CLEF 2011 LABs and Workshops. Amsterdam, The Netherlands (2011) * (28) Habash, N., Rambow, O.: Arabic Tokenization, Part-of-Speech Tagging and Morphological Disambiguation in One Fell Swoop. In: Proceedings of ACL (2005) * (29) Habash, N., Rambow, O., Roth, R.: MADA+TOKAN: A Toolkit for Arabic Tokenization, Diacritization, Morphological Disambiguation, POS Tagging, Stemming and Lemmatization. In: Proceedings of the Second International Conference on Arabic Language Resources and Tools (2009) * (30) Hall, D.L.W., Durrett, G., Klein, D.: Less Grammar, More Features. In: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing, pp. 228–237 (2014) * (31) Hamilton, W.L., Leskovec, J., Jurafsky, D.: Diachronic Word Embeddings Reveal Statistical Laws of Semantic Change. In: Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 1489–1501. Association for Computational Linguistics (2016). DOI 10.18653/v1/P16-1141. URL http://aclanthology.coli.uni-saarland.de/pdf/P/P16/P16-1141.pdf * (32) Hammo, B., Yagi, S., Ismail, O., AbuShariah, M.: Exploring and exploiting a historical corpus for Arabic. Language Resources and Evaluation 50(4), 839–861 (2016). DOI 10.1007/s10579-015-9304-9. URL https://doi.org/10.1007/s10579-015-9304-9 * (33) Holes, C.: Modern Arabic: Structures, Functions, and Varieties. Georgetown University Press, Washington, D.C. (2004) * (34) Ji, M.: A corpus-based study of lexical periodization in historical Chinese. Literary and Linguistic Computing 25(2), 199–213 (2010) * (35) Joachims, T.: Training Linear SVMs in Linear Time. KDD ’06, pp. 217–226. ACM, New York, NY (2006) * (36) de Jong, F., Rode, H., Hiemstra, D.: Temporal language models for the disclosure of historical text. Royal Netherlands Academy of Arts and Sciences (2005) * (37) Kasprzak, J., Brandejs, M.: Improving the Reliability of the Plagiarism Detection System. Lab Report for PAN at CLEF 2010. In: Braschler and Harman [12] * (38) Kilgarriff, A., Rychly, P., Smrz, P., Tugwell, D.: The Sketch Engine. In: Proceedings of EURALEX (2004) * (39) Kim, Y., Chiu, Y.I., Hanaki, K., Hegde, D., Petrov, S.: Temporal Analysis of Language through Neural Language Models. In: Proceedings of the ACL 2014 Workshop on Language Technologies and Computational Social Science, pp. 61–65. Association for Computational Linguistics, Baltimore, MD, USA (2014). URL http://www.aclweb.org/anthology/W14-2517 * (40) Kulkarni, V., Al-Rfou, R., Perozzi, B., Skiena, S.: Statistically Significant Detection of Linguistic Change. In: Proceedings of the 24th International World Wide Web Conference, WWW ’15 (2015) * (41) Lane, E.W.: Arabic-English Lexicon. Willams & Norgate (1863) * (42) Lentin, J.: Middle Arabic. In: K. Versteegh (ed.) Encyclopedia of Arabic Language and Linguistics, vol. 1, online edition edn., pp. 87–96. Brill, Leiden (2006) * (43) Li, W.P.: Language Technologies for Understanding Law, Politics, and Public Policy. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (2016) * (44) Magidow, A.: A Digital Philological Investigation of the History of _hā hunā_ Constructions. Romano-Arabica 16, 239–256 (2016) * (45) Mikolov, T., Chen, K., Corrado, G., Dean, J.: Efficient Estimation of Word Representations in Vector Space. CoRR abs/1301.3781 (2013). URL http://arxiv.org/abs/1301.3781 * (46) Mikolov, T., Sutskever, I., Chen, K., Corrado, G., Dean, J.: Distributed Representations of Words and Phrases and their Compositionality. CoRR abs/1310.4546 (2013). URL http://arxiv.org/abs/1310.4546 * (47) Mikolov, T., tau Yih, W., Zweig, G.: Linguistic Regularities in Continuous Space Word Representations. In: Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (NAACL-HLT-2013). Association for Computational Linguistics (2013). URL http://research.microsoft.com/apps/pubs/default.aspx?id=189726 * (48) Mikolov, T., Yih, W., Zweig, G.: Linguistic Regularities in Continuous Space Word Representations. In: Human Language Technologies: Conference of the North American Chapter of the Association of Computational Linguistics, Proceedings, June 9-14, 2013, Westin Peachtree Plaza Hotel, Atlanta, Georgia, USA, pp. 746–751 (2013). URL http://aclweb.org/anthology/N/N13/N13-1090.pdf * (49) Muhr, M., Kern, R., Zechner, M., Granitzer, M.: External and Intrinsic Plagiarism Detection using a Cross-Lingual Retrieval and Segmentation System. In: Braschler and Harman [12] * (50) Newman, D.L.: The Arabic Literary Language: the nahḍa and beyond. In: J. Owens (ed.) The Oxford Handbook of Arabic Linguistics, pp. 472–494. Oxford University Press, Oxford (2013) * (51) Niculae, V., Zampieri, M., Dinu, L., Ciobanu, A.M.: Temporal Text Ranking and Automatic Dating of Texts. In: Proceedings of the 14th Conference of the European Chapter of the Association for Computational Linguistics, volume 2: Short Papers, pp. 17–21. Gothenburg, Sweden (2014). URL http://www.aclweb.org/anthology/E14-4004 * (52) Osama Hamed, T.Z.: A Survey and Comparative Study of Arabic Diacritization Tools. JLCL 32(1), 27–47 (2017) * (53) Pasha, A., Al-Badrashiny, M., Diab, M., Kholy, A.E., Eskander, R., Habash, N., Pooleery, M., Rambow, O., Roth, R.: MADAMIRA: A Fast, Comprehensive Tool for Morphological Analysis and Disambiguation of Arabic. In: Proceedings of the Ninth International Conference on Language Resources and Evaluation (LREC’14), pp. 1094–1101. Reykjavik, Iceland (2014). URL http://www.lrec-conf.org/proceedings/lrec2014/pdf/593_Paper.pdf * (54) Popescu, O., Strapparava, C.: SemEval 2015, Task 7: Diachronic Text Evaluation. In: Proceedings of the 9th International Workshop on Semantic Evaluation (SemEval 2015), pp. 870–878. Denver, Colorado (2015). URL http://www.aclweb.org/anthology/S15-2147 * (55) Potthast, M., Stein, B., Barrón-Cedeño, A., Rosso, P.: An Evaluation Framework for Plagiarism Detection. In: C.R. Huang, D. Jurafsky (eds.) Proceedings of the 23rd International Conference on Computational Linguistics (COLING 2010), pp. 997–1005. COLING 2010 Organizing Committee, Beijing, China (2010) * (56) Rashwan, M.A., Al-Badrashiny, M.A., Attia, M., Abdou, S.M., Rafea, A.: A Stochastic Arabic Diacritizer Based on a Hybrid of Factorized and Unfactorized Textual Features. Trans. Audio, Speech and Lang. Proc. 19(1), 166–175 (2011). DOI 10.1109/TASL.2010.2045240. URL http://dx.doi.org/10.1109/TASL.2010.2045240 * (57) Rehurek, R., Sojka, P.: Software Framework for Topic Modelling with Large Corpora. In: Proceedings of the LREC 2010 Workshop on New Challenges for NLP Frameworks, pp. 45–50. ELRA, Valletta, Malta (2010) * (58) Romanov, M.: Algorithmic Analysis of Medieval Arabic Biographical Collections. Speculum 92(S1), S226–S246 (2017) * (59) Romanov, M.G.: Computational Reading of Arabic Biographical Collections with Special Reference to Preaching in the Sunni World (661–1300 CE). Ph.D. thesis, University of Michigan, Ann Arbor, MI, USA (2013) * (60) Romeo, S., Da San Martino, G., Belinkov, Y., Barrón-Cedeño, A., Eldesouki, M., Darwish, K., Mubarak, H., Glass, J., Moschitti, A.: Language processing and learning models for community question answering in arabic. Information Processing & Management (2017). DOI https://doi.org/10.1016/j.ipm.2017.07.003 * (61) Sagi, E., Kaufmann, S., Clark, B.: Semantic Density Analysis : Comparing word meaning across time and phonetic space. In: Proceedings of the EACL 2009 Workshop on GEMS: Geometrical Models of Natural Language Semantics, pp. 104–111 (2009) * (62) Scherbinin, V., Butakov, S.: Using Microsoft SQL Server Platform for Plagiarism Detection. In: Stein et al. [68], pp. 36–37. Http://ceur-ws.org/Vol-502 * (63) Schneider, N., Mohit, B., Oflazer, K., Smith, N.A.: Coarse Lexical Semantic Annotation with Supersenses: An Arabic Case Study. In: Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics: Short Papers - Volume 2, ACL ’12, pp. 253–258. Association for Computational Linguistics, Stroudsburg, PA (2012). URL http://dl.acm.org/citation.cfm?id=2390665.2390726 * (64) Schönemann, P.H.: A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966). DOI 10.1007/BF02289451. URL https://doi.org/10.1007/BF02289451 * (65) Shmidman, A., Koppel, M., Porat, E.: Identification of Parallel Passages Across a Large Hebrew/Aramaic Corpus. arXiv preprint arXiv:1602.08715 (2016) * (66) Shoufan, A., Alameri, S.: Natural Language Processing for Dialectical Arabic: A Survey. In: Proceedings of the Second Workshop on Arabic Natural Language Processing, pp. 36–48. Beijing, China (2015). URL http://www.aclweb.org/anthology/W15-3205 * (67) Smith, D.A., Cordell, R., Dillon, E.M., Stramp, N., Wilkerson, J.: Detecting and Modeling Local Text Reuse. In: Proceedings of the 14th ACM/IEEE-CS Joint Conference on Digital Libraries, JCDL ’14, pp. 183–192. London, United Kingdom (2014). URL http://dl.acm.org/citation.cfm?id=2740769.2740800 * (68) Stein, B., Rosso, P., Stamatatos, E., Koppel, M., Agirre, E. (eds.): SEPLN 2009 Workshop on Uncovering Plagiarism, Authorship, and Social Software Misuse (PAN 2009), vol. 502. CEUR-WS.org, San Sebastian, Spain (2009). Http://ceur-ws.org/Vol-502 * (69) Wijaya, D.T., Yeniterzi, R.: Understanding semantic change of words over centuries. Proceedings of the 2011 international workshop on DETecting and Exploiting Cultural diversiTy on the social web - DETECT ’11 p. 35 (2011). DOI 10.1145/2064448.2064475. URL http://dl.acm.org/citation.cfm?doid=2064448.2064475 * (70) Wilkerson, J., Smith, D., Stramp, N.: Tracing the Flow of Policy Ideas in Legislatures: A Text Reuse Approach. American Journal of Political Science 59(4), 943–956 (2015). DOI 10.1111/ajps.12175. URL http://dx.doi.org/10.1111/ajps.12175 * (71) Zack, E., Schippers, A. (eds.): Middle Arabic and Mixed Arabic: Diachrony and Synchrony. Brill Academic Publishers, Leiden (2012) * (72) Zaghouani, W.: Critical Survey of the Freely Available Arabic Corpora. In: Proceedings of the Workshop on Free/Open-Source Arabic Corpora and Corpora Processing Tools (2014) * (73) Zemánek, P., Milička, J.: Ranking Search Results for Arabic Diachronic Corpora. Google-like search engine for (non)linguists. In: Proceedings of CITALA 2014 (5th International Conference on Arabic Language Processing). Association for Computational Linguistics (2014) * (74) Zemánek, P., Milička, J.: Quotations, Relevance and Time Depth: Medieval Arabic Literature in Grids and Networks. In: Proceedings of the 3rd Workshop on Computational Linguistics for Literature (CLFL), pp. 17–24. Gothenburg, Sweden (2014). URL http://www.aclweb.org/anthology/W14-0903 ## Appendix A Mathematical Notation \begin{tabular}{l l} \hline \hline Symbol & Meaning \\ \hline $K$ / $M$ / $G$ & $10^{3}$ / $10^{6}$ / $10^{9}$ \\ $\mathcal{T}$ & Collection of texts \\ $\mathcal{P}$ & List of texts organized in time periods \\ $P_{i}$ & Texts belonging to time period $i$ \\ $W_{1}$, $W_{2}$ & Word embedding matrices \\ $Q$, $U$, $V$, $\Sigma$ & Matrices \\ $I$ & Identity matrix \\ $Q^{T}$ & Transpose of $Q$ \\ $\|\|W\|\|$ & Matrix norm \\ $\|\|W\|\|_{F}$ & Frobenius norm \\ \hline \hline \end{tabular} ## Appendix B Arabic examples Boiler Plate MatchesThe following examples illustrate the boilerplate passages identified in the first step of the text reuse algorithm (Section 5). The first is a chain of transmission that occurs 2,747 times in the corpus. The second is a part of a Hadith that occurs 221 times. ``` Hdvny mHmd bn Emrw qAl vnA >bw EASm qAl vnA EysY wHdvny AlHArv qAl vnA AlHsn qAl vnA wrqA’ jmyEA En Abn >by njyH En mjAhd ``` _Muhammad bin Amer told me, saying, “Abu Asim told me that Isa told him that Al-Harith said, ‘Al-Hasan spoke to us, saying “Warqa told all of us, based on what the son of Abi Najih said, quoting Mujahid”’”_ ``` mn sy}At >EmAlnA mn yhdh Allh flA mDl lh wmn yDll flA hAdy lh w>$hd >n lA <lh <lA Allh wHdh lA $ryk lh w>$hd >n mHmdA Ebdh wrswlh ``` _[I seek refuge in god] from the evil of our deeds; he who God guides rightly cannot go astray; he who goes astray cannot be led aright; I witness that there is no God but God alone with no partner, and that Muhammad is His servant and His Prophet_ Near-matchesThe next example is of a near-match with several differences. The first is the original, from 0292Yacqubi.TarikhYacqubi.Shia003468Vols-ara1: ``` lys Tlby llElm TmEA fy blwg qASyth wAlAsjylA’ ElY gAyth wlkn Altms $y}A lA ysE jhlh wlA yHsn bAlEAql. ``` _My search for knowledge is not greed to reach its utmost, or to seize its aim. Rather, I seek something of which ignorance is widespread, and which the wise dare not contradict._ As identified in a later text (1371MuhsinCamili.AcyanShica.Shia003636Vols), differences between forward slashes: ``` lys Tlby llElm TmEA fy blwg qASyth wAlAsjylA’ ElY /nhAyth/ wlkn /mErfp ma/ lA ysE jhlh wlA yHsn bAlEAql. ``` _My search for knowledge is not greed to reach its end, or to seize its aim. Rather, knowledge of that of which ignorance is widespread and which the wise dare not contradict._
1703.02499	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 59420, "num_imgs": 10, "llama3_tokens_count": 17498 }	[ "content_image/1703.02499/x1.png", "content_image/1703.02499/x2.png", "content_image/1703.02499/x3.png", "content_image/1703.02499/x5.png", "content_image/1703.02499/x7.png", "content_image/1703.02499/x9.png", "content_image/1703.02499/x10.png", "content_image/1703.02499/x11.png", "content_image/1703.02499/x12.png", "content_image/1703.02499/x14.png" ]	# URV Factorization with Random Orthogonal System Mixing Stephen Becker, James Folberth, Laura Grigori ###### Abstract The unpivoted and pivoted Householder QR factorizations are ubiquitous in numerical linear algebra. A difficulty with pivoted Householder QR is the communication bottleneck introduced by pivoting. In this paper we propose using random orthogonal systems to quickly mix together the columns of a matrix before computing an _unpivoted_ QR factorization. This method computes a URV factorization which forgoes expensive pivoted QR steps in exchange for mixing in advance, followed by a cheaper, unpivoted QR factorization. The mixing step typically reduces the variability of the column norms, and in certain experiments, allows us to compute an accurate factorization where a plain, unpivoted QR performs poorly. We experiment with linear least-squares, rank-revealing factorizations, and the QLP approximation, and conclude that our randomized URV factorization behaves comparably to a similar randomized rank-revealing URV factorization, but at a fraction of the computational cost. Our experiments provide evidence that our proposed factorization might be rank-revealing with high probability. ## 1 Introduction The QR factorization of a matrix $A\in\mathbb{R}^{m\times n}$ is a widely used decomposition, with applications in least-squares solutions to linear systems of equations, eigenvalue and singular value problems, and identification of an orthonormal basis of the range of $A$. The form of the decomposition is $A=QR$, where $Q$ is $m\times m$ and orthogonal and $R$ is $m\times n$ and upper triangular. When $A$ is dense and has no special structure, Householder reflections are often preferred to Gram-Schmidt (and its variants) and Givens rotations, due to their precise orthogonality and computational efficiency via the (compact) WY representation [18, 7, 27], which can utilize level-3 BLAS. Indeed, Householder QR with a compact WY representation is implemented in the LAPACK routine _geqrf[1]. A common variant of the QR factorization is column pivoted QR, which computes the factorization $A\Pi=QR$, where $\Pi$ is a permutation matrix. At the $i$th stage of the decomposition, the column of the submatrix $A(i:m,i:n)$ (in matlab notation) with the largest norm is permuted to the leading position of $A(i:m,i:n)$ and then a standard QR step is taken. The LAPACK routine _geqp3 implements column pivoted Householder QR using level-3 BLAS [1]. However, it is typically much slower than the unpivoted _geqrf, as _geqp3 still suffers from high communication costs [11] and cannot be cast entirely in level-3 operations [24]. We refer to Householder QR without pivoting as unpivoted QR (QR), and Householder QR with column pivoting as QRCP. Improving on QRCP, recent works have used random projections to select blocks of pivots, emulating the behaviour of QRCP, while more fully utilizing level-3 BLAS [10, 24]. Another approach uses so called “tournament pivoting” to select blocks of pivots and is shown to minimize communication up to polylogarithmic factors [11]. In each of these cases, a pivoted QR factorization is produced. URV factorizations decompose $A$ as $A=URV$, where $U$ and $V$ have orthonormal columns and $R$ is upper triangular. One can think of URV factorizations as a relaxation of the SVD, where instead of a diagonal singular value matrix, we require only that $R$ is upper-triangular. Similarly, QRCP can be thought of as a URV factorization where $V$ is a permutation matrix, a special orthogonal matrix. In Section 3 we discuss how URV factorizations can be used to solve linear least-squares problems in much the same manner as QR factorizations or the SVD. For example, let $V$ be a random orthogonal matrix sampled from the Haar distribution on orthogonal matrices. The matrices $U$ and $R$ are computed with an _unpivoted_ QR factorization of $\hat{A}=AV^{T}$, and the resulting URV factorization is a strong rank-revealing factorization with high probability (see Subsection 2.1) [9]; we call this randomized factorization RURV_Haar. This demonstrates that one can forego column pivoting at the cost of mixing together the columns of $A$ and still have a safe factorization. However, taking $V$ to be a random, dense orthogonal matrix is not terribly computationally efficient, as $V$ is generated with an $n\times n$ unpivoted QR and must be applied with dense matrix multiplication. We propose mixing with an alternating product of orthogonal Fourier-like matrices (e.g., discrete cosine, Hadamard, or Hartley transforms) and diagonal matrices with random $\pm 1$ entries, forming a so-called random orthogonal system (ROS) [2, 29, 21, 25]. This provides mixing, but with a fast transform, as $V$ is never formed explicitly and can be applied with the FFT, or FFT-style algorithms (see Subsection 2.2). We call this randomized URV factorization with ROS mixing RURV_ROS. Numerical experiments with our implementation of RURV_ROS demonstrate that for large matrices (i.e., where communication is the bottleneck of QRCP), RURV_ROS runs slightly slower than _geqrf and significantly faster than _geqp3. Figure 1 shows the average runtimes of dgeqrf, dgeqp3, and RURV_ROS. We used MATLAB’s LAPACK [22] and the reference FFTW [15] with 1 and 16 threads on a desktop workstation with two Intel® Xeon® E5-2630 v3 CPUs running at $2.4$ GHz. See Subsection 2.2 for more details on our implementation of RURV_ROS. Around $n=1000$, we begin to see a sharp increase in the runtime of dgeqp3, owing to the communication bottleneck of column pivoting. In this region, dgeqp3 with 16 threads does not see an appreciable improvement over running just a single thread. In contrast, dgeqrf parallelizes much more nicely, as we can see an order of magnitude improvement in runtime when using 16 threads. When using RURV_ROS, we also see a noticeable improvement in runtime when using 16 threads versus 1 thread. We also run timing and accuracy experiments on over- and underdetermined linear least-squares problems in Section 3. In Subsection 4.1 we sample the rank-revealing conditions of [17, 16] for a variety of QR and URV factorizations, which suggest that RURV_ROS behaves similarly to RURV_Haar. This provides evidence suggesting that RURV_ROS is rank-revealing with high probability. We also examine using RURV_Haar and RURV_ROS in a QLP approximation to the SVD in Subsection 4.3. <figure><img src="content_image/1703.02499/x1.png"><figcaption>Figure 1: Average runtimes over five runs of dgeqrf, dgeqp3, and RURV_ROS onslightly tall-skinny matrices (n=m/2). Note that we do not include the time togenerate the orthogonal factor Q (labelled U for RURV_ROS), as all routineswould use dorgqr. For the run with 16 threads, the sharp increase in runtimesbeginning around size 2000×1000 matrices corresponds to the beginning of theregime where communication is the bottleneck of QRCP.</figcaption></figure> ## 2 Randomized URV Factorization ### Randomized URV Factorization via Haar Random Orthogonal Mixing Demmel et al. proposed in [9] a randomized URV factorization (RURV), which we call RURV_Haar, to use as part of eigenvalue and singular value decompositions. Their RURV of an $m\times n$ matrix $A$ is based on sampling from the Haar distribution on the set of orthogonal (or unitary) matrices [23], using that sampled matrix to mix the columns of $A$, and then performing an _unpivoted_ QR on the mixed $A$, resulting in the factorization $A=URV$. ``` 1:$A\in\mathbb{R}^{m\times n}$ 2:$U,R,V$ 3:Generate a random $n\times n$ matrix $B$ whose entries are i.i.d. $N(0,1)$. 4:$[V,\hat{R}]=\texttt{qr}(B)$$\triangleright$$V$ is Haar distributed; $\hat{R}$ is unused 5:$\hat{A}=AV^{T}$ 6:$[U,R]=\texttt{qr}(\hat{A})$ ``` Algorithm 1RURV_Haar - Randomized URV with Haar mixing from [9] Haar orthogonal matrices are known to smooth the entries of the vectors on which they operate. By multiplying $A$ on the right by a Haar orthogonal matrix $V^{T}$, we can mix together the columns of $A$, and reduce the variance of the column norms (see Figure 2). The intuition behind the mixing is that by reducing the variance of the column norms, we reduce the effect that column pivoting would have, and can get away with unpivoted QR. Indeed, in [9] it is shown that Algorithm 1 produces a rank-revealing factorization with high probability, and can be used for eigenvalue and SVD problems. It was further shown that Algorithm 1 produces a _strong_ rank-revealing factorization in [4]. Criteria for a (strong) rank-revealing factorization of the form $A=URV$ are as follows (taken from [17, 16], but slightly weaker conditions were used in [4]): 1. $U$ and $V$ are orthogonal and $R=\begin{bmatrix}R_{11}&R_{12}\\ 0&R_{22}\end{bmatrix}$ is upper-triangular, with $R_{11}$$k\times k$ and $R_{22}$$(n-k)\times(n-k)$; 2. For any $1\leq i\leq k$ and $1\leq j\leq\min(m,n)-k$, \[1\leq\dfrac{\sigma_{i}(A)}{\sigma_{i}(R_{11})},\dfrac{\sigma_{j}(R_{22})}{ \sigma_{k+j}(A)}\leq q(k,n),\] (1) where $q(k,n)$ is a low-degree polynomial in $k$ and $n$. 3. In addition, if \[\\|R_{11}^{-1}R_{12}\\|_{2}\] (2) is bounded by a low-degree polynomial in $n$, then the rank-revealing factorization is called strong. These conditions state that the singular values of $R_{11}$ and $R_{12}$ are not too far away from the respective singular values of $A$. Thus, by performing a rank-revealing factorization instead of an expensive SVD, we can still gain insight into the singular values of $A$. Both QR factorizations in Algorithm 1 are _unpivoted_, and thus can be considerably cheaper than the standard column-pivoted Householder QR, QRCP. However, a major drawback is the expense of generating and applying the random matrix $V$. To sample an $n\times n$ matrix $V$ from the Haar distribution on orthogonal matrices, we take the $Q$ factor from an unpivoted QR factorization of an $n\times n$ matrix $B$ whose entries are i.i.d. $N(0,1)$[23]. The dominant cost of this computation is the unpivoted QR factorization, which requires $\mathcal{O}(n^{3})$ FLOPs. We then compute $\hat{A}=AV^{T}$, which requires $\mathcal{O}(mn^{2})$ FLOPs, followed by the unpivoted QR factorization to find $U$ and $R$, which costs $\mathcal{O}(mn^{2})$ FLOPs. To reduce the cost of forming and applying $V$, we propose replacing $V$ with a product of random orthogonal systems, which can each be applied implicitly and quickly, although providing slightly worse mixing. ### Randomized URV Factorization via Fast Random Orthogonal Mixing Consider a real $m\times n$ matrix $A$ and a product of random orthogonal systems (ROS) of the form \[V=\Pi\left[\prod_{i=1}^{N}FD_{i}\right],\] (3) where each $D_{i}$ is a diagonal matrix of independent, uniformly random $\pm 1$ and $F$ is an orthogonal Fourier-like matrix with a fast transform. Just like in RURV_Haar, we mix together the columns of $A$ as $\hat{A}=AV^{T}$. The matrix $\Pi$ is a permutation matrix chosen so $\hat{A}\Pi^{T}$ sorts the columns of $\hat{A}$ in order of decreasing norm. Replacing the Haar matrix $V$ in Algorithm 1 with the ROS based $V$ in (3) yields the new algorithm we call RURV_ROS, shown in Algorithm 2. ``` 1:$A\in\mathbb{R}^{m\times n}$, number of mixing steps $N$, $\{D_{i}\}_{i=1}^{N}$ diagonal $\pm 1$ matrices 2:$U,R$$\triangleright$ The $V$ matrix is not output because it is never explicitly formed 3:$\hat{A}=A\prod_{i=N}^{1}(D_{i}F^{T})$ 4:$\hat{A}=\hat{A}\Pi^{T}$$\triangleright$ Sort the columns of $\hat{A}$ so they are in order of decreasing $\ell_{2}$ norm. 5:$[U,R]=\texttt{qr}(\hat{A})$ 6:$V=\Pi\prod_{i=1}^{N}FD_{i}$ ``` Algorithm 2RURV_ROS - Randomized URV with ROS mixing Each product $FD_{i}$ is referred to as a random orthogonal system (ROS) [2, 29, 21, 25]. Examples of real-to-real, orthogonal Fourier-like transforms are the discrete cosine transform (e.g., DCT-II and DCT-III), the discrete Hartley transform, and the discrete Hadamard transform. The Fourier-like matrix is never explicitly constructed, but rather is only used as an operator, for which we use a fast transform. This brings the FLOP count for computing $AV^{T}$ from $\mathcal{O}(mn^{2},n^{3})$ to $\mathcal{O}(mn\log n)$. In our experiments, we use the DCT-II and DCT-III for $F$ and $F^{T}$, as implemented in FFTW [15]. Figure 2 shows the effect of mixing with Haar matrices and ROS on the column norms of a random $250\times 250$ matrix $A$, formed in matlab with A = bsxfun(@times, randn(m,n)+exp(10rand(m,n)), exp(2rand(1,n))), followed by A = A/mean(sqrt(sum(A.A))), so that the mean column norm is one. The variance of the column norms is clearly decreased by the mixing, and notably, Haar and ROS (with $N=1$) affect the distribution of column norms in a similar manner. A theme of this paper is that RURV_ROS behaves similarly to RURV_Haar, which likely stems from their similar effect on the distribution of column norms. <figure><img src="content_image/1703.02499/x2.png"><figcaption>Figure 2: Mixing columns of A together with Haar orthogonal matrices and ROSreduces the variance of column norms, while keeping the mean column norm aboutthe same.</figcaption></figure> To mix together the columns of $A$, we compute $\hat{A}=A\prod_{i=N}^{1}(D_{i}F^{T})$. The permutation/pre-sort matrix $\Pi$ is chosen so the columns of $\hat{A}\Pi^{T}$ are sorted in decreasing order of column norm. The pre-sort is included to potentially enhance the accuracy and stability of RURV_ROS. The cost of this one-time, single sort is much smaller than the cost of the repeated column pivots in QRCP. A matlab implementation of RURV_ROS with $F$ taken to be the DCT-II is shown in Listing 1. For in-core computations, it is sometimes more efficient to compute the mixing on left of $A^{T}$ via: \[AV^{T}=(VA^{T})^{T}=\left(\Pi\prod_{i=1}^{N}\left(FD_{i}\right)A^{T}\right)^{T}.\] This “transpose trick” is used in Listing 1 for efficiency, and also to cleanly interface with matlab’s dct function, which applies the transform to the columns of its input. Listing 1 explicitly returns $U$ and $R$ from the factorization, but returns function handles for $V$ and $V^{T}$, which can be used to apply $V$ and $V^{T}$, respectively, to the left side of their input. The implementation used for our experiment is similar, but has performance-critical sections written in C using matlab’s MEX interface. The mixing step is performed in C using FFTW and the unpivoted QR is performed in C using LAPACK routines from matlab’s LAPACK [15, 1, 22]. The use of FFTW gives us great control over how the transform is applied (e.g., in blocks, multithreaded, perhaps not utilizing the “transpose trick”, etc.). More details on the use of FFTW for mixing are given in Subsection 3.2. ``` function [U,R,V,Vt] = RURV_ROS(A, n_its) % RURV_ROS RURV with ROS mixing for real matrices [m,n] = size(A); D_diags = sign(rand(n,n_its)-0.5); % diagonals of D_i; i.i.d. uniform +- 1 % ROS mixing Ahat = AV’ using transpose trick Ahat = apply_V(A’,D_diags)’; ␣␣␣%␣pre-sort ␣␣␣nrms␣=␣sqrt(sum(Ahat.^2,1)); ␣␣␣[nrms,p]␣=␣sort(nrms,␣2,␣’descend’); ␣␣␣p_inv(p)␣=␣1:numel(p); ␣␣␣Ahat␣=␣Ahat(:,p); ␣␣␣%␣unpivoted␣QR␣factorization ␣␣␣[U,R]␣=␣qr(Ahat,0); ␣␣␣%␣Return␣function␣handles␣to␣apply␣V␣and␣V^T␣on␣the␣left ␣␣␣V␣=␣@(A)␣apply_V(A,D_diags,p); ␣␣␣Vt␣=␣@(A)␣apply_Vt(A,D_diags,p_inv); end function␣[Ahat]␣=␣apply_V(A,␣D_diags,␣p) %␣apply_V␣␣Apply␣ROS␣mixing:␣VA ␣␣␣Ahat␣=␣A; ␣␣␣for␣i=1:size(D_diags,2) ␣␣␣␣␣␣Ahat␣=␣bsxfun(@times,␣Ahat,␣D_diags(:,i));␣%␣Ahat␣=␣D_iAhat ␣␣␣␣␣␣Ahat␣=␣dct(Ahat);␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣%␣Ahat␣=␣FAhat; ␣␣␣end ␣␣␣if␣nargin␣==␣3␣%␣apply␣sorting ␣␣␣␣␣␣Ahat␣=␣Ahat(p,:); ␣␣␣end end function␣[Ahat]␣=␣apply_Vt(A,␣D_diags,␣p_inv) %␣apply_Vt␣␣Apply␣transpose␣ROS␣mixing:␣V^TA ␣␣␣if␣nargin␣==␣3␣%␣apply␣sorting ␣␣␣␣␣␣Ahat␣=␣A(p_inv,:); ␣␣␣else ␣␣␣␣␣␣Ahat␣=␣A; ␣␣␣end ␣␣␣for␣i=size(D_diags,2):-1:1 ␣␣␣␣␣␣Ahat␣=␣idct(Ahat);␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣␣%␣Ahat␣=␣F^TAhat; ␣␣␣␣␣␣Ahat␣=␣bsxfun(@times,␣Ahat,␣D_diags(:,i));␣%␣Ahat␣=␣D_iAhat ␣␣␣end end ``` Listing 1: A matlab implementation of RURV_ROS ## 3 Applications to Least-Squares Problems ### Solving Least-Squares Problems with a URV Factorization A URV factorization can be used to solve least-squares problems in much the same manner as a QR factorization. Throughout this subsection we assume that $A$ is $m\times n$ and full-rank. We are interested in finding a solution to \[\min_{x}\\|Ax-b\\|_{2}\] for both the overdetermined case $m\geq n$ and the underdetermined case $m<n$. #### 3.1.1 Overdetermined Systems Consider first the case when $A$ is overdetermined. To find the least-squares solution with a QR factorization, we only need a thin QR factorization, where $Q$ is $m\times n$ and $R$ is $n\times n$[18]. Similarly, the internal QR factorization in RURV_ROS can be a thin QR. By computing $A=URV$ and using that $U$ has orthonormal columns, \[\min_{x}\\|Ax-b\\|_{2}=\min_{x}\\|URVx-b\\|_{2}=\min_{x}\\|RVx-U^{T}b\\|.\] The least-squares problem reduces to the non-singular $n\times n$ upper-triangular system $Ry=U^{T}b$ in the auxiliary variable $y=Vx$. The system $Ry=U^{T}b$ is solved implicitly for $y$ with backward substitution, and then the least-squares solution is found with $x=V^{T}y$. Note that we do not need to explicitly form $U$ to apply $U^{T}$ to $b$. When we call LAPACK’s _geqrf on $\hat{A}=AV^{T}$, the routine overwrites the upper-triangular part of $\hat{A}$ with $R$ and the Householder reflectors in the strictly lower triangular part of $\hat{A}$. By feeding the Householder reflectors into _ormqr, we can implicitly compute $U^{T}b$ in $\mathcal{O}(mn)$ FLOPs without ever accumulating $U$[1]. The dominant cost of using RURV_Haar to compute least-squares solutions is a mix of generating $V$, computing $\hat{A}=AV^{T}$, and the thin QR to find $U$ and $R$. The latter two operations cost $\mathcal{O}(mn^{2})$ FLOPs. The dominant cost of using RURV_ROS is also $\mathcal{O}(mn^{2})$, but the leading cost term only comes from the unpivoted QR, as the mixing $\hat{A}=AV^{T}$ is $\mathcal{O}(mn\log n)$. #### 3.1.2 Underdetermined Systems Now consider the underdetermined case. A full URV factorization $A=URV$ is of the following form: (4) Since $A$ is assumed to be full-rank, $\min_{x}\\|Ax-b\\|_{2}=0$ and we seek to solve $Ax=b$. As in the overdetermined case, make the change of variable $y=Vx$; we now consider solving the upper-trapezoidal system $Ry=U^{T}b$. Partitioning $y$ into $m\times 1$ and $(n-m)\times 1$ blocks results in the block system \[\begin{bmatrix}R_{11}&R_{12}\end{bmatrix}\begin{bmatrix}y_{1}\\ y_{2}\end{bmatrix}=U^{T}b,\] where $R_{11}$ is upper-triangular and full-rank. A particularly simple solution is found by setting $y_{2}=0$ and performing backward substitution to find $y_{1}$. Following [18], we call this the basic solution. Note that the basic solution has $m-n$ zeros in $y$, but after unmixing to find $x_{\text{basic}}=V^{T}y$, the zeros in $y_{2}$ are mixed with the nonzeros in $y_{1}$, destroying the sparsity of $x_{\text{basic}}$. While this is less than ideal, mixing and unmixing is fast, and sparsity in the mixed domain might still be applicable in certain problems. Notice that $R_{12}$ is not used to compute the basic solution. Since $R$ is computed from $\hat{A}=AV^{T}$, which mixes all the columns of $A$ together, we may compute $U$ and $R_{11}$ from the $QR$ factorization of $\hat{A}(:,1:m)$ (in matlab notation). This avoids the computation of $R_{12}$, leading to a faster solution. Mixing to find $\hat{A}=AV^{T}$ costs $\mathcal{O}(mn\log n)$; computing $R_{11}$ costs $\mathcal{O}(m^{3})$; and applying $U^{T}b$, backward substitution to find $y=R_{11}^{-1}U^{T}b$, and unmixing to find $x_{\text{basic}}=V^{T}y$ all cost a negligible amount for large $m$ and $n$. This brings the total cost to compute the basic solution to $\mathcal{O}(m^{3},mn\log n)$ FLOPs. Another common solution is the minimum norm solution. Since the solution set $\mathcal{X}=\{x\in\mathbb{R}^{n}\,\|\,Ax=b\}$ is closed and convex, there exists a unique minimum norm solution, which is a principal attraction to the minimum norm solution (a similar statement holds even when $A$ is rank deficient). Finding the minimum norm solution can be expressed as the problem \[\begin{array}[]{ll}\min&\\|x\\|^{2}\\ \text{s.t.}&Ax=b.\end{array}\] Let $\mathcal{L}(x,\nu)=x^{T}x+\nu^{T}(Ax-b)$ be the Lagrangian function. Slater’s condition for this problem is simply that the problem is feasible, which is of course satisfied since we assume $A$ is full-rank. Therefore, strong duality holds and the KKT conditions, \[\nabla_{x}\mathcal{L}=2x+A^{T}\nu=0,\quad Ax-b=0,\] give necessary and sufficient conditions for the solution [6]. Solving the KKT conditions gives $x_{\text{mn}}=A^{T}(AA^{T})^{-1}b=A^{\dagger}b$, where $A^{\dagger}$ is the (right) pseudoinverse of $A$. To use this closed-form solution efficiently, it is convenient to perform a QR factorization of $A^{T}$. Specifically, if we let $A^{T}=QR$, then $x_{\text{mn}}=QR^{-T}b$, where $R^{-T}b$ is computed implicitly with forward substitution. To find the minimum norm solution with mixing, we should mix the columns of $A^{T}$ in preparation for the unpivoted QR of $\hat{A}^{T}$. Let $\hat{A}^{T}=A^{T}V^{T}$ (which we may compute via $\hat{A}=VA$) and compute $\hat{A}^{T}=U^{T}L^{T}$ via unpivoted QR. We then have the factorization $A=V^{T}LU$, where $V$ is our fast ROS mixing matrix, $L$ is $m\times m$ lower triangular, and $U$ is $m\times n$ with orthonormal rows (i.e., $U^{T}$ is orthonormal). We call the algorithm to compute $A=V^{T}LU$RVLU_ROS in analogy with RURV_ROS. By multiplying $Ax=b$ on the left by $V$, we find $\hat{A}x=Vb$, and from the discussion above, the minimum norm solution is $x_{\text{mn}}=U^{T}L^{-1}Vb$. Again note that $L^{-1}$ is applied implicitly using forward substitution. The dominant cost of this approach is again the unpivoted QR factorization of $\hat{A}^{T}$, which costs $\mathcal{O}(mn^{2})$ FLOPs, which can be significantly higher than the $\mathcal{O}(m^{3},mn\log n)$ FLOPs for the basic solution. ### Timing Experiments Solving least-squares problems with RURV_ROS or RVLU_ROS factorizations will be slightly slower than using unpivoted QR; the additional cost comes almost entirely from the mixing steps in Algorithm 2. In our code, we use the DCT-II and DCT-III, as implemented in FFTW [15]. For improved performance, we cache FFTW “wisdom” in a file and load it the next time it is applicable. Finding the solution proceeds in three stages: mixing to find $\hat{A}$, performing unpivoted QR factorization of $\hat{A}$ or $\hat{A}^{T}$, and computing the final solution vector, which may involve mixing a single vector. For moderately large overdetermined problems, mixing to find $\hat{A}$ takes about $25\%$ of the total runtime; unpivoted QR factorization $75\%$ of the total time; and solving/mixing takes a negligible amount of time, since it is applied to only a single vector. We compare with BLENDENPIK, which uses mixing across rows and row sampling to find a good preconditioner for LSQR [3, 26]. The authors wrote most of their code in C for efficiency, calling LAPACK and FFTW libraries and providing their own implementation of LSQR. When we installed BLENDENPIK, we precomputed FFTW “wisdom” for the most patient planner setting, which results in the highest performance at run-time. In the underdetermined case, BLENDENPIK computes the minimum norm solution. With the exception of using DCT mixing, we used the default parameters provided in BLENDENPIK’s interface. It is worth noting that the well-known backslash (\) operator in matlab solves (rectangular) linear systems in the least-squares sense using a QR-based approach. matlab’s \ operator tends to be significantly slower than BLENDENPIK and RURV_ROS, but \ also supports the case of rank-deficient matrices [22]. LAPACK has a variety of least-squares routines, and can handle full-rank and rank-deficient matrices. The LAPACK routine _gels uses a simple rescaling and unpivoted QR or LQ to solve full-rank least-squares problems [1]. For highly overdetermined systems, BLENDENPIK is reported to beat QR-based solvers, including _gels, by large factors [3]. For the following timing experiments, we take $A$ to be a random matrix constructed by $A=U\Sigma V^{T}$ where $U$ and $V$ are random orthogonal matrices and $\Sigma$ is a diagonal matrix of singular values such that $\kappa_{2}(A)=10^{6}$ ($\kappa_{2}(A)=\\|A\\|_{2}\\|A^{-1}\\|_{2}$ is the spectral condition number of $A$). We take a single random right-hand side vector $b$ with entries sampled from $N(0,1)$ and solve the problem $\min_{x}\\|Ax-b\\|_{2}$. We link BLENDENPIK and our code against matlab’s LAPACK and the standard FFTW library. For timing results, we run each routine once to “warm-up” any JIT-compiled matlab code, and run a number of samples. Our code is designed to scale up to multiple threads on a single machine, using multi-threaded versions of LAPACK and FFTW, but BLENDENPIK currently uses only a single thread for their FFTW calls. We note that it would be straightforward to extend BLENDENPIK to use multi-threaded FFTW calls, but mixing is hardly the dominant cost of BLENDENPIK, so one would not expect see a large improvement in runtimes. Nevertheless, we perform the following timing experiments using only a single thread in order to compare fairly BLENDENPIK and RURV_ROS. Figures 3 and 4 show the average runtime for random $A$ with $\kappa_{2}(A)=10^{6}$ of various sizes. We consider moderately underdetermined, slightly underdetermined, slightly overdetermined, and moderately overdetermined examples. For underdetermined systems, computing the basic solution with RURV_ROS is slightly faster than BLENDENPIK, which computes the minimum norm solution. Using RVLU_ROS to compute the minimum norm solution is moderately slower than BLENDENPIK. Notice in Figure 3 that the average runtime for the basic solution is slightly jagged. This variance is due to FFTW’s planner finding a faster plan for certain sizes. We could potentially improve the runtime by optionally zero-padding $A$ and using transforms of a slightly larger size, which may allow FFTW’s planner to find a faster plan for the larger size. Using zero-padding would change the values in the mixed matrix $\hat{A}$, however, so for now we do not investigate using zero-padding. <figure><img src="content_image/1703.02499/x3.png"><figcaption>Figure 3: Average runtime for BLENDENPIK, RURV_ROS, and RVLU_ROS approacheson moderately and slightly underdetermined systems. The RVLU_ROS based minimumnorm solution is consistently slower than BLENDENPIK, which also computes theminimum norm solution. The basic solution computed with RURV_ROS, beingsimpler to compute, is a considerably faster than the RVLU_ROS based minimumnorm solution.</figcaption></figure> <figure><img src="content_image/1703.02499/x5.png"><figcaption>Figure 4: Average runtime for BLENDENPIK and the RURV_ROS approach onoverdetermined systems. RURV_ROS compares favorably to BLENDENPIK for m/nclose to one. For larger m/n (i.e., more highly overdetermined), BLENDENPIKperforms better than RURV_ROS, which we expect from [AMT10], which shows thatBLENDENPIK tends to outperform QR-based solvers on highly overdeterminedsystems.</figcaption></figure> Figure 5 shows the ratio of the runtimes for RURV_ROS and RVLU_ROS for both 1 and 16 threads. For slightly underdetermined systems, the speedup factor approaches 4 (i.e., for large $m$, using 16 threads runs about 4 times faster than using only 1 thread). For slightly overdetermined systems, the speedup factor increases for $m\geq 1000$ and approaches 6. Although we see speedup factors of less than ideal 16, our implementation does parallelize nicely. The speedup factors may very well continue to increase outside of the range of matrices we tested (until we run out of memory on the machine, that is). The larger speedup factor for overdetermined systems is likely due to the ratio of the work computed in the mixing stage and the factorization stage. For overdetermined systems, mixing occurs along the smaller dimension of the matrix, so there are many smaller transforms, compared to underdetermined systems. This gives a higher proportion of the work to the QR factorization, during which we can more effectively utilize additional cores. <figure><img src="content_image/1703.02499/x7.png"><figcaption>Figure 5: Ratios of average runtime for the RURV_ROS and RVLU_ROS approacheson slightly under- and overdetermined systems using 1 and 16 threads. We plotthe ratio of the runtime using 1 thread over the runtime using 16 threads, sospeedup factors greater than 1 correspond to an improvement when running inparallel.</figcaption></figure> Before we continue with our discussion of our experiments, we make a brief note on implementing RURV_ROS on a distributed memory machine. The two major steps of RURV_ROS are mixing and the unpivoted QR factorization, which can be handled by FFTW and ScaLAPACK, respectively. FFTW has a distributed memory implementation using MPI, and the interface is very similar to the shared memory interface. ScaLAPACK’s routine p_geqrf performs unpivoted Householder QR, but uses a suboptimal amount of communication. In [12] communication-avoiding QR (CAQR) was introduced. CAQR sends a factor of $\mathcal{O}(\sqrt{mn/P})$ fewer messages than p_geqrf (where $P$ is the total number of processors in the grid), and achieves the optimal amount of communication (up to polylogarithmic factors). The reduction in communication is predicted to result in significantly faster factorization runtimes. Using FFTW, ScaLAPACK, or other, existing codes as building blocks, we expect that RURV_ROS can be implemented efficiently and straightforwardly for distributed memory environments. ### Example - Correlated Columns and the Basic Solution The dominant cost of using a URV factorization to compute the basic solution to an underdetermined $m\times n$ system ($m<n$) is computing a QR factorization of $\hat{A}(:,1:m)$. Thus, it is asymptotically cheaper than the minimum norm solution, which uses an LQ factorization of the full $\hat{A}$. Specifically, computing the basic solution costs $\mathcal{O}(m^{3},mn\log n)$ FLOPs, while computing the minimum norm solution costs $\mathcal{O}(m^{2}n)$ FLOPs [18]. We have seen in Figure 3 that computing the basic solution is significantly faster than computing the minimum norm solution with RURV_ROS, and even slightly faster than BLENDENPIK, which also computes the minimum norm solution. The following simple example shows that finding the basic solution using unpivoted QR is numerically unstable for some least-squares problems. Consider \[A=\begin{bmatrix}1&0&0&0\\ 0&1&1&1\\ 0&0&\epsilon&1\end{bmatrix},\] with $\epsilon\ll 1$ and consider solving $\min_{x}\\|Ax-b\\|_{2}$. Note that $A$ is full-rank and that it can be analytically verified that $\kappa_{2}(A)\sim 1+\sqrt{2}$ as $\epsilon\to 0$, so $A$ is very well conditioned for $\epsilon\ll 1$. However, columns 2 and 3 are increasingly correlated as $\epsilon\to 0$. Since $A$ is already upper trapezoidal, an unpivoted QR factorization does not change $A$ when finding $R$. When finding the basic least-squares solution in this manner, it transpires that we solve the linear system \[R_{11}x_{1}=\begin{bmatrix}1&0&0\\ 0&1&1\\ 0&0&\epsilon\end{bmatrix}\begin{bmatrix}x_{1,1}\\ x_{1,2}\\ x_{1,3}\end{bmatrix}=Q^{T}b=b=\begin{bmatrix}b_{1}\\ b_{2}\\ b_{3}\end{bmatrix}.\] However, this system has $\kappa_{2}(R_{11})\sim 2/\epsilon$ as $\epsilon\to 0$, and so can become quite ill-conditioned as columns 2 and 3 become more correlated. Even for this small system, finding the basic solution $\hat{x}$ with an unpivoted QR factorization leads to a large residual $\\|A\hat{x}-b\\|_{2}$. This can be fixed by using QRCP on all of $A$ instead of unpivoted QR on $A(:,1:m)$. Note that using QRCP on $A(:,1:m)$ will encounter similar ill-conditioning problems, as doing so will not allow QRCP to pivot in column 4. Using a URV factorization which mixes in column 4 of $A$ also leads to a much better conditioned $R_{11}$, alleviating the issue of correlated columns when using the basic solution. This simple example can be extended to larger matrices, as we show next. Consider an $m\times(n-p)$ matrix $A$ with $m<n$ such that each element of $A$ is sampled from $N(0,1)$. We augment $A$ by adding $p$ randomly selected columns of $A$ to the end of $A$, making $A$$m\times n$. The augmented $A$ has $p$ perfectly correlated columns, so we add a small amount of $N(0,\sigma^{2})$ noise to the augmented $A$ so the correlation is not perfect. We then randomly shuffle the columns. Listing 2 gives matlab code to generate such a matrix, which tends to be well-conditioned. If the final permutation places a pair of highly correlated columns in the first $m$ columns of $A$, finding the basic solution $\hat{x}$ with unpivoted QR will produce an ill-conditioned $R_{11}$, leading to a large residual $\\|A\hat{x}-b\\|_{2}$. This can be solved by mixing with RURV_Haar or RURV_ROS, or computing the (more costly) minimum norm solution. ``` m = 1000; n = 1500; p = 10; e = 1e-4; A = randn(m,n-p); perm = randperm(n-p,p); A = [A A(:,perm)]; perm = randperm(n); A = A(:,perm) + erandn(m,n); ``` Listing 2: matlab code to generate a matrix with a few correlated columns Table 1 shows the residuals and runtimes on a matrix generated with Listing 2. We tested unpivoted QR, QRCP, BLENDENPIK, RURV_Haar, and RURV_ROS. We use unpivoted QR on only the first $m$ columns of $A$, so it produce a significantly larger residual than the other methods. Note that BLENDENPIK actually computes the minimum norm solution and is included for reference. RURV_Haar and RURV_ROS both compute basic solutions with acceptably small residuals. As expected, RURV_ROS is considerably faster than RURV_Haar, but slightly slower than plain, unpivoted QR. It is interesting to note that the norm of the mixed basic solution is considerably smaller than the unmixed basic solution. Table 2 shows the same comparison for $p=0$ correlated columns, where we see that the mixed and unmixed basic solutions have norms that are not unreasonably large. The norms of the mixed basic solutions are on the same order for the cases of $p=10$ and $p=0$ correlated columns, unlike QR and QRCP. Method \| Residual - ∥A^x−b∥2 \| Norm - ∥^x∥2 \| Time (s) ---\|---\|---\|--- QR \| 3.0×10−9 \| 1.3×105 \| 0.04 QRCP \| 2.5×10−13 \| 5.8×100 \| 0.19 BLENDENPIK \| 1.4×10−13 \| 1.4×100 \| 0.16 RURV_Haar \| 5.8×10−12 \| 1.5×102 \| 0.52 RURV_ROS \| 1.4×10−12 \| 4.3×101 \| 0.10 Table 1: Comparison of basic solution residuals for the 1000×1500 matrix from Listing 2 with p=10 correlated columns. As expected, unpivoted QR has a relatively large residual, while the other methods perform better. Note that BLENDENPIK computes the minimum norm solution. Method \| Residual - ∥A^x−b∥2 \| Norm - ∥^x∥2 \| Time (s) ---\|---\|---\|--- QR \| 5.4×10−13 \| 1.8×101 \| 0.04 QRCP \| 2.5×10−13 \| 6.1×100 \| 0.20 BLENDENPIK \| 1.3×10−13 \| 1.4×100 \| 0.15 RURV_Haar \| 4.8×10−12 \| 1.2×102 \| 0.52 RURV_ROS \| 1.3×10−12 \| 3.9×101 \| 0.10 Table 2: Comparison of basic solution residuals for the 1000×1500 matrix from Listing 2 with p=0 correlated columns. All methods perform well, and that the two URV-based methods compute mixed basic solutions with norms on the same order as in the previous case with correlated columns. ## 4 Experimental Comparison of RURV_Haar and Rurv_ros In this section we experiment with a variety of QR and URV factorizations, some of which are known to be rank-revealing. In Subsection 4.1 we experiment with how the rank-revealing conditions (1) and (2) scale with increasing $n$. Our chief interest here is the comparison of RURV_Haar and RURV_ROS. We can use RURV_ROS to form low-rank approximations by performing the mixing and pre-sort as usual, but only performing $k$ steps of the QR factorization, yielding a rank-$k$ approximation. The mixing step costs $\mathcal{O}(mn\log n)$ FLOPs as usual, but now the partial QR factorization costs only $\mathcal{O}(mnk)$ FLOPs. In Subsection 4.3, we investigate pairing QR and URV factorizations with Stewart’s QLP approximation to the SVD [28]. One can use the QLP approximation to obtain an improved rank-$k$ approximation by truncating $L$ factor. ### Scaling of Rank-Revealing Conditions It was shown in [9, 4] that RURV_Haar produces a strong rank-revealing factorization with high probability. RURV_ROS simply replaces Haar mixing with ROS mixing and adds a pre-sort before the unpivoted QR factorization, so we expect RURV_ROS to behave similarly to RURV_Haar. Specifically, we hope that RURV_ROS obeys the strong rank-revealing conditions (1) and (2) in a manner similar to RURV_Haar. We experimentally test the scaling of the ratios $\sigma_{i}(A)/\sigma_{i}(R_{11})$, $\sigma_{j}(R_{22})/\sigma_{k+j}(A)$, and the norm $\\|R_{11}^{-1}R_{12}\\|$ to determine if they appear to be bounded above by a slowly-growing polynomial. In Figure 6 we take $A$ to be a random $m\times m$ matrix of rank $k\approx m/2$. The matrix $A$ is formed as $A=U\Sigma V^{T}$, where $U$ and $V$ are Haar random orthogonal matrices, $\Sigma=\operatorname{diag}(\sigma_{1},...,\sigma_{m})$, and the $\sigma_{i}$ decay slowly until $\sigma_{m/2}$, where there is a gap of about $10^{-10}$, after which the $\sigma_{i}$ decay slowly again. We sample sizes $m$ from 10 to 1000; for each $m$, we generate five instantiations of the matrix $A$, perform a variety of factorizations for each $A$, and compute the conditions (1) and (2) for each factorization. For plotting, we plot the maximum over the five instantiations of $\max_{i}\sigma_{i}(A)/\sigma_{i}(R_{11})$, $\max_{j}\sigma_{j}(R_{22})/\sigma_{k+j}(A)$, and $\\|R_{11}^{-1}R_{12}\\|$. We use the highly-accurate LAPACK routine dgejsv to compute singular values of the test matrices (when the exact singular values are unknown) and in the computation of the ratios (1). dgejsv implements a preconditioned Jacobi SVD algorithm, which can be more accurate for small singular values [13, 14]. Specifically, if $A=DY$ (or $A=YD$), where $D$ is a diagonal weighting matrix and $Y$ is reasonably well-conditioned, dgejsv is guaranteed to achieve high accuracy. The relative error of the singular values computed with the preconditioned Jacobi method are $\mathcal{O}(\epsilon)\kappa_{2}(Y)$, whereas the relative errors as computed with a QR-iteration based SVD are $\mathcal{O}(\epsilon)\kappa_{2}(A)$[13, 11]. This fact is particularly relevant when we test with the Kahan matrix, which is discussed later in the section. Even when $A$ is not of the form $A=DY$, $A=YD$, or even $A=D_{1}YD_{2}$, it is expected that dgejsv returns singular values at least as accurate as a QR-iteration based SVD. We test QRCP, RURV_Haar, RURV_ROS, HQRRP from [24], which uses random projections to select blocks of pivots, and DGEQPX from [5], which is known to be a rank-revealing QR. Note that HQRRP is intended to cheaply produce a column-pivoted Householder QR; it is not a rank-revealing QR, but it tends to be rank-revealing in practice, like QRCP. Figure 6 shows the rank-revealing conditions for $A$ a random $m\times m$ matrix of rank $k\approx m/2$. The three QR factorizations we test, QRCP, HQRRP, and DGEQPX, perform very well, meaning that the sampled rank-revealing conditions appear to be bounded above by a slowly growing polynomial. Note that Figure 6 uses a log-log scale, on which polynomial growth appears linear. As we expect, RURV_ROS performs about as well as RURV_Haar. With the exception of a few points, RURV_Haar and RURV_ROS appear to be bounded above by a slowly growing polynomial, albeit a significantly larger polynomial than for the three QR factorizations. The exceptions may very well be points where RURV_Haar or RURV_ROS failed to produce a rank-revealing factorization for at least one of the five sampled $A$ matrices. Figure 7 shows the rank-revealing conditions with $A$ the $m\times m$ Kahan matrix and $k$ chosen to be $m-1$. The Kahan matrix is a well-known counterexample on which QRCP performs no pivoting in exact arithmetic [8]. We use the Kahan matrix (with perturbation) as described in [11]. The $m\times m$ Kahan matrix is formed as \[A=\begin{bmatrix}1&0&0&\cdots&\cdots&0\\ 0&s&0&\cdots&\cdots&0\\ 0&0&s^{2}&\ddots&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&\vdots&\cdots&\ddots&\ddots&0\\ 0&0&0&\cdots&0&s^{m-1}\end{bmatrix}\begin{bmatrix}1&-c&-c&\cdots&\cdots&-c\\ 0&1&-c&\cdots&\cdots&-c\\ 0&0&1&\ddots&\cdots&-c\\ \vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&\vdots&\cdots&\ddots&\ddots&-c\\ 0&0&0&\cdots&0&1\end{bmatrix},\] (5) where $s^{2}+c^{2}=1$ and $s,c\geq 0$. When using QRCP to compute the factorization \[A\Pi=QR=Q\begin{bmatrix}R_{11}&R_{12}\\ &R_{12}\end{bmatrix},\quad R_{11}\in\mathbb{R}^{k\times k},R_{12}\in\mathbb{R} ^{k\times(m-k)},R_{22}\in\mathbb{R}^{(m-k)\times(m-k)},\] it is known that $\sigma_{k}(A)/\sigma_{k}(R_{11})\geq\frac{1}{2}c^{3}(1+c)^{m-4}/s$ for $k=m-1$, and $\sigma_{k}(R_{11})$ can be much smaller than $\sigma_{k}(A)$[17]. That is, QRCP does not compute a rank-revealing factorization, as the first ratio in (1) grows exponentially for $i=k=m-1$. To prevent QRCP from pivoting on the Kahan matrix in finite arithmetic, we multiply the $j$th column by $(1-\tau)^{j-1}$, with $1\gg\tau\gg\epsilon$[8, 11]. In our tests, we pick $c=0.1$ and $\tau=10^{-7}$. The most apparent feature of Figure 7 is that the rank-revealing conditions for QRCP grow exponentially. This is a known feature of the Kahan matrix, and shows that QRCP is not _strictly speaking_ a rank-revealing QR (in practice, however, it is still used as a rank-revealing factorization). Moreover, the Kahan matrix is so bad for QRCP, we believe dgejsv cannot accurately compute the singular values in the ratios $\sigma_{i}(A)/\sigma_{i}(R_{11})$ and $\sigma_{j}(R_{22})/\sigma_{k+j}(A)$. As $m$ grows, the right-hand matrix in (5) becomes increasingly ill-conditioned, and we see the exponential growth in Figure 7 stop around $m=10^{3}$. In infinite precision arithmetic, the exponential growth should continue, so we stop testing at $m\approx 400$. As expected, the rank-revealing conditions for RURV_ROS scale in the same manner as RURV_Haar, giving credence to our thought that RURV_ROS is rank-revealing with high probability. <figure><img src="content_image/1703.02499/x9.png"><figcaption>Figure 6: Maximum of the sampled values of the rank-revealing conditions fromSubsection 2.1 for five random m×m matrices of numerical rank m/2. The threeQR factorizations exhibit growth that is clearly bounded by a slowly growingpolynomial (linear in a log-log plot). RURV_Haar and RURV_ROS also appear toexhibit bounded growth, with only a few exceptions; recall that RURV_Haarproduces a strong rank-revealing factorization with high probability, notdeterministically.</figcaption></figure> <figure><img src="content_image/1703.02499/x10.png"><figcaption>Figure 7: Maximum of the sampled values of the rank-revealing conditions fromSubsection 2.1 on the m×m Kahan matrix. As expected, QRCP performs verypoorly, with all conditions scaling exponentially. Again, we see thatRURV_Haar and RURV_ROS behave similarly.</figcaption></figure> ### Accuracy of R-Values Another test we perform involves the accuracy of $\|R(i,i)\|$ in predicting $\sigma_{i}(R)$ ($R(i,i)$ is the $i$th diagonal element of the upper-triangular factor from a QR or URV). Following [11], we call the $\|R(i,i)\|$ R-values. The R-values can be used as a rough estimate of the singular values. A better approximation is to use Stewart’s QLP factorization [28], which we discuss in Subsection 4.3. Nevertheless, it is descriptive to investigate the behavior of the R-values. We test QRCP, RURV_Haar, and RURV_ROS on the first 18 test matrices from Table 2 of [11] (most matrices are from [19, 17]). In Figure 8, we plot the minimum, median, and maximum of the ratios $\|R(i,i)\|/\sigma_{i}$ for the 18 test matrices. For each matrix, we let r and s be the vectors of R-values and singular values, respectively; we plot min(r./s), median(r./s), and max(r./s) (using matlab syntax). We see that QRCP produces ratios that are at most just over an order of magnitude away from one. RURV_Haar produces slightly worse ratios, which seem to be spread over about two orders of magnitude away from one. RURV_ROS with one mixing iteration produces ratios comparable to RURV_Haar, with the exception of matrix 15, SPIKES. For matrix 15, the extreme ratios are significantly larger than on the rest of the test set. Adding a second mixing iteration brings the ratios back down to a couple orders of magnitude away from one, but does not improve the ratios for the other matrices beyond what is accomplished with a single mixing. We can also find a bound for the ratios obtained with QR and URV factorizations. Let $D$ be the diagonal part of $R$ obtained from a QR or URV factorization, and define $Y$ via $R=DY^{T}$. This results in the factorization $A\Pi=QDY^{T}$ for QRCP and $A=UDY^{T}V$ for RURV_Haar and RURV_ROS. For QRCP, the diagonal elements of $R$ are non-negative and sorted in decreasing order; this is not guaranteed for RURV_Haar or RURV_ROS. It follows from the Courant-Fischer minimax theorem [18] that QRCP has the bounds \[\dfrac{1}{\\|Y\\|}\leq\dfrac{R(i,i)}{\sigma_{i}}\leq\\|Y^{-1}\\|.\] (6) For RURV_Haar and RURV_ROS, let $\rho_{i}$ be the $i$th largest (in absolute value) diagonal element of $R$. For RURV_Haar and RURV_ROS, we have the bounds \[\dfrac{1}{\\|Y\\|}\leq\dfrac{\|\rho_{i}\|}{\sigma_{i}}\leq\\|Y^{-1}\\|.\] In addition to the minimum, median, and maximum values of $\|R(i,i)\|/\sigma_{i}$ for each matrix, we plot the bounds (6) for both QRCP and the two RURV factorizations. Even though the two RURV factorizations are not guaranteed to be bound by (6), since it is a strong rank-revealing URV, we expect the R-values to somewhat closely approximate the singular values and approximately obey the QRCP bounds. With the exception of matrix 12, formed as A=2rand(n)-1 in matlab, we see this behavior in Figure 8, and we again see RURV_ROS behaving similarly to RURV_Haar. <figure><img src="content_image/1703.02499/x11.png"><figcaption>Figure 8: Ratios of \|R(i,i)\|/σi(R) for the 18 matrices in Table 2 of[DGGX15]. The abscissa is the index of the matrix in the test set. For matrix15 (SPIKES), RURV_Haar produces ratios on par with the rest of the test set.For RURV_ROS, however, using only 1 mixing step produces very bad max/minratios; using two two mixing steps produces better ratios, but more mixingsteps doesn’t appear to yield further improvements. For the other 17 matrices,RURV_ROS produces ratios comparable with RURV_Haar.</figcaption></figure> ### Experiments With the QLP Approximation The QLP factorization was introduced by G.W. Stewart as an approximation to the SVD in [28]. The idea of the pivoted QLP factorization is to use QRCP to find R-values, and then improve the accuracy (by a surprising amount) by performing another QRCP on $R^{T}$. This results in a factorization of the form $A=Q_{1}\Pi_{2}LQ_{2}^{T}\Pi_{1}$, where $L$ is lower triangular. Following [11], we call the diagonal elements of the $L$ matrix L-values. In Stewart’s original experiments, it was found that L-values approximate the singular values significantly more accurately than the R-values. Also, the accuracy seemed intimately tied to using QRCP for the first factorization, but that unpivoted QR could be used in the second QR factorization with only cosmetic differences. It was later shown that the QLP factorization can be interpreted as the first two steps of QR-style SVD algorithm [20]. We experiment with QLP-style factorizations by performing QR, QRCP, RURV_Haar, or RURV_ROS, and following up with an unpivoted QR to compute the L-values. We denote such a factorization as {factorization}+QLP (e.g., QRCP+QLP). For the RURV factorizations, this QLP-style factorization is of the form $A=ULQ^{T}V$. Figure 9 shows the singular values and L-values for a random matrix of the form $A=U\Sigma V^{T}$, where $U,V$ are Haar random orthogonal, and the singular values are chosen to decay slowly, have a gap of approximate width $10^{-1}$, and decay slowly again. We see that all QLP-style factorizations, including QR+QLP, identify both the location and magnitude of the gap quite accurately. Also shown in Figure 9 are the L-values for the Devil’s stairs matrix, which is a particularly difficult example for rank-revealing factorizations. The Devil’s stairs matrix is discussed in [28, 11], and is formed with $A=U\Sigma V^{T}$, with $U,V$ Haar random orthogonal and $\Sigma$ controlling the stair-step behavior. Of all the factorizations, QRCP+QLP performs the best, accurately identifying the location and size of the singular value gaps. QR+QLP, RURV_Haar+QLP, and RURV_ROS+QLP all provide evidence for the existence of singular value gaps, but none is able to identify the precise location and size of the gaps. Figure 10 shows the minimum, median, and maximum L-values for 25 realizations of the Devil’s stairs matrix. We again use QR, QRCP, RURV_Haar, and RURV_ROS, followed by QR to form the QLP factorization. It is clear that QRCP+QLP produces the best L-values; RURV_Haar+QLP and RURV_ROS+QLP generate L-values visually similar to those produced with QR+QLP. The L-values are smeared around the jumps for QR and the two RURV factorizations, but the L-values have a lower variance around the middle of the flat stairs. The variance of the L-values around the gaps appears visually similar for QR+QLP, RURV_Haar+QLP, and RURV_ROS+QLP. For QR+QLP, the variance is explained only by the Haar random orthogonal matrices used to construct the Devil’s stairs matrix; for RURV_Haar+QLP and RURV_ROS+QLP, the variance is a combination of the random Devil’s stairs matrix and the random mixing. <figure><img src="content_image/1703.02499/x12.png"><figcaption>Figure 9: L-values from various QLP factorizations on a random 128×128 matrixwith slowly decaying singular values and a small gap of approximate size 10−1and the Devil’s stairs with gaps of approximate size 10−1. In each case, thelegend name is of the form {factorization}+QLP, where the second factorizationis always unpivoted QR.</figcaption></figure> <figure><img src="content_image/1703.02499/x14.png"><figcaption>Figure 10: Min/Median/Max L-values for 25 runs of a randomly generated128×128 Devil’s stairs matrix with jumps of approximate size 10−1. QR,RURV_Haar, and RURV_ROS appear to predictably show the presence andapproximate location of the gaps, but are not accurate enough to estimate thesize of the gaps. QRCP performs very well, and accurately shows the locationand size of the gaps.</figcaption></figure> ## 5 Discussion We have modified RURV_Haar, a strong rank-revealing factorization with high probability, to use random orthogonal mixing (ROS) instead of Haar orthogonal matrix mixing. The new algorithm, RURV_ROS, applies the mixing matrix implicitly and quickly, as opposed to RURV_Haar, where the mixing matrix is generated with an unpivoted QR and applied with dense matrix-matrix multiplication. With both randomized URV factorizations, one of the principal attractions is the use of cheaper, unpivoted QR, instead of relying on the more expensive QRCP. The ansatz is that mixing reduces the variance of the column norms, reducing the effect that column pivoting would have, and so we can forgo pivoting and use a cheaper, unpivoted QR. A URV factorization can be used in many applications that call for a QR, and since the dominant asymptotic cost of RURV_ROS is the same as unpivoted QR, RURV_ROS has the potential to be used as a safer alternative to unpivoted QR. We have considered only real matrices, but the extension to complex matrices and transforms is natural. We experiment with using RURV_ROS to solve over- and underdetermined least-squares problems. Using a URV factorization to solve least-squares is very similar to using a QR factorization. Our implementation of RURV_ROS even performs comparably to BLENDENPIK, which uses mixing and row sampling to create a preconditioner for LSQR. When one wants a solution to an underdetermined system, but does not need the minimum norm solution, RURV_ROS can be used to find a basic solution slightly faster than BLENDENPIK, which computes the minimum norm solution. Additionally, if even a few of the columns of the $A$ matrix are highly correlated, using unpivoted QR, or QRCP on the first $m$ columns, can lead to an inaccurate basic solution; using RURV_ROS computes a mixed basic solution with an accurate residual and for which the norm of the solution is only an order of magnitude larger than the minimum norm solution. Finally, we experiment with the possible rank-revealing nature of RURV_ROS. We test the scaling of the rank-revealing conditions (1) and (2) for RURV_Haar, RURV_ROS, and a few other QR factorizations, one of which is rank-revealing. The prominent feature of the scaling tests is that RURV_ROS behaves very similarly to RURV_Haar, which leads us to suspect that RURV_ROS produces a strong rank-revealing factorization with high probability. We plan to investigate theoretically the apparent rank-revealing nature of RURV_ROS. ## References * [ABB${}^{+}$99] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. _LAPACK Users’ Guide_. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. * [AC06] Nir Ailon and Bernard Chazelle. Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In _Proceedings of the thirty-eighth annual ACM symposium on Theory of computing_, pages 557–563. ACM, 2006. * [AMT10] Haim Avron, Petar Maymounkov, and Sivan Toledo. BLENDENPIK: Supercharging LAPACK’s least-squares solver. _SIAM Journal on Scientific Computing_, 32(3):1217–1236, 2010. * [BDD10] Grey Ballard, James Demmel, and Ioana Dumitriu. Minimizing communication for eigenproblems and the singular value decomposition. _arXiv preprint arXiv:1011.3077_, 2010. * [BQO98] Christian H Bischof and Gregorio Quintana-Ortí. Algorithm 782: codes for rank-revealing QR factorizations of dense matrices. _ACM Transactions on Mathematical Software (TOMS)_, 24(2):254–257, 1998. * [BV04] Stephen Boyd and Lieven Vandenberghe. _Convex optimization_. Cambridge university press, 2004. * [BVL87] Christian Bischof and Charles Van Loan. The WY representation for products of Householder matrices. _SIAM Journal on Scientific and Statistical Computing_, 8(1):s2–s13, 1987. * [DB08] Zlatko Drmač and Zvonimir Bujanović. On the failure of rank-revealing QR factorization software–a case study. _ACM Transactions on Mathematical Software (TOMS)_, 35(2):12, 2008. * [DDH07] James Demmel, Ioana Dumitriu, and Olga Holtz. Fast linear algebra is stable. _Numerische Mathematik_, 108(1):59–91, 2007. * [DG15] Jed A Duersch and Ming Gu. True BLAS-3 performance QRCP using random sampling. _arXiv preprint arXiv:1509.06820_, 2015. * [DGGX15] James W Demmel, Laura Grigori, Ming Gu, and Hua Xiang. Communication avoiding rank revealing QR factorization with column pivoting. _SIAM Journal on Matrix Analysis and Applications_, 36(1):55–89, 2015. * [DGHL12] James Demmel, Laura Grigori, Mark Hoemmen, and Julien Langou. Communication-optimal parallel and sequential qr and lu factorizations. _SIAM Journal on Scientific Computing_, 34(1):A206–A239, 2012. * [DV08a] Zlatko Drmač and Krešimir Veselić. New fast and accurate Jacobi SVD algorithm. I. _SIAM Journal on matrix analysis and applications_, 29(4):1322–1342, 2008. * [DV08b] Zlatko Drmač and Krešimir Veselić. New fast and accurate Jacobi SVD algorithm. II. _SIAM Journal on Matrix Analysis and Applications_, 29(4):1343–1362, 2008. * [FJ05] Matteo Frigo and Steven G Johnson. The design and implementation of FFTW3. _Proceedings of the IEEE_, 93(2):216–231, 2005. * [GCD16] Laura Grigori, Sebastien Cayrols, and James W Demmel. _Low rank approximation of a sparse matrix based on LU factorization with column and row tournament pivoting_. PhD thesis, INRIA, 2016. * [GE96] Ming Gu and Stanley C Eisenstat. Efficient algorithms for computing a strong rank-revealing QR factorization. _SIAM Journal on Scientific Computing_, 17(4):848–869, 1996. * [GVL98] Gene H Golub and Charles F Van Loan. _Matrix computations_, volume 3. JHU Press, 1998. * [Han07] Per Christian Hansen. Regularization tools version 4.0 for matlab 7.3. _Numerical algorithms_, 46(2):189–194, 2007. * [HC03] David A Huckaby and Tony F Chan. On the convergence of Stewart’s QLP algorithm for approximating the SVD. _Numerical Algorithms_, 32(2-4):287–316, 2003. * [M${}^{+}$11] Michael W Mahoney et al. Randomized algorithms for matrices and data. _Foundations and Trends® in Machine Learning_, 3(2):123–224, 2011. * [Mat] Mathworks, Inc. MATLAB 8.6. https://www.mathworks.com/. * [Mez07] Francesco Mezzadri. How to generate random matrices from the classical compact groups. _Notices of the American Mathematical Society_, 54(5):592–604, 2007. * [MQOHvdG15] Per-Gunnar Martinsson, Gregorio Quintana-Orti, Nathan Heavner, and Robert van de Geijn. Householder QR factorization: Adding randomization for column pivoting. FLAME working note# 78. _arXiv preprint arXiv:1512.02671_, 2015. * [MSM14] Xiangrui Meng, Michael A Saunders, and Michael W Mahoney. LSRN: a parallel iterative solver for strongly over- or underdetermined systems. _SIAM Journal on Scientific Computing_, 36(2):C95–C118, 2014. * [PS82] Christopher C Paige and Michael A Saunders. LSQR: An algorithm for sparse linear equations and sparse least squares. _ACM transactions on mathematical software_, 8(1):43–71, 1982. * [QOSB98] Gregorio Quintana-Ortí, Xiaobai Sun, and Christian H Bischof. A BLAS-3 version of the QR factorization with column pivoting. _SIAM Journal on Scientific Computing_, 19(5):1486–1494, 1998. * [Ste99] GW Stewart. The QLP approximation to the singular value decomposition. _SIAM Journal on Scientific Computing_, 20(4):1336–1348, 1999. * [Tro11] Joel A Tropp. Improved analysis of the subsampled randomized Hadamard transform. _Advances in Adaptive Data Analysis_, 3(01n02):115–126, 2011.
1303.7382	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 11435, "num_imgs": 1, "llama3_tokens_count": 3241 }	[ "content_image/1303.7382/x1.png" ]	# Felinic principle and measurement of the Hubble parameter Yodovina Piškur FITA/IFAT, 7020 108th St, Apt 2U, Forest Hills, NY 11375 Bumbarija Medolin FITA/IFAT, 7020 108th St, Apt 2U, Forest Hills, NY 11375 April 1, 2013 ###### Abstract Intelligent life can only appear in Universes, whose physical laws support sufficient complexity to make evolution of intelligent beings possible. Even inside those Universes, intelligent life does not appear randomly, but in parts with realized complexity, e.g. around stars in our Universe. As a consequence, the observational point of an intelligent observer cannot be assumed to be random and one must correct for this selection effect. In this paper we calculate how direct measurements of the Hubble parameter are affected when subject to the condition that they are observed from a Milky Way-like galaxy. ## I Introduction Felinic principle is based on a simple observation that of all possible Universes, intelligent life can develop only in a subset with physical laws that are sufficiently complex to support evolution of life. For example, if the laws of physics were such that no atoms would form, there would be no chemistry and likely no intelligent life Deutsch (1962). Having the right physical laws is a necessary, but not sufficient condition. It is possible that the actual physical laws can support the required complexity, but the actual values of physical constants are such that life cannot develop. For example, one can imagine a Universe in which the cosmological constant is so large, that the exponential growth of the scale-factor becomes dominant before stars can form and hence the Universe soon becomes a homogeneous sea of rarefied matterLimber (1954). However, even in the Universes that have the necessary complexity and the right values of physical constants, the life does not appear everywhere, but in those parts of the Universe, where the complexity is realized. For example, while it is true, that dogs can be appear out of pure vacuum as result of zero-point quantum-mechanical fluctuations if one waits long enough, higher forms of life are exceedingly unlikely to do so. In our Universe, intelligent life requires existence of chemistry and can thus only be produced around second or higher generation stars, where atoms of a wide atomic weight range are available. Felinic principle has been applied to the human species with some success (see e.g. Hong et al. (2012); Křížek (2012); Gilmour and Middleton (2009); Maor et al. (2008); Peacock (2007); Pogosian and Vilenkin (2007); Bousso (2012); Bousso et al. (2011, 2008)). In that context, the principle is referred to as an anthropic principle and its application is justified to a certain degree, because opening of a can of cat-food clearly shows some degree of psychomotor ability and dexterity¹. In this work, anthropic and felinic principle amount to the same thing, since humans are, after all, necessary for doing the sleep-inducing slog of performing the dirty measurements. [FOOTNOTE:1][ENDFOOTNOTE] When making cosmological inferences, one must therefore correct for the fact that observations are conditioned on the existence of favorable conditions that allowed creation of intelligent life. In this work we make one such calculation, by estimating, how much a direct measurement of the Hubble parameter is biased when observed by an observer observing from a Milky Way-like galaxy as opposed to an observer observing from a random position in the Universe. The crux of the calculation is in the next section and we conclude in Section 3 of this paper. ## II Calculation In our model, we assume that Milky Way resides in a dark-matter halo of mass $M_{\rm MW}\sim 7\times 10^{11}M_{\odot}$. Such halos are biased tracers of the underlying structure with bias parameter $b_{g}\sim 0.78$ (Jeremy Tinker, private communication). To calculate the amount by which a direct measurement of the Hubble parameter is biased, we want to calculate the mean radial peculiar velocity of a source at distance $r$ from a biased tracer. We rotate the coordinate system, so that observer’s galaxy is at origin and the position of the observed source, at which the velocity measurement is performed is at at distance $r$ along the $z$-axis. Denoting $v_{z}$ as the $z$-component of the velocity field and measuring around galaxy field \[1+\delta_{g}(\mathbf{x})=\sum_{i}\delta(\mathbf{x}_{i}-\mathbf{x}),\] (1) where $\mathbf{x}_{i}$ are positions of galaxies, the quantity of interest is \[\left<v_{r}\ \|\ {\rm Milky\ Way}\right>=\left<v_{z}(r\hat{z})(1+\delta_{g}(0)) \right>=\left<v_{z}(r)\delta_{g}(0)\right>\] (2) We will calculate the above expression using the linear theory. This should be an excellent approximation since virial velocities due to motion in individual halos do not correlate across halos and therefore they contribute only noise, but not bias, to the measurement of the Hubble parameter. We can write \[\delta_{g}(\mathbf{x}) = b_{g}\frac{1}{(2\pi)^{3}}\int\delta_{k}(\mathbf{k})e^{i\mathbf{k }\mathbf{x})d^{3}\mathbf{k}}\] (3) \[v_{z}(\mathbf{x}) = \frac{D^{\prime}(z)}{(2\pi)^{3}}\int-i\frac{k_{z}}{k^{2}}\delta_{ k}(\mathbf{k})e^{i\mathbf{k}\mathbf{x}}d^{3}\mathbf{k},\] (4) where $\delta_{k}$ denotes modes of the dark-matter fluctuations in the Fourier space and we has assumed a simple linear biasing for $\delta_{g}$ and a continuity equation without vorticity for the $v_{z}$ field. The symbol $g^{\prime}$ denotes the derivative of the growth factor with respect to the conformal time $d\eta=dt/a$. Using above expressions to evaluate expression (2), one finds \[\left<v_{z}(r\hat{z})(1+\delta_{g}(0))\right>=\\ \frac{dg}{da}\frac{a^{2}b_{g}H_{0}\sqrt{\Omega_{\Lambda}+\Omega_{ m}a^{-3}}}{2\pi^{2}}\\ \times\int\frac{\cos(kr)kr-\sin(kr)}{(kr)^{2}}P(k)kdk\] (5) This expression is a function of comoving distance $r$ between the observer, but these can be converted to redshift using the usual expression (we are writing out all the equations out just in case any non-felines are reading this) \[r=c\int_{0}^{z}H(z)^{-1}dz.\] (6) Correction to the measurement of the Hubble parameter is thus given by \[\Delta H(z)=\frac{\left<v_{z}(r\hat{z})(1+\delta_{g}(0))\right>}{r(z)}\] (7) <figure><img src="content_image/1303.7382/x1.png"><figcaption>Figure 1: The mean bias for the Hubble parameter measurement at redshift zwhich has been conditioned on being observed from a halo of a Milky Way mass(solid savory salmon colored line). Greenies colored points emphasizeredshifts of a low-redshift direct measurement of the Hubble parameter(z∼0.05) and the Lyman-α forest measurement of Hubble parameter using BAO(z∼2.4). See text for discussion.</figcaption></figure> We now evaluate this for a concordant cosmology (flat $\Lambda CDM$ with $\Omega_{m}=0.3$, $H_{0}=70$km/s, $\sigma_{8}=0.8$ and plot results in the Figure 1. ## III Discussion & Conclusions Inspecting Figure 1 we see the expected behavior. The Felinic principle bias falls rapidly with redshift and becomes completely negligible by $z\sim 0.04$ when compared to the precision of current observations. At redshift $z\sim 0.05$, corresponding to the measurement of the Hubble parameter by the Riess et al. (2011), we see that the calculated effect is around $0.02$ km/s/Mpc. We see that Felinic principle cannot alleviate the tension with Planck satellite Planck Collaboration et al. (13, 14). Moreover, it has the wrong sign. While conveniently ignoring the fact that measurements of the Hubble parameter with the Lyman-$\alpha$ forest are performed using a completely different method, we note that the bias of the Hubble parameter measurement is $0.5\times 10^{6}$ smaller at the redshift of $z\sim 2.4$ compared to direct measurements at $z\sim 0.05$. This confirms that Lyman-$\alpha$ forest measurements of the Hubble parameter Busca et al. (2012); Slosar et al. (2013) are approximately $500,000$ times more glorious than the Hubble parameter measurement using Cepheid variables. More importantly, we note that the fact that the effect is negligible at all redshifts leads to an important corollary. Namely, the likelihoods cannot distinguish between us existing and not, but in the spirit of Bayesian evidence and applying _lex parsimoniae_, we can conclude, that posterior indeed prefers that we do not exist. It is true that other physical effects can affect the result. For example, if the local non-Gaussianity parameter $f_{\rm NL}\sim 10^{9}$, this would lead to observable effects. Just imagine what it would do! We also note that this paper is of an appropriate quality to considered for publication in _Nature_, and in fact, it would be one of the finer cosmology contributions to that respected publication. Finally, we conclude by reminding the reader, that the galaxy clusters are still believed to be the largest gravitationally bound structures in the Universe. ## Acknowledgments We acknowledge useful discussions with many civilized people. Donations of cat food can be sent to the correspondence address. Subjects not sending _Fancy Feast Savory Salmon_ need not apply. ## References * Deutsch (1962) M. Deutsch, _Cooperation and trust: Some theoretical notes_ (University of Nebraska Press, Lincoln, NE, 1962). * Limber (1954) D. N. Limber, Astrophys. J. 119, 655 (1954). * Hong et al. (2012) S. E. Hong, E. D. Stewart, and H. Zoe, Phys. Rev. D 85, 083510 (2012), eprint 1110.3119. * Křížek (2012) M. Křížek, New Astronomy 17, 1 (2012). * Gilmour and Middleton (2009) J. D. Gilmour and C. A. Middleton, Icarus 201, 821 (2009). * Maor et al. (2008) I. Maor, L. Krauss, and G. Starkman, Physical Review Letters 100, 041301 (2008), eprint 0709.0502. * Peacock (2007) J. A. Peacock, _Mon. Not. Roy. Astron. Soc._ 379, 1067 (2007), eprint 0705.0898. * Pogosian and Vilenkin (2007) L. Pogosian and A. Vilenkin, _JCAP_ 1, 025 (2007), eprint arXiv:astro-ph/0611573. * Bousso (2012) R. Bousso, Phys. Rev. D 86, 023532 (2012), eprint 1205.6634. * Bousso et al. (2011) R. Bousso, B. Freivogel, S. Leichenauer, and V. Rosenhaus, Physical Review Letters 106, 101301 (2011), eprint 1011.0714. * Bousso et al. (2008) R. Bousso, B. Freivogel, and I.-S. Yang, Phys. Rev. D 77, 103514 (2008), eprint 0712.3324. * Riess et al. (2011) A. G. Riess, L. Macri, S. Casertano, H. Lampeitl, H. C. Ferguson, A. V. Filippenko, S. W. Jha, W. Li, and R. Chornock, Astrophys. J. 730, 119 (2011), eprint 1103.2976. * Planck Collaboration et al. (2013a) Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, et al., ArXiv e-prints (2013a), eprint 1303.5062. * Planck Collaboration et al. (2013b) Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, et al., ArXiv e-prints (2013b), eprint 1303.5076. * Busca et al. (2012) N. G. Busca, T. Delubac, J. Rich, S. Bailey, A. Font-Ribera, D. Kirkby, J.-M. Le Goff, M. M. Pieri, A. Slosar, É. Aubourg, et al., ArXiv e-prints (2012), eprint 1211.2616. * Slosar et al. (2013) A. Slosar, V. Iršič, D. Kirkby, S. Bailey, N. G. Busca, T. Delubac, J. Rich, É. Aubourg, J. E. Bautista, V. Bhardwaj, et al., ArXiv e-prints (2013), eprint 1301.3459.
1305.6278	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 25622, "num_imgs": 3, "llama3_tokens_count": 6345 }	[ "content_image/1305.6278/x1.png", "content_image/1305.6278/x2.png", "content_image/1305.6278/x3.png" ]	# Bounds on Thermal Efficiency from Inference Ramandeep S. Johal rsjohal@iisermohali.ac.in Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Sector 81, S.A.S. Nagar, Manauli P.O., Mohali 140306, India Renuka Rai Department of Applied Sciences, University Institute of Engineering and Technology, Panjab University, Chandigarh-160014, India. Günter Mahler Universität Stuttgart, 1. Institut für Theoretische Physik, Pfaffenwaldring 57 // IV, 70550 Stuttgart, Germany ###### Abstract The problem of inference is applied to the process of work extraction from two constant heat capacity reservoirs, when the thermodynamic coordinates of the process are not fully specified. The information that is lacking, includes both the specific value of a temperature as well as the label of the reservoir to which it is assigned. The estimates for thermal efficiency reveal that uncertainty regarding the exact labels, reduces the maximal efficiency below the Carnot value, its minimum value being the well known Curzon-Ahlborn value. We also make an average estimate of the efficiency _before_ the value of the temperature is revealed. It is found that if the labels are known with certainty, then in the near-equilibrium limit the efficiency scales as 1/2 of Carnot value, while if there is maximal uncertainty in the labels, then the average estimate for efficiency drops to 1/3 of Carnot value. We also suggest how infered properties of the incomplete model can be mapped to a model with complete information but with an additional source of thermodynamic irreversibility. Inference; Uncertainty; Thermal efficiency; Irreversibility pacs: 05.70.−a, 05.70.Ln, 02.50.Cw ## I Introduction In many situations, we have to reason from incomplete information. Scientific inference refers to application of a consistent set of principles in these situations, which satisfy our rationality. In its initial stages, it was termed as the “the art of conjecturing” by Bernoulli [1]. Later refined into a technical tool, Laplace [2] made a successful use of what is now known as Bayes’ formula [3]. Later, Cox showed that the only set of consistent axioms justifying plausible reasoning were the already established axioms of the probability theory [4; 5]. Many authors have clarified the scope and meaning of inference [6; 7; 8; 9]. Following Jaynes, much of the development in thermodynamics and statistical mechanics may be regarded as an application of the principles of plausible reasoning. Usually, the physical mechanisms which generate thermodynamic irreversibility are ascribed to a finite rate of heat transfer, internal friction, the finite size of the reservoirs and so on. However, from the perspective of information theory, this irreversibility is also related to a loss of information about the system into the environment. In this paper, we consider reversible thermodynamic models but with incomplete information. The issue we address here is that from inference performed on such models, the estimated behavior exhibits some features of irreversibility. Thus in our case, irreversibility does not appear at the objective physical level, but only in the estimated behavior. The central feature of our approach is that the missing part of information is interpreted in a subjective manner, or in other words as the observer’s lack of knowledge about the system. Due to lack of complete information, the observer has to perform inference to estimate the characteristic quantities of the system. We show that a consistent use of prior information in reversible models leads to an estimate for maximal efficiency which is lower than the Carnot value. We also suggest that the resulting inferred behavior is analogous to that obtained by incorporating explicitly some thermodynamic irreversibility in the actual (reversible) process. The motivation for our approach comes from the connection between thermodynamics and information. This has prompted a fruitful discussion on the role of information theory in thermodynamic frameworks, for example the role of Maxwell’s demon in information processing [10] which is continuing to this day. Recently, it has also been explored in [11; 12; 13; 14], that the identification and inclusion of prior information in heat cycles with incomplete specification, leads to interesting analogies with irreversible models. In particular, many different efficiencies show up in the inference based approach which are found in the context of time-dependent cycles or with other sources of irreversibility [15]. Now the prior information, which is to be exploited in making inference, can be present in different forms, some even quite raw or qualitative. In this paper, we want to understand further how different kinds of prior information, impact our expectations about the performance of these heat cycles. In particular, we assume an uncertainty not only in the value of a parameter, but also an uncertainty in the exact subsystem to which it is assigned. The later kind of uncertainty will be addressed as _label uncertainty_. For example, a classical system may be specified by a set of quantities $\{X,Y,..\}$. In the case of a multi-partite system, we also label the subsystems, say with index $i$. Then the properties of all constituent subsystems are distinguished if our labels are refined as $\{X_{i},Y_{i},...\}$. In the following, we consider a situation where the values of the individual parameters ($X,Y,..$) are known, but we may be uncertain about the exact subsystem labels. The question is how do we estimate the performance of the system based on this incomplete information. For simplicity, we will consider only a bipartite set up. The paper is organised as follows. Section II discusses the model and has three subsections. In subsection A, we introduce a reversible thermodynamic model of work extraction, with complete specification of all parameters. In subsection B, we assume uncertainty in the final temperatures as well as the label uncertainty. The work performed and the thermal efficiency are estimated for a given measure of uncertainty. In subsection C, we draw analogy of the infered behavior with an explicitly irreversible model of heat engine. Finally, in Section III we define an average estimate of efficiency, which is calculated using a uniform prior distribution over the uncertain temperature. The behavior of this average value for near-equilibrium is evaluated, which leads to establishing two distinct classes for the expected efficiency, based on zero or complete label uncertainty. The last section IV presents the conclusions. ## II The Model ### The case of complete information It is sufficient to consider a textbook example of two _finite_ ideal gas systems with a constant heat capacity $C$, at initial temperatures $T_{+}$ and $T_{-}$ ($T_{+}>T_{-}$), serving as the heat source and the sink, respectively. They are coupled via a reversible work source, which by design extracts maximal work due to the available temperature gradient. At some stage in this process, the initially hot reservoir obtains a temperature $T_{1}$ and the initially cold reservoir is at temperature $T_{2}$. The amount of heat taken in by the engine and the heat rejected to the sink, are respectively given by $Q_{\rm in}=C(T_{+}-T_{1})$, and $Q_{\rm out}=C(T_{2}-T_{-})$, respectively. The total entropy change in the two reservoirs being zero, we have $\bigtriangleup S=C\ln{(T_{1}/T_{+})}+C\ln{(T_{2}/T_{-})}=0$. This yields \[T_{1}=\frac{T_{+}T_{-}}{T_{2}},\] (1) as the relation between the final reservoir temperatures. The work performed, $W=Q_{\rm in}-Q_{\rm out}$ is: $W=C(T_{+}+T_{-}-T_{1}-T_{2})$. In the following, we set $C=1$. Finally, using Eq. (1), the efficiency $\eta=W/Q_{\rm in}$, can be written as: \[\eta = 1-\frac{T_{2}}{T_{+}},\] (2) \[= 1-\frac{T_{-}}{T_{1}}.\] (3) We note that the maximum work is obtained if the final temperatures obtained are: $T_{1}=T_{2}=\sqrt{T_{+}T_{-}}$, and the efficiency at this optimal process is $\eta=\eta^{}=1-\sqrt{{T_{-}}/{T_{+}}}$. Now in the standard analysis, $T_{+}$ and $T_{-}$ are the fixed initial values of the temperatures, and due to relation (1), we may regard all the expressions as functions of only one of the two temperatures, $T_{1}$ or $T_{2}$. Thus the work performed, can be rewritten as \[W(T_{2})=\left(T_{+}+T_{-}-T_{2}-\frac{T_{+}T_{-}}{T_{2}}\right),\] (4) with a similar expression in terms of $T_{1}$. Now note that just from the work expression, Eq. (4), it is not obvious as to which temperature is chosen as the variable. We have to look at the expression for the heat exchanged to assess the label corresponding to a specific reservoir. So when we have an exact knowledge about the temperatures, this information has two parts: i) the individual values of the temperatures and ii) the labels for the reservoirs to which a value is assigned. Thus the symbol $T_{2}$ denotes the temperature value of the particular reservoir (label 2). ### Incomplete information Now let us imagine a controller of the process who knows the final thermodynamic coordinates, or the temperatures of the reservoirs. The controller invites us to play a game of guessing and promises to reveal one of the values of the temperatures, but _not_ the reservoir to which this value belongs. The task ahead of us is to infer the performance of the engine by making estimates about work performed, efficiency and so on. As mentioned above, the work expression does not reveal unambiguously the individual labels of the reservoirs. Thus given some temperature value $T$, the work expression (written devoid of reservoir labels) will be $W(T)=T_{+}+T_{-}-T-{T_{+}T_{-}}/{T}$. Next comes the issue of the range of possible values for the final temperature $T$. This should be fixed from the information that is actually available. In particular, we invoke the fact that the extracted work satisfies: $W\geq 0$, so that $T$ is allowed to take values in the range $[T_{-},T_{+}]$. Thus to some extent, we have removed the problem from its physical context which involves such notions as the flow of heat from a hot to cold temperature and so on. In the spirit of an inference based approach, we seek to quantify our beliefs focusing on the prior information. To illustrate how our estimates are affected as our beliefs change, suppose further that we have a reason to believe that the disclosed value of temperature belongs to a specific reservoir. We quantify this belief by assigning a probability with numerical value $\gamma$ ($0\leq\gamma\leq 1$) to the hypothesis that the disclosed value $T$ belongs to the initially hot reservoir (henceforth labeled A). The parameter $\gamma$ is assumed to be independent of the $T$ value. Now we know that _if_ one of the final temperatures is $T$, the corresponding value for the other reservoir definitely is $T_{+}T_{-}/T$. But as per our beliefs, the final temperature of reservoir A is: (i) $T$, with probability $\gamma$, and (ii) $T_{+}T_{-}/T$, with probability $1-\gamma$. Then upon knowing the value $T$, the expected final temperature of reservoir A may be defined as a weighted average over the two values: \[\overline{T}_{A}=\gamma T+(1-\gamma)\frac{T_{+}T_{-}}{T}.\] (5) This is our estimate for the final temperature of reservoir A, given the information that it is a maximum work process and, one of the final temperatures is $T$. Correspondingly, for reservoir B, we should have: \[\overline{T}_{B}=(1-\gamma)T+\gamma\frac{T_{+}T_{-}}{T}.\] (6) Now we use these values to estimate further other quantities, which are relevant to the performance of the engine. Our estimate of the heat absorbed by the engine from reservoir A, is given by \[Q_{\rm in}=T_{+}-\overline{T}_{A}.\] (7) Similarly, our estimate for the heat rejected to reservoir B, will be: \[Q_{\rm out}=\overline{T}_{B}-T_{-}.\] (8) The estimate for work defined as: $W=Q_{\rm in}-Q_{\rm out}$, turns out to be $W=T_{+}+T_{-}-T-{T_{+}T_{-}}/{T}$, i.e. equal to the actual work performed. In particular, the estimate for work is independent of the parameter $\gamma$, showing that the work is not affected by label uncertainty. For brevity, we now calibrate all the temperatures, relative to the initial temperature of reservoir A, and define $\theta=T_{-}/T_{+}$. Then the expected work for a given value $T$, is \[W=1+\theta-T-\frac{\theta}{T}.\] (9) Finally, we note that the estimate for the efficiency $\eta=W/Q_{\rm in}$, given by \[\eta_{\gamma}(T)=\frac{T+\theta T-T^{2}-\theta}{T-\gamma T^{2}-(1-\gamma) \theta},\] (10) is also affected by the label uncertainty. Let us now look at the behavior of the above efficiency for some special values of $\gamma$. When $\gamma=1$ ($0$), it corresponds to the case when we are certain that the disclosed value $T$ belongs to reservoir A (B). In these cases, Eq. (10) reduces to $\eta_{0}=1-T$ and $\eta_{1}=1-\theta/T$, respectively. On the other extreme, when we are maximally uncertain about the label for the temperature $T$, or which means equal probabilities for $T$ to belong to any of the two reservoirs, then we must assign $\gamma=1/2$. Then from Eq. (10), the efficiency is expected to be \[\eta_{\frac{1}{2}}(T)=\frac{2(T+\theta T-T^{2}-\theta)}{(2T-T^{2}-\theta)}.\] (11) Now there are two quantities of interest: a) _Maximum work_: this is obtained by setting $\partial W/\partial T=0$[16], which holds at $T=\sqrt{\theta}$. The efficiency at maximum work is the well-known Curzon-Ahlborn formula: $\eta_{\rm CA}=1-\sqrt{\theta}$. b) _Estimated maximum efficiency_: this is given by the condition: $\partial\eta_{\gamma}/\partial T=0$. From Eq. (10), the maximum value is: \[\eta^{}_{\gamma}=\frac{1-\theta}{1+\sqrt{4\gamma(1-\gamma)\theta}}.\] (12) <figure><img src="content_image/1305.6278/x1.png"><figcaption>Figure 1: A parametric plot between W(T) as in Eq. (9) and efficiency in Eq.(10), for θ=0.25. T takes values in the range [θ,1] and γ is kept fixed: 1/2(thick black, monotonic curve), 1/4 (dotted), 1/8 (thin), and 1/10000(dashed). The point for maximum work remains the same, while the maximum inthe efficiency shifts with γ. There is a finite work obtained at the maximumof efficiency, which reduces and goes to zero as γ→0 (or γ→1), whereby themaximum efficiency reaches Carnot value, 1−θ=0.75.</figcaption></figure> From Eq. (12), we see that in both cases of certainty about the labels, implying maximal information, the maximum efficiency is the Carnot value. But an uncertainty in the exact labels reduces the maximal efficiency ($\eta^{}_{\gamma}$) below the Carnot value, and it reduces to CA value, in the case of maximal uncertainty ($\gamma=1/2$). Thus the upper bound for efficiency is related here directly with our state of knowledge and the CA efficiency emerges from an entirely different perspective. Here we have a mechanism which shows how inference under incomplete information leads us to expect a lower value for the maximum efficiency obtainable from a thermal engine. This feature is further exhibited in Fig. 1, which shows the work versus efficiency curves, each with a fixed value of $\gamma$. In general, the maximum work and the maximum efficiency points are different. At $\gamma=0$ and $\gamma=1$, they are maximally separate, but tend to merge together as $\gamma\to 1/2$. <figure><img src="content_image/1305.6278/x2.png"><figcaption>Figure 2: Plot of Eq. (10) for θ=0.25. T takes values in the range [θ,1]. γvaries as 1/2 (thick black), 1/4 (dashed), 1/8 (thin), 3/4 (dotted) and 19/20(thick gray). The maximum value of efficiency for a given γ is given by Eq.(12). As γ→0 or γ→1, the maximal value approaches the carnot value, 1−θ=0.75.</figcaption></figure> ### Analogy with irreversible model The fact that the maximal efficiency drops below the Carnot efficiency, indicates that we are infering an irreversible behavior for an otherwise reversible model with incomplete information. In this section, we suggest an irreversible physical process which corresponds to the above picture obtained through inference. The reservoirs A and B are initially at (scaled) temperatures $1$ and $\theta$, respectively. Now imagine a two-step process. In the first step, work is extracted by coupling reservoirs with a reversible work source, so that at the end of this step, reservoir A reaches temperature $T$ and consequently, reservoir B is at temperature $\theta/T$. The work performed is given by Eq. (9). In the second step, the two reservoirs are detached from the work source and put in mutual thermal contact. Heat is exchanged between them, conserving their total internal energy. Let at the end of the second-step, the temperatures of the reservoirs be $\overline{T}_{A}$ and $\overline{T}_{B}$, as given by Eqs. (5) and (6). The second step is intrinsically irreversible. The net heat released by reservoir A is: $Q_{\rm in}=1-\overline{T}_{A}$ and the net heat rejected to reservoir B is: $Q_{\rm out}=\overline{T}_{B}-\theta$. These quantities are the same as obtained through inference in Section II.B. In the above two-step process, we are assigning definite temperatures to the reservoirs A and B. Due to the second law, we expect that overall the heat flows from hot to cold reservoir. So at the end of work extracting process, we require that \[T\geq\theta/T.\] (13) If $T=\theta/T$, then the second step is redundant. So if $T>\theta/T$ holds then by the end of second step, we also expect that \[\overline{T}_{A}\geq\overline{T}_{B}.\] (14) It can be easily seen that the above condition requires $\gamma\geq 1/2$, with $\overline{T}_{A}=\overline{T}_{B}$ implying $\gamma=1/2$. ## III Average estimate of efficiency So far, we have assumed that the values of final temperatures are pre-specified. For instance, Eq. (11) provides an estimate of efficiency for a given value $T$ but with complete ignorance about the reservoir labels. If we are only provided the value $T$, then we must take $\gamma=1/2$ and our estimate for efficiency will be Eq. (11). In this section, we extend the game of guessing further and estimate the efficiency _before_ we are provided the value $T$. So now we assume to be ignorant about the value $T$ also. To quantify our guess in the absence of the value $T$, we have to specify a prior distribution for $T$[13; 14], which takes into account our belief as to which value $T$ from the allowed range, the controller may be holding. If we do not have a reason to expect one value over another, then all allowed values are equally likely in the interval $[T_{-},T_{+}]\equiv[\theta,1]$. So we must adopt a uniform prior distribution for $T$. Our average estimate for efficiency, defined as the mean value over this uniform prior is then \[\overline{\eta}_{\frac{1}{2}}(\theta)=\int_{\theta}^{1}\eta_{\frac{1}{2}}(T)\! \frac{dT}{1-\theta}.\] (15) On using Eq. (11) and solving Eq. (15), we obtain \[\bar{\eta}_{\frac{1}{2}} = 2-\left(1+\frac{1}{\sqrt{1-\theta}}\right)\ln\left(1+\sqrt{1- \theta}\right)\] (16) \[-\left(1-\frac{1}{\sqrt{1-\theta}}\right)\ln\left(1-\sqrt{1- \theta}\right).\] For close to equilibrium situations, i.e. $1-\theta\approx 0$, the average value of efficiency behaves as follows \[\bar{\eta}_{\frac{1}{2}}=\frac{1-\theta}{3}-\frac{(1-\theta)^{2}}{10}+O[1- \theta]^{3}.\] (17) On the other extreme, consider the two cases of certainty about the reservoir labels. For the special case of $\gamma=0$, we have $\eta_{0}(T)=1-T$, whose average over the uniform prior is \[\overline{\eta}_{0}=\frac{1-\theta}{2}.\] (18) Also for $\gamma=1$, we have $\eta_{1}(T)=1-\theta/T$ and the average value over uniform prior, is given by \[\overline{\eta}_{1}=1+\frac{\theta\ln\theta}{1-\theta},\] (19) whose expansion behaves as: $\overline{\eta}_{1}\approx(1-\theta)/2+(1-\theta)^{2}/6+O[1-\theta]^{3}$. Thus, both of the above averages yield that for near-equilibrium conditions, the average efficiency is given by $(1-\theta)/2$. This result holds if the reservoir labels are known i.e. we know which reservoir temperature is chosen as the uncertain variable. In contrast, we obtain one-third of Carnot value, if we are maximally uncertain about the specific reservoir labels. This is the main result of the paper, that the expected efficiency near equilibrium falls under two different classes, determined by the state of our knowledge about the system. Fig. 3 shows this dependence on $\gamma$ in a more clear fashion. <figure><img src="content_image/1305.6278/x3.png"><figcaption>Figure 3: Plot of the average efficiency over uniform prior versus γ, forθ=0.85. The horizontal lines denote (1−θ)/2 and (1−θ)/3 which act as lowerbounds for the case γ=1, and γ=1/2, respectively. For γ=0, the efficiencyestimate is exactly (1−θ)/2.</figcaption></figure> ## IV Summary Inference is a kind of common sense reasoning in the face of incomplete information. It seeks to provide the most unbiased guess consistent with the constraints and laws obeyed by the system. We have considered a plausible use of this reasoning for classical thermodynamic machines operating under reversible conditions. An important new kind of uncertainty studied in this paper relates to incomplete knowledge about the labels of the uncertain parameters. We define a guessing game which combines features of subjective ignorance with the objective matters of fact about the physical system. Some interesting observations made are that the infered behavior has features of irreversibility, for example, the maximum efficiency of the engine is below Carnot limit. We also suggested an analog physical model which mimics the estimates of thermodynamic quantities in the infered model. It is not known if the mapping to an actual physical model is always guaranteed from a given inference based model. Further, we cannot be sure about the uniqueness of this mapping, and there may exist more than one physical processes which simulate the consequences of our game, or conversely, more than one games which simulate a physical process. However, in our opinion, the possibility of a mapping to an objective physical model indicates that inference anticipates a behavior which is allowed by the physical laws. Thus the mapping also reassures that the estimates are consistent with the laws of thermodynamics. We have estimated an average efficiency based on the prior probability distribution and found that the so called label uncertainty makes the efficiency drop to 1/3 of Carnot value in near-equilibrium regime. On the other hand, if there is no label uncertainty, the infered efficiency is equal to 1/2 of Carnot value. This is consistent with our expectation that more information we lack, less is the efficiency we expect. Still the goal of inference is not to predict the “true” behavior of physical models proper, but to make a rational guess based on incomplete information. In this paper we have defined a set of specific “games” representing additional subjective lack of information. This setting supplements the “objective ignorance” present already due to the laws of thermodynamics. Nevertheless, it turns out that the combined model may be equivalent to a “purely physical” model with some specific irreversible features – a remarkable non-uniqueness in terms of interpretation. It is hoped that the approach and conclusions of this paper may help to further develop procedures which incorporate different kinds of prior information, which then could shed light on the subtle interplay between subjective and objective ignorance presumed in the modeling of natural phenomena like, e.g., in the context of (rather complex) climate models. ## References (1) J. Bernoulli, _The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis_, translated by Edith Dudley Sylla, Johns Hopkins University Press (2006) Baltimore. * (2) P.S. Laplace, _Memoir on the Probabilities of the Causes of Events_, translated by S. M. Stigler, Stat. Sc. 1, 364 (1986). * (3) T. Bayes, Phil. Trans. Roy. Soc. 330 (1763). * (4) R. T. Cox, Am. J. Phys. 17, 1 (1946). * (5) R.T. Cox, _Algebra of Probable Inference_, The Johns Hopkins University Press (2001). * (6) H. Jeffreys, _Scientific Inference_, Cambridge University Press (1931). * (7) H. Jeffreys, _Theory of Probability_, Second edition, Clarendon Press, Oxford (1948). * (8) G. Polya, _Mathematics and Plausible Reasoning_, Vol. I and II, Princeton University Press (1954). * (9) E. T. Jaynes, _Probability Theory: The Logic of Science_ (Cambridge University Press, Cambridge, 2003). * (10) H. S. Leff and A. F. Rex, _Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing_ (Institute of Physics, Bristol, 2003). * (11) R.S. Johal, Phys. Rev. E 82 041119 (2010). * (12) G. Thomas and R.S. Johal, Phys. Rev. E 85 041146 (2012). * (13) P. Aneja and R.S. Johal, Proceedings of _Sigma-Phi International Conference on Statistical Physics_-2011, Cent. Eur. J. Phys. 10 (3) 708 (2012). * (14) G. Thomas, P. Aneja and R.S. Johal, Proceedings of _International Conference on Frontiers of Quantum and Mesoscopic Thermodynamics_-2011, Phys. Scr. T151 014031 (2012). * (15) See [11; 13] and references therein. * (16) H.B. Callen, _Thermodynamics and an Introduction to Thermostatistics_, Second edition, John Wiley & Sons (1985).
1411.7559	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 27948, "num_imgs": 1, "llama3_tokens_count": 9421 }	[ "content_image/1411.7559/x1.png" ]	# A Light-Front approach to the ${}^{3}$He spectral function Sergio Scopetta S. Scopetta Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy Tel.: +39-075-5852721 Fax: +39-075-44666 ¹A. Del Dotto Dipartimento di Fisica, Università di Roma Tre and INFN, Roma 3, Italy L. Kaptari Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia E. Pace Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Roma 2, Italy M. Rinaldi Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy G. Salmè INFN Sezione di Roma, Italy Alessio Del Dotto S. Scopetta Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy Tel.: +39-075-5852721 Fax: +39-075-44666 ¹A. Del Dotto Dipartimento di Fisica, Università di Roma Tre and INFN, Roma 3, Italy L. Kaptari Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia E. Pace Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Roma 2, Italy M. Rinaldi Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy G. Salmè INFN Sezione di Roma, Italy Leonid Kaptari S. Scopetta Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy Tel.: +39-075-5852721 Fax: +39-075-44666 ¹A. Del Dotto Dipartimento di Fisica, Università di Roma Tre and INFN, Roma 3, Italy L. Kaptari Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia E. Pace Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Roma 2, Italy M. Rinaldi Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy G. Salmè INFN Sezione di Roma, Italy Emanuele Pace S. Scopetta Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy Tel.: +39-075-5852721 Fax: +39-075-44666 ¹A. Del Dotto Dipartimento di Fisica, Università di Roma Tre and INFN, Roma 3, Italy L. Kaptari Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia E. Pace Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Roma 2, Italy M. Rinaldi Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy G. Salmè INFN Sezione di Roma, Italy Matteo Rinaldi S. Scopetta Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy Tel.: +39-075-5852721 Fax: +39-075-44666 ¹A. Del Dotto Dipartimento di Fisica, Università di Roma Tre and INFN, Roma 3, Italy L. Kaptari Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia E. Pace Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Roma 2, Italy M. Rinaldi Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy G. Salmè INFN Sezione di Roma, Italy Giovanni Salmè S. Scopetta Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy Tel.: +39-075-5852721 Fax: +39-075-44666 ¹A. Del Dotto Dipartimento di Fisica, Università di Roma Tre and INFN, Roma 3, Italy L. Kaptari Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia E. Pace Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Roma 2, Italy M. Rinaldi Dipartimento di Fisica e Geologia, Università di Perugia and INFN, Sezione di Perugia, Italy G. Salmè INFN Sezione di Roma, Italy [FOOTNOTE:2][ENDFOOTNOTE] Received: date / Accepted: date ###### Abstract The analysis of semi-inclusive deep inelastic electron scattering off polarized ${}^{3}$He at finite momentum transfers, aimed at the extraction of the quark transverse-momentum distributions in the neutron, requires the use of a distorted spin-dependent spectral function for ${}^{3}$He, which takes care of the final state interaction effects. This quantity is introduced in the non-relativistic case, and its generalization in a Poincaré covariant framework, in plane wave impulse approximation for the moment being, is outlined. Studying the light-front spin-dependent spectral function for a J=1/2 system, such as the nucleon, it is found that, within the light-front dynamics with a fixed number of constituents and in the valence approximation, only three of the six leading twist T-even transverse-momentum distributions are independent. † [FOOTNOTE:†][ENDFOOTNOTE] ∎ ## 1 Introduction Information on the three-dimensional proton structure can be obtained from the quark transverse momentum distributions (TMDs) [1], which can be accessed through semi-inclusive deep inelastic electron scattering (SIDIS). In particular the single spin asymmetries (SSAs) allow one to extract the Sivers and the Collins contributions, expressed in terms of different TMDs and fragmentation functions (ff) [1]. A large Sivers asymmetry was measured in ${{{\vec{p}}(e,e^{\prime}\pi)x}}$[2] and a small one in ${{{\vec{D}}(e,e^{\prime}\pi)x}}$[3], showing a strong flavor dependence of TMDs. To clarify the flavour dependence of the quark transverse momentum distributions, high precision experiments, involving both protons and neutrons, are needed [4]. In Ref. [5] an experiment to extract information on the neutron TMDs from experimental measurements of the SSAs on ${}^{3}$He, at JLab at 12 GeV, was proposed. To obtain a reliable information one has to take carefully into account the structure of ${}^{3}$He, the interaction in the final state (FSI) between the observed pion and the remnant debris, and the relativistic effects. The present paper reports on our efforts about these items. ## 2 SIDIS off ${}^{3}$He A polarized ${}^{3}$He is an ideal target to study the polarized neutron since, at a 90% level, a polarized ${}^{3}$He is equivalent to a polarized neutron. Dynamical nuclear effects in inclusive deep inelastic electron scattering (DIS) ${}^{3}\vec{He}(e,e^{\prime})X$ (DIS) were evaluated [6] with a realistic ${}^{3}{{\vec{He}}}$ spin-dependent spectral function, ${{P^{\tau}_{\sigma,\sigma{\prime}}({\bf p},E,S_{He})}}$, with $\bf p$ the initial nucleon momentum in the laboratory frame and $E$ the missing energy [7]. It was found that the formula \[{{A_{n}}}\simeq{1\over{{p_{n}}}f_{n}}\left({A^{exp}_{3}}-2{p_{p}}f_{p}~{}{{A^{ exp}_{p}}}\right)\] (1) can be safely adopted to extract the neutron information from ${}^{3}$He and proton data. This formula is actually widely used by experimental collaborations. The nuclear effects are hidden in the proton and neutron ”effective polarizations”, $p_{p(n)}$. With the AV18 nucleon-nucleon interaction [8] the values ${{p_{p}}}=-0.023$, ${{p_{n}}}=0.878$ were obtained [9]. The quantities $f_{p(n)}$ in Eq. (1) are the dilution factors. To investigate if an analogous formula can be used to extract the SSAs, in [9] the processes ${}^{3}\vec{He}(e,e^{\prime}\pi^{\pm})X$, with ${}^{3}$He transversely polarized, were evaluated in the Bjorken limit and in PWIA, i.e. the final state interaction (FSI) was considered only within the two-nucleon spectator pair which recoils. In such a framework, SSAs for ${}^{3}$He involve convolutions of ${{P^{\tau}_{\sigma,\sigma^{\prime}}({\bf p},E,S_{He})}}$, with TMDs and ffs. Ingredients of the calculations were: i) a realistic ${{P^{\tau}_{\sigma,\sigma^{\prime}}({\bf p},E,S_{He})}}$ for ${}^{3}$He, obtained using the AV18 interaction; ii) parametrizations of data for TMDs and ff, whenever available; iii) models for the unknown TMDs and ff. It was found that, in the Bjorken limit, the extraction procedure through the formula successful in DIS works nicely for the Sivers and Collins SSAs [9]. The generalization of Eq. (1) to extract Sivers and Collins asymmetries from ${}^{3}$He and proton asymmetries was recently used by experimental collaborations [10]. In SIDIS experiments off ${}^{3}$He, the relative energy between the spectator $(A-1)$ system and the system composed by the detected pion and the remnant debris (see Fig. 1) is a few GeV and FSI can be treated through a generalized eikonal approximation (GEA) [11]. The GEA was already succesfully applied to unpolarized SIDIS in Ref. [12]. The FSI effects to be considered are due to the propagation of the debris, formed after the $\gamma^{}$ absorption by a target quark, and the subsequent hadronization, both of them influenced by the presence of a fully interacting $(A-1)$ spectator system (see Fig. 1). <figure><img src="content_image/1411.7559/x1.png"><figcaption>Figure 1: Interaction between the (A−1) spectator system and the debrisproduced by the absorption of a virtual photon by a nucleon in the nucleus.</figcaption></figure> In this approach, the key quantity to introduce the FSI effects is the _distorted_ spin-dependent spectral function, whose relevant part in the evaluation of SSAs is: \[\quad\quad\quad{\cal P}_{\|\|}^{PWIA({{FSI}})}={\cal O}^{IA({{FSI}} )}_{\frac{1}{2}\frac{1}{2}}-{\cal O}^{IA({{FSI}})}_{-\frac{1}{2}-\frac{1}{2}}\] (2) with: \[{\cal O}^{IA}_{\lambda\lambda^{\prime}}(\vec{p},E)=\sum\!\!\!\!\! \!\!\!\int{~{}d\epsilon^{}_{A-1}}~{}\rho\left(\epsilon^{}_{A-1}\right)~{} \langle S_{A},{{{\bf P}_{A}}}\|\Phi_{\epsilon^{}_{A-1}},\lambda^{\prime},\vec{ p}\rangle\langle\Phi_{\epsilon^{}_{A-1}},\lambda,\vec{p}\|S_{A},{{{\bf P}_{A}} }\rangle~{}\delta\left(E-B_{A}-\epsilon^{}_{A-1}\right)~{},\] (3) and \[{\cal O}^{{FSI}}_{\lambda\lambda^{\prime}}(\vec{p},E) = \sum\!\!\!\!\!\!\!\!\int{~{}d\epsilon^{}_{A-1}}~{}\rho\left( \epsilon^{}_{A-1}\right)~{}\langle S_{A},{{{\bf P}_{A}}}\|({{\hat{S}_{Gl}}})\{ \Phi_{\epsilon^{}_{A-1}},\lambda^{\prime},\vec{p}\}\rangle\langle({{\hat{S}_{ Gl}}})\{\Phi_{\epsilon^{}_{A-1}},\lambda,\vec{p}\}\|S_{A},{{{\bf P}_{A}}}\rangle\] (4) \[\times \delta\left(E-B_{A}-\epsilon^{}_{A-1}\right)~{},\] where $\rho\left(\epsilon^{}_{A-1}\right)$ is the density of the $(A-1)$-system states with intrinsic energy $\epsilon^{}_{A-1}$, and $\|S_{A},{{{\bf P}_{A}}}\rangle$ is the ground state of the $A$-nucleons nucleus with polarization $S_{A}$. ${\hat{S}_{Gl}}$ is the Glauber operator: \[\,\,{{\hat{S}_{Gl}}}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3})=\prod_{ i=2,3}\bigl{[}1-\theta(z_{i}-z_{1}){{\Gamma}}({\bf b}_{1}-{\bf b}_{i},{z}_{1}- {z}_{i})\bigr{]}\] (5) and $\Gamma({{\bf b}_{1i}},z_{1i})$ the generalized profile function: \[\quad{{\Gamma({{\bf b}_{1i}},z_{1i})}}\,=\,\frac{(1-i\,\alpha)\,\,{{\sigma_{ eff}(z_{1i})}}}{4\,\pi\,b_{0}^{2}}\,\exp\left[-\frac{{\bf b}_{1i}^{2}}{2\,b_{0 }^{2}}\right]~{},\] (6) where ${\bf r}_{1i}=\{{\bf b}_{1i},{\bf z}_{1i}\}$ with ${\bf z}_{1i}={\bf z}_{1}-{\bf z}_{i}$ and ${\bf b}_{1i}={\bf b}_{1}-{\bf b}_{i}$. The models for the profile function, $\Gamma({{\bf b}_{1i}},z_{1i})$, and for the effective cross section, $\sigma_{eff}(z_{1i})$, as well as the values of the parameters $\alpha$ and $b_{0}$ are the ones exploited in Ref. [12] to nicely describe the JLab data corresponding to the unpolarized spectator SIDIS off the deuteron. It occurs that ${\cal P}_{\|\|}^{PWIA}$ and ${\cal P}_{\|\|}^{FSI}$ can be very different, but the relevant observables for the SSAs involve integrals, dominated by the low momentum region, where the FSI effects on ${\cal P}_{\|\|}$ are minimized and the spectral function is large [11]. As a consequence the effective nucleon polarizations change from ${{p_{p}}}=-0.023$, ${{p_{n}}}=0.878$ to ${{p_{p}^{FSI}}}=-0.026$, ${{p_{n}^{FSI}}}=0.76$, where \[{p^{FSI}_{p(n)}}=\int d\epsilon_{S}\int d{\bf p}~{}{{Tr[{\bf{S}}_{He}{\bf{ \sigma}}~{}{P}^{p(n)}_{FSI}({\vec{p}},E,S_{He})]}}~{},\] (7) with ${P}^{p(n)}_{FSI}(\vec{p},E,S_{He})$ the distorted spin-dependent spectral function, defined in terms of the overlaps of Eq. (4) [11]. Then $p_{p(n)}$ with and without the FSI differ by 10-15% . Actually, one has also to consider the effect of the FSI on dilution factors. We have found, in a wide range of kinematics typical for the experiments at JLAB [5; 10], that the product of polarizations and dilution factors changes very little [13]. Indeed the effects of FSI in the dilution factors and in the effective polarizations are found to compensate each other to a large extent and the usual extraction appears to be safe : \[{{A_{n}}}\simeq{1\over{{p_{n}^{FSI}}}f_{n}^{FSI}}\left({A^{exp}_{ 3}}-2{p_{p}^{FSI}f_{p}^{FSI}{{A^{exp}_{p}}}}\right)~{}\simeq{1\over{{p_{n}}}f_ {n}}\left({A^{exp}_{3}}-2{p_{p}f_{p}~{}{{A^{exp}_{p}}}}\right)\] (8) In [9] the calculation was performed within a non relativistic approach for the spectral function, but with the correct relativistic kinematics in the Bjorken limit. For an accurate description of SIDIS processes, the role played by relativity has to be fully investigated: it will become even more important with the upgrade of JLab @ 12 GeV. To study relativistic effects in the actual experimental kinematics, we adopted [14] the Relativistic Hamiltonian Dynamics (RHD) introduced by Dirac [15]. Indeed the RHD of an interacting system with a fixed number of on-mass-shell constituents, with the Bakamijan-Thomas construction of the Poincaré generators [16], is fully Poincaré covariant. The Light-Front (LF) form of RHD has a subgroup structure of the LF boosts, allows a separation of the intrinsic motion from the global one, which is very important for the description of DIS, SIDIS and deeply virtual Compton scattering (DVCS) processes, and allows a meaningful Fock expansion. The key quantity to consider in SIDIS processes is the LF relativistic spectral function, ${\cal P}^{\tau}_{\sigma^{\prime}\sigma}(\tilde{\bf\kappa},\epsilon_{S},S_{He})$, with $\tilde{\bf\kappa}$ an intrinsic nucleon momentum and $\epsilon_{S}$ the energy of the two-nucleon spectator system. The LF spectral function will be very useful also for other studies (e.g., for nuclear generalized parton distributions (GPDs), where final states have to be properly boosted, studied so far only within a non-relativistic spectral function [17; 18]). The LF nuclear spectral function, ${\cal P}^{\tau}_{\sigma^{\prime}\sigma}(\tilde{\bf\kappa},\epsilon_{S},S_{He})$, is defined in terms of LF overlaps [19] between the ground state of a polarized ${}^{3}$He and the cartesian product of an interacting state of two nucleons with energy $\epsilon_{S}$ and a plane wave for the third nucleon. Within a reliable approximation [19] it can be given in terms of the unitary Melosh Rotations, ${{D^{{1\over 2}}[{\cal R}_{M}({\bf{\tilde{\kappa}}})]}}$, and the usual instant-form spectral function ${{{P}^{\tau}_{\sigma^{\prime}_{1}\sigma_{1}}}}$: \[{{{\cal P}^{\tau}_{\sigma^{\prime}\sigma}({\bf{\tilde{\kappa}}}, \epsilon_{S},S_{He})}}\propto~{}\sum_{\sigma_{1}\sigma^{\prime}_{1}}{{D^{{1 \over 2}}[{\cal R}_{M}^{\dagger}({\bf{\tilde{\kappa}}})]_{\sigma^{\prime} \sigma^{\prime}_{1}}}}~{}{{{P}^{\tau}_{\sigma^{\prime}_{1}\sigma_{1}}({\bf p}, \epsilon_{S},S_{He})}}~{}{{D^{{1\over 2}}[{\cal R}_{M}({\bf{\tilde{\kappa}}})] _{\sigma_{1}\sigma}}}\] (9) \| protonNR \| protonLF \| neutronNR \| neutronLF ---\|---\|---\|---\|--- ∫dEd→p Tr(Pσz)→SA=ˆz \| -0.02263 \| -0.02231 \| 0.87805 \| 0.87248 ∫dEd→p Tr(Pσy)→SA=ˆy \| -0.02263 \| -0.02268 \| 0.87805 \| 0.87494 Table 1: Proton and neutron effective polarizations within the non relativistic appproach (NR) and preliminary results within the light-front relativistic dynamics approach (LF). First line : longitudinal effective polarizations; second line : transverse effective polarizations. Notice that the instant-form spectral funtion ${P}^{\tau}_{\sigma^{\prime}_{1}\sigma_{1}}({\bf p},\epsilon_{S},S_{He})$ is given in terms of three independent functions, ${{B_{0},B_{1},B_{2}}}$[7], once parity and t-reversal are imposed: \[{{{P}^{\tau}_{\sigma^{\prime}_{1}\sigma_{1}}({\bf p},\epsilon_{S} ,S_{He})}}=\left[{{B_{0,S_{He}}^{\tau}\hskip-2.845276pt(\|{\bf p}\|,E)}}+{\mbox{ \boldmath{$\sigma$}}}\cdot{\bf f}^{\tau}_{S_{He}}\hskip-2.845276pt({\bf p},E) \right]_{\sigma^{\prime}_{1}\sigma_{1}}\] (10) with \[{\bf f}^{\tau}_{S_{He}}({\bf p},E)={\bf S}_{He}{{B_{1,S_{He}}^{ \tau}\hskip-2.845276pt(\|{\bf p}\|,E)}}+{\bf\hat{p}}~{}{\bf(\hat{p}}\cdot{\bf S} _{He}){{B_{2,S_{He}}^{\tau}\hskip-2.845276pt(\|{\bf p}\|,E)}}\] (11) Adding FSI, more terms should be included. We are now evaluating the SSAs using the LF hadronic tensor, at finite values of $Q^{2}$. The preliminary results are quite encouraging, since, as shown in Table 1, LF longitudinal and transverse polarizations change little from the ones obtained within the NR spectral function and weakly differ from each other. Then we find that the extraction procedure works well within the LF approach as it does in the non relativistic case. Concerning the FSI, we plan to include in our LF approach the FSI between the jet produced from the hadronizing quark and the two-nucleon system through the Glauber approach of Ref. [11]. ## 3 The $J=1/2$ LF spectral function and the nucleon LF TMDs The TMDs for a nucleon with total momentum $P$ and spin $S$ are introduced through the q-q correlator \[{{\Phi(k,P,S)}}_{\alpha\beta} =\] (12) \[+\] where $k$ is the parton momentum in the laboratory frame, so that the six twist-2 T-even functions, $A_{i},~{}\widetilde{A}_{i}~{}(i=1,3)$, can be obtained by proper traces of $\Phi(k,P,S)$. Indeed, particular combinations of the functions $A_{i},\widetilde{A}_{i}~{}(i=1,3)$ can be obtained by the following traces of $\Phi(k,P,S)$ : \[\frac{1}{2P^{+}}\,{{\rm Tr(\gamma^{+}\Phi)}} = {{A_{1}}}\,,\] \[\frac{1}{2P^{+}}\,{{\rm Tr(\gamma^{+}\gamma_{5}\Phi)}} = S_{L}\,{{A_{2}}~{}+~{}\frac{1}{M}\,\vec{k}_{\perp}{\cdot}\vec{S} _{\perp}\,{{\widetilde{A}_{1}}}}\,,\] \[\frac{1}{2P^{+}}\,{{\rm Tr(i\sigma^{j+}\gamma_{5}\Phi)}} = S_{\perp}^{j}\,{{A_{3}}}~{}+~{}\frac{S_{L}}{M}\,k_{\perp}^{j}\,{ {\widetilde{A}_{2}}}~{}+\frac{1}{M^{2}}~{}\vec{k}_{\perp}{\cdot}\vec{S}_{\perp }~{}k_{\perp}^{j}~{}{{\widetilde{A}_{3}}}\,~{}~{}(j=x,y).\] (13) The six T-even twist-2 TMDs for the quarks inside a nucleon can be obtained by integration of the functions $~{}A_{i},~{}\widetilde{A}_{i}$ on $k^{+}$ and $k^{-}$ as follows \[f(x,\|{\bf k}_{\perp}\|^{2})=\frac{1}{2}\int\frac{d{k^{+}}d{k^{-}} }{(2\pi)^{4}}~{}\delta[k^{+}-xP^{+}]~{}2P^{+}A_{1}\,,\] \[\Delta f(x,\|{\bf k}_{\perp}\|^{2})=\frac{1}{2}\int\frac{d{k^{+}}d{ k^{-}}}{(2\pi)^{4}}~{}\delta[k^{+}-xP^{+}]~{}2P^{+}A_{2}\,,\] \[g_{1T}(x,\|{\bf k}_{\perp}\|^{2})=\frac{1}{2}\int\frac{d{k^{+}}d{k ^{-}}}{(2\pi)^{4}}~{}\delta[k^{+}-xP^{+}]~{}2P^{+}\widetilde{A}_{1}\,,\] \[\Delta^{\prime}_{T}f(x,\|{\bf k}_{\perp}\|^{2})=\frac{1}{2}\int \frac{d{k^{+}}d{k^{-}}}{(2\pi)^{4}}~{}\delta[k^{+}-xP^{+}]~{}2P^{+}\left({A}_{ 3}+{\|{\bf k}_{\perp}\|^{2}\over 2M^{2}}\widetilde{A}_{3}\right)\,,\] \[h^{\perp}_{1L}(x,\|{\bf k}_{\perp}\|^{2})=\frac{1}{2}\int\frac{d{k ^{+}}d{k^{-}}}{(2\pi)^{4}}~{}\delta[k^{+}-xP^{+}]~{}2P^{+}\widetilde{A}_{2}\,,\] \[h^{\perp}_{1T}(x,\|{\bf k}_{\perp}\|^{2})=\frac{1}{2}\int\frac{d{k ^{+}}d{k^{-}}}{(2\pi)^{4}}~{}\delta[k^{+}-xP^{+}]~{}2P^{+}\widetilde{A}_{3}~{}.\] (14) Let us consider the contribution to the correlation function from on-mass-shell fermions \[{{~{}\Phi_{p}(k,P,S)=~{}{(~{}{k\hskip-5.690551pt/}_{on}~{}+~{}m) \over 2m}~{}{{\Phi(k,P,S)}}~{}{(~{}{k\hskip-5.690551pt/}_{on}~{}+~{}m)\over 2m }}}=\] (15) \[=\sum_{\sigma\sigma^{\prime}}~{}u_{LF}({\tilde{k}},\sigma^{\prime })~{}\bar{u}_{LF}({\tilde{k}},\sigma^{\prime})~{}{{\Phi(k,P,S)}}~{}u_{LF}({ \tilde{k}},\sigma)\bar{u}_{LF}({\tilde{k}},\sigma)~{},\] and let us identify $\bar{u}_{LF}({\tilde{k}},\sigma^{\prime})~{}\Phi(k,P,S)~{}u_{LF}({\tilde{k}},\sigma)$, up to a kinematical factor $K$, with the LF nucleon spectral function, ${\cal P}_{\sigma^{\prime}\sigma}(\tilde{\bf\kappa},\epsilon_{S},S)$[19]: \[{{\bar{u}_{LF}({\tilde{k}},\sigma^{\prime})~{}\Phi(k,P,S)~{}u_{LF }({\tilde{k}},\sigma)}}={{K}}~{}{{{\cal P}_{\sigma^{\prime}\sigma}(\tilde{\bf \kappa},\epsilon_{S},S)}}~{}.\] (16) In a reference frame where ${\bf P}_{\perp}=0$, the following relation holds between the off-mass-shell minus component $k^{-}$ of the momentum of the struck quark and the spectator diquark energy $\epsilon_{S}$ : \[{{k^{-}}}={{M^{2}}\over{P^{+}}}~{}-~{}{{({{\epsilon_{S}}}+m)~{}4m +\|{\mbox{\boldmath{$k$}}}_{\perp}\|^{2}}\over{P^{+}-k^{+}}}~{}.\] (17) Let us approximate the full correlation function ${{\Phi(k,P,S)}}$ by its particle contribution ${{~{}\Phi_{p}(k,P,S)}}$. Then, through relations analogous to the ones of Eq. (13), which allow one to obtain the functions $A_{i},~{}\widetilde{A}_{i}$ from the traces of ${{\Phi(k,P,S)}}$, the valence approximations ${{~{}A_{i}^{V},~{}\widetilde{A}_{i}^{V}~{}(i=1,3)}}$ for the functions ${{~{}A_{i},~{}\widetilde{A}_{i}}}$ can be obtained by the traces $~{}[\gamma^{+}~{}\Phi_{p}(k,P,S)]$, $[\gamma^{+}~{}\gamma_{5}~{}~{}\Phi_{p}(k,P,S)]$, and $[{k}\hskip-5.690551pt/_{\perp}\gamma^{+}\gamma_{5}~{}\Phi_{p}(k,P,S)]$. However these same traces can be also expressed through the LF spectral function, since from Eqs. (15,16) one has \[{{~{}Tr\left[\gamma^{+}~{}\Phi_{p}(k,P,S)\right]}}~{}=~{}{k^{+} \over m}~{}K~{}{{Tr\left[{\cal P}(\tilde{\bf\kappa},\epsilon_{S},S)\right]}}\] (18) \[{{~{}Tr\left[\gamma^{+}~{}\gamma_{5}~{}~{}\Phi_{p}(k,P,S)\right]} }={k^{+}\over{m}}~{}K~{}{{Tr\left[\sigma_{z}~{}{\cal P}(\tilde{\bf\kappa}, \epsilon_{S},S)\right]}}\] (19) \[{{Tr\left[{k}\hskip-5.690551pt/_{\perp}\gamma^{+}\gamma_{5}~{} \Phi_{p}(k,P,S)\right]}}={k^{+}\over m}K{{Tr\left[{\mbox{\boldmath{$k$}}}_{ \perp}\cdot{\mbox{\boldmath{$\sigma$}}}~{}{\cal P}(\tilde{\bf\kappa},\epsilon_ {S},S)\right]}}\] (20) In turn, as in the ${}^{3}$He case, the traces ${\rm Tr}({\cal P}I)$, ${\rm Tr}({\cal P}\sigma_{z})$, ${\rm Tr}({\cal P}\sigma_{i})$ ($i=x,y$) can be expressed in terms of three scalar functions, $B_{0}$, $B_{1}$, $B_{2}$ and known kinematical factors. Then, within the LF approach with a fixed number of particles and in the valence approximations, the six leading twist T-even functions ${{~{}A_{i}^{V},~{}\widetilde{A}_{i}^{V}~{}(i=1,3)}}$ can be expressed in terms of the three independent scalar functions ${{B_{0},B_{1},B_{2}}}$. Therefore only three of the six T-even TMDs are independent [19]. We stress however that this result, not valid in general in QCD (see, e.g., Ref. [20]), is a prediction of our peculiar framework, i.e., hamiltonian light-front dynamics with a fixed number of constituents, finalized to yield a proper Poincaré covariant description of the nucleon in the valence approximation. Its experimental check would be therefore a test of the reliability of our scheme to describe SIDIS processes in the valence region. ## 4 Conclusions A realistic study of the DIS processes off ${}^{3}\vec{He}$ and in particular of the SSAs in the reaction ${}^{3}\vec{He}(e,e^{\prime}\pi^{\pm})X$ beyond the PWIA and the non relativistic approach is under way. The FSI effects have been evaluated through the GEA and the introduction of a distorted spin-dependent spectral function. The relativistic effects are studied through the analysis of a LF spectral function (up to now only in PWIA). Preliminary results are encouraging, in view of a sound extraction of the neutron information from experimental data. Nuclear effects in the extraction of the neutron information are found to be under control, even when the interaction in the final state is considered, and the relativistic effects appear to be small. An analysis at finite $Q^{2}$ with the LF spectral function is in progress. The next step is to complete this program. Then we will apply the LF spectral function to other processes (e.g., DVCS) to exploit other possibilities to use ${}^{3}$He as an effective neutron target and as a laboratory for light-front studies. The introduction of a LF spin-dependent spectral function for a nucleon allowed us to find relations among the six leading twist T-even TMDs, which are exact within LF dynamics with a fixed number of degrees of freedom, in the valence approximation. It can be shown that, in this case, only three of the six T-even TMDs are independent. The above relations, although not true in QCD, could be experimentally checked to test our LF description of SIDIS in the valence region. ## References * [1] Barone, V., Drago, A., and Ratcliffe, P. G., “Transverse polarisation of quarks in hadrons,” Phys. Rept. 359, 1 (2002). * [2] Airapetian, A. _et al._ [HERMES Collaboration], “Single-spin asymmetries in semi-inclusive deep-inelastic scattering on a transversely polarized hydrogen target,” Phys. Rev. Lett. 94, 012002 (2005) . * [3] Alexakhin, V. Y _et al._ [COMPASS Collaboration], “First measurement of the transverse spin asymmetries of the deuteron in semi-inclusive deep inelastic scattering,” Phys. Rev. Lett. 94, 202002 (2005). * [4] Signori, A., Bacchetta, A., Radici, M. and Schnell, G., “Investigations into the flavor dependence of partonic transverse momentum,” JHEP 1311, 194 (2013) . * [5] G. Cates et al., E12-09-018, JLAB approved experiment, hallaweb.jlab.org/collab/PAC/PAC38//E12-09-018-SIDIS.pdf * [6] Ciofi degli Atti, C. _et al._ “Nuclear effects in deep inelastic scattering of polarized electrons off polarized He-3 and the neutron spin structure functions,” Phys. Rev. C 48, 968 (1993). * [7] Ciofi degli Atti, C., Salmè, G., and Pace, E. “Spin dependent spectral function of He-3 and the asymmetry in the process He-3 (polarized) (e (polarized),e-prime) X,” Phys. Rev. C 46, 1591 (1992) ; Kievsky, A. et al., “Neutron electromagnetic form-factors and inclusive scattering of polarized electrons by polarized He-3 and He-3 targets,” Phys. Rev. C 56, 64 (1997) . * [8] Wiringa, R. B., Stoks, V. G. and Schiavilla, R. “An Accurate nucleon-nucleon potential with charge independence breaking,” Phys. Rev. C 51, 38 (1995). * [9] Scopetta, S., “Neutron single spin asymmetries from semi-inclusive deep inelastic scattering off transversely polarized He-3,” Phys. Rev. D 75, 054005 (2007) . * [10] Allada, K. _et al._ [JLab Hall A Collaboration], “Single Spin Asymmetries of Inclusive Hadrons Produced in Electron Scattering from a Transversely Polarized ${}^{3}$He Target,” Phys. Rev. C 89, 042201 (2014) . * [11] Kaptari, L. P., Del Dotto, A., Pace, E. Salmè, G. and Scopetta, S. “Distorted spin-dependent spectral function of an A=3 nucleus and semi-inclusive deep inelastic scattering processes,” Phys. Rev. C 89, 035206 (2014) . * [12] Ciofi degli Atti, C. and Kaptari, L.P., “Semi-inclusive Deep Inelastic Scattering off Few-Nucleon Systems: Tagging the EMC Effect and Hadronization Mechanisms with Detection of Slow Recoiling Nuclei,” Phys. Rev. C 83, 044602 (2011) . * [13] Del Dotto, A. _et al._, “Flavor decomposition of transverse momentum dependent parton distributions,” EPJ Web Conf. 73, 02019 (2014). * [14] Pace, E. _et al._, “Neutron Transverse-Momentum Distributions and Polarized ${}^{3}He$ within Light-Front Hamiltonian Dynamics,” Few Body Syst. 54, 1079 (2013) . * [15] Dirac, P.A.M., “Forms of Relativistic Dynamics”, Rev. Mod. Phys. 21, 392 (1949) . * [16] Bakamjian, B. and Thomas, L. H., “Relativistic particle dynamics. 2,” Phys. Rev. 92, 1300 (1953) . * [17] Scopetta, S. “Generalized parton distributions of He-3,” Phys. Rev. C 70, 015205 (2004) ; “Conventional nuclear effects on generalized parton distributions of trinucleons,” Phys. Rev. C 79, 025207 (2009) . * [18] Rinaldi, M., and Scopetta, S. “Neutron orbital structure from generalized parton distributions of 3He,” Phys. Rev. C 85 (2012) 062201 ; “Extracting generalized neutron parton distributions from ${}^{3}He$ data,” Phys. Rev. C 87, no. 3, 035208 (2013) . * [19] Del Dotto, A., Pace, E., Salmè, G., and Scopetta, S., to be published. * [20] C. Lorcè and B. Pasquini, “On the Origin of Model Relations among Transverse-Momentum Dependent Parton Distributions,” Phys. Rev. D 84 (2011) 034039 .
1604.03562	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 46605, "num_imgs": 7, "llama3_tokens_count": 14521 }	[ "content_image/1604.03562/x1.png", "content_image/1604.03562/x2.png", "content_image/1604.03562/x3.png", "content_image/1604.03562/x4.png", "content_image/1604.03562/x5.png", "content_image/1604.03562/x6.png", "content_image/1604.03562/x7.png" ]	# Identifying IGR J14091–6108 as a magnetic CV with a massive white dwarf using X-ray and optical observations John A. Tomsick${}^{1}$, Farid Rahoui${}^{2,3}$, Roman Krivonos${}^{4}$, Maïca Clavel${}^{1}$, Jay Strader${}^{5}$, and Laura Chomiuk${}^{5}$ ${}^{1}$Space Sciences Laboratory, 7 Gauss Way, University of California, Berkeley, CA 94720-7450, USA ${}^{2}$European Southern Observatory, Karl Schwarzschild-Strasse 2, 85748 Garching bei Munchen, Germany ${}^{3}$Department of Astronomy, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA ${}^{4}$Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia ${}^{5}$Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA E-mail: jtomsick@ssl.berkeley.edu (JAT) ###### Abstract IGR J14091–6108 is a Galactic X-ray source known to have an iron emission line, a hard X-ray spectrum, and an optical counterpart. Here, we report on X-ray observations of the source with _XMM-Newton_ and _NuSTAR_ as well as optical spectroscopy with ESO/VLT and NOAO/SOAR. In the X-rays, this provides data with much better statistical quality than the previous observations, and this is the first report of the optical spectrum. Timing analysis of the _XMM_ data shows a very significant detection of $576.3\pm 0.6$ s period. The signal has a pulsed fraction of $30\pm 3$% in the 0.3–12 keV range and shows a strong drop with energy. The optical spectra show strong emission lines with significant variability in the lines and continuum, indicating that they come from an irradiated accretion disk. Based on these measurements, we identify the source as a magnetic Cataclysmic Variable of Intermediate Polar (IP) type where the white dwarf spin period is 576.3 s. The X-ray spectrum is consistent with the continuum emission mechanism being due to thermal Bremsstrahlung, but partial covering absorption and reflection are also required. In addition, we use the IP mass (IPM) model, which suggests that the white dwarf in this system has a high mass, possibly approaching the Chandrasekhar limit. keywords: stars: individual(IGR J14091–6108), white dwarfs, X-rays: stars, accretion, stars: novae, catalysmic variables ## 1 Introduction The hard X-ray imaging by the _International Gamma-Ray Astrophysics Laboratory (INTEGRAL)_ satellite (Winkler et al., 2003) has led to the discovery of a large number of new or previously poorly studied “IGR” sources. The most recent published catalogs of 17–100 keV sources detected by _INTEGRAL_ include more than 900 sources for the whole sky (Bird et al., 2016) and $\sim$400 sources within $17.5^{\circ}$ of the Galactic plane (Krivonos et al., 2012). A large fraction of the sources have been identified as Active Galactic Nuclei (AGN), Cataclysmic Variables (CVs), High-Mass X-ray Binaries (HMXBs), Low-Mass X-ray Binaries (LMXBs), and Pulsar Wind Nebulae (PWNe), but 23% of the 939 sources listed in Bird et al. (2016) are still unidentified, in part due to the difficulty in finding counterparts at other wavelengths with the $\sim$1${}^{\prime}$–4${}^{\prime}$_INTEGRAL_ position uncertainties. In an effort to characterize the populations of hard X-ray sources in the Galaxy, we have been performing follow-up observations of sources in the Galactic Plane to identify the natures of as many IGR sources as possible. During 2013–2015, we observed ten IGR sources with the _Chandra X-ray Observatory_, and the results were reported in Tomsick et al. (2015) and Tomsick et al. (2016). In some cases, the information from the _Chandra_ observation, including accurate source positions that provided optical or IR identifications, immediately led to a determination of the nature of the source. For example, IGR J04059+5416 and IGR J08297–4250 were identified as AGN (Tomsick et al., 2015), and IGR J18088–2741 was identified as a CV (Tomsick et al., 2016). IGR J14091–6108 was one of the sources observed in this program, but the _Chandra_ observation did not allow us to definitively determine its nature, and we selected it for further study as described in this work. IGR J14091–6108 was discovered when it was detected in the _INTEGRAL_ 9-year Galactic Hard X-ray Survey (Krivonos et al., 2012). A _Swift_ X-ray counterpart was found by Landi et al. (2012) and then the source was observed with _Chandra_, leading to the detection of a strong iron K$\alpha$ emission line in the X-ray spectrum and the identification of an optical/IR counterpart (Tomsick et al., 2014, 2016). _Chandra_ showed that the source has a hard power-law continuum with a photon index of $\Gamma=0.6\pm 0.4$, suggesting an accreting compact object with a high magnetic field strength (Tomsick et al., 2016). While we favored a magnetic CV, we did not rule out the possibility that the source is an HMXB. In this work, we use new X-ray and optical observations to study IGR J14091–6108 in more detail. ## 2 Observations and Data Reduction Table 1 lists the observations that we used, including simultaneous X-ray observations with the _X-ray Multi-Mirror Mission (XMM-Newton)_ and the _Nuclear Spectroscopic Telescope Array (NuSTAR)_ as well as optical spectroscopy with the Very Large Telescope (VLT¹) and with the Southern Astrophysical Research Telescope (SOAR). The X-ray observations occurred on 2015 July 20-21, and the observation identifiers (ObsIDs), exact start and stop times, and exposure times are provided in Table 1. The X-ray data provide a large improvement in the statistical quality over the previous _Chandra_ and _INTEGRAL_ observations to give better constraints on the spectrum and to allow for a sensitive search for periodic signals. The optical data provide a first look at the spectrum of the source. In the following, we describe how the data from each facility were reduced. [FOOTNOTE:1][ENDFOOTNOTE] ### _Xmm_ For the EPIC/pn (Strüder et al., 2001) and EPIC/MOS (Turner et al., 2001) instruments, we reduced the data using the _XMM_ Science Analysis Software v14.0.0 to make images, light curves, and spectra. For pn, we made a full-field 10--12 keV light curve to look for times of high background, and filtered these times out for the final data products. We followed the same procedure for MOS and obtained slightly more time on-source (28.9 ks for MOS compared to 24.7 ks for pn). In addition to the time filtering, we used the standard event filtering described in procedures that are available on-line². We extracted source counts from a circular region with a radius of $40^{\prime\prime}$ centered on IGR J14091-6108 and estimated the background using a rectangular region on parts of the detectors with no sources. After background subtraction, the 0.3–12 keV count rates for pn, MOS1, and MOS2 are $0.207\pm 0.003$, $0.059\pm 0.015$, and $0.065\pm 0.016$ c/s, respectively. We included 1% systematic errors on the pn and MOS source spectra when fitting the energy spectra to account for calibration uncertainties. We examined the data from the Reflection Grating Spectrometer (RGS), but the count rate for this instrument is too low to be useful. [FOOTNOTE:2][ENDFOOTNOTE] ### _NuSTAR_ For _NuSTAR_(Harrison et al., 2013), we reduced the data using HEASOFT v6.17, which includes NUSTARDAS v1.5.1. The calibration files are from the 2015 March 16 version of the calibration database (CALDB). We ran nupipeline to produce event lists for the two _NuSTAR_ instruments: focal plane modules A and B (FPMA and FPMB). We examined the full band images, and extracted light curves and spectra with nuproducts using a circular source region with a radius of $60^{\prime\prime}$. Although the _NuSTAR_ images do not show any significant stray light, spatial variations in the background are visible, which is expected. Thus, we estimated the background using nuskybgd(Wik et al., 2014), which is a package of IDL programs that sample the background over the field of view outside of the source region to model the spatial variations. We produced the model and then used it to estimate the background in the source region. We included 4% systematic errors on the background spectrum. After background subtraction, the 3–79 keV source count rates are $0.076\pm 0.002$ and $0.075\pm 0.002$ c/s for FPMA and FPMB, respectively. Finally, when producing the final spectra for fitting, we grouped the energy bins to give a significance of greater than 5-$\sigma$. ### Optical Spectroscopy We obtained low-resolution spectroscopy of the IGR J14091–6108 counterpart identified in Tomsick et al. (2016), which is CXOU J140846.0–610754 and VVV J140845.99–610754.1. We used the FORS2 instrument with the 300V and 300I grisms combined with the GC435 and OG590 filters, respectively. In both cases, the slit-width was set to 1″, giving a $R\sim 600$ average resolution. Atmospheric conditions were medium-to-good, with a thin sky, seeing at 500 nm in the range 0 $\aas@@fstack{\prime\prime}$6-0 $\aas@@fstack{\prime\prime}$8, and an airmass between 1.1 and 1.3. The integration time of each individual frame was set to 600 s and 500 s for the 300V, and 300I grisms, respectively, and two exposures were taken in each grism. The A0V spectro-photometric standard star CD-32 9927 was observed in similar conditions for flux-calibration. We reduced the data using the dedicated pipeline (v. 5.1.4) implemented in the ESO data reduction environment Reflex v. 2.6, following the standard steps for optical spectroscopy reduction to produce a cleaned, flatfielded, wavelength- and flux-calibrated 1D spectrum. We also obtained several optical spectra on UT 2015 January 15 using the Goodman High-Throughput Spectrograph (Clemens et al., 2004) on the SOAR 4.1-m telescope. All data were taken with a $1.03\arcsec$ slit. We obtained one 600 sec exposure using the 400 l mm${}^{-1}$ grating (wavelength coverage $\sim$3000–7000 Å; resolution 5.7 Å) and two 600 sec exposures using the 1200 l mm${}^{-1}$ grating (wavelength coverage 5480–6740 Å; resolution 1.7 Å). These data were wavelength calibrated with an FeAr arc lamp and were bias corrected using the overscan region. The spectra were optimally extracted using routines in IRAF. ## 3 Results ### X-ray Timing We produced 0.3–12 keV event lists for the three _XMM_ instruments, including only the photons in a $40^{\prime\prime}$ circular region centered on the position of IGR J14091–6108. The times of the individual events were shifted to the solar system barycenter using the SAS tool barycen. We used the $Z_{1}^{2}$ (Rayleigh) test (Buccheri et al., 1983) to make periodograms from the event list. We searched for a periodic signal in the pn data by making a periodogram with 20000 time bins between 1 and 2000 s. One significant peak is present with a peak value of $S=134$ (see Figure 1a). The false alarm probability (FAP) is given by 0.5 $e^{-S/2}$ multiplied by the number of trials, corresponding to a FAP of $8.5\times 10^{-26}$. The period is $576.3\pm 0.6$ s, where the 1-$\sigma$ errors are given by the periods where the periodogram has fallen to $S$–1. The MOS1 and MOS2 instruments showed significant peaks at $575.9\pm 0.8$ s and $577.1\pm 1.3$ s, respectively, which are both consistent with the period measured by pn (Figure 1a). We produced a folded light curve, which is shown in Figure 1b. An epoch of zero phase (minimum in the folded light curve) is MJD TDB $57224.0971\pm 0.0003$. To align the _XMM_ and _NuSTAR_ phases, we barycentered the _NuSTAR_ event lists and folded the _NuSTAR_ photons on the _XMM_ ephemeris. We calculated the pulsed fraction of the signal by defining maximum and minimum phase ranges using the 0.3–12 keV folded light curve (Figure 1b). The pulsed fraction is defined as the absolute value of the difference between the maximum and minimum count rate divided by the sum of these quantities. In the 0.3–12 keV range, it is $30\pm 3$%. The folded light curves from 0.3 to 79 keV are shown in Figure 2, and the values for the pulsed fraction are indicated on Figure 2 and plotted in Figure 3. The _XMM_ measurements show a strong decrease in the pulsed fraction with energy with the largest drop being at 7 keV. The _NuSTAR_ measurements also show a drop with energy, with a pulsed fraction of $13\pm 4$% in the 3–12 keV band and $0\pm 6$% (no detection) in the 12–79 keV band. A period of 576 s is fairly typical for the spin period of a white dwarf in a CV of Intermediate Polar (IP) type. In the 2014 update of the Ritter & Kolb (2003) CV catalog³, there are 78 IP-type CVs (CV/IPs) with white dwarf spin periods between 128 and 12071 s, and 20 systems have periods faster than 576 s. The shortest CV/IP orbital period is 1552 s, and the median orbital period is 4.1 hr. Thus, if IGR J14091–6108 is a CV/IP, the detected period is much more likely to be the white dwarf spin period, and the orbital period is probably between several and $\sim$100 times longer (Hong et al., 2012) [FOOTNOTE:3][ENDFOOTNOTE] ### Optical Spectra Figure 4 displays the flux-calibrated SOAR/Goodman (magenta) and VLT/FORS2 (blue) spectra of IGR J14091$-$6108, on which all the detected spectroscopic features are marked. Likewise, Table 2 lists their main parameters, i.e. their central wavelengths ($\lambda_{\rm c}$), equivalent widths ($\mathring{W}$), full-widths at half-maximum (FWHM), and intrinsic fluxes ($F_{\rm Line}$), obtained through single-Gaussian fitting. The FWHMs were quadratically corrected for the instrumental broadening and the underlying continua were locally assessed with a first-order polynomial. The continuum level being the primary source of inaccuracy, each measurement was repeated several times with different placements within the same wavelength range to obtain a set of values that eventually averaged out. The listed uncertainties are therefore the scatter to the mean rather than just statistical. The optical spectrum is very rich, with a wealth of spectral features that includes the Balmer and Paschen series as well as several signatures of He1 and He2, all in emission. The Bowen Blend at 4640 Å, typical of irradiated accretion disks and/or companion stars in accreting systems, is also clearly observed with FORS2 and perhaps with Goodman. The spectrum also appears to be strongly variable, the continuum and emission lines being roughly fives and three times brighter in the Goodman spectrum, respectively. Besides features intrinsic to IGR J14091$-$6108, we also report two diffuse interstellar bands (DIBs) centered at 5780 Å and 6284 Å. DIBs are strongly correlated to the ISM extinction along the line-of-sight of the sources in which they are detected and we can assess the latter using the relationships between their equivalent widths and $E(B-V)$ given in Jenniskens & Desert (1994). Nonetheless, DIB6284 is likely contaminated with some atmospheric absorption troughs and we therefore rely on DIB5780 only, for which Jenniskens & Desert (1994) obtain $\frac{\mathring{W}}{E(B-V)}\,=\,0.647\pm 0.053$. We measure $\mathring{W}_{\rm 5780}\,=\,0.97\pm 0.11$ and $\mathring{W}_{\rm 5780}\,=\,1.11\pm 0.07$ in the Goodman and FORS2 spectra, respectively, which, once averaged out, leads to $E(B-V)\,=\,1.615\pm 0.166$. Using the relationship $A_{V}\,=\,R_{V}\times E(B-V)$ and an average total-to-selective extinction ratio $R_{V}=3.1$, we thus derive $A_{V}\,=\,5.01\pm 0.51$, which is roughly consistent with a distance between 3 and 4 kpc based on the 3D Galactic extinction map derived in Marshall et al. (2006) for a line-of-sight 9 $\aas@@fstack{\prime\prime}$4 away from that of IGR J14091$-$6108. With the knowledge of $A_{V}$, it is now possible to correct IGR J14091$-$6108 emission from the ISM extinction and Figure 5 displays the extinction-corrected Goodman and FORS2 spectra expressed in Hz vs mJy. The first result is that both continua are roughly best-fit with power laws with 2.29 and 2.09 spectral indices, respectively, typical of the Raleigh-Jeans tail of a black body emitter. We can also estimate the Balmer decrements and we find H$\alpha$/H$\beta$ = $1.03\pm 0.20$ from the FORS2 spectrum as well as H$\alpha$/H$\beta$ = $1.20\pm 0.23$, H$\gamma$/H$\beta$ = $1.07\pm 0.26$, H$\delta$/H$\beta$ = $1.03\pm 0.27$, H$\epsilon$/H$\beta$ = $0.86\pm 0.27$, and H$\eta$/H$\beta$ = $1.38\pm 0.32$ from the Goodman spectrum. It is clear that all the values are consistent with unity, which is typical of Cataclysmic Variables (CVs) and some microquasars (see, e.g., Williams, 1980; Williams & Shipman, 1988; Rahoui et al., 2014). Based on the wealth of emission lines, spectral variability, Raleigh-Jeans continuum, and flat Balmer decrements around unity, it is thus very likely that the optical emission of IGR J14091$-$6108 originates in an optically-thick irradiated accretion disk in a CV. ### X-ray Spectrum For spectral analysis, we jointly fitted the _XMM_ (pn, MOS1, and MOS2) and _NuSTAR_ (FPMA and FPMB) data in the 0.3–79 keV energy range. We used the XSPEC package (Arnaud, 1996) and performed the fitting with $\chi^{2}$ minimization. We fitted with an absorbed power-law model, using Wilms et al. (2000) abundances and Verner et al. (1996) cross sections for the absorption. We also allowed for normalization differences between instruments by introducing a multiplicative constant. Thus, the overall model in XSPEC notation was constanttbabspegpwrlw. This showed that the spectrum is hard with $\Gamma\sim 0.8$, but the fit was poor with $\chi^{2}=1242$ for 346 degrees of freedom (dof). The largest residuals appear in the iron line region, indicating a strong emission line. Adding a Gaussian with $E_{\rm line}=6.56\pm 0.04$ keV and $\sigma_{\rm line}=0.35\pm 0.05$ keV provides a large improvement in the fit (to $\chi^{2}/\nu=711/343$), but negative residuals in the high energy part of the spectrum indicate curvature in the spectrum and possibly a cutoff. We changed the continuum model from a power-law to a thermal Bremsstrahlung component to allow for curvature in the model and also because this is the emission mechanism that is thought to operate in CV/IPs. While the model constanttbabs(gaussian+bremss) does not provide a good fit to the data ($\chi^{2}/\nu=997/343$), partial covering absorption is typically used when fitting CV/IPs (Suleimanov et al., 2005; Mukai et al., 2015), and, in our case, using pcfabs with $N_{\rm H}=(1.4\pm 0.3)\times 10^{23}$ cm${}^{-2}$ and a covering fraction of $0.68^{+0.02}_{-0.03}$ improves the fit greatly to $\chi^{2}/\nu=516/341$. With this model, the Bremsstrahlung temperature is very high, $kT>167$ keV (90% confidence limit), but there are still residuals at the high energy end that we suspect are related to the presence of a reflection component. Reflection of the hard X-ray emission off the white dwarf surface is often included in models when fitting CV/IP spectra, and recent work with _XMM_ and _NuSTAR_ spectra show strong evidence for this component in three other CV/IPs (Mukai et al., 2015). For IGR J14091–6108, we added a reflection component using the reflect model in XSPEC. This model is based on Magdziarz & Zdziarski (1995), which is for reflection of direct emission from neutral material and includes the dependence on viewing angle. By convolving bremss with reflect, the model includes both a direct and a reflected Bremsstrahlung component where the strength of the reflected component depends on the amplitude parameter, $\Omega/2\pi$. In our case, if the reflection amplitude is left as a free parameter, it will increase to values above 1.0. However, assuming that we see 100% of the direct emission, values of $\Omega/2\pi$ above unity are not physically possible for reflection from the white dwarf surface. Thus, we fix the reflection amplitude to 1.0, resulting in a fit with $\chi^{2}/\nu=472/339$. The spectrum fit with this model is shown in Figure 6, and the parameters are given in Table 3. Although the Bremsstrahlung temperature drops when the reflection component is added, we still measure $kT=81^{+31}_{-20}$ keV, which, as we discuss in Section 4, is high for a CV/IP. For this fit, the 0.3–79 keV unabsorbed (i.e., with the interstellar but not the partial covering column density set to zero) flux is $1.1\times 10^{-11}$ erg cm${}^{-2}$ s${}^{-1}$. While the actual reflection component has three main features: the Compton hump above 10 keV, the iron edge at 7.1 keV (for neutral iron), and an iron fluorescence line at 6.4 keV (also for neutral iron), the Magdziarz & Zdziarski (1995) model only includes the first two features, which is one reason that we include the Gaussian in the model above. If the emission line was due only to reflection from the white dwarf surface, we would expect a relatively narrow line centered at 6.4 keV; however, as shown in Table 3, we measure a broad line with $\sigma_{\rm line}=0.28\pm 0.04$ keV at an energy above 6.4 keV ($E_{\rm line}=6.59\pm 0.04$ keV). It is very likely that this is due to contributions from higher ionization states coming from the hot material in the accretion column, and this is common for CV/IPs (Hellier & Mukai, 2004). We modified the model to include two lines with energies fixed to 6.4 keV (neutral iron) and 6.7 keV (He-like iron). This more physical representation of the emission lines provides an equivalently good fit ($\chi^{2}/\nu=473/339$), but the lines are still relatively broad ($\sigma_{\rm line}=0.24\pm 0.05$ keV). Adding a third line at 6.97 keV (H-like iron) improves the fit ($\chi^{2}/\nu=458/338$), and these lines are significantly narrower ($\sigma_{\rm line}=0.07\pm 0.04$ keV), which is consistent with line widths measured for other CV/IPs (Hellier & Mukai, 2004). The line normalizations in Table 3 include 90% confidence uncertainties, suggesting that all the lines are significantly detected. Thus, the three-line model is very likely the correct interpretation based on the match between measured and expected line widths as well as on statistical grounds. In this model, the equivalent widths of the three lines are $320\pm 60$, $160\pm 40$, and $170\pm 50$ eV for neutral, He-like, and H-like iron, respectively. To explore the possible physical cause of the high temperature derived from the fitting the continuum with a Bremsstrahlung model, we replaced the Bremsstrahlung component with the IP mass (IPM) model of Suleimanov et al. (2005). For the IPM model, the hardness and cutoff energy of the predicted spectrum are both set by the white dwarf mass, and the only two free parameters are $M_{\rm WD}$ and the normalization. Physically, this connection between the $M_{\rm WD}$ and the spectrum is related to the maximum shock temperature in the accretion column (Suleimanov et al. 2005; Hailey et al., submitted to ApJ). The emission mechanism is still Bremsstrahlung, but the model includes a range of temperatures. The results for a model including direct and reflected IPM components, partial covering, and three Gaussians are reported in Table 4. As before, we fix the reflection amplitude to 1.0, and the other reflection parameters are consistent to the values found in the Bremsstrahlung fits. Also, there is little or no change in the continuum absorption and the emission line parameters. The IPM model parameters imply a high white dwarf mass, $M_{\rm WD}>1.38$ $M_{\mathord{\odot}}$. The components of the spectrum are shown in Figure 7. We note that none of the models we have used provide formally acceptable fits, with the best one having a reduced-$\chi^{2}$ of 1.36 for 338 dof. Mukai et al. (2015) obtained similar fit qualities for his _XMM_+_NuSTAR_ spectra of CV/IPs and suggested that it is related to calibration differences between instruments, and this is also a possibility in our case. However, we explored some possibilities for improving the fits. For example, the fit improves if we start with the model given in Table 4 and allow $\Omega/2\pi$ to increase above 1.0. Although likely unphysical, the fit improves from a reduced-$\chi^{2}$ of 1.42 (for 338 dof) to 1.33 (for 337 dof) at $\Omega/2\pi=5.0$. Also, allowing the reflection component to dominate at high energies causes the IPM component to soften, and, for the extreme assumption that $\Omega/2\pi=5.0$, $M_{\rm WD}$ drops to 1.2 $M_{\mathord{\odot}}$. The fit can also be improved by modifying the modeling of the absorption. In some CV/IPs, there is evidence for complex absorption patterns, and other authors have modeled this with two partial coverers (e.g., Beardmore et al., 2000). Adding a second partial coverer to the 3 Gaussian Bremsstrahlung model in Table 3 improves the reduced-$\chi^{2}$ from 1.36 (for 338 dof) to 1.23 (for 336 dof). Although this is a significant improvement, this model leads to an interstellar column density of $N_{\rm H}=(0.16\pm 0.10)\times 10^{22}$ cm${}^{-2}$, which is much lower than the value implied by our measured optical extinction. Using the relation from Güver & Özel (2009), an $A_{V}$ of $5.0\pm 0.5$ magnitudes corresponds to $N_{\rm H}=(1.1\pm 0.1)\times 10^{22}$ cm${}^{-2}$. Thus, while both high reflection amplitude and additional partial covering absorption lead to improved fits, the former is not consistent with the physical scenario and the latter leads to unrealistically low values for the interstellar column density. To avoid the complexities of the low energy part of the spectrum, we also explored fitting just the _NuSTAR_ spectrum above 10 keV. Fitting with a Bremsstrahlung model gives a reduced-$\chi^{2}$ of 1.62 for 37 dof, and the 90% confidence lower limit on the temperature is 112 keV. Adding a reflection component with $\Omega/2\pi=1$, which makes the model constantreflectbremss, provides a good fit (reduced-$\chi^{2}$ = 1.14 for 35 dof), and the Bremsstrahlung temperature is $66^{+33}_{-22}$ keV. With constantreflectipm and $\Omega/2\pi=1$, we find $M_{\rm WD}>1.31$ $M_{\mathord{\odot}}$, which is consistent with the high mass values we obtain when fitting the entire spectrum. ## 4 Discussion The results indicate that we can identify IGR J14091–6108 as a CV/IP with a high degree of confidence. The $576.3\pm 0.6$ s periodicity shows that there is a magnetized compact object in the system, and the hardness of the X-ray spectrum is typical of only a CV/IP or an accreting pulsar in an HMXB. The optical spectrum is dominated by an optically-thick irradiated accretion disk, and no evidence for emission from the companion star is found, indicating that it must be a low-mass star and ruling out the HMXB possibility. There is also strong long-term optical variability in the continuum and emission lines, the origin of which is not clear. IPs are known to be strongly variable in the optical regime and often show large spectroscopic modulations associated with the spin period of the white dwarf and/or the orbital period (see e.g. Still et al., 1998; Belle et al., 2003; Scaringi et al., 2011). While this may be the case for IGR J14091$-$6108, especially for the spectral lines, we believe that the large brightness difference between the FORS2 and Goodman optical spectra is mainly due to different accretion rates within the accretion disk. The X-ray spectrum is also consistent with a CV/IP, and we confirm the presence of the strong iron line originally reported by Tomsick et al. (2016) using _Chandra_ measurements. With _XMM_, we find that the line emission is consistent with being a combination of lines from three ionization states with widths of $0.07\pm 0.04$ keV and equivalent widths in the 160–320 eV range. These values are typical of iron line complexes seen in CV/IPs (Hellier & Mukai, 2004). Also, combining the 0.3–79 keV flux of $1.1\times 10^{-11}$ erg cm${}^{-2}$ s${}^{-1}$ with our estimated distance of 3–4 kpc from the optical extinction, we obtain an X-ray luminosity of $1.6\times 10^{34}$ $d_{3.5\,\rm{kpc}}^{2}$ erg s${}^{-1}$, which is consistent with expectations for CV/IPs. In the Bird et al. (2016) catalog, there are 40 confirmed and seven candidate CV/IPs. Of the confirmed systems, 18 have IGR names, making IGR J14091–6108 the 19th confirmed CV/IP discovered by _INTEGRAL_(Bird et al., 2016). However, we note that IGR J14091–6108 itself is not in the Bird et al. (2016) catalog, suggesting the possible presence of a much larger number of CV/IPs close to the _INTEGRAL_ detection limit. The strong drop in the pulsed fraction with energy that we see for IGR J14091–6108 has been seen for other CV/IPs. Taylor et al. (1997) report this energy dependence for the CV/IPs AO Psc and V1223 Sgr. For V1223 Sgr, phase-resolved spectroscopy shows that the energy dependence is primarily related to changes in the column density of absorbing material local to the CV (Hayashi et al., 2011), and we suspect that this is also the case for IGR J14091–6108. While IGR J14091–6108 is typical of CV/IPs in many respects, it is a candidate system for having a higher than typical white dwarf mass. Using the IP Mass model that Suleimanov et al. (2005) applied to 14 CV/IPs, we obtain a mass in excess of 1.3 $M_{\mathord{\odot}}$ for IGR J14091–6108, while Suleimanov et al. (2005) find masses between $0.50\pm 0.05$ $M_{\mathord{\odot}}$ (for EX Hya) and $1.00\pm 0.20$ $M_{\mathord{\odot}}$ (for V1062 Tau). Yuasa et al. (2010) also used spectral fitting to estimate white dwarf masses for 17 CV/IPs, and they found masses as high as $\sim$1.2 $M_{\mathord{\odot}}$ for V709 Cas, PQ Gem, and NY Lup. Hailey et al. (submitted) fit spectra from several X-ray satellites including _NuSTAR_ for the CV/IP IGR J17303–0601, which has a relatively massive white dwarf. With the _NuSTAR_ spectrum, partial covering absorption, and reflection, they measured a white dwarf mass of $1.16\pm 0.12$ $M_{\mathord{\odot}}$. If they replaced the IPM model with a Bremsstrahlung model, they obtained a temperature of $34\pm 2$ keV. From this comparison, the temperature of $81^{+31}_{-20}$ keV (or $66^{+33}_{-22}$ keV for the $>$10 keV fit) that we measure for IGR J14091–6108 is suggestive of mass in excess of $\sim$1.2 $M_{\mathord{\odot}}$. Hailey et al. discuss the CV/IP spectra in the context of the diffuse hard X-ray emission from the Galactic center (Perez et al., 2015), which may be caused by an unresolved population of magnetic CVs with white dwarfs that are more massive than the average Galactic population (Krivonos et al., 2007). One reason for interest in massive white dwarfs is the question of whether the progenitors of type Ia supernovae (SNe) are merging white dwarfs or accreting white dwarfs that detonate when they reach the Chandrasekhar limit. Based on the relatively low soft X-ray luminosities from nearby elliptical galaxies and galaxy bulges, Gilfanov & Bogdán (2010) and Di Stefano (7) argue that $<$5% of type Ia SNe are from accreting white dwarfs (although subsequent works showed that the X-rays could be attenuated by stellar winds, accretion winds, or white dwarf atmospheres; Di Stefano, 8; Nielsen et al., 25). Along these same lines, pre-explosion imaging of individual nearby type Ia SNe place interesting constraints on accreting and nuclear-burning white dwarfs as the SN progenitors (Liu et al., 2012; Nielsen et al., 2012, 28, 2014; Graur et al., 2014). On the other hand, the white dwarfs in CVs are, on average, more massive than white dwarfs that have not undergone mass accretion from a companion (Zorotovic et al., 2011), implying that accretion leads to a significant increase in the masses of the white dwarfs. Given the possibility that IGR J14091–6108 may harbor a white dwarf with a mass very close to the Chandrasekhar limit, follow-up observations to measure $M_{\rm WD}$ using other techniques should be high priority. Based on the 576 s spin period, an orbital period of several hours would be expected, and optical photometry might be used to search for this period. It may then be possible to obtain a radial velocity curve, but this may need to be done in the near-IR where we might see absorption lines from the companion star’s photosphere. <figure><img src="content_image/1604.03562/x1.png"><figcaption>Figure 1: (a) Periodograms for 0.3–12 keV light curves from XMM pn (black),MOS1 (blue), and MOS2 (green), showing the detection of a signal at 576.3±0.6s. For the periodogram, the power is calculated using the Z21 test. (b) Foldedpn light curve in the 0.3–12 keV band. The horizontal solid lines indicate thephases used to determine the maximum and minimum rates. The difference betweenthe maximum and minimum count rates divided by the sum of the maximum andminimum rates is the pulsed fraction.</figcaption></figure> <figure><img src="content_image/1604.03562/x2.png"><figcaption>Figure 2: The X-ray light curves folded on the 576.3 s period. Panels (a)-(e)show folded light curves for five XMM pn energy bands, and panel (f) shows the12–79 keV band from NuSTAR. Each panel shows the calculated value of thepulsed fraction with its 1-σ uncertainty. The same values are plotted vs.energy in Figure 3.</figcaption></figure> <figure><img src="content_image/1604.03562/x3.png"><figcaption>Figure 3: The pulsed fraction of the 576.3 s period versus energy. The filledcircles mark the XMM pn measurements, and the diamonds mark the NuSTAR(FPMA+FPMB) measurements. The errors shown are 1-σ.</figcaption></figure> <figure><img src="content_image/1604.03562/x4.png"><figcaption>Figure 4: Two optical spectra of the IGR J14091–6108 counterpart (VVVJ140845.99–610754.1). The purple spectrum was taken at SOAR in 2015 Januarywith the Goodman spectrograph, and the light blue spectrum was taken in 2015April at VLT with the FORS2 spectrograph. The emission lines from the source,and the lines due to interstellar absorption are labeled. See Table 2 fordetailed line information and parameters.</figcaption></figure> <figure><img src="content_image/1604.03562/x5.png"><figcaption>Figure 5: The same SOAR and VLT optical spectra of IGR J14091–6108 as shown inFigure 4. The spectra are dereddened, shown as a function of frequency, andthe continua are each compared to a power-law function.</figcaption></figure> <figure><img src="content_image/1604.03562/x6.png"><figcaption>Figure 6: (a) XMM and NuSTAR energy spectrum (folded through the instrumentresponse) fitted with a model consisting ofconstanttbabspcfabs(gaussian+reflectbremss). The black, light green, andblue spectra are for the XMM pn, MOS1, and MOS2 instruments, respectively. Theorange and red spectra are for NuSTAR FPMA and FPMB, respectively. (b) Theresiduals in terms of the data-to-model ratio.</figcaption></figure> <figure><img src="content_image/1604.03562/x7.png"><figcaption>Figure 7: Unfolded XMM and NuSTAR energy spectrum fitted with the IP Massmodel fit shown in Table 4. The components are 3 Gaussians (dash-dottedlines), reflection (dotted line), and the IP Mass model (dashed). The spectraare plotted using the same colors as for Figure 6.</figcaption></figure> Observatory ObsID Instrument Start Time (UT) End Time (UT) Exposure Time (ks) XMM 0761940301 pn 2015 July 21, 2.22 h 2015 July 21, 10.03 h 24.7 ” ” MOS1 2015 July 21, 1.89 h 2015 July 21, 10.15 h 28.9 ” ” MOS2 ” ” ” NuSTAR 30101001002 FPMA 2015 July 20, 21.85 h 2015 July 21, 10.52 h 22.9 ” ” FPMB ” ” ” VLT 095.D-0972(A) FORS2 2015 April 13, 6.64 h 2015 April 13, 7.49 h 2.2 SOAR — Goodman 2015 January 15, 8.45 h 2015 January 15, 8.62 h 0.60 Spectrograph Table 1: Observations of IGR J14091–6108 FORS2 Goodman Element λc444Measured wavelength in Å ˚W555Equivalent widths in Å FWHM666Full-width at half-maximum in km s−1, quadratically corrected for instrumental broadening Fline777Intrinsic line flux in units of 10−15ergcm−2s−1 ˚W FWHM Fline Hη 3835 – – – −3.4±0.7 637±96 1.10±0.20 Hϵ 3968 – – – −2.4±1.0 579±307 0.84±0.25 Hδ 4101 – – – −2.3±0.6 281±78 0.83±0.20 Hγ 4341 – – – −5.4±1.2 690±145 2.00±0.37 BB888The Bowen Blend, which is the \ionC3+\ionN3 complex. 4645 –3.2±1.1 1090±135 0.31±0.12 – – – \ionHe2 4684 –9.7±2.8 666±72 0.93±0.23 −5.9±0.9 856±95 2.78±0.38 Hβ 4861 –10.1±0.8 539±51 1.02±0.07 −5.4±1.1 494±43 2.94±0.27 \ionHe1 4922 –1.1±0.3 523±48 0.12±0.04 – – – \ionHe2 5409 -1.6±0.4 551±72 0.25±0.04 – – – \ionHe1 5875 –4.9±0.9 353±57 0.83±0.05 −4.0±0.5 492±82 3.21±0.44 Hα 6561 –37.6±1.5 652±45 6.84±0.12 −26.5±1.8 707±19 23.67±1.90 \ionHe1 6678 –4.2±0.5 370±52 0.77±0.03 −3.3±0.4 589±53 3.30±0.36 \ionH1 7065 –3.4±0.3 354±43 0.67±0.04 – – – \ionH1 8446 –2.1±0.5 313±36 0.42±0.04 – – – \ionH1 8500 –1.5±0.1 362±19 0.34±0.03 – – – \ionH1 8543 –2.2±0.3 401±51 0.48±0.06 – – – \ionH1 8597 –2.0±0.4 377±40 0.46±0.06 – – – \ionH1 8663 –3.5±0.5 402±39 0.76±0.05 – – – \ionH1 8749 –3.7±0.5 437±30 0.82±0.03 – – – \ionH1 8862 –5.4±0.6 507±39 1.17±0.14 – – – \ionH1 9015 –3.8±0.7 420±51 0.83±0.34 – – – \ionH1 9228 –14.1±1.0 520±107 3.02±0.25 – – – \ionH1 9543 –19.0±3.9 515±36 4.23±0.29 – – – \ionH1 10050 –14.6±3.2 676±93 3.33±0.80 – – – \ionHe2 10118 –9.7±3.2 881±123 2.23±0.78 – – – \ionH1 10828 –35.4±7.0 606±100 7.20±0.88 – – – \ionH1 10938 –21.8±6.2 652±113 4.39±0.68 – – – Table 2: Optical lines in the IGR J14091−6108 FORS2 and Goodman spectra. Parameter999The errors on the parameters are 90% confidence. Units 1 Gaussian101010The full model in XSPEC is constanttbabspcfabs(gaussian+reflectbremss). 2 Gaussians111111This is the same model as the first column except for an additional Gaussian. Ditto marks indicate parameters that are consistent with the first column. 3 Gaussians121212This is the same model as the first column except for two additional Gaussians. Ditto marks indicate parameters that are consistent with the other two columns. NH131313The column density is calculated assuming Wilms et al. (2000) abundances and Verner et al. (1996) cross sections. Along this line of sight, the Galactic value is NH=1.8×1022 cm−2 (Kalberla et al., 2005). 1022 cm−2 0.45+0.09−0.08 ” ” NH,pc 1022 cm−2 8+3−2 ” ” pc fraction — 0.65+0.04−0.05 ” ” kT keV 81+31−20 ” ” Nbremss141414The normalization for the bremss model is equal to 3.02×10−154πd2∫nenidV, where d is the distance to the source in units of cm, ne and ni are the electron and ion densities in the plasma, and V is the volume of the region containing the plasma. — (4.4+0.4−0.3)×10−4 ” ” Ω/2π — 1.0151515Fixed. ” ” A161616The abundance of elements heavier than He relative to solar. — 0.24+0.33−0.16 ” ” AFe171717The abundance of iron relative to the abundances specified by A. — 1.0g ” ” cosi — >0.70 ” ” Eline1 keV 6.59±0.04 6.4g 6.4g σline1 keV 0.28±0.04 0.24±0.05 0.07±0.04 Nline1 ph cm−2 s−1 (2.3±0.3)×10−5 (0.86+0.27−0.29)×10−5 (1.0±0.2)×10−5 EWline1 eV 940±120 230±80 320±60 Eline2 keV — 6.7g 6.7g σline2 keV — 0.24181818Tied to σline1. 0.07j Nline2 ph cm−2 s−1 — (1.4±0.3)×10−5 (0.65±0.16)×10−5 EWline2 eV — 470±100 160±40 Eline3 keV — — 6.97g σline3 keV — — 0.07j Nline3 ph cm−2 s−1 — — (0.51±0.15)×10−5 EWline3 eV — — 170±50 Cpn — 1.0g ” ” CMOS1 — 0.96±0.05 ” ” CMOS2 — 1.01±0.05 ” ” CFPMA — 1.11±0.07 ” ” CFPMB — 1.18±0.07 ” ” χ2/dof — 472/339 473/339 458/338 Table 3: Spectral Results for Bremsstrahlung Fits Parameter191919The errors on the parameters are 90% confidence. Units Value202020The full model is constanttbabspcfabs(gaussian+gaussian+gaussian+reflectipm). NH212121The column density is calculated assuming Wilms et al. (2000) abundances and Verner et al. (1996) cross sections. Along this line of sight, the Galactic value is NH=1.8×1022 cm−2 (Kalberla et al., 2005). 1022 cm−2 0.63+0.09−0.08 NH,pc 1022 cm−2 12+3−2 pc fraction — 0.72±0.03 MWD M⊙ >1.38 NIPM — (1.63+0.23−0.68)×10−13 Ω/2π — 1.0222222Fixed. A232323The abundance of elements heavier than He relative to solar. — 0.4+0.5−0.2 AFe242424The abundance of iron relative to the abundances specified by A. — 1.0d cosi — >0.84 Eline1 keV 6.4d σline1 keV 0.07+0.04−0.03 Nline1 ph cm−2 s−1 (1.1±0.2)×10−5 Eline2 keV 6.7d σline2 keV 0.07252525Tied to σline1. Nline2 ph cm−2 s−1 (0.67±0.16)×10−5 Eline3 keV 6.97d σline3 keV 0.07g Nline3 ph cm−2 s−1 (0.51±0.15)×10−5 Cpn — 1.0d CMOS1 — 0.96±0.05 CMOS2 — 1.00±0.05 CFPMA — 1.13±0.07 CFPMB — 1.20±0.07 χ2/dof — 480/338 Table 4: Spectral Results for IP Mass Model Fits ## Acknowledgments We would like to thank D. Wik for help with using nuskybgd to produce a background spectrum for _NuSTAR_. We also thank C. Hailey, J. Hong, and F. Fornasini for useful discussions. FR thanks the ESO staff who performed the service observations. JAT acknowledges partial support from NASA under _XMM_ Guest Observer grant NNX15AW09G. MC acknowledges partial support under NASA Contract NNG08FD60C for work on the _NuSTAR_ mission. This work was partially supported by NASA _Fermi_ grant NNX15AU83G. RK acknowledges support from Russian Science Foundation (grant 14-22-00271). This work made use of data from the _NuSTAR_ mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the _NuSTAR_ Operations, Software and Calibration teams for support with the execution and analysis of these observations. This research has made use of the _NuSTAR_ Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA). Based on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia, e Inovação (MCTI) da República Federativa do Brasil, the U.S. National Optical Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU). ## References * Arnaud (1996) Arnaud K. A., 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, Jacoby G. H., Barnes J., eds., p. 17 * Beardmore et al. (2000) Beardmore A. P., Osborne J. P., Hellier C., 2000, MNRAS, 315, 307 * Belle et al. (2003) Belle K. E., Howell S. B., Sion E. M., Long K. S., Szkody P., 2003, ApJ, 587, 373 * Bird et al. (2016) Bird A. J. et al., 2016, arXiv:1601.06074 * Buccheri et al. (1983) Buccheri R. et al., 1983, A&A, 128, 245 * Clemens et al. (2004) Clemens J. C., Crain J. A., Anderson R., 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5492, Ground-based Instrumentation for Astronomy, Moorwood A. F. M., Iye M., eds., pp. 331–340 * Di Stefano (2010a) Di Stefano R., 2010a, ApJ, 712, 728 * Di Stefano (2010b) Di Stefano R., 2010b, ApJ, 719, 474 * Gilfanov & Bogdán (2010) Gilfanov M., Bogdán Á., 2010, Nature, 463, 924 * Graur et al. (2014) Graur O., Maoz D., Shara M. M., 2014, MNRAS, 442, L28 * Güver & Özel (2009) Güver T., Özel F., 2009, MNRAS, 400, 2050 * Harrison et al. (2013) Harrison F. A. et al., 2013, ApJ, 770, 103 * Hayashi et al. (2011) Hayashi T., Ishida M., Terada Y., Bamba A., Shionome T., 2011, PASJ, 63, S739 * Hellier & Mukai (2004) Hellier C., Mukai K., 2004, MNRAS, 352, 1037 * Hong et al. (2012) Hong J., van den Berg M., Grindlay J. E., Servillat M., Zhao P., 2012, ApJ, 746, 165 * Jenniskens & Desert (1994) Jenniskens P., Desert F.-X., 1994, A&AS, 106, 39 * Kalberla et al. (2005) Kalberla P. M. W., Burton W. B., Hartmann D., Arnal E. M., Bajaja E., Morras R., Pöppel W. G. L., 2005, A&A, 440, 775 * Krivonos et al. (2007) Krivonos R., Revnivtsev M., Churazov E., Sazonov S., Grebenev S., Sunyaev R., 2007, A&A, 463, 957 * Krivonos et al. (2012) Krivonos R., Tsygankov S., Lutovinov A., Revnivtsev M., Churazov E., Sunyaev R., 2012, A&A, 545, A27 * Landi et al. (2012) Landi R., Bassani L., Masetti N., Bazzano A., Fiocchi M., Bird A. J., Drave S., 2012, The Astronomer’s Telegram, 4165, 1 * Liu et al. (2012) Liu J., Di Stefano R., Wang T., Moe M., 2012, ApJ, 749, 141 * Magdziarz & Zdziarski (1995) Magdziarz P., Zdziarski A. A., 1995, MNRAS, 273, 837 * Marshall et al. (2006) Marshall D. J., Robin A. C., Reylé C., Schultheis M., Picaud S., 2006, A&A, 453, 635 * Mukai et al. (2015) Mukai K., Rana V., Bernardini F., de Martino D., 2015, ApJ, 807, L30 * Nielsen et al. (2013a) Nielsen M. T. B., Dominik C., Nelemans G., Voss R., 2013a, A&A, 549, A32 * Nielsen et al. (2014) Nielsen M. T. B., Gilfanov M., Bogdán Á., Woods T. E., Nelemans G., 2014, MNRAS, 442, 3400 * Nielsen et al. (2012) Nielsen M. T. B., Voss R., Nelemans G., 2012, MNRAS, 426, 2668 * Nielsen et al. (2013b) Nielsen M. T. B., Voss R., Nelemans G., 2013b, MNRAS, 435, 187 * Perez et al. (2015) Perez K. et al., 2015, Nature, 520, 646 * Rahoui et al. (2014) Rahoui F., Coriat M., Lee J. C., 2014, MNRAS, 442, 1610 * Ritter & Kolb (2003) Ritter H., Kolb U., 2003, A&A, 404, 301 * Scaringi et al. (2011) Scaringi S. et al., 2011, A&A, 530, A6 * Still et al. (1998) Still M. D., Duck S. R., Marsh T. R., 1998, MNRAS, 299, 759 * Strüder et al. (2001) Strüder L. et al., 2001, A&A, 365, L18 * Suleimanov et al. (2005) Suleimanov V., Revnivtsev M., Ritter H., 2005, A&A, 435, 191 * Taylor et al. (1997) Taylor P., Beardmore A. P., Norton A. J., Osborne J. P., Watson M. G., 1997, MNRAS, 289, 349 * Tomsick et al. (2015) Tomsick J. A., Krivonos R., Rahoui F., Ajello M., Rodriguez J., Barrière N., Bodaghee A., Chaty S., 2015, MNRAS, 449, 597 * Tomsick et al. (2016) Tomsick J. A., Krivonos R., Wang Q., Bodaghee A., Chaty S., Rahoui F., Rodriguez J., Fornasini F. M., 2016, ApJ, 816, 38 * Tomsick et al. (2014) Tomsick J. A., Rahoui F., Krivonos R., Rodriguez J., Bodaghee A., Chaty S., 2014, The Astronomer’s Telegram, 6793, 1 * Turner et al. (2001) Turner M. J. L. et al., 2001, A&A, 365, L27 * Verner et al. (1996) Verner D. A., Ferland G. J., Korista K. T., Yakovlev D. G., 1996, ApJ, 465, 487 * Wik et al. (2014) Wik D. R. et al., 2014, ApJ, 792, 48 * Williams & Shipman (1988) Williams G. A., Shipman H. L., 1988, ApJ, 326, 738 * Williams (1980) Williams R. E., 1980, ApJ, 235, 939 * Wilms et al. (2000) Wilms J., Allen A., McCray R., 2000, ApJ, 542, 914 * Winkler et al. (2003) Winkler C. et al., 2003, A&A, 411, L1 * Yuasa et al. (2010) Yuasa T., Nakazawa K., Makishima K., Saitou K., Ishida M., Ebisawa K., Mori H., Yamada S., 2010, A&A, 520, A25 * Zorotovic et al. (2011) Zorotovic M., Schreiber M. R., Gänsicke B. T., 2011, A&A, 536, A42
1901.07963	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31894, "num_imgs": 2, "llama3_tokens_count": 11210 }	[ "content_image/1901.07963/x1.png", "content_image/1901.07963/x2.png" ]	The threshold age of Keyfitz’ entropy José Manuel Aburtoa,b, Jesus-Adrian Alvareza, Francisco Villavicencioc,a [FOOTNOTE:][ENDFOOTNOTE] [FOOTNOTE:a][ENDFOOTNOTE] [FOOTNOTE:b][ENDFOOTNOTE] [FOOTNOTE:c][ENDFOOTNOTE] and James W. Vaupela,b February 25, 2024 ## Abstract BACKGROUND Indicators of relative inequality of lifespans are important because they capture the dimensionless shape of aging. They are markers of inequality at the population level and express the uncertainty at the time of death at the individual level. In particular, Keyfitz’ entropy $H$ represents the elasticity of life expectancy to a change in mortality and it has been used as an indicator of lifespan variation. However, it is unknown how this measure changes over time and whether a threshold age exists, as it does for other lifespan variation indicators. RESULTS The time derivative of $H$ can be decomposed into changes in life disparity $e^{\dagger}$ and life expectancy at birth $e_{o}$. Likewise, changes over time in $H$ are a weighted average of age-specific rates of mortality improvements. These weights reflect the sensitivity of $H$ and show how mortality improvements can increase (or decrease) the relative inequality of lifespans. Further, we prove that $H$ , as well as $e^{\dagger}$, in the case that mortality is reduced in every age, has a threshold age below which saving lives reduces entropy, whereas improvements above that age increase entropy. CONTRIBUTION We give a formal expression for changes over time of $H$ and provide a formal proof of the threshold age that separates reductions and increases in lifespan inequality from age-specific mortality improvements. ## 1 Relationship Keyfitz’ entropy is a dimensionless indicator of the relative variation in the length of life compared to life expectancy (Keyfitz 1977; Demetrius 1978). It is usually defined as \[\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(t)=-\frac{\int_{0}^{\infty}\ell(a,t)\,\ln\ell(a,t)\,da}{\int_{0 }^{\infty}\ell(a,t)\,da}=\int_{0}^{\infty}c(a,t)\,H(a,t)\,da=\frac{e^{\dagger} (t)}{e_{o}(t)}\;,\] where $e^{\dagger}(t)=-\int_{0}^{\infty}\ell(a,t)\,\ln\ell(a,t)\,da$ is the life disparity or number of life-years lost as a result of death (Vaupel and Canudas-Romo 2003), $e_{o}(t)=\int_{0}^{\infty}\ell(a,t)\,da$ is the life expectancy at birth at time $t$, $\ell(a,t)$ is the life table survival function, $c(a,t)=\ell(a)\,/\,\int_{0}^{\infty}\ell(x)\,dx$ is the population structure, and $H(a,t)=\int_{0}^{a}\mu(x,t)\,dx$ is the cumulative hazard to age $a$, where $\mu(x,t)$ is the force of mortality (hazard rate or risk of death) at age $x$ at time $t$. Note that $\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(t)$ can be interpreted as an average value of $H(a,t)$ in the population at time $t$. Goldman and Lord (1986) and Vaupel (1986) proved that \[e^{\dagger}(t)=\int_{0}^{\infty}d(a,t)\,e(a,t)\,da\;,\] where $d(a,t)$ represents the distribution of deaths and $e(a,t)=\int_{a}^{\infty}\ell(x,t)\,dx\,/\,\ell(a,t)$ the remaining life expectancy at age $a$. This provides an alternative expression for Keyfitz’ entropy: \[\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(t)=\frac{\int_{0}^{\infty}d(a,t)\,e(a,t)\,da}{\int_{0}^{\infty} \ell(a,t)\,da}\;.\] Let $\dot{\hbox{\vbox{\hrule height 0.5pt width 100 $\kern-0.5pt}}}}$ denote the partial derivative of $H$ with respect to time.¹ We define $\rho(x)=-\dot{\mu}(x)\,/\,\mu(x)$ as the age-specific rates of mortality improvements. Then, the relative derivative of $H$ can be expressed as a weighted average of age-specific rates of mortality improvement, [FOOTNOTE:1][ENDFOOTNOTE] (1) \[\dot{\hbox{\vbox{\hrule height 0.5pt width 100 $\kern-0.5pt}}}}\,/\,\hbox{\vbox{\hrule height 0.5pt width 100 \hbox{\kern-2.5pt$H$\kern-0.5pt}}}=\int_{0}^{\infty}\rho(x)\,w(x)\,W(x)\,dx\;,\] with weights \[w(x)=\mu(x)\,\ell(x)\,e(x)\qquad\text{and}\qquad W(x)=\frac{1}{e^{\dagger}}\, \big{(}H(x)+\hbox{\vbox{\hrule height 0.5pt width 100 2.5pt$H$\kern-0.5pt}}}(x)-1\big{)}-\frac{1}{e_{o}}\;.\] Function $\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(x)$ is Keyfitz’ entropy conditioned on surviving to age $x$, defined as \[\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(x)=\frac{e^{\dagger}(x)}{e(x)}=\frac{\int_{x}^{\infty}d(a)\,e(a )\,da}{\int_{x}^{\infty}\ell(a)\,da}\,.\] where $e^{\dagger}(x)=\int_{x}^{\infty}d(a)\,e(a)\,da\,/\,\ell(x)$ refers to life disparity above age $x$, and $e(x)$ is the remaining life expectancy at age $x$. Note that Keyfitz’ entropy $H$ is a measure of lifespan inequality. Thus, higher values represent more lifespan disparity, whereas lower values denote less variation of lifespans. If mortality improvements over time occur at all ages, there exists a unique threshold age $a^{H}$ that separates _positive_ from _negative_ contributions to Keyfitz’ entropy $H$ resulting from those mortality improvements. This threshold age $a^{H}$ is reached when (2) \[H\left(a^{H}\right)+\hbox{\vbox{\hrule height 0.5pt width 100 \hbox{\kern-2.5pt$H$\kern-0.5pt}}}\left(a^{H}\right)=1+\hbox{\vbox{\hrule heig ht 0.5pt width 100%\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}\;.\] ## 2 Proof Fernández and Beltrán-Sánchez (2015) showed that the relative derivative of $H$ can be expressed as (3) \[\dot{\hbox{\vbox{\hrule height 0.5pt width 100 $\kern-0.5pt}}}}\,/\,\hbox{\vbox{\hrule height 0.5pt width 100 \hbox{\kern-2.5pt$H$\kern-0.5pt}}}=\frac{\dot{e}^{\dagger}}{e^{\dagger}}-\frac {\dot{e}_{o}}{e_{o}}\;,\] This formula indicates that relative changes in $H$ over time are given by the difference between relative changes in $e^{\dagger}$ (dispersion component) and relative changes in $e_{o}$ (translation component). We will first provide expressions for $\dot{e}_{o}$ and $\dot{e}^{\dagger}$ to prove that (1) and (3) are equivalent. Next, we will prove the existence of threshold age for $H$ and its uniqueness. ### Relative changes over time in $H$ Vaupel and Canudas-Romo (2003) showed that changes over time in life expectancy at birth are a weighted average of the total rates of mortality improvements: (4) \[\dot{e}_{o}=\int_{0}^{\infty}\rho(x)\,w(x)\,dx\;,\] where $\rho(x)=-\dot{\mu}(x)\,/\,\mu(x)$ are the age-specific rates of mortality improvement, and $w(x)=\mu(x)\,\ell(x)\,e(x)=d(x)e(x)$ is a measure of the importance of death at age $x$. Since $d(x)=\mu(x)\,\ell(x)$ and $\ell(x)\,e(x)=\int_{x}^{\infty}\ell(a)\,da$, the partial derivative with respect to time of $e^{\dagger}=\int_{0}^{\infty}d(a)\,e(a)\,da$ can be expressed as \[\begin{split}\dot{e}^{\dagger}&=\int_{0}^{\infty} \dot{\mu}(x)\,\ell(x)\,e(x)\,dx+\int_{0}^{\infty}\mu(x)\int_{x}^{\infty}\dot{ \ell}(a)\,da\,dx\\ &=-\int_{0}^{\infty}\rho(x)\,w(x)\,dx+\int_{0}^{\infty}\dot{\ell} (a)\int_{0}^{a}\mu(x)\,dx\,da\\ &=-\int_{0}^{\infty}\rho(x)\,w(x)\,dx+\int_{0}^{\infty}\dot{\ell} (a)\,H(a)\,da\\ &=-\int_{0}^{\infty}\rho(x)\,w(x)\,dx-\int_{0}^{\infty}\int_{0}^{ a}\dot{\mu}(x)\,dx\,\ell(a)\,H(a)\,da\;,\end{split}\] where $H(a)$ is the cumulative hazard to age $a$. By reversing the order of integration and doing some additional manipulations, we get (5) \[\begin{split}\dot{e}^{\dagger}&=-\int_{0}^{\infty} \rho(x)\,w(x)\,dx-\int_{0}^{\infty}\dot{\mu}(x)\int_{x}^{\infty}\,\ell(a)\,H(a )\,da\,dx\\ &=-\int_{0}^{\infty}\rho(x)\,w(x)\,dx+\int_{0}^{\infty}\rho(x)\,w (x)\,\frac{\int_{x}^{\infty}\,\ell(a)\,H(a)\,da}{\ell(x)\,e(x)}\,dx\\ &=\int_{0}^{\infty}\rho(x)\,w(x)\left(\frac{\int_{x}^{\infty}\, \ell(a)\big{(}H(a)-H(x)+H(x)\big{)}\,da}{\ell(x)\,e(x)}-1\right)dx\\ &=\int_{0}^{\infty}\rho(x)\,w(x)\left(H(x)\,\frac{\int_{x}^{ \infty}\,\ell(a)\,da}{\ell(x)\,e(x)}+\frac{\int_{x}^{\infty}\,\ell(a)\big{(}H( a)-H(x)\big{)}\,da}{\ell(x)\,e(x)}-1\right)\,dx\\ &=\int_{0}^{\infty}\rho(x)\,w(x)\left(H(x)+\frac{\int_{x}^{\infty }\,\ell(a)\big{(}H(a)-H(x)\big{)}\,da}{\ell(x)\,e(x)}-1\right)\,dx\;.\end{split}\] In Proposition 1 in the Appendix, we prove that (6) \[e^{\dagger}(x)=\frac{1}{\ell(x)}\int_{x}^{\infty}d(a)\,e(a)\,da=\frac{1}{\ell( x)}\int_{x}^{\infty}\,\ell(a)\big{(}H(a)-H(x)\big{)}\,da\;.\] Replacing (6) into (5) yields (7) \[\begin{split}\dot{e}^{\dagger}&=\int_{0}^{\infty} \rho(x)\,w(x)\left(H(x)+\frac{e^{\dagger}(x)}{e(x)}-1\right)dx\\ &=\int_{0}^{\infty}\rho(x)\,w(x)\big{(}H(x)+\hbox{\vbox{\hrule he ight 0.5pt\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)-1\big{)}\,dx\;. \end{split}\] Finally, replacing the expressions of $\dot{e}_{o}$ and $\dot{e}^{\dagger}$ from (4) and (7) into (3), we get \[\begin{split}\dot{\hbox{\vbox{\hrule height 0.5pt\kern 1.29pt \hbox{\kern-2.5pt$H$\kern-0.5pt}}}}\,/\,\hbox{\vbox{\hrule height 0.5pt\kern 1 .29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}&=\frac{1}{e^{\dagger}} \int_{0}^{\infty}\rho(x)\,w(x)\left(H(x)+\hbox{\vbox{\hrule height 0.5pt\kern 1 .29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)-1\right)dx-\frac{1}{e_{o}}\int_{0}^ {\infty}\rho(x)\,w(x)\,dx\\ &=\int_{0}^{\infty}\rho(x)\,w(x)\left(\frac{1}{e^{\dagger}}\left( H(x)+\hbox{\vbox{\hrule height 0.5pt\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5 pt}}}(x)-1\right)-\frac{1}{e_{o}}\right)dx\\ &=\int_{0}^{\infty}\rho(x)\,w(x)\,W(x)\,dx\;,\end{split}\] which proves (1) and shows that relative changes over time in Keyfitz’ entropy are the average of the rates of mortality improvement weighted by the product $w(x)\,W(x)$. $\square$ ### The threshold age for $H$ Using (1), changes over time in Keyfitz’ entropy $H$ are given by the function (8) \[\dot{\hbox{\vbox{\hrule height 0.5pt width 100 $\kern-0.5pt}}}}=\hbox{\vbox{\hrule height 0.5pt width 100 \kern-2.5pt$H$\kern-0.5pt}}}\,\int_{0}^{\infty}\rho(x)\,w(x)\,W(x)\,dx\;.\] If $\dot{\hbox{\vbox{\hrule height 0.5pt width 100 $\kern-0.5pt}}}}>0$, lifespan inequality increases over time, whereas $\dot{\hbox{\vbox{\hrule height 0.5pt width 100 $\kern-0.5pt}}}}<0$ implies that variation of lifespans decrease over time. Because $\ell(x)$ is a positive function bounded between 0 and 1, Keyfitz’ entropy $\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}>0$. Moreover, assuming age-specific death rates $\mu(x)$ improve over time for all ages, then $\dot{\mu}(x)<0$ and $\rho(x)>0$ at any age $x$. Therefore, (8) implies that 1. Those ages $x$ in which $w(x)\,W(x)>0$ will contribute _positively_ to Keyfitz’ entropy $H$ and increase lifespan variation; 2. Those ages $x$ in which $w(x)\,W(x)<0$ will contribute _negatively_ to Keyfitz’ entropy $H$ and favor lifespan equality; 3. Those ages $x$ in which $w(x)\,W(x)=0$ will have no effect on the variation over time of $H$ . Our goal is to prove that if mortality improvements occur for all ages and $\rho(x)>0$, there exists a unique threshold age $a^{H}$ such that $w\left(a^{H}\right)\,W\left(a^{H}\right)=0$. That threshold age will separate _positive_ from _negative_ contributions to $H$ resulting from mortality improvements. The product $w(x)\,W(x)$ can be re-expressed as \[\begin{split} w(x)\,W(x)&=\mu(x)\,\ell(x)\,e(x)\, \left(\,\frac{1}{e^{\dagger}}\left(H(x)+\hbox{\vbox{\hrule height 0.5pt\kern 1 .29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)-1\right)-\frac{1}{e_{o}}\right)\\ &=\frac{\mu(x)\,\ell(x)\,e(x)}{e^{\dagger}}\left(H(x)+\hbox{\vbox {\hrule height 0.5pt\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)-\hbox{ \vbox{\hrule height 0.5pt\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}-1 \right)\;.\end{split}\] Since $\mu(x)$, $\ell(x)$, $e(x)$ and $e^{\dagger}$ are all positive functions, the threshold age of $H$ occurs when (9) \[g(x)=H(x)+\hbox{\vbox{\hrule height 0.5pt width 100 5pt$H$\kern-0.5pt}}}(x)-\hbox{\vbox{\hrule height 0.5pt width 100 \hbox{\kern-2.5pt$H$\kern-0.5pt}}}-1=0\;.\] When $x$ is close to 0, $g(x)$ takes negative values since \[g(0)=H(0)+\hbox{\vbox{\hrule height 0.5pt width 100 5pt$H$\kern-0.5pt}}}^{+}(0)-\hbox{\vbox{\hrule height 0.5pt width 100 29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}-1=0+\hbox{\vbox{\hrule height 0.5pt wid th 100 t 0.5pt width 100%\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}-1=-1<0\;.\] Likewise, $g(x)$ takes positive values when $x$ becomes arbitrary large. Note that $H$ does not depend on age, and therefore \[\lim_{x\to\infty}g(x)=\lim_{x\to\infty}\left(H(x)+\hbox{\vbox{\hrule height 0. 5pt width 100%\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)\right)=\infty\;,\] because $\lim_{x\to\infty}H(x)=\infty$. By definition, $\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(x)\geq 0$ for all $x$, so regardless of the behavior of $\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(x)$ when $x$ is arbitrarily large, the limit of $g(x)$ tends to infinity. Hence, given that $g(0)=-1$ and $\lim_{x\to\infty}g(x)=\infty$, in a continuous framework the intermediate value theorem guarantees the existence of at least one age $a^{H}$ at which $g(a^{H})=0$. Moreover, as shown in Propostion 2 in the Appendix, $g(x)$ is an increasing function. Therefore, there is a unique threshold age $a^{H}$ that separates _positive_ from _negative_ contributions to Keyfitz’ entropy $H$ , and that threshold age is reached when \[w(x)\,W(x)=0\Longleftrightarrow g(x)=0\Longleftrightarrow H(x)+\hbox{\vbox{ \hrule height 0.5pt width 100 )=1+\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}\;,\] which proves (2). $\square$ ## 3 Related results Demographers have developed a battery of indicators to measure how lifespans vary in populations (Colchero et al. 2016; van Raalte and Caswell 2013). The most used indexes are the variance (Edwards and Tuljapurkar 2005; Tuljapurkar and Edwards 2011), standard deviation (van Raalte, Sasson, and Martikainen 2018), or coefficient of variation (Aburto et al. 2018) of the age at death distribution , the Gini coefficient (Shkolnikov, Andreev, and Begun 2003; Archer et al. 2018; Gigliarano, Basellini, and Bonetti 2017), Theil index (Smits and Monden 2009) and years of life lost (Vaupel, Zhang, and van Raalte 2011; Aburto and van Raalte 2018) among others. However, only few studies have analytically derived formulas for _threshold age_ below and above which mortality improvements respectively decrease and increase lifespan variation. Zhang and Vaupel (2009) showed that the threshold age $(a^{\dagger})$ for life disparity $(e^{\dagger})$ occurs when $H(x)+\bar{H}(x)=1$. Similarly, Gillespie, Trotter, and Tuljapurkar (2014) determined a threshold age for the variance of the age at death distribution. van Raalte and Caswell (2013) also showed that it is possible to determine the threshold age by performing an empirical sensitivity analysis of lifespan variation indicators. In this article, we contribute to the lifespan variation literature by deriving the threshold age $a^{H}$ for Keyfitz’ entropy. This age separates negative from positive contributions of age-specific mortality improvements. We analytically proved its existence and demonstrated in Section 4 that it differs from the threshold age of $e^{\dagger}$. ## 4 Applications ### Numerical findings Figure 1 depicts the threshold ages for the two related measures: $e^{\dagger}$ and $H$ . Calculations were performed using data from the Human Mortality Database (2018) for females in the United States and Italy in 2005. The blue line represents $g(x)$ from Equation (9). The threshold age $a^{H}$ occurs when $g(x)$ crosses zero. The red and grey line display the same functions that Zhang and Vaupel (2009) used to find the threshold age for $e^{\dagger}$. The intersection of these two lines denotes the threshold age $a^{\dagger}$. Finally, the dashed black line depicts the life expectancy at birth. Vaupel, Zhang, and van Raalte (2011) noted that $a^{\dagger}$ tends to fall just below life expectancy. The threshold age for Keyfitz’ entropy $a^{H}$ is greater than $a^{\dagger}$ and is very close above life expectancy for these countries. Not the similarity of the formulas for $a^{\dagger}$ given by $H(a^{\dagger})+\bar{H}(a^{\dagger})=1$ and $a^{H}$ given by $H(a^{H})+\hbox{\vbox{\hrule height 0.5pt width 100 pt$H$\kern-0.5pt}}}(a^{H})=1+\hbox{\vbox{\hrule height 0.5pt width 100 .29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}$. <figure><img src="content_image/1901.07963/x1.png"><figcaption>Figure 1: Derivation of threshold ages for e† (a†) and Keyfitz’ entropy (aH)from life table functions for United States and Italy in 2005. Source: HumanMortality Database (2018)</figcaption></figure> Panel A and B of Figure 2 illustrate the evolution of the threshold ages for $e^{\dagger}$ and $H$ for females in France and Sweden respectively. We chose these countries because they portray large series of reliable data available through the Human Mortality Database (2018). Values for $a^{\dagger}$ are close to life expectancy throughout the period. However, after around 1950 there is a crossover between $a^{\dagger}$ and $e_{o}$ such that $a^{\dagger}$ remained close to life expectancy but below it. This result shows that the threshold age $a^{\dagger}$ being below life expectancy is a modern feature of ageing populations with high life expectancy. From the beginning of the period of observation to the 1950s, the threshold age for Keyfitz’ entropy was above life expectancy for both countries. During some periods $a^{\dagger}$ was roughly constant whereas life expectancy trended upwards. After the 1950s, $a^{\dagger}$ converged towards life expectancy. The code and data to reproduce these results are publicly available through this repository [link not given to avoid identification of authors]. <figure><img src="content_image/1901.07963/x2.png"><figcaption>Figure 2: Threshold age for e† and Keyftiz’ entropy H for females in Franceand Sweden over time. Source: Human Mortality Database (2018)</figcaption></figure> ### Decomposition of the relative derivative of $H$ The relative derivative of $H$ defined in Equation (1) can be decomposed between components before and after the threshold age $a^{H}$ as follows: (10) \[\begin{split}\dot{\hbox{\vbox{\hrule height 0.5pt\kern 1.29pt \hbox{\kern-2.5pt$H$\kern-0.5pt}}}}\,/\,\hbox{\vbox{\hrule height 0.5pt\kern 1 .29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}&=\int_{0}^{\infty}\rho(x) \,w(x)\,W(x)\,dx\\ &=\int_{0}^{a^{H}}\rho(x)\,w(x)\,W(x)\,dx+\int_{a^{H}}^{\infty} \rho(x)\,w(x)\,W(x)\,dx\\ &=\underbrace{\left\{\frac{\dot{e}^{\dagger}[x\|x<a^{H}]}{e^{ \dagger}}-\frac{\dot{e}_{o}[x\|x<a^{H}]}{e_{o}}\right\}}_{\textit{Early life component}}+\underbrace{\left\{\frac{\dot{e}^{\dagger}[x\|x>a^{H}]}{e^{\dagger} }-\frac{\dot{e}_{o}[x\|x>a^{H}]}{e_{o}}\right\}}_{\textit{Late life component}} \end{split}\] If mortality reductions occur at every age, the _early life component_ in Equation (10) is always positive (contributing to reduce entropy) while the _late life component_ is negative (contributing to increasing entropy). Thus, it is clear that a negative relationship between life expectancy and entropy over time occurs if the early life component outpaces the late life component. This decomposition is based on the additive properties of the derivatives of life expectancy and $e^{\dagger}$ as previously shown in Vaupel and Canudas-Romo (2003) and Fernández and Beltrán-Sánchez (2015). ## 5 Conclusion Several authors have been interested in decomposing changes over time in life expectancy (Arriaga 1984; Vaupel 1986; Pollard 1988; Vaupel and Canudas-Romo 2003; Beltrán-Sánchez, Preston, and Canudas-Romo 2008; Beltrán-Sánchez and Soneji 2011). Recently, authors have also investigated how life disparity fluctuations over time can be decomposed (Wagner 2010; Zhang and Vaupel 2009; Shkolnikov et al. 2011; Aburto and van Raalte 2018). In this paper, we bring both perspectives together and shed light on the dynamics behind changes in Keyfitz’ entropy. Keyfitz (1977) first proposed $H$ as a life table function that measures _the change in life expectancy at birth consequent on a proportional change in age-specific rates_. Since then, several authors have been interested in this measure and its use (Demetrius 1978, 1979; Mitra 1978; Goldman and Lord 1986; Vaupel 1986; Hakkert 1987; Hill 1993; Fernández and Beltrán-Sánchez 2015). Keyfitz’ entropy has become an appropriate indicator of lifespan variation that permits comparison of the shape of ageing across different species and over time (Baudisch et al. 2013; Wrycza, Missov, and Baudisch 2015). In this article, we uncover the mathematical regularities behind the changes over time in Keyfitz’ entropy. In particular, this study contributes to the existing literature by showing that (1) Keyfitz’ entropy can be decomposed as a weighted average of rates of mortality improvements and (2) that there exists a threshold age that separates positive and negative contributions of reductions in mortality over time. ## References * Aburto and van Raalte (2018) Aburto, J.M. and van Raalte, A.A. (2018). Lifespan dispersion in times of life expectancy fluctuation: The case of Central and Eastern Europe. _Demography_ 55(6): 2071–2096. * Aburto et al. (2018) Aburto, J.M., Wensink, M., van Raalte, A., and Lindahl-Jacobsen, R. (2018). Potential gains in life expectancy by reducing inequality of lifespans in Denmark: An international comparison and cause-of-death analysis. _BMC Public Health_ 18(1): 831. * Archer et al. (2018) Archer, C.R., Basellini, U., Hunt, J., Simpson, S.J., Lee, K.P., and Baudisch, A. (2018). Diet has independent effects on the pace and shape of aging in Drosophila melanogaster. _Biogerontology_ 19(1): 1–12. * Arriaga (1984) Arriaga, E.E. (1984). Measuring and explaining the change in life expectancies. _Demography_ 21(1): 83–96. * Baudisch et al. (2013) Baudisch, A., Salguero-Gómez, R., Jones, O.R., Wrycza, T., Mbeau-Ache, C., Franco, M., and Colchero, F. (2013). The pace and shape of senescence in angiosperms. _Journal of Ecology_ 101(3): 596–606. * Beltrán-Sánchez, Preston, and Canudas-Romo (2008) Beltrán-Sánchez, H., Preston, S.H., and Canudas-Romo, V. (2008). An integrated approach to cause-of-death analysis: Cause-deleted life tables and decompositions of life expectancy. _Demographic Research_ 19: 1323–1350. * Beltrán-Sánchez and Soneji (2011) Beltrán-Sánchez, H. and Soneji, S. (2011). A unifying framework for assessing changes in life expectancy associated with changes in mortality: The case of violent deaths. _Theoretical Population Biology_ 80(1): 38–48. * Colchero et al. (2016) Colchero, F., Rau, R., Jones, O.R., Barthold, J.A., Conde, D.A., Lenart, A., Németh, L., Scheuerlein, A., Schoeley, J., Torres, C., …, Alberts, S.C., and Vaupel, J.W. (2016). The emergence of longevous populations. _Proceedings of the National Academy of Sciences_ 113(48): E7681–E7690. * Demetrius (1978) Demetrius, L. (1978). Adaptive value, entropy and survivorship curves. _Nature_ 275: 213–214. * Demetrius (1979) Demetrius, L. (1979). Relations between demographic parameters. _Demography_ 16(2): 329–338. * Edwards and Tuljapurkar (2005) Edwards, R.D. and Tuljapurkar, S. (2005). Inequality in life spans and a new perspective on mortality convergence across industrialized countries. _Population and Development Review_ 31(4): 645–674. * Fernández and Beltrán-Sánchez (2015) Fernández, O.E. and Beltrán-Sánchez, H. (2015). The entropy of the life table: A reappraisal. _Theoretical Population Biology_ 104: 26–45. * Gigliarano, Basellini, and Bonetti (2017) Gigliarano, C., Basellini, U., and Bonetti, M. (2017). Longevity and concentration in survival times: The log-scale-location family of failure time models. _Lifetime Data Analysis_ 23(2): 254–274. * Gillespie, Trotter, and Tuljapurkar (2014) Gillespie, D.O., Trotter, M.V., and Tuljapurkar, S.D. (2014). Divergence in age patterns of mortality change drives international divergence in lifespan inequality. _Demography_ 51(3): 1003–1017. * Goldman and Lord (1986) Goldman, N. and Lord, G. (1986). A new look at entropy and the life table. _Demography_ 23(2): 275–282. * Hakkert (1987) Hakkert, R. (1987). Life table transformations and inequality measures: Some noteworthy formal relationships. _Demography_ 24(4): 615–622. * Hill (1993) Hill, G. (1993). The entropy of the survival curve: An alternative measure. _Canadian Studies in Population_ 20(1): 43–57. * Human Mortality Database (2018) Human Mortality Database (2018). University of California, Berkeley, and Max Planck Institute for Demographic Research, Rostock. URL http://www.mortality.org. * Keyfitz (1977) Keyfitz, N. (1977). What difference would it make if cancer were eradicated? An examination of the Taeuber paradox. _Demography_ 14(4): 411–418. * Mitra (1978) Mitra, S. (1978). A short note on the Taeuber paradox. _Demography_ 15(4): 621–623. * Pollard (1988) Pollard, J.H. (1988). On the decomposition of changes in expectation of life and differentials in life expectancy. _Demography_ 25(2): 265–276. * Shkolnikov et al. (2011) Shkolnikov, V.M., Andreev, E.M., Zhang, Z., Oeppen, J., and Vaupel, J.W. (2011). Losses of expected lifetime in the United States and other developed countries: Methods and empirical analyses. _Demography_ 48(1): 211–239. * Shkolnikov, Andreev, and Begun (2003) Shkolnikov, V.M., Andreev, E.E., and Begun, A.Z. (2003). Gini coefficient as a life table function: Computation from discrete data, decomposition of differences and empirical examples. _Demographic Research_ 8(11): 305–358. * Smits and Monden (2009) Smits, J. and Monden, C. (2009). Length of life inequality around the globe. _Social Science & Medicine_ 68(6): 1114–1123. * Tuljapurkar and Edwards (2011) Tuljapurkar, S. and Edwards, R.D. (2011). Variance in death and its implications for modeling and forecasting mortality. _Demographic Research_ 24(21): 497–526. * van Raalte and Caswell (2013) van Raalte, A.A. and Caswell, H. (2013). Perturbation analysis of indices of lifespan variability. _Demography_ 50(5): 1615–1640. * van Raalte, Sasson, and Martikainen (2018) van Raalte, A.A., Sasson, I., and Martikainen, P. (2018). The case for monitoring life-span inequality. _Science_ . * Vaupel (1986) Vaupel, J.W. (1986). How change in age-specific mortality affects life expectancy. _Population Studies_ 40(1): 147–157. * Vaupel and Canudas-Romo (2003) Vaupel, J.W. and Canudas-Romo, V. (2003). Decomposing change in life expectancy: A bouquet of formulas in honor of Nathan Keyfitz’s 90th birthday. _Demography_ 40(2): 201–216. * Vaupel, Zhang, and van Raalte (2011) Vaupel, J.W., Zhang, Z., and van Raalte, A.A. (2011). Life expectancy and disparity: An international comparison of life table data. _BMJ Open_ bmjopen–2011–000128. * Wagner (2010) Wagner, P. (2010). Sensitivity of life disparity with respect to changes in mortality rates. _Demographic Research_ 23(3): 63–72. * Wrycza, Missov, and Baudisch (2015) Wrycza, T.F., Missov, T.I., and Baudisch, A. (2015). Quantifying the shape of aging. _PLoS ONE_ 10(3): e0119163. * Zhang and Vaupel (2009) Zhang, Z. and Vaupel, J.W. (2009). The age separating early deaths from late deaths. _Demographic Research_ 20(29): 721–730. ## Appendix Proposition 1.: _Let $e^{\dagger}(x)=\int_{x}^{\infty}d(a)\,e(a)\,da\,/\,\ell(x)$ be a measure of lifespan disparity above age $x$, where $d(a)$ accounts for the distribution of deaths, $e(a)$ the remaining life expectancy at age $a$, and $\ell(x)$ is the probability of surviving from birth to age $x$. Then,_ (A1) _where $H(x)$ is the cumulative hazard to age $x$._ Proof.: Note that \[\frac{1}{\ell(x)}\int_{x}^{\infty}\,\ell(a)\big{(}H(a)-H(x)\big{)}\,da=\frac{1 }{\ell(x)}\int_{x}^{\infty}\,\ell(a)\int_{x}^{a}\mu(y)\,dy\,da\;,\] where function $\mu(y)$ is the force of mortality or hazard rate. By reversing the order of integration, and using that $e(y)=\int_{y}^{\infty}\ell(a)\,da\,/\,\ell(y)$ and $d(y)=\mu(y)\,\ell(y)$, we get \[\begin{split}\frac{1}{\ell(x)}\int_{x}^{\infty}\,\ell(a)\int_{x}^ {a}\mu(y)\,dy\,da&=\frac{1}{\ell(x)}\int_{x}^{\infty}\mu(y)\int_{ y}^{\infty}\ell(a)\,da\,dy\\ &=\frac{1}{\ell(x)}\int_{x}^{\infty}\mu(y)\,\ell(y)\,e(y)\,dy\\ &=\frac{1}{\ell(x)}\int_{x}^{\infty}d(y)\,e(y)\,dy\\ &=e^{\dagger}(x)\;,\end{split}\] which proves (A1). ∎ Proposition 2.: _Let $\ell(x)$ be the probability of surviving from birth to age $x$. Let $H$ be Keyfitz’ entropy and $\hbox{\vbox{\hrule height 0.5pt width 100 \kern-0.5pt}}}(x)=e^{\dagger}(x)\,/\,e(x)$ Keyfitz’ entropy conditioned on reaching age $x$. Let $H(x)$ be the cumulative hazard to age $x$. Then, $g(x)=H(x)+\hbox{\vbox{\hrule height 0.5pt width 100 5pt$H$\kern-0.5pt}}}(x)-1-\hbox{\vbox{\hrule height 0.5pt width 100 pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}$ is an increasing function._ Proof.: In order to demonstrate that $g(x)$ is an increasing function it is sufficient to show that its first derivative is always positive. Hence, we must prove that (A2) \[\frac{\partial}{\partial x}\,g(x)=\frac{\partial}{\partial x}\left(H(x)+\hbox{ \vbox{\hrule height 0.5pt width 100 pt}}}(x)-1-\hbox{\vbox{\hrule height 0.5pt width 100 .5pt$H$\kern-0.5pt}}}\right)=\frac{\partial}{\partial x}\,H(x)+\frac{\partial} {\partial x}\hbox{\vbox{\hrule height 0.5pt width 100 2.5pt$H$\kern-0.5pt}}}(x)\geq 0\] for all ages $x$. By the fundamental theorem of calculus, (A3) \[\frac{\partial}{\partial x}\,H(x)=\frac{\partial}{\partial x}\int_{0}^{x}\mu(a )\,da=\mu(x)\;,\] whereas \[\frac{\partial}{\partial x}\,\hbox{\vbox{\hrule height 0.5pt width 100 .29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)=\frac{\partial}{\partial x}\,\left( \frac{e^{\dagger}(x)}{e(x)}\right)=\frac{1}{e(x)^{2}}\left(e(x)\,\frac{ \partial}{\partial x}\,e^{\dagger}(x)-e^{\dagger}(x)\,\frac{\partial}{\partial x }\,e(x)\right)\;.\] First, note that (A4) \[\begin{split}\frac{\partial}{\partial x}\,e^{\dagger}(x)& =\frac{\partial}{\partial x}\left(\frac{1}{\ell(x)}\int_{x}^{ \infty}d(a)\,e(a)\,da\right)\\ &=\frac{1}{\ell(x)^{2}}\left(\ell(x)\,\frac{\partial}{\partial x} \left(\int_{x}^{\infty}d(a)\,e(a)\,da\right)-\int_{x}^{\infty}d(a)\,e(a)\,da\, \frac{\partial}{\partial x}\,\ell(x)\right)\\ &=\frac{1}{\ell(x)^{2}}\left(\ell(x)\,\big{(}-d(x)\,e(x)\big{)}- \int_{x}^{\infty}d(a)\,e(a)\,da\,\big{(}-\mu(x)\,\ell(x)\big{)}\right)\\ &=-\frac{\mu(x)\,\ell(x)\,e(x)}{\ell(x)}+\mu(x)\,\frac{\int_{x}^{ \infty}d(a)\,e(a)\,da}{\ell(x)}\\ &=\mu(x)\left(e^{\dagger}(x)-e(x)\right)\;.\end{split}\] On the other hand, (A5) \[\begin{split}\frac{\partial}{\partial x}\,e(x)&= \frac{\partial}{\partial x}\left(\frac{1}{\ell(x)}\int_{x}^{\infty}\ell(a)\,da \right)\\ &=\frac{1}{\ell(x)^{2}}\,\left(\ell(x)\,\frac{\partial}{\partial x }\left(\int_{x}^{\infty}\ell(a)\,da\right)-\int_{x}^{\infty}\ell(a)\,da\,\frac {\partial}{\partial x}\,\ell(x)\right)\\ &=\frac{1}{\ell(x)^{2}}\,\left(\ell(x)\,\big{(}-\ell(x)\big{)}- \int_{x}^{\infty}\ell(a)\,da\,\big{(}-\mu(x)\,\ell(x)\big{)}\right)\\ &=e(x)\,\mu(x)-1\;.\end{split}\] Therefore, using (A4) and (A5), we get (A6) \[\begin{split}\frac{\partial}{\partial x}\,\hbox{\vbox{\hrule heig ht 0.5pt\kern 1.29pt\hbox{\kern-2.5pt$H$\kern-0.5pt}}}(x)&=\frac{ 1}{e(x)^{2}}\,\bigg{(}e(x)\,\mu(x)\left(e^{\dagger}(x)-e(x)\right)-e^{\dagger} (x)\big{(}e(x)\,\mu(x)-1\big{)}\bigg{)}\\ &=\frac{1}{e(x)^{2}}\left(e^{\dagger}(x)\,e(x)\,\mu(x)-e(x)^{2}\, \mu(x)-e^{\dagger}(x)\,e(x)\,\mu(x)+e^{\dagger}(x)\right)\\ &=\frac{e^{\dagger}(x)}{e(x)^{2}}-\mu(x)\;.\end{split}\] Finally, replacing (A3) and (A6) in (A2) yields \[\frac{\partial}{\partial x}\,g(x)=\mu(x)+\frac{e^{\dagger}(x)}{e(x)^{2}}-\mu(x )=\frac{e^{\dagger}(x)}{e(x)^{2}}\geq 0\;,\] for all ages $x$, which proves that $g(x)$ is an increasing function. ∎
1305.6681	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 27005, "num_imgs": 2, "llama3_tokens_count": 7639 }	[ "content_image/1305.6681/x1.png", "content_image/1305.6681/x5.png" ]	# Large eddy simulation of a muffler with the high-order spectral difference method Matteo Parsani Current Institution: Computational Aerosciences Branch, NASA Langley Research Center, Hampton, VA 23681, USA (matteo.parsani@nasa.gov)Division of Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, Thuwal, 23955-6900, KSA Michael Bilka University of Notre Dame, Department of Aerospace and Mechanical Engineering, 365 Fitzpatrick Hall, Notre Dame, IN 46556-5637, USA (michael.bilka.1@nd.edu) Chris Lacor Department of Mechanical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium (chris.lacor@vub.ac.be) ###### Abstract The combination of the high-order accurate spectral difference discretization on unstructured grids with subgrid-scale modelling is investigated for large eddy simulation of a muffler at $Re=4.64\cdot 10^{4}$ and low Mach number. The subgrid-scale stress tensor is modelled by the wall-adapting local eddy-viscosity model with a cut-off length which is a decreasing function of the order of accuracy of the scheme. Numerical results indicate that although the high-order solver without subgrid-scale modelling is already able to capture well the features of the flow, the coupling with the wall-adapting local eddy-viscosity model improves the quality of the solution. ## 1 Introduction Throughout the past two decades, the development of high-order accurate spatial discretization has been one of the major fields of research in computational fluid dynamics (CFD), computational aeroacoustics (CAA), computational electromagnetism (CEM) and in general computational physics characterized by linear and nonlinear wave propagation phenomena. High-order accurate discretizations have the potential to improve the computational efficiency required to achieve a desired error level. In fact, compared with low order schemes, high order methods offer better wave propagation properties and increased accuracy for a comparable number of degrees of freedom (DOFs). Therefore, it may be advantageous to use such schemes for problems that require very low numerical dissipation and small error levels [1]. Moreover, since computational science is increasingly used as an industrial design and analysis tool, high accuracy must be achieved on unstructured grids which are required for efficient meshing. These needs have been the driving force for the development of a variety of higher order schemes for unstructured meshes such as the Discontinuous Galerkin (DG) method [2, 3], the Spectral Volume (SV) method [4], the Spectral Difference (SD) method [5, 6], the Energy Stable Flux Reconstruction [7] and many others. In this study we focus on a SD solver for unstructured hexahedral grids (tensorial cells). The SD method has been proposed as an alternative high order collocation-based method using local interpolation of the strong form of the equations. Therefore, the SD scheme has an important advantage over classical DG and SV methods, that no integrals have to be evaluated to compute the residuals, thus avoiding the need for costly high-order accurate quadrature formulas. Although the formulation of high-order accurate spatial discretization is now fairly mature, their application for the simulation of general turbulent flows implies that particular attention has still to be paid to subgrid-scale (SGS) models. So far, the combination of the SD method with SGS models for LES has not been widely investigated. In 2010, Parsani et al. [8] reported the first implementation in study of a two-dimensional (2D) third-order accurate SD solver coupled with the Wall-Adapting Local Eddy-viscosity (WALE) model [14] and a cut-off length which is a decreasing function of the order of accuracy. A successful extension of that approach to a three-dimensional (3D) second-order accurate SD solver has been reported in [12]. Very recently, Lodato and Jameson [13] have presented an alternative technique to model the unresolved scales in the flow field: A structural SGS approach with the WALE Similarity Mixed model (WSM), where constrained explicit filtering represents a key element to approximate subgrid-scale interactions. The performance of such an algorithm has been also satisfactory. In this study, we couple for the first time the approach proposed in [8] with a 3D fourth-order accurate SD solver, for the simulation of the turbulent flow in an industrial-type muffler at $Re=4.64\cdot 10^{4}$. The goal is to investigate if the coupling of a high-order SD scheme with a sub-grid closure model improves the quality of the results when the grid resolution is relatively low. The latter requirement is often desirable when a high-order accurate spatial discretization is used. ## 2 Physical model and numerical algorithm In this study the system of the Navier-Stokes equations for a compressible flow are discretized in space using the SD method and the subgrid-scale stress tensor is modelled by the WALE approach. ### Filtered Navier-Stokes equations The three physical conservation laws for a general Newtonian fluid, i.e., the continuity, the momentum and energy equations, are introduced using the following notation: $\rho$ for the mass density, $\vec{u}\in\mathbb{R}^{dim}$ for the velocity vector in a physical space with $dim$ dimensions, $P$ for the static pressure and $E$ for the specific total energy which is related to the pressure and the velocity vector field by $E=\frac{1}{\gamma-1}\frac{P}{\rho}+\frac{\|\vec{u}\|^{2}}{2}$, where $\gamma$ is the constant ratio of specific heats and it is $1.4$ for air in standard conditions. The system, written in divergence form and equipped with suitable initial-boundary conditions, is \[\frac{\partial\textbf{w}}{\partial t}+\vec{\nabla}\cdot\left(\vec{\textbf{f}}_ {C}\left(\textbf{w}\right)-\vec{\textbf{f}}_{D}\left(\textbf{w},\vec{\nabla} \textbf{w}\right)\right)=\frac{\partial\textbf{w}}{\partial t}+\vec{\nabla} \cdot\vec{\textbf{f}}=0,\] (1) where $\textbf{w}=\left(\overline{\rho},\overline{\rho}\tilde{\vec{u}},\overline{\rho }\tilde{E}\right)^{T}$ is the vector of the filtered conservative variables and $\vec{\textbf{f}}_{C}=\vec{\textbf{f}}_{C}\left(\textbf{w}\right)$ and $\vec{\textbf{f}}_{D}=\vec{\textbf{f}}_{D}\left(\textbf{w},\vec{\nabla}\textbf{ w}\right)$ represent the convective and the diffusive fluxes, respectively. Here the symbols $(\overline{\cdot})$ and $(\tilde{\cdot})$ represent the spatially filtered field and the Favre filtered field defined as $\tilde{\vec{u}}=\overline{\rho\vec{u}}/\overline{\rho}$. In a general 3D ($dim=3$) Cartesian space, $\vec{x}=\left[x_{1},x_{2},x_{3}\right]^{T}$, the components of the flux vector $\vec{\textbf{f}}\left(\textbf{w},\vec{\nabla}\textbf{w}\right)=\left[\textbf{f }_{1},\textbf{f}_{2},\textbf{f}_{3}\right]^{T}$ are given by \[\textbf{f}_{1}=\left(\begin{array}[]{c}\overline{\rho}\tilde{u}_{1}\\ \overline{\rho}\tilde{u}_{1}^{2}+\overline{P}-\tilde{\sigma}_{11}+\tau_{11}^{ sgs}\\ \overline{\rho}\tilde{u}_{1}\tilde{u}_{2}-\tilde{\sigma}_{21}+\tau_{21}^{sgs} \\ \overline{\rho}\tilde{u}_{1}\tilde{u}_{3}-\tilde{\sigma}_{31}+\tau_{31}^{sgs} \\ \tilde{u}_{1}\left(\overline{\rho}\tilde{E}+\overline{P}\right)-\tilde{u}_{1} \left(\tilde{\sigma}_{11}-\tau_{11}^{sgs}\right)-\tilde{u}_{2}\left(\tilde{ \sigma}_{21}-\tau_{21}^{sgs}\right)-\tilde{u}_{3}\left(\tilde{\sigma}_{31}- \tau_{31}^{sgs}\right)-c_{P}\frac{\mu}{Pr}\frac{\partial\tilde{T}}{\partial x_ {1}}+q_{1}^{sgs}\end{array}\right),\] \[\textbf{f}_{2}=\left(\begin{array}[]{c}\overline{\rho}\tilde{u}_{2}\\ \overline{\rho}\tilde{u}_{1}\tilde{u}_{2}-\tilde{\sigma}_{12}+\tau_{12}^{sgs} \\ \overline{\rho}\tilde{u}_{2}^{2}+\overline{P}-\tilde{\sigma}_{22}+\tau_{22}^{ sgs}\\ \overline{\rho}\tilde{u}_{2}\tilde{u}_{3}-\tilde{\sigma}_{32}+\tau_{32}^{sgs} \\ \tilde{u}_{2}\left(\overline{\rho}\tilde{E}+\overline{P}\right)-\tilde{u}_{1} \left(\tilde{\sigma}_{12}-\tau_{12}^{sgs}\right)-\tilde{u}_{2}\left(\tilde{ \sigma}_{22}-\tau_{22}^{sgs}\right)-\tilde{u}_{3}\left(\tilde{\sigma}_{32}- \tau_{32}^{sgs}\right)-c_{P}\frac{\mu}{Pr}\frac{\partial\tilde{T}}{\partial x_ {2}}+q_{2}^{sgs}\end{array}\right),\] \[\textbf{f}_{3}=\left(\begin{array}[]{c}\overline{\rho}\tilde{u}_{3}\\ \overline{\rho}\tilde{u}_{1}\tilde{u}_{3}-\tilde{\sigma}_{13}+\tau_{13}^{sgs} \\ \overline{\rho}\tilde{u}_{2}\tilde{u}_{3}-\tilde{\sigma}_{23}+\tau_{23}^{sgs} \\ \overline{\rho}\tilde{u}_{3}^{2}+\overline{P}-\tilde{\sigma}_{33}+\tau_{33}^{ sgs}\\ \tilde{u}_{3}\left(\overline{\rho}\tilde{E}+\overline{P}\right)-\tilde{u}_{1} \left(\tilde{\sigma}_{13}-\tau_{13}^{sgs}\right)-\tilde{u}_{2}\left(\tilde{ \sigma}_{23}-\tau_{23}^{sgs}\right)-\tilde{u}_{3}\left(\tilde{\sigma}_{33}- \tau_{33}^{sgs}\right)-c_{P}\frac{\mu}{Pr}\frac{\partial\tilde{T}}{\partial x_ {3}}+q_{3}^{sgs}\end{array}\right),\] where $c_{P}$, $\mu$, $Pr$ and $T$ represent respectively the specific heat capacity at constant pressure, the dynamic viscosity, the Prandtl number and the temperature of the fluid. Moreover, $\sigma_{ij}$ represents the $ij-$component of the resolved viscous stress tensor [15]. Both momentum and energy equations differ from the classical fluid dynamic equations only for two terms which take into account the contributions from the unresolved scales. These contributions, represented by the specific subgrid-scale stress tensor $\tau_{ij}^{sgs}$ and by the subgrid heat flux vector defined $q_{i}^{sgs}$, appear when the spatial filter is applied to the convective terms [15]. The interactions of $\tau_{ij}^{sgs}$ and $q_{i}^{sgs}$ with the resolved scales have to be modeled through a subgrid-scale closure model because they cannot be determined using only the resolved flow field w. #### 2.1.1 The wall-adapted local eddy-viscosity closure model The smallest scales present in the flow field and their interaction with the resolved scales have to be modeled through the subgrid-scale term $\tau_{ij}^{sgs}$. The most common approach to model such a tensor is based on the eddy-viscosity concept in which one assumes that the residual stress is proportional to a measure of the filtered local strain rate [15], which is defined as follows: \[\tau_{ij}^{sgs}-\tau_{kk}^{sgs}\delta_{ij}=-2\,\overline{\rho}\,\nu_{t}\left( \tilde{S}_{ij}-\frac{\delta_{ij}}{3}\tilde{S}_{kk}\right).\] (2) In the WALE model, it is assumed that the eddy-viscosity $\nu_{t}$ is proportional to the square of the length scale of the cut-off length (or width of the grid filter) and the filtered local rate of strain. Although the model was originally developed for incompressible flows, it can also be used for variable density flows by giving the formulation as follows \[\nu_{t}=\left(C\Delta\right)^{2}\left\|\tilde{S}\right\|.\] (3) Here $\left\|\tilde{S}\right\|$ is defined as \[\left\|\tilde{S}\right\|=\frac{\left[\tilde{S}_{ij}^{d}\,\tilde{S}_{ij}^{d} \right]^{3/2}}{\left[\tilde{S}_{ij}\,\tilde{S}_{ij}\right]^{5/2}+\left[\tilde{ S}_{ij}^{d}\,\tilde{S}_{ij}^{d}\right]^{5/4}},\] (4) where $\tilde{S}_{ij}^{d}$ is the traceless symmetric part of the square of the resolved velocity gradient tensor $\tilde{g}_{ij}=\frac{\partial\tilde{u}_{i}}{\partial x_{j}}$. Note that in Equation (3) $\Delta$, i.e., the cut-off length, is an unknown function. Often the cut-off length is taken proportional to the smallest resolvable length scale of the discretization. In the present work, the definition of the grid filter function is given in Section 2.2, where the SD method is discussed. ### Spectral difference method Consider a problem governed by a general system of conservation laws given by Equation (1) and valid on a domain $\Omega\subset\mathbb{R}^{dim}$ with boundary $\partial\Omega$ and completed with consistent initial and boundary conditions. The domain is divided into $N$ non-overlapping cells, with cell index $i$. In order to achieve an efficient implementation of the SD method, all hexahedral cells in the physical domain are mapped into cubic elements using local coordinates $\vec{\xi}=\left[\xi_{1},\xi_{2},\xi_{3}\right]^{T}$. Such a transformation is characterized by the Jacobian matrix $\vec{\vec{\left.\mathrm{J}\right.}}_{i}$ with determinant $det(\vec{\vec{\left.\mathrm{J}\right.}}_{i})$. Therefore, system (1) can be written in the mapped coordinate system as \[\frac{\partial\mathbf{w}_{i}^{\hskip 1.13811pt\vec{\xi}}}{\partial t}=-\frac{ \partial\mathbf{f}_{1,i}^{\hskip 1.13811pt\vec{\xi}}}{\partial\xi_{1}}-\frac{ \partial\mathbf{f}_{2,i}^{\hskip 1.13811pt\vec{\xi}}}{\partial\xi_{2}}-\frac{ \partial\mathbf{f}_{3,i}^{\hskip 1.13811pt\vec{\xi}}}{\partial\xi_{3}}=-\vec{ \nabla}^{\hskip 1.13811pt\vec{\xi}}\cdot\vec{\textbf{f}}^{\hskip 1.13811pt\vec {\xi}}_{i},\] (5) where $\mathbf{w}^{\hskip 1.13811pt\vec{\xi}}_{i}\equiv det(\vec{\vec{\left.\mathrm{J }\right.}}_{i})\,\mathbf{w}$ and $\vec{\nabla}^{\hskip 1.13811pt\vec{\xi}}$ are the conserved variables and the generalized divergence differential operator in the mapped coordinate system, respectively. For a $\left(p+1\right)$-th-order accurate $dim$-dimensional scheme, $N^{s}$_solution collocation points_ with index $j$ are introduced at positions $\vec{\xi}_{j}^{s}$ in each cell $i$, with $N^{s}$ given by $N^{s}=\left(p+1\right)^{dim}$. Given the values at these points, a polynomial approximation of degree $p$ of the solution in cell $i$ can be constructed. This polynomial is called the _solution polynomial_ and is usually composed of a set of Lagrangian basis polynomial $L_{j}^{s}\left(\vec{\xi}\right)$ of degree $p$: \[\mathbf{W}_{i}\left(\vec{\xi}\right)=\sum_{j=1}^{N^{s}}\mathbf{W}_{i,j}\,L_{j} ^{s}\left(\vec{\xi}\right).\] (6) Therefore, the unknowns of the SD method are the interpolation coefficients $\mathbf{W}_{i,j}=\mathbf{W}_{i}\left(\vec{\xi}_{j}^{s}\right)$ which are the approximated values of the conserved variables $\mathbf{w}_{i}$ at the solution points. The divergence of the mapped fluxes at the solution points is computed by introducing a set of $N^{f}$ flux collocation points with index $l$ and at positions $\vec{\xi}_{l}^{f}$, supporting a polynomial of degree $p+1$. The evolution of the mapped flux vector $\vec{\textbf{f}}^{\hskip 1.13811pt\vec{\xi}}$ in cell $i$ is then approximated by a flux polynomial $\vec{\textbf{F}}^{\hskip 1.13811pt\vec{\xi}}_{i}$, which is obtained by reconstructing the solution variables at the flux points and evaluating the fluxes $\vec{\textbf{F}}^{\hskip 1.13811pt\vec{\xi}}_{i,l}$ at these points. The flux is also represented by a Lagrange polynomial: \[\vec{\textbf{F}}^{\hskip 1.13811pt\vec{\xi}}_{i}\left(\vec{\xi}\right)=\sum_{l =1}^{N^{f}}\vec{\textbf{F}}^{\hskip 1.13811pt\vec{\xi}}_{i,l}\,L_{l}^{f}\left( \vec{\xi}\right),\] (7) where the coefficients of the flux interpolation are defined as \[\vec{\textbf{F}}^{\hskip 1.13811pt\vec{\xi}}_{i,l}=\begin{cases}\vec{\textbf{F }}^{\hskip 1.13811pt\vec{\xi}}_{i}\left(\vec{\xi}_{l}^{f}\right),&\quad\vec{ \xi}_{l}^{f}\in\Omega_{i},\\ \vec{\textbf{F}}_{\mathrm{num}}^{\hskip 1.13811pt\vec{\xi}}\left(\vec{\xi}_{l} ^{f}\right),&\quad\vec{\xi}_{l}^{f}\in\partial\Omega_{i}.\end{cases}\] (8) Here $\vec{\textbf{F}}_{\mathrm{num}}^{\hskip 1.13811pt\vec{\xi}}$ is the numerical flux vector at the cell interface. In fact, the solution at a face is in general not continuous and requires the solution of a Riemann problem to maintain conservation at a cell level (i.e., the flux component normal to a face $\vec{\textbf{F}}_{\mathrm{num}}^{\hskip 1.13811pt\vec{\xi}}\cdot\vec{n}^{ \hskip 1.13811pt\vec{\xi}}$ must be continuous between two neighboring cells). Approximate Riemann solvers are typically used (e.g. Rusanov Riemann solver). The tangential component of $\vec{\textbf{F}}_{\mathrm{num}}^{\hskip 1.13811pt\vec{\xi}}$ is usually taken from the interior cell. Taking the divergence of the flux polynomial in the solution points results in the following modified form of (5), describing the evolution of the conservative variables at the solution points: \[\frac{d\mathbf{W}_{i,j}}{dt}=-\left.\vec{\nabla}\cdot\vec{\textbf{F}}_{i} \right\|_{j}=-\frac{1}{J_{i,j}}\left.\vec{\nabla}^{\hskip 1.13811pt\vec{\xi}} \cdot\vec{\textbf{F}}^{\hskip 1.13811pt\vec{\xi}}_{i}\right\|_{j}=\mathbf{R}_{i ,j},\] (9) where $\vec{\textbf{F}}_{i}$ is the flux polynomial vector in the physical space whereas $\mathbf{R}_{i,j}$ is the SD residual associated with $\mathbf{W}_{i,j}$. This is a system of ODEs, in time, for the unknowns $\mathbf{W}_{i,j}$. In this work, the optimized explicit eighteen-stages fourth-order Runge-Kutta schemes presented in [16] is used to solve such a system at each time step. #### 2.2.1 Solution and flux points distributions In 2007, Huynh [9] showed that for quadrilateral and hexahedral cells, tensor product flux point distributions based on a one-dimensional flux point distribution consisting of the end points and the Legendre-Gauss quadrature points lead to stable schemes for arbitrary order of accuracy. In 2008, Van den Abeele et al. [10] showed an interesting property of the SD method, namely that it is independent of the positions of its solution points in most general circumstances. This property implies an important improvement in efficiency, since the solution points can be placed at flux point positions and thus a significant number of solution reconstructions can be avoided. Recently, this property has been proved by Jameson [11]. #### 2.2.2 Cut-off length $\Delta$ In Section 2.1.1 we have seen that in the WALE model the cut-off length $\Delta$ is used to compute the turbulent eddy-viscosity $\nu_{t}$, i.e., Equation (3). Following the approach presented in [8], for each cell with index $i$ and each flux points with index $l$ and positions $\boldsymbol{\xi}_{l}^{f}$, we use the following definition of filter width \[\Delta_{i,l}=\left[\frac{1}{N^{s}}det\left(\left.\vec{\vec{\left.\mathrm{J} \right.}}_{i}\right\|_{\boldsymbol{\xi}_{l}^{f}}\right)\right]^{1/dim}=\left( \frac{det(J_{i,l})}{N^{s}}\right)^{1/dim}.\] (10) Notice that the cell filter width is not constant in one cell, but it varies because the Jacobian matrix is a function of the positions of the flux points. Moreover, for a given mesh, the number of solution points depends on the order of the SD scheme, so that the grid filter width decreases by increasing the polynomial order of the approximation. ## 3 Numerical results The main purpose of this section is to evaluate the accuracy and the reliability of the fourth-order SD-LES solver for simulating a 3D turbulent flow in an industrial-type muffler. The results are compared with the particle image velocimetry (PIV) measurement performed at the Department of Environmental and Applied Fluid Dynamics of the von Karman Institute for Fluid Dynamics [17]. In Figure 1, the geometry of the muffler and its characteristic dimensions are illustrated, where the flow is from left to right. [FIGURE:S3.F1][ENDFIGURE] At the inlet, mass density and velocity profiles are imposed. The inlet velocity profile in the $x_{3}$ direction is given by \[u_{3}=u_{max}\left\{\frac{1}{2}-\frac{1}{2}\tanh{\left[2.2\left(\frac{r}{d/2}- \frac{d/2}{r}\right)\right]}\right\}.\] At the outlet only the pressure is prescribed. In accordance to the experiments, the inlet Mach number and the Reynolds number, based on maximum velocity at the inlet $u_{max}$ and the diameter of the inlet/outlet $d$ ($d=4\,cm$), are set respectively to $M_{inlet}=0.05$ and $Re=4.64\cdot 10^{4}$. The flow is computed using fourth-order ($p=3$) SD scheme on a grid with $36,612$ hexahedral elements which was generated with the open source software Gmsh [18]. Second-order boundary elements are used to approximate the curved geometry. The total number of DOFs is approximately $2.3\cdot 10^{6}$ (i.e., $36,612\cdot(p+1)^{3}$). The maximum CFL number used for the computations started from $0.1$ and increased up to $0.65$. After the flow field was fully developed, time averaging is performed for a period corresponding to about 25 flow-through times. The computation is validated on the center plane of the expansion coinciding with the center planes of the inlet and outlet pipes using the PIV results from [17]. All of the measurements are taken on the symmetrical center plane of the muffler. The reference cross section corresponds to the entrance of the expansion chamber. It should be noted that the circular nature of the geometry acts as a lens causing a change in magnification in the radial direction ($x_{2}$) which prevents from capturing images close to the wall. It is found that outside $~{}1\,cm$ from the wall the magnification effect is negligible and as the mean stream-wise direction is in the direction of constant magnification and has only little effect on the particle correlations no corrections are deemed necessary. In Figure 2, the non-dimensional mean velocity profile in the axial direction $\langle\tilde{u}_{3}\rangle/u_{max}$ is shown for four different cross sections in the expansion chamber, where the PIV measurements were done. In this figure, the PIV data are also plotted for comparison. Figure 3 shows the non-dimensional Reynolds stress $\langle u_{2}^{\prime}u_{3}^{\prime}\rangle/{u_{max}^{2}}$ at the same cross sections. Although the high-order implicit LES is already able to capture well the features of the flow field, the use of the WALE model improves the results. In particular, when the SGS model is active, the local extrema of the time-averaged velocity profiles and the second-order statistical moment (which get fairly oscillatory by moving far away from the inlet pipe) are better captured. ## 4 Conclusions The fourth-order SD method in combination with the WALE model and the variable filter width performed well. The numerical results confirm that the model is correctly accounting for the unresolved shear stress computed from the resolved field, for the present internal flow. However, it should be noted that the SD discretization without subgrid-scale modelling also worked rather well, at least for the grid resolution used in this study. Work is currently under way to test both approaches for different orders of accuracy, grid resolutions and other realistic turbulent flows. We believe that the flexibility of the high-order SD scheme on unstructured grids together with the development of robust sub-grid closure models for highly separated flows and efficient grid generators for high-order accurate schemes will allow to perform LES of industrial-type flows in the near future. <figure><img src="content_image/1305.6681/x1.png"><figcaption>(a) 1d downstream.</figcaption></figure> <figure><img src="content_image/1305.6681/x5.png"><figcaption>(a) 1d downstream.</figcaption></figure> ## Acknowledgement The authors would like to thank Professor David I. Ketcheson and Professor Mark H. Carpenter for their support. This research used partially the resources of the KAUST Supercomputing Laboratory and was supported in part by an appointment to the NASA Postdoctoral Program at Langley Research Center, administered by Oak Ridge Associates Universities. These supports are gratefully acknowledged. ## References * [1] Wang, Z. J.: Adaptive High-order Methods in Computational Fluid Dynamics (Advances in Computational Fluid Dynamics), World Scientific Publishing Company (2011). * [2] Busch, K., König, M., and Niegemann, J.: Discontinuous Galerkin methods in nanophotonics. Laser Photonics Rev. 5(6), 773–809 (2011). * [3] Hesthaven, Jan S., and Warburton, Tim: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Texts in Applied Mathematics, 54. Computational Science & Engineering, Springer Publishing Company, Incorporated (2007). * [4] Sun, Y., Wang, Z. J., and Liu Y.: Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow. J. of Comput. Phys., 215(1), 41–58 (2006). * [5] May, G., and Jameson, A.: A spectral difference method for the Euler and Navier-Stokes equations on unstructured meshes. _AIAA paper_ 2006-304. 44th AIAA Aerospace Sciences Meeting, Reno, Nevada, USA, January 9-12, 2006. * [6] Sun, Y., Wang, Z. J., and Liu, Y.: High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys, 2(2), 310–333 (2007). * [7] Castonguay, P., Vincent P., and Jameson, A.: Application of High-Order Energy Stable Flux Reconstruction Schemes to the Euler Equations. _AIAA paper_ 2011-686. 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, USA, January 4-7, 2011. * [8] Parsani, M., Ghorbaniasl, G., Lacor C., and Turkel, E.: An implicit high-order spectral difference approach for large eddy simulation. J. Comput. Phys., 229(14), 5373–5393 (2010). * [9] Huynh, H. T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. _AIAA paper_ 2007-4079. 18th AIAA Computational Fluid Dynamics Conference, Miami, Florida, USA, June 25-28, 2007. * [10] Van den Abeele, K., Lacor, C., and Wang, Z. J.: On the stability and accuracy of the spectral difference method, J. Sci. Comput., 37(2), 162–188 (2008). * [11] Jameson, A.: A proof of the stability of the spectral difference method for all orders of accuracy, J. Sci. Comput., 45(1-3), 348–358 (2010). * [12] Parsani, M., Ghorbaniasl, G., and Lacor C.: Validation and application of an high-order spectral difference method for flow induced noise simulation. J. Comput. Acoust., 19(3), 241–268 (2011). * [13] Lodato, G., and Jameson, A.: LES modeling with high-order flux reconstruction and spectral difference schemes. _ICCFD paper_ 2201. 7th ICCFD Conference, Big Island, Hawaii, July 9-13, 2012. * [14] Nicoud, F., and Ducros, F.: Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turbul. Combust., 62(3), 183–200 (1999). * [15] Pope, Stephen B.: Turbulent flows. Cambridge University Press (2003). * [16] Parsani, M., Ketcheson, David I., and Deconinck, W.: Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems, SIAM J. Sci. Comput, 35(2):A957-A986 (2013). * [17] Bilka, M., and Anthoine, J.: Experimental investigation of flow noise in a circular expansion using PIV and acoustic measurements. _AIAA paper_ 2008-2952. 14th AIAA/CEAS Aeroacoustics Conference, Vancouver, British Columbia, Canada, May 5-6, 2008. * [18] Geuzaine, C., and Remacle J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng., 79(11), 1309–1331 (2009).
0811.0106	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 54256, "num_imgs": 1, "llama3_tokens_count": 20433 }	[ "content_image/0811.0106/x1.png" ]	# Entire solutions to equivariant elliptic systems with variational structure Nicholas D. Alikakos¹ and Giorgio Fusco² [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] ###### Abstract In the present paper we consider the system $\Delta u-W_{u}(u)=0$, where $u:\mathbb{R}^{n}\to\mathbb{R}^{n}$, for a class of potentials $W:\mathbb{R}^{n}\to\mathbb{R}$ that possess several global minima and are invariant under a general finite reflection group $G$. We establish existence of nontrivial entire solutions connecting the global minima of $W$ along certain directions at infinity. ## 1 Introduction We consider the system (1.1) \[\Delta u-W_{u}(u)=0,\text{ for }u:\mathbb{R}^{n}\to\mathbb{R}^{n},\] where $W:\mathbb{R}^{n}\to\mathbb{R}$ and the gradient $W_{u}:=\big{(}\frac{\partial W}{\partial u_{1}},\cdots,\frac{\partial W}{ \partial u_{n}}\big{)}^{\top}$; the system above is the Euler–Lagrange equation corresponding to the action (1.2) \[J(u)=\int_{\mathbb{R}^{n}}\Big{\{}\frac{1}{2}\|\nabla u\|^{2}+W(u)\Big{\}} \operatorname{d\!}x.\] One of the obstructions in the study of (1.1) is that the action is infinite for the class of solutions we are interested in, for dimensions $n\geq 2$. We begin by introducing and explaining the hypotheses on the potential $W$. Hypothesis 1 ($N$ nondegenerate global minima).: _The potential $W$ is of class $C^{2}$, satisfying $W=0$ on $A=\{a_{1},\ldots,a_{N}\}$ and $W>0$ in $\mathbb{R}^{n}\setminus A$. Furthermore, $\partial^{2}W(u)\geq c^{2}\mathrm{Id}$ for $\|u-a_{i}\|\leq r_{0}$, with $r_{0}>0$ fixed, and for $i=1,\ldots,N$._ We recall some of the known examples that have been studied in the past. The case $n=1$, $N=2$ is textbook material and the corresponding solution is known as the _heteroclinic connection_. In the work of Dang, Fife, and Peletier [8], a _saddle solution_ was constructed on the plane under a symmetry assumption on $W$. There, the solution is scalar but can be trivially embedded in our setup by taking the second component equal to zero, resulting in a problem of the form (1.1) with $n=2$, $N=2$. A genuine vector extension of the Dang–Fife–Peletier solution can be found in Alama, Bronsard, and Gui [1]. In [5], Bronsard, Gui, and Schatzman constructed a solution for $n=2$, $N=3$, while recently in [19], Gui and Schatzman constructed one for $n=3$, $N=4$; these last two solutions are known as the _triple-junction solution_ on the plane and the _quadruple-junction solution_ in space respectively and they were also obtained under symmetry hypotheses on $W$. They are related to the geometric evolution of interfaces evolving by mean curvature and satisfying Plateau angle conditions; see the work of Mantegazza, Novaga, and Tortorelli [23] and the recent works of Freire [14, 15], Mazzeo and Sáez [24], Schnürer and Schulze [30], Schnürer _et al._ [31], and Ikota and Yanagida [22]. These solutions have been formally linked to the singular limit in sharp-interface evolution problems (see Bronsard and Reitich [6] for the case $n=2$ and Rubinstein, Sternberg, and Keller [27] for related previous work) and were subsequently studied in [5, 19]. The $\Gamma$-limit of a rescaled action $J$ over a bounded domain under a volume constraint has been studied by Baldo in [4]. For rigorous linking results for the evolution problem, see the work of Sáez Trumper [28, 29]. Hypothesis 2 (Symmetry).: _The potential $W$ is invariant under a finite reflection group $G$ in $\mathbb{R}^{n}$ (Coxeter group), that is,_ (1.3) \[W(gu)=W(u),\text{ for all }g\in G\text{ and }u\in\mathbb{R}^{n}.\] _Moreover, we assume that $W(u)\geq\max_{\partial C_{0}}W$, for $u$ outside a certain bounded, $G$-invariant, convex set $C_{0}$._ We seek _equivariant_ solutions to system (1.1), that is, solutions satisfying (1.4) \[u(gx)=gu(x),\text{ for all }g\in G\text{ and }x\in\mathbb{R}^{n}.\] For notation and algebraic background we refer to [18]. In the past, the following groups have been employed: in [8]$G=\mathcal{H}^{2}_{2}$, the dihedral group on the plane, with four elements, in [5]$G=\mathcal{H}^{3}_{2}$, the group of symmetries of the equilateral triangle, with six elements, and in [19]$G=\mathcal{T}^{}$, the group of symmetries of the tetrahedron, with twenty four elements. Hypothesis 3* (Location and number of global minima).: _Let $F\subset\mathbb{R}^{n}$ be a fundamental region of $G$. We assume that $\bar{F}$ (the closure of $F$) contains a single global minimum of $W$, say $a_{1}$, and let $\mathrm{Stab}(a_{1})$ be the subgroup of $G$ that leaves $a_{1}$ fixed. Then,_ (1.5) \[N:=\frac{\|G\|}{\|\mathrm{Stab}(a_{1})\|}.\] We give here some examples. For $\mathcal{H}^{3}_{2}$ on the plane, we can take as $F$ the $\frac{\pi}{3}$ sector. If $a_{1}\in F$, then $N=6$, while if $a_{1}$ is on the walls, then $N=3$. Finally, if $a_{1}$ is placed at the origin, then $N=1$. In higher dimensions, we have more options since we can place $a_{1}$ in the interior of $\bar{F}$, in the interior of a face, on an edge, and so on. For example, if $G=\mathcal{W}^{}$, the group of symmetries of the cube in three-dimensional space, then $\|G\|=48$. If the cube is situated with its center at the origin and its vertices at the eight points $(\pm 1,\pm 1,\pm 1)$, then we can take as $F$ the simplex generated by $s_{1}=e_{1}+e_{2}+e_{3}$, $s_{2}=e_{2}+e_{3}$, and $s_{3}=e_{3}$, where the $e_{i}$’s are the standard basis vectors. We have then the following options: 1. On the edge $s_{3}$, $N=6$. 2. On the edge $s_{1}$, $N=8$. 3. On the edge $s_{2}$, $N=12$. 4. In the interior of a face, $N=24$. 5. In the interior of the fundamental region, $N=48$. 6. At the origin, $N=1$. Hypothesis 4* ($Q$-monotonicity).: _Let_ (1.6) \[D:=\mathrm{Int}\big{(}{\bigcup_{g\in\mathrm{Stab}(a_{1})}}g\bar{F} \big{)}.\] _We restrict ourselves to potentials $W$ for which there is a function $Q:\bar{D}\to\mathbb{R}$ with the following properties:_ (1.7a) \[\kern-113.811024ptQ\text{ is convex},\] (1.7b) \[\kern-113.811024ptQ(u)>0\text{ and }Q_{u}(u)\neq 0,\text{ on } \bar{D}\setminus\{a_{1}\},\] (1.7c) \[\kern-113.811024ptQ(u+a_{1})=\|u\|+\mathrm{o}\big{(}\|u\|\big{)},\] _and_ (1.8) \[Q_{u}(u)\cdot W_{u}(u)\geq 0,\text{ in }D\setminus\{a_{1}\}.\] For $n=1$ and odd symmetry, for a double-well potential $W$, and $D=\{u>0\}$, $Q$-monotonicity implies that $W$ is monotone in $D$ along the ray emanating from $a_{1}$. For $G=\mathcal{H}^{3}_{2}$ on the plane, $F$ the $\frac{\pi}{3}$ sector, and $a_{1}=(1,0)$, it can be verified that the triple-well potential \[W(u_{1},u_{2})=\|u\|^{4}+2u_{1}u_{2}^{2}-\frac{2}{3}u_{1}^{3}-\|u\|^{2}+\frac{2}{3}\] satisfies the $Q$-monotonicity condition in $D=\big{\{}(r,\theta)\mid r>0,\theta\in(-\frac{\pi}{3},\frac{\pi}{3})\big{\}}$, with $Q(u)=\|u-a_{1}\|$, where $u=(u_{1},u_{2})$. For $n=3$, $G=\mathcal{T}^{}$, $F$ the simplicial cone generated by and \[{a_{1}=\big{(}\sqrt{\frac{2}{3}},\,0,\,\frac{1}{\sqrt{3}}\big{)},}\] we can take as an example the quadruple-well potential \[W(u_{1},u_{2},u_{3})=\|u\|^{4}-\frac{4}{\sqrt{3}}(u^{2}_{1}-u^{2}_{2})u_{3}- \frac{2}{3}\|u\|^{2}+\frac{5}{9},\] with $Q(u)=\|u-a_{1}\|$, where $u=(u_{1},u_{2},u_{3})$, and $D$ the simplicial cone generated by $\big{(}0,\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}}\big{)}$, $\big{(}0,-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}}\big{)}$, and $\big{(}\sqrt{\frac{2}{3}},0,-\frac{1}{\sqrt{3}}\big{)}$. Next we explain how the $Q$-monotonicity is utilized in the proof. Let $u$ be a $C^{2}$ solution to (1.1); then, utilizing the smoothness of convex functions [12], (1.9) \[\Delta Q\big{(}u(x)\big{)}=\operatorname{tr}\big{\{}(\partial^{2}Q)(\nabla u)( \nabla u)^{\top}\big{\}}+Q_{u}\big{(}u(x)\big{)}\Delta u(x).\] If now $u$ has the property (1.10) \[u:\bar{D}\to\bar{D},\] then, from (1.9) and convexity it follows that (1.11) \[\Delta Q\big{(}u(x)\big{)} \geq Q_{u}\big{(}u(x)\big{)}\Delta u(x)\] \[\overset{\eqref{system}}{=} Q_{u}\big{(}u(x)\big{)}\cdot W_{u}\big{(}u(x)\big{)}\] \[\overset{\eqref{q-monotonicity}}{\geq} 0.\] Subharmonicity then provides a global first estimate on $u$. Hence, a key step is to show that the candidate solution $u$ is a _positive_ map, that is, it satisfies the positivity property (1.10). We now proceed with the statement of the main results. Theorem 1.1.: _Under Hypotheses 1–4, there exists an equivariant classical solution to system (1.1) such that:_ 1. $\|u(x)-a_{1}\|\leq K\mathrm{e}^{-kd(x,\partial D)}$_, for some positive constants_ $k,K$ _and for_ $x\in D$_,_ 2. $u(D)\subset D$_._ _In particular, $u$ connects the $N=\|G\|/\|\mathrm{Stab}(a_{1})\|$ global minima of $W$:_ \[\lim_{\lambda\to+\infty}u(\lambda ga_{1})=ga_{1},\text{ for all }g\in G.\] The proof of Theorem 1.1 is based on a family of constrained minimization problems (1.12a) \[\min J_{\Omega_{R}},\text{ where }J_{\Omega_{R}}(u)=\int_{\Omega_{R}}\Big{\{} \frac{1}{2}\|\nabla u\|^{2}+W(u)\Big{\}}\operatorname{d\!}x,\] under a constraint enforcing the desirable behavior at infinity, (1.12b) \[\|u(x)-a_{1}\|\leq\bar{q},\text{ for }x\in C_{R}\subset D\cap\Omega_{R}.\] Here, $\bar{q}$ is a fixed positive number with $0<\bar{q}\leq r_{0}$ (cf. Hypothesis 1), $\{\Omega_{R}\}$ is an appropriate class of homothetic symmetric domains (for example, $\Omega_{R}=\{x\in\mathbb{R}^{n}\mid\|x\|<R\}$) over which the action $J_{\Omega_{R}}$ is finite, and $C_{R}$ is a ball $B(x_{R},L)\subset D\cap\Omega_{R}$ with $x_{R}=\frac{R}{2}x_{0}$ and $L>0$, fixed, independent of $R$, and sufficiently large. Problems (1.12) provide a family of minimizers $\{u_{R}\}$; we seek then to construct the solution by taking the limit, that is, (1.13) \[u(x)=\lim_{R\to\infty}u_{R}(x).\] For carrying out this procedure successfully, we need uniform estimates in $R$. The following estimates are obtained in the course of the proof of Theorem 1.1. Theorem 1.2* (Uniform $R$-estimates).: _Under Hypotheses 1–4, there is an $R_{0}>0$, such that, for $R>R_{0}$, there exists an equivariant local minimizer $u_{R}$ of $J_{\Omega_{R}}$ which is a positive map and satisfies the estimates_ (1.14a) \[J_{\Omega_{R}}(u)\leq c_{0}R^{n-1},\] (1.14b) \[\|u_{R}(x)-a_{1}\|\leq K\mathrm{e}^{-kd(x,\partial D_{R})},\text{ for }x\in D_{R }:=D\cap\Omega_{R},\] _where $k$, $K$ are positive constants independent of $R$._ The estimate (1.14b) is inherited in the limit (1.13). Note that this estimate excludes the trivial solution $u\equiv 0$. Our proof consists of a geometric part and a PDE part. The geometric part is concerned with positivity; it utilizes the gradient flow (1.15) \[\left\{\begin{array}[]{l}u_{t}=\Delta u-W_{u}(u),\text{ in }\Omega_{R}\times(0 ,\infty),\\ \dfrac{\partial u}{\partial n}=0,\text{ on }\partial\Omega_{R}\times(0,\infty) .\end{array}\right.\] in the Sobolev space of equivariant maps $W^{1,2}_{\mathrm{E}}(\Omega_{R};\mathbb{R}^{n})$. We establish that the set of positive maps (1.16) \[\big{\{}f\in W^{1,2}(\Omega_{R};\mathbb{R}^{n})\mid f(\,\overline{D\cap\Omega_ {R}}\,)\subset\bar{D}\big{\}}\] is strongly (positively) invariant under the flow (1.15). With the help of this invariance, we establish that the minimization problem (1.12) for $R>R_{0}$ has a solution that satisfies the Euler–Lagrange equation $\Delta u-W_{u}(u)=0$ in $\Omega_{R}$, which is positive as well. We remark that the gradient flow is not used further in the rest of the paper. Concerning positivity we also note that the property $u:\bar{F}\to\bar{F}$ ($F$-positivity) together with $G$-equivariance, implies $u:\bar{D}\to\bar{D}$ ($D$-positivity), but not vice versa. Theorem 1.1 always guarantees the existence of an $F$-positive (and hence, $D$-positive), $G$-equivariant solution. In the event, however, that $D$ is the fundamental region of a symmetry subgroup $\mathcal{H}_{D}$ of $G$, then it also provides another potentially distinct solution with less symmetry that is $D$-positive (but not necessarily $F$-positive) and $\mathcal{H}_{D}$-equivariant (but not necessarily $G$-equivariant). For example, for the triple-junction problem on the plane, the relevant group is $\mathcal{H}^{3}_{2}$ and in this case, if $a_{1}$ is placed on an edge of $F$ (cf. Hypothesis 3), $D$ is the $\frac{2\pi}{3}$ sector and Theorem 1.1 provides a solution that is $\mathcal{H}^{3}_{2}$-equivariant; here there is no $\mathcal{H}_{D}$. On the other hand, for the group of symmetries of the square $\mathcal{H}^{4}_{2}$, and under analogous conditions for $a_{1}$, we obtain via Theorem 1.1, in addition to an $\mathcal{H}^{4}_{2}$-equivariant solution, an $\mathcal{H}^{2}_{2}$-equivariant solution. We remark in passing that the normality of $\mathrm{Stab}(a_{1})$ in $G$ is not a necessary condition for $D$ to be a fundamental region of a reflection subgroup of $G$. The PDE part of the proof is concerned with estimate (1.14b). By $D$-positivity (1.10), (1.11), (1.17) \[\Delta Q\big{(}u_{R}(x)\big{)}\geq 0,\text{ in }D\cap\Omega_{R}.\] On the other hand, by the nondegeneracy condition in Hypothesis 1, (1.18) \[\Delta p_{R}(x)\geq c^{2}p_{R}(x),\text{ in }C_{R},\] where \[p_{R}(x):=\|u(x)-a_{1}\|\ (\leq\bar{q}\text{ in }C_{R}).\] In the case where $Q(u)=\|u-a_{1}\|$, estimate (1.17) provides a first global bound on $p_{R}(x)$ in $D\cap\Omega_{R}$ while estimate (1.18) implies a stronger exponential bound on $p_{R}(x)$ in $C_{R}$. By alternating between (1.17) and (1.18), we can keep improving the range of validity of the exponential estimate and we can establish that it holds on a large domain, independent of $R$. For general $Q$ though, we need a global change of coordinates first. Our understanding of the problem owes a great deal to the fundamental works [5, 1]. However, our approach is generally different and, in particular, our assumptions are not generalizations of the ones in those works. In [5] and [19] they proceed via Dirichlet problems and build up a higher-dimensional object out of lower-dimensional solutions. We instead construct a nontrivial solution via Neumann problems, which can later be dissected (not done in the present paper). The paper [2] contains some seeds of the present work. Symmetry is a rather restrictive assumption. A possible approach for relaxing it could be to establish the stability of the constructed solution(s) in the class of general compact perturbations. This is reasonable for at least some of these solutions which enjoy extra minimality properties (as, for example, the triple-junction solution). Sáez Trumper [28, 29] has extended the results in [5] to the nonsymmetric case utilizing the gradient flow. Finally, in light of [2], uniqueness should not be expected in general. The scalar problem related to (1.1), for $u:\mathbb{R}^{n}\to\mathbb{R}$, and without any symmetry hypotheses on the potential, has been the object of intensive investigation for many years, with the De Giorgi conjecture and the related contributions at the center of this activity (see the expository article of Farina and Valdinoci [13]). It is well known that the vector nature of (1.1) is of significance. On the physical side, we note that for describing coexistence of three or more phases ($N\geq 3$), a vector-order parameter $u$ is needed. A triple-well in $\mathbb{R}^{2}$ or a quadruple-well in $\mathbb{R}^{3}$ would be appropriate for modeling coexistence of three or four phases correspondingly, with the origin $x=0$ representing the coexistence point. On the geometric side, the rescaled solution $u_{\varepsilon}(x):=u(x/\varepsilon)$ in the triple- and quadruple-well cases is expected to converge, as $\varepsilon\to 0$, to the solution of the corresponding partitioning problem (see Baldo [4]). The boundaries of the partitioning sets form a system of minimal surfaces meeting each other along free-boundary curves called ‘liquid edges’, and liquid edges meet at ‘supersingular’ points which coincide with the coexistence points mentioned above (cf. Dierkes _et al._[9, §4.10.7]). The paper is structured as follows. In Section 2 we establish the strong positivity property of the semigroup that (1.15) generates. In Section 3 we introduce the Q-coordinate system and in Section 4 we state and prove the comparison lemmas related to estimates (1.17) and (1.18). Finally, in Section 5 we give the proofs of Theorems 1.1 and 1.2 together. ## 2 The positivity property ### Algebraic preliminaries The material in this subsection is adapted from [18]. Let $G$ be a _Coxeter group_, that is, a finite effective subgroup of the orthogonal group $O(\mathbb{R}^{n})$, generated by a set of reflections. The reflection with respect to the hyperplane $\mathcal{P}_{r}=\{x\in\mathbb{R}^{n}\mid\langle x,r\rangle=0\}$ is the linear transformation (2.1) \[S_{r}u=u-2\langle u,r\rangle r,\text{ for }u\in\mathbb{R}^{n}\text{ and }\|r\|=1,\] where $\langle\cdot\,,\cdot\rangle$ is the Euclidean inner product in $\mathbb{R}^{n}$. Every finite subgroup of $O(\mathbb{R}^{n})$ has a _fundamental region_, that is, a subset $F\subset\mathbb{R}^{n}$ with the following properties: 1. $F$ is open and convex, 2. $F\cap g(F)=\varnothing$ if $\operatorname{Id}\neq g\in G$, where $\operatorname{Id}$ is the identity, 3. $\mathbb{R}^{n}=\bigcup\big{\{}\,\overline{g(F)}\mid g\in G\big{\}}$. The two unit vectors $\pm r$ that are perpendicular to $\mathcal{P}_{r}$ are called _roots_ of $G$. The set of all roots of $G$ is denoted by $\Delta$. For a fixed $t\in\mathbb{R}^{n}$ such that $\langle t,r\rangle\neq 0$ for every root $r$, the set of roots of $G$ is partitioned into two subsets \[\Delta^{+}=\big{\{}r\in\Delta\mid\langle t,r\rangle>0\big{\}},\quad\Delta^{-}= \big{\{}r\in\Delta\mid\langle t,r\rangle<0\big{\}}.\] A subset $\Pi$ of $\Delta^{+}$ that is minimal with respect to the property that every $r\in\Delta^{+}$ is a linear combination of elements of $\Pi$, with all coefficients nonnegative, is called a $t$_-base_ for $\Delta$. It can be shown that the $t$-base is unique and that it is also a basis for $\mathbb{R}^{n}$, hence $\Pi=\{r_{1},\ldots,r_{n}\}$. The roots $r_{i}$, $i=1,\ldots,n$, are called _fundamental_ and the corresponding reflections $S_{r_{i}}$ are the _fundamental reflections_ and generate $G$. Moreover, (2.2) \[\bar{F}={\bigcap\limits_{i=1}^{n}}\big{\{}u\in\mathbb{R}^{n}\mid \langle u,r_{i}\rangle\geq 0\big{\}}.\] The hyperplanes $\mathcal{P}_{r}$ which determine the boundary of $F$ are called the _walls_ of $F$. It can be shown that (2.3) \[\langle r_{i},r_{j}\rangle\leq 0,\text{ for }i\neq j.\] Given an open set $\Omega\subset\mathbb{R}^{n}$ invariant under $G$, that is, $g(\Omega)=\Omega$ for all $g\in G$, we define as a _fundamental domain_ of $G$ in $\Omega$ the set $F_{\Omega}:=\Omega\cap F.$ ### Parabolic flows and the equivariant class In the following, for simplicity we assume hypotheses that are sufficient for our purposes but stronger than we actually need. We take $W$ to be a $C^{2}$ potential satisfying the global bound (2.4) \[\|\partial^{2}_{u_{i}u_{j}}W(u)\|<C,\text{ in }\mathbb{R}^{n}.\] Hypothesis (2.4) can be imposed without loss of generality because of the _a priori_ pointwise bound (5.2). Moreover, we assume that the potential is invariant under $G$, that is, (2.5) \[W:\mathbb{R}^{n}\to\mathbb{R}^{n}\cup\{0\},\text{ with }W(gu)=W(u)\text{ for all }g\in G\text{ and }u\in\mathbb{R}^{n}.\] For a domain $\Omega$ invariant under $G$ and with smooth boundary, we introduce the following evolution problem in the class of maps $u:\Omega\to\mathbb{R}^{n}$ that are $G$-equivariant. (2.6) \[\left\{\begin{array}[]{l}\dfrac{\partial u}{\partial t}=\Delta u-W_{u}(u), \text{ in }\Omega\times(0,\infty),\\ \dfrac{\partial u}{\partial n}=0,\text{ on }\partial\Omega\times(0,\infty), \text{ where }\dfrac{\partial}{\partial n}\text{ is the normal derivative},\\ u(x,0)=u_{0}(x),\text{ in }\Omega.\end{array}\right.\] We will consider (2.6) in the equivariant Sobolev space $W^{1,2}_{\mathrm{E}}(\Omega;\mathbb{R}^{n})$ and denote the solution by $u(x,t;u_{0})$. A related evolution problem is obtained by considering the equation on a fundamental domain $F_{\Omega}$ with boundary conditions on the walls induced by the symmetries of the group. (2.7) \[\left\{\begin{array}[]{l}\dfrac{\partial\hat{u}}{\partial t}=\Delta\hat{u}-W_{ u}(\hat{u}),\text{ in }F_{\Omega}\times(0,\infty),\\ \langle\hat{u},r_{i}\rangle=0\text{ and }\dfrac{\partial\hat{u}}{\partial r_{i }}=\big{\langle}\dfrac{\partial\hat{u}}{\partial r_{i}},r_{i}\big{\rangle}r_{i },\text{ for }x\in\partial F_{\Omega}\cap P_{i},\text{ and for }i=1,\ldots,n, \\ \dfrac{\partial\hat{u}}{\partial n}=0,\text{ on }\partial F_{\Omega}\setminus \partial F,\\ \hat{u}(x,0)=u_{0}(x),\text{ on }F_{\Omega}.\end{array}\right.\] Clearly, solutions of (2.6) with sufficient smoothness satisfy (2.7), and conversely, solutions of (2.7) can be extended to solutions of (2.6) via equivariance by utilizing the invariance of the Laplacian under orthogonal transformations and the generation of the group by the fundamental reflections. ### Invariance of the class of positive maps Definition.: For $\Omega=B_{R}:=\{x\in\mathbb{R}^{n}\mid\|x\|<R\}$, we define the set of maps (2.8) \[P:=\big{\{}v\in W^{1,2}_{\mathrm{E}}(B_{R};\mathbb{R}^{n})\mid v(\bar{F}_{B_{R }})\subset\bar{F}\big{\}}\] which we call _positive_. A positive map is called _strongly positive_ if (2.9) \[v(F_{B_{R}})\subset F.\] The set of strongly positive maps is denoted by $P_{0}$. Theorem 2.1.: _Suppose $W$ satisfies properties (2.4) and (2.5). Then, (2.6) with $\Omega=B_{R}$ leaves the positive class $P$ invariant, that is,_ \[P\ni u_{0}\mapsto u(\cdot,t;u_{0})\in P,\] _and, moreover,_ \[\big{(}P\cap\{v\in W^{1,2}_{\mathrm{E}}(B_{R};\mathbb{R}^{n})\mid v(F_{B_{R}}) \nsubseteq\partial F\}\big{)}\ni u_{0}\mapsto u(\cdot,t;u_{0})\in P_{0},\text{ for }t>0.\] We begin with a lemma. Lemma 2.2.: _Let $u:B_{R}\to\mathbb{R}^{n}$ be an equivariant map. Then, $u$ is a positive map if and only if_ (2.10) \[u(\,\overline{\mathcal{P}^{+}_{r}\cap B_{R}}\,)\subset\mathcal{P}^{+}_{r}, \text{ for all roots }r.\] Here, $\mathcal{P}^{+}_{r}=\{x\in\mathbb{R}^{n}\mid\langle x,r\rangle\geq 0\}$. Proof.: Suppose that (2.10) holds. Then, by (2.2), \[u(\bar{F}_{B_{R}})=u\big{(}{\bigcap\limits_{i=1}^{n}}\overline{ \mathcal{P}_{r_{i}}^{+}\cap B_{R}}\,\big{)}\subset{\bigcap\limits_{i =1}^{n}}u(\,\overline{\mathcal{P}_{r_{i}}^{+}\cap B_{R}}\,)\subset{ \bigcap\limits_{i=1}^{n}}\mathcal{P}_{r_{i}}^{+}=\bar{F}.\] Hence, $u$ is positive. Conversely, suppose that $u$ is a positive equivariant map on $B_{R}$. Then, equivalently, $u_{\mathrm{e}}$ defined by (2.11) \[u_{\mathrm{e}}(x):=\left\{\begin{array}[]{cl}u(x),&\text{for }x\in B_{R}\\ 0,&\text{for }x\in\mathbb{R}^{n}\setminus B_{R}\end{array}\right.\] is a positive equivariant map on $\mathbb{R}^{n}$. For any $g\in G$, we have from equivariance and positivity, \[u_{\mathrm{e}}\big{(}g(\bar{F})\big{)} =g\big{(}u_{\mathrm{e}}(\bar{F})\big{)}\] (2.12) \[\subset g(\bar{F}).\] Now pick a root $r$ and take an $x\in\mathcal{P}_{r}^{+}$ and fix it. There is a $g\in G$, denoted by $g_{x}$, such that $x\in g_{x}(\bar{F})$ and since $g_{x}(F)$ is also a fundamental domain and $\Pi$ is a positive $t$-base, (2.13) \[g_{x}(\bar{F})\subset\mathcal{P}_{r}^{+}.\] Thus, by (2.3), $u_{\mathrm{e}}(\mathcal{P}_{r}^{+})\subset\mathcal{P}_{r}^{+}$, and so (2.10) follows. ∎ We continue with the _Proof of Theorem 2.1._ Consider (2.6) with $\Omega=B_{R}$ and $u_{0}\in W_{\mathrm{E}}^{1,2}(B_{R};\mathbb{R}^{n})$. By the regularizing property of the equation, the solution is classical for $t>0$, and by (2.4), it is global in time (see [21]). By taking the inner product of the equation with $r$, an arbitrary root, we obtain from (2.6) (2.14) \[\left\{\begin{array}[]{l}\dfrac{\partial\phi(u)}{\partial t}=\Delta\phi(u)- \dfrac{\langle W_{u}(u),r\rangle}{\phi(u)}\phi(u),\text{ in }B_{R}^{+}\times(0 ,\infty),\\ \phi\big{(}u(x,0)\big{)}=\phi\big{(}u_{0}(x)\big{)},\end{array}\right.\] where \[\phi(u):=-\langle u,r\rangle,\text{ for }u\in\mathbb{R}^{n}.\] For establishing positivity, it is sufficient, by Lemma 2.2, to show that $\phi(u_{0}(x))\leq 0$ implies $\phi(u(x,t))\leq 0$, for $t\geq 0$. We will accomplish this by applying the maximum principle to (2.14). Fix a $\tau>0$. By smoothness, $\phi(u(x,t))=0$ on $\mathcal{P}_{r}\cap[\tau,\infty)$ and \[\frac{\partial}{\partial n}\phi\big{(}u(x,t)\big{)}=0,\text{ on }(\partial B_{ R}^{+}\setminus\mathcal{P}_{r})\times[\tau,\infty)\quad(\text{cf.\ \eqref{ related-evolution-problem}})\] By the smoothness of $W$ and the symmetry of $u$ we have (2.15) \[\langle W_{u}(u),r\rangle=\langle u-\xi,r\rangle\,\big{\langle}W_{uu}\big{(} \xi+\mathrm{Proj}(u-\xi)+\hat{\delta}r\big{)}r,\,r\big{\rangle},\] for $\xi\in\mathcal{P}_{r}$ and $0\leq\hat{\delta}\leq\langle u-\xi,r\rangle$, where Proj stands for the projection on the plane $\mathcal{P}_{r}$. Thus, the coefficient of $\phi(u)$ in (2.14) is bounded (actually continuous) on $B_{R}^{+}$, hence both the maximum principle and the boundary lemma hold and render that $\phi(u(x,t))\leq 0$, for $t\geq\tau$ (note that the corner is not an issue since $\phi(u)=0$ there). Finally, the strong maximum principle applies for $t\geq\tau>0$ to render that \[\phi\big{(}u(x,t)\big{)}<0,\text{ in }B_{R}^{+}\times(0,\infty),\] unless $\phi(u(x,\tau))\equiv 0$, hence unless $\phi(u_{0}(x))\equiv 0$. But the hypothesis $u_{0}(F_{B_{R}})\not\subseteq\partial F$ excludes this second option. ∎ In [3], utilizing a different technique, we establish the positivity property for the more general evolution problem (2.16) for $K_{B_{R}}=K\cap B_{R}$, $K$ a simplex, not necessarily related to a reflection group, defined via \[K:={\bigcap\limits_{i=1}^{n}}\big{\{}u\in\mathbb{R}^{n}\mid\langle u ,\xi_{i}\rangle\geq 0\big{\}},\] where $\{\xi_{i},\ldots,\xi_{n}\}$ is a set of linearly independent vectors satisfying the inequalities \[\langle\xi_{i},\xi_{j}\rangle\leq 0,\text{ for }i\neq j\text{ (cf.\ \eqref{ rirj})},\] and with a potential $W$ satisfying \[\langle W_{u}(u),\xi_{i}\rangle=0,\text{ for }u\in\mathcal{P}_{\xi_{i}}\text{ and }i=1,\dots,n.\] Note that first-order derivatives can be included in (2.16) without harm. ## 3 The coordinate system Lemma 3.1.: _Suppose that $Q:\bar{D}\to\mathbb{R}$ is a function with the following properties:_ (3.1a) \[\kern-113.811024ptQ\text{ is convex},\] (3.1b) \[\kern-113.811024ptQ(u)>0\text{ and }Q_{u}(u)\neq 0,\text{ on } \bar{D}\setminus\{a_{1}\},\] (3.1c) \[\kern-113.811024ptQ(u+a_{1})=\|u\|+\mathrm{o}\big{(}\|u\|\big{)}.\] _Then, for each $\nu\in\mathbb{S}^{n-1}$, the ODE system_ (3.2) \[\dot{u}=\frac{Q_{u}(u)}{\langle Q_{u}(u),Q_{u}(u)\rangle},\text{ for }u\in \mathbb{R}^{n}\setminus\{a_{1}\},\] _has a unique global solution $\tilde{u}:(0,\infty)\to\mathbb{R}^{n}$, such that_ (3.3) \[\lim_{q\to 0+}\frac{\tilde{u}(q;\nu)}{\|\tilde{u}(q;\nu)\|}=\nu.\] _Moreover, the maps defined through the solution_ \[(q,\nu) \mapsto\tilde{u}(q;\nu),\text{ from }\mathbb{R}^{n}\times\mathbb{ S}^{n-1}\to\mathbb{R}^{n}\setminus\{a_{1}\},\] \[\nu \mapsto\tilde{u}(q;\nu),\text{ from }\mathbb{S}^{n-1}\to\{u\mid Q (u)=q\},\text{ for all }q\in(0,\infty),\] _are global diffeomorphisms onto their range._ Proof.: Here, for simplicity, we present the proof under the stronger hypothesis \[Q(u+a_{1})=\|u\|,\text{ for }\|u\|\leq r_{0}\text{ with }r_{0}>0\text{ and small}.\] For the general case we refer to [3]. From (3.2) we have that \[\dfrac{\operatorname{d\!}}{\operatorname{d\!}q}Q\big{(}u(q)\big{)}=1.\] This implies that the left extremum of the interval of existence of $u$ is $q=0$ and, furthermore, that (3.4) \[\lim_{q\to 0+}u(q)=a_{1}.\] Moreover, for $\|u-a_{1}\|\leq r_{0}$, \[\dfrac{\operatorname{d\!}}{\operatorname{d\!}q}\frac{u(q)}{\|u(q)\|}=\frac{1}{\|u \|\langle Q_{u},Q_{u}\rangle}\Big{(}Q_{u}-\frac{u}{\|u\|}\big{\langle}Q_{u},\frac {u}{\|u\|}\big{\rangle}\Big{)}=0,\] hence, the existence of the limit in (3.3) follows. Finally, note the identity (3.5) \[Q\big{(}\tilde{u}(q;\nu)\big{)}=q.\] The rest of the lemma follows by standard ODE facts. ∎ Lemma 3.2.: _Consider the mapping $(q,\nu)\mapsto\tilde{u}(q;\nu)$ as defined in Lemma 3.1. Then, for any fixed vector $t\perp\nu$, the quadratic form_ (3.6) \[\omega(\alpha,\beta)=-\langle\tilde{u}_{qq},\tilde{u}_{q}\rangle\alpha^{2}+ \langle\tilde{u}_{q\nu}t,\tilde{u}_{\nu}t\rangle\beta^{2}-2\langle\tilde{u}_{q \nu}t,\tilde{u}_{q}\rangle\alpha\beta,\text{ for }\alpha,\beta\in\mathbb{R}\] _is positive semidefinite._ Proof.: From (3.5), differentiating in $q$, we obtain (3.7) \[\langle Q_{u},\tilde{u}_{q}\rangle=1.\] On the other hand, differentiating in $\nu$, we obtain (3.8a) \[\langle Q_{u},\tilde{u}_{\nu}t\rangle=0\Leftrightarrow\langle\tilde{u}_{q}, \tilde{u}_{\nu}t\rangle=0;\] (3.8b) \[\langle\tilde{u}_{q\nu}t,\tilde{u}_{\nu}t\rangle+\langle\tilde{u}_{q},\tilde{u }_{\nu\nu}(t,t)\rangle=0.\] Now, differentiating (3.7) in $q$ yields (3.9a) \[\langle Q_{uu}\tilde{u}_{q},\tilde{u}_{q}\rangle+\langle Q_{u},\tilde{u}_{qq} \rangle=0\Leftrightarrow\langle\tilde{u}_{qq},\tilde{u}_{q}\rangle=-\frac{ \langle Q_{uu}\tilde{u}_{q},\tilde{u}_{q}\rangle}{\langle\tilde{u}_{q},\tilde{ u}_{q}\rangle},\] while differentiating in \[\nu\] yields (3.9b) \[\langle Q_{uu}\tilde{u}_{\nu}t,\tilde{u}_{q}\rangle+\langle Q_{u},\tilde{u}_{q \nu}t\rangle=0\Leftrightarrow\langle\tilde{u}_{q\nu}t,\tilde{u}_{q}\rangle=- \frac{\langle Q_{uu}\tilde{u}_{\nu}t,\tilde{u}_{q}\rangle}{\langle\tilde{u}_{q },\tilde{u}_{q}\rangle}.\] Finally, differentiating ( 3.8 ) in \[q\] yields (3.9c) \[\langle Q_{uu}\tilde{u}_{q},\tilde{u}_{\nu}t\rangle+\langle Q_{u},\tilde{u}_{q \nu}t\rangle=0\Leftrightarrow\langle\tilde{u}_{q\nu}t,\tilde{u}_{q}\rangle=- \frac{\langle Q_{uu}\tilde{u}_{q},\tilde{u}_{\nu}t\rangle}{\langle\tilde{u}_{q },\tilde{u}_{q}\rangle},\] while differentiating in \[\nu\] yields (3.9d) \[\langle Q_{uu}\tilde{u}_{\nu}t,\tilde{u}_{\nu}t\rangle+\langle Q_{u},\tilde{u} _{\nu\nu}(t,t)\rangle=0\Leftrightarrow\langle\tilde{u}_{\nu\nu}(t,t),\tilde{u} _{q}\rangle=-\frac{\langle Q_{uu}\tilde{u}_{\nu}t,\tilde{u}_{\nu}t\rangle}{ \langle\tilde{u}_{q},\tilde{u}_{q}\rangle}.\] The convexity of $Q$ implies (3.10) \[\langle Q_{uu}v,v\rangle\geq 0,\text{ for all }v\in\mathbb{R}^{n}.\] From this, (3.8b), and (3.9d), we obtain (3.11) \[\langle\tilde{u}_{q\nu}t,\tilde{u}_{\nu}t\rangle\geq 0,\] while from (3.10) and (3.9) we obtain (3.12) \[-\langle\tilde{u}_{qq},\tilde{u}_{q}\rangle\geq 0.\] From (3.10), by the same argument that proves the Schwarz inequality, we have (3.13) \[\langle Q_{uu}v,w\rangle^{2}\leq\langle Q_{uu}v,v\rangle\langle Q_{uu}w,w \rangle,\text{ for all }v,w\in\mathbb{R}^{n}.\] Thus, from (3.9) and (3.13), \[-\langle\tilde{u}_{qq},\tilde{u}_{q}\rangle\langle\tilde{u}_{q\nu}t,\tilde{u}_ {\nu}t\rangle-\langle\tilde{u}_{q\nu}t,\tilde{u}_{q}\rangle^{2}\geq 0,\] from which, via (3.11) and (3.12), we obtain (3.6). ∎ The diffeomorphism defined in Lemma 3.1 associates a ‘polar’ representation to a given map $v(x)$ in the following way: (3.14) \[v(x)\leftrightarrow\big{(}q^{v}(x),\nu^{v}(x)\big{)},\text{ where }v(x)=\tilde {u}\big{(}q^{v}(x);\nu^{v}(x)\big{)}.\] Dropping the superscripts, we calculate \[v_{x_{i}}=\tilde{u}_{q}q_{i}(x)+\tilde{u}_{\nu}\nu_{x_{i}}(x),\] (3.15) \[\|\nabla v\|^{2}=\langle\tilde{u}_{q},\tilde{u}_{q}\rangle\|\nabla q(x)\|^{2}+\sum _{j=1}^{n}\langle\tilde{u}_{\nu}\nu_{x_{j}},\tilde{u}_{\nu}\nu_{x_{j}}\rangle,\] and thus, we obtain that for $v\in W^{1,2}(\Omega;\mathbb{R}^{n})$ there holds \[q\in W^{1,2}(\Omega;\mathbb{R})\text{ and }\nu\in W^{1,2}(\Omega;\mathbb{S}^{n -1}).\] Given a $C^{2}$ potential $W:\mathbb{R}^{n}\to\mathbb{R}$, we define a $C^{2}$ function $V:(0,\infty)\times\mathbb{S}^{n-1}\to\mathbb{R}$ via (3.16) \[V(q,\nu):=W\big{(}\tilde{u}(q;\nu)\big{)}.\] Therefore, the action takes the form \[J_{\Omega}(v) =\int_{\Omega}\Big{\{}\frac{1}{2}\|\nabla v\|^{2}+W(v)\Big{\}} \operatorname{d\!}x\] \[=\int_{\Omega}\Big{\{}\frac{1}{2}\big{(}\langle\tilde{u}_{q}, \tilde{u}_{q}\rangle\|\nabla q(x)\|^{2}+\sum_{j=1}^{n}\langle\tilde{u}_{\nu}\nu_ {x_{j}},\tilde{u}_{\nu}\nu_{x_{j}}\rangle\big{)}+V(q,\nu)\Big{\}}\operatorname {d\!}x.\] For a fixed $v\in W^{1,2}(\Omega;\mathbb{R}^{n})$ and an open set $A\in\Omega$, we introduce the following functionals: (3.17) \[\mathcal{K}_{A}(\rho):=\int_{A}\frac{1}{2}\Big{\{}\langle\tilde{u}_{q}(\rho, \nu),\tilde{u}_{q}(\rho,\nu)\rangle\|\nabla\rho\|^{2}+\sum_{j=1}^{n}\langle \tilde{u}_{\nu}(\rho,\nu)\nu_{x_{j}},\tilde{u}_{\nu}(\rho,\nu)\nu_{x_{j}} \rangle\Big{\}}\operatorname{d\!}x,\] (3.18) \[\mathcal{V}_{A}(\rho):=\int_{A}V(\rho,\nu)\operatorname{d\!}x,\] (3.19) \[\mathcal{E}_{A}(\rho):=\mathcal{K}_{A}+\mathcal{V}_{A}.\] We remark that (1.8) in Hypothesis 4 implies, via (3.16) and (3.2), (3.20) \[\frac{\partial V}{\partial q}(q,\nu)\geq 0.\] On the other hand, just Hypothesis 1 implies (3.21) \[\frac{\partial V}{\partial q}(q,\nu)\geq c^{2}\langle\tilde{u}_{q},\tilde{u}_{ q}\rangle p,\text{ for }0\leq p\leq q\leq r_{0}.\] ## 4 Comparison lemmas Let $A\subset\Omega$ be an open, connected set with Lipschitz boundary. In the lemma next, we utilize the following as a comparison problem (4.1) \[\left\{\begin{array}[]{l}\Delta\phi=0,\text{ in }A,\\ \phi=g,\text{ on }\partial A,\end{array}\right.\] for a function $g:\partial A\to\mathbb{R}$. Lemma 4.1 below requires global hypotheses (as shown in parenthesis) and provides global estimates. Lemma 4.1 (Hypotheses 1,3,4).: _Let $u\in W^{1,2}(\Omega;\mathbb{R}^{n})$ with the following properties:_ 1. $J_{\Omega}(u)<\infty$_,_ 2. $u(\Omega)\subset D$_,_ 3. $q^{u}\|_{\partial A}\leq g$_._ _Then, there is a $v\in W^{1,2}(\Omega;\mathbb{R}^{n})$ such that_ 1. $q^{v}\leq\phi$_, a.e. in_ $A$_, and_ $\nu^{v}=\nu^{u}$_, a.e. in_ $A$_,_ 2. $v\|_{\Omega\setminus A}=u\|_{\Omega\setminus A}$_,_ 3. $J_{\Omega}(v)\leq J_{\Omega}(u)$_._ Proof.: Consider $\tilde{\rho}\mapsto\mathcal{K}_{A}(\tilde{\rho})$ in (3.17) as a functional in $\tilde{\rho}$, for a fixed $\nu=\nu^{u}$, and minimize it in the class of $W^{1,2}(A;\mathbb{R})$ functions with Dirichlet values (4.2) \[\tilde{\rho}=q^{u},\text{ on }\partial A.\] Since $Q$ is locally Lipschitz, there exists a minimizer $\rho$ which satisfies (4.3) \[\begin{split}\int_{A}\Big{\{}\langle\tilde{u}_{qq}(\rho,\nu), \tilde{u}_{q}(\rho,&\,\nu)\rangle\|\nabla\rho\|^{2}+\sum_{j=1}^{n} \langle\tilde{u}_{q\nu}(\rho,\nu)\nu_{x_{j}},\tilde{u}_{\nu}(\rho,\nu)\nu_{x_{ j}}\rangle\Big{\}}\eta\,\operatorname{d\!}x\\ &+\int_{A}\langle\tilde{u}_{q}(\rho,\nu),\tilde{u}_{q}(\rho,\nu) \rangle\nabla\rho\nabla\eta\,\operatorname{d\!}x=0,\text{ for all }\eta\in W^{ 1,2}_{0}(A).\end{split}\] Taking $\omega=\omega_{j}$ in (3.6), with $\alpha=\rho_{x_{j}}$, $\beta=1$, and $t=\nu_{x_{j}}$, we obtain, for $\eta\geq 0$, (4.4) \[\Big{(}\sum_{j=1}^{n}\omega_{j}\Big{)}\eta=\Big{(}\!-\langle\tilde{u}_{qq}, \tilde{u}_{q}\rangle\|\nabla\rho\|^{2}+\sum_{j=1}^{n}\langle\tilde{u}_{q\nu}\nu_ {x_{j}},\tilde{u}_{\nu}\nu_{x_{j}}\rangle-2\sum_{j=1}^{n}\langle\tilde{u}_{q \nu}\nu_{x_{j}},\tilde{u}_{q}\rangle\rho_{x_{j}}\Big{)}\eta\geq 0.\] Subtracting (4.4) from (4.3), after integrating gives (4.5) \[\int_{A}\nabla\rho\nabla\big{(}\langle\tilde{u}_{q},\tilde{u}_{q}\rangle\eta \big{)}\operatorname{d\!}x\leq 0,\text{ for }\eta\in W_{0}^{1,2}(A)\text{ with }\eta\geq 0,\] or, equivalently, $\Delta\rho\langle\tilde{u}_{q},\tilde{u}_{q}\rangle\leq 0$, or, (4.6) \[\Delta\rho\leq 0,\text{ in }W_{0}^{1,2}(A).\] By the maximum principle, (c) and (1) imply (See Figure 1.) (4.7) \[\rho\leq\phi,\text{ a.e.\ in }A.\] <figure><img src="content_image/0811.0106/x1.png"><figcaption>Figure 1: The functions ρ and ϕ.</figcaption></figure> Define (4.8) \[\left\{\begin{array}[]{l}q^{v}(x)=\min\{\rho(x),q^{u}(x)\},\text{ for }x\in A, \\ q^{v}(x)=q^{u}(x),\text{ for }x\in\Omega\setminus A,\\ \nu^{v}(x)=\nu^{u}(x),\text{ for }x\in\Omega.\end{array}\right.\] Statements (i) and (ii) follow by (4.7). For (iii) we argue as follows. Let \[J_{\Omega}(v)=J_{A}(v)+J_{\Omega\setminus A}(u).\] But, \[J_{A}(v) =J_{\Sigma}(v)+J_{A\setminus\Sigma}(v)\quad(\text{where }\Sigma:= \{x\in A\mid q^{u}>\rho(x)\})\] \[=J_{\Sigma}(v)+J_{A\setminus\Sigma}(u)\quad(\text{since }v=u\text { on }\Sigma\setminus A)\] \[=\mathcal{K}_{\Sigma}(\rho)+\mathcal{V}_{\Sigma}(\rho)+J_{A \setminus\Sigma}(u)\] \[\leq\mathcal{K}_{\Sigma}(q^{u})+\mathcal{V}_{\Sigma}(q^{u})+J_{A \setminus\Sigma}(u)\quad(\text{by the definition of }\rho\text{ and }\eqref{v- pos})\] \[=J_{\Sigma}(u)+J_{A\setminus\Sigma}(u)\] \[=J_{A}(u).\qed\] Next, we utilize the following as a comparison problem (4.9) \[\left\{\begin{array}[]{l}\Delta\psi=c^{2}\psi,\text{ in }A,\\ \psi=h,\text{ on }\partial A,\end{array}\right.\] for a function $h:\partial A\to\mathbb{R}$ and $c$ as in Hypothesis 1. The following lemma is based only on local properties. Lemma 4.2 (Hypothesis 1).: _Let $u\in W^{1,2}(\Omega;\mathbb{R}^{n})$ with the following properties:_ 1. $J_{\Omega}(u)<\infty$_,_ 2. $q^{u}(x)\leq\bar{q}<r_{0},$ _for_ $x\in A$ _(cf. Hypothesis 1),_ 3. $q^{u}(x)\leq h(x)\leq\bar{q},$ _for_ $x\in\partial A$_._ _Then, there is a $v\in W^{1,2}(\Omega;\mathbb{R}^{n})$ such that_ 1. $q^{v}\leq\psi$_, a.e. in_ $A$_, and_ $\nu^{v}=\nu^{u}$_, a.e. in_ $A$_,_ 2. $v\|_{\Omega\setminus A}=u\|_{\Omega\setminus A}$_,_ 3. $J_{\Omega}(v)\leq J_{\Omega}(u)$_._ Proof.: Let $\rho$ be a minimizer of $\tilde{\rho}\mapsto\mathcal{E}_{A}(\tilde{\rho})$ in (3.19) for $\nu=\nu^{u}$ fixed. Then, as in the proof of Lemma 4.1, (4.10) \[\int_{A}\Big{\{}\nabla\rho\nabla\big{(}\langle\tilde{u}_{q}(\rho,\nu),\tilde{u }_{q}(\rho,\nu)\rangle\eta\big{)}+V_{q}(\rho,\nu)\eta\Big{\}}\operatorname{d\! }x\leq 0,\text{ for }\eta\in W_{0}^{1,2}(A)\text{ with }\eta\geq 0,\] or, equivalently, (4.11) \[\Delta\rho\langle\tilde{u}_{q},\tilde{u}_{q}\rangle+V_{q}(\rho,\nu)\leq 0, \text{ in }W_{0}^{1,2}(A).\] Utilizing (3.21) and the maximum principle, we obtain (4.12) \[\rho\leq\psi,\text{ a.e.\ in }A.\] The remaining of the proof is analogous to the proof of Lemma 4.1. ∎ ## 5 Proof of Theorems 1.1 and 1.2 We begin by introducing various sets and relevant notation. Let $D_{B_{R}}=D\cap B_{R}$, with $D$ as in Hypothesis 4 and $B_{R}=B(0;R)$, and let \[P_{D_{B_{R}}}=\big{\{}u\in W^{1,2}_{\mathrm{E}}(B_{R};\mathbb{R}^{n})\mid u( \bar{D}_{B_{R}})\subset\bar{D}\big{\}}.\] Fix a $\bar{q}$, with $0<\bar{q}<r_{0}$, where $r_{0}$ is as in Hypothesis 1. Fix an $x_{0}$ in the interior of $D$, set $x_{R}=\frac{R}{2}x_{0}$, with $B(x_{R};L)$ standing for the ball with center $x_{R}$ and radius $L>0$, $L,l$ fixed, independent of $R$, and to be selected later; $B(x_{R};L+l)\subset D_{B_{R}}$ for $R>R_{0}$. Furthermore, let (5.1) \[PU^{\mathrm{c}}=\big{\{}u\in P_{D_{B_{R}}}\mid q^{u}(x)\leq\bar{q},\text{ for }x\in\overline{B(x_{R};L)}\,\big{\}}.\] The proof will be presented in three steps. _Step 1. There exists a minimizer $u_{R}\in W^{1,2}_{\mathrm{E}}(B_{R};\mathbb{R}^{n})$ of $J_{B_{R}}$on $PU^{\mathrm{c}}$, for $R>R_{0}$. Moreover, there exist constants $b>0$, $C>0$, independent of $R$, such that_ (5.2) \[\|u_{R}(x)\|<b,\text{ for }x\in B_{R},\text{ and }J_{B_{R}}(u_{R})<CR^{n-1}.\] Define (5.3) \[u_{\mathrm{aff}}(x):=\left\{\begin{array}[]{cl}d(x;\partial D)a_{1},&\text{for }x\in D_{B_{R}}\text{ and }d(x;\partial D)\leq 1\\ a_{1},&\text{for }x\in D_{B_{R}}\text{ and }d(x;\partial D)\geq 1\end{array}\right.\] and extend it equivariantly on $B_{R}$. Clearly, $u_{\mathrm{aff}}\in PU^{\mathrm{c}}$ for $R\geq R_{0}$. By the nonnegativity of $W$ and a simple calculation, (5.4) \[0\leq\inf_{PU^{\mathrm{c}}}J_{B_{R}}<J_{B_{R}}(u_{\mathrm{aff}})<CR^{n-1},\] for some constant $C$ independent of $R$. Let $\{u_{k}\}$ be a minimizing sequence. By Hypothesis 2, without loss of generality we can assume that $u_{k}(x)\in C_{0}$, and therefore, $\|u_{k}(x)\|<b$, for some $b>0$ independent of $R$, for $x\in B_{R}$. We have the following easy estimates (5.5) \[\frac{1}{2}\int_{B_{R}}\|\nabla u_{k}\|^{2}\operatorname{d\!}x<J_{B_{R}}(u_{ \mathrm{aff}})<CR^{n-1}\text{ and }\int_{B_{R}}\|u_{k}\|^{2}\operatorname{d\!}x< C(R),\] where $C(R)$ denotes a constant depending on $R$. By standard arguments, we obtain, along possibly a subsequence, (5.6) \[u_{k}\to u_{R},\text{ a.e.},\] where $u_{R}$ is a minimizer. Clearly, $q^{u_{R}}(x)\leq\bar{q}$ and $\|u_{R}(x)\|<b$ on $\overline{B(x_{R};L)}$. This finishes the proof of Step 1. _Step 2. There exist $L_{0}(\bar{q},b,N)>0$ and $\delta(\bar{q},b,N)>0$, independent of $R$, such that for $L>L_{0}$ and $R>R_{0}$ we have the estimates_ \[q^{u_{R}}(x)<\bar{q}(1-k\delta),\text{ on }B(x_{R};L+\delta)\setminus B(x_{R}; L-\delta),\] _for some $k=k(\bar{q},b,N)>0$, independent of $R$._ We introduce the following three comparison functions $\Psi_{\mathrm{I}}$, $\Psi_{\mathrm{II}}$, and $\Psi_{\mathrm{III}}$. \[\left\{\begin{array}[]{l}\dfrac{\partial^{2}\Psi_{\mathrm{I}}}{ \partial r^{2}}+\dfrac{N-1}{r}\dfrac{\partial\Psi_{\mathrm{I}}}{\partial r}-c^ {2}\Psi_{\mathrm{I}}=0,\text{ for }r=\|x-x_{R}\|\text{ and }x\in B(x_{R};L),\\ \Psi_{\mathrm{I}}^{\prime}(0)=0,\ \Psi_{\mathrm{I}}(L)=\bar{q};\end{array}\right.\] \[\left\{\begin{array}[]{l}\dfrac{\partial^{2}\Psi_{\mathrm{II}}}{ \partial r^{2}}+\dfrac{N-1}{r}\dfrac{\partial\Psi_{\mathrm{II}}}{\partial r}=0 ,\text{ for }r=\|x-x_{R}\|\text{ and }x\in B(x_{R};L+l)\setminus B(x_{R};L),\\ \Psi_{\mathrm{II}}(L)=\bar{q},\ \Psi_{\mathrm{II}}(L+l)=b;\end{array}\right.\] \[\left\{\begin{array}[]{l}\dfrac{\partial^{2}\Psi_{\mathrm{III}}}{ \partial r^{2}}+\dfrac{N-1}{r}\dfrac{\partial\Psi_{\mathrm{III}}}{\partial r}= 0,\text{ for }r=\|x-x_{R}\|\text{ and }x\in B(x_{R};L+\delta)\setminus B(x_{R};L -\delta),\\ \Psi_{\mathrm{III}}(L-\delta)=\Psi_{\mathrm{I}}(L-\delta),\ \Psi_{\mathrm{III} }(L+\delta)=\Psi_{\mathrm{II}}(L-\delta);\end{array}\right.\] where $L,l,\delta$ are positive numbers to be specified. By standard facts about Bessel functions, (5.7) \[\dfrac{\partial\Psi_{\mathrm{I}}}{\partial r}(L)=c\bar{q}\big{(}1+\mathrm{o}(1 )\big{)},\text{ as }L\to\infty,\text{ with }\mathrm{o}(1)\to 0,\text{ as }L\to\infty.\] By an explicit calculation we have the estimates (5.8) \[\frac{b-\bar{q}}{l}\leq\dfrac{\partial\Psi_{\mathrm{II}}}{\partial r}(L)\leq(b -\bar{q})\frac{(L+l)^{N-1}}{lL^{N-1}},\] (5.9) with $\mathrm{O}(\delta)$ uniformly in $L$ and $l$. We need to satisfy the inequalities (5.10) \[\dfrac{\partial\Psi_{\mathrm{II}}}{\partial r}(r)\leq\dfrac{\partial\Psi_{ \mathrm{III}}}{\partial r}(r)\leq\dfrac{\partial\Psi_{\mathrm{I}}}{\partial r} (r),\text{ for }r=\|x-x_{R}\|,\ x\in B(x_{R};L+\delta)\setminus B(x_{R};L-\delta),\] which will follow from certain estimates. It is sufficient to satisfy (5.11) \[\frac{1}{2}\Big{(}\frac{b-\bar{q}}{l}+c\bar{q}\big{(}1+\mathrm{o}(1)\big{)}\! \Big{)}+\mathrm{O}(\delta)>(b-\bar{q})\frac{(L+l)^{N-1}}{lL^{N-1}},\] (5.12) \[\frac{1}{2}\Big{(}(b-\bar{q})\frac{(L+l)^{N-1}}{lL^{N-1}}+c\bar{q}\big{(}1+ \mathrm{o}(1)\big{)}\!\Big{)}+\mathrm{O}(\delta)<c\bar{q}\big{(}1+\mathrm{o}(1 )\big{)}.\] We let now $l=L$. Clearly, given a fixed $\lambda\in(0,\frac{1}{2})$ and taking $\delta$ small, there exists an $L_{0}$, which deteremines $R_{0}$, such that for $L\geq L_{0}$, \[\dfrac{\partial\Psi_{\mathrm{I}}}{\partial r}-\dfrac{\partial\Psi_{\mathrm{III }}}{\partial r}\geq\big{(}\frac{1}{2}-\lambda\big{)}c\bar{q}\ \text{ and }\ \dfrac{\partial\Psi_{\mathrm{III}}}{\partial r}-\dfrac{\partial\Psi_{\mathrm{ II}}}{\partial r}\geq\big{(}\frac{1}{2}-\lambda\big{)}c\bar{q}.\] From these it follows, by taking $\delta$ smaller if necessary, that \[\bar{q}\Big{(}1-\big{(}\frac{1}{2}-\lambda\big{)}c\delta\Big{)}\geq\Psi_{ \mathrm{III}},\text{ on }B(x_{R};L+\delta)\setminus B(x_{R};L-\delta).\] The proof of Step 2 is completed by applying in succession Lemma 4.2 once for $\Omega=B_{R}$, $A=B(x_{R};L)$, and Lemma 4.1 twice, for $\Omega=B_{R}$, $A=B(x_{R};L+l)\setminus B(x_{R};L)$, and also for $\Omega=B_{R}$, $A=B(x_{R};L+\delta)\setminus B(x_{R};L-\delta)$. _Step 3. (Conclusion)_ The estimate provided by Step 2 above has two immediate consequences. First, it implies that there is a minimizer $u_{R}$ of $J_{B_{R}}$ which does not realize the pointwise constraint $q^{u_{R}}(x)\leq\bar{q}$ on $\overline{B(x_{R};L)}$. Second, by reapplying the estimate on displaced balls, we conclude that the constraint ‘propagates’ on a much larger set, (5.13) \[q^{u_{R}}(x)\leq\bar{q},\text{ on }D_{B_{R}}\setminus N(\partial D_{B_{R}}),\] where $N(\partial D_{B_{R}})$ is a neighborhood of $\partial D_{B_{R}}$ determined by $L$, independent of $R$, and the angles of the simplex $D$. In particular, (5.13) holds on a set independent of $R$. Next, we switch to the parabolic flow and apply Theorem 2.1 with $F_{B_{R}}$ replaced by $D_{B_{R}}$; $\{a_{1}\}$ is in the interior of $D$. Thus, by (5.13), $u_{R}(D_{B_{R}})\nsubseteq\partial D$, and so, by strong positivity, $u(\cdot,t;u_{R})$ maps $D_{B_{R}}$ into $D$ for any $t>0$. By choosing $t>0$ and small, via (5.13) we secure that (5.14) \[q^{u(\cdot,t;\,u_{R})}(x)<\bar{q},\text{ on }\overline{B(x_{R};L)},\] that is, the open pointwise constraint is satisfied. Fix such a $t$. By the gradient property (5.15) \[J_{B_{R}}\big{(}u(\cdot,t;u_{R})\big{)}\leq J_{B_{R}}(u_{R}),\] and since $u(\cdot,t;u_{R})$ is free of all constraints on $D_{B_{R}}$, it satisfies the Euler–Lagrange equation (5.16) \[\Delta u-W_{u}(u)=0,\text{ on }D_{B_{R}},\text{ for }R>R_{0},\] together with boundary conditions analogous to those in (2.7). By the comments following (2.7), $u$ can be extended to a solution of (5.16) on $B_{R}$, satisfying Neumann conditions on $\partial B_{R}$. Finally, we pass to the limit along a subsequence in $R$ and capture a function (5.17) \[u(x)=\lim_{R_{n}\to\infty}u(\cdot,t;u_{R_{n}}).\] By (5.13) and the comments following it, we conclude that $u(x)$ is a nontrivial solution, that is, $u\not\equiv 0$, to (5.18) \[\Delta u-W_{u}(u)=0,\text{ on }\mathbb{R}^{n}.\] By taking the limit in (5.13), we obtain the estimate (5.19) \[q^{u}(x)<\bar{q},\text{ on }D\setminus N(\partial D).\] Consider now one of the hyperplanes determining $D$, $\mathcal{P}_{r_{i}}=\{x\in\mathbb{R}^{n}\mid\langle x,r_{i}\rangle=0\}$. By (5.19), we can assume that there is an $\eta>0$ such that $q^{u}(x)<\bar{q}$ for $d(x,\mathcal{P}_{r_{i}})\geq\eta$. By Lemma 4.2, (5.20) \[q^{u}(x)\leq\psi(x),\text{ in }\{x\in\mathbb{R}^{n}\mid d(x,\mathcal{P}_{r_{i} })\geq\eta\},\] where (5.21) \[\left\{\begin{array}[]{l}\Delta\psi=c^{2}\psi,\text{ in }\{x\in\mathbb{R}^{n} \mid d(x,\mathcal{P}_{r_{i}})\geq\eta\},\\ \psi=\bar{q},\text{ on }\{x\in\mathbb{R}^{n}\mid d(x,\mathcal{P}_{r_{i}})=\eta \}.\end{array}\right.\] By rotating coordinates, we may assume that the hyperplane $P_{r_{i}}$ coincides with $\{x\in\mathbb{R}^{n}\mid x_{1}=0\}$ in the new coordinate system. We simply note that (5.22) \[\psi(x)=\bar{q}\mathrm{e}^{-c(x_{1}-\eta)}\] satisfies (5.21). This concludes the proofs of Theorems 1.1 and 1.2.∎ ## Acknowledgments NDA was partially supported by Kapodistrias Grant No. 70/14/5622 at the University of Athens. NDA would like to express his gratitude to the group attending the Applied Analysis and PDE’s seminar at the Department of Mathematics of the University of Athens, in particular to Vassilis Papanicolaou and Achilles Tertikas, for useful discussions on the contents of the paper. We are also in debt to Peter Bates for his numerous suggestions that improved the presentation. Finally, special thanks are also due to Apostolos Damialis and Nikolaos Katzourakis. ## References * [1] S. Alama, L. Bronsard, and C. Gui. Stationary layered solutions in $\mathbb{R}^{2}$ for an Allen–Cahn system with multiple well potential. _Calc. Var._5 (1997), pp. 359–390. * [2] N. D. Alikakos and G. Fusco. On an elliptic system with symmetric potential possessing two global minima. arXiv:0810.5009. * [3] N. D. Alikakos and G. Fusco. Entire solutions for elliptic systems with symmetric multiwell potentials. To appear in _Singularities in nonlinear evolution phenomena and applications_, M. Novaga and G. Orlandi eds. Proceedings of the workshop held at the Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy, May 26–30, 2008. * [4] S. Baldo. Minimal interface criterion for phase transitions in mixtures of Cahn–Hilliard fluids. _Ann. Inst. Henri Poincaré, Anal. Non Linéaire_7 No. 2 (1990), pp. 67–90. * [5] L. Bronsard, C. Gui, and M. Schatzman. A three-layered minimizer in $\mathbb{R}^{3}$ for a variational problem with a symmetric three-well potential. _Comm. Pure. Appl. Math._49 No. 7 (1996), pp. 677–715. * [6] L. Bronsard and F. Reitich. On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation. _Arch. Rat. Mech. Anal._124 No. 4 (1993), pp. 355–379. * [7] K. N. Chueh, C. C. Conley, and J. A. Smoller. Positively invariant regions for systems of nonlinear diffusion equations. _Ind. Univ. Math. J._26 (1977), pp. 373–392. * [8] H. Dang, P. C. Fife, and L. A. Peletier. Saddle solutions of the bistable diffusion equation. _Z. Angew. Math. Phys._43 No. 6 (1992), pp. 984–998. * [9] U. Dierkes, S. Hildenbrandt, A. Küster, and O. Wohlrab. _Minimal surfaces I. Boundary value problems_. Grundlehren der mathematischen Wissenschaften 295, Springer-Verlag, Berlin, 1992. * [10] U. Dierkes, S. Hildenbrandt, A. Küster, and O. Wohlrab. _Minimal surfaces II. Boundary regularity_. Grundlehren der mathematischen Wissenschaften 296, Springer-Verlag, Berlin, 1992. * [11] L. C. Evans. _Partial differential equations_. Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998. * [12] L. C. Evans and R. F. Gariepy. _Measure theory and fine properties of functions_. CRC Press, Boca Raton, FL, 1992. * [13] A. Farina and E. Valdinoci. The state of the art for a conjecture of De Giorgi and related problems. In _Recent progress on reaction-diffusion systems and viscosity solutions_, Y. Du, H. Ishii, and W.-Y. Lin eds. World Scientific, Hackensack, NJ, 2008. * [14] A. Freire. The existence problem for Steiner networks in strictly convex domains. To appear in _Arch. Rat. Mech. Anal._ * [15] A. Freire. Mean curvature motion of graphs with constant contact angle and moving boundaries. arXiv:0805.4592. * [16] A. Friedman. _Partial differential equations of parabolic type_. Prentice-Hall, Englewood Cliffs, NJ, 1964. * [17] D. Gilbarg and N. S. Trudinger. _Elliptic partial differential equations of second order_. Grundlehren der mathematischen Wissenschaften 224, Springer-Verlag, Berlin, revised second edition, 1998. * [18] L. C. Grove and C. T. Benson. _Finite reflection groups_. Graduate Texts in Mathematics 99, Springer-Verlag, Berlin, second edition, 1985. * [19] C. Gui and M. Schatzman. Symmetric quadruple phase transitions. _Ind. Univ. Math. J._57 No. 2 (2008), pp. 781–836. * [20] S. Heinze. _Travelling waves for semilinear parabolic partial differential equations in cylindrical domains_. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg, 1989. * [21] D. Henry. _Geometric theory of semilinear parabolic equations_. Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-Heidelberg-New York, 1981. * [22] R. Ikota and E. Yanagida. Stability of stationary interfaces of binary-tree type. _Calc. Var._22 No. 4 (2005), pp. 375–389. * [23] C. Mantegazza, M. Novaga, and V. M. Tortorelli. Motion by curvature of planar networks. _Ann. Sc. Norm. Sup. Pisa Cl. Sci._ Ser. V 3 No. 2 (2004), pp. 235–324. * [24] R. Mazzeo and M. Sáez. Self-similar expanding solutions for the planar network flow. To appear in _Séminaires et Congrès_, Société Mathématique de France. * [25] R. Osserman. _A survey of minimal surfaces_. Dover, New York, 1986. * [26] M. H. Protter and H. F. Weinberger. _Maximum principles in differential equations_. Prentice-Hall, Englewood Cliffs, NJ, 1967. * [27] J. Rubinstein, P. Sternberg, and J. B. Keller. Fast reaction, slow diffusion, and curve shortening. _SIAM J. Appl. Math._49 No. 1 (1989), pp. 116–133. * [28] M. Sáez Trumper. Existence of a solution to a vector-valued Ginzburg–Landau equation with a three well potential. arXiv:math/0702662. * [29] M. Sáez Trumper. Relaxation of the flow of triods by curve shortening flow via the vector-valued parabolic Allen–Cahn equation. To appear in _J. Reine Angew. Math._ * [30] O. Schnürer and F. Schulze. Self-similarly expanding networks to curve shortening flow. _Ann. Sc. Norm. Sup. Pisa Cl. Sci._ Ser. V 6 No. 4 (2007), pp. 511–528. * [31] O. Schnürer, A. Azouani, M. Georgi, J. Hell, N. Jangle, A. Koeller, T. Marxen, S. Ritthaler, M. Sáez, F. Schulze, and B. Smith. Evolution of convex lens-shaped networks under curve shortening flow. To appear in _Trans. Amer. Math. Soc._ * [32] J. Smoller. _Shock waves and reaction-diffusion equations_. Grundlehren der Mathematischen Wissenschaften 258, Springer-Verlag, Berlin, second edition, 1994. * [33] H. Weinberger. Invariant sets for weakly coupled parabolic and elliptic systems. _Rend. Math._ Ser. VI 8 (1975), pp. 295–310.
1404.6576	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 55089, "num_imgs": 5, "llama3_tokens_count": 16084 }	[ "content_image/1404.6576/x1.png", "content_image/1404.6576/x3.png", "content_image/1404.6576/x5.png", "content_image/1404.6576/x7.png", "content_image/1404.6576/x8.png" ]	# Dynamical detection of three triple stellar systems in open clusters J. F. González${}^{1}$, M. E. Veramendi${}^{1}$, and C. R. Cowley${}^{2}$ ${}^{1}$Instituto de Ciencias Astronómicas, de la Tierra y del Espacio, Casilla de Correo 49, 5400 San Juan, Argentina ${}^{2}$Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1042, USA E-mail:fgonzalez@icate-conicet.gob.ar Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11 ###### Abstract We present a kinematic analysis of three triple stellar systems belonging to two open clusters: CPD$-$60${}^{\circ}$961 and HD 66137 in NGC 2516, and HD 315031 in NGC 6530. All three systems are hierarchical triples with a close binary bound to a third body in a wider orbit, whose presence is detected through velocity variations of the close binary barycentre. Orbital parameters are derived from radial velocity curves. Absolute parameters for all stars are estimated assuming cluster membership. Some dynamical and evolutionary aspects of these systems are discussed, particularly the possible influence of Kozai cycles. The two systems of NGC 2516 have similar orbital configurations with inner periods of 11.23 d and 8.70 d and outer periods of 9.79 yr and 9.24 yr. The young system HD 315031 in the cluster NGC 6530 has an inner binary with a period of 1.37 d and a very eccentric ($e$=0.85) outer orbit with a period of 483 d. We report also radial velocity measurements of the components of the visual binary CPD$-$60${}^{\circ}$944 in NGC 2516. Including results from previous works, this cluster would harbor 5 hierarchical triples. Possible dynamical evolutionary scenarios are discussed. Long-term radial velocity monitoring is highlighted as strategy for the detection of subsystems with intermediate separations, which are hard to cover with normal spectroscopic studies or visual techniques. keywords: binaries: spectroscopic – stars: chemically peculiar – open clusters and associations: individual: NGC 2516 – open clusters and associations: individual: NGC 6530. ## 1 Introduction The frequency and the observed properties of multiple systems constitute key evidence for models of stellar formation and evolution. Presently, our statistical knowledge of early-type high-order multiple systems is insufficient for a useful test of theoretical models. In fact, in most of the catalogued multiple systems, the hierarchical configuration, the physical link between the companions, and even the number of stellar components has not been reliably established. Indeed, in our spectroscopic study of stellar components in 30 early-type multiples (Veramendi & González, 2014), we found that a significant fraction of the systems have a number of components that is different from those previously known. Early-type multiple stars are also crucial for the understanding of B and A-type chemically peculiar stars, whose physics is not completely understood. On the one hand, peculiar stars that occur in multiple systems and clusters may be assigned an age, which makes it possible to approach the time development of chemical peculiarities (Bailey et al., 2014). The study of the companions gives eventually the opportunity to know the original composition of the peculiar star. On the other hand, statistically some types of chemical peculiarities appear predominantly in binary or multiple stars. This is the case of HgMn stars for example, whose multiplicity frequency might be as high as 90% (Schöller et al., 2010). Finally, there is observational evidence suggesting that stellar companions might influence the generation of surface regions with abnormal composition (Hubrig et al., 2006; Hubrig et al., 2012; Makaganiuk et al., 2011). Open clusters provide large samples of coeval stellar objects. However, the studies of multiplicity in open clusters are surprisingly scattered in the literature (Duchêne & Kraus, 2013). Searches for spectroscopic and visual binaries have been carried out only in the nearest associations (Taurus-Aurigae, the Hyades, Praesepe, and the Pleiades), for which frequencies and some properties of multiple systems have been derived. Currently, hydrodynamic simulations of cluster formation (Bate, 2012) predict fractions of multiplicity increasing with the stellar mass, with values in agreement with the observational results for solar- or low-mass stars, including very-low-mass objects. According to Leigh & Sills (2011), dynamical encounters involving triple systems should be common in open clusters and they would significantly contribute to the population of blue-stragglers. Leigh & Geller (2013) confirmed these predictions considering the fractions of binaries and triples empirically determined in Taurus-Aurigae, the Hyades, Praesepe, and the Pleiades. These authors concluded that in such clusters encounters involving triple systems are as frequent as, or even more, than encounters involving single stars and binaries. These results suggest that models of open cluster evolution should include high-order multiples in their initial population and allow these systems to evolve dynamically through the cluster evolution and to influence the dynamical evolution of the latter through energy exchange (Leigh & Geller, 2013). From the observational point of view, these results highlight the need for a more detailed knowledge of the binary and especially triple population in clusters (Leigh & Sills, 2011). The detection and exhaustive characterization of hierarchical multiples represent an observational challenge, since different instrumental techniques have to be combined to cover all the range of separation between the components of subsystems, which span several orders of magnitudes from a few solar radii to thousands of astronomical units. Spectroscopy is the main tool for studying binary subsystems with periods shorter than tens or a few hundreds of days, while astrometric techniques and high-resolution imaging are used for the widest systems. Particularly tough is the intermediate range of separations, with periods on the order of a few thousands of days (a$\sim$10 AU). In the present paper we report the discovery of triple systems through the detection of long-period variations in the centre-of-mass velocity of spectroscopic binaries, demonstrating the usefulness of long-term spectroscopic monitoring for the study of this intermediate range of separation. In the long-term spectroscopic survey of stars in open clusters carried out at ICATE (Levato et al., 2004) several new spectroscopic binaries have been discovered. Eventually, some of the systems showed run-to-run variations of the barycentric velocity and from then on were monitored until the whole cycle of those slow variations was covered. The cluster NGC 2516 is particularly interesting for its content of binary and peculiar stars. Early spectroscopic works of this cluster reported about ten chemically peculiar late-B and early-A type stars (Abt & Morgan, 1969; Dachs, 1972; Hartoog, 1976) and nine single-lined spectroscopic binaries (Abt & Levy, 1972; Gieseking & Karimie, 1982). However, only a few of these objects have been studied subsequently and good quality orbits have been published for only two of them: the eclipsing binary V392 Car (Debernardi & North, 2001) and HD 65949 (Cowley et al., 2010). On the other hand, González & Lapasset (2000) reported three new double-lined spectroscopic binaries (SB2s) and one radial velocity variable (CPD$-$60${}^{\circ}$944A). The stars studied in the present paper, HD 66137 and CPD$-$60${}^{\circ}$961 (stars 19 and 2 in the cluster numbering by Cox, 1955), are two of these three SB2s. The third one is HD66066A, a short period binary studied by González & Lapasset (2003), who mentioned that would be a triple star, since it has a visual companion to which it could be dynamically bound (Dachs & Kabus, 1989). On the other hand, no spectroscopic study of stars HD 66137 and CPD$-$60${}^{\circ}$961 has been published since they were reported as binaries, although preliminary results of the present investigation, were presented by Veramendi & González (2010). In addition to these two objects, in the present paper we present a few observations of CPD$-$60${}^{\circ}$944A, the star reported as variable in González & Lapasset (2000), confirming its binary nature. Since this star has (at least) one visual companion (Dachs & Kabus, 1989) this would be also a multiple star. HD 315031 is an early-B type star member of the young open cluster NGC 6530. It was studied by González & Lapasset (2003), who measured the radial velocity of both components using two-dimensional cross-correlations (Zucker & Mazeh, 1994), and determined the spectroscopic orbit. Already in that first study, the measurements from different runs were found to be inconsistent with a constant binary centre-of-mass velocity, which was interpreted as due to the presence of a third body. However, the available data were not sufficient to secure the period of the outer orbit. In Sect. 2 and 3 we describe the orbital analysis and derive physical properties of the systems studied. In Sect 4 we discuss some dynamical and evolutionary aspects. Finally, our main conclusions are summarised in Sect. 5. ## 2 Observations and orbital analysis In the present paper we analyse spectroscopic observations taken over 15 years with the 2.15 m telescope and the REOSC echelle spectrograph at Complejo Astronómico El Leoncito (CASLEO), San Juan, Argentina. The spectra cover the wavelength range 3700–5900 Å with a resolving power R=13 300. The spectra were reduced by using standard procedures with the NOAO/IRAF package. In the case of HD 315031 our analysis includes remeasurement of most of the spectra of González & Lapasset (2003). Our spectra show, in principle, two sets of spectral lines belonging to the companions of the close binary system. We will identify these stars with letters A and B, being A the most massive of this pair. The third companion in the outer orbit will be called star C, regardless of its mass. We applied the spectral separation method by González & Levato (2006) to measure radial velocities and reconstruct the spectra of both visible stellar components. For the cross-correlations used by this method in the RV calculation, standard templates taken from the Pollux database (Palacios et al., 2010) were used. Table 1 (available electronically) lists all measured radial velocities for the three spectroscopic binaries. Columns 3 and 4 are orbital phases for the inner and outer orbits, calculated from the time of periastron passage. In the case of the inner orbit of HD 315031, which is circular, the phase origin is at the primary conjunction. Columns 5 to 8 list the measured radial velocities and their errors. Columns 9 and 10 are the calculated velocities for the two close-binary companions with respect to its barycentre, while the last column lists the calculated radial velocity for the barycentre of the binary, $V_{\mathrm{o}}$. Object \| HJD \| ϕ \| ϕo \| VA \| errA \| VB \| errB \| (VA−Vo)cal \| (VB−Vo)cal \| Vo ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| \| km s−1 \| km s−1 \| km s−1 \| km s−1 \| km s−1 \| km s−1 \| km s−1 CPD−60∘961 \| 2451587.7233 \| 0.9437 \| 0.4843 \| 16.4 \| 0.6 \| 39.6 \| 0.6 \| -9.7 \| 14.0 \| 26.8 CPD−60∘961 \| 2451590.6421 \| 0.2035 \| 0.4851 \| 47.1 \| 0.6 \| -5.3 \| 0.7 \| 21.4 \| -30.8 \| 26.8 CPD−60∘961 \| 2451939.7566 \| 0.2838 \| 0.5819 \| 46.9 \| 0.5 \| -1.2 \| 0.5 \| 19.4 \| -28.0 \| 26.6 CPD−60∘961 \| 2452270.7007 \| 0.7465 \| 0.6736 \| 6.3 \| 0.8 \| 56.0 \| 1.1 \| -20.6 \| 29.7 \| 26.2 CPD−60∘961 \| 2452271.7615 \| 0.8409 \| 0.6739 \| 5.1 \| 0.8 \| 54.6 \| 0.9 \| -20.3 \| 29.3 \| 26.1 … \| \| \| \| \| \| \| \| \| \| Table 1: Radial velocities. This is a sample of the full table. Radial velocity data were fitted with a double Keplerian orbit. In this scheme, the two visible components are assumed to move in a two mass-points orbit whose centre-of-mass follows in turn a Keplerian orbit. In the orbital phase calculation, observation times are corrected by the light-time effect due to the variation of the distance between the close binary and the observer. To find the best solution our orbit fitting program seeks in the parameter space the minimum of the quantity $\chi^{2}$. Consequently data points are weighted according to their formal measurement errors. The parameter errors are calculated considering the parameter variation for an increment $\Delta\chi^{2}$=1 while all remaining parameters are allowed to vary to minimize $\chi^{2}$, taking into account in this way correlations between parameters. In this manner, we fitted simultaneously 12 orbital parameters: the six elements of the inner orbit (orbital period $P$, time of periastron passage $T_{\pi}$, radial velocity amplitudes $K_{\mathrm{A}}$ and $K_{\mathrm{B}}$, argument of periastron $\omega$, and eccentricity $e$) and six parameters for the outer orbit that the binary barycentre describes around the centre-of-mass of the whole triple system (systemic velocity $\gamma_{o}$, period $P_{\mathrm{o}}$, time of periastron passage $T_{\pi\mathrm{,o}}$, radial velocity amplitude $K_{\mathrm{o}}$, argument of periastron $\omega_{\mathrm{o}}$, and eccentricity $e_{\mathrm{o}}$). Table 2 lists the orbital parameters for the three systems including, in addition to the mentioned orbital elements, the time of primary conjunction $T_{\mathrm{I}}$, projected orbital semiaxis $a\sin i$, and minimum masses $M\sin^{3}i$. Note that the subindex “o” does not correspond to the orbit of star C but to the observed movement of the barycentre of the spectroscopic binary. In particular, the semiaxis of the outer relative orbit is $a_{\mathrm{out}}=a_{\mathrm{o}}\cdot(M_{\mathrm{A}}+M_{\mathrm{B}}+M_{\mathrm{ C}})/M_{\mathrm{C}}$. For notation simplicity in the parameters of the inner binary concerning both components ($P$, $e$, $\omega$, $q$, etc.) we use no subindex. <figure><img src="content_image/1404.6576/x1.png"><figcaption>Figure 1: Radial velocity curves of CPD−60∘961\. The upper panel shows theradial velocity curves of the inner binary over the 11.23 day period. At thebottom of the same panel residuals observed-minus-calculated are plotted. Thelower panel shows the longer time variation of the centre-of-mass velocity.Filled (open) circles correspond to the primary (secondary) star.</figcaption></figure> Parameter \| Units \| CPD−60∘961 \| HD 66137 \| HD 315031 ---\|---\|---\|---\|--- P \| d \| 11.23259 ± \| 0.00013 \| 8.70355 ± \| 0.00004 \| 1.377436 ± \| 0.000003 TI \| d \| 2 454 705.43 ± \| 0.04 \| 2 454 446.884 ± \| 0.014 \| 2 454 484.623 ± \| 0.005 Tπ \| d \| 2 454 688.55 ± \| 0.13 \| 2454448.142 ± \| 0.017 \| - KA \| km s−1 \| 21.4 ± \| 0.3 \| 64.4 ± \| 0.7 \| 66.2 ± \| 1.1 KB \| km s−1 \| 30.9 ± \| 0.3 \| 64.7 ± \| 0.6 \| 89.0 ± \| 1.6 ω \| \| 4.70 ± \| 0.07 \| 3.236 ± \| 0.014 \| - e \| \| 0.124 ± \| 0.009 \| 0.380 ± \| 0.005 \| 0.00 q \| \| 0.694 ± \| 0.013 \| 0.995 ± \| 0.013 \| 0.744 ± \| 0.018 aAsini \| R⊙ \| 4.72 ± \| 0.07 \| 10.24 ± \| 0.10 \| 1.80 ± \| 0.03 aBsini \| R⊙ \| 6.80 ± \| 0.07 \| 10.29 ± \| 0.10 \| 2.42 ± \| 0.04 asini \| R⊙ \| 11.52 ± \| 0.10 \| 20.53 ± \| 0.14 \| 4.22 ± \| 0.05 MAsin3i \| M⊙ \| 0.096 ± \| 0.003 \| 0.769 ± \| 0.016 \| 0.306 ± \| 0.012 MBsin3i \| M⊙ \| 0.067 ± \| 0.002 \| 0.765 ± \| 0.016 \| 0.228 ± \| 0.008 γo \| km s−1 \| 24.54 ± \| 0.19 \| 24.64 ± \| 0.24 \| 6.8 ± \| 0.8 Po \| d \| 3575 ± \| 65 \| 3376 ± \| 39 \| 482.9 ± \| 0.5 TI,o \| d \| 2 456 385 ± \| 55 \| 2 456 138 ± \| 46 \| 2 454 513 ± \| 13 Tπ,o \| d \| 2 460 611 ± \| 148 \| 2 454 865 ± \| 125 \| 2 455 339 ± \| 2 Ko \| km s−1 \| 4.2 ± \| 0.4 \| 10.1 ± \| 0.4 \| 37.6 ± \| 1.6 ωo \| \| 3.2 ± \| 0.2 \| 5.4 ± \| 0.3 \| 5.15 ± \| 0.13 eo \| \| 0.47 ± \| 0.06 \| 0.17 ± \| 0.03 \| 0.85 ± \| 0.03 aosinio \| R⊙ \| 258 ± \| 21 \| 660 ± \| 24 \| 190 ± \| 10 n \| \| 34 \| 36 \| 71 σA \| km s−1 \| 0.58 \| 1.17 \| 4.4 σB \| km s−1 \| 0.61 \| 1.23 \| 7.7 Table 2: Orbital parameters. Parameters with subindex ”o” refers to the outer orbit described by the barycentre of the spectroscopic binary. The last three rows are the number of observations and the RMS of the residuals. For both triples of NGC 2516, CPD$-$60${}^{\circ}$961 and HD 66137, cluster membership is confirmed kinematically by the agreement between barycentric velocity of the two triples and the cluster velocity: 22.0 km s${}^{-1}$(González & Lapasset, 2000), 23.08 km s${}^{-1}$(Mermilliod et al., 2008). <figure><img src="content_image/1404.6576/x3.png"><figcaption>Figure 2: Radial velocity curves of HD 66137. The upper panel shows the radialvelocity curves of the inner binary. At the bottom of the same panel theresiduals observed-minus-calculated are plotted. The lower panel shows thetime variation of the centre-of-mass velocity. Filled (open) circlescorresponds to the primary (secondary) star.</figcaption></figure> <figure><img src="content_image/1404.6576/x5.png"><figcaption>Figure 3: Radial velocity curves of HD 315031. The upper panel shows theradial velocity curves of the inner binary. At the bottom of the same panelthe residuals observed-minus-calculated are plotted. The lower panel shows thetime variation of the centre-of-mass velocity. Filled (open) circlescorresponds to the primary (secondary) star.</figcaption></figure> Radial velocity curves are shown in Figures 1, 2, and 3. In these figures upper panels show the velocity of the spectroscopic binary companions with respect to the barycentric velocity of the binary, while the lower panels show the velocity variations of the binary barycentre. ## 3 Physical parameters Cluster membership allows us to know the age of these multiple systems and to estimate absolute stellar parameters from their position in the cluster color-magnitude diagram. Nevertheless, the third body might contribute significantly to the integrated light of the system, making these calculations less reliable. Therefore, we adopted primarily estimated masses from the spectral types. Using line ratios measured in the separated spectra, we determined spectral types for the six studied stars. We then derived temperatures using the Schmidt-Kaler (1982) calibration and estimated masses and other parameters interpolating in the isochrone corresponding to the cluster age in the stellar model grids of the Geneva group (Ekström et al., 2012; Mowlavi et al., 2012; Georgy et al., 2013). Uncertainties in these parameters were estimated from the range of spectral types considered compatible with the observed spectrum. We used the spectroscopic parameters of the outer orbit and the estimated masses of the visible companions to derive a lower limit to the mass of the third body. Combining the expression for the radial velocity semiamplitude of the wide orbit: with the Kepler equation for the outer orbit we obtain: \[\frac{M_{\mathrm{C}}}{(M_{\mathrm{A}}+M_{\mathrm{B}}+M_{\mathrm{C}})^{2/3}}= \frac{K_{\mathrm{o}}P_{\mathrm{o}}^{1/3}\sqrt{1-e_{\mathrm{o}}^{2}}}{(2\pi G)^ {1/3}\sin i_{\mathrm{o}}}\geq\frac{K_{\mathrm{o}}P_{\mathrm{o}}^{1/3}\sqrt{1-e _{\mathrm{o}}^{2}}}{(2\pi G)^{1/3}}.\] (1) On the other hand, when it was possible we estimated an upper limit to $M_{\mathrm{C}}$ considering the intensity that its spectral lines would have in the spectrum. This upper limit to $M_{\mathrm{C}}$ was used to obtain a lower limit to the outer orbit inclination $i_{\mathrm{o}}$ using eq. 1. Obtaining at least rough estimates for the masses of all three stars, allow us to get, in combination with the spectroscopic parameters, information about the mutual inclination between the inner and the outer orbits. In fact, the mutual inclination between both orbital planes ($i_{\mathrm{m}}$) is given by: \[\cos(i_{\mathrm{m}})=\cos(i)\cos(i_{\mathrm{o}})+\sin(i)\sin(i_{\mathrm{o}}) \cos(\Delta\Omega),\] where $\Delta\Omega$ is the angle between the lines of nodes of the two orbits. Therefore, \[\cos(i_{\mathrm{m}})\leq\cos(i-i_{\mathrm{o}}),\] and finally, taking into account that both $i_{\mathrm{m}}$ and $\left\|i-i_{\mathrm{o}}\right\|$ are in the interval (0, $\pi$), we obtain \[i_{\mathrm{m}}\geq\left\|i-i_{\mathrm{o}}\right\|.\] In short, from estimates of stellar masses it is possible to obtain a lower limit for the inclination between inner and outer orbits, which is an important parameter for the dynamics of the triple. In Table 3 we summarise these results, which are discussed for each system in the following sections. Parameter \| Units \| CPD−60∘961 \| HD 66137 \| HD 315031 ---\|---\|---\|---\|--- Teff(A) \| K \| 10 500 ± \| 1000 \| 10 100 ± \| 900 \| 28 000 ± \| 2000 Teff(B) \| K \| 8450 ± \| 250 \| 10 100 ± \| 900 \| 25 400 ± \| 2000 MA \| M⊙ \| 2.6 ± \| 0.4 \| 2.3 ± \| 0.3 \| 12.9 ± \| 1.5 MB \| M⊙ \| 1.8 ± \| 0.2 \| 2.3 ± \| 0.3 \| 9.6 ± \| 1.1 i \| deg \| 19.5 ± \| 1.0 \| 43.5 ± \| 2.6 \| 16.7 ± \| 0.6 MC111Lower limit from the spectroscopic mass-function. \| M⊙ \| >0.8 \| >2.6 \| >7.0 MC222Estimated from detection/non-detection of lines in the spectrum. \| M⊙ \| <1.6 \| ∼3.2±0.5 \| <10 io \| deg \| \ga72 \| ∼77±7 \| \ga75 im \| deg \| \ga52 \| \ga34 \| \ga58 Table 3: Estimated absolute masses and inferred orbital inclinations. ### Cpd$-$60${}^{\circ}$961 For the components of this spectroscopic binary we derived spectral types B9 V and A4 V. In our spectra both stars have sharp lines (full-with half-maximum of about 25 km s${}^{-1}$) with rotational velocity too low to be measured with our spectral resolution. As already noted in the preliminary work by Veramendi & González (2010), star A shows a peculiar spectrum with a notable $\lambda$3984 Hg ii line and other lines typical of HgMn stars (strong Y ii, Sr ii, P ii, and Pt ii). However, strikingly, Mn ii lines are not present. We analyzed 884 wavelength measurements by the method of wavelength coincidence statistics and the results shows very low probability for the presence of Mn. P ii is clearly present but not Ga ii. Highly significant results were obtained for Sr ii, Y ii, and Pt ii. Interestingly, the presence of several lanthanide spectra is very likely, particularly lines of Nd iii and Pr iii. All these characteristics, which altogether are not typical of any group of peculiar stars, resemble the spectrum of the unique star HD 65949 studied by Cowley et al. (2006); Cowley et al. (2010), which is also a spectroscopic binary of the same cluster NGC 2516. From the spectral types we estimated temperatures $T_{\mathrm{eff}}(\mathrm{A})$=10 500$\pm$1000 K and $T_{\mathrm{eff}}(\mathrm{B})$=8 450$\pm$250 K. We then interpolated in the Geneva model grid to find stellar models that are consistent with the temperature estimates, the spectroscopic mass-ratio, and the cluster age. We adopted for the cluster $\log(\mathrm{age})$=8.0–8.3, which is a conservative range consistent with the published values: 8.04 (Dachs & Kabus, 1989), 8.15 (Meynet et al., 1993), 8.2$\pm$0.1 (Sung et al., 2002). We obtained masses $M_{\mathrm{A}}=2.6\pm 0.4$ M${}_{\odot}$ and $M_{\mathrm{B}}=1.8\pm 0.2$ M${}_{\odot}$, and visual absolute magnitudes $M_{V}(\mathrm{A})=0.9\pm 0.5$ mag and $M_{V}(\mathrm{B})=2.3\pm 0.2$ mag. The comparison of these masses with the spectroscopic parameters gives a low orbital inclination ($i$=19$\aas@@fstack{\circ}$5$\pm$1$\aas@@fstack{\circ}$0) and for $M_{\mathrm{C}}$ a lower limit of 0.8 M${}_{\odot}$. An upper limit can be established considering that there is no trace of its lines in the spectrum. Assuming this third body is a main-sequence star, we estimate that its relative flux is at least 30% lower than the flux of the secondary star, even if its lines were rotationally broadened. Therefore, $M_{\mathrm{C}}$ is expected to be less than about 1.6 M${}_{\odot}$. Assuming an absolute magnitude corresponding to a star of 0.8–1.6 M${}_{\odot}$ for component C, we obtained an integrated absolute magnitude for the system of about $M_{\mathrm{V}}$(ABC)=+0.6$\pm$0.4 mag, which corresponds to an apparent distance modulus $V-M_{\mathrm{V}}=8.2\pm 0.4$ mag. This value is consistent with the photometric values determined for the cluster by Dachs & Kabus (1989, 8.54 mag), Sung et al. (2000, 8.12 mag), Terndrup et al. (2002, 8.30 mag), and Loktin et al. (1994, 8.19), as well as the one derived from the Hipparcos parallaxes ($V_{\mathrm{o}}-M_{\mathrm{V}}=7.70\pm 0.16$ mag, Robichon et al., 1999). Approximate values for the orbital inclination of both the inner and outer orbits can be derived if the masses of the three stars are estimated. For the inner orbit we obtained $i\approx$19$\aas@@fstack{\circ}$5. For the outer orbit, assuming an upper limit of 1.6 M${}_{\odot}$ for star C, we calculated a lower limit $i_{\mathrm{o}}\ga 72^{\circ}$. Therefore, the mutual inclination of the inner and outer orbits is $i_{\mathrm{m}}\ga 52^{\circ}$. Dynamically the system contains a close binary with two stars about 2.6 M${}_{\odot}$ and 1.8 M${}_{\odot}$ in an orbit with a semiaxis of about 35 R${}_{\odot}$ and moderate eccentricity. This close pair is bound to a third star, less massive, orbiting in an eccentric orbit with a semiaxis of about 8 AU. The triple is markedly hierarchical with a semiaxis ratio of $\sim$50. ### Hd 66137 The spectral morphology of components A and B are very similar to each other: both have spectral type B9-A0 V and low rotational velocity. Their temperatures would be approximately 9700$\pm$400 K, which in the cluster isochrone corresponds to a mass of 2.3$\pm$0.3 M${}_{\odot}$ and an absolute visual magnitude of 1.35$\pm$0.45 mag. The lower limit for the mass of star C is high. Using eq. 1 we deduced that star C contains at least 35–37% of the total mass of the triple, being probably the most massive star of the three. Likewise, the intensity of spectral lines of the two visible components suggests that the flux contribution of star C is considerable. Indeed, the equivalent width of all spectral lines in stars A and B are smaller than expected for stars of the same spectral type by a factor 0.3–0.4. In absence of a third light, this factor would be 0.5 for both stars. Therefore, all three stars seem to have comparable brightness. <figure><img src="content_image/1404.6576/x7.png"><figcaption>Figure 4: Stacked spectra of HD 66137 ordered by orbital phase. Shadows atλ4471 and λ4481 are broad spectral lines belonging to the fast rotating starC. Several sharp lines of the close binary are visible: the strong line Mg ii4481 and weak lines of Ti ii (4468 and 4501) and Fe ii (4489 and 4491).</figcaption></figure> The lines of star C are in fact present in the spectrum, but pass unnoticed due to its high rotation. In order to make its spectrum more visible, we built the gray-scale image shown in Fig. 4, in which all observed spectra (excluding a few low-S/N spectra) are stacked ordered by orbital phase. Besides the sharp lines of the spectroscopic binary, which nicely trace its orbital motion, broad spectral lines are visible as shadows at the position of lines He i$\lambda$4471 and Mg ii$\lambda$4481. A few other similar broad lines can be distinguished in the spectrum, including He i lines at $\lambda\lambda$4089, 4144, 4388, He i-Fe ii blends at $\lambda$4922-24 and $\lambda$5016-18, and the Si ii doublet $\lambda$4128-30. Since He i$\lambda$4471 and Mg ii$\lambda$4481 have similar intensity, we estimate a spectral type B7-B8 for star C, somewhat earlier than its companions. Its rotational velocity is on the order of 200-250 km s${}^{-1}$. If these broad lines belong indeed to the same object that is responsible for the barycentric movement of the close binary, its radial velocity is expected to vary with a semiamplitude of the order of 10 km s${}^{-1}$. These variations, however, are very hard to measure with such shallow and broad lines. The integrated magnitude of the triple allow us to make an estimate of the mass of star C. From the apparent magnitude of the triple ($V=7.85$, Dachs & Kabus, 1989), the apparent distance modulus, and the estimated magnitudes of the components of the inner binary, we obtain for star C an absolute magnitude $M_{\mathrm{V}}(\mathrm{C})\approx$+0.2 mag, which corresponds to a mass $M_{\mathrm{C}}\sim$3.2 M${}_{\odot}$. From the estimated mass of star C we derived $i_{\mathrm{o}}\approx 77^{\circ}\pm 7^{\circ}$ and a lower limit $i_{\mathrm{m}}\ga 34\degr$ for the mutual inclination of the orbits, although considering the uncertainties in $i_{\mathrm{o}}$ and $i$ this lower limit might be as low as 24°. In short, this triple is structured in two hierarchical levels. The outer binary subsystem has a period of 9.2 yr and a semiaxis of about 8–9 AU. The less massive component of this subsystem is a 3 M${}_{\odot}$ star, while the primary is a close binary of 8.7 d period with twin components of about 2.3 M${}_{\odot}$. ### Hd 315031 We determined for the visible components of this system’s spectral types B0.5 IV-V and B1 V, which correspond to temperatures $T_{\mathrm{eff,A}}=28\,000\pm 2000$ K and $T_{\mathrm{eff,B}}=25\,400\pm 2000$ K, according to the Schmidt-Kaler (1982) calibration. The low values of the spectroscopic minimum masses in comparison with the masses corresponding to the spectral types, indicate a low orbital inclination. The eccentricity of the orbit is indistinguishable from zero, as is expected as a consequence of tidal friction for a binary with a period as short as this (P=1.377 d). Hence, in the final calculation of the orbital parameters we fixed the eccentricity at zero. Furthermore, both stellar components are expected to rotate synchronously with the orbital motion, since the timescale for synchronization is shorter than for circularization. Under this hypothesis, the projected radii can be calculated from the projected rotational velocities. We determined $v\sin i$ by applying the method by Díaz et al. (2011) on the reconstructed spectra of stars A and B. We obtained $v_{\mathrm{A}}\sin i=48\pm 4$ km s${}^{-1}$and $v_{\mathrm{B}}\sin i=42\pm 4$, from which we calculated $R_{\mathrm{A}}\sin i=1.31\pm 0.11\:\mathrm{R}_{\odot}$ and $R_{\mathrm{B}}\sin i=1.14\pm 0.12\:\mathrm{R}_{\odot}$. Even though the orbital inclination is in principle unknown, the projected radius can be combined with the minimum mass obtained from the orbital analysis to get the mean stellar densities: \[\left(\frac{\rho}{\rho_{\odot}}\right)_{j}=0.01343\ \left[\frac{K_{\mathrm{A}} +K_{\mathrm{B}}}{(v\sin i)_{j}}\right]^{3}\cdot\frac{q^{j-1}}{P^{2}\:(1+q)}\] (2) where $j$=1 ($j$=2) for star A (B), the period is in days, and the numerical constant is $(1\:\mathrm{R}_{\odot}/1\:\mathrm{AU})^{3}\cdot(1\:\mathrm{yr}/1\:\mathrm{d})^ {2}$. Using this equation we obtained: $\rho_{\mathrm{A}}=0.137\pm 0.035\:\rho_{\odot}$ and $\rho_{\mathrm{B}}=0.137\pm 0.035\:\rho_{\odot}$. These values correspond to stars very close of the zero-age main-sequence. Fig. 5 shows the position of both companions in the Color-Magnitude diagram. The colored regions mark the stellar models that are consistent with all observational information, essentially spectroscopic mass-ratio, densities, and temperatures. In short, this binary is formed by two unevolved main-sequence stars of 12.9$\pm$1.5 and 9.6$\pm$1.1 M${}_{\odot}$ with radii of about 4.6$\pm$0.4 and 3.9$\pm$0.3 R${}_{\odot}$, corresponding to an age of about 1 Myr. The inclination of the orbit is about 16$\aas@@fstack{\circ}$7. The spectroscopic minimum mass for the third star is a fraction 0.24$\pm$0.01 of the total mass. This value corresponds to 0.7-0.8 times the mass of star B. Considering that the lines of star C are not clearly visible in the spectrum, its mass should be close to this lower limit. Even if it is a fast rotator, a conservative higher limit for its mass would be 9-10 M${}_{\odot}$. The integrated absolute visual magnitude derived from the estimated stellar parameters is in agreement with the cluster membership. Although NGC 6530 has been subject of several photometric studies, the distance to this cluster is not well known. Published values for the distance modulus range from $V_{0}-M_{\mathrm{V}}\approx 11.3$ to $V_{0}-M_{\mathrm{V}}\approx 10.5$(Sung et al., 2000; Arias et al., 2006; Prisinzano et al., 2005), while the $E(B-V)$ reddening value would be between 0.20 and 0.35 mag. The disagreement between different authors or even different star samples within the same work, might be related to variable extinction, an abnormal extinction law, or the existence of several stellar groups at different distances (Arias et al., 2006). The location of this triple in the cluster color-magnitude diagram corresponds to $M_{\mathrm{V}}\approx-2.9$ for $V_{0}-M_{\mathrm{V}}$=10.5, $E(B-V)$=0.2 and $M_{\mathrm{V}}\approx-4.1$ for $V_{0}-M_{\mathrm{V}}$=11.25, $E(B-V)$=0.35. Our estimated absolute parameters correspond to an integrated absolute magnitude of the binary HD 315031AB of $M_{\mathrm{V}}=-3.16\pm 0.25$. Assuming $M_{\mathrm{C}}$ is in the range 7–10 M${}_{\odot}$, the total absolute magnitude of the triple would be $M_{\mathrm{V}}\approx-3.4$, which is consistent with the cluster distance. <figure><img src="content_image/1404.6576/x8.png"><figcaption>Figure 5: Position of the stellar components of HD 315031 in the Colormagnitude diagram. Black heavy lines are the zero-age and terminal-age main-sequences; dotted lines are the isochrones logτ=6.0, 6.5, 7.0, and 7.5; dashedlines are isotherms from 20 000 K (right) to 32 000 K (left); thin continuouslines are curves corresponding to densities ρ=0.137ρ⊙ and ρ=0.152ρ⊙. The gray(red and blue in the color version) areas mark the stellar models compatiblewith the observed temperatures and densities.</figcaption></figure> The total mass A+B+C is of the order of 31 M${}_{\odot}$, and therefore the outer orbital semiaxis is about 800 $R_{\odot}$. However, since the orbit is very eccentric, at periastron the third star is at only 120 $R_{\odot}$ from the close binary. This distance is about 8 times larger than the separation of the companions of the inner pair. The high inclination of the outer orbit and low inclination of the inner orbit ($i\approx 17^{\circ}$) assure a high relative inclination of the two orbital planes ($i_{\mathrm{m}}\ga 58$°). ## 4 Discussion ### Detection of long period spectroscopic subsystems We reported in the present paper the dynamical discovery of two triples in the cluster NGC 2516 with outer periods between 9 and 10 years. Our time baseline was long enough to cover in both cases about one and a half orbital cycles of the outer subsystems. At the distance of the cluster (380 pc), the projected semiaxis of the outer subsystems in angular units is, in both triples, about 20–25 mas. This separation range is still out of the reach of visual techniques like speckle interferometry or adaptive optic imaging, showing the difficulty of a complete multiplicity survey, covering all separation ranges. Even in close well-studied clusters like the Pleiades and Praesepe our knowledge of multiplicity at intermediate separations (5–15 AU) is presently very poor (Leigh & Geller, 2013). Interestingly, the semiaxis of the outer orbit for our two triples of NGC 2516 are in the middle of this separation range, showing the importance of long time-baseline radial velocity monitoring in multiplicity surveys. Most long-period spectroscopic binaries in open clusters have been discovered by studies focussed on late-type stars, like the systematic survey of giant stars of Mermilliod & Mayor (1989)(see also Mermilliod et al., 2007). The small-amplitude velocity variations of wide binaries are easier to detect in late-type stars where higher precision can be reached in radial velocity measurements. By contrast, our knowledge of early-type wide binary systems is much poorer, as is evident from the content of the Ninth Catalogue of Spectroscopic Binary Orbits (Pourbaix et al., 2009). Among the 254 binaries with periods above 3000 days in this catalog, 231 are late-type (F-M) dwarfs or giants, while only 14 binaries have A-type primaries and 9 have spectral types O or B. Binaries and multiples with periods of several years are virtually impossible to discover in fast rotating early-type stars. However, errors of 1 km s${}^{-1}$can be obtained in O-B-A stars of low $v\sin i$, and the monitoring such objects, particularly in young clusters, would alleviate the lack of statistical information on massive long period spectroscopic binaries. ### Triple stars of NGC 2516 In this section we show, through the analysis of published and new information, that the cluster NGC 2516 would contain five hierarchical triples among its early-type main-sequence stars. The two triple systems detected dynamically in the present paper are hierarchical with similar orbital configuration: an outer subsystem with a semiaxis of about 8 AU and an inner subsystem with $a\approx 30\:\mathrm{R}_{\odot}$. However, in the triple CPD$-$60${}^{\circ}$961 the most distant star (component C) is the least massive of the three, while in HD 66137 it is the most massive. Additionally, we present here spectroscopic material for one visual-spectroscopic triple. The visual binary CPD$-$60${}^{\circ}$944AB is very probably a physical pair (Dachs & Kabus, 1989), and its primary was reported as radial velocity variable by González & Lapasset (2000) on the basis of two radial velocity measurements. We present in Table 4 velocity measurements obtained in the last years for both visual companions. This observations definitely confirm the binarity of the visual primary. The available observations, however, are not sufficient to fit the orbit reliably. We found two possible orbits with periods of 121.6 and 182.5 days and eccentricity $e\approx 0.4$ in both cases. A notable fact of this triple is that both visible components are chemically peculiar. CPD$-$60${}^{\circ}$944A was reported as B8pSi by Hartoog (1976). In our spectra Si ii is clearly enhanced, Cr ii appears weak, and Eu ii lines at $\lambda\lambda$ 4130, 4205, and 4436 Å are clearly visible. Wavelength coincidence statistics showed that several other rare earth elements are also present. With a wavelength tolerance of 0.06 Å, highly significant results were obtained for ions Si ii, Ca ii, Ti ii, Fe ii, Pr ii, Nd iii, Eu ii, and Ho ii. At somewhat lower confidence level Cr ii, Nd ii, Dy ii, and Er iii would be also present. Even though significance levels change slightly using different tolerance values or line lists, the presence of at least three lanthanide (Eu, Dy, Ho) is a robust result. On the other hand, CPD$-$60${}^{\circ}$944B, classified as B9.5IVp(Si) by Dachs & Kabus (1989), is in fact a HgMn star, which exhibits strong lines of Hg ii, Mn ii, P ii, Ga ii, and Xe ii. This system, therefore, would be one of the few known multiple system formed by peculiar stars of different type. CPD−60∘944A HJD RV km s−1 2 451 590.6287 22.7 2 451 592.7282 24.3 2 451 939.7416 18.7 2 452 270.7174 09.8 2 454 905.5859 28.3 2 455 572.6178 09.8 2 456 695.7500 22.8 CPD−60∘944B HJD RV km s−1 2 453 761.7224 24.3 2 454 835.6489 23.9 2 454 905.5928 23.9 2 454 907.5665 23.3 2 455 566.6242 24.2 2 455 572.6046 23.2 2 455 674.5250 23.7 2 456 695.7389 22.8 2 456 696.6007 21.8 Table 4: Radial velocities of visual pair CPD−60∘944AB. Uncertainties are about 1.2 km s−1 for component A and 0.9 km s−1 for component B. A fourth triple would be HD 65949, a chemically-peculiar single-lined spectroscopic binary studied by Cowley et al. (2010), who detected a small variation in the centre-of-mass velocity, which was interpreted as due to the presence of a third body. The chemical pattern of this star is not typical of any group of peculiar stars, presenting several lines typical of HgMn stars, but lacking Mn lines. Strikingly this morphology is very similar to the primary of CPD$-$60${}^{\circ}$961, one of the triples analysed in this paper. Finally, the short period spectroscopic binary HD66066A (González & Lapasset, 2003) has a visual companion that could be dynamically bound (Dachs & Kabus, 1989), being therefore also a hierarchical triple. ### Dynamical evolution of the observed triples We will comment in this section about three aspects of the system dynamics: tidal effects in the close binary, orbital stability of the triple, and the possible occurrence of Kozai oscillations. In close binaries, tidal interaction in combination with energy dissipation mechanisms (radiative damping in the case of early-type stars) tends to synchronize stellar rotation with orbital motion and circularize the orbit (Zahn, 1977; Hut, 1981). In order to estimate the circularization timescales of the close pair in our triples we used the binary evolution code developed by Hurley et al. (2002)³. Inner binaries in both triple systems of NGC 2516 have periods long enough for circularization not to be reached until after the end of the main-sequence. By contrast, HD 315031 would have been circularized in about 1.0-1.5 Myr, which is comparable with the cluster age. [FOOTNOTE:3][ENDFOOTNOTE] According to Mardling & Aarseth (2001), a triple or higher-order star system is stable if the periastron distance of the third star (r${}_{\pi}^{\mathrm{out}}$) and the inner semiaxis ($a$) orbits obey the criterion: \[\frac{r_{\pi}^{\mathrm{out}}}{a}>2.8\ \left[(1+q_{\mathrm{out}})\cdot\frac{1+e _{\mathrm{o}}}{(1-e_{\mathrm{o}})^{1/2}}\right]^{2/5}\cdot\left(1-0.3\frac{i}{ \pi}\right),\] which as a function of the spectroscopic parameter $a_{\mathrm{o}}$ can be written: \[\frac{a_{\mathrm{o}}}{a}>2.8\ \frac{q_{\mathrm{out}}}{(1+q_{\mathrm{out}})^{3/ 5}}\cdot\frac{(1+e_{\mathrm{o}})^{2/5}}{(1-e_{\mathrm{o}})^{6/5}}\cdot\left(1- 0.3\frac{i}{\pi}\right).\] For the three triples studied here this relation is satisfied. In the two triples of NGC 2516 the estimated semiaxis ratio is at least 10 times larger than this minimum value for stability, while in the more eccentric system HD 315031 the observed value would be about twice the limit value. In a hierarchical triple system, both the eccentricity of the inner binary and the mutual inclination execute periodic oscillations known as Kozai cycles (Kozai, 1962; Harrington, 1968; Ford et al., 2000). The amplitude of the eccentricity variations are significant when the relative inclination between the orbits is higher than $\arcsin(\sqrt{2/5})$=39$\aas@@fstack{\circ}$2, being maximum when orbital planes are perpendicular to each other. On the other hand, the eccentricity amplitude does not depend on either the mass of the third body or the outer semiaxis. The duration of the Kozai cycle, however, does depend on the outer period and the mass of the distant star. In practice, tidal effects between the companions of the inner binary, rotational deformation of the stars, or relativistic terms can detune the Kozai effect in systems with large outer-to-inner semiaxis ratio. According to Makarov & Eggleton (2009) the Kozai cycling is suppressed if $P_{\mathrm{o}}(\mathrm{yr})\gtrsim[P(\mathrm{days}]^{1.4}$ . In the three analysed systems the outer period is lower than this limit. The period of Kozai cycles is given approximately by (Ford et al., 2000; Mazeh & Shaham, 1979): \[P_{\mathrm{e}}\approx\beta\frac{P}{q_{\mathrm{out}}}\left(\frac{a_{\mathrm{ou} }}{a_{\mathrm{in}}}\right)^{3}(1-e^{2}_{\mathrm{o}})^{3/2},\] (3) where $\beta$ is a factor of order unity that depends on the initial values for mutual inclination, eccentricity, and argument of periastron of the inner orbit. For the two triples of NGC 2516 the periods of the eccentricity oscillations, which depend mainly on orbital periods $P$ and $P_{\mathrm{o}}$, are similar to each other: $\sim$8–14 $\times 10^{3}$yr for CPD$-$60${}^{\circ}$961 and $\sim$8–10 $\times 10^{3}$yr for HD 66137. These values are much shorter than the timescales for tidal effects or nuclear evolution. The short time-scale for the variation of orbital elements and the high mutual inclination ($i_{\mathrm{m}}\ga 52^{\circ}$, see Sect. 3.1) assure that the Kozai mechanism would be working efficiently in CPD$-$60${}^{\circ}$961. The maximum eccentricity, which is a function of orbital inclination (Kinoshita & Nakai, 1999), would be $e_{\mathrm{max}}\ga 0.60$. In the case of HD 66137 the lower limit for the mutual inclination ($\ga 34\degr$) is slightly lower than the critical value, so the occurrence of Kozai cycles in this system is not certain, although probable. In fact, in both triples, mutual inclinations above 80°–85°are still compatible with the spectroscopic parameters. If high-eccentricity configurations take place periodically, the inner orbit could have been (or is being) shrunk by tidal interactions. However, if during the eccentricity cycles the maximum values are not very high ($e_{\mathrm{max}}\la 0.7$), then strong binary interactions (tidal circularization, mass-transfer) would not take place before the end of the main-sequence stage. If the inner orbit is not dynamically modified, in both triples the primary star will overflow its Roche lobe during giant branch ascent before the core He-burning, giving place to case B mass-transfer. HD 315031 has a very eccentric outer subsystem ($e_{\mathrm{o}}$=0.85) and a very close inner subsystem ($P$=1.38 d). The angular momentum exchange between the inner binary and a third body in hierarchical triple systems has been proposed to play a key role in formation of short-periods binaries (Tokovinin, 1997). Statistics of binary and triples among solar-type stars supports this scenario (Tokovinin et al., 2006). Even though the mutual inclination of the inner and outer orbits is high, we consider unlikely that the inner binary has reduced its size through the combination of Kozai cycles and tidal friction. The reason is that the high eccentricity of the outer orbit (which remains constant in Kozai cycles), leaves little room for the size or the original inner orbit. For example, for an inner orbit with $P\sim 6$ d the system would not be stable. A possible explanation is that the present configuration is not the result of the isolated evolution of the triple system, but the outcome of the dynamical decay of a higher order multiple system. In fact, dynamical simulations of small-N clusters produce frequently triples with relatively small period-ratio and high outer eccentricity (Sterzik & Tokovinin, 2002), as is the case of HD 315031. This system is expected to experience mass-transfer during the main-sequence (case A mass-transfer). In fact, the primary Roche lobe radius is currently $\sim$6 R${}_{\odot}$, while the terminal-age main-sequence radius for a star with the primary mass is close to 12 R${}_{\odot}$. According to Hurley’s evolutionary code, the secondary star will become a blue straggler star at about 14 Myr. ## 5 Summary and Conclusions The long-term spectroscopic monitoring of three double-lined spectroscopic binaries members of two open clusters, led to the discovery of the triple nature of these systems. All three systems are hierarchical with a close pair and a third object in a wide orbit. The ratio of the semiaxis of the outer and inner orbits are larger than 50, while the period ratio is larger than 350. Besides the two triples of NGC 2516 analysed in detail in the present paper (CPD$-$60${}^{\circ}$961 and HD 66137), spectroscopic data for the visual-spectroscopic triple CPD$-$60${}^{\circ}$944 of the same cluster are reported. NGC 2516 harbors five known hierarchical triples with inner binaries in the spectroscopic separation range, with periods between 1.7 and a few hundred days. Three of these systems contains at least one chemically peculiar star. The triple HD 66137 (and probably also CPD$-$60${}^{\circ}$961) in NGC 2516 might be experiencing Kozai cycles with inner eccentricity oscillating in timescales of $\sim 10^{4}$yr. However, the orbits of the inner binaries have not been circularized yet. Without a precise knowledge of the mutual inclination between the inner and outer orbit, it is not clear if significant tidal effects are taking place at epochs of high inner eccentricity. If the eccentricity remains lower than $\sim$0.7, strong tidal interaction is not expected until the end of the main sequence. Even without orbit shrinking, the inner binaries of both systems would experience case-B mass transfer, giving origin eventually to exotic objects. On the other hand, HD 315031 contains a short-period massive binary and a third star orbiting in a very eccentric orbit. Due to the proximity of the third star at periastron the inner binary cannot have evolved from a significantly wider orbit. We speculate therefore that the present configuration is the result of a dynamical decay of a non-hierarchical multiple. The inner binary is expected to suffer mass transfer during the main-sequence stage and the secondary star will become a cluster blue straggler star. ## Acknowledgments This work was partially supported by a grant from FONCyT-UNSJ PICTO-2009-0125. ## References * Abt & Levy (1972) Abt H. A., Levy S. G., 1972, ApJ, 172, 355 * Abt & Morgan (1969) Abt H. A., Morgan W. W., 1969, AJ, 74, 813 * Arias et al. (2006) Arias J. I., Barbá R. H., Maíz Apellániz J., Morrell N. I., Rubio M., 2006, MNRAS, 366, 739 * Bailey et al. (2014) Bailey J. D., Landstreet J. D., Bagnulo S., 2014, A&A, 561, A147 * Bate (2012) Bate M. R., 2012, MNRAS, 419, 3115 * Cowley et al. (2006) Cowley C. R., Hubrig S., González G. F., Nuñez N., 2006, A&A, 455, L21 * Cowley et al. (2010) Cowley C. R., Hubrig S., Palmeri P., Quinet P., Biémont É., Wahlgren G. M., Schütz O., González J. F., 2010, MNRAS, 405, 1271 * Cox (1955) Cox A. N., 1955, ApJ, 121, 628 * Dachs (1972) Dachs J., 1972, A&A, 21, 373 * Dachs & Kabus (1989) Dachs J., Kabus H., 1989, A&AS, 78, 25 * Debernardi & North (2001) Debernardi Y., North P., 2001, A&A, 374, 204 * Díaz et al. (2011) Díaz C. G., González J. F., Levato H., Grosso M., 2011, A&A, 531, A143 * Duchêne & Kraus (2013) Duchêne G., Kraus A., 2013, ARA&A, 51, 269 * Ekström et al. (2012) Ekström S., Georgy C., Eggenberger P., Meynet G., Mowlavi N., Wyttenbach A., Granada A., Decressin T., Hirschi R., Frischknecht U., Charbonnel C., Maeder A., 2012, A&A, 537, A146 * Ford et al. (2000) Ford E. B., Kozinsky B., Rasio F. A., 2000, ApJ, 535, 385 * Georgy et al. (2013) Georgy C., Ekström S., Granada A., Meynet G., Mowlavi N., Eggenberger P., Maeder A., 2013, A&A, 553, A24 * Gieseking & Karimie (1982) Gieseking F., Karimie M. T., 1982, A&AS, 49, 497 * González & Lapasset (2000) González J. F., Lapasset E., 2000, AJ, 119, 2296 * González & Lapasset (2003) González J. F., Lapasset E., 2003, A&A, 404, 365 * González & Levato (2006) González J. F., Levato H., 2006, A&A, 448, 283 * Harrington (1968) Harrington R. S., 1968, AJ, 73, 190 * Hartoog (1976) Hartoog M. R., 1976, ApJ, 205, 807 * Hubrig et al. (2012) Hubrig S., González J. F., Ilyin I., Korhonen H., Schöller M., Savanov I., Arlt R., Castelli F., Lo Curto G., Briquet M., Dall T. H., 2012, A&A, 547, A90 * Hubrig et al. (2006) Hubrig S., González J. F., Savanov I., Schöller M., Ageorges N., Cowley C. R., Wolff B., 2006, MNRAS, 371, 1953 * Hurley et al. (2002) Hurley J. R., Tout C. A., Pols O. R., 2002, MNRAS, 329, 897 * Hut (1981) Hut P., 1981, A&A, 99, 126 * Kinoshita & Nakai (1999) Kinoshita H., Nakai H., 1999, Celestial Mechanics and Dynamical Astronomy, 75, 125 * Kozai (1962) Kozai Y., 1962, AJ, 67, 591 * Leigh & Sills (2011) Leigh N., Sills A., 2011, MNRAS, 410, 2370 * Leigh & Geller (2013) Leigh N. W. C., Geller A. M., 2013, MNRAS, 432, 2474 * Levato et al. (2004) Levato H., González J. F., Malaroda S., Grosso M., 2004, in Allen C., Scarfe C., eds, Revista Mexicana de Astronomia y Astrofisica Conference Series Vol. 21 of Revista Mexicana de Astronomia y Astrofisica Conference Series, Spectroscopic binaries in southern open clusters. pp 141–142 * Loktin et al. (1994) Loktin A. V., Matkin N. V., Gerasimenko T. P., 1994, Astronomical and Astrophysical Transactions, 4, 153 * Makaganiuk et al. (2011) Makaganiuk V., Kochukhov O., Piskunov N., Jeffers S. V., Johns-Krull C. M., Keller C. U., Rodenhuis M., Snik F., Stempels H. C., Valenti J. A., 2011, A&A, 529, A160 * Makarov & Eggleton (2009) Makarov V. V., Eggleton P. P., 2009, ApJ, 703, 1760 * Mardling & Aarseth (2001) Mardling R. A., Aarseth S. J., 2001, MNRAS, 321, 398 * Mazeh & Shaham (1979) Mazeh T., Shaham J., 1979, A&A, 77, 145 * Mermilliod et al. (2007) Mermilliod J.-C., Andersen J., Latham D. W., Mayor M., 2007, A&A, 473, 829 * Mermilliod & Mayor (1989) Mermilliod J.-C., Mayor M., 1989, A&A, 219, 125 * Mermilliod et al. (2008) Mermilliod J. C., Mayor M., Udry S., 2008, A&A, 485, 303 * Meynet et al. (1993) Meynet G., Mermilliod J.-C., Maeder A., 1993, A&AS, 98, 477 * Mowlavi et al. (2012) Mowlavi N., Eggenberger P., Meynet G., Ekström S., Georgy C., Maeder A., Charbonnel C., Eyer L., 2012, A&A, 541, A41 * Palacios et al. (2010) Palacios A., Gebran M., Josselin E., Martins F., Plez B., Belmas M., Lèbre A., 2010, A&A, 516, A13 * Pourbaix et al. (2009) Pourbaix D., Tokovinin A. A., Batten A. H., Fekel F. C., Hartkopf W. I., Levato H., Morell N. I., Torres G., Udry S., 2009, VizieR Online Data Catalog, 1, 2020 * Prisinzano et al. (2005) Prisinzano L., Damiani F., Micela G., Sciortino S., 2005, A&A, 430, 941 * Robichon et al. (1999) Robichon N., Arenou F., Mermilliod J.-C., Turon C., 1999, A&A, 345, 471 * Schmidt-Kaler (1982) Schmidt-Kaler T., 1982, in Schaifers K., Voigt H. H., eds, Landolt-Bornstein, New Series, Group VI, Vol. 2b Berlin: Springer * Schöller et al. (2010) Schöller M., Correia S., Hubrig S., Ageorges N., 2010, A&A, 522, A85 * Sterzik & Tokovinin (2002) Sterzik M. F., Tokovinin A. A., 2002, A&A, 384, 1030 * Sung et al. (2002) Sung H., Bessell M. S., Lee B.-W., Lee S.-G., 2002, AJ, 123, 290 * Sung et al. (2000) Sung H., Chun M.-Y., Bessell M. S., 2000, AJ, 120, 333 * Terndrup et al. (2002) Terndrup D. M., Pinsonneault M., Jeffries R. D., Ford A., Stauffer J. R., Sills A., 2002, ApJ, 576, 950 * Tokovinin et al. (2006) Tokovinin A., Thomas S., Sterzik M., Udry S., 2006, A&A, 450, 681 * Tokovinin (1997) Tokovinin A. A., 1997, Astronomy Letters, 23, 727 * Veramendi & González (2010) Veramendi M. E., González J. F., 2010, in Prša A., Zejda M., eds, Binaries - Key to Comprehension of the Universe Vol. 435 of Astronomical Society of the Pacific Conference Series, Orbital Analysis of Two Triple Systems in the Open Cluster NGC 2516. p. 107 * Veramendi & González (2014) Veramendi M. E., González J. F., 2014, A&A, in press * Zahn (1977) Zahn J.-P., 1977, A&A, 57, 383 * Zucker & Mazeh (1994) Zucker S., Mazeh T., 1994, ApJ, 420, 806
1910.10665	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 64835, "num_imgs": 0, "llama3_tokens_count": 21473 }	[]	# Mimicking Networks Parameterized by Connectivity Parinya Chalermsook Aalto University, Finland. Syamantak Das Indraprastha Institute of Information Technology Delhi, India Bundit Laekhanukit Shanghai University of Finance and Economics, China. Daniel Vaz Technische Universität München, Germany. Work done while at Max-Planck-Institut für Informatik, Germany. ###### Abstract Given a graph $G=(V,E)$, capacities $w(e)$ on edges, and a subset of terminals ${\mathcal{T}}\subseteq V:\|{\mathcal{T}}\|=k$, a mimicking network for $(G,{\mathcal{T}})$ is a graph $(H,w^{\prime})$ that contains copies of ${\mathcal{T}}$ and preserves the value of minimum cuts separating any subset $A,B\subseteq{\mathcal{T}}$ of terminals. Mimicking networks of size $2^{2^{k}}$ are known to exist and can be constructed algorithmically, while the best known lower bound is $2^{\Omega(k)}$; therefore, an exponential size is required if one aims at preserving cuts exactly. In this paper, we study mimicking networks that preserve connectivity of the graph exactly up to the value of $c$, where $c$ is a parameter. This notion of mimicking network is sufficient for some applications, as we will elaborate. We first show that a mimicking of size $3^{c}\cdot k$ exists, that is, we can preserve cuts with small capacity using a network of size linear in $k$. Next, we show an algorithm that finds such a mimicking network in time $2^{O(c^{2})}\operatorname{poly}(m)$. ## 1 Introduction _Graph compression_ is a basic information-theoretic and computational question of the following nature: Given an $n$-node graph $G$ (imagine $n$ to be very large), can we compute a “compact” (much smaller) representation of $G$ that preserves information that is important for us? Many such objects have been central in algorithm designs. For instance, graph spanners aim at preserving the distance between nodes in the graphs, while vertex sparsifiers focus on preserving the cut sizes among designated nodes. In this paper, we focus on _mimicking networks_: Given a graph $G=(V,E)$, capacities $w(e)$, and a subset of terminals ${\mathcal{T}}\subseteq V:\|{\mathcal{T}}\|=k$, our goal is to find a smaller capacitated graph $(H,w^{\prime})$ that contains copies of ${\mathcal{T}}$ and preserves the value of minimum cuts separating any subset $A,B\subseteq{\mathcal{T}}$. In this case, we say that $H$ is a mimicking network of $G$. This question was introduced by Hagerup et al. [7] where they presented a mimicking network of size $2^{2^{k}}$ which depends only on $k$ but not $n$ (see also an improvement by Khan and Raghavendra [9]). Krauthgamer and Rika [10] showed that the exponential dependence on $k$ is needed; they presented a lower bound of $2^{\Omega(k)}$. It remains an intriguing open problem to close this gap. While for some applications (e.g. for cut and flow problems), it is desirable to preserve the cut values exactly for every cut, this is not the case for connectivity problems. For instance, if we want to keep only information about $c$-connected subgraphs, it would be enough to treat all cuts of size larger than $c$ as having value exactly $c$. Therefore, we initiate the study of _connectivity-$c$ mimicking network_ and present some application in fast computation in graphs of low treewidth. The following is our main theorem. Theorem 1: _There is a connectivity-$c$ mimicking network of size $3^{c}\cdot c\cdot k$. Moreover, there exists a deterministic algorithm to find such a network in time $O\bigl{(}2^{O(c^{2})}\cdot k^{2}\cdot m^{2}\bigr{)}$._ Our main theorem shows that the exponential lower bound in $k$ does not apply in the “low-connectivity” setting. Related work:For special graph classes, better bounds are known. For instance, Krauthgamer and Rika [10] presented an upper bound of $k^{2}2^{2k}$ for planar graphs, and this was proved to be almost tight [8]. When all terminals lie on the same face, the exponential lower bound does not apply and mimicking networks of size $O(k^{2})$ are known [6]. For bounded treewidth graphs, a the upper bound of $O(k)$ is known [3]. If we allow graph $H$ to only approximate the cut size, such an object is known under the name of cut sparsifiers (introduced in [13, 11]): For $q\geq 1$, a quality-$q$ cut sparsifier preserves cut values between terminals up to a factor of $q$. (in this language, a mimicking network is simply a quality-$1$ cut sparsifier.) Please refer to [5, 1] and references therein for discussions on cut sparsifiers and most recent results. ## 2 Preliminaries For simplicity, we view any capacitated graph $(G,w)$ as a multi-graph $G$ obtained by making $\min(w(e),c)$ copies of parallel edges of $e$. Hence, we will only be dealing with uncapacitated, multi-graphs from now on. For any graph $G$ and two disjoint subsets $A,B\subseteq V(G)$, denote by $E_{G}(A,B)$ the edges with one endpoint in $A$ and the other one in $B$, while, by $\operatorname{mincut}_{G}(A,B)$, the value of the minimum cut separating the sets $A$ from $B$. If either set is empty, the mincut has a value of $0$. We also need the notion of thresholded minimum cut; that is, for an integer $C$, denote by $\operatorname{mincut}_{G}^{c}(A,B)=\min\{\operatorname{mincut}_{G}(A,B),c\}$. Bounded-Connectivity Mimicking Network:For any graph $G$ and terminal set ${\mathcal{T}}$, we say that graph $H$ is a _connectivity-$c$ mimicking network_ for $G$ if the following holds: * $V(H)$ contains at least one copy of each terminal in ${\mathcal{T}}$. * For any pair of disjoint subsets $A,B\subseteq{\mathcal{T}}$ of terminals, the thresholded minimum cuts are preserved in $H$, i.e. \[\operatorname{mincut}_{H}^{c}(A,B)=\operatorname{mincut}^{c}_{G}(A,B)\] We will assume that every terminal has degree exactly one. This assumption can be made by creating $c$ auxiliary vertices for each terminal $t\in{\mathcal{T}}$, and connecting them to $t$; each of these auxiliary vertices becomes a new terminal. Notice that this increases the number of terminals by a factor of $c$, and the bounds for the size of the connectivity-$c$ mimicking network correspondingly. From now on, we assume an input graph $G=(S\cup{\mathcal{T}},E)$ where each node in ${\mathcal{T}}$ has degree one and is attached to the set $S$. We interchangeably refer to terminals as either (1) the nodes in ${\mathcal{T}}$ or (2) the edges connecting ${\mathcal{T}}$ to $S$. For a set of vertices $X\subseteq S$, we use the notation $\partial(X)$ to denote the set of boundary edges $E_{G}(X,V(G)-X)$. By definition, $\partial(S)={\mathcal{T}}$. Notice that for $X\subseteq S$, it could be that $\partial(X)$ contains edges that are not in $\partial(S)$. Contraction-based mimicking network:We are interested in mimicking networks with a specific structure. Let $X\subseteq S$. Denote by $G/X$ the graph $G$ obtained after contracting every edge in $G[X]$. In particular, given an input $G=(S\cup\partial(S),E)$, our mimicking network is always obtained by contracting disjoint subsets $S_{1},\ldots,S_{\ell}\subseteq S$. Well-linked sets:A standard tool for studying flows and cuts is the notion of well-linkedness. We extend this notion so as to capture our bounded connectivity setting. A set $X\subseteq S$ is said to be a _connectivity-$c$ linked set_ in $G$ if for every pair of disjoint sets $A,B\subseteq X$, we have that: \[\|E_{G}(A,B)\|\geq\min(\|\partial(A)\cap\partial(X)\|,\|\partial(B)\cap\partial(X)\| ,c)\] When $X$ is not connectivity-$c$ linked, a cut $(A,B)$ that violates the linkedness condition is referred to as a _violating cut_ for $X$. A low-connectivity linked set is desirable for us since it can be contracted without changing the connectivity, as formalized in the following lemma. Lemma 2: _Let $X$ be a connectivity-$c$ linked set in $G$. Then $G/X$ is a connectivity-$c$ mimicking network for $G$._ Proof.: Let $G^{\prime}=G/X$ be the contracted graph and $v_{X}$ be the contracted vertex in $G^{\prime}$ that is obtained by contracting $G[X]$. Since we do not contract the terminals, it suffices to show that, for any two subsets $X_{A},X_{B}\subseteq{\mathcal{T}}$, we have $\operatorname{mincut}^{c}_{G^{\prime}}(X_{A},X_{B})=\operatorname{mincut}^{c}_ {G}(X_{A},X_{B})$. Starting with $\operatorname{mincut}^{c}_{G^{\prime}}(X_{A},X_{B})\geq\operatorname{mincut}^{ c}_{G}(X_{A},X_{B})$, we can see that all the edges in $G^{\prime}$ are also in $G$, which implies that any cutset in $G^{\prime}$ is also in $G$. We conclude that the size of the minimum cut in $G$ must be at most the size of the minimum cut in $G^{\prime}$, for any pair of terminals sets. In general, we can say that contraction of edges only ever increases connectivity, which implies the above. Let us now show the converse, that is, $\operatorname{mincut}^{c}_{G^{\prime}}(X_{A},X_{B})\leq\operatorname{mincut}^{ c}_{G}(X_{A},X_{B})$. Since we are in the unweighted setting, it is sufficient to consider $\|X_{A}\|,\|X_{B}\|\leq c$. Suppose that $\operatorname{mincut}^{C}_{G^{\prime}}(X_{A},X_{B})=\ell\leq c$. Then there must be $\ell$ disjoint paths connecting $X^{\prime}_{A}\subseteq X_{A}$ to $X^{\prime}_{B}\subseteq X_{B}$ such that $\|X^{\prime}_{A}\|=\|X^{\prime}_{B}\|=\ell$. Denote the set of such paths in $G^{\prime}$ by ${\mathcal{P}}^{\prime}$. We will construct the set of edge-disjoint paths ${\mathcal{P}}$ in $G$ connecting $X^{\prime}_{A}$ to $X^{\prime}_{B}$, thus implying that $\operatorname{mincut}^{c}_{G}(X_{A},X_{B})\geq\ell$. We write ${\mathcal{P}}^{\prime}$ as ${\mathcal{P}}^{\prime}={\mathcal{P}}^{\prime}_{1}\cup{\mathcal{P}}^{\prime}_{2}$ where ${\mathcal{P}}^{\prime}_{1}$ are the paths that do not go through the contracted vertex $v_{X}$. We add the paths in ${\mathcal{P}}^{\prime}_{1}$ to ${\mathcal{P}}$, since they correspond to edge disjoint paths in the original graph $G$. For paths in ${\mathcal{P}}^{\prime}_{2}$, we will need to specify their behavior inside the contracted set $G[X]$. Let $E_{in}\subseteq\partial(X)$ be the set of boundary edges of $X$ that paths in ${\mathcal{P}}^{\prime}_{2}$ use to enter $v_{X}$; analogously, we define $E_{out}\subseteq\partial(X)$. Notice that $\|E_{in}\|=\|E_{out}\|=\|{\mathcal{P}}^{\prime}_{2}\|\leq c$. Since $X$ is connectivity-$c$ linked, there is a collection of disjoint paths ${\mathcal{P}}_{X}$ connecting $E_{in}$ to $E_{out}$. We stitch the three parts of the paths in ${\mathcal{P}}^{\prime}_{2}\cup{\mathcal{P}}_{X}$ together to add to ${\mathcal{P}}$: (1) a subpath of some path $P\in{\mathcal{P}}^{\prime}_{2}$ from a node in $X^{\prime}_{A}$ to $E_{in}$, (2) a path in ${\mathcal{P}}_{X}$ from such edge in $E_{in}$ to an edge in $E_{out}$, and (3) a subpath of some path $Q\in{\mathcal{P}}^{\prime}_{2}$ from such an edge in $E_{out}$ to a node in $X^{\prime}_{B}$. We remark that, even though ${\mathcal{P}}$ contains $\ell$ edge-disjoint paths connecting $X^{\prime}_{A}$ to $X^{\prime}_{B}$, the pairing induced by ${\mathcal{P}}$ and ${\mathcal{P}}^{\prime}$ may be different. ∎ ## 3 Constructing Bounded Connectivity Mimicking Network In this section, we present an algorithm that produces a connectivity-$c$ mimicking network of size at most $3^{c}\cdot k$. We show how to efficiently implement our algorithm in the next section. ### Warmup: Connectivity Two Let $G[S]\cup\partial(S)$ be a graph where $\|\partial(S)\|=k$ and $G[S]$ is connected. In this section, we familiarize readers with the arguments we use by showing how to construct a connectivity-$2$ mimicking network of size at most $2k-2$. Lemma 3: _If $k\geq 3$, set $S$ can be decomposed into at most $k-2$ sets that are connectivity-$2$ linked._ As discussed in the previous section, contracting such clusters gives us the desired mimicking network containing $2w-2$ vertices for all $k\geq 3$. The proof relies on two simple observations: Observation 4: _Consider a graph $G[S]\cup\partial(S)$. Let $X\subseteq S$ be a $2$-connected component in $G[S]$. Then $X$ is connectivity-$2$ linked._ Observation 5: _Consider a graph $G[S]\cup\partial(S)$. Let $uv\in E(G)$ where $u,v\in S$ and $deg_{G}(u)=2$. Then $\{u,v\}$ is connectivity-$2$ linked._ We can now prove the lemma. Proof.: There are two steps. In the first step, we contract all $2$-connected components in $G[S]$ into nodes. The graph $G^{\prime}$ obtained after contraction is a forest. Next, whenever there is an edge $uv\in E(G^{\prime})$ where both $u$ and $v$ are not terminals and $u$ is of degree two, we contract edge $uv$. We are left with the forest $G^{\prime\prime}$ such that leafs correspond to terminals, and each internal node has degree at least $3$, except for when it is only adjacent to terminals. A simple counting argument implies that the number of internal nodes is at most $k-2$: Each leaf receives one token; it sends its token to the parent; each internal node receives at least two tokens from its children, and it keeps one for itself and passes along the rest; in the end, this process leaves at least one token on each internal node and at least $3$ tokens at the root since the root must have degree at least $3$; we conclude that the number of internal nodes is at most $k-2$¹. [FOOTNOTE:1][ENDFOOTNOTE] Notice that each internal node in $G^{\prime\prime}$ corresponds to a connectivity-$2$ linked set $W\subseteq S$. Therefore, this procedure in fact computes a collection of disjoint connectivity-$2$ linked clusters and contracts them. ∎ ### General Case In this section, we generalize the arguments above to show that we can decompose $S$ into a relatively small number of connectivity-$q$ linked clusters. Our recursive procedure takes set $X\subseteq S$ and connectivity parameter $q$ and is supposed to mark connectivity-$q$ linked clusters inside $X$. In particular, the procedure MarkClusters($X$, $q$) performs the following steps: * (Base:) If $X$ is connectivity-$q$ linked, mark $X$ as a tentative cluster and return. Or if $q=2$, then we use the procedure described in Lemma 3 to mark tentative clusters in $X$ and return. * (Inductive:) if $\|\partial(X)\|\leq 2q-1$, call MarkClusters($X$, $q-1$). Else, we find a violating cut $X=A\cup B$ and make recursive calls to MarkClusters($A,q$) and MarkClusters($B,q$). The following observation follows trivially. Observation 6: _When the procedure MarkClusters($X,q$) returns, the clusters in $X$ form a partition of $X$._ It is also easy to see that the procedure always terminates. Lemma 7: _Assume that a violating cut can be found in time $f(k,c){O}(m)$. Then MarkClusters($X,c$) terminates in time $O(f(k,c)){O}(m^{2})$._ Proof.: At each recursive call, either the value of $q$ decreases or the number of edges in the induced subgraph of $X$ decreases. The work done outside the recursive calls involves finding a violating cut in $X$, if there exists one, which takes time $f(k,c)\tilde{O}(\|E_{G}(X,X)\|)$. Hence, the total work done at each level of the recursion tree is $f(k,c)\tilde{O}(m)$, since the subsets on which recursion is called at each level are disjoint. This gives an overall runtime of $f(k,c)\tilde{O}(m^{2})$. ∎ Next, we argue that, when the procedure MarkClusters($X,q$) returns, we have at most $3^{q}\cdot\|X\|$ tentative clusters that are contained in $X$ and each such cluster is connectivity-$q$ linked. After contracting such clusters, we obtain the desired weak mimicking network. Lemma 8: _Let $W\subseteq X$ be a tentative cluster marked when calling MarkClusters($X,q$). Then $W$ is connectivity-$q$ linked._ Proof.: We prove this by induction. The base case when $X$ is connectivity-$q$ linked is obvious. The other base case is when $q=2$, where we can use Lemma 3. For the inductive case, there are two possible subcases. The first subcase is when $\|\partial(X)\|\leq 2q-1$. Let $W\subseteq X$ be a tentative cluster marked by MarkClusters($X,q-1$), so $W$ is connectivity-$(q-1)$ linked. We claim that it is also connectivity-$q$ linked. Indeed, consider any cut $A\cup B=W$ where $\|\partial(A)\cap\partial(W)\|\leq\|\partial(B)\cap\partial(W)\|$. Since $\|\partial(X)\leq 2q-1\|$, then $\|\partial(A)\cap\partial(W)\|\leq q-1$. Thus, $(A,B)$ is a violating cut for connectivity-$q$ if and only if it is a violating cut for connectivity-$(q-1)$. The second subcase follows from definition. ∎ Theorem 9: _The number of tentative clusters (and therefore the size of mimicking network) is at most $3^{c}\|{\mathcal{T}}\|$._ Proof.: We again prove this by induction. Let $N_{q}(k)$ be the maximum number of tentative clusters when running the procedure on $(X,q)$ where $\|\partial(X)\|=k$. We will prove that: \[N_{q}(k) \leq 3^{q-2}(k-2(q-1)) \text{if }k>2(q-1)\] \[N_{q}(k) \leq 3^{q-2} \text{if }k\leq 2(q-1)\] The base case when $X$ is already linked is trivial since we have one cluster in $X$. Notice that if $k=1$, the set $X$ is also linked, so we only need to consider the other base case when $k\geq 2$. The other base case is when we have $q=2$ and $k\geq 2$. In such case, $N_{2}(2)=1$, and when $k\geq 3$, we have that $N_{2}(k)\leq k-2$, from Lemma 3. For the inductive step, if $\|\partial(X)\|\leq 2q-1$, then the procedure simply reduces the value of $q$ to $q-1$, and the induction hypothesis applies. Therefore, we consider the case when $\|\partial(X)\|>2q-1$, and a violating cut is found, resulting in the calls to MarkClusters($A,q$) and MarkClusters($B,q$). Notice that the boundary edges $\partial(A)$ and $\partial(B)$ are the edges from $\partial(X)$ as well as the edges in the violating cut $E_{G}(A,B)$. Assume that $\|E_{G}(A,B)\|=\ell$, $\|\partial(X)\cap\partial(A)\|=k_{1}$ and $\|\partial(X)\cap\partial(B)\|=k_{2}$. Also, assume that $k_{1}\leq k_{2}$. Notice that since this is a violating cut, we have that $k_{1}>\ell$. There are two possibilities. In the first case, suppose that $\|\partial(A)\|=k_{1}+\ell>2(q-1)$. In such case, we have that, \[N_{q}(k) \leq N_{q}(k_{1}+\ell)+N_{q}(k_{2}+\ell)\] \[\leq 3^{q-2}\big{(}k_{1}+\ell-2(q-1)\big{)}+3^{q-2}\big{(}k_{2}+ \ell-2(q-1)\big{)}\] \[\leq 3^{q-2}\big{(}k_{1}+k_{2}-2(q-1)+2\ell-2(q-1)\big{)}\] \[\leq 3^{q-2}\big{(}k-2(q-1)\big{)}\] Otherwise, using the fact that $k_{1}>\ell$, \[N_{q}(k) \leq N_{q}(k_{1}+\ell)+N_{q}(k_{2}+\ell)\] \[\leq N_{q}(2q-1)+N_{q}(k-k_{1}+\ell)\] \[\leq N_{q-1}(2q-1)+N_{q}(k-1)\] \[\leq 3^{q-3}\big{(}2q-1-2(q-2)\big{)}+3^{q-2}\big{(}k-1-2(q-1) \big{)}\] \[\leq 3^{q-3}\cdot 3+3^{q-2}\big{(}k-1-2(q-1)\big{)}\] \[\leq 3^{q-2}\big{(}k-2(q-1)\big{)}\] ∎ ## 4 Efficiently Finding a Violating Cut In order to speed up our computation, it is sufficient to compute a violating cut efficiently in the subroutine MarkClusters($X,q$). In this section, we present an algorithm that either finds a violating cut in $X$ or certifies that $X$ is connectivity-$q$ linked. Observe first that a violating cut can be found in time $k^{O(c)}\operatorname{poly}(n)$ by simply computing all possible minimum cuts separating any disjoint subset of terminals ${\mathcal{T}}_{0},{\mathcal{T}}_{1}\subseteq{\mathcal{T}}$ of size $q\leq c$, whose minimum cut contains less than $q$ edges. We will refer to the two sides of the cuts as _zero side_ and _one side_ respectively. We are, however, aiming at running time $f(c)\operatorname{poly}(n)$, so we cannot afford to do this enumeration to find the “correct” ${\mathcal{T}}_{0}$ and ${\mathcal{T}}_{1}$. Our algorithm will actually solve a more general problem. We say that a cut $(A_{0},A_{1})$ of $G$ is a valid $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut if * $Q_{0}\subseteq A_{0}\setminus{\mathcal{T}}$ and $Q_{1}\subseteq A_{1}\setminus{\mathcal{T}}$. * $\|A_{j}\cap{\mathcal{T}}\|\geq c_{j}$ for $j=0,1$. * $E_{G}(A_{0},A_{1})$ contains at most $\ell$ edges. In words, $Q_{0}$ and $Q_{1}$ are the non-terminals that are “constrained” to be on different sides. The values of $c_{0}$ and $c_{1}$ are the minimum required number of terminals on the sides of $A_{0}$ and $A_{1}$ respectively. Observation 10: _Given a sub-routine that finds a valid $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut in time given by some function $T(m,k,\max(c_{0},c_{1},\ell))$, we can compute a violating cut in $G$ or report that such a cut does not exist in time $O(cT(m,k,c))$._ In the rest of the section, we shall describe an algorithm that finds a valid $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut. Let $c=\max(c_{0},c_{1},\ell)$. Our algorithm has two steps, encapsulated in the following two lemmas. Lemma 11 (Reduction): _There is an algorithm that runs in time $2^{O(c^{2})}\cdot k^{2}\cdot m$, and reduces the problem of finding a valid $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut to at most $2^{O(c^{2})}$ instances of finding valid $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut where $\min(c^{\prime}_{0},c^{\prime}_{1})=0$._ We remark that each such generated instance may have different constrained parameters. The only property we guarantee is the fact that $\min(c^{\prime}_{0},c^{\prime}_{1})=0$, that, is, the fact that there is only a one-sided terminal requirement. Lemma 12 (Base case): _For $\ell\leq c$, there is an algorithm that finds a valid $(Q_{0},Q_{1},0,c,\ell)$-constrained cut (and analogously, $(Q_{0},Q_{1},c,0,\ell)$-constrained) in time $2^{O(c^{2})}\cdot k^{2}\cdot m$._ The following theorem follows in a straightforward manner, since every violating cut is also $(\emptyset,\emptyset,\ell+1,\ell+1,\ell)$-constrained, for some $\ell\in[c-1]$. Theorem 13: _There is an algorithm that runs in time $2^{O(c^{2})}\cdot k^{2}\cdot m$ and either returns a violating cut or reports that such a cut does not exist._ ### The reduction to the base case In this subsection, we prove Lemma 11. The main ingredient for doing so is the following lemma. Lemma 14: _There is a reduction from $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut to solving at most $2^{O(c)}$ instances of finding valid $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut where $(c^{\prime}_{0}+c^{\prime}_{1})<(c_{0}+c_{1})$._ In other words, this lemma allows us to reduce the number of required terminals on at least one of the sides by one. Applying Lemma 14 recursively will allow us to turn an input instance of $(Q_{0},Q_{1},c_{0},c_{1},\ell)$ constrained cut into at most $2^{O(c^{2})}$ instances of the base problem: This follows from the fact that at every recursive call, the value of at least one of $c_{0}$ and $c_{1}$ decreases by at least one. Therefore the depth of the recursion is at most $2c$, and the “degree” of the recursion tree is at most $2^{O(c)}$ as guaranteed by the above lemma. Let $(G,{\mathcal{T}})$ be an input. We now proceed to prove Lemma 14, that is, we show how to compute a $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut in $(G,{\mathcal{T}})$. Our algorithm:Let $(A^{\prime}_{0},A^{\prime}_{1})$ be a minimum cut in $G$ such that $Q_{0}\subseteq A^{\prime}_{0}$ and $Q_{1}\subseteq A^{\prime}_{1}$ and each side contains at least one terminal. This cut can be found by a standard minimum $s$-$t$ cut algorithm. Observe that the value of this cut is at most $\ell$ if there is a valid constrained cut. Such a cut can be used for our recursive approach to solve smaller sub-problems by recursing on $G[A^{\prime}_{i}]$ as follows. Denote by ${\mathcal{T}}_{i}=A^{\prime}_{i}\cap{\mathcal{T}}$ for $i=0,1$. By definition, each set ${\mathcal{T}}_{i}$ is non-empty, and this is crucial for us. If $\|E_{G}(A^{\prime}_{0},A^{\prime}_{1})\|>\ell$, the procedure terminates and reports no valid solution. Or, if $\|{\mathcal{T}}_{i}\|\geq k_{i}$ for all $i=0,1$, then we have found our desired constrained cut. Otherwise, assume that $\|{\mathcal{T}}_{0}\|<k_{0}$ (the other case is symmetric). We create a collection of $2^{O(c)}$ instances of smaller sub-problems as follows. Sub-Instances. First, we guess the “correct” way to partition terminals in ${\mathcal{T}}_{0}$ into ${\mathcal{T}}_{0}={\mathcal{T}}_{0}^{0}\cup{\mathcal{T}}_{0}^{1}$. There are at most $2^{c}$ possible guesses. Second, we guess the “correct” partition of the (non-terminal) boundary vertices in $V(E_{G}(A^{\prime}_{0},A^{\prime}_{1}))-{\mathcal{T}}$, into $B_{0}\cup B_{1}$ where $B_{0}$ and $B_{1}$ are the vertices supposed to be on the zero-side and one-side respectively. Let $\tilde{E}=E_{G}(B_{0},B_{1})$. There are $2^{c}$ possible guesses. Now we will solve sub-problems in $G[A^{\prime}_{0}]$ and $G[A^{\prime}_{1}]$. Notice that $G[A^{\prime}_{0}]$ has small number of terminals, so we could solve it by brute force. For $G[A^{\prime}_{1}]$ we will solve it recursively. Let $E_{0}$ be the minimum cut in $G[A^{\prime}_{0}]$ that separates $S_{0}=Q_{0}\cup(B_{0}\cap A^{\prime}_{0})\cup{\mathcal{T}}_{0}^{0}$ and $T_{0}=(B_{1}\cap A^{\prime}_{0})\cup{\mathcal{T}}_{0}^{1}$. Next, we solve an instance of valid $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut in $G[A^{\prime}_{1}]$ with terminal set ${\mathcal{T}}_{1}$, where $Q^{\prime}_{0}=(B_{0}\cap A^{\prime}_{1})$, $Q^{\prime}_{1}=Q_{1}\cup(B_{1}\cap A^{\prime}_{1})$, $c^{\prime}_{0}=\max(c_{0}-\|{\mathcal{T}}_{0}^{0}\|,0)$, $c^{\prime}_{1}=\max(c_{1}-\|{\mathcal{T}}_{0}^{1}\|,0)$, and $\ell^{\prime}=\ell-\|\tilde{E}\|-\|E_{0}\|$. Let $E_{1}$ be a $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut. Our algorithm outputs $E_{0}\cup E_{1}\cup\tilde{E}$. Analysis.Clearly, $c^{\prime}_{0}+c^{\prime}_{1}<c_{0}+c_{1}$. The following lemma will finish the proof. Lemma 15: _There is a $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut in $(G,{\mathcal{T}})$ if and only if there exist correct guesses $(B_{0},B_{1},{\mathcal{T}}_{0}^{0},{\mathcal{T}}_{0}^{1})$ such that a $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut exists in $(G[A^{\prime}_{1}],{\mathcal{T}}_{1})$._ Proof.: We argue the “if” part. Suppose that there exists such a guess $(B_{0},B_{1},{\mathcal{T}}_{0}^{0},{\mathcal{T}}_{0}^{1})$. We claim that $E_{0}\cup E_{1}\cup\tilde{E}$ is actually a $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut that we are looking for. Observe that the size of the cut is at most $\ell$. We argue that there are two subsets of terminals $\widetilde{{\mathcal{T}}}_{0}$ of size $c_{0}$ and $\widetilde{{\mathcal{T}}}_{1}$ of size $c_{1}$ that are separated after removing $E_{0}\cup E_{1}\cup\tilde{E}$. Let ${\mathcal{T}}_{1}^{0}$ and ${\mathcal{T}}_{1}^{1}$ be the sets of terminals in ${\mathcal{T}}_{1}$ that are on the side of $Q^{\prime}_{0}$ and $Q^{\prime}_{1}$ respectively (in particular, ${\mathcal{T}}_{1}^{0}$ cannot reach $Q^{\prime}_{1}$ in $G[A^{\prime}_{1}]$ after removing $E_{1}$). Notice that $\|{\mathcal{T}}_{0}^{0}\cup{\mathcal{T}}_{1}^{0}\|\geq c_{0}$ and $\|{\mathcal{T}}_{0}^{1}\cup{\mathcal{T}}_{1}^{1}\|\geq c_{1}$. The following claim completes the proof of the “if” part. ∎ Claim 16: $Q_{0}\cup{\mathcal{T}}_{0}^{0}\cup{\mathcal{T}}_{1}^{0}$ _and $Q_{1}\cup{\mathcal{T}}_{0}^{1}\cup{\mathcal{T}}_{1}^{1}$ are not connected in $G$ after removing $\tilde{E}\cup E_{0}\cup E_{1}$._ Proof.: Let us consider a path $P$ from $Q_{0}$ to $Q_{1}$ in $G$; we view it such that the first vertex starts in $Q_{0}$ and so on until the last vertex on the path is in $Q_{1}$. Let $u$ be the last vertex the path from the start lies completely in $G[A^{\prime}_{0}]$ and $v$ be the first vertex such that the path from $v$ to the end lies completely in $G[A^{\prime}_{1}]$. Break path $P$ into $P_{1}P_{2}P_{3}$ where $P_{1}$ is the path from the first vertex to $u$, $P_{2}$ is the path from $u$ to $v$, and $P_{3}$ the path from $v$ to the last vertex of $P$ in $Q_{1}$. If $\|\{u,v\}\cap B_{0}\|=1$, we would be done, since $P_{2}$ contains some edge in $\tilde{E}$. So it must be that (i) $u,v\in B_{0}$ or (ii) $u,v\in B_{1}$. In case (i), we have $v\in Q^{\prime}_{0}$ while the last vertex of $P$ is in $Q_{1}\subseteq Q^{\prime}_{1}$, so path $P_{3}$ is path in $G[A^{\prime}_{1}]$ connecting $Q^{\prime}_{0}$ to $Q^{\prime}_{1}$. Hence, $P_{3}$ contains an edge in $E_{1}$. In case (ii), we have that $u\in T_{0}$, while the first vertex in $P$ is in $Q_{0}\subseteq S_{0}$. Therefore, path $P_{1}$ is a path in $G[A^{\prime}_{0}]$ connecting $S_{0}$ to $T_{0}$, which must be cut by $E_{0}$. Similar analysis can be done when considering path $P$ that connects $Q_{0}$ and ${\mathcal{T}}_{1}^{1}$, or between ${\mathcal{T}}_{0}^{0}$ and $Q_{1}\cup{\mathcal{T}}_{1}^{1}$. The only (somewhat) different case is when path $P$ connects $Q_{0}$ to ${\mathcal{T}}^{1}_{0}$. Assume that $P$ is not completely contained in $G[A^{\prime}_{0}]$, otherwise it’s trivial. Let $u$ be the last vertex on $P$ such that the path from the start to $u$ lies completely inside $G[A^{\prime}_{0}]$; and $v$ be the first vertex on $P$ such that the path from $v$ to the end of $P$ lies completely inside $G[A^{\prime}_{0}]$. Again, we break $P$ into three subpaths $P_{1}P_{2}P_{3}$ similarly to before. If $u\in B_{1}$, we are done because $P_{1}$ would then contain an edge in $E_{0}$; or if $v\in B_{0}$, we are also done since $P_{3}$ would contain an edge in $E_{0}$; therefore, $u\in B_{0}$ and $v\in B_{1}$, so $P_{2}$ must contain an edge in $\tilde{E}$. ∎ To prove the “only if” part, assume that $(A_{0},A_{1})$ is a valid $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut. We argue that there is a choice of guess such that the sub-problem also finds a valid $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut. We define $B_{i}=V(E_{G}(A_{0},A_{1}))\cap A_{i}$ for $i=0,1$, and ${\mathcal{T}}_{0}^{i}={\mathcal{T}}_{0}\cap A_{i}$ for $i=0,1$. With these choices, we have determined the values of $Q^{\prime}_{0}$, $Q^{\prime}_{1}$, $c^{\prime}_{0}$ and $c^{\prime}_{1}$. The following claim will finish the proof. Claim 17: _There exists a cut $E_{0}$ that separates $S_{0}$ and $T_{0}$ in $G[A^{\prime}_{0}]$ and a cut $E_{1}$ that is a $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell^{\prime})$-constrained cut._ Proof.: First, we remark that $\|E_{G}(A_{0},A_{1})\|\leq\ell$ and \[E_{G}(A_{0},A_{1})=E_{G}(B_{0},B_{1})\cup E_{G}(A^{\prime}_{0}\cap A_{0},A^{ \prime}_{0}\cap A_{1})\cup E_{G}(A^{\prime}_{1}\cap A_{0},A^{\prime}_{1}\cap A _{1})\] To complete the proof of the claim, it suffices to show that $E_{G}(A^{\prime}_{0}\cap A_{0},A^{\prime}_{0}\cap A_{1})$ is an $(S_{0},T_{0})$ cut in $G[A^{\prime}_{0}]$ and that $E_{G}(A^{\prime}_{1}\cap A_{0},A^{\prime}_{1}\cap A_{1})$ is a valid constrained cut in $G[A^{\prime}_{1}]$. The first claim is simple: Since $S_{0}\subseteq A_{0}$ and $T_{0}\subseteq A_{1}$, any path from $S_{0}$ to $T_{0}$ in $G[A^{\prime}_{0}]$ must contain an edge in $E_{G}(A^{\prime}_{0}\cap A_{0},A^{\prime}_{0}\cap A_{1})$. The second claim is also simple: (i) $Q^{\prime}_{0}\subseteq A_{0}$ and $Q^{\prime}_{1}\subseteq A_{1}$, so the edge set $E_{G}(A^{\prime}_{1}\cap A_{0},A^{\prime}_{1}\cap A_{1})$ separates $Q^{\prime}_{0}$ and $Q^{\prime}_{1}$, (ii) For $i=0,1$, the number of terminals on the $Q^{\prime}_{i}$-side must be at least $c_{i}-\|{\mathcal{T}}_{0}^{i}\|$, for otherwise this would contradict the fact that $(A_{0},A_{1})$ is a $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut. ∎ Lemma 18: _Let $c=\max\{\ell,c_{0},c_{1}\}$. Then, the algorithm to reduce the problem of finding a $(Q_{0},Q_{1},c_{0},c_{1},\ell)$-constrained cut with $\min{c_{0},c_{1}}>0$ to an instance to find a $(Q^{\prime}_{0},Q^{\prime}_{1},c^{\prime}_{0},c^{\prime}_{1},\ell)$-constrained cut with $\min{c_{1},c_{0}}=0$ terminates in time $2^{O(c^{2})}\cdot k^{2}\cdot{O}(m)$._ Proof.: Lemma 14 implies that the depth of the recursion tree is at the most $2c$ and that each recursive step reduces to solving $2^{O(c)}$ sub-instances. Hence, the total number of nodes in the recursion tree is $2^{O(c^{2})}$. The total runtime outside the recursive calls is dominated by a minimum $s-t$-cut computation. However, we observe that we are only interested in minimum cuts that are of value at the most $c$. Hence, such a cut can be found in time $O(mc)$ using any standard augmentation path based algorithm. Also, recall that we are looking for cuts that have at least one terminal on each side and hence we need to make $k^{2}$ guesses. The total runtime for this procedure is $k^{2}\cdot{O}(mc)$ and we have the lemma. ∎ ### Handling the base case In this subsection, we prove Lemma 12, i.e. we present an algorithm that finds a $(Q_{0},Q_{1},c_{0},0,\ell)$-constrained cut $(A^{\prime}_{0},A^{\prime}_{1})$. We first consider the case of $c_{0}=0$: since neither side of the cut must contain any terminals, we can simply compute a minimum-cut between $Q_{0}$ and $Q_{1}$. If one of these is empty (say $Q_{1}$), we take $A^{\prime}_{0}=V(G)$, $A^{\prime}_{1}=\emptyset$. In any case, let $E_{1}$ be the edges of the cut. Now, there are two possibilities: if $\|E_{1}\|\leq\ell$, our cut is a solution to the subproblem; if $\|E_{1}\|>\ell$, then there is no cut separating $Q_{0}$ from $Q_{1}$ with at most $\ell$ edges, and therefore, there is no valid constrained cut. We can now focus on the case where $c_{0}>0$. We can further assume that $\|{\mathcal{T}}\|\geq c_{0}$; otherwise, there is no feasible solution. For simplification, we also assume that $Q_{0}$ is connected; if it is not, we can add fake edges to make it connected in the run of the algorithm, which we can remove afterwards (these edges will never be cut, since $Q_{0}\subseteq A^{\prime}_{0}$). Important Cuts.The main tool we will be using is the notion of important cuts, introduced by Marx [12] (see [4] and references within for other results using this concept). Definition 19 (Important cut): _Let $G$ be a graph and $X,Y\subseteq V(G)$ be disjoint subsets of vertices of $G$._ _A cut $(S_{X},S_{Y})$, $X\subseteq S_{X}$, $Y\subseteq S_{Y}$ is an important cut if it has (inclusion-wise) maximal reachability (from $X$) among all cuts with at most as many edges. In other words, there is no cut $(S^{\prime}_{X},S^{\prime}_{Y})$, $X\subseteq S^{\prime}_{X}$, $Y\subseteq S^{\prime}_{Y}$, such that $\|E(S^{\prime}_{X},S^{\prime}_{Y})\|\leq\|E(S_{X},S_{Y})\|$ and $S_{X}\subsetneq S^{\prime}_{X}$._ Proposition 20 ([4]): _Let $G$ be an undirected graph and $X,Y\subseteq V(G)$ two disjoint sets of vertices._ _Let $(S_{X},S_{Y})$ be an $(X,Y)$-cut. Then there is an important $(X,Y)$-cut $(S^{\prime}_{X},S^{\prime}_{Y})$ (possibly $S_{X}=S^{\prime}_{X}$) such that $S_{X}\subseteq S^{\prime}_{X}$ and $\|E(S^{\prime}_{X},S^{\prime}_{Y})\|\leq\|E(S_{X},S_{Y})\|$._ Theorem 21 ([4]): _Let $G$ be an undirected graph, $X,Y\subseteq V(G)$ be two disjoint sets of vertices and $k\geq 0$ be an integer. There are at most $4^{k}$ important $(X,Y)$-cuts of size at most $k$._ Proposition 22: _Let $G$ be an undirected graph and $X,Y\subseteq V(G)$ two disjoint sets of vertices, and let $(S_{X},S_{Y})$ be an important $(X,Y)$-cut._ _Then $(S_{X},S_{Y})$ is also an important $(X^{\prime},Y)$-cut for all $X^{\prime}\subseteq S_{X}$._ Proof.: Assume that the statement is false for contradiction. Then there is an important cut $(S^{\prime}_{X},S^{\prime}_{Y})$ for $(X^{\prime},Y)$, with $\|E(S^{\prime}_{X},S^{\prime}_{Y})\|\leq\|E(S_{X},S_{Y})\|$ and $S_{X}\subsetneq S^{\prime}_{X}$ by Proposition 20. But then, $X\subseteq S_{X}\subseteq S^{\prime}_{X}$, which means $(S^{\prime}_{X},S^{\prime}_{Y})$ is an $(X,Y)$-cut, and therefore $(S_{X},S_{Y})$ is not an important cut for $(X,Y)$, which is a contradiction. ∎ Cut profile vectors.In order to make the exposition of the algorithm clearer, we introduce the concept of cut profile vectors. Definition 23: _Let $c,\ell\geq 0$. A cut profile vector is a vector of $\lambda\leq c$ pairs of numbers $\{(\kappa_{i},\ell_{i})\}_{i\in[\lambda]}$, with $\kappa_{i}\in[c-1]$, $\ell_{i}\in[\ell]$, satisfying_ \[c\leq\sum_{i=1}^{\lambda}\kappa_{i}\leq 2c,\quad\quad\sum_{i=1}^{\lambda}\ell_ {i}\leq\ell\] _Each of the pairs $(\kappa_{i},\ell_{i})$ is called a slot of this profile. We say a cut $(A,B)$ is compatible with a slot $(\kappa_{i},\ell_{i})$ if $\|A\cap{\mathcal{T}}\|=\kappa_{i}$ and $\|E(A,B)\|=\ell_{i}$_ Observation 24: _There are at most $c^{c}\cdot\ell^{c}$ different cut profile vectors._ Given a cut $(A,B)$, a cut profile vector represents the bounds for terminals covered and cut edges for each of the components of $G[A]$: there are $\lambda$ connected components, and component $C_{i}$ contains $\kappa_{i}$ terminals and has $\ell_{i}$ cut edges. Our algorithm will enumerate all the possible cut profile vectors and, for each of them, try to find a solution that fits the constraints given by the input. If there is a solution to the problem, there must be a corresponding profile vector, and therefore the algorithm finds a solution. We refer to Figure 1 for a formal description of the algorithm. We will now show that, if there is a solution to the problem, our algorithm always finds a solution. This implies that, when we output “No Valid Solution”, there is no solution. From now on, we assume that there is a solution to the problem. Let $(A,B)$ be a solution that minimizes the number of connected components of $G[A]$. Let ${\mathcal{C}}_{0}$ be the set of all important cuts $(C,\bar{C})$ for $(Q_{0},Q_{1})$, and let ${\mathcal{C}}$ be the set of all important cuts $(C,\bar{C})$ for $(t,Q_{1})$, for any $t\in{\mathcal{T}}$. Lemma 25: _There is a solution $(A^{\prime},B^{\prime})$ such that every connected component $C$ of $G[A^{\prime}]$ corresponds to an important cut $(C,\bar{C})$ in ${\mathcal{C}}_{0}$ or ${\mathcal{C}}$. Furthermore, the number of connected components of $G[A^{\prime}]$ is not greater than that of $G[A]$._ Proof.: We will show an iterative process that turns a solution $(A,B)$ into a solution $(A^{\prime},B)$ where every component corresponds to an important cut as above. Let $C$ be a component of $G[A]$ that does not correspond to an important cut in ${\mathcal{C}}_{0}$ or ${\mathcal{C}}$. Notice that $C$ cannot contain a proper non-empty subset of $Q_{0}$, since $Q_{0}\subseteq A$ and we assume that $Q_{0}$ is connected. If $C$ does not contain any terminals or $Q_{0}$, we move $C$ to $B$ (resulting in the cut $(A\setminus C,B\cup C)$). Since $C$ is a connected component of $G[A]$, all of the neighbors of $C$ are in $B$, and therefore moving $C$ to $B$ does not add any cut edges. In the remaining case, $C$ contains a terminal $t\in{\mathcal{T}}$ or $Q_{0}$, but is not an important cut. By Proposition 20, there is an important cut $(C^{\prime},\bar{C}^{\prime})$, with at most as many cut edges as $(C,\bar{C})$ and $C\subsetneq C^{\prime}$. We can replace $C$ by a component corresponding to an important cut by taking the cut $(A\cup C^{\prime},B\setminus C^{\prime})$. This is still a valid solution, since all terminals contained in $A$ are contained in $A\cup C^{\prime}$, and $Q_{0}\subseteq A$, $Q_{1}\subseteq B\setminus C^{\prime}$. Additionally, the number of edges crossing the cut does not increase: since $\|E(C^{\prime},\bar{C}^{\prime})\|\leq\|E(C,\bar{C})\|$, the number of edges added to the cutset is at most the number of edges removed. We can apply the operations above until the constraints in the lemma are satisfied. Notice that when applying the operations above, the number of components of $G[A]$ never increases and the number of vertices in $A$ connected to terminals in $G[A]$ never decreases. Furthermore, each operation changes at least one of the two quantities above, so this process must finish after a finite number of operations. ∎ Due to Lemma 25, we can assume that every connected component of $G[A]$ corresponds to an important cut. Now, let $C^{}_{0},C^{}_{1},\ldots,C^{}_{\lambda}$ be the connected components of $G[A]$, with $C^{}_{0}\in{\mathcal{C}}_{0}$ being the component that contains $Q_{0}$. Let $\left\{(\kappa_{i},\ell_{i})\right\}_{\lambda}$ be the cut profile vector corresponding to the cuts $(C^{}_{i},\bar{C}^{}_{i})$ for $i\in\left\{1,\ldots,\lambda\right\}$ (excluding $C^{}_{0}$), meaning that $\kappa_{i}$, $\ell_{i}$ are the number of terminals in $C^{}_{i}$ and the number of edges in the cutset, $E(C^{}_{i},\bar{C}^{}_{i})$, respectively. Notice that, if $C^{}_{0}$ or $C^{}_{0}\cup C^{}_{i}$ (for some $i\in[\lambda]$) contain at least $c$ terminals, then we can remove all the other components of $A$. In this case, the algorithm finds $C_{0}\in{\mathcal{C}}_{0}$ or $C_{0}\in{\mathcal{C}}_{0}$, $C_{1}\in{\mathcal{C}}$ by enumeration and returns a valid solution. Otherwise, all the components contain at most $c-1$ terminals each (and thus $A$ induces a slot vector as in Definition 23). Consider the iteration of the algorithm in which the cut profile vector defined above is considered and $C_{0}=C^{}_{0}$. The next part of the algorithm (Lines 25–32) greedily fills the slots with compatible important cuts from ${\mathcal{C}}$, while making sure that each set contains a disjoint set of terminals from the others. Though it seems that our goal at this stage is to obtain a feasible solution, what we intend is to obtain a set of terminals, denoted $S$, such that the set of important cuts for terminals in $S$ contains a feasible solution. For instance, if $S$ contains at least one terminal from each $C^{}_{i}$, $i\in[\lambda]$, our goal is achieved. The above considerations motivate the following definition. We say a slot $i$ is _hit_ by $S$ if $S\cap C^{}_{i}\neq\emptyset$. Notice that slot $i$ is hit by $S$ if $C_{ji}=C^{}_{i}$ for some $j$, since the terminals in $C^{}_{i}$ is added to $S$. Slot $i$ is also hit by $S$ if, for some $j$, we cannot find a set $C_{ji}$, since that implies that $C_{ji}=C^{}_{i}$ is not a valid choice, and thus $S\cap C^{}_{i}\neq\emptyset$. Furthermore, if slot $i$ is not hit by $S$, then $C_{ji}$ is found in all $(c+1)$ rounds. Let ${\mathcal{C}}_{S}\subseteq{\mathcal{C}}$ be the subset of important cuts containing terminals in $S$ (by Proposition 22 these are the important $(t,Q_{1})$-cuts for $t\in S$). It is now suficient to show that there is a sequence of $\lambda$ cuts $\left\{(C_{i},\bar{C}_{i})\right\}_{i\in\lambda}$ from ${\mathcal{C}}_{S}$, such that all $C_{i}$ contain disjoint sets of terminals (also disjoint with the terminals in $C_{0}$), and such that $(C_{i},\bar{C}_{i})$ is compatible with $(\kappa_{i},\ell_{i})$. Taking $C=C_{0}\cup\bigcup_{i=1}^{\lambda}C_{i}$, we obtain a feasible solution $(C,\bar{C})$, which may be different from $(A,B)$, but has the same number of connected components as $G[A]$, and the same numbers of terminals contained in each component and cut edges separating each component from the other side of the cut. Since the algorithm enumerates all such sequences of $\lambda$ sets, it will find either $(C,\bar{C})$ or a different feasible solution. We now define the sets $C_{i}$: if a slot $i$ is hit by $S$ we can set $C_{i}=C^{}_{i}$, since there is $t\in C^{}_{i}\cap S$, and therefore, $(C^{}_{i},\bar{C}^{}_{i})\in{\mathcal{C}}_{S}$. This cut is trivially compatible with $(\kappa_{i},\ell_{i})$, and is disjoint to all other sets defined similarly. Let $I_{H}$ be the set of all $i\in[\lambda]$ such that slot $i$ is hit by $S$, and let $C^{}=\bigcup\left\{C_{i}:i\in I_{H}\right\}$. All that is left to prove is that, for every slot $i$ that is not hit by $S$, there is an important cut $(C_{i},\bar{C}_{i})\in{\mathcal{C}}_{S}$, which contains terminals not in any previous $C_{i^{\prime}}$, $i^{\prime}\leq i$, or in $C^{}$. Notice that we have covered at most $c$ terminals so far (if we covered more, then the components so far are sufficient and therefore the number of components of $G[A]$ is not minimal). Since there are $c+1$ important cuts $(C_{ji},\bar{C}_{ji})$, $j\in[c+1]$, all compatible with slot $i$ and containing disjoint sets of terminals (since the terminals of $C_{ji}$ are added to $S$ after being picked), there must be one set $C_{ji}$ that does not contain any of the at most $c$ terminals in $\bigcup_{i^{\prime}<i}C_{i^{\prime}}$, or in $C^{}$, and we can set $C_{i}=C_{ji}$. Therefore, a sequence $\left\{(C_{i},\bar{C}_{i})\right\}_{i\in\lambda}$ exists, and the algorithm outputs a feasible solution. ## 5 Survivable Network Design on Bounded-Treewidth In this section, we consider the rooted survivable network design problem (rSNDP), in which we are given a graph $G$ with edge-costs $w$, as well as $h$ demands $(v_{i},d_{i})\in V\times\mathbb{Z}$, $i\in[h]$, and a root $r\in V$. The goal is to find a minimum-cost subgraph that contains, for every demand $(v_{i},d_{i})$, $i\in[h]$, $d_{i}$ edge-disjoint paths connecting $r$ to $v_{i}$. We will show how to solve rSNDP optimally in running time $f(c,\operatorname{tw}(G))n$, where $c=\max_{i}d_{i}$ is the maximum demand, and $\operatorname{tw}(G)$ is the treewidth of $G$. Our algorithm uses the ideas of Chalermsook et al. [2] together with connectivity-$c$ mimicking networks. Our running time is $\exp(9^{c}\operatorname{tw}(G)^{2})m$ which is double-exponential in $c$, but only single-exponential in $\operatorname{tw}(G)$ (whereas the result by Chalermsook et al. [2] is double-exponential in both $c$ and $\operatorname{tw}(G)$). Let $(T,X)$ be a tree decomposition of $G$ satisfying the following properties (see [2]): (i)$T$has height $O(\log n)$; (ii)$\|X_{t}\|\leq O(\operatorname{tw}(G))$for all $t\in T$; (iii)every leaf bag contains no edges ($E_{t}=\emptyset$ for all leaves $t\in T$); (iv)every non-leaf has exactly $2$ children. Additionally, we add the root $r$ to every bag $X_{t}$, $t\in{\mathcal{T}}$. The main idea of our algorithm is to assign, to each $t\in T$, a state representing the connectivity of the solution restricted to $X_{t}$. By assigning these states in a manner that is consistent across $T$, we can piece together the solution by looking at the states for each individual node. We will show that representing connectivity by two connectivity-$c$ mimicking networks is sufficient for our purposes, and that we can achieve consistency across $T$ by using very simple local rules between the state for a node $t$ and the states for its children $t_{1}$, $t_{2}$. These rules can be applied using dynamic programming to compute the optimum solution. ### Local Connectivity Rules In this section, we will introduce the local connectivity rules which will allow us to assign states in a consistent manner to the nodes of $T$. The states we will consider consist of two connectivity-$c$ mimicking networks roughly corresponding to the connectivity of the solution in $E(G_{t})$ and $E\setminus E(G_{t})$. We then present some rules that make these states consistent across $T$, while only being enforced for a node and its children. We remark that this notation deviates from the one used by Chalermsook et al. [2], in which states represent connectivity in $E(G_{t})$ and $E$. We do so because taking the union of overlapping mimicking networks would lead to overcounting of the number of edge-disjoint paths. The following local definition of connectivity introduces the desired consistency rules that we can use to define a dynamic program for the problem. Lemma 26 shows that a collection of mimicking networks satisfy the local definition if and only if they represent the connectivity in $G$ with terminals given by a bag. Definition 26* (Local Connectivity): _We say that the pairs of weak-mimicking networks $\{(\mathcal{H}^{\prime}_{t},\mathcal{H}_{t})\}_{t\in V(T)}$ satisfy the local connectivity definition if_ \[\mathcal{H}^{\prime}_{t} \equiv^{c}_{X_{t}}(X_{t},\emptyset)\] _for every leaf node \[t\] of \[T\]_ \[\mathcal{H}_{\operatorname{root}(T)} \equiv^{c}_{X_{t}}(X_{t},\emptyset)\] _and for every internal node $t\in V(T)$ with children $t_{1}$ and $t_{2}$,_ \[\mathcal{H}^{\prime}_{t} \equiv^{c}_{X_{t}}(X_{t},E_{t})\cup\mathcal{H}^{\prime}_{t_{1}} \cup\mathcal{H}^{\prime}_{t_{2}}\] \[\mathcal{H}_{t_{1}} \equiv^{c}_{X_{t}}(X_{t},E_{t})\cup\mathcal{H}^{\prime}_{t_{2}} \cup\mathcal{H}_{t}\] _where $A\equiv^{c}_{X_{t}}B$ means that $\operatorname{mincut}^{c}_{A}(S_{1},S_{2})=\operatorname{mincut}^{c}_{B}(S_{1} ,S_{2})$ for all disjoint sets $S_{1},S_{2}\subseteq X_{t}$._ Lemma 27: _Let $G=(V,E)$ be a graph, and $(\mathcal{T},X)$ its tree decomposition satisfying [the usual properties]. For every $t\in V(T)$, let $(\mathcal{H}^{\prime}_{t},\mathcal{H}_{t})$ be a pair as in Definition 26._ _Then, the pairs $\left\{(\mathcal{H}^{\prime}_{t},\mathcal{H}_{t})\right\}_{t\in T}$ satisfy the local definitions iff for every $t\in V(\mathcal{T})$,_ \[\mathcal{H}^{\prime}_{t} \equiv^{c}_{X_{t}}G_{t}\] \[\mathcal{H}_{t} \equiv^{c}_{X_{t}}G\setminus E(G_{t})\] _where $A\equiv^{c}_{X_{t}}B$ means that $\operatorname{mincut}^{c}_{A}(S_{1},S_{2})=\operatorname{mincut}^{c}_{B}(S_{1} ,S_{2})$ for all disjoint sets $S_{1},S_{2}\subseteq X_{t}$._ Proof.: We start by proving the statement for $\mathcal{H}^{\prime}$ by bottom-up induction, and then the one for $\mathcal{H}$ by top-down induction. We will show that Let $t\in\mathcal{T}$ be a leaf of the tree decomposition. Then $E(G_{t})=\emptyset$, so the statement immediately follows. Consider now an internal node $t$ with children $t_{1}$, $t_{2}$, and assume that the claim follows for $t_{1}$, $t_{2}$. We will define $H^{\prime}_{t}=(X_{t},E_{t})\cup\mathcal{H}^{\prime}_{t_{1}}\cup\mathcal{H}^{ \prime}_{t_{2}}$, and prove that $H^{\prime}_{t}\equiv^{c}_{X_{t}}G_{t}$. That implies that $\mathcal{H}^{\prime}_{t}\equiv^{c}_{X_{t}}H^{\prime}_{t}$ (that is, $\mathcal{H}^{\prime}_{t}$ satisfies the local connectivity definition) if and only if $\mathcal{H}^{\prime}_{t}\equiv^{c}_{X_{t}}G_{t}$. Let $S_{1},S_{2}\subseteq X_{t}$, and $F$ be the cutset for a mincut between $S_{1}$ and $S_{2}$ in $E(G_{t})$. We will use $c_{G}(S_{1},S_{2})=\operatorname{mincut}^{c}_{G}(S_{1},S_{2})$ for conciseness (in this proof only). Then \[c_{G_{t}} (S_{1},S_{2})\] \[=\min(c,\|F\|)\] \[=\min(c,\|F\cap E_{t}\|+\|F\cap E(G_{t_{1}})\|+\|F\cap E(G_{t_{2}})\|)\] \[\geq\min\bigl{(}c,c_{E_{t}}(S_{1},S_{2})+c_{E(G_{t_{1}})}(S_{1} \cap X_{t_{1}},S_{2}\cap X_{t_{1}}\bigr{)}+c_{E(G_{t_{2}})}(S_{1}\cap X_{t_{2} },S_{2}\cap X_{t_{2}}))\] \[=\min\bigl{(}c,c_{E_{t}}(S_{1},S_{2})+c_{\mathcal{H}^{\prime}_{t_ {1}}}(S_{1}\cap X_{t_{1}},S_{2}\cap X_{t_{1}}\bigr{)}+c_{\mathcal{H}^{\prime}_ {t_{2}}}(S_{1}\cap X_{t_{2}},S_{2}\cap X_{t_{2}}))\] \[\geq c_{H^{\prime}_{t}}(S_{1},S_{2})\] The third inequality follows because each of the three terms corresponds to a min-cut between $S_{1}$ and $S_{2}$ for the respective edge sets. The fourth inequality follows by induction hypothesis, and the final one follows by definition of $H^{\prime}_{t}$. For this last step, we crucially use that $X_{t_{1}}\cap X_{t_{2}}\subseteq X_{t}$, which means that any cut for $E_{t}$, $\mathcal{H}^{\prime}_{t_{1}}$ and $\mathcal{H}^{\prime}_{t_{2}}$ uses disjoint edges and disjoint vertices outside of $X_{t}$. These edges provide an upper bound for the cut $c_{H^{\prime}_{t}}$. Analogously, we can prove that $c_{G_{t}}\leq c_{H^{\prime}_{t}}$, by taking a set of edges $F^{\prime}$ of $H^{\prime}_{t}$ that realizes the minimum cut in that graph. The same steps then apply to prove the desired inequality. \[c_{H^{\prime}_{t}}(S_{1},S_{2}) =\min(c,\|F^{\prime}\|)\] \[=\min(c,\|F^{\prime}\cap E_{t}\|+\|F^{\prime}\cap E(\mathcal{H}^{ \prime}_{t_{1}})\|+\|F^{\prime}\cap E(\mathcal{H}^{\prime}_{t_{2}})\|)\] \[\geq\min\bigl{(}c,c_{E_{t}}(S_{1},S_{2})+c_{\mathcal{H}^{\prime}_ {t_{1}}}(S_{1}\cap X_{t_{1}},S_{2}\cap X_{t_{1}}\bigr{)}+c_{\mathcal{H}^{ \prime}_{t_{2}}}(S_{1}\cap X_{t_{2}},S_{2}\cap X_{t_{2}}))\] \[=\min\bigl{(}c,c_{E_{t}}(S_{1},S_{2})+c_{G_{t_{1}}}(S_{1}\cap X_{ t_{1}},S_{2}\cap X_{t_{1}}\bigr{)}+c_{G_{t_{2}}}(S_{1}\cap X_{t_{2}},S_{2}\cap X _{t_{2}}))\] \[\geq c_{G_{t}}(S_{1},S_{2})\] This concludes the first part of the proof. For the second part of the proof, we will use top-down induction. For $t=r$, notice that $E\setminus E(G_{t})=\emptyset$, so the statement follows. We now prove the equality for a node $t_{1}$ with parent $t$ and sibling $t_{2}$. Let $H_{t_{1}}=(X_{t},E_{t})\cup\mathcal{H}^{\prime}_{t_{2}}\cup\mathcal{H}_{t}$, and prove that $H_{t}\equiv^{c}_{X_{t}}G\setminus G_{t}$. This implies the statement, as it shows that $\mathcal{H}_{t}\equiv^{c}_{X_{t}}H_{t}$ (that is, $\mathcal{H}_{t}$ satisfies the local connectivity definition) if and only if $\mathcal{H}_{t}\equiv^{c}_{X_{t}}G\setminus G_{t}$. Let $S_{1},S_{2}\subseteq X_{t_{1}}$, and $F$ be the cutset for a mincut between $S_{1}$ and $S_{2}$ in $E\setminus E(G_{t_{1}})$. Then \[c_{E\setminus E(G_{t_{1}})}(S_{1},S_{2}) =\min(c,\|F\|)\] \[=\min(c,\|F\cap E_{t}\|+\|F\cap(E\setminus E(G_{t}))\|+\|F\cap E(G_{t_ {2}})\|)\] \[\geq\begin{aligned} \min(c,c_{E_{t}}(S_{1}\cap X_{t} ,S_{2}\cap X_{t})&+c_{E\setminus E(G_{t})}(S_{1}\cap X_{t},S_{2} \cap X_{t})\\ &+c_{E(G_{t_{2}})}(S_{1}\cap X_{t_{2}},S_{2}\cap X_{t_{2}}))\end{aligned}\] \[=\begin{aligned} \min(c,c_{E_{t}}(S_{1}\cap X_{t},S_ {2}\cap X_{t})&+c_{\mathcal{H}_{t}}(S_{1}\cap X_{t},S_{2}\cap X_{ t})\\ &+c_{\mathcal{H}^{\prime}_{t_{2}}}(S_{1}\cap X_{t_{2}},S_{2}\cap X _{t_{2}}))\end{aligned}\] \[\geq c_{H_{t_{1}}}(S_{1},S_{2})\] Similarly to the proof above, we use the fact that $F\cap E_{t}$, $F\cap(E\setminus E(G_{t}))$, $F\cap E(G_{t_{2}})$ are cuts in the subgraphs $E_{t}$, $G\setminus E(G_{t})$, $E(G_{t_{2}})$ respectively. The last step follows from the fact that the three terms correspond to cuts in $E_{t}$, $\mathcal{H}_{t}$ and $\mathcal{H}^{\prime}_{t_{2}}$, and therefore their union forms a cut in $\mathcal{H}_{t}\cup E_{t}\cup\mathcal{H}^{\prime}_{t_{2}}$. Since $H_{t_{1}}\equiv^{c}_{X_{t_{1}}}\mathcal{H}_{t}\cup E_{t}\cup\mathcal{H}^{ \prime}_{t_{2}}$, the inequality follows. The converse follows similarly: \[c_{H_{t_{1}}}(S_{1},S_{2}) =\min(c,\|F\|)\] \[=\min(c,\|F\cap E_{t}\|+\|F\cap E(\mathcal{H}_{t})\|+\|F\cap E( \mathcal{H}^{\prime}_{t_{2}})\|)\] \[\geq\begin{aligned} \min(c,c_{E_{t}}(S_{1}\cap X_{t} ,S_{2}\cap X_{t})&+c_{\mathcal{H}_{t}}(S_{1}\cap X_{t},S_{2}\cap X _{t})\\ &+c_{\mathcal{H}^{\prime}_{t_{2}}}(S_{1}\cap X_{t_{2}},S_{2}\cap X _{t_{2}}))\end{aligned}\] \[=\begin{aligned} \min(c,c_{E_{t}}(S_{1}\cap X_{t},S_ {2}\cap X_{t})&+c_{E\setminus E(G_{t})}(S_{1}\cap X_{t},S_{2}\cap X _{t})\\ &+c_{E(G_{t_{2}})}(S_{1}\cap X_{t_{2}},S_{2}\cap X_{t_{2}}))\end{aligned}\] \[\geq c_{E\setminus E(G_{t_{1}})}(S_{1},S_{2})\] This completes the proof. ∎ ### Dynamic Program for rSNDP In this section, we present an algorithm for rSNDP on bounded-treewidth graphs, which uses dynamic programming to compute a solution bottom-up. Our goal is to assign two mimicking networks $\mathcal{H}^{\prime}_{t}$, $\mathcal{H}_{t}$ to each node $t\in T$, corresponding to the connectivity of the solution in $E(G_{t})$ and $E\setminus E(G_{t})$. We argue that any solution for $G_{t}$, $t\in T$ that is compatible with a state $(\mathcal{H}^{\prime}_{t},\mathcal{H}_{t})$ can be interchangeably used, which implies that the dynamic program will obtain the minimum-cost solution. We define a dynamic programming table $D$, with entries $D[t,\mathcal{H}^{\prime},\mathcal{H}]$, $t\in T$, $\mathcal{H}^{\prime}$, $\mathcal{H}$ connectivity-$c$ mimicking networks with terminal set $X_{t}$. The entry $D[t,\mathcal{H}^{\prime},\mathcal{H}]$ represents the minimum cost of a solution $F$ that is consistent with $\mathcal{H}^{\prime}$ (i.e. $F\equiv^{c}_{X_{t}}\mathcal{H}^{\prime}$), such that $F\cup\mathcal{H}_{t}$ satisfies all the demands contained in $G_{t}$. We compute $D[t,\mathcal{H}^{\prime},\mathcal{H}]$ as follows: * For any leaf $t$, set $D[t,\emptyset,\mathcal{H}]=0$ and $D[t,\mathcal{H}^{\prime},\mathcal{H}]=+\infty$ for $\mathcal{H}^{\prime}\neq\emptyset$; * For the root node $\operatorname{root}(T)$, set $D[\operatorname{root}(T),\mathcal{H}^{\prime},\mathcal{H}]=+\infty$ if $\mathcal{H}\neq\emptyset$; * For any demand $(v_{i},d_{i})$, and $t\in T$ such that $v_{i}\in X_{t}$, set $D[t,\mathcal{H}^{\prime},\mathcal{H}]=+\infty$ if $\mathcal{H}^{\prime}\cup\mathcal{H}$ contain fewer than $d_{i}$ edge-disjoint paths connecting $r$ to $v_{i}$. For all other entries of $T$, compute it recursively as: \[D[t,\mathcal{H}^{\prime},\mathcal{H}]=\min\Big{\{}w(Y)+D[t_{1}, \mathcal{H}^{\prime}_{1},\mathcal{H}_{1}] +D[t_{2},\mathcal{H}^{\prime}_{2},\mathcal{H}_{2}]:Y\subseteq E_{ t},\] \[\mathcal{H}^{\prime}\equiv^{c}_{X_{t}}Y\cup\mathcal{H}^{\prime}_{ 1}\cup\mathcal{H}^{\prime}_{2},\] We now want to prove that the dynamic program is feasible, that is, that the entries $D[\operatorname{root}(T),\mathcal{H}^{\prime},\emptyset]$ correspond to feasible solutions; and that it is optimal, meaning that we will obtain the optimum solution to the problem. To prove that the dynamic program is feasible, notice that, by definition, any solution obtained induces a choice of $Y_{t}$, $\mathcal{H}^{\prime}_{t}$, $\mathcal{H}_{t}$ for each $t\in T$. Let $Y=\cup_{t\in T}$. The recursion formula of the dynamic program implies that the pairs $\left\{(\mathcal{H}^{\prime}_{t},\mathcal{H}_{t})\right\}_{t\in T}$ satisfy the local connectivity definition with regard to the graph $(V,Y)$. By Lemma 27, this implies that \[\mathcal{H}^{\prime}_{t}\equiv^{c}_{X_{t}}G_{t}[Y],\mathcal{H}_{t }\equiv^{c}_{X_{t}}G[Y]\setminus E(G_{t}),\] and hence, $\mathcal{H}^{\prime}_{t}\cup\mathcal{H}_{t}\equiv^{c}_{X_{t}}G[Y]$. Let $(v_{i},d_{i})$ be a demand and $t\in T$ be a node such that $v_{i}\in X_{t}$. Since we know that $\mathcal{H}^{\prime}_{t}\cup\mathcal{H}_{t}$ contains $d_{i}$ edge-disjoint paths from $r$ to $v_{i}$ (otherwise $D[t,\mathcal{H}^{\prime}_{t},\mathcal{H}_{t}]=+\infty$), then we know that the minimum cut separating $r$ from $v_{i}$ has at least $d_{i}$ edges, which implies that $Y$ must also contain $d_{i}$ edge-disjoint paths connecting $r$ and $v_{i}$. For the converse, we will prove that any feasible solution $F$ can be captured by the dynamic program. Given $F$, it is sufficient to take $\mathcal{H}^{\prime}_{t}$, $\mathcal{H}_{t}$ to be connectivity-$c$ mimicking networks for $G_{t}[F]$, $G[F]\setminus E(G_{t})$, respectively. By Lemma 27 (applied to graph $(V,F)$), we know that $\left\{(\mathcal{H}^{\prime}_{t},\mathcal{H}_{t})\right\}_{t\in T}$ satisfy the local connectivity definition for $(V,F)$, and therefore $D[t,\mathcal{H}^{\prime}_{t},\mathcal{H}_{t}]$ can be computed recursively from $D[t_{1},\mathcal{H}^{\prime}_{t_{1}},\mathcal{H}_{t_{1}}]$, $D[t_{2},\mathcal{H}^{\prime}_{t_{2}},\mathcal{H}_{t_{2}}]$, $Y_{t}=F\cap E_{t}$. Let $(v_{i},d_{i})$ be a demand and $t\in T$ be a node such that $v_{i}\in X_{t}$. Since $F$ is a feasible solution, it contains $d_{i}$ edge-disjoint paths from $r$ to $v_{i}$, and therefore $\operatorname{mincut}^{c}_{F}(\left\{r\right\},\left\{v_{i}\right\})\geq d_{i}$. This implies that , and thus $\mathcal{H}^{\prime}_{t}\cup\mathcal{H}_{t}$ contains $d_{i}$ edge-disjoint paths from $r$ to $v_{i}$ (and is a valid entry of $T$). We conclude that the dynamic program above computes an optimum solution for rSNDP. By Theorem 1, there is a weak-mimicking network for any graph, with $w$ terminals, of size $O(3^{c}cw)$. Since the number of edges is at most the square of the number of vertices, there are $O(\exp(9^{c}c^{2}w^{2}))$ possible states for each node, which completes the proof. [FIGURE:S5.F1][ENDFIGURE] Acknowledgement:Parinya Chalermsook has been supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 759557) and by Academy of Finland Research Fellows, under grant number 310415. Bundit Laekhanukit has been partially supported by the 1000-talent award by the Chinese government. ## References * [1] A. Andoni, A. Gupta, and R. Krauthgamer. Towards (1+ $\varepsilon$)-approximate flow sparsifiers. In _Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms_, pages 279–293. Society for Industrial and Applied Mathematics, 2014. * [2] P. Chalermsook, S. Das, G. Even, B. Laekhanukit, and D. Vaz. Survivable network design for group connectivity in low-treewidth graphs. In E. Blais, K. Jansen, J. D. P. Rolim, and D. Steurer, editors, _Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA_, volume 116 of _LIPIcs_, pages 8:1–8:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. * [3] S. Chaudhuri, K. Subrahmanyam, F. Wagner, and C. D. Zaroliagis. Computing mimicking networks. _Algorithmica_, 26(1):31–49, 2000. * [4] M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. _Parameterized Algorithms_. Springer, 2015. * [5] M. Englert, A. Gupta, R. Krauthgamer, H. Racke, I. Talgam-Cohen, and K. Talwar. Vertex sparsifiers: New results from old techniques. _SIAM Journal on Computing_, 43(4):1239–1262, 2014. * [6] G. Goranci, M. Henzinger, and P. Peng. Improved guarantees for vertex sparsification in planar graphs. In _25th Annual European Symposium on Algorithms (ESA 2017)_. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. * [7] T. Hagerup, J. Katajainen, N. Nishimura, and P. Ragde. Characterizing multiterminal flow networks and computing flows in networks of small treewidth. _Journal of Computer and System Sciences_, 57(3):366–375, 1998. * [8] N. Karpov, M. Pilipczuk, and A. Zych-Pawlewicz. An exponential lower bound for cut sparsifiers in planar graphs. _Algorithmica_, pages 1–14, 2018. * [9] A. Khan and P. Raghavendra. On mimicking networks representing minimum terminal cuts. _Information Processing Letters_, 114(7):365–371, 2014. * [10] R. Krauthgamer and I. Rika. Mimicking networks and succinct representations of terminal cuts. In _Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms_, pages 1789–1799. SIAM, 2013. * [11] F. T. Leighton and A. Moitra. Extensions and limits to vertex sparsification. In _Proceedings of the forty-second ACM symposium on Theory of computing_, pages 47–56. ACM, 2010. * [12] D. Marx. Parameterized graph separation problems. _Theor. Comput. Sci._, 351(3):394–406, 2006. * [13] A. Moitra. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In _2009 50th Annual IEEE Symposium on Foundations of Computer Science_, pages 3–12. IEEE, 2009.
0710.3364	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31858, "num_imgs": 0, "llama3_tokens_count": 10474 }	[]	# Spectral properties of the nonspherically decaying radiation generated by a rotating superluminal source Houshang Ardavan University of Cambridge Arzhang Ardavan University of Oxford John Singleton jsingle@lanl.gov Joseph Fasel Andrea Schmidt Los Alamos National Laboratory ###### Abstract The focusing of the radiation generated by a polarization current with a superluminally rotating distribution pattern is of a higher order in the plane of rotation than in other directions. Consequently, our previously published asymptotic approximation to the value of this field outside the equatorial plane breaks down as the line of sight approaches a direction normal to the rotation axis, _i.e.,_ is nonuniform with respect to the polar angle. Here we employ an alternative asymptotic expansion to show that, though having a rate of decay with frequency ($\mu$) that is by a factor of order $\mu^{2/3}$ slower, the equatorial radiation field has the same dependence on distance as the nonspherically decaying component of the generated field in other directions: it, too, diminishes as the inverse square root of the distance from its source. We also briefly discuss the relevance of these results to the giant pulses received from pulsars: the focused, nonspherically decaying pulses that arise from a superluminal polarization current in a highly magnetized plasma have a power-law spectrum (_i.e.,_ a flux density $S\propto\mu^{\alpha}$) whose index ($\alpha$) is given by one of the values $-2/3$, $-2$, $-8/3$, or $-4$. ## I Introduction Radiation by polarization currents whose distribution patterns move faster than light _in vacuo_ has been the subject of several theoretical and experimental studies in recent years Bessarab et al. (2004); Ardavan et al. (2); Bessarab et al. (2006); Bolotovskii and Serov (2006); Bolotovskii and Bykov (1990); Ardavan (1998); Ardavan et al. (7); Ardavan et al. (2007). When the motion of its source is accelerated, this radiation exhibits features that are not shared by any other known emission. In particular, the radiation from a rotating superluminal source consists, in certain directions, of a collection of subbeams whose azimuthal and polar widths narrow (as $R_{\rm P}^{-3}$ and $R_{\rm P}^{-1}$, respectively) with distance $R_{\rm P}$ from the source Ardavan et al. (2007). Being composed of tightly focused wave packets that are constantly dispersed and reconstructed out of other waves, these subbeams neither diffract nor decay in the same way as conventional radiation beams. The field strength within each subbeam diminishes as $R_{\rm P}^{-1/2}$, instead of $R_{\rm P}^{-1}$, with increasing $R_{\rm P}$Ardavan (1998); Ardavan et al. (7); Ardavan et al. (2007). In earlier treatments Ardavan et al. (7); Ardavan et al. (2007), we evaluated the field of a superluminally rotating extended source by superposing the fields of its constituent volume elements, _i.e.,_ by convolving its density with the familiar Liénard-Wiechert field of a rotating point source. This Liénard-Wiechert field is described by an expression essentially identical to that which is encountered in the analysis of synchrotron radiation, except that its value at any given observation time receives contributions from more than one retarded time. The multivalued nature of the retarded time gives rise to the formation of caustics. The wave fronts emitted by each constituent volume element of a superluminally moving accelerated source possess a cusped envelope on which the field is infinitely strong (see Figs. 1 and 4 of Ref. Ardavan et al. (2007)). Correspondingly, the Green’s function for the problem is nonintegrably singular for those source elements that approach the observer along the radiation direction with the speed of light and zero acceleration at the retarded time (see Fig. 3 of Ref. Ardavan et al. (2007)): the cusp of the envelope of wave fronts emanating from each such element is a spiraling curve extending into the far zone that passes through the position of the observer. When the source oscillates at the same time as rotating, the Hadamard finite part of the divergent integral that results from convolving the Green’s function with the source density has a rapidly oscillating kernel for a far-field observation point. The stationary points of the phase of this kernel turn out to have different orders depending on whether the observer is located in or out of the equatorial plane. To reduce the complications posed by the higher-order stationary points of this phase, we restricted the asymptotic evaluation of the radiation integral thus obtained in Refs. Ardavan et al. (7); Ardavan et al. (2007) to observation points outside the plane of rotation, _i.e.,_ to spherical polar angles $\theta_{\rm P}$ that do not equal $\pi/2$. The purpose of this paper is to evaluate the field of a superluminally rotating extended source also for the smaller class of observers at polar coordinate $\theta_{\rm P}=\pi/2$ and to obtain, thereby, a more global description of the nonspherically decaying radiation that is generated by such a source. The asymptotic expansion presented in Refs. Ardavan et al. (7); Ardavan et al. (2007) breaks down in the case of a subbeam that is perpendicular to the rotation axis because there is a higher-order focusing associated with the waves emitted by those source elements whose actual speeds (rather than the line-of-sight components of their speeds) equal the speed of light as they approach the observer with zero acceleration. Here, we present a brief account of the background material on the radiation field of a rotating superluminal source in Section 2, and the asymptotic evaluation of this field for an equatorial observer in Section 3. In Section 4, we give a description of the spectral properties of the nonspherically decaying component of this radiation in the light of the present results and those obtained in Refs. Ardavan et al. (7); Ardavan et al. (2007), and discuss the relevance of these properties to pulsar observations. ## II Background: radiation field of a rotating superluminal source We base our analysis on a generic superluminal source distribution Ardavan et al. (7); Ardavan et al. (2007), which has been created in the laboratory Ardavan et al. (2). This source comprises a polarization current density ${\bf j}=\partial{\bf P}/\partial t$ for which \[P_{r,\varphi,z}(r,\varphi,z,t)=s_{r,\varphi,z}(r,z)\cos(m{\hat{\varphi}})\cos( \Omega t),\qquad-\pi<{\hat{\varphi}}\leq\pi,\] (1) with \[{\hat{\varphi}}\equiv\varphi-\omega t,\] (2) where $P_{r,\varphi,z}$ are the components of the polarization ${\bf P}$ in a cylindrical coordinate system based on the axis of rotation, ${\bf s}(r,z)$ is an arbitrary vector that vanishes outside a finite region of the $(r,z)$ space, and $m$ is a positive integer. For fixed $t$, the azimuthal dependence of the density (1) along each circle of radius $r$ within the source is the same as that of a sinusoidal wave train, of wavelength $2\pi r/m$, whose $m$ cycles fit around the circumference of the circle smoothly. As time elapses, this wave train both propagates around each circle with the velocity $r\omega$ and oscillates in its amplitude with the frequency $\Omega$. This is a generic source: one can construct any distribution with a uniformly rotating pattern $P_{r,\varphi,z}(r,{\hat{\varphi}},z)$ by the superposition over $m$ of terms of the form $s_{r,\varphi,z}(r,z,m)\cos(m{\hat{\varphi}})$. The electromagnetic fields \[{\bf E}=-{\bf\nabla}_{\rm P}A^{0}-\partial{\bf A}/\partial(ct_{\rm P}),\quad{ \bf B}={\bf\nabla}_{\rm P}{\bf\times A}\] (3) arising from such a source are given, in the absence of boundaries, by the following classical expression for the retarded four-potential: \[A^{\mu}({\bf x}_{\rm P},t_{\rm P})=c^{-1}\int{\rm d}^{3}x{\rm d}t\,j^{\mu}({ \bf x},t)\delta(t_{\rm P}-t-R/c)/R,\quad\mu=0,\cdots,3.\] (4) Here, $({\bf x}_{\rm P},t_{\rm P})=(r_{\rm P},\varphi_{\rm P},z_{\rm P},t_{\rm P})$ and $({\bf x},t)=(r,\varphi,z,t)$ are the space-time coordinates of the observation point and the source points, respectively, $R$ stands for the magnitude of ${\bf R}\equiv{\bf x}_{\rm P}-{\bf x}$, and $\mu=1,2,3$ designate the spatial components, ${\bf A}$ and ${\bf j}$, of $A^{\mu}$ and $j^{\mu}$ in a Cartesian coordinate system. To find the retarded field that follows from Eq. (4) for the source described in Eq. (1), we first calculated in Ref. Ardavan et al. (7) the Liénard-Wiechert field arising from a circularly moving point source with a speed $r\omega>c$, _i.e.,_ a generalization of the synchrotron radiation to the superluminal regime. We then evaluated the integral representing the retarded field of the extended source (1) by superposing the fields generated by the constituent volume elements of this source, _i.e.,_ by using the generalization of the synchrotron field as the Green’s function for the problem. In the superluminal regime, this Green’s function has extended singularities arising from the constructive intereference of the emitted waves on the envelope of wave fronts and its cusp. Labeling each element of the extended source (1) by its Lagrangian coordinate ${\hat{\varphi}}$ and performing the integration with respect to $t$ and ${\hat{\varphi}}$ (or equivalently $\varphi$ and ${\hat{\varphi}}$) in the multiple integral implied by Eqs. (1)–(4), we showed in Ref. Ardavan et al. (7) that the resulting expression for the radiation field ${\bf B}$ (or ${\bf E}$) consists of two parts: one that decays spherically (as $R_{\rm P}^{-1}$, as in a conventional radiation field) and another, ${\bf B}^{\rm ns}$ (with ${\bf E}^{\rm ns}={\hat{\bf n}}{\bf\times}{\bf B}^{\rm ns}$), that decays nonspherically (as $R_{\rm P}^{-1/2}$) within the conical shell $\arcsin(1/{\hat{r}}_{>})\leq\theta_{\rm P}\leq\arcsin(1/{\hat{r}}_{<})$ in the far zone. Here, $(R_{\rm P},\theta_{\rm P},\varphi_{\rm P})$ are the spherical polar coordinates of the observation point ${\rm P}$, ${\hat{r}}$ stands for $r\omega/c$, ${\hat{\bf n}}\equiv{\bf R}/R$ is a unit vector in the radiation direction, and ${\hat{r}}_{<}>1$ and ${\hat{r}}_{>}>{\hat{r}}_{<}$ denote the radial boundaries of the support of the source density ${\bf s}$. The expression for the nonspherically decaying component of the field within this conical shell, in the far zone, is \[\begin{split}{\bf B}^{\rm ns}\simeq&-{4 \over 3}{\rm i}\exp[{\rm i}(\Omega/\omega)(\varphi_{\rm P}+3\pi/2)]\sum_{\mu= \mu_{\pm}}\mu\exp(-{\rm i}\mu{\hat{\varphi}}_{\rm P})\\ &\times\sum_{j=1}^{3}{\bar{q}}_{j}\int_{\Delta\geq 0}{\hat{r}}{ \rm d}{\hat{r}}\,{\rm d}{\hat{z}}\,\Delta^{-1/2}{\bf u}_{j}\exp(-{\rm i}\mu \phi_{-}),\end{split}\] (5) where $\mu_{\pm}\equiv(\Omega/\omega)\pm m$, ${\hat{\varphi}}_{\rm P}\equiv\varphi_{\rm P}-\omega t_{\rm P}$, \[{\bar{q}}_{j}\equiv(1\qquad-{\rm i}\Omega/\omega\qquad{\rm i}\Omega/\omega),\] (6) \[{\bf u}_{1}\equiv s_{r}\cos\theta_{\rm P}{\hat{\bf e}}_{\parallel}+s_{\varphi} {\hat{\bf e}}_{\perp},\quad{\bf u}_{2}\equiv-s_{\varphi}\cos\theta_{\rm P}{ \hat{\bf e}}_{\parallel}+s_{r}{\hat{\bf e}}_{\perp},\quad{\bf u}_{3}\equiv-s_{ z}\sin\theta_{\rm P}{\hat{\bf e}}_{\parallel},\] (7) \[\Delta\equiv({\hat{r}}_{\rm P}^{2}-1)({\hat{r}}^{2}-1)-({\hat{z}}-{\hat{z}}_{ \rm P})^{2},\] (8) \[\phi_{\pm}\equiv{\hat{R}}_{\pm}+\varphi_{\pm}-\varphi_{\rm P},\] (9) \[\varphi_{\pm}=\varphi_{\rm P}+2\pi-\arccos[(1\mp\Delta^{1/2})/({\hat{r}}{\hat{ r}}_{\rm P})],\] (10) and \[{\hat{R}}_{\pm}\equiv[({\hat{z}}-{\hat{z}}_{\rm P})^{2}+{\hat{r}}^{2}+{\hat{r} }_{\rm P}^{2}-2(1\mp\Delta^{1/2})]^{1/2};\] (11) see Eq. (47) of Ref. Ardavan et al. (7). In this expression, $({\hat{r}},{\hat{z}};{\hat{r}}_{\rm P},{\hat{z}}_{\rm P})$ stand for $(r\omega/c,z\omega/c;r_{\rm P}\omega/c,z_{\rm P}\omega/c)$, and ${\hat{\bf e}}_{\parallel}\equiv{\hat{\bf e}}_{z}\times{\hat{\bf n}}/\|{\hat{\bf e }}_{z}\times{\hat{\bf n}}\|$ (which is parallel to the plane of rotation) and ${\hat{\bf e}}_{\perp}\equiv{\hat{\bf n}}{\bf\times}{\hat{\bf e}}_{\parallel}$ comprise a pair of unit vectors normal to the radiation direction ${\hat{\bf n}}$ (${\hat{\bf e}}_{z}$ is the base vector associated with the coordinate $z$). The domain of integration consists of the part of the source distribution ${\bf s}(r,z)$ that falls within $\Delta\geq 0$ (see Fig. 4 of Ref. Ardavan et al. (2007)). Both derivatives, $\partial\phi_{-}/\partial{\hat{r}}$ and $\partial\phi_{-}/\partial{\hat{z}}$, of the function $\phi_{-}({\hat{r}},{\hat{z}})$ that appears in the phase of the integrand in Eq. (5) vanish at the point ${\hat{r}}=1$, ${\hat{z}}={\hat{z}}_{\rm P}$, where the cusp curve of the bifurcation surface is tangent to the light cylinder (see Figs. 3 and 4 of Ref. Ardavan et al. (2007)). However, $\partial^{2}\phi_{-}/\partial{\hat{r}}^{2}$ diverges at this point, so that neither the phase nor the amplitude of the kernel of the integral in Eq. (5) are analytic at ${\hat{r}}=1,{\hat{z}}={\hat{z}}_{\rm P}$. Only for an observer who is located outside the plane of rotation, _i.e.,_ whose coordinate $z_{\rm P}$ does not match the coordinate $z$ of any source element, is the function $\phi_{-}(r,z)$ analytic throughout the domain of integration. To take advantage of the simplifications offered by the analyticity of $\phi_{-}$ as a function of $r$, we restricted the analyses in Refs. Ardavan et al. (7); Ardavan et al. (2007) to observation points for which $\theta_{\rm P}\neq\pi/2$. In the calculation that follows, we find an asymptotic approximation to the integral \[{\cal I}\equiv\int_{\Delta\geq 0}{\hat{r}}{\rm d}{\hat{r}}\,{\rm d}{\hat{z}}\, \Delta^{-1/2}{\bf u}_{j}\exp(-{\rm i}\mu\phi_{-})\] (12) in Eq. (5) that is valid in the plane of rotation, _i.e.,_ for $\theta_{\rm P}=\pi/2$. We shall treat only the case of positive $\mu$; ${\cal I}(\mu)$ for negative $\mu$ can then be obtained via ${\cal I}(-\mu)={\cal I}(\mu)^{}$. ## III Asymptotic value of the field for an equatorial observer in the far zone Since the main contribution toward the value of the field at $\theta_{\rm P}=\pi/2$ is made by the source elements that lie in the vicinity of the critical point ${\hat{r}}=1$, ${\hat{z}}={\hat{z}}_{\rm P}$, the first step in the asymptotic evaluation of ${\bf B}^{\rm ns}$ is to replace $({\hat{r}},{\hat{z}})$ by a new pair of variables $(\rho,\sigma)$ for which the phase function $\phi_{-}(\rho,\sigma)$ is rendered analytic at this point: \[{\hat{r}}=(1+\rho^{2}\cosh^{2}\sigma)^{1/2},\] (13) \[{\hat{z}}={\hat{z}}_{\rm P}+({\hat{r}}_{\rm P}^{2}-1)^{1/2}\rho\sinh\sigma\] (14) This transformation replaces ${\hat{r}}\Delta^{-1/2}{\rm d}{\hat{r}}{\rm d}{\hat{z}}$ by $\rho\cosh\sigma{\rm d}\rho{\rm d}\sigma$ and yields \[\begin{split}\phi_{-}(\rho,\sigma)=&[{\hat{r}}_{\rm P }^{2}-1-2({\hat{r}}_{\rm P}^{2}-1)^{1/2}\rho+({\hat{r}}_{\rm P}^{2}\sinh^{2} \sigma+1)\rho^{2}]^{1/2}+2\pi\\ &-\arccos\{{\hat{r}}_{\rm P}^{-1}(1+\rho^{2}\cosh^{2}\sigma)^{-1/ 2}[1+({\hat{r}}_{\rm P}^{2}-1)^{1/2}\rho]\},\end{split}\] (15) which is analytic at $\rho=\sigma=0$. In the plane of rotation, _i.e.,_ for $\sigma=0$, the two critical points designated as ${\rm C}$ and ${\rm S}$ in Refs. Ardavan et al. (7); Ardavan et al. (2007) coalesce, and both derivatives, $\partial\phi_{-}/\partial\rho$ and $\partial^{2}\phi_{-}/\partial\rho^{2}$, of the resulting function $\phi_{-}(\rho,0)$ vanish at $\rho=0$ [see Eqs. (15) and (24) below], so that the function $\phi_{-}(\rho,\sigma)$ is stationary at $\rho=\sigma=0$. To see that applying the method of stationary phase to the integral in Eq. (5) results in a valid asymptotic approximation for large ${\hat{R}}_{\rm P}$, let us begin by casting the $\sigma$ dependence of the phase $\phi_{-}$ into a canonical form Borovikov (1994). Since $\sigma=0$ is an isolated stationary point of $\phi_{-}$ (when regarded as a function of the single variable $\sigma$), we may employ the following transformation: \[\phi_{-}=\phi_{-}\|_{\sigma=0}+{1\over 2}b\zeta^{2},\] (16) in which \[b\equiv{\partial^{2}\phi_{-}\over\partial\sigma^{2}}\Big{\|}_{\sigma=0}={\rho^{ 2}[{\hat{r}}_{\rm P}^{2}-1-({\hat{r}}_{\rm P}^{2}-1)^{1/2}\rho+{\hat{r}}_{\rm P }^{2}\rho^{2}]\over(1+\rho^{2})[({\hat{r}}_{\rm P}^{2}-1)^{1/2}-\rho]}.\] (17) Equation (16) expresses $\sigma$ as a function of $\zeta$ implicitly. Repeated differentiations of this equation with respect to $\zeta$ result in \[{\partial\phi_{-}\over\partial\sigma}{\partial\sigma\over\partial\zeta}=b\zeta,\] (18) \[{\partial\phi_{-}\over\partial\sigma}{\partial^{2}\sigma\over\partial\zeta^{2} }+{\partial^{2}\phi_{-}\over\partial\sigma^{2}}\Big{(}{\partial\sigma\over \partial\zeta}\Big{)}^{2}=b,\] (19) and so on, which when evaluated at $\zeta=0$ supply the coefficients $\partial\sigma/\partial\zeta\|_{\sigma=0}$, $\partial^{2}\sigma/\partial\zeta^{2}\|_{\sigma=0}$, etc., in the Taylor expansion of $\sigma$ in powers of $\zeta$. The integral ${\cal I}$ in Eq. (12) can therefore be written as \[{\cal I}=\int{\rm d}\rho{\rm d}\zeta Q(\rho,\zeta)\exp(-{\rm i}\beta\zeta^{2}),\] (20) where \[Q(\rho,\zeta)=\rho\cosh\sigma{\bf u}_{j}\exp(-{\rm i}\mu\phi_{-}\|_{\sigma=0}) \partial\sigma/\partial\zeta,\] (21) \[{\partial\sigma\over\partial\zeta}={b\zeta{\hat{R}}_{-}(1+\rho^{2}\cosh^{2} \sigma)\over\rho^{2}\sinh\sigma\cosh\sigma}\big{[}{\hat{r}}_{\rm P}^{2}-1-({ \hat{r}}_{\rm P}^{2}-1)^{1/2}\rho+{\hat{r}}_{\rm P}^{2}\rho^{2}\cosh^{2}\sigma \big{]}^{-1},\] (22) and $\beta\equiv{1\over 2}\mu b$. The limits of integration are determined by the image of $\Delta\geq 0$ under transformation (16). The parameter $b$ that multiplies the phase of the integrand in Eq. (20) has a large value in the far zone: \[b\simeq\rho^{2}{\hat{r}}_{\rm P},\qquad{\hat{R}}_{\rm P}\gg 1,\] (23) [see Eq. (17)]. The asymptotic value of the integral ${\cal I}$ for large ${\hat{R}}_{\rm P}$ therefore receives its leading contribution from the immediate vicinity of $\zeta=0$, where the phase of its integrand is stationary Borovikov (1994). Replacing $Q(\rho,\zeta)$ in Eq. (20) by $Q(\rho,0)$ and extending the range of integration with respect to $\zeta$ to $(-\infty,\infty)$, we obtain \[\begin{split}{\cal I}&\simeq(2\pi/\mu)^{1/2}\int{\rm d }\rho\,\rho{\bf u}_{j}\|_{\sigma=0}b^{-1/2}\exp[-{\rm i}(\mu\phi_{-}\|_{\sigma=0 }+\pi/4)](\partial\sigma/\partial\zeta)_{\sigma=0}\\ &\simeq(2\pi/\mu)^{1/2}{\hat{r}}_{\rm P}^{-1/2}\exp\{-{\rm i}[\mu ({\hat{r}}_{\rm P}+3\pi/2)+\pi/4]\}\\ &\quad\times\int{\rm d}\rho\,{\bf u}_{j}\|_{{\hat{r}}=(1+\rho^{2}) ^{1/2},{\hat{z}}={\hat{z}}_{\rm P}}\exp[-{\rm i}\mu(\arctan\rho-\rho)],\qquad{ \hat{R}}_{\rm P}\gg 1,\end{split}\] (24) where the integation extends over all values of $\rho$ for which the source density ${\bf s}\|_{{\hat{r}}=(1+\rho^{2})^{1/2},{\hat{z}}={\hat{z}}_{\rm P}}$ is nonzero [see Eq. (7)]. In deriving this expression, we have inferred the value $\partial\sigma/\partial\zeta\|_{\sigma=0}=1$ of the indeterminate Jacobian that appears in $Q(\rho,0)$ from Eq. (19) [or, equivalently, from Eq. (22) and l’Hôpital’s rule], and expressed $\phi_{-}\|_{\sigma=0}$ and $b$ in terms of their far-field values by means of Eqs. (15) and (23). The contribution ${\bf B}^{\rm ns}$ toward the magnetic field ${\bf B}$ of the radiation is made by those volume elements of the source that approach the observation point ${\rm P}$, along the radiation direction, with the speed of light and zero acceleration at the retarded time, _i.e.,_ by the source elements for which $\Delta=0$. Hence, the amplitude of the integrand in Eq. (5) has already been approximated by its leading term in powers of $\Delta^{1/2}=({\hat{r}}_{\rm P}^{2}-1)^{1/2}\rho$ (see Refs. Ardavan et al. (7); Ardavan et al. (2007)). To be consistent, we must also approximate the amplitude of the integrand in Eq. (24) by its value for $\rho\ll 1$: \[\begin{split}{\cal I}\simeq&(2\pi/\mu)^{1/2}{\hat{r} }_{\rm P}^{-1/2}{\bf u}_{j}\|_{{\hat{r}}=1,{\hat{z}}={\hat{z}}_{\rm P}}\exp\{-{ \rm i}[\mu({\hat{r}}_{\rm P}+3\pi/2)+\pi/4]\}\\ &\times\int_{0}^{({\hat{r}}_{>}^{2}-1)^{1/2}}{\rm d}\rho\,\exp[-{ \rm i}\mu(\arctan\rho-\rho)],\end{split}\] (25) where ${\hat{r}}_{>}$ denotes the radial extent of the support of the source density ${\bf s}$. This reduces to \[{\cal I}=3^{-2/3}\Gamma({{1\over 3}})(2\pi)^{1/2}\mu^{-5/6}{\bf u}_{ j}\|_{{\hat{r}}=1,{\hat{z}}={\hat{z}}_{\rm P}}\exp\{-{\rm i}[\mu({\hat{r}}_{\rm P }+3\pi/2)+\pi/12]\}{\hat{r}}_{\rm P}^{-1/2}\] (26) in the regime $\mu\gg 1$, where we can approximate $\arctan\rho-\rho$ in the argument of the exponential by $-{1\over 3}\rho^{3}$ and replace the upper limit of integration by $\infty$Borovikov (1994). Thus, Eqs. (5), (12) and (25) jointly yield the following expression for the leading term in the asymptotic expansion, for ${\hat{R}}_{\rm P}\gg 1$, of the magnetic field of the radiation close to the plane $\theta_{\rm P}=\pi/2$: \[\begin{split}{\bf B}\simeq&-{4\over 3}{\rm i}(2\pi)^ {1/2}{\hat{R}}_{\rm P}^{-1/2}\csc^{1/2}\theta_{\rm P}\exp[{\rm i}(\Omega/ \omega)(\varphi_{\rm P}+3\pi/2)]\\ &\times\sum_{\mu=\mu_{\pm}}\|\mu\|^{1/2}{\rm sgn}(\mu)\exp\{-{\rm i }[\mu({\hat{\varphi}}_{\rm P}+{\hat{r}}_{\rm P}+3\pi/2)+{\pi\over 4}{\rm sgn}( \mu)]\}\\ &\times\sum_{j=1}^{3}{\bar{q}}_{j}{\bf u}_{j}\big{\|}_{{\hat{r}}=1 ,{\hat{z}}={\hat{z}}_{\rm P}}{\cal J},\end{split}\] (27) where \[{\cal J}\equiv\int_{0}^{({\hat{r}}_{>}^{2}-1)^{1/2}}{\rm d}\rho\,\exp[-{\rm i} \mu(\arctan\rho-\rho)];\] (28) for, the contribution ${\bf B}^{\rm ns}$ toward the magnetic field ${\bf B}$ of the radiation is larger by a factor of the order of ${\hat{R}}_{\rm P}^{1/2}$ than the spherically decaying contribution. This is the counterpart of Eq. (55) of Ref. Ardavan et al. (7) and Eq. (61) of Ref.Ardavan et al. (2007) (the electric-field vector of this radiation is given by ${\hat{\bf n}}{\bf\times B}$ as in any other radiation). Note that the remaining integral in the above expression reduces to \[{\cal J}\simeq 3^{-2/3}\Gamma({{1\over 3}})\exp({\rm i}\pi/6)\mu^{-1 /3}\] (29) in the limit $\|\mu\|\gg 1$ [see Eq. (26)]. ## IV Spectrum of the nonspherically decaying radiation: relevance to pulsar observations Eq. (27) shows that the radiation field of a rotating superluminal source diminishes as ${\hat{R}}_{\rm P}^{-1/2}$ with the distance ${\hat{R}}_{\rm P}$ also in the equatorial plane $\theta_{\rm P}=\pi/2$. This differs from the corresponding result for $\theta_{\rm P}\neq\pi/2$ [Eq. (55) of Ref. 7] mainly in its dependence on frequency. The Fourier transform ${\bar{\bf s}}$ in Eq. (57) of Ref. Ardavan et al. (7) has the asymptotic dependence $\mu^{-1}$ on $\mu$ for a source density ${\bf s}(r,z)$ that is a smooth function of $z$. Therefore, when ${\bf s}(z)$ is smooth and the radiation frequency $\|\mu\omega\|$ appreciably exceeds the rotation frequency $\omega$, the field in the plane of rotation decays more slowly with frequency, by a factor of order $\mu^{2/3}$, than does the field outside this plane. Since the azimuthal width of the generated subbeams (and hence the duration of the narrow signals that constitute the overall pulse) is independent of frequency, the flux density $S$ of such signals (_i.e.,_ the power propagating across a unit area per unit frequency) is proportional to $\|{\bf B}\|^{2}/\Delta\mu$, where $\Delta\mu\sim\|\mu\|$ is the bandwidth of the radiation. The flux density of the emission described by Eq. (27) thus depends on frequency as $S\propto\mu^{-2/3}{{\bar{q}}_{j}}^{2}\|{\bf s}(\mu)\|^{2}$ for $\|\mu\|\gg 1$. Here, $\|{\bf s}(\mu)\|$ designates the frequency dependence of the factor ${\bf s}$ that enters the expression for the polarization ${\bf P}$ and the definitions of ${\bf u}_{j}$ [see Eqs. (1) and (7)]. The flux density of the corresponding emission outside the equatorial plane depends on frequency as $S\propto\mu^{-2}{{\bar{q}}_{j}}^{2}\|{\bf s}(\mu)\|^{2}$ for $\|\mu\|\gg 1$ since, apart from the dependence ${\bf s}(\mu)$ of a mutiplicative factor (such as electric susceptibility) in ${\bf s}$, the Fourier transform ${\bar{\bf s}}$ would decay as $\mu^{-1}$ for a smooth ${\bf s}(z)$. To compare the predictions of Eq. (27) [and Eq. (55) of Ref. 7] with the observed spectra of the giant pulses from pulsars, we therefore need to estimate the frequency dependence of the electric susceptibility (contained in the factor ${\bf s}$) for the magnetospheric plasma of a pulsar. The simple classical model of propagation of electromagnetic disturbances in a cold magnetized plasma yields a dielectric tensor, and hence an electric susceptibility, whose components decay with frequency as $(\mu\omega)^{-1}$ when the frequency $\mu\omega$ of the disturbance that polarizes the medium is much lower than the gyration frequency of the electrons in the magnetized plasma; see, _e.g.,_ Eq. (7.67) of Jackson (1999). For a magnetic field as strong as that of a pulsar ($\sim 10^{12}$ G), the Larmor frequency of an electron exceeds the highest radio frequencies at which the pulses are observed by a factor of order $10^{6}$, so that ${\bf s}(\mu)\propto\mu^{-1}$ for pulsars. Using this result, we obtain $S\propto{{\bar{q}}_{j}}^{2}\mu^{-8/3}$ for $\theta_{\rm P}=\pi/2$, and $S\propto{{\bar{q}}_{j}}^{2}\mu^{-4}$ for $\theta_{\rm P}\neq\pi/2$. Depending on whether the modulation frequency $\Omega$ in the expression for ${\bar{q}}_{j}$ [Eq. (6)] is comparable to or much smaller than the frequency $m\omega$ of the sinusoidal wave train characterizing the spatial distribution of the source, therefore, the spectral density of the nonspherically-decaying radiation is given by \[S\propto\mu^{-2/3},\qquad\theta_{\rm P}=\pi/2,\quad\Omega/\omega\simeq\|\mu\|,\] (30) \[S\propto\mu^{-2},\qquad\theta_{\rm P}\neq\pi/2,\quad\Omega/\omega\simeq\|\mu\|,\] (31) \[S\propto\mu^{-8/3},\qquad\theta_{\rm P}=\pi/2;\quad\Omega/\omega\ll\|\mu\|\quad{ \rm or}\quad j=1,\] (32) or \[S\propto\mu^{-4},\qquad\theta_{\rm P}\neq\pi/2;\quad\Omega/\omega\ll\|\mu\|\quad {\rm or}\quad j=1.\] (33) In other words, the spectral index of the pulses portraying the subbeams can have any of the values $-2/3$, $-2$, $-8/3$, or $-4$. The range of spectral indices ($-4\leq\alpha\leq-2/3$) implied by Eq. (27) and its counterpart, Eq. (57) of Ref. Ardavan et al. (7), is consistent with that which characterizes the observed power-law spectra of the giant pulses from pulsars Sallmen et al. (1999); Kinkhabwala and Thorsett (2000); Popov et al. (2006). For radio pulsars, the rotation frequency $\omega$ of the distribution pattern of the radiating polarization current is of the order of $1$ rad/sec, and the oscillation frequency $\mu\omega/2\pi$ of the source density, of the order of $100$ MHz, so that $\mu$ has a large value of the order of $10^{9}$. The coherent component of the radiation, _i.e.,_ the sharply focused subbeams that decay as ${\hat{R}}_{\rm P}^{-1/2}$, are emitted at the frequency $\mu\omega$Ardavan et al. (7). The spherically decaying, incoherent component of the radiation arising from the polarization current described in Eq. (1), on the other hand, contains frequencies that are higher than $\mu\omega$ by a factor of order $(\Omega/\omega)^{2}$Ardavan et al. (2003). In pulsars, $(\Omega/\omega)^{2}\sim 10^{18}$ when $\Omega/\omega$ is comparable to $\mu$, _i.e.,_ when the frequency $m\omega$ that characterizes the spatial fluctuations of the emitting plasma is of the order of, or smaller than, its modulation frequency $\Omega$. Hence, not only the power-law indices of the coherent component, but the unusually broad spectral distribution of the incoherent component of this radiation, too, is consistent with the observational data from certain pulsars. The pulsed emission from the Crab pulsar, for example, extends over 53 octaves of the electromagnetic spectrum from radio waves to $\gamma$-rays Lorimer and Kramer (2005). We note, finally, that neither the asymptotic expansion presented here nor that which was obtained in Refs. Ardavan et al. (7); Ardavan et al. (2007) are uniform with respect to the parameter $\theta_{\rm P}$. The present approximation receives contributions only from the volume elements in the vicinity of the single source point ${\hat{r}}=1$, ${\hat{z}}={\hat{z}}_{\rm P}$ at which the cusp curve $\Delta=0$ of the bifurcation surface touches the light cylinder (see Figs. 3 and 4 of Ref. Ardavan et al. (7)). This is in sharp contrast to the asymptotic expansion of the field outside the equatorial plane, where the leading term receives contributions from a filamentary locus of source elements, _i.e.,_ from the intersection of the cusp curve of the bifurcation surface with the entire volume of the source. Comparison of Eq. (27) with its counterpart, Eq. (57) of Ref. Ardavan et al. (7), shows that the (smooth) transition from the nonequatorial to the equatorial regime occurs across $\theta_{\rm P}\simeq\arccos(\mu^{-2/3})$. However, the derivation of a uniform asymptotic approximation to integral ${\cal I}$ that would determine the field in the transition region is a challenging mathematical problem that remains open. ## Acknowledgemens H. A. thanks Janusz Gil for helpful conversations. A. A. is supported by the Royal Society. J. S., J. F., and A. S. are supported by U.S. Department of Energy grant LDRD 20050540ER. ## References Bessarab et al. (2004) A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, IEEE Trans. Plasma Sci. 32, 1400 (2004), ISSN 0093-3813. * Ardavan et al. (2004a) A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, J. Appl. Phys. 96, 7760 (2004a), ISSN 0021-8979, corrected version of 96(8), 4614–4631. * Bessarab et al. (2006) A. V. Bessarab, S. P. Martynenko, N. A. Prudkoi, A. V. Soldatov, and V. A. Terekhin, Radiation Physics and Chemistry 75, 825 (2006), ISSN 0969-806X. * Bolotovskii and Serov (2006) B. M. Bolotovskii and A. V. Serov, Radiation Physics and Chemistry 75, 813 (2006), ISSN 0969-806X. * Bolotovskii and Bykov (1990) B. M. Bolotovskii and V. P. Bykov, Sov. Phys. Usp. 33, 477 (1990), ISSN 0038-5670. * Ardavan (1998) H. Ardavan, Phys. Rev. E 58, 6659 (1998), ISSN 1063-651X. * Ardavan et al. (2004b) H. Ardavan, A. Ardavan, and J. Singleton, J. Opt. Soc. Am. A 21, 858 (2004b), ISSN 1084-7529. * Ardavan et al. (2007) H. Ardavan, A. Ardavan, J. Singleton, J. Fasel, and A. Schmidt, J. Opt. Soc. Am. A 24, 2443 (2007). * Borovikov (1994) V. A. Borovikov, _Uniform Stationary Phase Method_ (Institution of Electrical Engineers, Stevenage, U.K, 1994). * Jackson (1999) J. D. Jackson, _Classical Electrodynamics_ (Wiley, New York, 1999), 3rd ed. * Popov et al. (2006) A. V. Popov, A. D. Kuz’min, O. M. Ul’yanov, A. A. Deshpande, A. A. Ershov, V. V. Zakharenko, V. I. Kondrat’ev, S. V. Kostyuk, B. Y. Losovskii, and V. A. Soglansnov, Astron. Rep. 50, 562 (2006), ISSN 1063-7729. * Kinkhabwala and Thorsett (2000) A. Kinkhabwala and S. E. Thorsett, Astrophys. J. 535, 365 (2000). * Sallmen et al. (1999) S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, Astrophys. J. 517, 460 (1999), ISSN 0004-637X. * Ardavan et al. (2003) H. Ardavan, A. Ardavan, and J. Singleton, J. Opt. Soc. Am. A 20, 2137 (2003), ISSN 0740-3232. * Lorimer and Kramer (2005) D. Lorimer and M. Kramer, _Handbook of Pulsar Astronomy_ (Cambridge U. Press, Cambridge, U.K, 2005).
1801.08534	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 64820, "num_imgs": 16, "llama3_tokens_count": 17560 }	[ "content_image/1801.08534/Fig1.png", "content_image/1801.08534/Fig2.png", "content_image/1801.08534/Fig3.png", "content_image/1801.08534/Fig4.png", "content_image/1801.08534/FigS1.png", "content_image/1801.08534/FigS2.png", "content_image/1801.08534/FigS3.png", "content_image/1801.08534/FigS4.png", "content_image/1801.08534/FigS5.png", "content_image/1801.08534/FigS6.png", "content_image/1801.08534/FigS7.png", "content_image/1801.08534/FigS7E.png", "content_image/1801.08534/FigS8.png", "content_image/1801.08534/FigS9.png", "content_image/1801.08534/FigS10.png", "content_image/1801.08534/FigS11.png" ]	# Electrical generation and detection of spin waves in a quantum Hall ferromagnet Di S. Wei John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 Toeno van der Sar Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Seung Hwan Lee Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Kenji Watanabe Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan Takashi Taniguchi Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan Bertrand I. Halperin Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Amir Yacoby John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 Department of Physics, Harvard University, Cambridge, Massachusetts 02138 ###### Abstract Spin waves are collective excitations of magnetic systems. An attractive setting for studying long-lived spin-wave physics is the quantum Hall (QH) ferromagnet, which forms spontaneously in clean two-dimensional electron systems at low temperature and in a perpendicular magnetic field. We used out-of-equilibrium occupation of QH edge channels in graphene to excite and detect spin waves in magnetically ordered QH states. Our experiments provide direct evidence for long distance spin wave propagation through different ferromagnetic phases in the N=0 Landau level, as well as across the insulating canted antiferromagnetic phase. Our results will enable experimental investigation of the fundamental magnetic properties of these exotic two-dimensional electron systems. Quantum Hall (QH) ferromagnetism arises from the interaction of electrons in massively degenerate, quantized energy levels known as Landau levels (LLs) [1]. When disorder is low enough for Coulomb interactions to manifest, the electrons in partially filled LLs spin-polarize spontaneously to minimize their exchange energy, with the single-particle Zeeman effect dictating their polarization axis [2; 3]. In graphene, these phenomena give rise to ferromagnetic phases when the N=0 LL is at quarter- and three-quarter-filling [4; 5; 6; 7; 8]. Such QH ferromagnets have an insulating topological bulk and spin-polarized edge states. Furthermore, a canted antiferromagnetic (CAF) state is believed to emerge at one-half filling, with a canting angle determined by the competing valley anisotropy and Zeeman energy [9; 10]. Spin waves, also known as magnons, are the lowest energy excitation in both the QH ferromagnet and CAF [1; 11; 12], and could provide crucial information about these topologically non trivial magnetic states. In our experimental setup, we generate magnons by creating an imbalance of chemical potential between two edge states of opposite spin that run along the boundary of a QH magnet. If this imbalance is smaller than the energy required for generating magnons in the QH magnet (and there are no thermal magnons already present in the system), scattering between these two edge states is forbidden because the change in angular momentum of a scattered electron cannot be absorbed by the system. Indeed, previous measurements have shown that oppositely spin-polarized edge channels do not equilibrate as long as the imbalance is small [13; 14]. However, we find edge channel equilibration commences when the imbalance exceeds the minimum energy required for exciting magnons in the QH ferromagnet. Because the magnetization of the QH ferromagnet is extremely dilute, there are negligible demagnetizing fields and the minimum energy to excite magnons is given by the Zeeman energy $E_{\mathrm{Z}}=g\mu_{B}B$[1; 15], where $g$ is the electron g-factor, $\mu_{B}$ is the Bohr magneton, and $B$ is the external magnetic field. Although magnon generation does not directly affect the conductance of the system, the reverse process of magnon absorption by far-away edge states does, allowing us to detect the propagation of magnons electrically, in close analogy to the conventional detection of magnons in insulators via the inverse spin Hall effect [16; 17; 18; 19]. <figure><img src="content_image/1801.08534/Fig1.png"><figcaption>Figure 1: Magnons in a quantum Hall ferromagnet. (A-C) A chemical potentialdifference (μ) is applied between the left and right leads. Edge channels withhigh and low chemical potential are labeled “hot” and “cold”, respectively.Spin-up and spin-down polarization is denoted by the green and orange arrows,respectively. The central region is tuned to ν = 1 and adjacent regions aretuned to ν = 2. (A) The chemical potential difference (μ) between the spin-upand spin-down edge channel is less than the Zeeman energy (EZ), and scatteringis suppressed. (B) μ≥EZ: Electrons have enough energy to flip their spins andtransfer spin angular momentum (magnons) into the bulk (at the encircled minussign). These magnons are absorbed at distant corners, causing electrons toflip from spin-up into spin-down channels. (C) μ≤−EZ: Magnons are generated atthe location denoted by the encircled plus sign. (D) Bulk spin polarizationbefore and after magnon creation, conserving total spin angular momentum. (E)Optical micrograph of device 1; graphene is outlined in white. TG, top gate.(F) A dc voltage (Vdc) and a 50-μV ac excitation voltage (Vac) are applied tothe left contact and the differential conductance (dI/dV, where V=Vac+Vdc) ismeasured through the right contact (Bperp = 4 T, VTG = -0.18 V, VBG = 3 V).Conductance is quantized to e2/h until \|μ\|≥≥EZ. (G) dI/dV as a function ofbias and magnetic field. The blue dashed line is the Zeeman energy, EZ⊥=gμBB⊥calculated using the perpendicular (total) magnetic field B⊥; The black dashedline is the Zeeman energy, EZT=gμBBT calculated using the total magnetic fieldBT). Both the top gate (VTG) and back gate (VBG) are swept to stay at nu=1throughout the device from 7T (VTG = 0.16 V, VBG = 0.73 V) to 5T (VTG = 0.12V, VBG = 0.44 V) The decrease in conductance from e2/h evolves linearly withthe magnetic field coinciding with EZT rather than EZ⊥. Right inset: Asaturated color plot (from 0.98 to 1.02 e2/h) of the region enclosed by theyellow box. All measurement are conducted in a cryostat with a basetemperature of 20 mK.</figcaption></figure> To demonstrate spin wave propagation, we begin with a dual-gated monolayer graphene device (device $1$) where the central region can be tuned to a different filling factor than the adjacent regions (Fig. 1A). Connecting the two leads is a chiral edge state that carries spin-polarized electrons aligned with the magnetic field, which we call spin-up. We tune the central region to a three-quarters-filled LL ($\nu=1$), whereas the outer regions are tuned to a non-magnetic fully filled LL ($\nu=2$). We apply a source-drain voltage $V_{\mathrm{dc}}$ to induce a difference in chemical potential $\mu=-eV_{\mathrm{dc}}$ between the edge channels emerging from the two contacts, where $e$ is the electron charge. Once $\|\mu\|\geq E_{\mathrm{Z}}$, an electron traveling in a high-energy (“hot”), spin-down edge state can relax into a low-energy (“cold”), spin-up edge state by emitting a magnon into the ferromagnetic bulk (Fig. 1, B and C). Because equilibration must occur close to the ferromagnetic bulk in order to launch magnons, the edge states must equilibrate over short length scales at localized “hot spots” where the hot and cold edges meet. This makes graphene an ideal platform to observe this phenomenon, where edge state equilibration can occur over length scales $<1\mu$m [13; 20; 21] (See [22] for further discussion). Because only spin-down angular momentum can be propagated into the spin-up bulk, magnon generation occurs at the location denoted by an encircled minus sign when $\mu\geq E_{\mathrm{Z}}$ (Fig. 1B) and at the location denoted by an encircled plus sign when $\mu\leq-E_{\mathrm{Z}}$ (Fig. 1C). These magnons propagate through the insulating QH ferromagnet and can be absorbed by the reverse process between other edge channels (Fig. 1B-C), which causes a deviation in the conductance from a well-quantized $\nu=1$ QH state. When we measure the conductance of the graphene device (Fig. 1E, atomic force microscopy image in fig. S3) as a function of $V_{\mathrm{dc}}$, we find that the $\nu=1$ QH ferromagnet remains precisely quantized at the expected value of $e^{2}/h$, and then changes once the applied bias reaches the Zeeman threshold ($V_{\mathrm{dc}}=\pm V_{\mathrm{EZ}}=\mp E_{\mathrm{Z}}/e$), as expected from our model (Fig. 1F). Interestingly, we find that thanks to contact doping [22; 23] we can tune the entire device to $\nu=1$ and find the same phenomenon of conductance deviation at the Zeeman threshold (fig. S4). <figure><img src="content_image/1801.08534/Fig2.png"><figcaption>Figure 2: Effect of relative magnon absorption on conductance. (A) Opticalmicrograph of device 2. Graphene is outlined in white. (B) Schematic of a two-terminal conductance measurement using leads L2 and L1 where hot and coldedges are colored red and blue, respectively, for both μ≥EZ (left) and μ≤−EZ(right), and the magnon generation site is labeled by the encircled plus orminus sign indicating positive or negative bias. μ≥EZ : magnon absorption atε1 transfers chemical potential from a forward-moving edge to a backward-moving edge, causing the particle current (IP=−I/e) to decrease. Conversely,magnon absorption at ε2 transfers chemical potential from a backward-movingedge to a forward-moving edge, increasing IP. μ≤−EZ : Magnon absorption at ε1causes an increase in \|−IP\|; absorption at ε2 causes a decrease in \|−IP\|. (C)The effects of ε1 and ε2 at μ≥EZ and μ≤−EZ. The current changes caused by ε1are dominant and are circled in red. The purple arrows indicate an increase(up) or decrease (down) in the magnitude of the signed particle current.(D)Conductance from L2 to L1 (g21 = dI/dV = dIP/dμ) decreases at Vdc=−VEZ andincreases at Vdc=VEZ, indicating that ε1 has a larger effect than ε2 (B = 8 T,VBG = 4 V). See WeiSupp2018 for full circuit analysis. (E-F) Conductance fromL3 and L2 (g32) where the entire device is tuned to ν=1 (VBG = 4V, TG1 = 0V isnot shown). At positive bias, ε2>ε1, and at negative bias, ε1>ε2, resulting ina conductance drop for both biases.(G-H) Conductance from L3 to L2 (g32) whereTG1 is tuned to νTG1=1 (TG1=-0.36 V) while the regions outside are set toνbg=2 (VBG = 6.5V). At positive bias, ε1>ε2, and at negative bias, ε2>ε1,resulting in a conductance rise for both biases. See fig. S5. for a detailedanalysis.</figcaption></figure> <figure><img src="content_image/1801.08534/Fig3.png"><figcaption>Figure 3: Non-local voltage signal due to magnon absorption. Shown are thedata from device 2. (A) Schematic circuit configuration for measuring a non-local voltage in device 2. The filling factor under TG1 (νTG1) = 1 for allmeasurements while the filling factor under TG2 (νTG2) is swept from -2 to 2,and the rest of the device is kept at νbg = 1 (VBG = 4 V). The bottom panelhighlights the magnetic properties of different cases of νTG2: non-magnetic(NM), ferromagnetic (FM), or canted antiferromagnetic (CAF). (B) SNL (purple)superimposed onto dI/dV (green) as a function of Vdc when νTG2 = 1 (B = 8 T).The onset of SNL is slightly offset in bias from the decrease in conductance,indicating that magnon generation needs to reach a threshold before beingabsorbed in distant contacts. (C) A pronounced SNL signal when νTG2 = 1 andνTG2 = -1 (See Fig. S8 for similar measurements using TG1). Tuning TG2 to thenonmagnetic QH phases (νTG2 = 2 and νTG2 = -2), as well the νTG2 = 0 CAFstate, strongly suppresses SNL. There is a small finite background SNL whenedge states pass through TG2, discussed in fig. S7, B. Solid brown lineindicates where νTG2 = 0, 1, and 2 (fig. S7, C and D). (D) The spatialvariation of the LLs at a ν = 1/ν = -1 junction, with the expected valley andspin polarizations of each level labeled.</figcaption></figure> By tilting the external magnetic field with respect to the sample-plane normal axis, we verify that the change in conductance occurs when the applied chemical potential exceeds the bare Zeeman energy $E_{\mathrm{Z}}=g\mu_{\mathrm{B}}B_{\mathrm{T}}$ ($g$=2), which is given by the total field $B_{\mathrm{T}}$ (Fig. 1G – sample is tuned entirely to $\nu=1$). In contrast, previous transport studies of spin and valley excitations in graphene and GaAs have only found excitations related to the exchange energy gap [2; 3; 24], which depends on the component of the field perpendicular to the sample plane ($B_{\perp}$). Our tilted-field measurements therefore corroborate our magnon-based interpretation of the observed change in sample conductance. All subsequent experiments described in this work are done at perpendicular field. The conductance change at $E_{\mathrm{Z}}$ can either be positive or negative, depending on the number of magnons absorbed at each contact. To examine this, we use different sets of leads in the same device (Fig. 2A, device 2) to perform two-terminal conductance measurements. We start with leads L${}_{2}$ and L${}_{1}$ in Fig. 2B. We label the amount of redistributed chemical potential at each of the absorption sites $\varepsilon_{i}$, with $i$ indexing the absorption site (note that $\varepsilon_{i}$ = 0 for $-E_{\mathrm{Z}}<\mu<+E_{\mathrm{Z}}$), where $\varepsilon_{i}$ is proportional to the number of magnons absorbed at site $i$. Absorption at $\varepsilon_{1}$ and $\varepsilon_{2}$ have opposite effects on the conductance, as magnon absorption transfers chemical potential from the outer edge to the inner edge. Therefore, for $\mu\geq E_{\mathrm{Z}}$, magnon absorption at $\varepsilon_{1}$ decreases the particle current ($I_{\mathrm{P}}=-I/e$ where $I$ is the charge current) whereas magnon absorption at $\varepsilon_{2}$ increases $I_{\mathrm{P}}$ (Fig. 2B). For $\mu\leq-E_{\mathrm{Z}}$, the hot and cold reservoirs are reversed, and we now consider the change to the negative particle current $-I_{\mathrm{P}}$. Although $\varepsilon_{1}$ still decreases the particle current, $I_{\mathrm{P}}$ is now negative, and so $\varepsilon_{1}$ actually increases the magnitude of the particle current ($\|-I_{\mathrm{P}}\|$); similarly, for $\mu\leq-E_{\mathrm{Z}}$, $\varepsilon_{2}$ decreases $\|-I_{\mathrm{P}}\|$ (Fig. 2C). We can quantify this using current conservation to formulate the differential conductance as a function of $\varepsilon_{i}$ and $\mu$: \[\frac{\mathrm{d}I}{\mathrm{d}V}=\frac{\mathrm{d}I_{\mathrm{P}}}{\mathrm{d}\mu} =\frac{1}{R_{\mathrm{Q}}}\Big{(}1+\frac{\mathrm{d}\varepsilon_{2}}{\mathrm{d} \mu}-\frac{\mathrm{d}\varepsilon_{1}}{\mathrm{d}\mu}\Big{)}\] (1) where $R_{\mathrm{Q}}=h/e^{2}$ is the resistance quantum, $V=V_{\mathrm{ac}}+V_{\mathrm{dc}}$, and we have neglected contact resistance (see [22] for a derivation of Eq. 1, which takes contact resistance into account). We find that the conductance decreases at negative bias and increases at positive bias (Fig. 2D) – indicating that $\varepsilon_{1}>\varepsilon_{2}$ for both positive and negative bias. This implies that more magnons are absorbed at $\varepsilon_{1}$ than at $\varepsilon_{2}$. Because our contacts have all been fabricated identically, we conclude this is because $\varepsilon_{1}$ is closer to magnon generation than $\varepsilon_{2}$ (for both positive and negative bias, see Fig. 2, A and B). Using different sets of contacts and top gates (Fig. 2, E to H) we can change the relative distances of $\varepsilon_{i}$ to the locations of magnon generation. We confirm that for each configuration, the conductance values after $E_{\mathrm{Z}}$ correspond to a greater number of magnons absorbed at the site closer to magnon generation. This change to the conductance is not a consequence of QH breakdown. Conductance deviations after the Zeeman threshold that depend on the sign of $V_{\mathrm{dc}}$ are not explained by any current breakdown theories [25]. Additionally, we find that the threshold voltage bias does not depend on the lead configuration (Fig. 2), the size of the $\nu=1$ region (fig. S4), or the density of the $\nu=1$ region (fig. S6) – which is all inconsistent with trivial QH breakdown, but consistent with our magnon model. In total, we have measured this $\nu=1$ conductance deviation occurring at the Zeeman energy for eight devices of widely varying geometries (figs. S3, S4, and S11). <figure><img src="content_image/1801.08534/Fig4.png"><figcaption>Figure 4: Non-local voltage signal due to magnon propagation through the ν=0CAF. (A) Schematic of the circuit used to measure SNL in device 2 across a ν=0region. νTG1=1 for all measurements while νTG2 is swept from -1 to 1 (νBG = 1,VBG = 4 V). (B) Top: Postulated spatial variation of the LLs and spinarrangement in a ν=1/ν=0/ν=1 geometry. Close to the interface between ν=1 andν=0, spins in the two filled Landau levels prefer to be in an alignedantiferromagnetic (AF) arrangement. Deeper into the ν=0 region, the spinsslowly rotate into the canted antiferromagnetic phase. Because the minimummagnon energy in the aligned AF region is higher than EZ, it should present abarrier for incident magnons close to the energy threshold. Bottom: Energybarrier seen by the magnons as a function of position, where EINT is theenergy barrier of the interface. (C) When magnons are generated, we seeanother onset of SNL at energies exceeding ±VEZ (B = 8 T), indicating thathigher energy magnons have overcome EINT and have propagated through the νTG2= 0 region. Purple dashed lines indicate a region where vertical line cutswere taken and averaged to obtain the line trace in (D). (D) A clear onset ofSNL is shown at biases exceeding ±VEZ when νTG2 = 0. It is not presentlyunderstood why the signal is asymmetric both in energy of onset and strengthof signal. The zoomed-in region shows a clear increase in SNL at -VEZ and asignal consistent with a decrease, slightly offset from +VEZ, indicating thatmagnons can tunnel through the interface barrier at lower energies (SNL isoffset by 0.01 μV at Vdc = 0 and is manually corrected for).</figcaption></figure> Thus far we have established that we are able to generate and absorb magnons at current carrying contacts. If these chargeless excitations propagate through the insulating bulk, we also expect to see signatures of magnon propagation and absorption via non-local voltage measurements (d$V_{\mathrm{NL}}$/d$V$—referred to as nonlocal signal $S_{\mathrm{NL}}$), away from the source-drain current. To measure $S_{\mathrm{NL}}$ we use L${}_{3}$ and L${}_{2}$ in device 2 as source-drain contacts, and use contacts L${}_{4}$ and L${}_{5}$ as voltage probes (Fig. 3A). These contacts are separated from the source-drain contacts by a top gate (TG2) which we tune between $\nu_{\mathrm{TG2}}$ = -2 and $\nu_{\mathrm{TG2}}$ = 2, where all other regions are tuned to $\nu=1$. The conductance between L${}_{3}$ and L${}_{2}$ drops at $V_{\mathrm{EZ}}$ in accordance with our model (Fig. 3B), whereas magnon generation is largely unaffected by TG2 (fig. S7, A). At $\nu_{\mathrm{TG2}}$ = 1 we measure a change in $S_{\mathrm{NL}}$ at $\pm V_{\mathrm{EZ}}$ due to the relative absorption at each magnon absorption site ($\varepsilon_{i}$). The sign of S${}_{\mathrm{NL}}$ indicates that there is more magnon absorption at sites closer to where magnon generation occurs. Through current conservation ([22]) we find that the measured differential voltage (unitless) is: \[\frac{\mathrm{d}V_{\mathrm{NL}}}{\mathrm{d}V}=\Big{(}\frac{\mathrm{d} \varepsilon_{4}}{\mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_{5}}{\mathrm{d}\mu }\Big{)}\] (2) The site labeled by $\varepsilon_{4}$ is closer to magnon generation than $\varepsilon_{5}$ for both negative and positive bias, so $\|\mathrm{d}\varepsilon_{4}\|>\|\mathrm{d}\varepsilon_{5}\|$. However, the differential change in voltage ($\mathrm{d}\varepsilon_{i}/\mathrm{d}\mu$) is negative for $V_{\mathrm{dc}}\geq V_{\mathrm{EZ}}$ and positive for $V_{\mathrm{dc}}\leq-V_{\mathrm{EZ}}$, corresponding to an overall negative value for $S_{\mathrm{NL}}$ at $V_{\mathrm{dc}}\geq V_{\mathrm{EZ}}$ and a positive value at $V_{\mathrm{dc}}\leq-V_{\mathrm{EZ}}$ (Fig. 3C). The device geometry used for our non-local measurements allows us to tune TG2 away from $\nu_{\mathrm{TG2}}=1$, and thereby examine magnon transmission through different filling factors. We make two surprising observations. We observe that when $\nu_{\mathrm{TG2}}=-1$ the signal $s_{\mathrm{NL}}$ is almost identical signal to when $\nu_{\mathrm{TG2}}=1$ (Fig. 3C and fig. S8). This signal arises in the absence of any charge leakage across the $\nu_{\mathrm{TG2}}=-1$ region (fig. S9), so that changes in $S_{\mathrm{NL}}$ can be attributed to magnon transport through the $\nu_{\mathrm{TG2}}=-1$ ferromagnet. This suggest that there is neither spin nor valley mismatch between the ferromagnetic states on either side of the boundary. We therefor propose an ordering of the LLs that does not require a spin or valley flip for magnons to travel across the interface between $\nu_{\mathrm{BG}}=1$ and $\nu_{\mathrm{TG2}}=-1$ (Fig. 3D; see [22] for a theoretical discussion.) In addition, we unexpectedly find that $S_{\mathrm{NL}}$ is suppressed at $\pm V_{\mathrm{EZ}}$ when $\nu_{\mathrm{TG2}}=0$. For non-magnetic regions such as $\nu_{\mathrm{TG2}}=2$, it is expected that magnons will be blocked from passing through, as experimentally confirmed in Fig. 3C (the non-local signal occurring at the transition between $\nu=1$ and $\nu=2$ is explained in Fig. S7E). However, $\nu=0$ is purportedly a canted antiferromagnet which is theoretically capable of hosting even zero-energy magnons [12]. It appears that the probability for an incident magnon to be transmitted across the junction between the $\nu=0$ and $\nu=1$ regions is very small for energies close to $E_{\mathrm{Z}}$. This may be caused by, in part, the mismatch in propagation velocities in the two phases, or a barrier due to the complex nature of the interface region. Close to the boundary with a $\nu=1$ phase, the ground state of the $\nu=0$ phase may not have canted spins but may instead be in an aligned antiferromagnet state, where spins are parallel to the magnetic field on one sublattice and antiparallel on the other. Eventually, far from the boundary, we may expect the local spin arrangement to rotate into the CAF orientation (Fig. 4B). In the transition region, the minimum magnon energy will be larger than $E_{\mathrm{Z}}$ due to effects of the valley-dependent interaction terms [9], which were initially responsible for the antiferromagnet arrangement to be favored over the ferromagnetic arrangement. In order to cross from the $\nu=1$ region to the CAF region, a magnon with energy close to $E_{\mathrm{Z}}$ would have to tunnel through the barrier region, and we would expect the transmission rate to be low. If the magnons have enough energy to overcome this barrier, they should be able to more easily enter the CAF region. Fig. 4C shows that we can experimentally exceed this barrier, where we see non-local signals at higher $\|V_{\mathrm{dc}}\|$ with signs in agreement with our magnon model. The onset of this magnon signal is unaffected by any charge transport across the $\nu_{\mathrm{TG2}}=0$ region (fig. S10). Closely examining the signal at $\nu_{\mathrm{TG2}}=0$, we see signals commencing at $\pm V_{\mathrm{EZ}}$ which we attribute to tunneling events across this $\nu_{\mathrm{BG}}=1/\nu_{\mathrm{TG2}}=0$ barrier (Fig. 4D). Note that all non-local signals (occurring at $\nu_{\mathrm{TG2}}$ =-1, 0, and 1) appear only in a finite band of $V_{\mathrm{dc}}$. This suppression of the differential voltage signal indicates that either magnon generation is suppressed, or alternatively, that the differently-spaced contacts begin to see identical amounts of magnon absorption once the system has reached a certain magnon density threshold. We further speculate that this cut-off could be related to the magnon bandwidth, but leave this to a future investigation. The experiments presented here introduce a method of using magnons to probe the SU(4) spin and valley anisotropies of graphene QH systems, whichc can be used to probe highly correlated states such as the fractional QH regime [26], or the quantum-spin Hall phase of monolayer graphene [10]. Owing to the theoretical prediction for spin superfluidity in the CAF state [12], this study paves the way for exploring and realizing dissipationless spin waves in a Bose-Einstein condensate (BEC) of magnons. Such condensates should result in a coherent precession of the spin in the QH magnet, which may be probed through emitted microwave radiation. Furthermore, coherent spin waves associated with a BEC may be able to propagate long distances with negligible dissipation, which could be tested by careful length dependence measurements. ## I Methods ### Sample Fabrication All devices consist of graphene encapsulated by two layers of hexagonal boron nitride (hBN) on doped Si chips with a 285 nm layer of SiO${}_{2}$ that acts as a dielectric for the Si back gate. Graphene is mechanically exfoliated from bulk graphite obtained from NGS Naturgraphit GmbH using 1009R tape from Ultron Systems and subsequently encapsulated in hexagonal boron nitride (hBN) using a dry transfer process [27]. Before the first metal deposition step, we annealed the devices in vacuum at $500^{\circ}$C to improve device quality. We then created top gates using electron-beam lithography and thermal evaporation of Cr/Au. We etched the devices into the desired geometry by reactive ion etching in $\mathrm{O}_{2}/\mathrm{CHF}_{3}$ using a PMMA/HSQ bilayer of resist (patterned by electron-beam lithography) as the etch mask. To fabricate edge-contacts to the graphene we etched through the entire hBN/graphene stack. We then created edge contacts by thermally evaporating Cr/Au while rotating the sample using a tilted rotation stage. ### Measurement Our measurements were performed in a Leiden dry dilution refrigerator with a base temperature of 20 mK. Measurements of differential conductance were performed using a lock-in amplifier with an a.c. excitation voltage of 50 $\mu$V at 17.77 Hz. All measurements of differential conductance were corrected for contact/line resistances, which were independently determined by lining up the robust $\nu=2$ QH conductance plateau with $2e^{2}/h$. ## References * (1) Girvin, S. M. The Quantum Hall Effect : Novel Excitations and Broken Symmetries. In Comtet, A., Jolicœur, T., Ouvry, S. & David, F. (eds.) _Les Houches Lecture Notes, in: Topological Aspects of Low Dimensional Systems_, vol 69, 53–175 (Springer-Verlag, 2000). * (2) Young, A. F. _et al._ Spin and valley quantum Hall ferromagnetism in graphene. _Nature Physics_8, 550–556 (2012). * (3) Sondhi, S. L., Karlhede, A., Kivelson, S. A. & Rezayi, E. H. Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies. _Physical Review B_47, 16419–16426 (1993). * (4) Alicea, J. & Fisher, M. P. A. Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes. _Physical Review B_74, 075422 (2006). * (5) Yang, K., Das Sarma, S. & MacDonald, A. H. Collective modes and skyrmion excitations in graphene SU (4) quantum Hall ferromagnets. _Physical Review B_74, 075423 (2006). * (6) Zhang, Y. _et al._ Landau-level splitting in graphene in high magnetic fields. _Phys. Rev. Lett._96, 136806 (2006). * (7) Nomura, K. & Macdonald, A. H. Quantum Hall ferromagnetism in graphene. _Phys. Rev. Lett._96, 256602 (2008). * (8) Goerbig, M. Electronic properties of graphene in a strong magnetic field. _Reviews of Modern Physics_83, 1193–1243 (2011). * (9) Kharitonov, M. Phase diagram for the $\nu=0$ quantum hall state in monolayer graphene. _Phys. Rev. B_85, 155439 (2012). * (10) Young, A. F. _et al._ Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. _Nature_505, 528–32 (2014). * (11) Green, A. G. & Cooper, N. R. Dissipative transport in quantum Hall ferromagnets by spin-wave scattering. _Phys. Rev. B_65, 125329 (2002). * (12) Takei, S., Yacoby, A., Halperin, B. I. & Tserkovnyak, Y. Spin Superfluidity in the $\nu$=0 Quantum Hall State of Graphene. _Physical Review Letters_116, 216801 (2016). * (13) Amet, F., Williams, J. R., Watanabe, K., Taniguchi, T. & Goldhaber-Gordon, D. Selective equilibration of spin-polarized quantum Hall edge states in graphene. _Phys. Rev. Lett._112, 196601 (2014). * (14) Wei, D. S. _et al._ Mach-Zehnder interferometry using spin- and valley-polarized quantum Hall edge states in graphene. _Science Advances_3, 1–8 (2017). * (15) Kittel, C. On the theory of ferromagnetic resonance absorption. _Physical Review_73, 155–161 (1948). * (16) Kajiwara, Y. _et al._ Transmission of electrical signals by spin-wave interconversion in a magnetic insulator. _Nature_464, 262–266 (2010). * (17) Cornelissen, L. J., Liu, J., Duine, R. A., Youssef, J. B. & Wees, B. J. V. Long-distance transport of magnon spin information in a magnetic insulator at room temperature. _Nature Physics_11, 1022–1026 (2015). * (18) Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics. _Nature Physics_11, 453–461 (2015). * (19) Wesenberg, D., Liu, T., Balzar, D., Wu, M. & Zink, B. L. Long-distance spin transport in a disordered magnetic insulator. _Nature Physics_13, 987–993 (2017). * (20) Williams, J. R., Dicarlo, L. & Marcus, C. M. Quantum Hall effect in a gate-controlled p-n junction of graphene. _Science_317, 638–640 (2007). * (21) Özyilmaz, B. _et al._ Electronic transport and quantum Hall effect in bipolar graphene p-n-p junctions. _Phys. Rev. Lett._99, 166804 (2007). * (22) Supplementary Materials . * (23) Giovannetti, G. _et al._ Doping graphene with metal contacts. _Physical Review Letters_101, 026803 (2008). * (24) Schmeller, A., Eisenstein, J.P., Pfeiffer, L.N., West, K. Evidence for skyrmions and single spin flips in the integer quantized Hall effect. _Physical Review Letters_75, 4290–4293 (1995). * (25) Nachtwei, G. Breakdown of the quantum Hall effect. _Physica E_4, 79–101 (1999). * (26) Macdonald, A. H. & Palacios, J. J. Magnons and skyrmions in fractional Hall ferromagnets. _Physical Review B_58, 171–174 (1998). * (27) Wang, L. _et al._ One-dimensional electrical contact to a two-dimensional material. _Science_342, 614–7 (2013). * (28) Van Wees, B. J. _et al._ Quantum ballistic and adiabatic electron transport studied with quantum point contacts. _Physical Review B_43, 12431–12453 (1991). * (29) Alphenaar, B. W., McEuen, P. L., Wheeler, R. G. & Sacks, R. N. Selective equilibration among the current-carrying states in the quantum hall regime. _Physical Review Letters_64, 677–680 (1990). * (30) Komiyama, S., Hirai, H., Sasa, S. & Hiyamizu, S. Violation of the integral quantum Hall effect: Influence of backscattering and the role of voltage contacts. _Physical Review B_40, 12566–12569 (1989). * (31) Chklovskii, D., Shklovskii, B. & Glazman, L. Electrostatics of edge channels. _Physical Review B_46, 4026–4034 (1992). * (32) Ahlswede, E., Weis, J., Van Klitzing, K. & Eberl, K. Hall potential distribution in the quantum Hall regime in the vicinity of a potential probe contact. _Physica E: Low-Dimensional Systems and Nanostructures_12, 165–168 (2002). * (33) Li, G., Luican-Mayer, A., Abanin, D., Levitov, L. & Andrei, E. Y. Evolution of Landau levels into edge states in graphene. _Nature communications_4, 1744 (2013). eprint 1203.5540. * (34) Müller, G. _et al._ Equilibration length of electrons in spin-polarized edge channels. _Phys. Rev. B_45, 3932–3935 (1992). * (35) Burkard, G., Loss, D. & DiVincenzo, D. P. Coupled quantum dots as quantum gates. _Physical Review B_59, 2070–2078 (1999). * (36) Jung, J. & MacDonald, A. H. Theory of the magnetic-field-induced insulator in neutral graphene sheets. _Physical Review B_80, 235417 (2009). * (37) Abanin, D. A., Feldman, B. E., Yacoby, A. & Halperin, B. I. Fractional and integer quantum Hall effects in the zeroth Landau level in graphene. _Physical Review B_88, 115407 (2013). * (38) Pientka, F., Waissman, J., Kim, P. & Halperin, B. I. Thermal Transport Signatures of Broken-Symmetry Phases in Graphene. _Physical Review Letters_119, 027601 (2017). * (39) Feldman, B. E. _et al._ Fractional quantum hall phase transitions and four-flux states in graphene. _Physical Review Letters_111, 076802 (2013). ## II Acknowledgements ### General We thank A. H. Macdonald, J. D. Sanchez-Yamagishi, S. L. Tomarken, and S. P. Harvey for helpful discussions and feedback, X. Liu for fabrication help, and P. Kim for providing the transfer setup. Funding: Supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4531; the U.S. Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering under award DE-SC0001819 (D.S.W. and T.v.d.S.); NSF Graduate Research Fellowship grant DGE1144152 (D.S.W.); the STC Center for Integrated Quantum Materials, NSF grant DMR-1231319 (B.I.H.); and the Elemental Strategy Initiative conducted by MEXT, Japan, and JSPS KAKENHI grant JP15K21722 (K.W. and T.T.). Nanofabrication was performed at the Center for Nanoscale Systems at Harvard, supported in part by NSF NNIN award ECS-00335765. Author contributions: D.S.W., T.v.d.S., B.I.H., and A.Y. conceived and designed the experiments; D.S.W. fabricated the devices; D.S.W. and A.Y. performed the experiments; D.S.W., T.v.d.S., S.H.L., B.I.H., and A.Y. analyzed the data and wrote the paper; and K.W. and T.T. synthesized the hexagonal boron nitride crystals. Competing interests: The authors declare no competing financial interests. Data and materials availability: All measured data are available in the supplementary materials. ## III Correspondence Correspondence and requests for materials should be addressed to A.Y. (email: yacoby@physics.harvard.edu). ## IV Supplementary Information Supplementary Note 1. Equilibration of edge states at ‘hot spots’ One question that arises from this study is why, after decades of experimental investigation into QH ferromagnets, has this phenomenon not been observed in GaAs quantum wells? We posit that this is due to the readiness of edge states in graphene to equilibrate over small length scales due to the sharp confining potentials. Because there is a limited spatial range over which the ‘hot spot’ magnon generation can occur adjacent to the $\nu=1$ ferromagnetic bulk, spin-flip induced edge equilibration must occur over short lengths. Past studies have shown that in graphene edges of the same spin are able to fully equilibrate over length scales $<1$$\mu$m [13; 20; 21], while similar studies done in GaAs found typical lengths of around tens of microns, and sometimes up to 200 $\mu$m [28; 29; 30]. In order to experimentally verify that magnons are generated at these corner ‘hot spots’, we have fabricated a device with gated corners showing conductance changes at $E_{\mathrm{Z}}$ in accordance with our magnon model (Fig. S11). The difference in experimentally-determined equilibration lengths between graphene and GaAs is likely due to the sharper confining potentials in graphene, which allow for small spatial edge channel separation—increasing the likelihood of inter-channel scattering. Additionally, a smooth potential may allow for edge reconstruction [31], which if present, could also affect the inter-channel scattering rate and limit magnon generation. Experiments in GaAs based systems have shown that edge recondsturction plays an important role due to the smooth confining potential [32]. The graphene devices investigated in this work have gate electrodes located at just a few tens of nanometers distance away, likely limiting the amount of edge reconstruction. Furthermore, the close proximity of the metal gates to the graphene may screen the electric fields that cause edge reconstruction, which is another potential difference with GaAs-based systems [33]. However, interestingly, we note that although an electrostatically-defined confinement potential is able to suppress the magnon signal, it does not eliminate it completely (Fig. S11). This suggests that strong edge disorder is not required for the $\nu$ = 2 and $\nu$ = 1 edge channels to equilibrate, and that it is possible that magnons could be generated in GaAs devices with a sharp electrostatic confinement potential. We note, however, that while magnon generation may be possible, magnon propagation may not be as efficient in GaAs systems due to large spin-orbit coupling [34] and more nuclear spins [35] relative to graphene, which could facilitate magnon dissipation. Such dissipative processes would be important because, as we describe in the main text, magnon generation itself does not affect the sample conductance – only when magnons are able to propagate and are absorbed in by electrons in other edge channels do we detect a change in sample conductance. Additionally, we note that even when we do not explicitly add an extra edge state near the contacts (by gating the side regions to $\nu$ = 2), contact doping of the graphene by the Cr/Au leads [23] introduces additional spin-down edge states—which also leads to magnon generation at EZ. Supplementary Note 2. Calculating $V_{\mathrm{dc}}$ necessary to exceed $V_{\mathrm{EZ}}$ given a finite contact resistance In a two-terminal measurement, the applied bias voltage ($V_{\mathrm{dc}}$) drops over both the contact resistances at both the source and the drain. The d.c. current is therefore \[I_{\mathrm{dc}}=\frac{V_{\mathrm{dc}}}{2R_{\mathrm{C}}+R_{\mathrm{Q}}}\] (S1) where $R_{\mathrm{Q}}$ is the quantum resistance of an edge channel, and where $R_{\mathrm{C}}$ includs both the contact resistance at each lead as well as the filtering on the lines. The filtering on each line is 4.5k$\Omega$, and the contact resistance at each lead of a typical device is about 500$\Omega$. The actual d.c. voltage that drops over the edge channel ($V_{\mathrm{dc}}^{\prime}$) is therefore given by \[V_{\mathrm{dc}}^{\prime}=I_{\mathrm{dc}}R_{\mathrm{Q}}\] (S2) Therefore: \[V_{\mathrm{dc}}=\frac{V_{\mathrm{dc}}^{\prime}\Big{(}2R_{\mathrm{C}}+R_{ \mathrm{Q}}\Big{)}}{R_{\mathrm{Q}}}\] (S3) In our figures, we use ‘$V_{\mathrm{EZ}}$’ ($V_{\mathrm{EZ}}$ = -$E_{\mathrm{Z}}/e$) to denote the bias at which -$eV_{\mathrm{dc}}^{\prime}$ reaches the Zeeman energy ($E_{\mathrm{Z}}=g\mu_{B}B_{T}$). Supplementary Note 3. Circuit analysis for two-terminal conductance measurement. <figure><img src="content_image/1801.08534/FigS1.png"><figcaption>Figure S1: Schematic of a two-terminal device where a voltage is sourced atthe left contact and drained at the right. The negative and positive signsdenote magnon generation for negative and positive Vdc respectively. ε1 and ε2denote locations where magnon absorption occurs. Arrows indicate how thechemical potential redistributes after magnon generation, and the chemicalpotential of the edge states after magnons are absorbed are labeled. theelectrochemical potential applied by the voltage source is defined as μ. μ1and μ2 are the chemical potential reservoirs connected to the source and drainvia a contact resistance RC (assumed to be identical for both contacts).</figcaption></figure> In Fig. S1 The (particle) current conservation equation at the source reservoir (labeled $\mu_{1}$) is: \[\frac{2\mu_{1}}{R_{\mathrm{Q}}}=\frac{\mu_{2}-\varepsilon_{2}}{R_{\mathrm{Q}}} +\frac{\mu_{1}+\varepsilon_{1}}{R_{\mathrm{Q}}}+\frac{\mu-\mu_{1}}{R_{\mathrm{ C}}}\] (S4) where $R_{\mathrm{Q}}$ is the resistance quantum. As described in the main text, $\varepsilon_{i}$ denotes the chemical potential redistributed between edge states at the $i$th contact. Additionally, although there are indeed also spin-flips occurring at the negative-bias magnon generation location when positive bias magnons are being generated (and for the reverse case), we ignore these in our analysis because they do not contribute to changes in the conductance. The equation at the drain contact is: \[\frac{\mu_{2}+\varepsilon_{2}}{R_{\mathrm{Q}}}+\frac{\mu_{1}-\varepsilon_{1}}{ R_{\mathrm{Q}}}=\frac{2\mu_{2}}{R_{\mathrm{Q}}}+\frac{\mu_{2}}{R_{\mathrm{C}}}\] (S5) Solving for $\mu_{2}$ we find \[\mu_{2}=\frac{\mu+\varepsilon_{2}-\varepsilon_{1}}{2+\frac{R_{\mathrm{Q}}}{R_{ \mathrm{C}}}}\] (S6) Using the chemical potential of the voltage source as $\mu=-eV_{\mathrm{dc}}$, and the charge current $I=-eI_{\mathrm{P}}$ (where $I_{\mathrm{P}}$ is defined as $\frac{1}{e^{2}}$$\frac{\mu_{2}}{R_{\mathrm{C}}}$ to normalize the units) the differential conductance measured is \[\begin{split}\frac{\mathrm{d}I}{\mathrm{d}V_{\mathrm{dc}}}& =\frac{\mathrm{d}I_{\mathrm{P}}}{\mathrm{d}\mu}=\frac{\big{(} \mathrm{d}\mu_{2}/R_{\mathrm{C}}\big{)}}{\mathrm{d}\mu}\end{split}\] (S7) \[\begin{split}\frac{\mathrm{d}I}{\mathrm{d}V_{\mathrm{dc}}}=\frac{ 1}{2R_{\mathrm{C}}+R_{\mathrm{Q}}}\Big{(}1+\frac{\mathrm{d}\varepsilon_{2}}{ \mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_{1}}{\mathrm{d}\mu}\Big{)}\end{split}\] (S8) This becomes Equation 1 (main text) in the absence of contact resistance ($R_{\mathrm{C}}$ =0). Supplementary Note 4. Circuit analysis for non-local voltage measurements. <figure><img src="content_image/1801.08534/FigS2.png"><figcaption>Figure S2: Schematic circuit diagram of the multi-terminal device Device 2 -optical micrograph shown in Fig. 2A) that is used to measure SNL. μ1, μ2, μ3,μ4 and μ5 are the chemical potential reservoirs connected to each contact(L1-L5) by a contact resistance RC. The electrochemical potential applied bythe voltage source is defined as μ. Voltage is sourced at L3 and drained fromL2. L1 is floating. SNL is measured between L4 and L5. The negative andpositive signs denote magnon generation for negative and positive Vdcrespectively. ε1, ε2, ε3, ε4, and ε5 label locations of magnon absorption.</figcaption></figure> In Fig. S2 the chemical potential of the edge states after magnons are absorbed are labeled as $\mu_{i}-\varepsilon_{i}$ and $\mu_{i}+\varepsilon_{i}$ where ‘$i$’ denotes the contact where the chemical potential originates. We calculate the conductance expected after the Zeeman energy has been reached. This device has 5 contacts in total. We write a current conservation equation at each contact: \[\mathrm{L}_{1}:\mu_{2}-\varepsilon_{2}+\mu_{1}+\varepsilon_{1}=2\mu_{1}\] (S9) \[\mathrm{L}_{2}:\mu_{3}-\varepsilon_{3}+\mu_{2}+\varepsilon_{2}=2\mu_{2}+\mu_{2 }\frac{R_{\mathrm{Q}}}{R_{\mathrm{C}}}\] (S10) \[\mathrm{L}_{3}:(\mu-\mu_{3})\frac{R_{\mathrm{Q}}}{R_{\mathrm{C}}}+\mu_{3}+ \varepsilon_{3}+\mu_{4}-\varepsilon_{4}=2\mu_{3}\] (S11) \[\mathrm{L}_{4}:\mu_{5}-\varepsilon_{5}+\mu_{4}+\varepsilon_{4}=2\mu_{4}\] (S12) \[\mathrm{L}_{5}:\mu_{1}-\varepsilon_{1}+\mu_{5}+\varepsilon_{5}=2\mu_{5}\] (S13) Solving for $\mu_{2}$ we find \[\mu_{2}=\frac{\mu+\varepsilon_{2}-\varepsilon_{3}}{2+\frac{R_{\mathrm{Q}}}{R_{ \mathrm{C}}}}\] (S14) Therefore, \[\begin{split}\frac{\mathrm{d}I}{\mathrm{d}V_{\mathrm{dc}}}=\frac{ \mathrm{d}I_{\mathrm{P}}}{\mathrm{d}\mu}=\frac{\mathrm{d}\big{(}\mu_{2}/R_{ \mathrm{C}}\big{)}}{\mathrm{d}\mu}=\frac{1}{2R_{\mathrm{C}}+R_{\mathrm{Q}}} \Big{(}1+\frac{\mathrm{d}\varepsilon_{2}}{\mathrm{d}\mu}-\frac{\mathrm{d} \varepsilon_{3}}{\mathrm{d}\mu}\Big{)}\end{split}\] (S15) The non-local voltage measured is: \[S_{\mathrm{NL}}=\frac{\mathrm{d}V_{\mathrm{NL}}}{\mathrm{d}V}=\Big{(}\frac{ \mathrm{d}\varepsilon_{4}}{\mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_{5}}{ \mathrm{d}\mu}\Big{)}\] (S16) By defining $S_{\mathrm{NL}}$ as the difference between two voltage probes, any edge current which reaches the two voltage probes should not affect the measurement — although we do see some small background voltage which is explained in Fig. S7, B. A similar circuit analysis can be done for any of the configurations found in the main text or supplementary materials. Supplementary Note 5. Theoretical Notes. We first note that the energy levels shown in the Figure 3D are only schematic. The actual Landau levels will be broadened due to electron-electron interactions and, perhaps, disorder. The curves represent more accurately the energy in the middle of the Landau level, and the ordering of the levels is more meaningful than the actual energies. Our ordering of levels was guided by the following observations. For a uniform graphene system at $\nu$ = 0, it is believed that the valley anisotropy energy is large compared to the Zeeman energy, and that the ground state is a canted antiferromagnet state [10; 9]. In this half-filled N=0 Landau level, there is one electron per flux quantum on each sublattice, with spins oriented predominantly in opposite directions. In the absence of Zeeman coupling the antiferromagnetic axis could point equally well in any direction, with no difference in the energy [36]. In the presence of the Zeeman field, there is a small energy gain for the antiferromagnetic axis to line up in the x-y plane, perpendicular to the Zeeman field, allowing the spins on both sublattices to cant slightly in the direction of the Zeeman field. The energy gain for this is of order $E_{Z}^{2}$/$E_{A}$, where $E_{A}$ is the valley anisotropy energy. For a general filling fraction in the range $-1<\nu<0$, if one calculates the ground state energy in a restricted Hartree-Fock approximation, which assumes that only two of the possible spin-valley states are occupied by electrons, one generally finds that one valley, say the K valley, has one electron per flux quantum, while the other valley has occupancy $1+\nu<1$. For fillings very close to $\nu$ = 0, the system may remain in a canted configuration, but for $\|\nu\|$ exceeding a critical value, of order $E_{Z}/E_{A}$, it will be more favorable for the antiferromagnetic axis to align in the z-direction, so that the majority spin is fully aligned with the Zeeman field. (See, e.g. the discussion in [37]) Similarly, for $0<\nu<1$, we would find the antiferromagnetic spin axis to be aligned with the magnetic field, except for a small region close to $\nu=0.$ In a situation where the electron density varies rapidly in space, the spin and valley orientationsshould be determined by the dominant exchange energy, arising from the long-range part of theCoulomb interaction, which is indifferent to the specific orientation of the occupied levels in spin-valley space, but disfavors any rapid changes or discontinuities in the occupations. In a boundary between $\nu=-1$ and $\nu=1$, we are forced to have two discontinuities in occupancy, but we can avoid any other discontinuities, if we choose to fill the levels in the order suggested in Fig. 3D. Moreover, it is likely that in a relatively steep boundary, the canted orientation will be completely suppressed, and that spins will remain quantized along the z-axis. We have seen in a previous study an absence of mixing between spin states at a $\nu=1/\nu=-1$ interface, supporting our assumption of spin alignment in the present case [14]. By contrast, when the filling fraction is $\nu=0$ under the center of our gate, it is likely that the system will assume the canted orientation near the center of the gate. At the same time, there should be a strip on either side of the gate, where the filling fraction is intermediate between $\nu=1$ and $\nu=0$, where the antiferromagnetic axis is in the z-direction. An interval where the filling fraction is between $\nu=1$ and $\nu=-1$ , with spin axes parallel to z, will act as a barrier, to a spin wave incident from a region where $\nu=1$, as the energy at the bottom of the spin wave band will be raised by an amount of order the valley anisotropy energy (This should be small compared to the Coulomb exchange energy, but larger than the Zeeman energy) [38]. In the case where the filling under the gate is $\nu=-1$, we would expect the barrier regions at the two sides to be relatively thin, and it is plausible that the spin waves can tunnel rather easily through the barrier region. When the filling at the gate center is ν = 0, we would expect the barriers to be much thicker, and tunneling through the barriers should be reduced accordingly. In a bulk region where the filling is very close to ν = 1, we expect that the unoccupied spin state will have its spin opposite to the magnetic field, but it will have no particular preference for either the K or K’ valley or an arbitrary linear combination of them. Different valley polarizations may be selected near the physical boundaries of the sample, but we expect that the valley orientation in the vicinity of a gate where the charge density varies rapidly should be determined by energy considerations under the gate. It should cost relatively little energy for the valley orientation to vary smoothly between the sample edges and the gate, and we would not expect spin-wave propagation to be affected by such variations. Our analysis, based on a Hartree-Fock approximation, ignores correlation effects, which can lead to fractional quantized Hall states, varying spin polarization, and transitions between different spin states in uniform graphene sample [37; 39]. However, we would not expect such correlation effects to be important in the present case, where the charge density varies considerably on a sub-micron scale. <figure><img src="content_image/1801.08534/FigS3.png"><figcaption>Figure S3: Atomic force microscope (AFM) images of device 1 and device 2 (A-B)AFM images of the hBN/graphene/hBN heterostructures used for device 1 anddevice 2. Leads are illustrated in solid yellow and top gates are intransparent yellow. Dashed white lines outline the graphene flake. Scale bar:1 μm.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS4.png"><figcaption>Figure S4: Comparison of spin-reservoirs from contact doping and spinreservoirs from the ν=2 edge (device 1). (A) Schematic of device 1 where boththe top and back-gated regions are set to ν=1 and the magnon generation andabsorption occurs at the contacts (See Fig. 1 for optical micrograph, and Fig.S3 for AFM image). The chemical potential redistribution at each magnonabsorption site ‘i’ is labeled by εi (see discussion of εi in the main text).(B) Schematic where the top-gated region is set to ν=1 and the back-gatedregions are set to ν=2. Magnon generation and absorption occurs at theinterface between ν=1 and ν=2. (C) Two-terminal conductance measurement at B =4 T where a constant d.c. voltage (Vdc) and a 50 μV a.c. excitation voltage(Vac) are applied to the left contact and the differential conductance (dI/dV,where V = Vdc+Vac) is measured through the right contact. The two cases arecompared, one in which contact doping provides an opposite-spin reservoir asshown in (A) (VBG =1.24 V and VTG= 0.12 V) – the other where the ν=2 providesthe opposite-spin reservoir as shown in (B) (VBG =3 V and VTG = -0.18V. Thebreakdown of the quantized ν=1 plateau occurs at identical values of ±Vdc,although the value of the conductance decrease changes dramatically. This islikely due to the changes to the relative magnitudes of magnon absorption atthe different corners. The magnitudes of ε1 and ε2 may change as the ν=1 areachanges. There may also be some effects on the magnitudes of ε1 and ε2 thatarise from changing the nature of the spin reservoirs.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS5.png"><figcaption>Figure S5: Effect of relative magnon absorption on conductance using differentlead configurations. Schematic illustrations of different two-terminalconductance measurement using leads L3 and L2 where hot (cold) edges arecolored red (blue), for both μ≥EZ and μ≤−EZ . The magnon generation site islabeled by the plus (minus) sign for positive (negative) bias (See Fig. 2A foroptical micrograph, and Fig. S3 for AFM image). (A) Measurement where theentire device is tuned to ν=1, so TG1 is not shown. (Upper panel) μ≥EZ(−eVdc): magnon absorption at ε1 transfers chemical potential from a forwardmoving edge to a backwards moving edge — causing the particle current (IPwhere IP = -I/e) to decrease. Conversely, magnon absorption at ε2 transferschemical potential from a backward moving edge to a forward moving edge,increasing IP. (Lower panel) μ≤−EZ : magnon absorption at ε1 (ε2) causes anincrease (decrease) in \|−IP\|. (B) A summary of the effects of ε1 and ε2 atμ≥EZ and μ≤−EZ. For μ≥EZ, ε1 is closer to the magnon generation site, so thecurrent change caused by ε1 is predicted to be dominant and is circled in red.For μ≤−EZ, ε2 is closer to the magnon generation site, so the current changecaused by ε2 is predicted to be dominant and is circled in red. (C)Measurement where the region under TG1 (νTG1) is tuned to νTG1 = 1 while theregions outside, tuned by the back gate (νBG), are set to νBG = 2, providing aspin-down reservoir in the inner edge channel. (Left panel) μ≥EZ (−eVdc):magnon absorption at ε1 transfers chemical potential from a forward movingedge to a backwards moving edge — causing the particle current (IP where IP =−I/e) to decrease. Conversely, magnon absorption at ε2 transfers chemicalpotential from a backward moving edge to a forward moving edge, increasing IP.(Right panel) μ≤−EZ : magnon absorption at ε1 (ε2) causes an increase(decrease) in \|−IP\|. (D) A summary of the effects of ε1 and ε2 at μ≥EZ andμ≤−EZ. For μ≥EZ, ε2 is closer to the magnon generation site, so the currentchange caused by ε2 is predicted to be dominant and is circled in red. Forμ≤−EZ, ε1 is closer to the magnon generation site, so the current changecaused by ε1 is predicted to be dominant and is circled in red.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS6.png"><figcaption>Figure S6: Comparison of the breakdown of ν=0 and ν=1 LLs as a function ofdensity (device 2). (A) The region under top gate 1 (TG1) is at νTG1=0 whileoutside regions, gated only by the back gate, are at νBG=2. (B) νTG1=1 andνBG=2. (C) Two-terminal conductance measurement at B = 3 T where a constantd.c. voltage (Vdc) and a 50 μV a.c. excitation voltage are applied to L3 andthe differential conductance (dI/dV) is measured through L2. TG1 is swept fromνTG1=0 to νTG1=2, and νBG=2(VBG = 1.8V). The horizontal black dashed linesdenote ±VEZ and the location of the line cut taken in (D) is shown by thevertical purple dashed line. The bias at which νTG1=0 breaks down appearsheavily dependent on the density under TG1 while the bias at which νTG1=1breaks down is relatively independent of the density under TG1, occurring at±VEZ across the plateau. (D) The dependence of dI/dV on Vdc shows a sharpincrease at ±VEZ. (E) The dependence of dI/dV on νTG1 at Vdc=0 shows wellquantized quantum Hall plateaus at νTG1 = 0, 1, and 2.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS7.png"><figcaption>Figure S7: The conditions under which non-local voltage (SNL) is measured. (A)The conductance between L3 and L2 as a function of Vdc and VTG2 (B = 8 T).Horizontal dashed black lines indicate ±VEZ. Vertical green dashed line iswhere the line cut in Fig. 3B is taken. We see a sharp drop in conductancewhen \|Vdc\|>VEZ due to magnon generation. This drop is largely unaffected whentop gate 2 (TG2) is changed. Features at \|Vdc\|>VEZ coinciding with νTG2=−1 andνTG2=1 indicate that magnons absorbed at the non-local voltage contacts affectthe amount of magnons absorbed at the drain contact. (B) SNL is measuredbetween L4 and L5 at Vdc = 0, showing a small negative voltage when the topgate is tuned from ν=0 to ν=1. This indicates a small number of bulk carriersthat give a resistance between the two contacts — a quantity which gives asmall background to the SNL signal, which can be subtracted out whencalculating the value of SNL when \|Vdc\|>VEZ. (C) Two-terminal conductancemeasured between L3 and L2 as a function of the gate voltage on TG2 (VTG2) andon the back gate (VBG) (Vdc = 0). The line cut in Fig. 3C (main text) is meantto show the corresponding filling factors under TG2 for the voltage range onthe x-axis, with VBG = 4V (bulk at ν = 1). However, a line cut at VBG = 4Vdoes not show the transition between ν=1 and ν=2 because there is noequilibration between the ν=1 and ν=2 edges due to opposite spin polarizationAmet2014 ; Wei2017 . We therefore use a line cut taken at VBG = 6.5V (νBG=2),where the step between ν=1 and ν=2 is clear, in order to estimate the steps infilling factor at VBG = 4V. In order to account for the extra contribution indensity due to the additional 2.5V applied by the back gate, we take thevoltage interval of VTG2 at VBG = 4V and shift it up by the slope of the hallplateaus (indicated by the black arrows pointing from the red-dashed line at4V to the red-dashed line at 6.5V). (D) Conductance over the voltage range ofVTG2 indicated by the red-dashed line in (C) at fixed VBG = 6.5V.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS7E.png"><figcaption>Figure S8: (E) Comparison of a two-terminal conductance measurement across atop gate (left) to a non-local magnon-transmission measurement across the sametop gate(right), as the density in the top-gated region is tuned from ν=1 toν=2. Panel I: (left) the two edge states in the outer regions are not yet ableto enter the top-gated nu = 1 region, resulting in a conductance of e2/h. Inthe corresponding non-local measurement (right), magnons are able topropagate, yielding a non-local voltage. Panel II: (left) As the density isincreased further, the developing ν=2 region under the top gate connects withthe outer ν=2 regions, changing the measured conductance to 2e2/h. However,some ν=1 regions under the top gate remain present. In the corresponding non-local measurement (right), these remaining ν=1 regions under the top gatestill allow magnon transport. In the non-local measurements shown in Fig. 3C,we expect these regions to be responsible for the non-local voltage signalseen when the region under the top gate is transitioning from νTG2 = 1 to νTG2= 2. Panel III: Once the density is increased sufficiently, the topgatedregion consists almost entirely of ν=2, yielding a conductance of 2e2/h and anear complete suppression of magnon transport in the non-local measurement.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS8.png"><figcaption>Figure S9: Dependence of SNL on filling factors under TG1 and TG2. (A) Acircuit configuration for measuring a non-local voltage in device 2(schematic). The filling factor under TG1 (νTG1) and under TG2 (νTG2) are bothswept from -2 to 2, while the outside regions are maintained by a fixed back-gate voltage at νBG =1 (VBG = 4V, B = 8 T). The bottom panel highlights thecase of νTG1=-1: Edge states in both regions co-propagate along the boundary,but do not equilibrate because of their opposite spin-polarization Wei2017 .(B) Setting μ>EZ (VDC = -2.8 mV), and measuring SNL between L2 and L1 we findstrong non-local signals in four quadrants around ν=0. Strips where the signalis highly suppressed coincide with where the charge neutrality point occurs indensity measurements of TG1 (TG2) at B = 0 T, shown by the superimposed lightblue (dark blue) line cuts. We see similar signals in all four quadrants,indicating that magnons are not suppressed by the ν=−1 regions.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS9.png"><figcaption>Figure S10: Absence of current leakage when spin transport is mediated by theν=−1 ferromagnet. (A) A circuit used to measure SNL in device 2, as well as aleakage current across TG2 when it is tuned to νTG2=−1 (schematic). νTG1=1 forall measurements while νTG2 is swept from -2 to 2. (B) Non-local voltage (SNL)measured between L3 and L2 as a function of Vdc and VTG1. Horizontal dashedblack lines indicate ±VEZ. We note a delay in the onset of the non-localsignal for positive bias, which we tentatively attribute to the fact that theabsorption of magnon generation for positive bias is far from both the non-local leads and is mostly absorbed at ε5 and ε4, with only enough magnons togenerate a non-local signal at larger energies. (C) Conductance into L1 withthe color scale saturated. This measures the current not drained at L4 due tothe contact resistance RC. Black dashed lines indicate ±VEZ. White line cut istaken from the plot shown in (B) (over the same span of VTG1) at fixed Vdc =-1.8mV, and overlaid onto the conductance map. When we see an increase in SNLthere is a negligible amount of leakage current (g51<0.01 e2/h) measured atL1. Additionally, when we see an increase in the leakage current (g51>0.01e2/h), there is no corresponding effect on SNL. This is expected because anedge current should bring L3 and L2 to the same chemical potential. From thiswe conclude that the SNL we measure is not due to leakage current.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS10.png"><figcaption>Figure S11: Absence of current leakage when spin transport is mediated by theν=0 CAF. (A) The circuit used to measure SNL in device 2 across a ν=0 region(schematic). L1 is grounded in order to measure the amount of residual chargethat leaks through to the other side of νTG2=0. (B) When magnons are generatedin the νBG, νTG1 = 1 region and νTG2 = 0, we see an onset of SNL at energiesexceeding ±VEZ. This indicates that higher energy magnons have overcome theinterface barriers and have propagated through the νTG2 = 0 region. At morepositive gate voltages, we see the effects of the residual current on SNL (dueto the finite contact resistance of L4) which passes through when νTG2>0. Weobserve that these effects disappear once νTG2>0 and do not play a role in theSNL signal measured in this region. (C) Residual current measured at L1indicating that residual leakage does not correlate with the appearance of theνTG2=0 signal.</figcaption></figure> <figure><img src="content_image/1801.08534/FigS11.png"><figcaption>Figure S12: Verifying positive and negative bias magnon generation locations(device 3). (A) Optical micrograph of device 3. The outline of the graphene isshown by the dotted white line, and the scale bar is 1 μm. For this device, anextra BN dielectric (10nm) was used between the top gates and side gates toelectrically isolate them. There are 4 leads (L1-L4), one top gate (TG), andfour side gates (SG1 - SG4). (B) AFM image of device 3. (C) Schematic ofdevice 3 depicting a two-terminal conductance measurement between L1 (source)and L3 (drain) with L2 and L4 floating. The leads are yellow, the top gate(TG) is orange and the side gates are light blue. The regions outside of thetop-gated region (including the side gates) are tuned to ν=2 and the regionunder the top gated region is tuned to ν=1. Chiral edge states are shown bythe lines with arrows and edges with higher (lower) chemical potential arecolored red (blue) and labeled hot (cold). The side gates can be used to pushthe edge states away from the physical edge of the device (as illustrated inthe inset, for SG2). (D) Two-terminal conductance measurement at B = 7 T wherea constant d.c. voltage (Vdc) and a 50 μV a.c. excitation voltage are appliedto L1 (source) and the differential conductance (dI/dV) is measured through L3(drain). Magnons are generated when a spin-down hot edge meets a spin-up coldedge at EZ. For this configuration, we expect magnons to be generated underSG2 for positive Vdc only. When we reach +VEZ we see a change in theconductance while at −VEZ we see almost no change, as expected. (E-G) Similaranalysis for different lead configurations shows magnons are generated inaccordance with our model predictions. The effect of SG2 is stronger than SG3for unknown reasons. The exact change in conductance is difficult to predictbecause we are changing both the nature of scattering between the two edgestates Wei2017 as well as the distance between magnon generation andabsorption, so here we note only qualitative changes.</figcaption></figure>
1108.5304	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 132801, "num_imgs": 19, "llama3_tokens_count": 36007 }	[ "content_image/1108.5304/x1.png", "content_image/1108.5304/x2.png", "content_image/1108.5304/x3.png", "content_image/1108.5304/x4.png", "content_image/1108.5304/x6.png", "content_image/1108.5304/x7.png", "content_image/1108.5304/x8.png", "content_image/1108.5304/x9.png", "content_image/1108.5304/x10.png", "content_image/1108.5304/x11.png", "content_image/1108.5304/x13.png", "content_image/1108.5304/x14.png", "content_image/1108.5304/x15.png", "content_image/1108.5304/x16.png", "content_image/1108.5304/x17.png", "content_image/1108.5304/x18.png", "content_image/1108.5304/x19.png", "content_image/1108.5304/x20.png", "content_image/1108.5304/x21.png" ]	# Solar magnetism eXplorer (SolmeX) Exploring the magnetic field in the upper atmosphere of our closest star H. Peter H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy L. Abbo H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy V. Andretta H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy F. Auchère H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy A. Bemporad H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy F. Berrilli H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy V. Bommier H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy A. Braukhane H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy R. Casini H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy W. Curdt H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy J. Davila H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy H. Dittus H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy S. Fineschi H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy A. Fludra H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy A. Gandorfer H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy D. Griffin H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy B. Inhester H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy A. Lagg H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy E. Landi Degl’Innocenti H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy V. Maiwald H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy R. Manso Sainz H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy V. Martínez Pillet H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy S. Matthews H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy D. Moses H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy S. Parenti H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy A. Pietarila H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy D. Quantius H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy N.-E. Raouafi H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy J. Raymond H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy P. Rochus H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy O. Romberg H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy M. Schlotterer H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy U. Schühle H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy S. Solanki H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy D. Spadaro H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy L. Teriaca H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy S. Tomczyk H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy J. Trujillo Bueno H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy J.-C. Vial H. Peter, W. Curdt, A. Gandorfer, B. Inhester, A. Lagg, U. Schühle, S. Solanki, L. Teriaca, Max Planck Institute for Solar System Research, 37191 Katlenburg-Lindau, Germany ¹L. Abbo, A. Bemporad, S. Fineschi, INAF Osservatorio Astronomico di Torino, Italy V. Andretta, INAF Osservatorio Astronomico di Capodimonte, Napoli, Italy F. Auchère, J.-C. Vial, Institut d’Astrophysique Spatiale, Orsay, France F. Berrilli, Università degli Studi di Roma “Tor Vergata”, Italy V. Bommier, Observatoire de Paris-Meudon, France A. Braukhane, H. Dittus, V. Maiwald, D. Quantius, O. Romberg, M. Schlotterer, DLR Institute of Space Systems, Bremen, Germany R. Casini, S. Tomczyk, NCAR / High Altitude Observatory, Boulder, CO, USA J. Davila, NASA / GSFC, Greenbelt, MD, USA A. Fludra, D. Griffin, STFC Rutherford Appleton Laboratory, Oxon, UK E. Landi Degl’Innocenti, Università degli Studi di Firenze, Italy R. Manso Sainz, V. Martínez Pillet, J. Trujillo Bueno, Instituto de Astrofísica de Canarias, Tenerife, Spain S. Matthews, Mullard Space Science Laboratory, Surrey, UK D. Moses, Naval Research Laboratory, Washington, DC, USA S. Parenti, Royal Observatory of Belgium, Brussels, Belgium A. Pietarila, National Solar Observatory, Tucson, AZ, USA N.-E. Raouafi, Johns Hopkins University / APL, Laurel, USA J. Raymond, Smithsonian Astrophysical Observatory, Cambridge, USA P. Rochus, Centre Spatial de Liège, Université de Liège, Belgium D. Spadaro, INAF Osservatorio Astrofisico di Catania, Italy [FOOTNOTE:2][ENDFOOTNOTE] Version: August 26, 2011 ###### Abstract The magnetic field plays a pivotal role in many fields of Astrophysics. This is especially true for the physics of the solar atmosphere. Measuring the magnetic field in the upper solar atmosphere is crucial to understand the nature of the underlying physical processes that drive the violent dynamics of the solar corona – that can also affect life on Earth. SolmeX, a fully equipped solar space observatory for remote-sensing observations, will provide the first comprehensive measurements of the strength and direction of the magnetic field in the upper solar atmosphere. The mission consists of two spacecraft, one carrying the instruments, and another one in formation flight at a distance of about 200 m carrying the occulter to provide an artificial total solar eclipse. This will ensure high-quality coronagraphic observations above the solar limb SolmeX integrates two spectro-polarimetric coronagraphs for off-limb observations, one in the EUV and one in the IR, and three instruments for observations on the disk. The latter comprises one imaging polarimeter in the EUV for coronal studies, a spectro-polarimeter in the EUV to investigate the low corona, and an imaging spectro-polarimeter in the UV for chromospheric studies. SOHO and other existing missions have investigated the emission of the upper atmosphere in detail (not considering polarization), and as this will be the case also for missions planned for the near future. Therefore it is timely that SolmeX provides the final piece of the observational quest by measuring the magnetic field in the upper atmosphere through _polarimetric_ observations. Keywords:Sun: atmosphere Magnetic fields Space vehicles: instruments Techniques: polarimetic ESA Cosmic Vision † [FOOTNOTE:†][ENDFOOTNOTE] ∎ ## 1 Introduction The Sun is our closest star and the only one for which we can resolve details on the surface, in its atmosphere, and in the astrosphere surrounding it. The direct impact the Sun has on the Earth and human life makes it the most important object in the sky. A total eclipse of the Sun is still one of the most breathtaking and captivating wonders of nature. While in bygone ages it frightened whole societies, today scientists are intrigued by the beauty of the physics hidden behind the aesthetic pictures taken during these rare events. During a total eclipse we see the dilute hot outer atmosphere of the Sun. Composed of million K hot plasma permeated by magnetic field, it presents us with a unique plasma physics laboratory, representing a parameter space not accessible on Earth. The magnetic field confines the coronal structures, drives the plasma in the upper solar atmosphere, provides the energy released in the form of thermal and kinetic energy, and controls the acceleration of particles. Therefore, it is of pivotal importance to know the state of the magnetic field to reveal the nature of the governing processes in the outer solar atmosphere. So far measurements of the solar magnetic field are mostly restricted to the low layers of the solar atmosphere. Extrapolation techniques containing numerous assumptions are used to estimate the magnetic field in the transition region from the chromosphere to the corona and in the corona itself. Until now, few facilities provide measurement access to the magnetic field in the upper atmosphere, and each only represents a partial aspect and suffers from significant constraints and deficiencies. The _measurement objective_ of SolmeX is to fill this gap of the observational puzzle and to present the first comprehensive set of magnetic field measurements of the upper solar atmosphere. Such observations are the key to address the main _science objectives_ of SolmeX which are summarized through the following questions: * What is the magnetic structure of the outer solar atmosphere? * What is the nature of the changes of the magnetic field over the solar cycle? * What drives large-scale coronal disruptions such as flares and coronal mass ejections? * How do magnetic processes drive the dynamics and heat the outer solar atmosphere? * How does the magnetic field couple the solar atmosphere from the photosphere to the corona? These scientific goals of SolmeX are closely related to two of the four main questions of the _Cosmic Vision_ plan, namely “_how does the solar system work?_” and “_what are the conditions for planet formation and the emergence of life?_” Besides contributing to the understanding of the upper solar atmosphere as an astrophysical object in its own right, SolmeX will provide essential information on the origin of “space weather”, which affects life and the habitability on Earth as well as other solar system bodies. Unveiling the nature of the outer atmosphere of the Sun is the most important step in understanding the magnetised atmospheres of other cool stars. This is of particular interest for younger stars, because the magnetic field surrounding the star plays a decisive role in the formation of planets around them. To reach the objective to measure the magnetic field in the upper atmosphere, SolmeX will investigate the emission and its polarization from the extreme ultraviolet to the infrared. The magnetic field in the source region of the emission modifies the polarization through the Zeeman and Hanle effects. Measuring not only the spectral intensity, but also the polarization state, SolmeX will provide the essential data to decipher the imprint of the magnetic field. Through inversion procedures and a comparison with numerical forward models SolmeX gives access to the magnetic field in the upper solar atmosphere. The key for the coronagraphic observations is to provide an artificial eclipse. To minimize stray light for the delicate polarimetric observations and to reach the required high spatial resolution, the occulter that blocks the light from the solar disk has to be far removed from the entrance aperture. SolmeX will achieve this by mounting an occulting disk on a separate spacecraft that serves as an “artificial Moon” for the coronagraphs on the master spacecraft, which carries all science instruments. The spacecraft carry out a formation flight, separated by 200 m.. We plan an orbit around the Lagrange point between Sun and Earth (L1), which provides an unobstructed view of the Sun at constant thermal conditions with the smallest possible impact on the formation flight. The anticipated mission lifetime spans a nominal phase of three years and an extended phase of three years. The nominal phase will cover the rise of solar activity, while the extended phase will monitor the solar activity maximum, assuming a launch in 2022. SolmeX comprises five instruments dedicated to study five prime target regions off-limb and on-disk. VIRCOR and ChroME are imaging spectro-polarimeters, CUSP and SUSP are slit spectro-polarimeters. EIP provides imaging polarimetry data. \begin{tabular}{} \end{tabular} The SolmeX mission is based on significant heritage from the Compass proposal [1] to a previous M-class call by ESA. The concept of CUSP, VIRCOR and SUSP follows the Compass proposal, but EIP and especially ChroME provide new developments. While the basic idea for the mission remains similar, the two spacecraft for the formation flight are a completely new design and the mission profile has been developed again from scratch. The scientific objectives are summarized in Sect. 2 and the basic measurement techniques are briefly presented in Sect. 3. In Sect. 4 the five prime target structures and the relevant diagnostics to measure the magnetic field are described. The mission profile with two spacecraft in formation flight is presented in Sect. 5. Section 6 describes the details of the five payload instruments and Sect. 7 gives some details on the required spacecraft. Section 8 concludes the paper. ## 2 Scientific objectives One of the greatest challenges in space astrophysics in the immediate future will be the empirical investigation of the magnetic field vector in a variety of astrophysical plasmas, such as the solar corona, circumstellar envelopes, accreting systems, etc. In particular, we need to decipher the three-dimensional magnetic structure of the solar upper atmosphere. This is because the magnetic field (a) is the source for the energy to heat the corona, (b) is governing the transformation of magnetic into thermal energy and (c) is driving and channelling the dynamics in the upper atmosphere. The upper atmosphere of the Sun is highly structured, which is evident in contrast-enhanced eclipse photographs (cf. Fig. 1, left panel) and from images in the extreme ultraviolet. In fact, this is due to the magnetic field, but our knowledge of the magnetic field in the upper solar atmosphere mainly depends on _static_ extrapolations from the solar surface (cf. Fig. 1, right panel). While this technique is useful for a number of science questions, there is the acute need to actually _measure_ the magnetic field and its evolution in the _dynamic_ solar upper atmosphere. <figure><img src="content_image/1108.5304/x1.png"><figcaption>Figure 1: _Left panel:_ Total solar eclipse of 11 Jul 2010 (courtesyDruckmüller, Dietzel, Habbal, Rušin;<http://www.zam.fme.vutbr.cz/>∼druck/eclipse/). The image is contrastenhanced. _Right panel:_ Magnetic field lines as following from a global MHDmodel for the same moment as the eclipse observation on the left. On the solarsurface a magnetogram of the photospheric field is shown (courtesy Linker,Mikić, Lionello, Riley, Titov;<http://www.predsci.com/corona/jul10eclipse/jul10eclipse.html>).</figcaption></figure> Our current lack of measurements of the magnetic field in the upper chromosphere, transition region, and corona is a major obstacle to our understanding of the outer solar atmosphere. Consequently, the main scientific goal of the proposed mission is to provide the first comprehensive set of measurements of the magnetic field in the chromosphere, the transition region to the corona, and the corona itself through remote-sensing techniques over a broad range of spatial and temporal scales. In this section we will limit the discussion of the scientific objectives to a summary in tabular form and a few examples. Table 1 shows how the five major science questions are subdivided into more detailed questions and how these map to the measurements of the SolmeX instruments (Sect. 6) of the prime target regions (Sect. 4). The combined measurements on different temporal and spatial scales will investigate the magnetic coupling of various solar structures. This is central to achieve the scientific goals. ### Science example 1: Magnetic structure of coronal loops Imaging observations show loops in great detail that reach high into the corona, up to altitudes of the order of 100 Mm ($\approx$10% of the solar radius) or more (Fig. 2). It is generally accepted that these outline magnetic field lines, which are loaded with hot plasma as a consequence of the coronal heating process. Furthermore, the magnetic structure of coronal loops is unknown, e.g., their twist and their magnetic connection to the lower atmosphere are open issues. <figure><img src="content_image/1108.5304/x2.png"><figcaption>Figure 2: The corona at 106 K seen in the extreme ultraviolet. The right panelshows an enlargement of the framed region in the left panel. SolmeX will notonly provide intensity images like these, but also data on the magnetic fielddirection, which is important to understand, e.g., the nature of coronal loops(arrow in right panel). Image courtesy SDO/AIA team.</figcaption></figure> SolmeX will measure not only the appearance and dynamical evolution of the loops in coronal extreme UV emission, but will also measure the strength and direction of the magnetic field along the loop and at its base where it is rooted to the chromosphere. Furthermore, coronagraphic observations with SolmeX will provide eclipse-like images (see Fig. 1 for an eclipse image) with unprecedented quality very close to the limb, and this will enable us to study the evolution of coronal loops, e.g., above active regions, in hitherto unknown detail. ### Science example 2: Spicules and their coupling to the corona The Sun is highly structured and very dynamic on smaller scales of the order of 1 Mm in the chromosphere and in the transition region to the corona. A multitude of small-scale short-lived phenomena can be seen, such as spicules, fibrils, explosive events, etc. An example of spicules seen on the disk and above the limb is shown in Fig. 3. Spicules have been extensively studied with Hinode in the chromospheric emission of Ca II. Indications of Alfvén waves that travel along the spicules have been found [2]. Recent observations provide evidence of high-velocity upflows in type-II spicules, which feed mass into the hot corona and thus are an integral part of the coronal heating problem [3; 4]. However, all these conclusions are based on intensity images alone, and with ground-based telescopes only low-resolution measurements of the magnetic field in spicules could be performed [5, and references therein]. At somewhat higher temperatures, there are small transition region loops found crossing the chromospheric network. These are puzzling because there appears to be no connection from these small network loops to the underlying photospheric magnetic field [6]. <figure><img src="content_image/1108.5304/x3.png"><figcaption>Figure 3: Spicules on the disk (dark, left) and at the limb (bright, right).ChroME on SolmeX will deliver similar images, but now including measurementsof the magnetic field in these thin structures. Left panel: courtesy of A.Pietarila, right panel from [7].</figcaption></figure> SolmeX will provide key data to address these questions by inferring the magnetic field from the observed polarization in lines formed in the chromosphere and the transition region. Measurements of the magnetic field in Mg II k (279.6 nm), Ly-$\alpha$, and C IV (154.8 nm) will yield magnetic information on spicules on the disk and above the limb, and on small transition region loops above the chromospheric network. Such observations will uncover the role of such features in heating the plasma. The coronagraphic instruments will provide the crucial connection to the structures in the corona. ### Science example 3: magnetic structure of a CME during an eruption Once a coronal mass ejection (CME) lifts off, it expands, and in the process some CMEs reveal their internal structure (Fig. 4). Our current information is based on intensity images, thus no information on the magnetic field is available. This is unfortunate because competing models for CME eruption make detailed predictions on their internal magnetic structure, e.g., breakout, flux rope catastrophe, or kink instability [8]. As CMEs often lift off with speeds of the order of 1000 km/s, they cross a field of view of one solar radius in 10 minutes, which sets minimum requirements for the cadence of observations. <figure><img src="content_image/1108.5304/x4.png"><figcaption>Figure 4: Coronal mass ejection (CME) with a helical structure observed byLasco C2 on SOHO. Through spectro-polarimetric observations SolmeX will allowus to measure the magnetic field in these structures for the first time. (Thethin white line outlines the solar limb).</figcaption></figure> Through high-cadence infrared coronagraphic spectro-polarimetry SolmeX will provide outstanding data that will not only reveal the density structure of CMEs, but also return the magnetic field vector. This includes the phases before, during and after an eruption, as the spectro-polarimetric coronagraph can reach a cadence of down to 30 s, well below the crossing time of a CME through the field-of-view. This will decide the debate of the different CME models and show the true physics of the CME eruption. ![](x5.png) Table 1: Summary of science goals and their relation to the prime target structures and observational techniques and the measurement requirements for the instruments (Sects. 4 and 6). ## 3 Observable parameters and measurement techniques The only means of investigating the upper atmosphere of the Sun at distances of below a couple of solar radii above the surface is through remote sensing. We can detect the direction, flux, energy, and state of polarization of the incoming photons. To observe the corona in the visible and infrared (IR), one has to block the light from the solar disk by an occulter (creating an artificial eclipse) because otherwise it would outshine the faint emission from the corona (see Sect. 5). In contrast, observations in the ultraviolet (UV) and extreme UV (EUV) range also show the transition region and the corona in front of the solar disk, because the cool photosphere is relatively dark at these short wavelengths. Therefore, UV/EUV instruments and coronagraphs are complementary. The emission of the solar coronal plasma in the EUV (and X-rays), observable only from space, has been studied extensively by, e.g., SOHO, TRACE, Hinode, and SDO. These missions provided high-resolution intensity images and spectra. They have revolutionised our understanding of the outer solar atmosphere, because they allowed quantitative measurements of the kinetic temperature, composition, density, and dynamics of the coronal plasma. However, the information on the coronal magnetic field is encoded in the polarization of the spectral lines, and was therefore inaccessible to previous space instruments. For the first time the SolmeX instruments will measure the polarization of light from the chromosphere and the corona from space, on-disk as well as off-limb. Thus, they will provide a comprehensive set of measurements of the coronal magnetic field. Besides this unique capability, all instruments perform as well or better than previous and existing space-borne instruments of their category when operated in non-polarimetric mode. The details of these instruments are discussed in Sect. 6 and summarized in Table 2 The Stokes vector,$(I,Q,U,V)$, is used to describe the polarization state of the light [9]. Here $V$ is the intensity difference between right and left circular polarization, $Q$ is the intensity difference between linear polarization parallel and perpendicular to a given reference direction, and $U$ is the intensity difference between linear polarization at $+45^{\circ}$ and $-45^{\circ}$. $I$ is the total intensity at the wavelength under consideration. SolmeX will measure the Stokes vectors of near IR, optical and EUV spectral lines and use the Zeeman and Hanle effects to determine the vector magnetic field. The Zeeman effectis a manifestation of the interaction of the magnetic moment of the atom and the magnetic field. In the presence of a magnetic field, the spectral lines split into components with circular and linear polarization, depending on the angle between the line-of-sight and the magnetic field vector. The Zeeman effect is more prominent in circular polarization (longitudinal Zeeman effect), and particularly in the infrared (IR). It is a valuable tool for coronagraphic observations above the limb [10]. Circular polarization is measurable in some chromospheric and low coronal ultraviolet (UV) lines also when pointing to active regions on the solar disk, although with smaller amplitudes. Previous attempts to measure polarization in UV lines did not achieve the necessary signal-to-noise ratio (see Sect. 4.4). The most recent attempt with the SUMI rocket (summer 2010) unfortunately did not provide polarimetric data. SolmeX is designed to overcome the previous shortcomings. The Hanle effectis the magnetic modification of atomic level polarization (population imbalances and quantum coherences among the sublevels of degenerate atomic levels), which results from the absorption of anisotropic radiation. The (magnetically modified) atomic level polarization leads to linear (scattering) polarization of the emitted light. Measuring this polarization gives access to the magnetic field. Depending on the scattering geometry, the modification consists of a reduction or enhancement of the linear polarization amplitude and in a rotation of the direction of linear polarization. In general, the Hanle effect in ultraviolet (UV) permitted lines (e.g., Ly-$\alpha$) is sensitive to the strength and orientation of the magnetic field vector. In coronal forbidden lines, the Hanle effect saturates and is sensitive to the field orientation only [11]. This is one of the reasons why the Hanle effect diagnostics in permitted UV and forbidden IR lines are complementary and their combination gives access to a large range of magnetic field strengths. ## 4 Prime target structures and magnetic field diagnostics with SolmeX SolmeX will use five different observational techniques to derive the magnitude and direction of the magnetic field. Each of these techniques is best suited for one prime target structure as detailed below. Table 1 shows how the observational techniques relate to the prime targets of the SolmeX instruments and the science questions. While all instruments for SolmeX will also provide traditional diagnostics well known from previous space missions, e.g., density or velocity diagnostics, we will concentrate in the following on the new diagnostic tools to measure the magnetic field in the prime target structures. ### Large-scale corona above the limb <figure><img src="content_image/1108.5304/x6.png"><figcaption>Figure 5: Average magnetic field in the corona above the limb as derived fromdifferent sources. Over-plotted in colour are minimum field strengthsensitivities of the emission lines to be observed by CUSP on SolmeX.</figcaption></figure> The large-scale corona above the limb, e.g. at the boundaries between coronal holes and quiet Sun or in coronal streamers, can be expected to show only weak magnetic fields of 1 G to 100 G (Fig. 5). This requires the use of the Hanle effect in 90${}^{\circ}$ scattering geometry, which also has the advantage that it is sensitive to magnetic fields that are randomly oriented within the observational spatio-temporal resolution element (contrary to the Zeeman effect polarization). The best-suited spectral lines are found in the UV-range, the lines of the Lyman series of H and O VI at 103.2 nm [e.g. 12]. Above the limb these show high linear polarization (up to 25%), which is modified by the magnetic field [13]. Using a roll of the SOHO spacecraft, SUMER was used to estimate the linear polarization of the O VI 103.2 nm line via intensity modulation as a function of the roll angle [Raouafi et al. 1999]. 3G were found at 0.3 R above the limb, which proved the measurement concept [11]. Using a roll of the SOHO spacecraft, SUMER was used to estimate the linear polarization of the O VI 103.2 nm line via intensity modulation as a function of the roll angle[14]. The derived polarisation corresponded to 3 G at 0.3 $R_{\odot}$ above the limb, which proved the measurement concept [15]. The full potential of this method can only be reached by a dedicated UV spectro-polarimeter for off-limb observations that occults the UV light from the solar disk. Only these coronagraphic observations allow a sufficiently low level of instrumental scattered light. This will be done by CUSP on SolmeX, reaching signal-to-noise ratios of 100 in polarimetric measurements of UV lines, thus providing sufficient sensitivity for the expected magnetic field strengths (Fig. 5). ### Off-limb corona above active regions This target is best measured by an infrared (IR) coronagraph with an imaging spectro-polarimeter. At the high magnetic field strengths of active regions, the IR lines show a large splitting of the Zeeman components. This allows us to deduce the direction and the strength of the magnetic field, if the linear polarization signal produced by scattering processes is also taken into account [16]. Because the corona above an active region (when located near the limb) is bright above the limb, temporal changes on the time scale of minutes can be detected making it possible to study Alfvén waves, i.e., distortions of the magnetic field. These measurements have been performed on the ground, e.g., with the COMP [17]. They showed some 30 G at 0.05 $R_{\odot}$ above the limb [18] and indicate that the polarimetric accuracy has to be about $10^{-4}$ (circular and linear polarization compared to the total intensity) to obtain reliable values for the magnetic field. Ground-based observations suffer from a high level of stray light in the Earth’s atmosphere, limiting the signal-to-noise ratio, and thus the polarimetric accuracy. This prevents observations higher than ${\approx}0.15\,R_{\odot}$ above the limb [19], and underlines the pressing need for space-based observations. <figure><img src="content_image/1108.5304/x7.png"><figcaption>Figure 6: Predicted linear polarization signal Q/I of the coronal Fe X line at17.4 nm for on-disk observations for different heights h above the surface.With densities of some 108 cm−3, a polarimetric signal Q/I of well above 10−3can be expected for typical coronal loops, which will be detectable with EIPon SolmeX. [See 20]. The left panel shows a TRACE coronal image similar towhat we expect to see in the intensity channel of EIP.</figcaption></figure> ### Coronal structures on the disk On-disk coronal structures, including those located close to the solar limb, are best observed in permitted extreme UV (EUV) lines produced by the $10^{6}$ K coronal plasma. This was done with great success by EIT/SOHO, TRACE, or most recently AIA/SDO by high-resolution imaging. But they provide information on the intensity only. It has been shown that permitted EUV lines of Fe X (formed around $10^{6}$ K) should display linear polarization signals that can be used to infer the direction of the magnetic field in coronal loops and arcades [20]. The ground level of Fe X is strongly polarized by anisotropic radiation pumping in a forbidden-line transition at visible wavelengths (where the solar disk is very bright). A significant fraction of this atomic level polarization can be transferred to the upper level of the permitted Fe X line at 17.4 nm by collisional excitation. The spontaneous emission from the thus polarized upper level then generates linearly polarized emission in the EUV line of Fe X at 17.4 nm (i.e., a wavelength at which the underlying solar disk is practically dark). For magnetic fields above 0.01 G the lower level of the EUV line is in the Hanle saturation regime. Therefore, we can derive the direction of the magnetic field from the linear polarization signal (but not its strength). A first estimate of the ratio of radiative to collisional excitations in the Fe X 17.4 nm line suggests that the linear polarization caused by scattering in the EUV line itself is less significant than that caused by the polarizing mechanism explained above. For typical coronal structures with densities up to $10^{9}$ cm${}^{-3}$ the linear polarization signal of the Fe X line at 17.4 nm is expected to be a fraction of a percent (cf. Fig. 6), which is detectable with the imaging polarimeter EIP proposed here for SolmeX. <figure><img src="content_image/1108.5304/x8.png"><figcaption>Figure 7: Predicted linear polarization signal of Ly-α for on-diskobservations. For different strengths of a horizontal magnetic field the expected Q/Iprofile is plotted for a close-to-the-limb observation (left) and for diskcentre (right). The dashed line shows the polarimetric accuracy of SUSP forSolmeX. Figures from [21]. See also [22].</figcaption></figure> ### Structures of the transition region and low corona on the disk This target is best observed with a UV spectro-polarimeter. This instrument has to combine good spectral coverage and spectral purity with the ability to image an appropriate region. For on-disk observations two suitable lines are Ly-$\alpha$ at 121.5 nm and C IV at 154.8 nm. Both are formed in the transition region from the chromosphere to the corona at different heights, with individual structures reaching well into the corona up to some 30 Mm. Scattering processes in the solar transition region plasma are expected to produce linear polarization in Ly-$\alpha$, whose modification via the Hanle effect reveals the magnetic field [see 21]. While close to the limb depolarization is expected, in the forward-scattering geometry of a disk centre observation the Hanle effect of inclined magnetic fields creates linear polarization [23]. In both cases we can expect linear polarization of the order of 0.5% for Ly-$\alpha$ (see Fig. 7). Measuring the Hanle effect in Ly-$\alpha$ is the main objective of the recently-proposed CLASP sounding rocket experiment [24]. <figure><img src="content_image/1108.5304/x9.png"><figcaption>Figure 8: Circular polarization signal in Stokes V for C IV (1548 Å)synthesized from a 3D MHD coronal model. The left panels show the verticalmagnetic field in the photosphere and the synthesized emission in part of a 3DMHD coronal model, looking at the computational domain from straight above.This part shows the surroundings of a pore. The synthesized Stokes V profileshown to the right is acquired by averaging in time over 12 minutes and inspace over 1.5 Mm×1.5 Mm corresponding to 2′′×2′′ as outlined by the smallboxes in the left panels. The red curve shows the synthesized signal withoutnoise. The bars represent the signal as it would be sampled with a noise of10−3 and a spectral resolution of about 40 mÅ, which is comparable to theSUMER instrument. Data based on [25].</figcaption></figure> Concerning the C IV line at 154.8 nm, in regions with sufficiently strong magnetic field this line shows a measurable signature in circular polarization through the Zeeman effect. The level of polarization is also about a fraction of a percent. This can be expected in active regions, but also in smaller structures such as pores (see Fig. 8) and possibly even in patches of strong network magnetic field in the quiet Sun. Previous measurements with uvsp/smm a polarimetric accuracy just below 1% [26; 27] gave no conclusive results except in sunspot umbrae [28]. Based on the new more sensitive SolmeX observations and with the help of models for the emission from the corona [e.g. 29], the structure of the magnetic field can be inverted. ### Chromospheric magnetic structures These structures can be revealed by spectro-polarimetric imaging in strong chromospheric lines. Among the most promising spectral lines for these diagnostics are the Mg II h and k lines near 280 nm. Both lines are Zeeman-sensitive (effective Landé factors $g_{\rm{eff}}{=}1.17$ and 1.33). In addition, the Mg II k line is expected to show strong linear polarization signals due to scattering processes, which through the Hanle effect are sensitive to the magnetic field of the upper solar chromosphere [see 22]. For Mg II k one would measure the full Stokes vector, i.e., linear and circular polarization are recorded. This allows one to derive the full vector of the magnetic field (direction and strength). The strong absorption owing to the ozone band in the Earth’s atmosphere requires this line to be studied in space outside the geocorona. <figure><img src="content_image/1108.5304/x10.png"><figcaption>Figure 9: Circular polarization signal expected for the Mg II k line at 279.6nm. Only the very centre of the line profile is shown. The colored curvesrepresent the Stokes V/I signal for different magnetic fields along the line-of-sight. The curves are already degraded to show a synthetic profile similarto what can be expected from ChroME on SolmeX.</figcaption></figure> Based on 1D atmospheric models one can estimate the polarization signal from the Mg II lines. As an example, Fig. 9 shows the circular polarization produced by the Zeeman effect assuming a magnetic field parallel to the line-of-sight (close to disk centre). For the typical magnetic fields of the upper chromosphere of active regions the expected circular polarization of Mg II k is of the order of 1%. In the quiet Sun chromosphere the linear polarization can be expected to be a few percent, which is sensitive to the weaker fields there via the Hanle effect [22]. Because the chromospheric magnetic structures are rooted in the photosphere, co-temporal and co-spatial measurements of the photospheric magnetic field vector are required. This true for the comparison with the chromospheric channel, but also for the interpretation of SUSP and EIP data. Such measurements can be achieved by recording the full Stokes vector, e.g., in the Fe I line at 525 nm. ## 5 Mission profile: Two spacecraft in formation flight The measurement objectives require coronagraphic observations at _low scattered light_ level in the infrared, visible, and UV at _high resolution_ and _close to the limb_. These three requirements, which are essential for the ambitious spectro-polarimetric observations in the solar corona above the limb, can only be achieved with an external occulter removed far from the entrance aperture of the instruments, because only this produces a true artificial eclipse. _This results in a mission profile with two spacecraft_, where one carries the instruments, and the other is the “artificial Moon”, i.e. the occulter. A similar profile is currently under investigation for the Proba-3 technology mission with its coronagraph Aspiics[e.g. 30; 31] and was employed for the proposal to the previous ESA M-class call DynaMICCS [32]. However, the major science objectives of these missions are quite different. As a technology mission Proba-3 contains only a limited set of instruments, and DynaMICCS was proposed to aim mainly at processes in the solar interior. SolmeX as well as it predecessor Compass [1] are focussing on the processes in the upper solar atmosphere, most noticeably on the role of the magnetic field investigated through measurements of the polarization from the EUV to the IR. _The scattered light level_ we aim for with SolmeX is about four orders of magnitude lower than for the leading ground-based coronagraph at Mauna Loa, Hawaii. The stray-light level of the Hawaii coronagraph of ${\approx}5{\cdot}10^{-6}$ is a factor of 5 better than what is envisaged for the high-resolution ground-based solar 4 m telescope ATST in its coronagraphic mode. Unlike the ATST, which will enjoy its lowest stray-light level for only a short fraction of the time, SolmeX will achieve its astonishingly low stray-light level of $10^{-10}$ virtually all the time. This is possible through the combination of the external occulter, the refractive optics and the internal occulter and baffling. Thus SolmeX will provide a data quality surpassing that of planned ground-based facilities. This is essential to observe not only the brightest coronal structures close to the limb, but also, e.g., faint large loops outside of active regions with sufficient signal-to noise for the polarimetric observations. _High-resolution coronagraphic observations_ are possible only for large occulter distances, because only then do the diffraction patterns induced at the occulter become sharp. This is illustrated in Fig. 10, which shows the spatial resolution (as a function of distance from disk centre) for various occulter distances. Very close to the limb the resolution is poor because of the reduced effective aperture close to the occulter. <figure><img src="content_image/1108.5304/x11.png"><figcaption>Figure 10: Spatial resolution that is possible for coronagraphic observationsas a function of radial distance from the disk centre for different distancesto the external occulter. An occulter distance of 200 m ensures a resolutionof 2” down to 0.07 solar radii or 50 Mm above the limb. (Courtesy of J.Davila).</figcaption></figure> _Observations close to the limb_ are therefore difficult, and only possible at large occulter distances – graphically speaking, the shadow has to have a distinct edge. These coronagraphic observations close to the limb are essential for understanding the magnetic coupling through the atmosphere and for achieving a connection to the on-disk observations, especially in the chromosphere. For high-resolution observations (resolution element of 2”) of coronal loops that reach some 10% of the solar radius into the corona, one has to observe at least as close as 50 Mm to the limb. For an occulter removed about 200 m from the entrance aperture this can be achieved (cf. Fig. 10), which also ensures the low stray-light level anticipated for SolmeX. Because an occulter removed 200 m from the spacecraft cannot be kept sufficiently stable by a boom, the occulter and the telescopes of the coronagraphs need to be placed separately on two spacecraft. ## 6 Instruments to measure the magnetic field in the upper solar atmosphere ![](x12.png) Table 2: Summary of the instruments discussed in Sects. 6.1 to 6.5. The suite of instruments for SolmeX consists of five instruments. The two coronagraphs for off-limb observations use the 200 m distant occulting disk on the separate occulter spacecraft to eclipse the solar disk. The three on-disk instruments investigate the chromosphere, transition region and corona in front of the solar disk in the UV and extreme UV light. An overview of the technical details of the instruments can be found in Table 2. The field of view (FOV) of the SolmeX instruments is depicted in Fig. 11. While the on-disk instruments either see the full disk or can point anywhere on the disk, the off-limb instruments need a simultaneous roll of the SolmeX spacecraft to observe different parts of the corona. <figure><img src="content_image/1108.5304/x13.png"><figcaption>Figure 11: Fields of view of the SolmeX instruments. For the slit instrumentsSUSP and CUSP a typical raster area is shown. SUSP can point anywhere on thedisk and up to 1 R⊙ above the limb, ChroME anywhere on the disk. CUSP andVIRCOR can point to different parts of the corona through a roll of thespacecraft.</figcaption></figure> ### Coronal UV Spectro-Polarimeter (CUSP) CUSP will measure the linear polarization of the solar UV radiation in the range of 95 nm to 125 nm in two channels, using an all-reflective polarization unit. The linear polarization signal is recorded in Ly-$\alpha$, $\beta$, and $\gamma$, and the O VI doublet at 103 nm. This provides access to the magnetic field through the Hanle effect. <figure><img src="content_image/1108.5304/x14.png"><figcaption>Figure 12: Schematic opto-mechanical layout of the Coronal UV Spectro-Polarimeter (CUSP).</figcaption></figure> CUSP is an all-reflecting externally occulted coronagraph with the 200 m long occultation baseline provided by the formation-flying occulter spacecraft. It uses the linear part of the occulter (Sect. 7) to minimize the effects of diffraction patterns. Light from the solar corona enters the square entrance aperture and is imaged by a parabolic telescope mirror on the entrance slit of the spectrometer (Fig. 12). Inside the spectrometer, the polarization is modulated by a rotating linear polarization analyser that consists of four reflecting plates at the Brewster angle [33], see inset of Fig. 12. An all-reflective design is necessary because transmission optics such as those used for SUSP (Sect. 6.4) work only above 105 nm. This polarimeter configuration maintains the output beam coaxial with the input beam and can be retracted from the light beam. A toroidal varied line-spaced (TVLS) grating disperses the coronal radiation on the detector. The baseline detector is an Intensified CCD camera (ICCD) that is based on the suborbital experiment herschel. A phosphorous screen is optically coupled to an image sensor, generally a fast-scan frame transfer CCD. Coupling is achieved with a lens or a fiber optic taper which also provides the scaling required to match the CCD format to the phosphorous screen. The sampling is 5”/pixel in the spatial and 9 pm/pixel in the spectral direction. The spatial resolution is sufficient to investigate large-scale structures in the off-limb corona, such as streamers, or the boundary from the quiet Sun to coronal holes. The spectral resolution allows us to resolve the broad spectral profiles of coronal lines in detail. The 10” wide slit covers 0.7${}^{\circ}$, corresponding to $2.5\,R_{\odot}$. The slit can be positioned to perform a raster scan by tilting the telescope mirror. A typical map will have a FOV of $0.4^{\circ}\times 0.7^{\circ}$. CUSP will record the best-suited UV lines for magnetic field diagnostics: the H-Lyman series and O VI (Sect. 4.1). Because the expected degree of polarization is 1% to 25%, a signal-to-noise ratio of at least 100 has to be achieved. The required exposure times (for different spatial binning) are shown in Fig. 13[e.g. 34; 35]. They are short ($<$1 min) in the low corona, but go up to a fraction of a day higher up. As the faint large-scale corona shows only slow variation these long exposure times still contain valuable information. <figure><img src="content_image/1108.5304/x15.png"><figcaption>Figure 13: Typical acquisition time for CUSP for the prime target lines. Thisincludes three exposures in three polarization states until a signal-to-noiseratio of 100 is reached. A spatial binning of 5” was assumed for Ly-α and OVI, 10” for Ly-β, and 25” up to 100” for Ly-γ.</figcaption></figure> The key resources of CUSP are summarized in Table 2. To estimate the data rate, a full spectro-polarimetric frame (3 images) was assumed to be downlinked every 5 minutes, resulting in 150 kbit/s. Nonetheless, a fast digital processing electronics (implementing pattern recognition and event centroiding) is required to analyse the frames produced by the image sensor in real time. To achieve the required stray light rejection levels, the required absolute pointing (pitch/yaw) of the spacecraft should be $<$5’ and the pointing stability (pitch/yaw jitter) 1” in 15 minutes. The possibility of removing the polarization unit allows the normal polarimetric mode (pol) as well as a non-polarimetric mode (n-p). In n-p mode CUSP will be able to operate in a similar fashion to UVCS/SOHO, but with superior spatial and temporal resolution, because of the increased aperture and throughput. ### Visible light and IR Coronagraph (VIRCOR) VIRCOR consists of two channels that provide maps of the magnetic field vector, the structure, and the electron density in the corona above the limb. In the infrared (IR) channel the full Stokes vector is recorded, i.e., linear and circular polarization. Using a tunable narrow-band filter, the spectral profiles are recorded. The visible light (VL) channel captures the K-corona with a tunable broad-band filter. <figure><img src="content_image/1108.5304/x16.png"><figcaption>Figure 14: Schematic optical layout of the Visible light and IR Coronagraph(VIRCOR).</figcaption></figure> VIRCOR will be a complete instrument including mechanical structure, optical system, and local electronics. The Carbon Fiber Reinforced Plastic (CFRP) optical bench facesheets, baffles, and structural panels ensure that the stiffness and thermal stability requirements are met while achieving a low-mass structure. The optical system is shown in Fig. 14. The entrance aperture will be 20 cm in diameter, large enough to achieve the required resolution and signal-to-noise ratio. The IR coronal magnetograph will fully profit from the unique observation conditions of SolmeX in space because of the absence of seeing-induced polarization cross-talk and of atmosphere-induced intensity fluctuations. The large distance to the external occulter, located on the occulting spacecraft, enables high spatial resolution to be achieved in both the IR and the VL channel. The IR spectro-polarimeter design is based on the HAO/NCAR Coronal Multi-channel Polarimeter [CoMP; 36], which is now a part of HAO’s Mauna Loa Solar Observatory in Hawaii. The 2.5 $R_{\odot}$$\times$ 2.5 $R_{\odot}$ FOV (Fig. 11) is imaged onto a Teledyne Imaging HgCdTe 1024$\times$1024 pixels detector with 18 $\mu$m square pixels (2.3”/pixel). It records the intensity as well as the the linear and circular polarization of the forbidden lines of Fe XIII at 1074.7 nm and 1079.8 nm, and of the He I line at 1083 nm. This provides information on the full vector of the coronal magnetic field. The IR channel measures the line-of-sight plasma velocity from Doppler observations and the plasma density from the ratio of the Fe XIII lines at 1074.7 nm and 1079.8 nm. Liquid Crystal Variable Retarder technology is used for both the polarimetry analysis and tunable wavelength selection. Post-focus instrumentation includes a narrow-band tunable filter to obtain precise polarimetry across the emission lines over the entire FOV with modest spectral resolution. A six-stage birefringent filter will be used to attain the wavelength range. The visible light (VL) channel has a high spatial sampling of 1.18”/pixel and a more extended 5 $R_{\odot}$$\times$ 5 $R_{\odot}$ FOV imaged on a 4k $\times$ 4k CCD. This provides context for the fine-scale features and takes full advantage of the high flux and excellent straylight rejection provided by the large external occulter distance. To satisfy the mission objectives, it provides total and polarized brightness images of the K-corona, i.e., of light scattered by free electrons in the corona. Using a broad-band filter ensures a high photon flux at the detector plane to allow short exposures down to 1 s and the highest possible spatial resolution, surpassing ground-based eclipse imaging. Tuning the broad-band filter in wavelength provides access to temperature and flow diagnostics as required by the science objectives. The key resources are summarized in Table 2. A data rate of 300 kbps is required to downlink the full Stokes parameters every 30 s and the white light K-corona images every 5 minutes using compression. To achieve the required stray light rejection levels, the absolute pointing of the spacecraft has to be ${<}1^{\prime}$. The pointing stability must ensure that the same region on the Sun is imaged on each detector pixel within the sequence of spectro-polarimetric exposures. VIRCOR will operate mainly in a synoptic mode. To investigate rapid variations, high-cadence observations will be performed for limited times. Then either only part of the FOV will be transmitted or the instrument will observe at low temporal cadence until the burst of high-cadence data is downlinked. To point to different parts of the corona, a spacecraft roll will have to be coordinated with the other instruments. ### EUV Imaging Polarimeter (EIP) EIP is a normal incidence EUV full-disk telescope that measures the linear polarization of the Fe X line at 17.4 nm in order to map the magnetic field orientation in the corona. To derive the Stokes $I$, $Q/I$ and $U/I$ parameters, the intensity is measured at three orientations (0${}^{\circ}$, 60${}^{\circ}$, 120${}^{\circ}$) by rotating the focal plane assembly (FPA) that is composed of a polarizing mirror and the detector. The amplitude of the expected polarization signal is of the order of $10^{-3}$ (Sect. 4.3, Fig. 6). A 3$\sigma$ detection of this signal requires about $10^{7}$ photons per exposure. The main design driver is therefore to collect enough photons within exposure times shorter than the typical evolution time scale of the observed structures. <figure><img src="content_image/1108.5304/x17.png"><figcaption>Figure 15: Optical design of the EUV Imaging Polarimeter (EIP) and itsrotating focal plane assembly (FPA). </figcaption></figure> EIP will be a Ritchey-Chrétien telescope (Fig. 15). The linear polarizer is a 45${}^{\circ}$ folding mirror located close to the detector (${\approx}$10 mm), and this focal plane assembly (FPA) is rotated as a whole to provide polarization measurements. This single mirror polarizing system maximizes the throughput, and ensures a stable image on the detector, allowing us to measure Stokes $Q/I$ and $U/I$ independent of the flatfield. To keep the image stable within a pixel, the rotation axis of the mechanism must be maintained within 1 arcmin, which is achievable with careful design. The rotating FPA imposes that cooling lines and electrical connectors follow the movements of the detector. The full disk (0.6${}^{\circ}$x0.6${}^{\circ}$) is imaged on a 4k x 4k detector with 0.55”/pixel, providing a spatial resolution of 1.1”. The detector is a back-thinned EUV sensitive CCD with 12 $\mu$m square pixels (e.g., E2V CCD203), or equivalent APS. In order to minimize the detector dark current during the exposure times, the array will be cooled to about $-80^{\circ}$ C, so that the measurements are effectively shot-noise limited. EIP uses state-of-the-art low-roughness Al/B${}_{4}$C/Mo multilayer coatings with 50% reflectivity. They are optimized for maximum throughput at 17.4 nm and maximum rejection of nearby lines of Fe IX, x, and xi, some of which are predicted to show weaker or null polarization (e.g., Fe IX at 17.1 nm). It is of critical importance to reject these other lines because they constitute stray light in the polarization measurement and degrade the measurement signal-to-noise. We achieve the required high spectral purity by using the 2nd order of multilayers tuned to twice the 17.4 nm wavelength, because the 2nd order has a narrower passband. The 1st order is suppressed by the Zr filter at the focal plane. The resulting passband is only ${\approx}0.35$ nm wide (Fig. 16) and a spectral analysis reveals a spectral pureness of about 70%, which is twice as good as for traditional coatings. The thicknesses of the coatings can be controlled with a precision to ensure centering of the passband within 0.1 nm. <figure><img src="content_image/1108.5304/x18.png"><figcaption>Figure 16: Spectral response of EIP for the s- and p-polarizations (solid anddashed) and their average (red), including the efficiencies of all opticalelements. The EIT/SOHO response (blue) is shown for comparison.</figcaption></figure> Multilayer coatings are also used for the 45${}^{\circ}$ polarizer. They are optimized to reflect only the s-polarization at 45${}^{\circ}$ of incidence. The p-polarization is suppressed by about a factor of 100 (Fig. 16). Nonetheless, the relative s-polarization can be measured with a high accuracy of $10^{-3}$[37], because a non-zero p-polarization is only reducing the amplitude of the response when rotating the polarizing mirror (with the whole FPA). Thin film ($\approx$150 nm) metallic filters are used at the entrance of the telescope and at the focal plane. The entrance Al filter suppresses the visible and infrared light. The focal plane filter is an Al/Zr/Al sandwich. It suppresses the 1st order of the multilayer coatings and provides redundancy against pinholes developing in the front filter during the mission lifetime. We computed the expected signal at the focal plane for the quiet Sun and for active regions with the measured radiances and the polarization levels from Sect. 4.3 and Fig. 6. From this we estimated the exposure times for EIP using a 2”x2” binning. For an active region the required polarization level of $7{\cdot}10^{-4}$ can be reached within a 43 s exposure (1$\sigma$ detection). Because three exposures (at 0${}^{\circ}$, 60${}^{\circ}$, 120${}^{\circ}$ polarizer angle) have to be taken, a magnetic field measurement can be performed in 3 min at 1$\sigma$ detection level. Our spectral analysis shows that the other lines in the band contribute only 4% to the polarization when compared to the EIP target line. The key instrument parameters are given in Table 2. The pointing requirements are derived from the need to keep the image stable on the detector within a fraction of a pixel over the typical exposure times. If the spacecraft pointing stability does not meet the instrument requirements, actuators on the secondary mirror will be used to correct small jitter errors. The observation programme will consist of repeated sequences of three polarization images at 0${}^{\circ}$, 60${}^{\circ}$, and 120${}^{\circ}$. The best compromise between signal level, resolution (binning) and exposure times will be determined during commissioning. Keeping the polarizer at a fixed position, EIP can take data at full resolution with a cadence of 3 s to study the fast evolution of coronal structures. ### Scanning UV Spectro-Polarimeter (SUSP) SUSP will measure linear and circular polarization in the UV range from 115 nm to 155 nm, i.e., the full Stokes vector, with the prime lines for magnetic field diagnostics being Ly-$\alpha$ and C IV using the Hanle and the Zeeman effect (cf. Sect. 4.4). Other lines in this wavelength range cover a wide temperature range from $10^{4}$ to well above $10^{6}$ K, allowing plasma diagnostics also in the hot parts of the atmosphere. <figure><img src="content_image/1108.5304/x19.png"><figcaption>Figure 17: Scanning UV Spectro-Polarimeter (SUSP). A single mirror telescopeand a concave grating spectrometer image part of the UV spectrum on a systemof three detectors. The inset shows a magnification of the polarizer and slitunit — rotated by 90∘ around the beam axis for clarity. The polarizer unit canbe removed from the light beam for non-polarimetric observations.</figcaption></figure> SUSP is an efficient normal incidence UV slit spectrometer with very high throughput. It consists of a telescope with a single parabolic mirror and a grating spectrograph with a concave variable line spacing (VLS) grating (Fig. 17). In the focal plane a large format multi-channel plate (MCP) covers the entire spectral and spatial FOV (120 mm x 30 mm). The image produced by the anode phosphorous coating of the visible-blind MCP intensifier is transferred by fiber optical couplers to three separate CMOS-APS sensors (2k x 2k pixels) that are read out in photon-counting mode. The telescope mirror mechanism allows us to point anywhere on the solar disk and up to $1\,R_{\odot}$ above the limb, providing an overlap with the coronagraphic instruments CUSP and VIRCOR. The transmissive polarization optics are placed between the entrance slit and the grating. The design of the polarizing unit follows [38], who measured 14% transmission for the complete unit. It consists of a rotating MgF${}_{2}$ retarder plate and a double set of Brewster-angle MgF${}_{2}$ plates, with the Brewster angle perpendicular to the dispersion plane (Fig. 17). The sampling is 1” per pixel in the spatial and 6.6 pm per pixel in the spectral direction, the latter is sufficient to measure flows with 2 km/s accuracy (by line centroiding). Good image quality can be achieved over 360” along the slit. To reach the required polarimetric sensitivity of $10^{-3}$ (cf. Sect. 4.4, Figs. 7, 8), a signal-to-noise level of $10^{3}$ has to be achieved, i.e., at least $10^{6}$ photons have to be detected (Poisson statistics). This can be achieved with a photon-counting detector. The photon budget for the whole instrument shows that for quiet Sun observations $10^{6}$ counts are reached in 7 s for Ly-$\alpha$ in one pixel and in about 6 min for C IV with 2”x2” binning. With such such a binning a polarization signal of more than 0.1% can be expected (see Fig. 8) Active region observations or larger binning would shorten the necessary exposure times. The key resources of SUSP are listed in Table 2. The data rate was estimated assuming uninterrupted observations downlinking wavelength windows for selected lines and lossless compression. The pointing stability has to be sufficient to keep the slit on the same structure. Spacecraft absolute pointing must ensure to find the target well within the FOV provided by the raster scan. SUSP will have a polarimetric (pol) and a non-polarimetric (n-p) observing mode. The polarization optics can be retracted from the optical path entirely (inset of Fig. 17). In the pol mode the instrument observes different polarization states, i.e., at different rotation angles of the retarder plate, in rapid succession. For longer exposure times, which are needed for C IV, the polarization states are then summed separately over the data sequence, e.g., 12 min $=$ 2 x 6 min for quiet Sun C IV, and analysed to return magnetic field information. This is the fundamental reason why a photon counting detector is required. In the n-p mode the polarizer unit is removed from the light beam and the UV spectra can be acquired in unprecedented time cadence for plasma diagnostics. In the pol and n-p modes either a time series of spectra or raster maps can be acquired. ### Chromospheric Magnetic Explorer (ChroME) ChroME will provide chromospheric and photospheric vector magnetic field maps using spectro-polarimetric measurements in the core of the chromospheric Mg II k 279.6 nm line and a photospheric line. The combined action of atomic level polarization and the Hanle and Zeeman effects creates and modifies polarization signatures, allowing us to retrieve the direction and strength of the magnetic field. <figure><img src="content_image/1108.5304/x20.png"><figcaption>Figure 18: Conceptual design of the Chromospheric Magnetic Explorer (ChroME).</figcaption></figure> The conceptual design of ChroME is sketched in Fig. 18. The 25 cm aperture Gregory-type telescope uses an active mirror (M2) for internal image stabilization and refocussing. The polarimetry unit is located before any inclined optical element to minimize instrumental polarization. The polarizing beam splitter acts as the analyser of the polarimetry unit. At the same time it feeds light to the correlation tracker sensor. The filtergraph consists of a double Fabry-Pérot interferometer, mounted in a collimated setup to keep a constant spectral resolution of 50 mÅ at 280 nm over the whole FOV. Behind the filtergraph a dichroic beam splitter distributes the light to the chromospheric and the photospheric channels. The chromospheric channel is equipped with a $\approx$15 Å wide pre filter. The photospheric channel is currently designed to use the Fe I line at 525 nm, as there is no sufficiently strong high-$g$ unblended line at shorter wavelengths. ChroME delivers chromospheric vector magnetograms at a spatial resolution of 0.30” (with 0.15”/pixel sampling) and a spectral resolution of 50 mÅ. The FOV is 300${}^{\prime\prime}$$\times$300${}^{\prime\prime}$ and the nominal cadence is 30 s. Photon budget considerations, based on measurements of the solar irradiance [39], on theoretical calculations of the Zeeman signal of the Mg II k line, and the scattering polarization signal modified by the Hanle effect [22] yield a noise level for the polarimetric measurements of $3{\cdot}10^{-3}$ of the intensity, sufficient to fulfill the requirement for measuring the chromospheric Hanle and Zeeman signals over network and active regions. On-board spatial binning and/or longer exposure times, resulting in a polarimetric precision of $10^{-4}$, allow for chromospheric magnetic field measurements also in quiet Sun areas. The Fe I line for the photospheric channel permits reliable photospheric vector magnetograms and detailed magnetic field investigations [e.g. 40]. The photospheric channel uses the same filtergraph as the chromospheric channel. Simultaneous observations in the chromospheric and the photospheric channel can be achieved by adjusting the free spectral range (FSR) of the etalons accordingly. Table 2 summarizes the baseline properties of ChroME. The full performance of ChroME is exploited if simultaneous measurement of the chromospheric and the photospheric magnetic field vector are acquired within 30 s over the full FOV. This would result in a very high data rate of 15 MByte/s. However, through an efficient on-board data selection (smaller FOV, lower cadence at times of low activity) and compression it is possible to reduce the maximum data rate to 700 kBit/s. The required pointing accuracy of ChroME is 1/20 pixel on the science detector ($7.5\times 10^{-3}$ arcsec). This will be achieved by an internal image stabilization system (correlation tracker), and reduces the pointing requirement for the spacecraft to 10${}^{\prime\prime}$ per minute. Repointing of the FOV should allow for observations on the entire solar disk. The default operation mode of ChroME will deliver full Stokes images at 15 wavelength positions simultaneously in the Mg II k line and the photospheric line. The 15 Å wide prefilter of the chromospheric channel allows for an extended observing mode that additionally covers the non-Hanle-sensitive Mg II h line, removing ambiguities in the magnetic field determination. A fast observing mode using fewer wavelength points and/or only circular or linear polarization states will be implemented to allow the investigation of wave phenomena and reconnection events. ## 7 Model spacecraft, mass and power To obtain a feasible design for the spacecraft system and its components and to derive reliable estimates for the mass, power, and costs for the SolmeX mission, we performed a concurrent engineering (CE) study at the DLR Institute of Space Systems in Bremen, Germany. While the basic idea for the formation flight is similar to previously proposed missions such as DynaMICCS [32] and especially Compass [1], the spacecraft concept and related issues such as telemetry or the formation flight concept have been newly developed from scratch during the CE study. A short summary of the study report’s results are provided in the following. <figure><img src="content_image/1108.5304/x21.png"><figcaption>Figure 19: _The left panel_ shows both spacecraft stacked on top of each otherin the launch configuration. The circular main body of the instrumentspacecraft (blue) is mounted on the circular launch adapter of the SoyuzFregat. The occulter spacecraft also has a circular main body (yellow) and ismounted on the top. _The right panel_ shows the flight configuration. The umbra of the occultercovers only the entrance apertures of the coronagraphic instruments, while theon-disk instruments and the solar panels are outside the penumbra. The Sunwould be found towards the top. (Spacecraft distance not to scale).</figcaption></figure> The design of the spacecraft structure is driven by the requirement that the apertures of the coronagraphic instruments (CUSP, VIRCOR) have to be in the umbra of the occulter, and that the on-disk instruments (EIP, SUSP, ChroME) have to be outside the penumbra. Based on the science requirements, the distance to the occulter should be 200 m (Sect. 5, Fig. 10). As a consequence the dimensions of the occulter have to be about 2.2 m x 2.5 m. The radius of the penumbra at the instrument spacecraft is then about 2 m, which sets the separation between the coronagraphic and the on-disk instruments (cf. Fig. 19). _The instrument spacecraft_ is built around a circular main body with a diameter matching the circular-shaped launch adapter of the Soyuz Fregat. This carbon fibre tube main body will accommodate most spacecraft subsystem components. Exceptions are, e.g., the star trackers. The instruments are then mounted on the main body through a light-weight frame structure, which also holds the solar panels (Fig. 19, right panel). _The occulter spacecraft_ is also built around a circular body fitting on top of the instrument spacecraft in order to have a compact and stable launch configuration (Fig. 19, left panel). Here, the subsystems are again accommodated in the central tube. The occulter plate is mounted on top of the tube and carries at its Sun-directed side solar panels sufficient to power the occulter spacecraft. In the currently proposed design all instruments are housed in the instrument spacecraft: the instruments for on-disk (outside the penumbra) and for above-limb coronagraphic observations (inside the umbra; see Fig. 19). This has the advantage that the demands on the occulter spacecraft are quite relaxed, e.g., in terms of pointing stability or power consumption. For example it is sufficient to place the solar cells for the occulter spacecraft at the front side of the occulter, without the need for external panels (see Fig. 19). This way the SolmeX system is a combination of an instrument spacecraft with a complexity common for solar space observatories and a relatively simple occulter spacecraft. _The launch configuration_ has maximum extensions of 3.1 m and 2.2 m in the horizontal directions and 1.9 m in height. Because the launch configuration is not rectangular (Fig. 19, left panel), it easily fits into the Soyuz Fregat with its 3.86 m diameter. The occulter is not rectangular, but has one round edge to allow imaging observations very close to the limb with the coronagraphic imaging spectro-polarimeter VIRCOR. The slit instrument CUSP uses the linear part of the occulter. Both VIRCOR and CUSP will be able to observe simultaneously. For observations of different parts of the corona both the occulter and the instrument spacecraft have to roll. The positioning of the instruments implies that the roll axis intersects the instrument spacecraft at its side (i.e., at the entrance aperture of the coronagraphs). Therefore, the instrument spacecraft has to roll around a point not identical to its centre of mass. This is achieved through propulsion, i.e., cold gas micro thrusters. The estimates for the overall resources are based on the concurrent engineering (CE) study for the mission profile and spacecraft. They contain margins between 10% and 20% depending on the technology readiness level. The instrument masses (cf. Table 2) contain similar margins. Additionally a system margin of 20% was added to the total dry mass. The total launch mass of the instrument spacecraft (including launch adapter and propellant) is about 1400 kg including the launch adapter, while the occulter spacecraft is just below 700 kg. Thus the total mass is below the limit of 2.1 t for a Soyuz Fregat launch to L1. In the framework of the CE study the power consumption of all subsystems was estimated for a number of operating modes. The _science mode_ with the acquisition of the science data through the payload instruments will be used most of the time. The _maneuver mode_ for changing the attitude, spacecraft rolls, and de-saturation of the reaction wheels will be activated about once a day for about 30 min. These modes drive the requirements for power consumption and, thus, the size of the solar cell elements. The _safe mode_ for vehicle rescue is the driver for the battery power. It is assumed that the the spacecraft can run in safe mode for 1.5 days on battery power. The peak power required for the instrument S/C is about 800 W (including spacecraft and instruments), the occulter S/C demands a peak power of 370 W. Each S/C has its own power system, and the solar cells are designed assuming a degradation of 3.75% per year. The required area of the solar cells is 5.5 m${}^{2}$ for the instrument S/C and 2.6 m${}^{2}$ for the occulter S/C. The communication is realized with a data downlink of 7 Mbit/s employing a 60 W transmitter using a 0.7 m diameter dish antenna. This provides sufficient bandwidth to ensure the downlink of the science data with an average rate of 2 Mbit/s (cf. Table 2). ## 8 Summary and conclusions The information on the magnetic field of the outer solar atmosphere is encoded in the polarization of ultraviolet and infrared spectral lines [e.g., 41], and was not accessible to previous space instruments. Because the level of polarization is generally low, the required signal-to-noise ratio for the observations has to be as high as $10^{4}$ for some measurements. Thus, the instruments have to provide a combination of large aperture and high throughput. Previous solar space missions did not include the capability of polarimetric observations, or it turned out that the signal-to-noise ratio was not sufficient to provide reliable results (e.g., uvsp/smm). SolmeX will overcome these limitations and will provide the first measurements of the magnetic field in the upper solar atmosphere. Space observatories such as SOHO [42], TRACE [43], Hinode [44], and SDO [45] have provided us with a new picture of the state of the plasma and its dynamics in the upper solar atmosphere, and Solar Orbiter will exceed this with increased spatial resolution from vantage points not explored before. While these observatories provided (or will provide, in the case of Solar Orbiter) information on the flux and energy of the photons originating in the solar atmosphere, e.g., spectral radiance, SolmeX will for the first time also explore their state of polarization, giving access to the magnetic field. This will open a new window to investigate the interaction of the magnetic field with the plasma and to address the leading question of the nature of the upper solar atmosphere. To reach this goal, SolmeX will employ imaging polarimetric and spectro-polarimetric instruments operating at extreme UV, UV and infrared wavelengths. These will observe the upper solar atmosphere of the Sun on the disk and above the limb. For the coronagraphic instruments an external occulter will be mounted on a second spacecraft in formation flight with the spacecraft carrying the instruments. Modern solar physics started with the first magnetic field measurement in sunspots by G. E. Hale in 1908. This revolution has remained incomplete, however, because the measurements of the magnetic field have been restricted mainly to the solar surface. SolmeX will complete these achievements by providing the first comprehensive measurements of the magnetic field in the outer atmosphere of our Sun. ## References * [1] S. Fineschi, S. Solanki, & COMPASS team, COMPASS: Coronal Magnetism, Plasma and Activity Studies from Space: a formation flying mission to measure the solar magnetism. Proposal to ESA Cosmic Vision 2015-2025 (2007) * [2] B. De Pontieu, S.W. McIntosh, M. Carlsson, et al., Science 318, 1574 (2007) * [3] S.W. McIntosh, B. De Pontieu, ApJ 707, 524 (2009) * [4] H. Peter, A&A 521, A51 (2010) * [5] R. Centeno, J. Trujillo Bueno, A. Asensio Ramos, ApJ 708, 1579 (2010) * [6] U. Feldman, I.E. Dammasch, K. Wilhelm, ApJ 558, 423 (2001) * [7] P. Judge, M. Carlsson, ApJ 719, 469 (2010) * [8] T.G. Forbes, J.A. Linker, J. Chen, et al., Space Sci. Rev. 123, 251 (2006) * [9] M. Stix, _The Sun_ (2nd ed., Springer, Berlin, 2002) * [10] J.W. Harvey, _Magnetic Fields Associated with Solar Active-Region Prominences_ (PhD thesis, Univ. Colorado, Boulder, 1969) * [11] P. Charvin, Annales d’Astrophysique 28, 877 (1965) * [12] S. Fineschi, ASP Conf. Series 248, 597 (2001) * [13] V. Bommier, S. Sahal-Brechot, Solar Phys. 78, 157 (1982) * [14] N.E. Raouafi, P. Lemaire, S. Sahal-Bréchot, A&A 345, 999 (1999) * [15] N.E. Raouafi, S. Sahal-Bréchot, P. Lemaire, A&A 396, 1019 (2002) * [16] R. Casini, P.G. Judge, ApJ 522, 524 (1999) * [17] S. Tomczyk, S.W. McIntosh, S.L. Keil, et al., Science 317, 1192 (2007) * [18] H. Lin, M.J. Penn, S. Tomczyk, ApJ 541, L83 (2000) * [19] H. Lin, J.R. Kuhn, R. Coulter, ApJ 613, L177 (2004) * [20] R. Manso Sainz, J. Trujillo Bueno, in _Solar Polarization 5_, ed. by S.V. Berdyugina, et al. (ASP Conf. Series, Vol. 405, 2009), pp. 423–428 * [21] J. Trujillo Bueno, J. Štěpán, R. Casini, ApJ 738, L11 (2011) * [22] J. Trujillo Bueno, in _Solar Polarization 6_, ed. by J. Kuhn, et al. (ASP Conf. Series, vol. 437, 2011), pp. 83–98 * [23] J. Trujillo Bueno, E. Landi Degl’Innocenti, M. Collados, et al., Nature 415, 403 (2002) * [24] K. Kobayashi, S. Tsuneta, J. Trujillo Bueno, et al., AGU Fall Meeting Abstracts p. B1632 (2010) * [25] H. Peter, B. Gudiksen, Å. Nordlund, ApJ 638, 1086 (2006) * [26] W. Henze, Jr., E. Tandberg-Hanssen, M.J. Hagyard, et al., Solar Phys. 81, 231 (1982) * [27] M.J. Hagyard, D. Teuber, E.A. West, et al., Solar Phys. 84, 13 (1983) * [28] B. Lites, in _Magnetic fields across the Hertzsprung-Russell diagram_, ed. by G. Mathys, S.K. Solanki, D.T. Wickramasinghe (ASP Conf. Series Vol. 248, 2001), pp. 553–561 * [29]H. Peter, B. Gudiksen, Å. Nordlund, ApJ 617, L85 (2004) * [30] S. Vivès, P. Lamy, P. Levacher, et al., (2006), _SPIE Conference Series_, vol. 6265, pp. 24–1 * [31] P. Lamy, S. Vivès, L. Damé, S. Koutchmy, (2008), _SPIE Conference Series_, vol. 7010, pp. 1H–1 * [32] S. Turck-Chièze, P. Lamy, C. Carr, et al., Exp. Astron. 23, 1017 (2009) * [33] M. Romoli, S. Fineschi, L.D. Gardner, J.L. Kohl, SPIE Conference Series 2283, 288 (1994) * [34] A. Khan, L. Belluzzi, E. Landi Degl’Innocenti, et al., A&A 529, A12 (2011) * [35] A. Khan, E. Landi Degl’Innocenti, A&A 532, A70 (2011) * [36] S. Tomczyk, G.L. Card, T. Darnell, Solar Phys. 247, 411 (2008) * [37] J.C. del Toro Iniesta, M. Collados, Applied Optics 39, 1637 (2000) * [38] H. Winter, H.W. Ortjohann, Rev. Sci. Instrum. 58(3), 359 (1987) * [39] R.P. Cebula, G.O. Thuillier, M.E. Vanhoosier, et al., Geophys. Res. Letters 23, 2289 (1996) * [40] S.K. Solanki, P. Barthol, S. Danilovic, et al., ApJ 723, L127 (2010) * [41] J. Trujillo Bueno, E. Landi Degl’Innocenti, R. Casini, V. Martínez Pillet, in _Trends in Space Science and Cosmic Vision 2020_, ed. by F. Favata, et al. (ESA SP-588, 2005), p. 203 * [42] V. Domingo, A.I. Fleck, B. und Poland, Solar Phys. 162, 1 (1995) * [43]B.N. Handy, et al., Solar Phys. 187, 229 (1999) * [44] T. Kosugi, K. Matsuzaki, T. Sakao, et al., Solar Phys. 243, 3 (2007) * [45] W.D. Pesnell, B.T. Thompson, P.C. Chamberlin, Solar Phys., submitted (2011)
1502.01838	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 43733, "num_imgs": 5, "llama3_tokens_count": 10288 }	[ "content_image/1502.01838/x1.png", "content_image/1502.01838/x1.png", "content_image/1502.01838/x1.png", "content_image/1502.01838/x1.png", "content_image/1502.01838/x3.png" ]	# Distributed Verification of Rare Properties using Importance Splitting Observers Cyrille Jegourel INRIA Rennes – Bretagne Atlantique Axel Legay INRIA Rennes – Bretagne Atlantique Sean Sedwards and Louis-Marie Traonouez INRIA Rennes – Bretagne Atlantique ###### Abstract Rare properties remain a challenge for statistical model checking (SMC) due to the quadratic scaling of variance with rarity. We address this with a variance reduction framework based on lightweight importance splitting observers. These expose the model-property automaton to allow the construction of score functions for high performance algorithms. The confidence intervals defined for importance splitting make it appealing for SMC, but optimising its performance in the standard way makes distribution inefficient. We show how it is possible to achieve equivalently good results in less time by distributing simpler algorithms. We first explore the challenges posed by importance splitting and present an algorithm optimised for distribution. We then define a specific bounded time logic that is compiled into memory-efficient observers to monitor executions. Finally, we demonstrate our framework on a number of challenging case studies. ## 1 Introduction The ‘state explosion problem’ [6] associated with probabilistic model checking has been well addressed by statistical model checking (SMC) [28]. SMC includes a number of approximative techniques based on Monte Carlo sampling [24], which only generate states on the fly during simulation. The performance of SMC is typically independent of the size of the state space [25], while simulation cost may be divided linearly on parallel computation architectures. Rare properties pose a problem because the standard and relative errors scale quadratically with rarity [13, 26]. For example, 4000 simulations would be sufficient to estimate a probability of $0.1\pm 10\%$ with $95\%$ confidence, whereas $4\times 10^{13}$ simulations would be necessary to estimate a probability of $10^{-6}\pm 10\%$ with the same confidence. Since quantifying rare properties is often important to certify the reliability of complex critical systems, we seek to enhance SMC with variance reduction techniques, such as importance sampling and importance splitting [21, 13, 26], while still taking advantage of the easy distribution that SMC typically affords. Importance sampling weights the executable model of a system so that the rare property occurs more frequently in simulations. The proportion of simulations that satisfy the property using the weighted model overestimates the true probability, but the estimate may be exactly compensated by the weights. It is generally not feasible to implement a perfectly weighted executable model for importance sampling because (_i_) the perfect model may not actually exist as a re-parametrisation of the original model and (_ii_) a perfect re-parametrisation typically requires an iteration over all the transitions, defeating the benefits of sampling. Practical approaches tend to use a low dimensional vector of parameters to weight the model [17, 16]. Given such a parametrisation, importance sampling can be implemented with minimal memory and may be distributed efficiently on parallel computational architectures. The principal limitation of importance sampling is that without a guarantee that the simulation model is perfect, it is difficult to formally bound the error of estimates. In contrast, useful confidence intervals have been defined for importance _splitting_[5, 4]. Importance splitting divides a rare property into a set of less rare sub-properties that correspond to an increasing sequence of disjoint levels: the initial state corresponds to the lowest level, while states that satisfy the rare property corresponds to the final level. Importance splitting algorithms use a series of easy simulation experiments to estimate the conditional probabilities of going from one level to the next. Since relatively few simulations fail to satisfy the sub-properties, the overall simulation budget may be reduced. Each experiment comprises simulations initialised with the terminal states of previous simulations that reached the current level. The overall probability is the product of the estimates, with the best performance (lowest variance) achieved with many levels of equal conditional probability. Importance splitting poses several challenges for optimisation and distribution. In the context of SMC, importance splitting algorithms repeatedly initialise simulations with states of the model-property product automaton. For arbitrary properties this may have size proportional to the length of a simulation trace. At the same time, increasing the number of levels to maximise performance reduces the number of simulation steps in each simulation experiment. The cost of sending the model-property state across slow communication channels may be significantly greater than the cost of short simulations. In addition, to specify levels with equal conditional probabilities it is necessary to define a ‘score function’ that maps the states of the product automaton to a value. This cannot easily be automated, so a syntactic description of the property automaton must be accessible for the user to construct a score function manually. To address the above challenges we present an importance splitting framework for SMC, specifically considering the problems of distribution. We first discuss the problems of distributing importance splitting algorithms and present a fixed level algorithm optimised for distribution. We then define an expressive bounded time temporal logic and describe the system of efficient lightweight observers that implement it. These make the product automaton (_i_) accessible to the user, (_ii_) efficient to construct, (_iii_) efficient to distribute and (_iv_) efficient to execute. Finally, we demonstrate the performance and flexibility of our framework on a number of case studies that are intractable to numerical methods. ### Related Work There have been many ad hoc implementations of importance splitting based on the original ideas of [21, 22]. The algorithm of [27] is a relatively recent example that is often cited. The work of [5, 4] is novel because the authors define efficient adaptive importance splitting algorithms that also include confidence intervals. To our knowledge, [18] is the first work to explicitly link importance splitting to arbitrary logical properties, while the present work is the first to describe a practical importance splitting framework for SMC. The present work is thus the first to consider the problems of distributing importance splitting for SMC. SMC tools construct an automaton (a monitor) to accept traces that satisfy a temporal logic formula, typically based on a time bounded variant of temporal logic. The proportion of independent simulations of a stochastic model that satisfy the property is then used to estimate the probability of the property or to test hypotheses about the probability. There is considerable intersection between runtime verification (RV) and SMC, with few concepts unique to either. In particular, there have been many works that construct RV monitors from temporal logic (e.g., [10, 12, 14, 9, 2]). Such monitors typically comprise tableau-based automata [11] whose states represent the combinations of subformulas of the overall property. While some have considered timed properties (e.g., [2]), the focus is predominantly unbounded LTL properties interpreted on finite paths [8]. In contrast, SMC typically checks formulas with explicit time bounds (see, e.g., (1)), which are inherently defined on finite traces. To avoid the combinatorial explosion of subformulas caused by including time in this way, the monitors used by [17, 3] and other tools are compact “programs” that generate the states of an automaton on the fly and do not store them. We adapt this “lightweight” approach to allow importance splitting for SMC to be efficiently distributed on high performance parallel computational architectures. ## 2 Technical Background Our SMC tools [17, 3] implement a bounded linear temporal logic having the following typical syntactic form: \[\phi=\mathbf{X}^{k}\phi\mid\mathbf{F}^{k}\phi\mid\mathbf{G}^{k}\phi\mid\phi \mathbf{U}^{k}\phi\mid\neg\phi\mid\phi\vee\phi\mid\phi\wedge\phi\mid\phi \Rightarrow\phi\mid\alpha\] (1) This syntax allows arbitrary combinations and nesting of temporal and atomic properties (i.e., those which may be evaluated in a single state and denoted by $\alpha$). The time bound $k$ may denote discrete steps or continuous time, but in this work we consider only discrete time semantics. Given a finite trace $\omega$, comprising sequence of states $\omega_{0}\omega_{1}\omega_{2}\cdots$, $\omega^{(i)}$ denotes the suffix $\omega_{i}\omega_{i+1}\omega_{i+2}\cdots$. The semantics of the satisfaction relation $\models$ is constructed inductively as follows \[\small\begin{split}\omega^{(i)}\models& true\\ \omega^{(i)}\models&\alpha\iff\alpha\textnormal{ is }\mathit{true}\textnormal{ in state }\omega_{i}\\ \omega^{(i)}\models&\neg\varphi\iff\omega^{(i)} \models\varphi\not\in\,\models\\ \omega^{(i)}\models&\varphi_{1}\vee\varphi_{2}\iff \omega^{(i)}\models\varphi_{1}\textnormal{ or }\omega^{(i)}\models\varphi_{2} \\ \omega^{(i)}\models&\mathbf{X}^{k}\varphi\iff\omega^ {(k+i)}\models\varphi\\ \omega^{(i)}\models&\varphi_{1}\mathbf{U}^{k}\varphi _{2}\iff\exists j\in\{i,\dots,i+k\}:\omega^{(j)}\models\varphi_{2}\\ &\wedge(j=i\vee\forall l\in\{i,\dots,j-1\}:\omega^{(l)}\models \varphi_{1})\end{split}\] (2) Other elements of the relation are constructed using the equivalences $\mathit{false}\equiv\neg\mathit{true}$, $\phi\wedge\phi\equiv\neg(\neg\phi\vee\neg\phi)$, $\mathbf{F}^{k}\phi\equiv\mathit{true}\mathbf{U}^{k}\phi$, $\mathbf{G}^{k}\phi\equiv\neg(\mathit{true}\mathbf{U}^{k}\neg\phi)$. Hence, given a property $\varphi$, with syntax according to (1), $\omega\models\varphi$ is evaluated by $\omega^{(0)}\models\varphi$. ### Importance Splitting and Score Functions The neutron shield model of [20, 21] is illustrative of how importance splitting works. The distance travelled by a neutron in the shield defines a monotonic sequence of levels $0=s_{0}<s_{1}<s_{2}<\cdots<s_{m}=\mathit{shield\,thickness}$, such that reaching a given level implies having reached all the lower levels. While the overall probability $\gamma$ of passing through the shield is small, the probability of passing from one level to another can be made arbitrarily close to 1 by reducing the distance between levels. Denoting the abstract level of a neutron as $s$, the probability of a neutron reaching level $s_{i}$ can be expressed as $\mathrm{P}(s\geq s_{i})=\mathrm{P}(s\geq s_{i}\mid s\geq s_{i-1})\mathrm{P}(s \geq s_{i-1})$. Defining $\gamma=\mathrm{P}(s\geq s_{m})$ and $\mathrm{P}(s\geq s_{0})=1$, \[\gamma=\prod_{i=1}^{m}\mathrm{P}(s\geq s_{i}\mid s\geq s_{i-1}).\] (3) Each term of (3) is necessarily greater than or equal to $\gamma$, making their estimation easier. By writing $\gamma_{i}=\mathrm{P}(s\geq s_{i}\mid s\geq s_{i-1})$ and denoting the estimates of $\gamma$ and $\gamma_{i}$ as respectively $\hat{\gamma}$ and $\hat{\gamma}_{i}$, [18] defines the unbiased confidence interval (4) Confidence is specified via $z_{\alpha}$, the $1-\alpha/2$ quantile of the standard normal distribution, while $n$ is the per-level simulation budget. We infer from (4) that for a given $\gamma$ the confidence is maximised by making both the number of levels $m$ and the simulation budget large, with all $\gamma_{i}$ equal. The concept of levels can be generalised to arbitrary systems and properties in the context of SMC, treating $s$ and $s_{i}$ in (3) as values of a score function over the model-property product automaton. Intuitively, a score function discriminates good paths from bad, assigning higher scores to paths that more nearly satisfy the overall property. Since the choice of levels is crucial to the effectiveness of importance splitting, various ways to construct score functions from a temporal logic property are proposed in [18]. Formally, given a set of finite trace prefixes $\omega\in\Omega$, an ideal score function $S:\Omega\rightarrow\mathbb{R}$ has the characteristics $S(\omega)>S(\omega^{\prime})\iff\mathrm{P}(\models\varphi\mid\omega)>\mathrm{P }(\models\varphi\mid\omega^{\prime})$, where $\mathrm{P}(\models\varphi\mid\omega)$ is the probability of eventually satisfying $\varphi$ given prefix $\omega$. Intuitively, $\omega$ has a higher score than $\omega^{\prime}$ iff there is more chance of satisfying $\varphi$ by continuing $\omega$ than by continuing $\omega^{\prime}$. The minimum requirement of a score function is $S(\omega)\geq s_{\varphi}\iff\omega\models\varphi$, where $s_{\varphi}$ is an arbitrary value denoting that $\varphi$ is satisfied. Any trace that satisfies $\varphi$ must have a score of at least $s_{\varphi}$ and any trace that does not satisfy $\varphi$ must have a score less than $s_{\varphi}$. In what follows we assume that (3) refers to scores. ## 3 Distributing Importance Splitting Simple Monte Carlo SMC may be efficiently distributed because once initialised, simulations are executed independently and the result is communicated at the end with just a single bit of information (i.e., whether the property was satisfied or not). By contrast, the simulations of importance splitting are dependent because scores generated during the course of each simulation must be processed centrally. The amount of central processing can be minimised by reducing the number of levels, but this generally reduces the overall performance. Alternatively, entire instances of the importance splitting algorithm may be distributed and their estimates averaged, with each instance using a proportionally reduced simulation budget. We use this approach to generate some of the results in Section 6, but note that if the budget is reduced too far, the algorithm will fail to pass from one level to the next and no valid estimate will be produced. Distribution of importance splitting is thus possible, but its efficiency is dependent on the particular problem. In this work we therefore provide the framework to explore different approaches. In Section 3.1 we first describe the concept of an adaptive importance splitting algorithm and then explain why this otherwise optimised technique is unsuitable for distribution. In Section 3.2 we motivate the use of a fixed level algorithm for “lightweight” distribution and provide a suitable algorithm. The results we present in Section 6 demonstrate that this simpler approach can be highly effective. ### The Adaptive Algorithm The basic notion of importance splitting described in Section 2 can be directly implemented in a so-called fixed level algorithm, i.e., an algorithm in which the levels are pre-defined by the user. With no a priori information, such levels will typically be chosen to subdivide the maximum score equally. In general, however, this will not equally divide the conditional probabilities of the levels, as required by (4) to maximise performance. In the worst case, one or more of the conditional probabilities will be too low for the algorithm to pass between levels. Finding good or even reasonable levels by trial and error may be computationally expensive and has prompted the development of adaptive algorithms that discover optimal levels on the fly [5, 18, 19]. Instead of pre-defining levels, the user specifies the proportion of simulations to retain after each iteration. This proportion generally defines all but the final conditional probability in (3). The adaptive importance splitting algorithm first performs a number of simulations until the overall property is decided, storing the resulting traces of the model-property automaton. Each trace induces a sequence of scores and a corresponding maximum score. The algorithm finds a level that is less than or equal to the the maximum score of the desired proportion of simulations to retain. The simulations whose maximum score is below this current level are discarded. New simulations to replace the discarded ones are initialised with states corresponding to the current level, chosen at random from the retained simulations. The new simulations are continued until the overall property is decided and the procedure is repeated until a sufficient proportion of simulations satisfy the overall property. The principal advantage of the adaptive algorithm is that by simply rejecting the minimum number of simulations at each level it is possible to maximise confidence for a given score function. The principal disadvantage is that it stores simulation traces, severely limiting the size of model and simulation budget. The use of lightweight computational threads is effectively prohibited. Moreover, minimising the number of rejected simulations reduces the number of simulations performed between levels, thus reducing the possibility to perform computations in parallel. Minimising the rejected simulations also maximises the number of levels, which in turn minimises the number of simulation steps between each level. This further limits the feasibility of dividing the algorithm, since sending a model-property state over a slow communication channel may be orders of magnitude more costly than performing a short simulation locally. ### A Fixed Level Algorithm for Distribution In contrast to the adaptive algorithm, the fixed level importance splitting algorithm does not need to store traces, making it lightweight and suitable for distribution. Scores are calculated on the fly and only the states that achieve the desired level are retained for further consideration. While the choice of levels remains a problem, an effective strategy is to first use the adaptive algorithm with a relatively high rejection rate to find good fixed levels. An estimate with high confidence can then be generated efficiently by distributing the fixed level algorithm. Algorithm 1 is our fixed level importance splitting algorithm optimised for distribution. We use the terms server and client to refer to the root and leaf nodes of a network of computational devices or to mean independent computational threads on the same machine. In essence, the server manages the job and the clients perform the simulations. The server initially sends compact representations of the model and property to each client. Thereafter, only the state of the product automaton is communicated. In general, each client returns terminal states of simulations that reached the current level and the server distributes these as initial states for the next round of simulations. Algorithm 1 optimises this. The server requests and distributes only the number of states necessary to restart the simulations that failed to reach the current level, while maintaining the randomness of the selection. Despite this optimisation, however, the performance of this and other importance splitting algorithms will be confounded by the combination of large state size and properties having short time bounds. Under such circumstances it may be preferable to distribute entire instances of the algorithm, as described above. The memory requirements of Algorithm 1 are minimal. Each client need only store the state of $n$ simulations. As such, it is conceivable to distribute simulations on lightweight computational threads, such as those provided by GPGPU (general purpose computing on graphics processing units). ``` input:$s_{1}<s_{2}<\cdots<s_{m}$ is a sequence of scores, with $s_{m}=s_{\varphi}$ the score necessary to satisfy property $\varphi$ 1$\hat{\gamma}\gets 1$ is the initial estimate of $\gamma=\mathrm{P}(\omega\models\varphi)$ server sends compact description of model and observer to $k$ clients each client initialises $n$ simulations for $s\gets s_{1},\dots,s_{m}$ do 2 each client continues its $n$ simulations from their current state simulations halt as soon as their scores reach $s$$\forall$ clients, client $i$ sends server the number of traces $n_{i}$ that reached $s$ server calculates $\hat{\gamma}\leftarrow\hat{\gamma}n^{\prime}/kn$, where $n^{\prime}=\sum n_{i}$for $j\gets 1,\dots,kn-n^{\prime}$ do 3 server chooses client $i$ at random, with probability $n_{i}/n^{\prime}$ client $i$ sends server a state chosen uniformly at random from those that reached $s$ server sends state to client corresponding to failed simulation $j$, as initial state of new simulation to replace simulation $j$ 4 output:$\hat{\gamma}$ ``` Algorithm 1Distributed Fixed Level Importance Splitting ## 4 Linear Temporal Logic for Importance Splitting High performance SMC tools, such as [17, 3], avoid the complexity of standard model checking by compiling the property to a program of size proportional to the formula and memory proportional to the maximum sum of nested time bounds. This program implicitly encodes the model checking automaton, but is exponentially smaller. For example, the property $\mathbf{X}^{k}\phi$ can be implemented as a loop that generates $k$ simulation steps before returning the truth of $\phi$ in the last state; the property $\psi\mathbf{U}^{k}\phi$ can be implemented as a loop that generates up to $k$ simulation steps while $\psi$ is true and $\phi$ is not true, returning the value of $\phi$ in the last state otherwise. If $\psi$ and $\phi$ are atomic, the programs require just $\mathcal{O}(\log k)$ bits of memory to hold a loop counter. In contrast, the nested property $\mathbf{F}^{k1}(\psi\vee\mathbf{G}^{k2}\phi)$ has an $\mathcal{O}(k2)$ memory requirement. If $\psi$ is not true on step $i<k1$ it may be necessary to simulate up to step $i+k2$ to decide subformula $\mathbf{G}^{k2}\phi$. If $\psi\vee\mathbf{G}^{k2}\phi$ turns out to be false on step $i$, it will then be necessary to consider the truth of $\psi$ on step $i+1$, noting that the last simulated step could be $i+k2$. To evaluate this formula it is effectively necessary to remember the truth of $\psi$ on $\mathcal{O}(k2)$ simulation steps. Similar requirements can arise when the until operator ($\mathbf{U}$) is a subformula of a temporal operator. In all such cases, the sequence of stored truth values become part of the state of the property automaton. SMC using importance splitting requires that simulations are repeatedly and frequently initialised with the state of the model-property product automaton. If the size of this state is proportional to the time bounds of temporal operators, initialisation may have comparable complexity to simulation. This becomes especially problematic if the state is to be transmitted across relatively slow communication channels for the purposes of distribution. We therefore define a subset of (1), the size of whose automata is not dependent on the bounds of temporal operators: \[\begin{split}\phi=&\mathbf{X}^{k}\phi \mid\psi\mathbf{U}^{k}\psi\mid\neg\phi\mid\phi\vee\phi\mid\phi\wedge\phi\mid \phi\Rightarrow\phi\mid\psi\\ \psi=&\mathbf{X}^{k}\psi\mid\mathbf{F}^{k}\psi\mid \mathbf{G}^{k}\psi\mid\alpha\end{split}\] (5) The semantics of (5) is the same as (1), but (5) restricts how temporal operators may be combined. In particular, $\mathbf{U}$ may not be the subformula of a temporal operator other than $\mathbf{X}$ and temporal operators that are subformulas of other temporal operators may not be combined with Boolean connectives. Temporal operators containing other temporal operators as subformulas may, however, be combined. This logic expresses many useful properties, including nested bounded temporal properties that are not implemented in Prism. ## 5 Lightweight Observers for Importance Splitting To facilitate the construction of score functions we implement the logic given by (5) as a set of nested observers. Each observer corresponds to either a temporal operator, a Boolean operator acting on temporal operators, or as a predicate describing an atomic property. In our implementation observers are written in a syntax based on the commonly used reactive modules language [1], using the notion of ‘guarded commands’ [7] with sequential semantics. An observer comprises a set of guarded commands, any number of which may be enabled and executed on a given simulation step. Updates are performed in syntactic order after all guards have been evaluated, hence the update of one command does not affect the guards of commands in the same observer. In general, the output of one observer is the input to another and observers are therefore executed in reverse order of their nesting. Observers evaluate states as they are generated by the simulation. Since it may not be possible to decide a property before seeing a certain number of states, observers implement a three valued logic. In Figs. 1, 2 and 3 we use the symbols ?, $\top$ and $\bot$ to denote the three values _undecided_, _true_ and _false_, respectively. The state of an observer changes only when at least on of its inputs is decided. An observer may reach a deadlock state (no commands enabled) once its output is decided and cannot be changed by further input. A simulation terminates when the output of the root observer is decided, i.e., the property is decided. Simulations may also be paused by the importance splitting algorithm if the score reaches a desired level. Observers implementing the same temporal operator behave differently according to their level of nesting within a formula. We therefore distinguish _outer_ and _inner_ temporal observers. The temporal operators closest to the root of any branch of the syntax tree induced by a formula are implemented by outer observers. Their output proceeds from _undecided_ to either _true_ or _false_ and then does not change. Inner observers encode temporal operators that are the subformulas of other temporal operators. Their output proceeds from _undecided_ to a possibly alternating sequence of _true_, _false_ and _undecided_ values because their enclosing operator(s) cause them to evaluate a moving widow of states. The inner and outer variants of $\mathbf{X}$, $\mathbf{F}$ and $\mathbf{G}$ are closely related—outer observers are essentially simplified inner observers. When $\mathbf{U}$ is a subformula of $\mathbf{X}$, however, the $\mathbf{X}$ is implemented as a delay within the $\mathbf{U}$ observer. In what follows we describe the important aspects of the various observers that implement (5). The accompanying figures include a diagrammatic representation of how the observers work and a set of commands written in the form $\mathit{predicate}:\mathit{update}$. Each observer has Boolean output variables $o$ and $d$ to indicate respectively the result and whether the property has been decided (observers for atomic formulas omit $d$). Observers for temporal operators take discrete time bound $k$ as a parameter and use a counter variable $w$ ($\mathbf{U}$ uses counter variables $w^{\prime}$ and $w^{\prime\prime}$). Inner temporal operators make use of an additional counter, $t$ ($\mathbf{U}$ uses $t^{\prime}$ and $t^{\prime\prime}$). The inputs of observers are Boolean variables $o^{\prime}$ and $o^{\prime\prime}$, with corresponding decidedness $d^{\prime}$ and $d^{\prime\prime}$. #### Connective Observers These observers implement Boolean connectives at syntactic level $\phi$ in (5) and take advantage of the equivalences $\mathit{false}\wedge\textsf{?}=\mathit{false}$, $\mathit{true}\vee\textsf{?}=\mathit{true}$, $\mathit{false}\Rightarrow\textsf{?}=\mathit{false}$ and $\textsf{?}\Rightarrow\mathit{true}=\mathit{true}$, for any truth value of ?. Figure (a)a describes the observer for conjunction and Fig. (b)b describes the observer for implication. The observer for disjunction may be derived from that of conjunction by negating all instances of $o^{\prime}$ and $o^{\prime\prime}$, and by exchanging $o\leftarrow\mathit{true}$ and $o\leftarrow\mathit{false}$. Negation is implemented by inverting the truth assignment of the observer to which it applies, i.e., by exchanging $o\leftarrow\mathit{true}$ and $o\leftarrow\mathit{false}$. The connectives may be combined with themselves and outer temporal operators. Boolean connectives that apply only to atomic properties (i.e., syntactic level $\alpha$) are implemented directly in formulas within observers for atomic properties. <figure><img src="content_image/1502.01838/x1.png"><figcaption>Figure 4: Leader election.</figcaption></figure> #### Inner Temporal Observers These observers act on a moving window of states created by an enclosing temporal operator. The output may pass from one decided value to the other and also become undecided. Figure (a)a describes the observer for $\mathbf{X}^{k}$. Command 1 counts decided input states until bound $k$ is reached. Thereafter command 2 sets the output decided and equal to the value of the input. Figure (b)b describes the observer for $\mathbf{F}^{k}$. While decided inputs are not _true_, command 1 increments $w$ from 0 to $k$. If at any time the input is _true_, command 2 sets the output to _true_ and the “true-counter” $t$ is set to $w$. Command 5 decrements $t$ on subsequent false inputs. The output remains true while $t>0$. If $w$ reaches $k$ while $t=0$, command 3 sets the output to _false_. The observer for $\mathbf{G}^{k}$ may be derived from that of $\mathbf{F}^{k}$ by negating all instances of $o^{\prime}$ and $\neg o^{\prime}$, and by exchanging $o\leftarrow\mathit{true}$ and $o\leftarrow\mathit{false}$. <figure><img src="content_image/1502.01838/x1.png"><figcaption>Figure 4: Leader election.</figcaption></figure> #### Outer Temporal observers The outer observers for $\mathbf{X}^{k}$ and $\mathbf{F}^{k}$ are not illustrated but may be derived from their respective inner observers given in Fig. 2. For $\mathbf{X}^{k}$, command 3 is removed and the guard of command 2 is strengthened with $\neg d$. For $\mathbf{F}^{k}$, commands 4, 5 and 6, together with all references to counter $t$, are removed, while the guards of commands 2 and 3 are strengthened by $\neg d$. The outer observer for $\mathbf{G}^{k}$ can be derived from that of $\mathbf{F}^{k}$ in the same way as described for inner temporal observers. Figure 3 describes the observer for properties of the form $\mathbf{X}^{k_{\mathbf{X}}}(\psi\mathbf{U}^{k}\phi)$ and $\psi\mathbf{U}^{k}\phi$. Since $\psi$ and $\phi$ may be temporal formulas that are satisfied on different simulation steps in arbitrary order, the observer employs variables $w^{\prime}$ and $w^{\prime\prime}$ to respectively count the sequences of $\neg\phi$ and $\psi$ (commands 3 and 5). Variable $t^{\prime}$ then records the position of the first $\phi$ (command 4), while $t^{\prime\prime}$ records the position of the last $\psi$ (command 5). Using $t^{\prime}$ and $t^{\prime\prime}$, commands 7 and 8 are able to determine if the property is satisfied or falsified, respectively. Properties of the form $\mathbf{X}^{k_{\mathbf{X}}}(\psi\mathbf{U}^{k}\phi)$ are implemented by simply initialising variables $w^{\prime}$ and $w^{\prime\prime}$ to $-k_{\mathbf{X}}$, forcing the observer to ignore the first $k_{\mathbf{X}}$ decided values of $\psi$ and $\phi$. If the property is not of this form, $w^{\prime}$ and $w^{\prime\prime}$ are initialised to 0 and the automaton may be simplified by removing commands 1 and 2 and all instances of expressions $w^{\prime}\geq 0$ and $w^{\prime\prime}\geq 0$. <figure><img src="content_image/1502.01838/x1.png"><figcaption>Figure 4: Leader election.</figcaption></figure> ## 6 Case Studies We have implemented our importance splitting framework in Plasma[3] and demonstrate its use on three case studies whose state space is intractable to numerical model checking. The following results do not seek to promote a particular methodology (adaptive or fixed level algorithm, distributed or single machine), but serve to illustrate the flexibility of our platform. The software, models and observers can be downloaded from our website¹. The leader election and dining philosophers models are also illustrated on the Prism case studies website². [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] For each model we performed a number of experiments to compare the performance of the fixed and adaptive importance splitting algorithms with and without distribution, using different budgets and levels. Our results are illustrated in the form of empirical cumulative probability distributions of $100$ estimates, noting that a perfect (zero variance) estimator distribution would be a single step. The results are also summarised in Table 6. The probabilities we estimate are all close to $10^{-6}$ and are marked on the figures with a vertical line. Since we are not able to use numerical techniques to calculate the true probabilities, we use the average of $200$ low variance estimates as our best overall estimate. As a reference, we applied the adaptive algorithm to each model using a single computational thread. We chose parameters to maximise the number of levels and thus minimise the variance for a given score function and budget. The resulting distributions, sampled at every tenth percentile, are plotted with circular markers in the figures. Over these points we superimpose the results of applying a single instance of the fixed level algorithm with just a few levels. We also superimpose the average estimates of five parallel threads running the fixed level algorithm, using the same levels. The figures confirm our expectation that the fixed level algorithm with few levels is outperformed by the adaptive algorithm. The figures also demonstrate that the average of parallel instances of the fixed level algorithm are very close to the performance of the adaptive algorithm. The timings given in Table 6 show that the distributed approach achieves these results in less time. For comparison we also include the estimated time of using a simple Monte Carlo (MC) estimator to achieve the same standard deviation. Importance splitting gives more than three orders of magnitude improvement in all cases. All results were generated using an Intel Core i7-3740 CPU with 4 cores running at 2.7 GHz. In the remainder of this section we briefly describe our models and their associated properties and score functions. <figure><img src="content_image/1502.01838/x1.png"><figcaption>Figure 4: Leader election.</figcaption></figure> #### Leader Election Our leader election case study is based on the Prism model of the synchronous leader election protocol of [15]. With $N=20$ processes and $K=6$ probabilistic choices the model has approximately $1.2\times 10^{18}$ states. We consider the probability of the property $\mathbf{G}^{420}\neg\mathit{elected}$, where $\mathit{elected}$ denotes the state where a leader has been elected. Our chosen score function uses the time bound of the $\mathbf{G}$ operator to give nominal scores between $0$ and $420$. The model constrains these to only $20$ actual levels, but with evenly distributed probability. For the fixed level algorithm we use scores of $70,140,210,280,350$ and $420$. #### Dining Philosophers Our dining philosophers case study extends the Prism model of the fair probabilistic protocol of [23]. With 150 philosophers our model contains approximately $2.3\times 10^{144}$ states. We consider the probability of the property $\mathbf{F}^{30}\textit{Phil eats}$, where $\mathit{Phil}$ is the name of an arbitrary philosopher. The adaptive algorithm uses the heuristic score function described in [19], which includes the five logical levels used by the fixed level algorithm. The heuristic favours short paths, based on the assumption that as time runs out the property is less likely to be satisfied. <figure><img src="content_image/1502.01838/x3.png"><figcaption>Figure 6: Dependent counters.</figcaption></figure> #### Dependent Counters Our dependent counters case study comprises ten counters, initially set to zero, that with some probability dependent on the values of the other counters are either incremented or reset to zero. This can be viewed as modelling an abstract computational process, a set of reservoirs of finite capacity, or as the failure and repair of ten different types of components in a system, etc. With a maximum count of $10$, the model has approximately $2.6\times 10^{10}$ states. We consider the probability of the property $\mathbf{X}^{1}(\neg\mathit{init}\mathbf{U}^{1000}\mathit{complete})$, where $\mathit{init}$ and $\mathit{complete}$ denote the initial state and the state where all counters have reached their maximum value. Our score function ranges over values between 0 and 99, but the probabilities are not evenly distributed. With a budget of $500$, uniformly distributed fixed scores fail to produce traces that satisfy the property until the difference between the last two levels is about $5$. Note that our budget is limited to only 500 simulations due to the length of the traces that must be stored by the adaptive algorithm. We maintain this budget for the fixed level algorithm to simplify comparison. After a small amount of trial and error, we adopted fixed scores of $80,90,95$ and $99$. ## 7 Challenges and Prospects Our results demonstrate the effectiveness and flexibility of our framework with discrete time properties applied to standard case studies. Future challenges include industrial scale examples and the implementation of continuous time properties. We also intend to provide proofs of the correctness of our observers and of our logic’s memory requirements. Although the manual construction of score functions adds to the overall cost of using importance splitting, we believe that distribution relaxes the need for these to be highly optimised. We also expect that it will be possible to construct good score functions automatically using statistical learning techniques. ## References * [1] R. Alur and T. A. Henzinger. Reactive modules. _Formal Methods in System Design_, 15(1):7–48, 1999. * [2] A. Bauer, M. Leucker, and C. Schallhart. Monitoring of real-time properties. In _FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science_, pages 260–272. Springer, 2006. * [3] B. Boyer, K. Corre, A. Legay, and S. Sedwards. PLASMA-lab: A flexible, distributable statistical model checking library. In K. Joshi, M. Siegle, M. Stoelinga, and P. R. D’Argenio, editors, _Quantitative Evaluation of Systems_, volume 8054 of _LNCS_, pages 160–164. Springer, 2013. * [4] F. Cérou, P. Del Moral, T. Furon, and A. Guyader. Sequential Monte Carlo for rare event estimation. _Statistics and Computing_, 22:795–808, 2012. * [5] F. Cérou and A. Guyader. Adaptive multilevel splitting for rare event analysis. _Stochastic Analysis and Applications_, 25:417–443, 2007. * [6] E. M. Clarke, E. A. Emerson, and J. Sifakis. Model checking: algorithmic verification and debugging. _Commun. ACM_, 52(11):74–84, November 2009. * [7] E. W. Dijkstra. Guarded commands, nondeterminacy and formal derivation of programs. _Commun. ACM_, 18(8):453–457, August 1975. * [8] C. Eisner, D. Fisman, J. Havlicek, Y. Lustig, A. McIsaac, and D. Van Campenhout. Reasoning with temporal logic on truncated paths. In _Computer Aided Verification_, pages 27–39. Springer, 2003. * [9] B. Finkbeiner and H. Sipma. Checking finite traces using alternating automata. _Formal Methods in System Design_, 24(2):101–127, 2004. * [10] M. C. W. Geilen. On the construction of monitors for temporal logic properties. _Electronic Notes in Theoretical Computer Science_, 55(2):181–199, 2001. * [11] R. Gerth, D. Peled, M. Y. V. Vardi, and P. Wolper. Simple on-the-fly automatic verification of linear temporal logic. In _In Protocol Specification Testing and Verification_, pages 3–18. Chapman & Hall, 1995. * [12] D. Giannakopoulou and K. Havelund. Automata-based verification of temporal properties on running programs. In _Proceedings of 16th Annual International Conference on Automated Software Engineering_, pages 412–416. IEEE, Nov 2001. * [13] J. M. Hammersley and D. C. Handscomb. _Monte Carlo Methods_. Methuen & Co., 1964. * [14] K. Havelund and G. Roşu. Synthesizing monitors for safety properties. In J.-P. Katoen and P. Stevens, editors, _Tools and Algorithms for the Construction and Analysis of Systems_, volume 2280 of _Lecture Notes in Computer Science_, pages 342–356. Springer, 2002. * [15] A. Itai and M. Rodeh. Symmetry breaking in distributed networks. _Information and Computation_, 88(1):60–87, 1990. * [16] C. Jegourel, A. Legay, and S. Sedwards. Cross-entropy optimisation of importance sampling parameters for statistical model checking. In P. Madhusudan and S. A. Seshia, editors, _Computer Aided Verification_, volume 7358 of _LNCS_, pages 327–342. Springer, 2012. * [17] C. Jegourel, A. Legay, and S. Sedwards. A platform for high performance statistical model checking – PLASMA. In C. Flanagan and B. König, editors, _Tools and Algorithms for the Construction and Analysis of Systems_, volume 7214 of _LNCS_, pages 498–503. Springer, 2012. * [18] C. Jegourel, A. Legay, and S. Sedwards. Importance splitting for statistical model checking rare properties. In _Computer Aided Verification_, volume 8044 of _LNCS_, pages 576–591. Springer, 2013. * [19] C. Jegourel, A. Legay, and S. Sedwards. An effective heuristic for adaptive importance splitting in statistical model checking. In T. Margaria and B. Steffen, editors, _Leveraging Applications of Formal Methods, Verification and Validation. Specialized Techniques and Applications_, volume 8803 of _LNCS_, pages 143–159. Springer, 2014. * [20] H. Kahn. Random sampling (Monte Carlo) techniques in neutron attenuation problems. _Nucleonics_, 6(5):27, 1950. * [21] H. Kahn and T. E. Harris. Estimation of particle transmission by random sampling. In _Applied Mathematics_, volume 5 of _series 12_. National Bureau of Standards, 1951. * [22] H. Kahn and A. W. Marshall. Methods of reducing sample size in Monte Carlo computations. _Operations Research_, 1(5):263–278, November 1953. * [23] D. Lehmann and M. O. Rabin. On the advantage of free choice: A symmetric and fully distributed solution to the dining philosophers problem. In _Proc. $8^{\mathit{th}}$Ann. Symposium on Principles of Programming Languages_, pages 133–138, 1981. * [24] N. Metropolis and S. Ulam. The Monte Carlo method. _Journal of the American Statistical Association_, 44(247):335–341, September 1949. * [25] H. Niederreiter. _Random Number Generation and Quasi-Monte Carlo Methods_. Society for Industrial and Applied Mathematics, 1992. * [26] G. Rubino and B. Tufin (eds.). _Rare Event Simulation using Monte Carlo Methods_. John Wiley & Sons, Ltd, 2009. * [27] M. Villén-Altamirano and J. Villén-Altamirano. RESTART: A method for accelerating rare event simulations. In J. W. Cohen and C. D. Pack, editors, _Queueing, Performance and Control in ATM_, pages 71–76. Elsevier, 1991. * [28] H. S. Younes, M. Kwiatkowska, G. Norman, and D. Parker. Numerical vs. statistical probabilistic model checking. _International Journal on Software Tools for Technology Transfer_, 8(3):216–228, 2006.
1610.02239	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 57060, "num_imgs": 6, "llama3_tokens_count": 15361 }	[ "content_image/1610.02239/x1.png", "content_image/1610.02239/x3.png", "content_image/1610.02239/x6.png", "content_image/1610.02239/x8.png", "content_image/1610.02239/x10.png", "content_image/1610.02239/x13.png" ]	# Surface Conductivity of Si(100) and Ge(100) Surfaces Determined from Four-Point Transport Measurements Using an Analytical N-Layer Conductance Model Sven Just Peter Grünberg Institut (PGI-3) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany Helmut Soltner Central Institute of Engineering, Electronics and Analytics (ZEA-1), Forschungszentrum Jülich, 52425 Jülich, Germany Stefan Korte Peter Grünberg Institut (PGI-3) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany Vasily Cherepanov Peter Grünberg Institut (PGI-3) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany Bert Voigtländer b.voigtlaender@fz-juelich.de Peter Grünberg Institut (PGI-3) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany February 22, 2024 ###### Abstract An analytical N-layer model for charge transport close to a surface is derived from the solution of Poisson’s equation and used to describe distance-dependent electrical four-point measurements on the microscale. As the N-layer model comprises a surface channel, multiple intermediate layers and a semi-infinite bulk, it can be applied to semiconductors in combination with a calculation of the near-surface band-bending to model very precisely the measured four-point resistance on the surface of a specific sample and to extract a value for the surface conductivity. For describing four-point measurements on sample geometries with mixed 2D-3D conduction channels often a very simple parallel-circuit model has so far been used in the literature, but the application of this model is limited, as there are already significant deviations, when it is compared to the lowest possible case of the N-layer model, i.e. the 3-layer model. Furthermore, the N-layer model is applied to published distance-dependent four-point resistance measurements obtained with a multi-tip scanning tunneling microscope (STM) on Germanium(100) and Silicon(100) with different bulk doping concentrations resulting in the determination of values for the surface conductivities of these materials. pacs: Valid PACS appear here † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction Due to the downscaling of modern nanoelectronic devices the surface-to-volume ratio increases continuously and the surface becomes increasingly important as an additional conductance channel for charge transport. To assess the influence of this surface channel on the device performance or even be able to use it as a functional unit, a reliable value for the two-dimensional surface conductivity has to be known. However, the determination of the surface conductivity from electrical four-point measurements is quite a challenging task, as the main difficulty is to separate the 2D conductance at the surface from the conductance through other channels, e.g. the bulk and the space charge layer. Often indirect measurement methods are used for the separation of the 2D conductance at the surface, but these methods have special requirements on the material and the preparation of the sample under study. For example, one method for separating the surface conductivity is based on the comparison of measurements before and after quenching the surface states by adsorption of atoms or molecules Hasegawa and Ino (1992); Hasegawa _et al._ (1996); Petersen _et al._ (1997); D’angelo _et al._ (2009); Martins _et al._ (2014). The adsorption species has to be chosen specifically for the material under study and for the quenched system several conditions have to be carefully confirmed. First, all of the surface states have to be quenched and, secondly, the conductivity of the near-surface space charge region has to remain unchanged under the influence of the adsorbed surface layer. Thirdly, no additional surface conductance has to be induced by the adsorbed layer. If one of these conditions is not fulfilled, the experiments based on the difference method can result in underestimated values for the surface conductivity. Here, we present a generic N-layer conductance model, free of such requirements, for describing the measured four-point resistance on samples consisting of a surface channel, a space charge region due to the near-surface band-bending and a semi-infinite bulk. No special sample preparation is necessary and the model can directly be applied to the raw data, which in combination with a calculation of the conductivity profile in the space charge region permits to extract the value for the surface conductivity from distance-dependent four-point measurements. First, we compare a very simple model often used to describe measurements on samples with mixed 2D-3D conduction channels, the parallel-circuit model, to the N-layer model, and point out that the application of the parallel-circuit model is very limited, as there are already significant deviations if the N-layer model is reduced to the simplest case of a 3-layer model ($N=3$). Secondly, we apply the N-layer model to different distance-dependent four-point measurements from the literature obtained with a multi-tip scanning tunneling microscope on the semiconductors Ge(100) and Si(100) with different types and concentrations of doping, and determine values for the surface conductivity of these materials. The analytical derivation of the N-layer model is shown in detail in the appendix. ## II Mixed 2D-3D conduction channels <figure><img src="content_image/1610.02239/x1.png"><figcaption>Figure 1: (Color online) (a) Calculated four-point resistance of theSi(111)-(7×7) surface with a bulk conductivity of σB=0.14S/m and a surfaceconductivity of σS=5.14⋅10−6S/□ as a function of the equidistant probedistance s and with the ratio σSC/σB between the conductivities of the spacecharge layer and the bulk as additional parameter (colored curves). The orangecurve located between the two limiting cases of pure 2D and pure 3Dconductance (dotted blue and red curves) is based on measurements Just _etal._ (2015), while the magenta, green, blue and red curves correspond tovariations of the ratio σSC/σB over several orders of magnitude. The blackcurve results from the description by the parallel-circuit model withoutconsidering an additional space charge layer between surface and bulk. In theinset, the equidistant linear tip arrangement with the outer current-injectingtips and the inner voltage-measuring tips is shown. (b) Calculated percentageof surface current Isurf as function of the ratios σSz−1S/σB between thesurface conductivity and the bulk (zS=3\AA), and σSC/σB between theconductivity of the space charge layer and the bulk. The colored pointscorrespond to the position of the curves in (a). Inside the region marked bythe two dotted lines the parallel-circuit model can be applied for describingthe four-point resistance on the surface with an error of less than 10%.</figcaption></figure> <figure><img src="content_image/1610.02239/x3.png"><figcaption>Figure 2: (Color online) Color plots of the absolute value of the in-linecomponent of the current density j(x,y,z) in the xz-plane as a function ofdepth z into the sample and lateral distance x along the tip positioning line.The current density is calculated from the 3-layer model for a distance3s=150μm of the current-injecting tips, and for a sample with a bulkconductivity σB=0.14S/m, a surface conductivity σS=5.14⋅10−6S/□ and an averagethickness z2=2.5μm of the intermediate space charge layer. The averageconductivity of the intermediate space charge layer is varied in the threecases (a) - (c) showing the significant influence of the space charge regionon the vertical current distribution in the sample. According to the 3-layermodel the red dashed lines indicate the interfaces between the surface, thespace charge layer and the bulk. The black dotted vertical lines mark theposition of the current-injecting tips on the surface. (a) In the case of avery low conducting space charge layer with σSC≪σB (σSC=2.5⋅10−4S/m) themajority of the current flows through the surface even if the bulk is highlyconductive, as the space charge region acts as a blockade for the injectioninto the bulk and an enhanced 2D transport can be observed. (b) If σSC=σB,there is effectively no space charge region and the current flow through thebulk takes place according to the bulk conductivity. In this case the four-point resistance on the surface can be approximated by the parallel-circuitmodel. (c) If the space charge layer is highly conductive with σSC≫σB(σSC=2.5⋅102S/m), the current flows not only through the surface, but alsoequally through the space charge layer, while the current in the bulk is againreduced.</figcaption></figure> For pure 2D or pure 3D charge transport, there exist simple analytic relations between the measured four-point resistance and the conductivity. For an equidistant probe setup with a distance $s$ between the tips, the following equations are obtained for a 2D sheet and a 3D half-space Schroder (2006), respectively \[R^{4p}_{2D}=\frac{\ln 2}{\pi\sigma_{2D}},\quad\mathrm{and}\quad R^{4p}_{3D}= \frac{1}{2\pi\sigma_{3D}}\cdot s^{-1}\] (1) with the 2D surface conductivity $\sigma_{2D}$ and the 3D bulk conductivity $\sigma_{3D}$. The equation for the 2D case shows a constant four-point resistance, independent of the probe spacing, while the conductance through a 3D channel depends on the spacing $s$. Due to this characteristic probe-spacing dependency, it is possible to distinguish between 2D and 3D channels from distance-dependent four-point measurements. However, if a sample consists of a mixed 2D-3D geometry, e.g. a conducting sheet on a conducting substrate, these two equations cannot be applied any more. Often, a simple approximation of a parallel-circuit consisting of the four-point resistance of the surface and the bulk according to Eq.(1) is used Perkins _et al._ (2013); Wells _et al._ (9) \[R^{4p}_{\parallel}(s)=\left(\frac{1}{R^{4p}_{2D}}+\frac{1}{R^{4p}_{3D}(s)} \right)^{-1}\,\mathrm{,}\] (2) but this approach has restrictions and shortcomings, as it can be seen in the following. In the parallel-circuit model a complete separation of the surface conductance channel and the bulk is assumed. The splitting of the injection current between the surface and the bulk only takes place at the injection points and depends on the ratio of the four-point resistances of the two individual layers. However, the two-point resistance, and not the four-point resistance, should determine, which amount of current flows through the surface channel and which part through the bulk Polley _et al._ (2012). Therefore, the exact current path through the sample depends also on details of the injection, e.g. the tip diameter, which are not included in the parallel-circuit model. The most important point, however, is the fact that in the approximation of the parallel-circuit model the current is injected equally into the surface channel and the bulk, and any influence of a possible near-surface space charge region, which particularly exists in semiconductors, is neglected. But especially this space charge region has a significant influence on the charge transport through the sample, as it will be discussed in the following. A different approach presented in Durand _et al._ (2016) uses an approximation for the surface current to solve the current continuity equations for 2D and 3D resulting in a combination of both 2D and 3D conduction channels. This approach removes the artificial separation between surface and bulk and uses a real injection geometry with extended tips, but it takes only into account a two-layer structure consisting of the surface and the bulk, so that the results are very similar to the parallel-circuit model. Any additional conductivity distribution between the surface and the bulk caused by a space charge region is neglected, which is also the major restriction in the parallel-circuit model. For this reason, the model can only be applied, if no near-surface band-bending occurs and a sharp transition between surface and bulk exists. Another approach published in Lis _et al._ (2015) attempts to describe the deviation from a pure 3D conductance behavior caused by an additional 2D channel with an expansion of distance-dependent terms, and introduces an effective conductivity consisting of the bulk conductivity and a value for the deviation from the pure 3D case. However, although this model may also be able to treat deviations caused by a near-surface space charge region, it is not suitable to determine a value for the surface conductivity, as the deviations from the pure 3D conductance are only indicated by one numerical value, which cannot be easily interpreted as a physical quantity. In Wells _et al._ (13) a computational method is described using no longer an analytical model for the four-point resistance but a finite element calculation for approximating the different conduction channels in the sample. In this case, also the near-surface space-charge layer between the surface channel and the 3D bulk can be taken into account. However, as the surface channel has only a depth of several Å, while the space-charge layer may be extended up to several $\mu m$, very different length scales are involved, so that the finite element calculation of the complete sample geometry can be very sophisticated and computationally time consuming. The best way to point out the important role of the space charge region, which is especially important for semiconductors, and the limited applicability of a two-layer model, like the parallel-circuit model, is a comparison of the four-point resistance with the lowest N-layer model including the influence of the space charge region, i.e. the 3-layer model. Apart from the surface layer and the bulk region this 3-layer model uses only one additional layer to approximate the space charge region, but despite this quite rough approximation it is able to describe four-point resistance measurement values much better than the parallel-circuit model and was successfully applied to determine the surface conductivity of the Si(111)-($7\times 7)$ surface Just _et al._ (2015). In Fig. 1 (a) the calculated distance-dependent four-point resistance for the Si(111)-($7\times 7)$ surface on an n-doped substrate ($700\,\mathrm{\Omega cm}$) is shown (orange line) located between the two limiting cases of pure surface conductance (dotted blue line) and pure bulk conductance (dotted red line). The calculation is based on the 3-layer model with parameters obtained in Just _et al._ (2015) and assumes an equidistant linear tip configuration with a tip spacing $s$. Using the same parameters for surface and bulk conductivity the four-point resistance expected from the parallel-circuit model according to Eq. 2 is plotted as solid black line, which exhibits a very strong deviation from the curve based on the 3-layer model. The major reason for this behavior is the absence of the additional space charge layer between surface and bulk in the parallel-circuit model. In the case of the Si(111)-($7\times 7)$ surface on an n-doped Si substrate with $\sigma_{B}=0.14\,\mathrm{S/m}$ the ratio between the average conductivity of the space charge region $\sigma_{SC}$ and the bulk can be estimated as $\sigma_{SC}/\sigma_{B}=0.002$Just _et al._ (2015). For smaller values of this ratio, the deviation of the 3-layer model from the parallel-circuit model increases and the calculated four-point resistance approaches the 2D case (magenta curve). On the other hand, if the ratio becomes larger, the deviation between the two models decreases (green and blue curves). But only if the ratio $\sigma_{SC}/\sigma_{B}$ is close to 1 (red curve), the deviation between both models is so small, that the parallel-circuit model can be used as approximation without a large error. This error is smallest, if the near-surface space charge region vanishes completely, and in this case the parallel-circuit model is a suitable simple approach to approximate the four-point resistance of a two-layer structure consisting of a 2D and a 3D conduction channel. The significant influence of the space charge region can also be deduced from the amount of current flowing through the surface compared to the totally injected current. In Fig. 1 (b) the calculated percentage of surface current is shown in dependence of the conductivity ratios between space charge layer and bulk $\sigma_{SC}/\sigma_{B}$ and the surface and bulk $\sigma_{S}\,z_{S}^{-1}/\sigma_{B}$ (thickness of surface layer $z_{S}\approx 3\,\mathrm{\AA}$) for a constant tip distance of $s=50\,\mu\mathrm{m}$. The calculation is again based on the 3-layer model and on parameters obtained in Just _et al._ (2015) for the measurements of the Si(111)-($7\times 7$) surface. For a vanishing space charge layer, i.e. $\sigma_{SC}/\sigma_{B}\approx 1$, the amount of surface current approximately only depends on the ratio $\sigma_{S}\,z_{S}^{-1}/\sigma_{B}$ and increases with an increasing ratio. However, if the influence of the space charge layer becomes larger, i.e. if the ratio $\sigma_{SC}/\sigma_{B}$ deviates from 1, the contour lines in the plot get distorted, so that for large ratios the amount of surface current is reduced and for small ratios enhanced. The reason for this behavior is that the conductivity of the space charge layer controls the current injection into the bulk below. If the near-surface band-bending leads to a depletion zone or an inversion zone so that the average conductivity in the space charge region is significantly reduced compared to the bulk, then this region behaves as a blocking region preventing the injected current to flow through the bulk, even if it has a very high conductivity. This results in an enhanced surface domination of charge transport, which cannot be considered in the parallel-circuit model. In Fig. 2 this behavior is visualized by the depth-dependent current density inside the sample. The absolute value of the in-line component of the current density $\mathbf{j}(x,y,z)$ in the xz-plane is plotted as function of depth $z$ into the sample and lateral distance $x$ along the tip positioning line. The calculation is based on the 3-layer model with the same parameters as used in Fig. 1(b) and a distance of $3s=150\,\mu\mathrm{m}$ for the current injecting tips. For the first case in Fig. 2(a), a very low conducting space charge layer with $\sigma_{SC}\ll\sigma_{B}$ (thickness $z_{SC}=2.5\,\mathrm{\mu m}$) is used for the calculation, and the result shows that the majority of the current flows through the surface layer (thickness $z_{S}=3\,\mathrm{\AA}$), whereas only a very small amount of current is injected through the space charge layer into the bulk. The current density inside the bulk material is one order of magnitude lower than in the case of a vanishing near-surface band-bending, where the space charge layer coincides with the bulk ($\sigma_{SC}\approx\sigma_{B}$), which is depicted in Fig. 2 (b). On the other hand, if an accumulation zone is formed near the surface with a high conductivity compared to the bulk, this region can act as an additional conductance channel totally surpassing the current flow through the bulk and also reducing the current through the surface states. In this case shown in Fig. 2 (c), where $\sigma_{SC}\gg\sigma_{B}$, the current flow through the bulk is again reduced by an order of magnitude, while not only transport through the surface states but also through the space charge region is now preferred equally. As the space charge layer has a finite thickness, the current transport may seem to be purely 2-dimensional for larger probe spacings and the usage of the parallel-circuit model for the four-point resistance on such a system would result in a largely overestimated value for the surface conductivity. In conclusion, the parallel-circuit model has only a very limited applicability within a certain range of conductivity parameters, where the space charge region does not play a significant role for the current transport. In Fig. 1(b) the dotted lines indicate the region, inside which the parallel-circuit model can be applied to four-point resistance measurements with an error of less than 10%. Inside this region, the contour lines of the color plot are approximately perpendicular to the x-axis indicating that the surface current is nearly independent of the ratio $\sigma_{SC}/\sigma_{B}$, which is an essential requirement for the application of the parallel-circuit model. For comparison, the four colored points indicate the positions of the resistance curves from Fig. 1 (a). Only the red curve, which is very close to the parallel-circuit model, is located inside the dotted region, while the orange curve representing a measurement of the Si(111)-($7\times 7$) surface on an n-doped substrate is clearly outside the region. Although the 3-layer model is obviously better suitable to describe measurement data over a wider range of conductivity parameters than the parallel-circuit model, it still has a basic restriction: the very rough description of the space charge region by only a single layer. Especially for semiconductors, which can have a very strong band-bending near the surface, this can be a major drawback. For this reason, the 3-layer model should be refined by introducing more layers resulting in an N-layer model, which is discussed in the following section. ## III The N-layer model The 3-layer model offers only a rough approximation of the space charge region described by only a single layer with an average conductivity and average thickness. However, the conductivity profile in this region can exhibit a very strong dependence on the z-position, and, especially, if an inversion layer is formed in the near-surface region, the description by a single layer is not sufficient any more. Therefore, we try to approximate the space charge region by more than one layer and present an N-layer model for charge transport consisting of a thin surface layer, $N-2$ layers for the near-surface space charge region, and a semi-infinite bulk. Such a multi-layer model was first proposed by Schumann and Gardner Schumann and Gardner (14, 15); Gardner and Schumann (1965) and primarily applied to the method of spreading resistance measurements Leong _et al._ (1977); Berkowitz and Lux (1979); Clarysse _et al._ (2000), but also extended to four-point measurements Wang _et al._ (2005) for determining individual sheet conductivities. However, as far as we know, it has not yet been used for obtaining the conductivity of surface states of semiconductors in combination with a calculated conductivity profile of the space charge region as input. A detailed description and mathematical derivation of the N-layer model is shown in the appendix A. In the following section, the application of the N-layer model is demonstrated and it is used to obtain values for the surface conductivity of the Ge(100) and Si(100) surfaces. ## IV Application of the N-layer model The advantage of the N-layer model is that it can be used for evaluation of all distance-dependent four-probe resistance measurements without the need of any special sample preparation before the measurement, e.g. in order to quench the surface states D’angelo _et al._ (2009); Hasegawa and Ino (1992); Hasegawa _et al._ (1996); Petersen _et al._ (1997), or special measurement conditions, e.g. varying the temperature Wells _et al._ (2006); Tanikawa _et al._ (2003); Yoo and Weitering (2002). For this reason, we apply the N-layer model to already published data of the semiconductor surfaces Ge(100) and Si(100), which were described previously by either pure 2D or pure 3D conductance, but not by a mixed transport channel. In combination with the N-layer model, it is now possible to take into account simultaneously the current transport through the 2D surface and through the 3D bulk both influenced by the presence of the near-surface space charge layer, and to determine values for the surface conductivities of the materials from these measurements. ### Germanium(100) <figure><img src="content_image/1610.02239/x6.png"><figcaption>Figure 3: (Color online) (a) Four-point resistance of a p-doped Ge(100) sample(nominal bulk resistivity (0.1−0.5)Ωcm) as function of probe distance sbetween the inner voltage-measuring tips Wojtaszek _et al._ (2014). Differentcolored data points correspond to different distances D in the symmetriclinear tip configuration shown in the inset. The solid lines represent onesingle fit to all data points using the N-layer model for charge transport,which results in a value for the surface conductivity of σS=(2.9±0.6)⋅10−4S/□and for the bulk resistivity of ρB=(0.22±0.01)Ωcm. The dotted lines indicatethe expected four-point resistances for a vanishing surface conductancechannel, i.e. σS=0, taking into account only the space charge region and thebulk. (b) The calculated conductivity profile of the space charge layer asfunction of the depth z into the sample starting from the surface. Thisprofile is approximated with N=20 layers and used as input for the N-layermodel. The band diagram in the inset shows the surface pinning of the Fermilevel EF (red) located 0.11eV above the valence band edge and the resultingnear-surface band-bending of the conduction band EC (green) and the valenceband EV (blue).</figcaption></figure> Distance-dependent four-point transport measurements on the Ge(100) surface were published by Wojtaszek _et al._Wojtaszek _et al._ (2014). They used a room-temperature, ultra-high vacuum multi-tip STM and carried out four-point resistance measurements on Ge(100) substrates with different bulk doping concentration and type. A symmetric linear probe configuration was used, where the outer current-injecting tips have a distance $D$ and the inner voltage-measuring tips are separated by the distance $s$. The complete setup is symmetric with respect to the centre plane of the tip positioning line. In Fig. 3(a), the experimental data for a p-type Ga-doped sample with a nominal bulk resistivity of $0.1-0.5\,\Omega\mathrm{cm}$ are shown Wojtaszek _et al._ (2014). The measured four-point resistance is plotted as a function of the spacing $s$ between the voltage-measuring tips and with the distance $D$ between the current-injecting tips as additional parameter. In the framework of the publication Wojtaszek _et al._ (2014), these data were described by a pure 3D conductance channel. However, it was mentioned that there were some systematic deviations from the 3D model, which increasingly appear, if the voltage-measuring tips approach the positions of the current-injecting tips, i.e. $s/D\geq 0.7$, but the origin of these deviations could not be explained quantitatively. In fact, for the symmetric linear tip configuration, it is particularly the region with a ratio $s/D$ close to 1, where the setup is most sensitive to surface transport and a possible surface conductance channel would have the most influence on the measured four-point resistance. So, it is reasonable to assume that the observed deviations are caused by an additional 2D conductance channel through the surface states of the Ge(100)-(2$\times$1) surface, which cannot be considered by the pure 3D model. <figure><img src="content_image/1610.02239/x8.png"><figcaption>Figure 4: (Color online) (a) Four-point resistance of an n-type doped, almostintrinsic Ge(100) sample (nominal bulk resistivity ∼45Ωcm) as function ofprobe distance s between the inner voltage-measuring tips Wojtaszek _et al._(2014). Different colored data points correspond to different distances D inthe symmetric linear tip configuration (inset in Fig. 3 (a)). The solid linesrepresent a single fit to all data points using the N-layer model for chargetransport (N=20), which results in a value for the surface conductivity ofσS=(3.4±0.2)⋅10−4S/□ and for the bulk resistivity of ρB=(45±22)Ωcm. The dottedlines correspond to the expected four-point resistances without any surfacechannel (σS=0) taking into account only the bulk and the space charge region.(b) Calculated conductivity profile of the space charge region as function ofthe depth z from the surface (red line). The approximated profile (green line)is used as input for the N-layer model. In the upper inset, the complete rangeof the conductivity profile of the space charge region exhibiting a shape ofan inversion layer is shown. The lower inset depicts the surface pinning ofthe Fermi level EF (red) and the induced near-surface band-bending of theconduction band EC (green) and the valence band EV (blue).</figcaption></figure> In order to describe this additional 2D transport channel more quantitatively, we evaluate the existing data with the N-layer model. First, the near-surface band-bending of the p-type Ge(100) sample is calculated by solving Poisson’s equation and using a Fermi level pinning at the surface of $\sim 0.11\,\mathrm{eV}$ above the valence band Tsipas and Dimoulas (2009); Dimoulas _et al._ (2006); Broqvist _et al._ (2008). Fig 3 (b) shows the resulting depth-dependent conductivity profile of the space charge region consisting of a near-surface accumulation layer. This conductivity profile is approximated by a step function of $(N-2)$ steps ($N=20$) determining the values for $\sigma_{n}$ and $z_{n}$ to be used as input for the N-layer model (details in the appendix). For the symmetric linear tip setup the four-point resistance according to the N-layer model can be expressed as function of $s$ and $D$ by the equation \[R^{4p}(s,D)= \frac{2}{I}\int_{0}^{\infty}\left[a_{0}(k)+a_{1}(k)\right]\cdot \left[J_{0}\left(k\cdot\frac{D-s}{2}\right)\right.\] \[\left.-J_{0}\left(k\cdot\frac{D+s}{2}\right)\right]\,\mathrm{d}k \,\mathrm{,}\] (3) which is fitted to the measurement data resulting in the colored solid curves shown in Fig. 3 (a). All four curves for the different values for the distance $D$ correspond to only a single fit with the surface conductivity $\sigma_{S}$ and the bulk conductivity $\sigma_{B}$ confined close to the range of the nominal values as free parameters. As the conductivity profile of the space charge region also depends on the bulk conductivity, an iterative fitting process is applied, which includes the calculation of the space charge region and the fit to the data in each iteration. For values of $\sigma_{S}=(2.9\,\pm\,0.6)\cdot 10^{-4}\mathrm{S}/\square$ and $\sigma_{B}=(460\,\pm\,11)\mathrm{S}/\mathrm{m}$ the iterative process converges and the best fit is obtained describing the data very precisely throughout the complete measurement range without any systematic deviations. A further advantage is the resulting single value for each of the parameters $\sigma_{S}$ and $\sigma_{B}$, which is sufficient to describe precisely all four resistance curves for the different distances $D$. In the case of a pure 3D model, as it is used for the fitting process in Wojtaszek _et al._ (2014), it is not possible to model all four data sets with only one value for the bulk conductivity $\sigma_{B}$. The 3D fit has to be applied separately to each curve resulting in different values for $\sigma_{B}$ spreading by a relative deviation of $\sim$ 25%. However, the measured bulk conductivity should not change during the variation of the tip configuration by the distance $D$ on the same substrate. This reveals that, even if the transport in the sample is mostly 3D dominated due to the highly conductive bulk and the weak accumulation zone near the surface, a description of the data by a pure 3D model is not sufficient and an additional 2D channel has to be taken into account. For validating the results for the additional surface conductance channel and ensuring that the observed amount of two-dimensional conductance is not merely caused by the near-surface accumulation layer, the dotted colored curves in Fig. 3 (a) correspond to the expected four-point resistance for a vanishing surface channel. In these curves, only the bulk conductivity and the conductivity profile of the space charge region according to Fig. 3 (b) are taken into account, while the value for the surface conductivity $\sigma_{S}$ is set to zero. The clearly visible deviation of the dotted curves from the measurement data verifies that an additional 2D surface conductance channel is necessary for describing the measured four-point resistance, and, therefore, proves the existence of conducting surface states. Fig. 4 (a) shows similar distance-dependent four-point resistance measurements on an n-type doped, almost intrinsic Ge(100) sample with a nominal bulk resistivity of $\sim 45\,\Omega\mathrm{cm}$Wojtaszek _et al._ (2014). As the measurement data show an enhanced two-dimensional character of conductance, a pure 2D model was used in Wojtaszek _et al._ (2014), which was justified by the presence of a near-surface inversion layer totally preventing the current to be injected into the bulk and acting as a 2D channel, which confines the current close to the surface. However, any possible presence of an additional 2D surface channel caused by surface states was neglected. In this case, a further disentanglement between the conductivity of the near-surface p-type part of the inversion layer and the surface conductivity would be required. So, we try again to describe the measurement data with the N-layer model. The calculated conductivity profile of the space charge region shows the expected inversion layer depicted in Fig. 4 (b). For the calculation, the transition region between p-type and n-type of conduction has not been taken into account and only the absolute value of the conductivity is considered, but, as the majority of the current flows through the near-surface p-type part of the inversion layer and through the surface channel, this approximation should be suitable in the present case. The conductivity profile is described by a step function (green line) and used in combination with the N-layer model for a fit to the data according to Eq. 3. In Fig. 4 (a), the two solid curves result from a single fit with the parameters $\sigma_{S}=(3.4\,\pm\,0.2)\cdot 10^{-4}\,\mathrm{S}/\square$ and $\sigma_{B}=(2.2\,\pm\,1.1)\,\mathrm{S}/\mathrm{m}$ and describe the data very precisely. For verification, the dotted lines shown in Fig. 4 (a) again represent the expected four-point resistance without any additional surface channel ($\sigma_{S}=0$). The very strong deviation from the measurement data indicates clearly that the observed transport behavior cannot only be caused by the enhanced conductivity close to the surface due to the inversion layer, but that there has to be an additional surface conductance channel also on the n-type sample. If the results for the p-type and n-type Ge(100) samples are compared, the values for the obtained surface conductivity coincide within the error limits. This is expected, as the surface states should not be influenced by the doping type of the substrate. Thus, this is another confirmation that really the conductivity of the surface states was determined. By combining the results of the p- and n-type sample, a more precise value for the surface conductivity of the Ge(100)-(2$\times$1) surface of $\sigma_{S,Ge(100))}=(3.1\,\pm\,0.6)\cdot 10^{-4}\,\mathrm{S}/\square$ can be obtained. ### Silicon(100) <figure><img src="content_image/1610.02239/x10.png"><figcaption>Figure 5: (Color online) (a) Four-point resistance of a p-doped (red datapoints) and an n-doped (blue data points) Si(100)-(2×1) sample (nominal bulkresistivity (1−10)Ωcm) as function of the equidistant probe distance sreproduced from Polley _et al._ (2012). Fits to the data (solid lines) basedon the N-layer model result in a surface conductivity of σS=(1.9±1.4)⋅10−4S/□and in a bulk resistivity of ρB=(7.5±0.9)Ωcm for the p-doped case, and inσS=(1.6±0.4)⋅10−4S/□ and ρB=(10±7.5)Ωcm, respectively, for the n-doped sample.The inset shows the equidistant tip configuration. (b),(c) Calculatedconductivity profiles of the space charge region for the p- and n-dopedsamples (red curves). The approximation by N=20 layers (green curves) is usedfor the N-layer model. In the insets, the near-surface band-bending of theconduction band EC (green) and the valence band EV (blue) caused by thesurface pinning of the Fermi level EF (red) due to the surface states located≈0.31eV above the valence band edge is shown.</figcaption></figure> Distance-dependent four-point resistance measurements on p-type and n-type doped Si(100) substrates were carried out by Polley _et al._Polley _et al._ (2012). For the measurements, a room temperature, ultra-high vacuum multi-tip STM was used with a linear equidistant tip configuration with spacing $s$ between adjacent tips. The current was injected by the outer tips and the potential drop between the inner tips was measured. In Fig. 5 (a), the measured four-point resistance is shown as a function of the tip distance $s$ for an n-type (blue points) and a p-type (red points) Si(100) substrate both with a nominal bulk resistivity of $(1-10)\,\Omega\mathrm{cm}$. Although the bulk doping concentrations of p- and n-type sample are similar, the observed transport behavior is completely different. In the p-type case, a 3D conduction channel is more dominant, while in the n-type case the majority of current flows through a 2D transport channel. Again, this was explained by the presence of an inversion layer in the n-type sample preventing the current to flow through the bulk. So, the measured data were described in Polley _et al._ (2012) by a pure 3D conductance model for the p-type substrate and by a pure 2D model in the n-type case. However, this approach cannot consider any possible mixed 2D-3D conductance channels through the space charge region and the bulk in both samples, and, especially, neglects the two-dimensional surface state, which should be present on the Si(100)-(2$\times$1) surface Mårtensson _et al._ (1986). For refining the description of the measured data on the Si(100) substrates and for determining a value for the conductivity of the Si(100)-(2$\times$1) surface state, we use the N-layer model. Fig. 5 (b) and (c) show the corresponding conductivity profiles of the space charge region for the p-type and n-type Si(100) substrates, respectively. For the calculation, a Fermi level pinning of the surface states of $\sim 0.31\,\mathrm{eV}$ above the valence band is used Yoo and Weitering (2002); Mårtensson _et al._ (1986); Himpsel _et al._ (1980). In the p-type case, a depletion zone is formed close to the surface, while in the n-type case an inversion layer separates the bulk from the near-surface region. Again, the pn-transition is not considered for the inversion layer, as the n-type bulk does not contribute significantly to current transport. The approximation of the conductivity profiles (green curves) is used as input for fitting the respective measurement data in Fig. 5 (a) according to Eq. 20. The results are depicted as solid curves in Fig. 5 (a) and correspond to fitparameters for the surface conductivity of $\sigma_{S}=(1.9\,\pm\,1.4)\cdot 10^{-4}\,\mathrm{S}/\square$ and for the bulk conductivity, which is confined to the range of the nominal value, of $\sigma_{B}=(13.3\,\pm\,1.7)\,\mathrm{S}/\mathrm{m}$ for the p-type sample, and to values of $\sigma_{S}=(1.6\,\pm\,0.4)\cdot 10^{-4}\,\mathrm{S}/\square$ and $\sigma_{B}=(10\,\pm\,7.5)\,\mathrm{S}/\mathrm{m}$ in the n-type case. As the four-point resistance measurement for the p-type sample in the chosen tip distance range is not very surface sensitive, the determined value for the surface conductivity has quite a large error, even if the curve fits quite well to the data. The fitted curve for the n-type substrate shows some larger deviations due to a larger spread and a slight increasing behavior of the data, which might be caused by tip positioning errors or influence of the sample edges. However, the obtained value for the surface conductivity is more precise, as the transport behavior in the n-type sample is now more dominated by the near-surface region. So, as both values are still consistent within the error limits, the value resulting from the n-type sample can describe the conductivity of the Si(100)-(2$\times$1) surface state more precisely as $\sigma_{S,Si(100)}=(1.6\,\pm\,0.4)\cdot 10^{-4}\,\mathrm{S}/\square$. ## V Conclusion Surface reconstruction \| Surface conductivity σS ---\|--- Si(100)-(2×1) \| (1.6±0.4)⋅10−4S/□ Ge(100)-(2×1) \| (3.1±0.6)⋅10−4S/□ Si(111)-(7×7) \| (8.6±1.9)⋅10−6S/□ Just _et al._ (2015) Bi/Si(111)-(√3×√3)R30∘ \| (1.4±0.1)⋅10−4S/□ Just _et al._ (2015) Ag/Si(111)-(√3×√3)R30∘ \| (3.1±0.4)⋅10−3S/□ Lüpke _et al._ (2015) Table 1: Values for the surface conductivity of different reconstructed and passivated surfaces of silicon and germanium. In conclusion, we applied an analytically derived N-layer model for current transport through multiple layers of different conductivity including the calculation of the near-surface band-bending to interpret distance-dependent four-point resistance measurements on semiconductor surfaces. First, the important role of the space charge region for the current distribution in the sample was discussed and it was shown that already the lowest case of the N-layer model, i.e. the 3-layer model, can describe measured four-point resistance data much better than the often used parallel-circuit model, which completely neglects the space charge region. The derivation of the N-layer model and its usage for multi-probe distance-dependent four-point resistance measurements on surfaces was presented. Finally, the N-layer model was used for describing published distance-dependent four-point measurements on Ge(100) and Si(100) surfaces and values for the conductivities of the surface states of these materials could be determined as summarized in Tab. 1. For comparison, values for the surface conductivities of differently reconstructed and passivated Si(111) surfaces are also listed. In total, the presented method is quite generic and can easily be used for many other materials to determine values for the surface conductivity. * ## Appendix A Derivation of the analytical N-layer conductance model <figure><img src="content_image/1610.02239/x13.png"><figcaption>Figure 6: (Color online) The N-layer model consists of a layered samplestructure with N layers described by the conductivities σn and the positionsof the interfaces zn (n=1,…,N−2), respectively. The first layer 0 and the lastlayer N−1 represent the surface layer and the semi-infinite bulk,respectively. The other layers in between are used to approximate thez-dependent conductivity profile of the space charge region. The current I isinjected by a cylindrical tip of radius rt at the origin on the surface layer.</figcaption></figure> The N-layer model uses a structure shown in Fig. 6 to describe the sample properties. It consists of a thin surface layer, multiple intermediate layers for approximating the space charge region and a semi-infinite bulk characterized by their respective conductivities $\sigma_{0}$, $\sigma_{n}$ and $\sigma_{N-1}$, and positions of the interfaces $z_{0}$ and $z_{n}$ ($\mathrm{n}=1,\ldots,N-2$). At the surface a current $I$ is injected by a cylindrical tip with radius $r_{t}$. Due to calculation requirements, the surface layer cannot be two-dimensional, so that a finite thickness of one atomic layer ($3\,\mathrm{\AA}$) is assumed. As $\nabla\cdot\mathbf{j}=0$ for the current density $\mathbf{j}=\sigma\mathbf{E}=-\sigma\,\nabla\Phi$ inside the sample (excluding the injection point), the electrical potential $\Phi$ in this region can be determined by solving the Laplace equation \[\Delta\Phi =0\] (4) in cylindrical coordinates. Taking account of the angle-independent polar symmetry for one tip, a solution for the potential in the individual layers is Jackson (1999) \[\Phi_{0}(\rho,z)=\int_{0}^{\infty}\left[a_{0}(k)\,e^{kz}+a_{1}(k) \,e^{-kz}\right]J_{0}(k\rho)\,\mathrm{d}k\,\mathrm{,}\] (5) \[\Phi_{n}(\rho,z)=\int_{0}^{\infty}\left[a_{2n}(k)\,e^{kz}+a_{2n+1 }(k)\,e^{-kz}\right]J_{0}(k\rho)\,\mathrm{d}k\,\mathrm{,}\] (6) \[\Phi_{N-1}(\rho,z)=\int_{0}^{\infty}a_{2N-2}(k)\,e^{-kz}\,J_{0}(k \rho)\,\mathrm{d}k\,\mathrm{,}\] (7) with $J_{0}$ denoting the Bessel function of the first kind. With the assumption of a uniform current flux beneath the tip contact the boundary conditions are \[\sigma_{0}\frac{\partial}{\partial z}\Phi_{0}(\rho,0) =-j_{0}\,H(r_{t}-\rho)\,\mathrm{,}\] (8) \[\Phi_{n-1}(\rho,z_{n-1}) =\Phi_{n}(\rho,z_{n-1})\,\mathrm{,}\] (9) \[\sigma_{n-1}\frac{\partial}{\partial z}\Phi_{n-1}(\rho,z_{n-1}) =\sigma_{n}\frac{\partial}{\partial z}\Phi_{n}(\rho,z_{n-1})\, \mathrm{,}\] (10) \[\Phi_{N-2}(\rho,z_{N-2}) =\Phi_{N-1}(\rho,z_{N-2})\,\mathrm{,}\] (11) \[\sigma_{N-2}\frac{\partial}{\partial z}\Phi_{N-2}(\rho,z_{N-2}) =\sigma_{N-1}\frac{\partial}{\partial z}\Phi_{N-1}(\rho,z_{N-2}) \,\mathrm{,}\] (12) resulting from the current injection (Eq. 8), as well as from the continuous transitions of the potential (Eq. 9 and Eq. 11) and the current density (Eq. 10 and Eq. 12) between the layers. In Eq. 8, the expression $H(r_{t}-\rho)$ denotes the Heaviside step function. According to the uniform flux condition the injected current density is described by $j_{0}=\frac{I}{\pi\,r_{t}^{2}}$ assuming a cylindrical tip with a tip radius of $r_{t}\approx 25\,\mathrm{nm}$, which seems reasonable for an STM tip. Nevertheless, it turns out that also other values for the tip radius in the range of $5\,\mathrm{nm}$ to $100\,\mathrm{nm}$ do not influence the results of the calculations in a considerable manner. Besides the uniform flux condition Leong _et al._ (1977), several other assumptions for the current density at the injection point have been presented in the literature, i.e. the variable flux condition based on the exact solution for a circular contact on an infinetely thick slab Schumann and Gardner (14) and the Dirac delta current distribution leading to a ring current density Berkowitz and Lux (1979); Wang _et al._ (2005). All approaches are used to approximate the exact surface boundary condition of constant potential beneath the probe, which would lead to a more difficult mixed boundary value problem. However, the differences between the three conditions are rather small Leong _et al._ (1978); Berkowitz and Lux (1979); Wang _et al._ (2005), and especially for small layer thicknesses compared to the radius of the probe contacts, as it applies for the highly conductive surface layer with a thickness of $3\,\mathrm{\AA}$, the uniform flux condition is the best approximation Leong _et al._ (1978), so that we use this condition for the calculation. Based on Eqs. 8$-$12, a matrix equation determining the coefficients $a_{0}(k),\dotsc,a_{2N-2}(k)$ is derived \[\begin{pmatrix}1&-1&0&0&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots &\ldots&\ldots&0\\ \omit\span\omit\span\omit\span\omit\hbox{\multirowsetup$\mathbf{A}^{0,1}$}&0&0 &\omit\span\omit\hbox{\multirowsetup$\ldots$}&\omit\span\omit\hbox{ \multirowsetup$\ldots$}&\omit\span\omit\hbox{\multirowsetup$\ldots$}&0\\ \omit\span\omit\span\omit\span\omit&0&0&\omit\span\omit&\omit\span\omit&\omit \span\omit&0\\ 0&0&\omit\span\omit\span\omit\span\omit\hbox{\multirowsetup$\mathbf{A}^{1,2}$} &0&0&\omit\span\omit\hbox{\multirowsetup$\ldots$}&\omit\span\omit\hbox{ \multirowsetup$\ldots$}&0\\ 0&0&\omit\span\omit\span\omit\span\omit&0&0&\omit\span\omit&\omit\span\omit&0 \\ 0&0&0&0&\omit\span\omit\span\omit\span\omit\hbox{\multirowsetup$\ddots$}&0&0& \omit\span\omit\hbox{\multirowsetup$\ldots$}&0\\ 0&0&0&0&\omit\span\omit\span\omit\span\omit&0&0&\omit\span\omit&0\\ 0&0&\omit\span\omit\hbox{\multirowsetup$\ldots$}&0&0&\omit\span\omit\span\omit \span\omit\hbox{\multirowsetup$\mathbf{A}^{n-1,n}$}&0&0&0\\ 0&0&\omit\span\omit&0&0&\omit\span\omit\span\omit\span\omit&0&0&0\\ 0&0&\omit\span\omit\hbox{\multirowsetup$\ldots$}&\omit\span\omit\hbox{ \multirowsetup$\ldots$}&0&0&\omit\span\omit\span\omit\span\omit\hbox{ \multirowsetup$\ddots$}&0\\ 0&0&\omit\span\omit&\omit\span\omit&0&0&\omit\span\omit\span\omit\span\omit&0 \\ 0&0&\omit\span\omit\hbox{\multirowsetup$\ldots$}&\omit\span\omit\hbox{ \multirowsetup$\ldots$}&\omit\span\omit\hbox{\multirowsetup$\ldots$}&0&0&\omit \span\omit\span\omit\hbox{\multirowsetup$\mathbf{B}$}\\ 0&0&\omit\span\omit&\omit\span\omit&\omit\span\omit&0&0&\omit\span\omit\span \omit\\ \end{pmatrix}\cdot\begin{pmatrix}a_{0}(k)\\ a_{1}(k)\\ a_{2}(k)\\ a_{3}(k)\\ a_{4}(k)\\ \vdots\\ a_{2n-2}(k)\\ a_{2n-1}(k)\\ a_{2n}(k)\\ a_{2n+1}(k)\\ \vdots\\ a_{2N-3}(k)\\ a_{2N-2}(k)\\ \end{pmatrix} =\begin{pmatrix}I(k,\sigma_{0})\\ 0\\ 0\\ 0\\ 0\\ \vdots\\ 0\\ 0\\ 0\\ 0\\ \vdots\\ 0\\ 0\end{pmatrix}\,\mathrm{,}\] (13) with the submatrices \[\mathbf{A}^{n-1,n}=\begin{pmatrix}\frac{\sigma_{n-1}}{\sigma_{n}} &-\frac{\sigma_{n-1}}{\sigma_{n}}\,e^{-2kz_{n-1}}&-1&e^{-2kz_{n-1}}\\ 1&e^{-2kz_{n-1}}&-1&-e^{-2kz_{n-1}}\end{pmatrix}\] (14) and \[\mathbf{B}=\begin{pmatrix}\frac{\sigma_{N-2}}{\sigma_{N-1}}&- \frac{\sigma_{N-2}}{\sigma_{N-1}}\,e^{-2kz_{N-2}}&e^{-2kz_{N-2}}\\ 1&e^{-2kz_{N-2}}&-e^{-2kz_{N-2}}\\ \end{pmatrix}\,\mathrm{,}\] (15) and the expression \[I(k,\sigma_{0})=-\frac{j_{0}}{\sigma_{0}}\,\int_{0}^{r_{t}}\rho \,J_{0}(k\rho)\,\mathrm{d}\rho\,\mathrm{.}\] (16) This equation can be solved by means of numerical matrix inversion of the $(2N-1)\times(2N-1)$ matrix. As the potential at the surface ($z=0$) can be expressed by \[\Phi_{\mathrm{surf}}(\rho)=\Phi_{0}(\rho,0) =\int_{0}^{\infty}\left[a_{0}(k)+a_{1}(k)\right]J_{0}(k\rho)\, \mathrm{d}k\,\mathrm{,}\] (17) only the coefficients $a_{0}(k)$ and $a_{1}(k)$ are relevant for the calculation. Introducing cartesian coordinates with $\mathbf{x}=\begin{pmatrix}x&y\end{pmatrix}^{T}$ and $\rho=\|\mathbf{x}-\mathbf{x_{0}}\|=\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}$ for a tip positioned at $\mathbf{x_{0}}$, the combined potential on the surface $\Phi_{\mathrm{surf},12}$ for a current source at position $\mathbf{x_{0_{1}}}$ and a current sink at position $\mathbf{x_{0_{2}}}$ results by superposition in \[\Phi_{\mathrm{surf},12}(\mathbf{x}) =\Phi_{\mathrm{surf},1}(\|\mathbf{x}-\mathbf{x_{0_{1}}}\|)-\Phi_{ \mathrm{surf},2}(\|\mathbf{x}-\mathbf{x_{0_{2}}}\|)\,\mathrm{.}\] (18) Finally, the four-point resistance $R^{4p}$ measured on the surface is determined by the quotient of the potential difference between the positions $\mathbf{x_{0_{3}}}$ and $\mathbf{x_{0_{4}}}$ of the voltage-measuring tips and the current $I$, resulting in \[R^{4p} =\frac{\Phi_{\mathrm{surf},12}(\mathbf{x_{0_{3}}})-\Phi_{\mathrm{ surf},12}(\mathbf{x_{0_{4}}})}{I}\] \[=\frac{1}{I}\int_{0}^{\infty}\left[a_{0}(k)+a_{1}(k)\right]\cdot \left[J_{0}(k\,\|\mathbf{x_{0_{3}}}-\mathbf{x_{0_{1}}}\|)\right.\] \[\left.-J_{0}(k\,\|\mathbf{x_{0_{3}}}-\mathbf{x_{0_{2}}}\|) \hskip 0.0pt-J_{0}(k\,\|\mathbf{x_{0_{4}}}-\mathbf{x_{0_{1}}}\|)\right.\] \[\left.+J_{0}(k\,\|\mathbf{x_{0_{4}}}-\mathbf{x_{0_{2}}}\|) \right]\,\mathrm{d}k\,\mathrm{.}\] (19) For the linear probe configuration with equidistant spacing $s$ between the four tips Eq. 19 simplifies to \[R^{4p}(s) =\frac{2}{I}\int_{0}^{\infty}\left[a_{0}(k)+a_{1}(k)\right]\cdot \left[J_{0}(ks)-J_{0}(2ks)\right]\,\mathrm{d}k\,\mathrm{.}\] (20) The integral over the Bessel functions in Eq. 20 can be evaluated numerically and the result can be fitted to the four-point measurement data. However, the conductivities $\sigma_{n}$ and interface positions $z_{n}$ of all N layers are far too many free parameters for being determined by a single fit. Therefore, the depth-dependent conductivity profile $\sigma(z)$ of the space charge region has to be calculated before, based on the solution of Poisson’s equation using basic material parameters like the Fermi level pinning of the surface states, the band-gap, the effective masses, the mobilities and the bulk doping concentration and type Lüth (2015). The approximation of this profile by a step function of $(N-2)$ - steps then determines the values for $\sigma_{1},\ldots,\sigma_{N-2}$ and $z_{1},\ldots,z_{N-2}$, which are used as input for the N-layer model. The thickness of the surface layer $z_{0}$ determines the vertical extension of the surface states and can be approximated by the thickness of one atomic layer. The value for the bulk conductivity $\sigma_{N-1}$ can be determined by macroscopic resistivity measurements and should be in agreement with the nominal doping concentration. Finally, only one free parameter remains to be determined by a fit to measurement data, which is the surface conductivity $\sigma_{0}$. ## References * Hasegawa and Ino (1992)S. Hasegawa and S. Ino, Phys. Rev. Lett. 68, 1192 (1992). * Hasegawa _et al._ (1996)Y. Hasegawa, I.-W. Lyo, and P. Avouris, Surf. Sci. 357, 32 (1996). * Petersen _et al._ (1997)C. L. Petersen, F. Grey, and M. Aono, Surf. Sci. 377, 676 (1997). * D’angelo _et al._ (2009)M. D’angelo, K. Takase, N. Miyata, T. Hirahara, S. Hasegawa, A. Nishide, M. Ogawa, and I. Matsuda, Phys. Rev. B 79, 035318 (2009). * Martins _et al._ (2014)B. V. C. Martins, M. Smeu, L. Livadaru, H. Guo, and R. A. Wolkow, Phys. Rev. Lett. 112, 246802 (2014). * Just _et al._ (2015)S. Just, M. Blab, S. Korte, V. Cherepanov, H. Soltner, and B. Voigtländer, Phys. Rev. Lett. 115, 066801 (2015). * Schroder (2006)D. K. Schroder, _Semiconductor material and device characterization_, 3rd ed. (Wiley, New York, 2006). * Perkins _et al._ (2013)E. Perkins, L. Barreto, J. Wells, and P. Hofmann, Rev. Sci. Instrum. 84, 033901 (2013). * Wells _et al._ (2008a)J. W. Wells, K. Handrup, J. F. Kallehauge, L. Gammelgaard, P. Boggild, M. B. Balslev, J. E. Hansen, P. R. E. Petersen, and P. Hofmann, J. Appl. Phys. 104, 053717 (2008a). * Polley _et al._ (2012)C. M. Polley, W. R. Clarke, J. A. Miwa, M. Y. Simmons, and J. W. Wells, Appl. Phys. Lett. 101, 262105 (2012). * Durand _et al._ (2016)C. Durand, X.-G. Zhang, S. M. Hus, C. Ma, M. A. McGuire, Y.Xu, H. Cao, I. Miotkowski, Y. P. Chen, and A.-P. Li, Nano Lett. 16, 2213 (2016). * Lis _et al._ (2015)J. Lis, M. Wojtaszek, R. Zuzak, B. Such, and M. Szymonski, Phys. Rev. B 92, 035309 (2015). * Wells _et al._ (2008b)J. W. Wells, J. F. Kallehauge, and P. Hofmann, Surf. Sci. 602, 1742 (2008b). * Schumann and Gardner (1969a)P. A. Schumann and E. E. Gardner, J. Electrochem. Soc. 116, 87 (1969a). * Schumann and Gardner (1969b)P. A. Schumann and E. E. Gardner, Solid State Electron. 12, 371 (1969b). * Gardner and Schumann (1965)E. E. Gardner and P. A. Schumann, Solid State Electron. 8, 165 (1965). * Leong _et al._ (1977)M. S. Leong, S. C. Choo, and C. C. Wang, Solid State Electron. 20, 255 (1977). * Berkowitz and Lux (1979)H. L. Berkowitz and R. A. Lux, J. Electrochem. Soc. 126, 1479 (1979). * Clarysse _et al._ (2000)T. Clarysse, W. Vandervorst, E. J. H. Collart, and A. J. Murrell, J. Electrochem. Soc. 147, 3569 (2000). * Wang _et al._ (2005)C.-W. Wang, A. M. Sastry, K. A. Striebel, and K. Zaghib, J. Electrochem. Soc. 152, A1001 (2005). * Wells _et al._ (2006)J. W. Wells, J. F. Kallehauge, T. M. Hansen, and P. Hofmann, Phys. Rev. Lett. 97, 206803 (2006). * Tanikawa _et al._ (2003)T. Tanikawa, K. Yoo, I.Matsuda, S. Hasegawa, and Y. Hasegawa, Phys. Rev. B 68, 113303 (2003). * Yoo and Weitering (2002)K. Yoo and H. H. Weitering, Phys. Rev. B 65, 115424 (2002). * Wojtaszek _et al._ (2014)M. Wojtaszek, J. Lis, R. Zuzak, B. Such, and M. Szymonski, Appl. Phys. Lett. 105, 042111 (2014). * Tsipas and Dimoulas (2009)P. Tsipas and A. Dimoulas, Appl. Phys. Lett. 94, 012114 (2009). * Dimoulas _et al._ (2006)A. Dimoulas, P. Tsipas, A. Sotiropoulos, and E. K. Evangelou, Appl. Phys. Lett. 89, 252110 (2006). * Broqvist _et al._ (2008)P. Broqvist, A. Alkauskas, and A. Pasquarello, Phys. Rev. B 78, 075203 (2008). * Mårtensson _et al._ (1986)P. Mårtensson, A. Cricenti, and G. V. Hansson, Phys. Rev. B 33, 8855 (1986). * Himpsel _et al._ (1980)F. J. Himpsel, P. Heimann, T. C. Chiang, and D. E. Eastman, Phys. Rev. Lett. 45, 1112 (1980). * Lüpke _et al._ (2015)F. Lüpke, S. Korte, V. Cherepanov, and B. Voigtländer, Rev. Sci. Instrum. 86, 123701 (2015). * Jackson (1999)J. D. Jackson, _Classical Electrodynamics_, 3rd ed. (Wiley, New York, 1999). * Leong _et al._ (1978)M. S. Leong, S. C. Choo, and L. S. Tan, Solid State Electron. 21, 933 (1978). * Lüth (2015)H. Lüth, _Solid Surfaces, Interfaces and Thin Films_, 6th ed. (Springer Verlag, Cham, 2015).
1807.03992	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 73757, "num_imgs": 4, "llama3_tokens_count": 26444 }	[ "content_image/1807.03992/x1.png", "content_image/1807.03992/x1.png", "content_image/1807.03992/x2.png", "content_image/1807.03992/x3.png" ]	# Real algebraic curves with large finite number of real points Erwan Brugallé Erwan Brugallé, Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, F-44322 Nantes Cedex 3, France erwan.brugalle@math.cnrs.fr Alex Degtyarev Alex Degtyarev, Bilkent University Department of Mathematics 06800 Ankara, Turkey degt@fen.bilkent.edu.tr Ilia Itenberg Ilia Itenberg, Institut de Mathématiques de Jussieu–Paris Rive Gauche Sorbonne Université 4 place Jussieu, 75252 Paris Cedex 5, France and Département de Mathématiques et Applications, Ecole Normale Supérieure 45 rue d’Ulm, 75230 Paris Cedex 5, France ilia.itenberg@imj-prg.fr Frédéric Mangolte Frédéric Mangolte, Laboratoire angevin de recherche en mathématiques (LAREMA), Université d’Angers, CNRS, 49045 Angers Cedex 01, France frederic.mangolte@univ-angers.fr http://www.math.univ-angers.fr/ mangolte ###### Abstract. We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled. ## 1. Introduction A _real algebraic variety_$(X,c)$ is a complex algebraic variety equipped with an anti-holomorphic involution $c\colon X\to X$, called a _real structure_. We denote by $\mathbb{R}X$ the real part of $X$, _i.e._, the fixed point set of $c$. With a certain abuse of language, a real algebraic variety is called _finite_ if so is its real part. Note that each real point of a finite real algebraic variety of positive dimension is in the singular locus of the variety. ### Statement of the problem In this paper we mainly deal with the first non-trivial case, namely, finite real algebraic curves in $\mathbb{C}P^{2}$. (Some of the results are extended to more general surfaces.) The degree of such a curve $C\subset\mathbb{C}P^{2}$ is necessarily even, $\deg C=2k$. Our primary concern is the number $\|\mathbb{R}C\|$ of real points of $C$. Problem 1.1.: For a given integer $k\geq 1$, what is the maximal number \[\delta(k)=\max\{\|\mathbb{R}C\|:\text{$C\subset\mathbb{C}P^{2}$ a finite real algebraic curve, $\deg C=2k$}\}?\] For given integers $k\geq 1$ and $g\geq 0$, what is the maximal number \[\delta_{g}(k)=\max\{\|\mathbb{R}C\|:\text{$C\subset\mathbb{C}P^{2}$ a finite real algebraic curve of genus $g$, $\deg C=2k$}\}?\] (See Section 2 for our convention for the genus of reducible curves.) The Petrovsky inequalities (see [16] and Remark 2.3) result in the following upper bound: \[\|\mathbb{R}C\|\leq\frac{3}{2}k(k-1)+1.\] Currently, this bound is the best known. Furthermore, being of topological nature, it is sharp in the realm of pseudo-holomorphic curves. Indeed, consider a rational simple Harnack curve of degree $2k$ in $\mathbb{C}P^{2}$ (see [14, 11, 1]); this curve has $(k-1)(2k-1)$ solitary real nodes (as usual, by a node we mean a non-degenerate double point, _i.e._, an $A_{1}$-singularity) and an oval (see Remark 2.3 for the definition) surrounding $\frac{1}{2}(k-1)(k-2)$ of them. One can erase all inner nodes, leaving the oval empty. Then, _in the pseudo-holomorphic category_, the oval can be contracted to an extra solitary node, giving rise to a finite real pseudo-holomorphic curve $C\subset\mathbb{C}P^{2}$ of degree $2k$ with $\|\mathbb{R}C\|=\frac{3}{2}k(k-1)+1$. ### Principal results For the moment, the exact value of $\delta(k)$ is known only for $k\leq 4$. The upper (Petrovsky inequality) and lower bounds for a few small values of $k$ are as follows: \[\begin{array}[]{c \| c\| c\| c\| c\| c\| c\| c\| c\| c\| c}k&1&2&3&4&5&6&7&8&9&10\\ \hline\delta(k)\leq&1&4&10&19&31&46&64&85&109&136\\ \hline\delta(k)\geq&1&4&10&19&30&45&59&78&98&123\end{array}\] The cases $k=1,2$ are obvious (union of two complex conjugate lines or conics, respectively). The lower bound for $k=6$ is given by Proposition 4.7, and all other cases are covered by Theorem 4.5. Asymptotically, we have \[\frac{4}{3}k^{2}\lesssim\delta(k)\lesssim\frac{3}{2}k^{2},\] where the lower bound follows from Theorem 4.5. A finite real sextic $C_{6}$ with $\|\mathbb{R}C_{6}\|=\delta(3)=10$ was constructed by D. Hilbert [8]. A finite real octic $C_{8}$ with $\|\mathbb{R}C_{8}\|=\delta(4)=19$ could easily be obtained by perturbing a quartuple conic, although we could not find such an octic in the literature. The best previously known asymptotic lower bound $\delta(k)\gtrsim\frac{10}{9}k^{2}$ is found in M. D. Choi, T. Y. Lam, B. Reznick [2]. With the genus $g=g(C)$ fixed, the upper bound \[\delta_{g}(k)\leq k^{2}+g+1\] is also given by a strengthening of the Petrovsky inequalities (see Theorem 2.5). In Theorem 4.8, we show that this bound is sharp for $g\leq k-3$. Most results extend to curves in ruled surfaces: upper bounds are given by Theorem 2.5 (for $g$ fixed) and Corollary 2.6; an asymptotic lower bound is given by Theorem 4.2 (which also covers arbitrary projective toric surfaces), and a few sporadic constructions are discussed in Sections 5, 6. ### Contents of the paper In Section 2, we obtain the upper bounds, derived essentially from the Comessatti inequalities. In Section 3, we discuss the auxiliary tools used in the constructions, namely, the patchworking techniques, bigonal curves and _dessins d’enfants_, and deformation to the normal cone. Section 4 is dedicated to curves in $\mathbb{C}P^{2}$: we recast the upper bounds, describe a general construction for toric surfaces (Theorem 4.2) and a slight improvement for the projective plane (Theorem 4.5), and prove the sharpness of the bound $\delta_{g}(k)\leq k^{2}+g+1$ for curves of small genus. In Section 5, we consider surfaces ruled over $\mathbb{R}$, proving the sharpness of the upper bounds for small bi-degrees and for small genera. Finally, Section 6 deals with finite real curves in the ellipsoid. ### Acknowledgments Part of the work on this project was accomplished during the second and third authors’ stay at the _Max-Planck-Institut für Mathematik_, Bonn. We are grateful to the MPIM and its friendly staff for their hospitality and excellent working conditions. We extend our gratitude to Boris Shapiro, who brought the finite real curve problem to our attention and supported our work by numerous fruitful discussions. We would also like to thank Ilya Tyomkin for his help in specializing general statements from [20] to a few specific situations. ## 2. Strengthened Comessatti inequalities Let $(X,c)$ be a smooth real projective surface. We denote by $\sigma^{\pm}_{\text{\rm inv}}(X,c)$ (respectively, $\sigma^{\pm}_{\text{\rm skew}}(X,c)$) the inertia indices of the invariant (respectively, skew-invariant) sublattice of the involution $\$$c_{}\colon H_{2}(X;\mathbb{Z})\to H_{2}(X;\mathbb{Z})$ induced by $c$. The following statement is standard. Proposition 2.1* (see, for example, [22]).: _One has_ \[\sigma^{-}_{\text{\rm inv}}(X,c)=\frac{1}{2}(h^{1,1}(X)+\chi(\mathbb{R}X))-1, \qquad\sigma^{-}_{\text{\rm skew}}(X,c)=\frac{1}{2}(h^{1,1}(X)-\chi(\mathbb{R} X)),\] _where $h^{\bullet,\bullet}$ are the Hodge numbers and $\chi$ is the topological Euler characteristic._ Corollary 2.2 (Comessatti inequalities).: _One has_ \[2-h^{1,1}(X)\leq\chi(\mathbb{R}X)\leq h^{1,1}(X).\] Remark 2.3.: Let $C\subset\mathbb{C}P^{2}$ be a smooth real curve of degree $2k$. Recall that an _oval_ of $C$ is a connected component $\mathfrak{o}\subset\mathbb{R}C$ bounding a disk in $\mathbb{R}P^{2}$; the latter disk is called the _interior_ of $\mathfrak{o}$. An oval $\mathfrak{o}$ of $C$ is called _even_ (respectively, _odd_) if $\mathfrak{o}$ is contained inside an even (respectively, odd) number of other ovals of $C$; the number of even (respectively, odd) ovals of a given curve $C$ is denoted by $p$ (respectively, $n$). The classical Petrovsky inequalities [16] state that \[p-n\leq\frac{3}{2}k(k-1)+1,\qquad n-p\leq\frac{3}{2}k(k-1).\] These inequalities can be obtained by applying Corollary 2.2 to the double covering of $\mathbb{C}P^{2}$ branched along $C\subset\mathbb{C}P^{2}$ (see _e.g._[22], [12, Th. 3.3.14]). The Comessatti and Petrovsky inequalities, strengthened in several ways (see, _e.g._, [21]), have a variety of applications. For example, for nodal finite real rational curves in $\mathbb{C}P^{2}$ we immediately obtain the following statement. Proposition 2.4.: _Let $C\subset\mathbb{C}P^{2}$ be a nodal finite rational curve of degree $2k$. Then, $\|\mathbb{R}C\|\leq k^{2}+1$._ Proof.: Denote by $r$ the number of real nodes of $C$, and denote by $s$ the number of pairs of complex conjugate nodes of $C$. We have $r+2s=(k-1)(2k-1)$. Let, further, $Y$ be the double covering of $\mathbb{C}P^{2}$ branched along the smooth real curve $C_{t}\subset\mathbb{C}P^{2}$ obtained from $C$ by a small perturbation creating an oval from each real node of $C$. The union of $r$ small discs bounded by $\mathbb{R}C_{t}$ is denoted by $\mathbb{R}P^{2}_{+}$; let $\bar{c}\colon Y\to Y$ be the lift of the real structure such that the real part projects onto $\mathbb{R}P^{2}_{+}$. Each pair of complex conjugate nodes of $C$ gives rise to a pair of $\bar{c}_{}$-conjugate vanishing cycles in $H_{2}(Y;\mathbb{Z})$; their difference is a skew-invariant class of square $-4$, and the $s$ square $-4$ classes thus obtained are pairwise orthogonal. Since $h^{1,1}(Y)=3k^{2}-3k+2$ (see, _e.g._[22]), Corollary 2.2 implies that \[\chi(\mathbb{R}Y)\leq h^{1,1}(Y)-2s=3k(k-1)+2-2s=k^{2}+1+r.\] Thus, $r\leq k^{2}+1$. ∎ The above statement can be generalized to the case of not necessarily nodal curves of arbitrary genus in any smooth real projective surface. Recall that the _geometric genus_$g(C)$ of an irreducible and reduced algebraic curve $C$ is the genus of its normalization. If $C$ is reduced with irreducible components $C_{1},\ldots,C_{n}$, the geometric genus of $C$ is defined by \[g(C)=g(C_{1})+\ldots+g(C_{n})+1-n.\] In other words, $2-2g(C)=\chi(\tilde{C})$, where $\tilde{C}$ is the normalization. Define also the _weight_$\\|p\\|$ of a solitary point $p$ of a real curve $C$ as the minimal number of blow-ups _at real points_ necessary to resolve $p$. More precisely, $\\|p\\|=1+\sum\\|p_{i}\\|$, the summation running over all real points $p_{i}$ over $p$ of the strict transform of $C$ blown up at $p$. For example, the weight of a simple node equals $1$, whereas the weight of an $A_{2n-1}$-type point equals $n$. If $\|\mathbb{R}C\|<\infty$, we define the _weighted point count_$\\|\mathbb{R}C\\|$ as the sum of the weights of all real points of $C$. The topology of the ambient complex surface $X$ is present in the next statement in the form of the coefficient of the Todd genus (see [9]). Theorem 2.5.: _Let $(X,c)$ be a simply connected smooth real projective surface with non-empty connected real part. Let $C\subset X$ be an ample reduced finite real algebraic curve such that $[C]=2e$ in $H_{2}(X;\mathbb{Z})$. Then, we have_ (1) \[\|\mathbb{R}C\|\leq\\|\mathbb{R}C\\|\leq e^{2}+g(C)-T_{2,1}(X)+\chi(\mathbb{R}X)-1.\] _Furthermore, the inequality is strict unless all singular points of $C$ are double._ Proof.: Since $[C]\in H_{2}(X;\mathbb{Z})$ is divisible by $2$, there exists a real double covering $\rho\colon(Y,\bar{c})\to(X,c)$ ramified at $C$ and such that $\rho(\mathbb{R}Y)=\mathbb{R}X$. By the embedded resolution of singularities, we can find a sequence of real blow-ups $\pi_{i}\colon X_{i}\to X_{i-1}$, $i=1,\ldots,n$, real curves $C_{i}=\pi^{}C_{i-1}\bmod 2\subset X_{i}$, and real double coverings $\rho_{i}=\pi_{i}^{}\rho_{i-1}\colon Y_{i}\to X_{i}$ ramified at $C_{i}$ such that the curve $C_{n}$ and surface $Y_{n}$ are nonsingular. (Here, a _real blow-up_ is either a blow-up at a real point or a pair of blow-ups at two conjugate points. By $\pi_{i}^{}C_{i-1}\bmod 2$ we mean the reduced divisor obtained by retaining the _odd multiplicity components_ of the divisorial pull-back $\pi_{i}^{}C_{i-1}$.) Using Proposition 2.1, we can rewrite (1) in the form \[e^{2}+g(C)+h^{1,1}(X)-T_{2,1}(X)+2\chi(\mathbb{R}X)-2\sigma^{-}_{\text{\rm inv }}(X,c)-\\|\mathbb{R}C\\|\geq 3.\] We proceed by induction and prove a modified version of the latter inequality, namely, (2) \[e_{i}^{2}+g(C_{i})+h^{1,1}(X_{i})-T_{2,1}(X_{i})+b_{1}^{-1}(Y_{i})+2\chi( \mathbb{R}X_{i})-2\sigma^{-}_{\text{\rm inv}}(X_{i},c)-\\|\mathbb{R}C_{i}\\|\geq 3,\] where $[C_{i}]=2e_{i}\in H_{2}(X_{i};\mathbb{Z})$ and $b_{1}^{-1}(\cdot)$ is the dimension of the $(-1)$-eigenspace of $\rho_{}$ on $H_{1}(\cdot;\mathbb{C})$. For the “complex” ingredients of (2), it suffices to consider a blow-up $\pi\colon\tilde{X}\to X$ at a singular point $p$ of $C$, not necessarily real, of multiplicity $O\geq 2$. Denoting by $C^{\prime}$ the strict transform of $C$, we have $\tilde{C}=\pi^{}C\bmod 2=C^{\prime}+\varepsilon E$, where $E=\pi^{-1}(p)$ is the exceptional divisor and $O=2m+\varepsilon$, $m\in\mathbb{Z}$, $\varepsilon=0,1$. Then, in obvious notation, \[e^{2}=\tilde{e}^{2}+m^{2},\qquad g(C)=g(\tilde{C})+\varepsilon,\qquad h^{1,1}( X)=h^{1,1}(\tilde{X})-1,\qquad T_{2,1}(X)=T_{2,1}(\tilde{X})+1.\] Furthermore, from the isomorphisms $H_{1}(\tilde{Y},\tilde{\rho}^{}E)=H_{1}(Y,p)=H_{1}(Y)$ we easily conclude that \[b_{1}^{-1}(Y)\geq b_{1}^{-}(\tilde{Y})-b_{1}^{-}(\tilde{\rho}^{}E)\geq b_{1}^ {-1}(\tilde{Y})-2(m-1).\] It follows that, when passing from $\tilde{X}$ to $X$, the increment in the first five terms of (2) is at least $(m-1)^{2}+\varepsilon-1\geq-1$; this increment equals $(-1)$ if and only if $p$ is a double point of $C$. For the last three terms, assume first that the singular point $p$ above is real. Then \[\chi(\mathbb{R}X)=\chi(\mathbb{R}\tilde{X})+1,\qquad\sigma^{-}_{\text{\rm inv} }(X_{i},c)=\sigma^{-}_{\text{\rm inv}}(\tilde{X}_{i},\tilde{c}),\qquad\\| \mathbb{R}C\\|=\\|\mathbb{R}\tilde{C}\\|+1,\] and the total increment in (2) is positive; it equals $0$ if and only if $p$ is a double point. Now, let $\pi\colon\tilde{X}\to X$ be a pair of blow-ups at two complex conjugate singular points of $C$. Then \[\chi(\mathbb{R}X)=\chi(\mathbb{R}\tilde{X}),\qquad\sigma^{-}_{\text{\rm inv}}( X_{i},c)=\sigma^{-}_{\text{\rm inv}}(\tilde{X}_{i},\tilde{c})-1,\qquad\\| \mathbb{R}C\\|=\\|\mathbb{R}\tilde{C}\\|,\] and, again, the total increment is positive, equal to $0$ if and only if both points are double. To establish (2) for the last, nonsingular, curve $C_{n}$, we use the following observations: $\chi(Y_{n})=2\chi(X_{n})-\chi(C_{n})$ (the Riemann-Hurwitz formula); * $\sigma(Y_{n})=2\sigma(X_{n})-2e_{n}^{2}$ (Hirzebruch’s theorem); * $b_{1}(Y_{n})-b_{1}(X_{n})=b_{1}^{-1}(Y_{n})$, as $b_{1}^{+1}(Y_{n})=b_{1}(X_{n})$_via_ the transfer map; * $\chi(\mathbb{R}Y_{n})=2\chi(\mathbb{R}X_{n})$, since $\mathbb{R}C_{n}=\varnothing$ and $\mathbb{R}Y_{n}\to\mathbb{R}X_{n}$ is an unramified double covering. Then, (2) takes the form \[\sigma^{-}_{\text{\rm inv}}(Y_{n},\bar{c}_{n})\geq\sigma^{-}_{\text{\rm inv}}( X_{n},c_{n}),\] which is obvious in view of the transfer map $H_{2}(X_{n};\mathbb{R})\to H_{2}(Y_{n};\mathbb{R})$: this map is equivariant and isometric up to a factor of $2$. Thus, there remains to notice that $b_{1}^{-1}(Y_{0})=0$. Indeed, since $C_{0}=C$ is assumed ample, $X\smallsetminus C$ has homotopy type of a CW-complex of dimension $2$ (as a Stein manifold). Hence, so does $Y\smallsetminus C$, and the homomorphism $H_{1}(C;\mathbb{R})\to H_{1}(Y;\mathbb{R})$ is surjective. Clearly, $b_{1}^{-1}(C)=0$. ∎ Corollary 2.6.: _Let $(X,c)$ and $C\subset X$ be as in Theorem 2.5. Then, we have_ \[2\|\mathbb{R}C\|\leq 3e^{2}-e\cdot c_{1}(X)-T_{2,1}(X)+\chi(\mathbb{R}X),\] _the inequality being strict unless each singular point of $C$ is a solitary real node of $\mathbb{R}C$._ Proof.: By the adjunction formula we have \[g(C)\leq 2e^{2}-e\cdot c_{1}(X)+1-\|\mathbb{R}C\|,\] and the result follows from Theorem 2.5. ∎ Remark 2.7.: The assumptions $\pi_{1}(X)=0$ and $b_{0}(\mathbb{R}X)=1$ in Theorem 2.5 are mainly used to assure the existence of a real double covering $\rho\colon Y\to X$ ramified over a given real divisor $C$. In general, one should speak about the divisibility by $2$ of the _real divisor class_$\|C\|_{\mathbb{R}}$, _i.e._, class of real divisors modulo real linear equivalence. (If $\mathbb{R}X\neq\varnothing$, one can alternatively speak about the set of real divisors in the linear system $\|C\|$ or a real point of $\operatorname{Pic}(X)$.) A necessary condition is the vanishing \[[C]=0\in H_{2n-2}(X;\mathbb{Z}/2\mathbb{Z}),\qquad[\mathbb{R}C]=0\in H_{n-1}( \mathbb{R}X;\mathbb{Z}/2\mathbb{Z}),\] where $n=\dim_{\mathbb{C}}(X)$ and $[\mathbb{R}C]$ is the homology class of the real part of (any representative of) $\|C\|$ (the sufficiency of this condition in some special cases is discussed in Lemma 3.4 below). If not empty, the set of double coverings ramified over $C$ and admitting real structure is a torsor over the space of $c^{}$-invariant elements of $H^{1}(X;\mathbb{Z}/2\mathbb{Z})$. The proof of the following theorem repeats literally that of Theorem 2.5. Theorem 2.8.: _Let $(X,c)$ be a smooth real projective surface and $C\subset X$ an ample finite reduced real algebraic curve such that the class $\|C\|_{\mathbb{R}}$ is divisible by $2$. A choice of a real double covering $\rho\colon Y\to X$ ramified over $C$ defines a decomposition of $\mathbb{R}X$ into two disjoint subsets $\mathbb{R}X_{+}=\rho(\mathbb{R}Y)$ and $\mathbb{R}X_{-}$ consisting of whole components. Then, we have_ \[\\|\mathbb{R}C\cap\mathbb{R}X_{+}\\|-\\|\mathbb{R}C\cap\mathbb{R}X_{-}\\|\leq e^{2 }+g(C)-T_{2,1}(X)+\chi(\mathbb{R}X_{+})-\chi(\mathbb{R}X_{-})-1,\] _the inequality being strict unless all singular points of $C$ are double. ∎_ ## 3. Construction tools ### Patchworking If $\Delta$ is a convex lattice polygon contained in the non-negative quadrant $(\mathbb{R}_{\geq 0})^{2}\subset\mathbb{R}^{2}$, we denote by ${\operatorname{Tor}}(\Delta)$ the toric variety associated with $\Delta$; this variety is a surface if $\Delta$ is non-degenerate. In the latter case, the complex torus $(\mathbb{C}^{})^{2}$ is naturally embedded in ${\operatorname{Tor}}(\Delta)$. Let $V\subset(\mathbb{R}_{\geq 0})^{2}\cap\mathbb{Z}^{2}$ be a finite set, and let $P(x,y)=\sum_{(i,j)\in V}a_{ij}x^{i}y^{j}$ be a real polynomial in two variables. The _Newton polygon_$\Delta_{P}$ of $P$ is the convex hull in $\mathbb{R}^{2}$ of those points in $V$ that correspond to the non-zero monomials of $P$. The polynomial $P$ defines an algebraic curve in the $2$-dimensional complex torus $(\mathbb{C}^{})^{2}$; the closure of this curve in ${\operatorname{Tor}}(\Delta_{P})$ is an algebraic curve $C\subset{\operatorname{Tor}}(\Delta_{P})$. If $Q$ is a quadrant of $(\mathbb{R}^{})^{2}\subset(\mathbb{C}^{})^{2}$ and $(a,b)$ is a vector in $\mathbb{Z}^{2}$, we denote by $Q(a,b)$ the quadrant \[\{(x,y)\in(\mathbb{R}^{})^{2}\ \|\ ((-1)^{a}x,(-1)^{b}y)\in Q\}.\] If $e$ is an integral segment whose direction is generated by a primitive integral vector $(a,b)$, we abbreviate $Q(e^{\perp}):=Q(b,-a)$. A real algebraic curve $C\subset{\operatorname{Tor}}(\Delta)$ is said to be $\frac{1}{4}$_-finite_ (respectively, $\frac{1}{2}$_-finite_) if the intersection of the real part $\mathbb{R}C$ with the positive quadrant $(\mathbb{R}_{>0})^{2}$ (respectively, the union $(\mathbb{R}_{>0})^{2}\cup(\mathbb{R}_{>0})^{2}(1,0)$ is finite. Fix a subdivision ${\mathcal{S}}=\{\Delta_{1},\ldots,\Delta_{N}\}$ of a convex polygon $\Delta\subset(\mathbb{R}_{\geq 0})^{2}$ such that there exists a piecewise-linear convex function $\nu\colon\Delta\to\mathbb{R}$ whose maximal linearity domains are precisely the non-degenerate lattice polygons $\Delta_{1},\ldots,\Delta_{N}$. Let $a_{ij}$, $(i,j)\in\Delta\cap\mathbb{Z}^{2}$, be a collection of real numbers such that $a_{ij}\neq 0$ whenever $(i,j)$ is a vertex of $\mathcal{S}$. This gives rise to $N$ real algebraic curves $C_{k}$, $k=1,\ldots,N$: each curve $C_{k}\subset{\operatorname{Tor}}(\Delta_{k})$ is defined by the polynomial \[P(x,y)=\sum_{(i,j)\in\Delta_{k}\cap\mathbb{Z}^{2}}a_{ij}x^{i}y^{j}\] with the Newton polygon $\Delta_{k}$. Commonly, we denote by $\mathop{\mathrm{Sing}}(C)$ the set of singular points of a curve $C$. If $C\subset{\operatorname{Tor}}(\Delta)$ and $e\subset\Delta$ is an edge, we put $T_{e}(C):=C\cap D(e)$, where $D(e)$ is the toric divisor corresponding to $e$. Assume that each curve $C_{k}$ is nodal and $\mathop{\mathrm{Sing}}(C_{k})$ is disjoint from the toric divisors of ${\operatorname{Tor}}(\Delta_{k})$ (but $C_{k}$ can be tangent with arbitrary order of tangency to some toric divisors). For each inner edge $e=\Delta_{i}\cap\Delta_{j}$ of $\mathcal{S}$, the toric divisors corresponding to $e$ in ${\operatorname{Tor}}(\Delta_{i})$ and ${\operatorname{Tor}}(\Delta_{j})$ are naturally identified, as they both are ${\operatorname{Tor}}(e)$. The intersection points of $C_{i}$ and $C_{j}$ with these toric divisors are also identified, and, at each such point $p\in{\operatorname{Tor}}(e)$, the orders of intersection of $C_{i}$ and $C_{j}$ with ${\operatorname{Tor}}(e)$ automatically coincide; this common order is denoted by $\operatorname{mult}p$ and, if $\operatorname{mult}p>1$, the point $p$ is called _fat_. Assume that $\operatorname{mult}p$ is even for each _fat_ point $p$ and that the local branches of $C_{i}$ and $C_{j}$ at each _real_ fat point $p$ are in the same quadrant $Q_{p}\subset(\mathbb{R}^{})^{2}$. Each edge $E$ of $\Delta$ is a union of exterior edges $e$ of $\mathcal{S}$; denote the set of these edges by $\{E\}$ and, given $e\in\{E\}$, let $k(e)$ be the index such that $e\subset\Delta_{k(e)}$. The toric divisor $D(E)\subset{\operatorname{Tor}}(\Delta)$ is a smooth real rational curve whose real part $\mathbb{R}D(E)$ is divided into two halves $\mathbb{R}D_{\pm}(E)$ by the intersections with other toric divisors of ${\operatorname{Tor}}(\Delta)$; we denote by $\mathbb{R}D_{+}(E)$ the half adjacent to the positive quadrant of $(\mathbb{R}^{})^{2}$. Similarly, the toric divisor $D(e)\subset{\operatorname{Tor}}(\Delta_{k(e)})$ is divided into $\mathbb{R}D_{\pm}(e)$. Theorem 3.1 (Patchworking construction; essentially, Theorem 2.4 in [19]).: _Under the assumptions above, there exists a family of real polynomials $P^{(t)}(x,y)$, $t\in\mathbb{R}_{>0}$, with the Newton polygon $\Delta$, such that, for sufficiently small $t$, the curve $C^{(t)}\subset{\operatorname{Tor}}(\Delta)$ defined by $P^{(t)}$ has the following properties:_ * _the curve_ $C^{(t)}$ _is nodal and_ $\mathop{\mathrm{Sing}}(C^{(t)})$ _is disjoint from_ _the toric divisors;_ * _if all curves_ $C_{1},\ldots,C_{N}$ _are_ $\frac{1}{2}$_-finite_ (_respectively,_ $\frac{1}{4}$_-finite_)_, then so is_ $C^{(t)}$_;_ * _there is an injective map_ \[\Phi\colon\coprod_{k=1}^{N}\mathop{\mathrm{Sing}}(C_{k})\to\mathop{\mathrm{ Sing}}(C^{(t)}),\] _such that the image_ _of each real point_ _is a real point_ _of the same type_ (_solitary/non-solitary_) _and_ _in the same quadrant of_ $(\mathbb{R}^{})^{2}$_, and the image of each imaginary point_ _is imaginary;_ _there is a partition_ \[\mathop{\mathrm{Sing}}(C^{(t)})\smallsetminus\text{\rm image of $\Phi$}= \coprod_{p}\Pi_{p},\] $p$ _running over all fat points, so that_ $\|\Pi_{p}\|=2m-1$ _if_ $\operatorname{mult}p=2m$_._ _The points in_ $\Pi_{p}$ _are imaginary if_ $p$ _is imaginary and real and solitary if_ $p$ _is real; in the latter case,_ $(m-1)$ _of these points lie in_ $Q_{p}$ _and the_ _others_ $m$ _points_ _lie in_ $Q_{p}(e_{p}^{\perp})$_, where_ $p\in{\operatorname{Tor}}(e_{p})$_;_ * _for each edge_ $E$ _of_ $\Delta$_, there is a bijective map_ \[\Psi_{E}\colon\coprod_{e\in\{E\}}T_{e}(C_{k(e)})\to T_{E}(C^{(t)})\] _preserving the intersection multiplicity and the position of points in_ $\mathbb{R}D_{\pm}(\cdot)$ _or_ $D(\cdot)\smallsetminus\mathbb{R}D(\cdot)$_._ Proof.: To deduce the statement from [19, Theorem 2.4], one can use Lemma 5.4(ii) in [18] and the deformation patterns described in [10], Lemmas 3.10 and 3.11 (_cf_. also the curves $C_{,0,0}$ in Lemma 3.2 below). ∎ ### Bigonal curves _via__dessins d’enfants_ We denote by $\Sigma_{n}$, $n\geq 0$, the Hirzebruch surface of degree $n$, _i.e._, $\Sigma_{n}=\mathbb{P}(\mathcal{O}_{\mathbb{C}P^{1}}(n)\oplus\mathcal{O}_{ \mathbb{C}P^{1}})$. Recall that $\Sigma_{0}=\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ and $\Sigma_{1}$ is the blow-up of $\mathbb{C}P^{2}$ at a point. The bundle projection induces a map $\pi\colon\Sigma_{n}\to\mathbb{C}P^{1}$, and we denote by $F$ a fiber of $\pi$; it is isomorphic to $\mathbb{C}P^{1}$. The images of $\mathcal{O}_{\mathbb{C}P^{1}}$ and $\mathcal{O}_{\mathbb{C}P^{1}}(n)$ are denoted by $B_{0}$ and $B_{\infty}$, respectively; these curves are sections of $\pi$. The group $H^{2}(\Sigma_{n};\mathbb{C})=H^{1,1}(X;\mathbb{C})$ is generated by the classes of $B_{0}$ and $F$, and we have \[[B_{0}]^{2}=n,\quad[B_{\infty}]^{2}=-n,\quad[F]^{2}=0,\quad B_{\infty}\sim B_{ 0}-nF,\quad c_{1}(\Sigma_{n})=2[B_{0}]+(2-n)[F].\] (If $n>0$, the _exceptional section_$B_{\infty}$ is the only irreducible curve of negative self-intersection.) In other words, we have $D\sim aB_{0}+bF$ for each divisor $D\subset\Sigma_{n}$, and the pair $(a,b)\in\mathbb{Z}^{2}$ is called the _bidegree_ of $D$. The cone of effective divisors is generated by $B_{\infty}$ and $F$, and the cone of ample divisors is $\{aB_{0}+bF\,\|\,a,b>0\}$. In this section, we equipp $\mathbb{C}P^{1}$ with the standard complex conjugation, and the surface $\Sigma_{n}$ with the real structure $c$ induced by the standard complex conjugation on $\mathcal{O}_{\mathbb{C}P^{1}}(n)$. Unless $n=0$, this is the only real structure on $\Sigma_{n}$ with nonempty real part. In particular $c$ acts on $H^{2}(\Sigma_{n};\mathbb{C})$ as $-\operatorname{Id}$, and so $\sigma^{-}_{\text{\rm inv}}(X,c)=0$. The real part of $\Sigma_{n}$ is a torus if $n$ is even, and a Klein bottle if $n$ is odd. In the former case, the complement $\mathbb{R}\Sigma_{n}\smallsetminus(\mathbb{R}B_{0}\cup\mathbb{R}B_{\infty})$ has two connected components, which we denote by $\mathbb{R}\Sigma_{n,\pm}$. Lemma 3.2.: _Given integers $n>0$, $b\geq 0$, and $0\leq q\leq n+b-1$, there exists a real algebraic rational curve $C=C_{n,b,q}$ in $\Sigma_{2n}$ of bidegree $(2,2b)$ such that (see Figure 1):_ 1. _all singular points of_ $C$ _are_ $2n+2b-1$ _solitary nodes;_ $n+b+q$ _of them lie_ _in_ $\mathbb{R}\Sigma_{2n,+}$_, and the other_ $n+b-q-1$ _lie_ _in_ $\mathbb{R}\Sigma_{2n,-}$_;_ 2. _the real part_ $\mathbb{R}C$ _has a single extra oval_ $\mathfrak{o}$_, which is contained in_ $\mathbb{R}\Sigma_{2n,-}\cup\mathbb{R}B_{0}\cup\mathbb{R}B_{\infty}$ _and does not contain any of the nodes in its interior;_ 3. _each intersection_ $p_{\infty}:=\mathfrak{o}\cap B_{\infty}$ _and_ $p_{0}:=\mathfrak{o}\cap B_{0}$ _consists of a single point, the multiplicity being_ $2b$ _and_ $4n+2b-2q$_, respectively; the points_ $p_{0}$ _and_ $p_{\infty}$ _are on the same fiber_ $F$_._ <figure><img src="content_image/1807.03992/x1.png"><figcaption>Figure 2. Decoration of a dessin</figcaption></figure> _This curve can be perturbed to a curve $\widetilde{C}_{n,b,q}\subset\Sigma_{2n}$ satisfying conditions (1) and (2) and the following modified version of condition (3):_ 1. _the oval_ $\mathfrak{o}$ _intersects_ $B_{\infty}$ _and_ $B_{0}$ _at, respectively,_ $b$ _and_ $2n+b-q$ _simple tangency points._ Note that $C_{n,b,q}$ intersects $B_{0}$ in $q$ additional pairs of complex conjugate points. Proof.: Up to elementary transformations of $\Sigma_{2n}$ (blowing up the point of intersection $C\cap B_{\infty}$ and blowing down the strict transforms of the corresponding fibers) we may assume that $b=0$ and, hence, $C$ is disjoint from $B_{\infty}$. Then, $C$ is given by $P(x,y)=0$, where (3) \[P(x,y)=y^{2}+a_{1}(x)y+a_{2}(x),\qquad\deg a_{i}(x)=2in.\] (Strictly speaking, $a_{i}$ are sections of appropriate line bundles, but we pass to affine coordinates and regard $a_{i}$ as polynomials.) We will construct the curves using the techniques of _dessins d’enfants_, cf. [15, 4, 3]. Consider the rational function $f\colon\mathbb{C}P^{1}\to\mathbb{C}P^{1}$ given by \[f(x)=\frac{a_{1}^{2}(x)-4a_{2}(x)}{a_{1}^{2}(x)}.\] (This function differs from the $j$-invariant of the trigonal curve $C+B_{0}$ by a few irrelevant factors.) The _dessin_ of $C$ is the graph $\mathcal{D}:=f^{-1}(\mathbb{R}P^{1})$ decorated as shown in Figure 2. <figure><img src="content_image/1807.03992/x1.png"><figcaption>Figure 2. Decoration of a dessin</figcaption></figure> In addition to $\times$-, $\circ$-, and $\bullet$-vertices, it may also have _monochrome_ vertices, which are the pull-backs of the real critical values of $f$ other that $0$, $1$, or $\infty$. This graph is real, and we depict only its projection to the disk $D:=\mathbb{C}P^{1}/x\sim\bar{x}$, showing the boundary $\partial D$ by a wide grey curve: this boundary corresponds to the real parts $\mathbb{R}C\subset\mathbb{R}\Sigma_{2n}\to\mathbb{R}P^{1}$. Assuming that $a_{1}$, $a_{2}$ have no common roots, the real special vertices and edges of $\mathcal{D}$ have the following geometric interpretation: a $\times$-vertex $x_{0}$ corresponds to a double root of the polynomial $P(x_{0},y)$; the curve is tangent to a fiber if $\operatorname{val}x_{0}=2$ and has a double point of type $A_{p-1}$, $p=\frac{1}{2}\operatorname{val}x_{0}$, otherwise; * a $\circ$-vertex $x_{0}$ corresponds to an intersection $\mathbb{R}C\cap\mathbb{R}B_{0}$ of multiplicity $\frac{1}{2}\operatorname{val}x_{0}$; * the real part $\mathbb{R}C$ is empty over each point of a solid edge and consists of two points over each point of any other edge; * the points of $\mathbb{R}C$ over two $\times$-vertices $x_{1}$, $x_{2}$ are in the same half $\mathbb{R}\Sigma_{2n,\pm}$ if and only if one has $\sum\operatorname{val}z_{i}=0\bmod 8$, the summation running over all $\bullet$-vertices $z_{i}$ in any of the two arcs of $\partial D$ bounded by $x_{1}$, $x_{2}$. (For the last item, observe that the valency of each $\bullet$-vertex is $0\bmod 4$ and the sum of all valencies equals $2\deg f=8n$; hence, the sum in the statement is independent of the choice of the arc.) Now, to construct the curves in the statements, we start with the dessin $\widetilde{\mathcal{D}}_{n,0,0}$ shown in Figure 3, left: it has $2n$$\bullet$-vertices, $2n$$\circ$-vertices, and $(2n+1)$$\times$-vertices, two bivalent and $(2n-1)$ four-valent, numbered consecutively along $\partial D$. <figure><img src="content_image/1807.03992/x2.png"><figcaption>Figure 3. The dessin ˜Dn,0,0 and its modifications</figcaption></figure> To obtain $\widetilde{\mathcal{D}}_{n,0,q}$, we replace $q$ disjoint embraced fragments with copies of the fragment shown in Figure 3, right; by choosing the fragments replaced around _even-numbered_$\times$-vertices, we ensure that the solitary nodes would migrate from $\mathbb{R}\Sigma_{2n,-}$ to $\mathbb{R}\Sigma_{2n,+}$. Finally, $\mathcal{D}_{n,0,q}$ is obtained from $\widetilde{\mathcal{D}}_{n,0,q}$ by contracting the dotted real segments connecting the real $\circ$-vertices, so that the said vertices collide to a single $(8n-4q)$-valent one. Each of these dessins $\mathcal{D}$ gives rise to a (not unique) equivariant topological branched covering $f\colon S^{2}\to\mathbb{C}P^{1}$ (cf. [15, 4, 3]), and the Riemann existence theorem gives us an analytic structure on the sphere $S^{2}$ making $f$ a real rational function $\mathbb{C}P^{1}\to\mathbb{C}P^{1}$. There remains to take for $a_{1}$ a real polynomial with a simple zero at each (double) pole of $f$ and let $a_{2}:=\frac{1}{4}a_{1}^{2}(1-f)$. ∎ Generalizing, one can consider a geometrically ruled surface $\pi\colon\Sigma_{n}(\mathcal{O}):=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}_{B})\to B$, where $B$ is a smooth compact real curve of genus ${\mathfrak{g}}\geq 1$ and $\mathcal{O}$ is a line bundle, $\deg\mathcal{O}=n\geq 0$. If $\mathcal{O}$ is also real, the surface $\Sigma_{n}(\mathcal{O})$ acquires a real structure; the sections $B_{0}$ and $B_{\infty}$ are also real and we can speak about $\mathbb{R}B_{0}$, $\mathbb{R}B_{\infty}$. The real line bundle $\mathcal{O}$ is said to be _even_ if the $GL(1,\mathbb{R})$-bundle $\mathbb{R}\mathcal{O}$ over $\mathbb{R}B$ is trivial (cf. Remark 2.7). In this case, the real part $\mathbb{R}\Sigma_{n}(\mathcal{O})$ is a disjoint union of tori, one torus $T_{i}$ over each real component $\mathbb{R}_{i}B$ of $B$, and each complement $T_{i}^{\circ}:=T_{i}\smallsetminus\left(\mathbb{R}B_{0}\cup\mathbb{R}B_{\infty }\right)$ is made of two connected components (open annuli). A smooth compact real curve $B$ of genus ${\mathfrak{g}}$ is called _maximal_ if it has the maximal possible number of real connected components: $b_{0}(\mathbb{R}B)={\mathfrak{g}}+1$. Lemma 3.3.: _Let $n$, ${\mathfrak{g}}$ be two integers, $n\geq{\mathfrak{g}}-1\geq 0$. Then there exists an even real line bundle $\mathcal{O}$ of degree $\deg\mathcal{O}=2n$ over a maximal real algebraic curve $B$ of genus ${\mathfrak{g}}$, and a nodal real algebraic curve $C_{n}({\mathfrak{g}})\subset\Sigma_{2n}(\mathcal{O})$ realizing the class $2[B_{0}]\in H_{2}(\Sigma_{2n}(\mathcal{O});\mathbb{Z})$ such that_ 1. $\mathbb{R}C_{n}({\mathfrak{g}})\cap T_{1}$ _consists of_ $2n$ _solitary nodes, all in the same connected component of_ $T_{1}^{\circ}$_;_ 2. $\mathbb{R}C_{n}({\mathfrak{g}})\cap T_{2}$ _is a smooth connected curve, contained in a single connected component of_ $T_{2}^{\circ}$ _except for_ $n$ _real points of simple tangency of_ $C_{n}$ _and_ $B_{0}$_;_ 3. $\mathbb{R}C_{n}({\mathfrak{g}})\cap T_{i}$_,_ $i\geq 3$_, is a smooth connected curve, contained in a single connected component of_ $T_{2}^{\circ}$ _except for one real point of simple tangency of_ $C_{n}$ _and_ $B_{0}$_._ Note that we can only assert the existence of a ruled surface $\Sigma_{2n}(\mathcal{O})$: the analytic structure on $B$ and line bundle $\mathcal{O}$ are given by the construction and cannot be fixed in advance. Proof.: We proceed as in the proof of Lemma 3.2, with the “polynomials” $a_{i}$ sections of $\mathcal{O}^{\otimes i}$ in (3) and half-dessin $\mathcal{D}_{n}({\mathfrak{g}})/c_{B}$ in the surface $D:=B/c_{B}$, which, in the case of maximal $B$, is a disk with ${\mathfrak{g}}$ holes; as above, we have $\partial D=\mathbb{R}B$. The following technical requirements are necessary and sufficient for the existence of a topological ramified covering $f\colon B\to\mathbb{C}P^{1}$ (see [4, 3]) with $B$ the orientable double of $D$: * each _region_ (connected component of $D\smallsetminus\mathcal{D}$) should admit an orientation inducing on the boundary the orientation inherited from $\mathbb{R}P^{1}$ (the order on $\mathbb{R}$), and * each _triangular_ region (_i.e._, one with a single vertex of each of the three special types $\times$, $\circ$, and $\bullet$ in the boundary) should be a topological disk. (For example, in the dessins $\widetilde{\mathcal{D}}_{n,0,q}$ in Figure 3 the orientations are given by a chessboard coloring and all regions are triangles.) The curve $C_{n}({\mathfrak{g}})$ as in the statement is obtained from the dessin $\mathcal{D}_{n}({\mathfrak{g}})$ constructed as follows. If ${\mathfrak{g}}=1$, then $\mathcal{D}_{n}(1)$ is the dessin in the annulus shown in Figure 4, left <figure><img src="content_image/1807.03992/x3.png"><figcaption>Figure 4. The dessin Dn(1) and its modifications</figcaption></figure> (which is a slight modification of $\widetilde{\mathcal{D}}_{n,0,n-1}$ in Figure 3): it has $2n$ real four-valent $\times$-vertices, $n$ inner four-valent $\bullet$-vertices, and $2n$$\circ$-vertices, $n$ real four-valent and $n$ inner bivalent. (Recall that each inner vertex in $D$ doubles in $B$, so that the total valency of the vertices of each kind sums up to $8n=2\deg f$, as expected.) This dessin is maximal in the sense that all its regions are triangles. To pass from $\mathcal{D}_{n}(1)$ to $\mathcal{D}_{n}(1+q)$, $q\leq n$, we replace small neighbourhoods of $q$ inner $\circ$-vertices with the fragments shown in Figure 4, right, creating $q$ extra boundary components. Each dessin $\mathcal{D}_{n}({\mathfrak{g}})$ satisfies the two conditions above and, thus, gives rise to a ramified covering $f\colon B\to\mathbb{C}P^{1}$. The analytic structure on $B$ is given by the Riemann existence theorem, and $\mathcal{O}$ is the line bundle $\mathcal{O}_{B}(\frac{1}{2}P(f))$, where $P(f)$ is the divisor of poles of $f$. (All poles are even.) Then, the curve in question is given by “equation” (3), with the sections $a_{i}\in H^{0}(B;\mathcal{O}^{\otimes i})$ almost determined by their zeroes: $Z(a_{1})=\frac{1}{2}P(f)$ and $Z(a_{2})=Z(1-f)$. Further details of this construction (in the more elaborate trigonal case) can be found in [4, 3]. ∎ Next few lemmas deal with the real lifts of the curves constructed in Lemma 3.3 under a ramified double covering of $\Sigma_{2n}(\mathcal{O})$. First, we discuss the existence of such coverings, _cf_. Remark 2.7. Lemma 3.4.: _Let $\Sigma_{n}(\mathcal{O})$ be a real ruled surface over a real algebraic curve $B$ such that $\mathbb{R}B\neq\varnothing$, and let $D$ be a real divisor on $X$. Then there exists a real divisor $E$ on $X$ such that $\|D\|_{\mathbb{R}}=2\|E\|_{\mathbb{R}}$ if and only if $[\mathbb{R}D]=0\in H_{1}(\mathbb{R}X;\mathbb{Z}/2\mathbb{Z})$._ Proof.: By [7, Proposition 2.3], we have \[\mbox{Pic}(\Sigma_{n}(\mathcal{O}))\simeq\mathbb{Z}B_{0}\oplus\mbox{Pic}(B),\] and this isomorphism respects the action induced by the real structures. Let \[\|D\|=m\|B_{0}\|+\|D_{0}\|.\] Then $m=[\mathbb{R}D]\circ[\mathbb{R}F]\bmod 2$, where $F$ is the fiber of the ruling over a real point $p\in\mathbb{R}B$, and $D_{0}=D\circ B_{\infty}$, so that $[\mathbb{R}D_{0}]=[\mathbb{R}D]\circ[\mathbb{R}B_{\infty}]$. There remains to observe that $\|D_{0}\|_{\mathbb{R}}$ is divisible by $2$ in $\mathbb{R}\operatorname{Pic}(B)$ if and only if $[\mathbb{R}D_{0}]=0\in H_{0}(B;\mathbb{Z}/2\mathbb{Z})$. The “only if” part is clear, and the “if” part follows from the fact that $D_{0}$ can be deformed, through real divisors, to $(\deg D_{0})p$. ∎ Lemma 3.5.: _Let $X:=\Sigma_{n}(\mathcal{O})$ be a real ruled surface over a real algebraic curve $B$ such that $\mathbb{R}B\neq\varnothing$, and let $C$ be a reduced real divisor on $X$ such that $[\mathbb{R}C]=0\in H_{1}(\mathbb{R}X;\mathbb{Z}/2\mathbb{Z})$. Then, for any surface $S\subset\mathbb{R}X$ such that $\partial S=\mathbb{R}C$, there exists a real double covering $Y\to X$ ramified over $C$ such that $\mathbb{R}Y$ projects onto $S$._ Proof.: Pick one covering $Y_{0}\to X$, which exists by Lemma 3.4, and let $S_{0}$ be the projection of $\mathbb{R}Y_{0}$. We can assume that $S_{0}\cap T_{1}=S\cap T_{1}$ for one of the components $T_{1}$ of $\mathbb{R}X$. Given another component $T_{i}$, consider a path $\gamma_{i}$ connecting a point in $T_{i}$ to one on $T_{1}$, and let $\tilde{\gamma}_{i}=\gamma_{i}+c_{}\gamma_{i}$; in view of the obvious equivariant isomorphism $H_{1}(Y;\mathbb{Z}/2\mathbb{Z})\simeq H_{1}(B;\mathbb{Z}/2\mathbb{Z})$, these loops form a partial basis for the space of $c_{}$-invariant classes in $H_{1}(X;\mathbb{Z}/2\mathbb{Z})$. Now, it suffices to twist $Y_{0}$ (_cf_. Remark 2.7) by a cohomology class sending $\tilde{\gamma}_{i}$ to $0$ or $1$ if $S\cap T_{i}$ coincides with $S_{0}\cap T_{i}$ or with the closure of its complement, respectively. ∎ Lemma 3.6.: _Let $n$, ${\mathfrak{g}}$ be two integers, $n\geq{\mathfrak{g}}-1\geq 0$, and let $B$, $\mathcal{O}$, and $C_{n}(\mathfrak{g})\subset\Sigma_{2n}(\mathcal{O})$ be as in Lemma 3.3. Then there exists a real double covering $\Sigma_{n}(\mathcal{O}^{\prime})\to\Sigma_{2n}(\mathcal{O})$ ramified along $B_{0}\cup B_{\infty}$ and such that the pullback of $C_{n}({\mathfrak{g}})$ is a finite real algebraic curve $C^{\prime}_{n}({\mathfrak{g}})\subset\Sigma_{n}(\mathcal{O}^{\prime})$ with_ \[\mathopen{\|}\mathbb{R}C^{\prime}_{n}({\mathfrak{g}})\mathclose{\|}=5n-1+{ \mathfrak{g}}.\] Proof.: By Lemma 3.5, there exists a real double covering $\Sigma_{n}(\mathcal{O}^{\prime})\to\Sigma_{2n}(\mathcal{O})$ ramified along the curve $B_{0}\cup B_{\infty}$, such that the pull back in $\Sigma_{n}(\mathcal{O}^{\prime})$ of the curve $C_{n}({\mathfrak{g}})$ from Lemma 3.3 is a finite real algebraic curve $C^{\prime}_{n}({\mathfrak{g}})$. Each node of $C_{n}({\mathfrak{g}})$ gives rise to two solitary real nodes of $C^{\prime}_{n}({\mathfrak{g}})$, and each tangency point of $C_{n}({\mathfrak{g}})$ and $\mathbb{R}B_{0}$ gives rise to an extra solitary node of $C^{\prime}_{n}({\mathfrak{g}})$. ∎ ### Deformation to the normal cone We briefly recall the deformation to normal cone construction in the setting we need here, and refer for example to [5] for more details. Given $X$ a non-singular algebraic surface, and $B\subset X$ a non-singular algebraic curve, we denote by $N_{B/X}$ the normal bundle of $B$ in $X$, its projective completion by $E_{B}=\mathbb{P}(N_{B/X}\oplus\mathcal{O}_{B})$, and we define $B_{\infty}=E_{B}\smallsetminus N_{B/X}$. Note that if both $X$ and $B$ are real, then so are $E_{B}$ and $B_{\infty}$. Let $\mathcal{X}$ be the blow up of $X\times\mathbb{C}$ along $B\times\{0\}$. The projection $X\times\mathbb{C}\to\mathbb{C}$ induces a flat projection $\sigma\colon\mathcal{X}\to\mathbb{C}$, and one has $\sigma^{-1}(t)=X$ if $t\neq 0$, and $\sigma^{-1}(0)=X\cup E_{B}$. Furthermore, in this latter case $X\cap E_{B}$ is the curve $B$ in $X$, and the curve $B_{\infty}$ in $E_{B}$. Note that if both $X$ and $B$ are real, and if we equip $\mathbb{C}$ with the standard complex conjugation, then the map $\sigma$ is a real map. Let $C_{0}=C_{X}\cup C_{B}$ be an algebraic curve in $X\cup E_{B}$ such that: 1. $C_{X}\subset X$ is nodal and intersects $B$ transversely; 2. $C_{B}\subset E_{B}$ is nodal and intersects $B_{\infty}$ transversely; let $a=[C_{B}]\circ[F]$ in $H_{2}(E_{B};\mathbb{Z})$; 3. $C_{X}\cap B=C_{B}\cap B_{\infty}=C_{X}\cap C_{B}$. In the following two propositions, we use [20, Theorem 2.8] to ensure the existence of a deformation $C_{t}$ in $\sigma^{-1}(t)$ within the linear system $\|C_{X}+aB\|$ of the curve $C_{0}$ in some particular instances. We denote by $\mathcal{P}$ the set of nodes of $C_{0}\smallsetminus(X\cap E_{B})$, and by $\mathcal{I}_{X}$ (resp. $\mathcal{I}_{B}$) the sheaf of ideals of $\mathcal{P}\cap X$ (resp. $\mathcal{P}\cap E_{B}$). Proposition 3.7.: _In the notation above, suppose that $X\subset\mathbb{C}P^{3}$ is a quadric ellipsoid, and that $B$ is a real hyperplane section. If $C_{0}$ is a finite real algebraic curve, then there exists a finite real algebraic curve $C_{1}$ in $X$ in the linear system $\|C_{X}+aB\|$ such that_ \[\|\mathbb{R}C_{1}\|=\|\mathbb{R}C_{0}\|.\] Proof.: One has the following short exact sequence of sheaves \[0\longrightarrow\mathcal{O}(C_{X})\otimes\mathcal{I}_{X}\longrightarrow \mathcal{O}(C_{X})\longrightarrow\mathcal{O}_{\mathcal{P}\cap X} \longrightarrow 0.\] (To shorten the notation, we abbreviate ${\mathcal{O}}(D)={\mathcal{O}}_{X}(D)$ for a divisor $D\subset X$ when the ambient variety $X$ is understood.) Since $H^{1}(X,\mathcal{O}(C_{X}))=0$, one obtains the following exact sequence \[0\longrightarrow H^{0}(X,\mathcal{O}(C_{X})\otimes\mathcal{I}_{X}) \longrightarrow H^{0}(X,\mathcal{O}(C_{X}))\longrightarrow H^{0}(\mathcal{P} \cap X,\mathcal{O}_{\mathcal{P}\cap X})\longrightarrow H^{1}(X,\mathcal{O}(C_{ X})\otimes\mathcal{I}_{X})\longrightarrow 0.\] The surface $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ is toric and it is a classical application of Riemann-Roch Theorem that $H^{0}(X,\mathcal{O}(C_{X})\otimes\mathcal{I}_{X})$ has codimension $\|\mathcal{P}\cap X\|$ in $H^{0}(X,\mathcal{O}(C_{X}))$ (see for example [17, Lemma 8 and Corollary 2]). Since $h^{0}(\mathcal{P}\cap X,\mathcal{O}_{\mathcal{P}\cap X})=\|\mathcal{P}\cap X\|$, we deduce that \[H^{1}(X,\mathcal{O}(C_{X})\otimes\mathcal{I}_{X})=0.\] The curve $B$ is rational, and the surface $E_{B}$ is the surface $\Sigma_{2}$. In particular, $E_{B}$ is a toric surface and $B_{\infty}$ is an irreducible component of its toric boundary. Hence we analogously obtain \[H^{1}(E_{B},\mathcal{O}(C_{B}-B_{\infty})\otimes\mathcal{I}_{B})=0.\] Hence by [20, Theorem 3.1], the proposition is now a consequence of [20, Theorem 2.8]. ∎ Recall that $H^{0}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B})$ is the set of elements of $H^{0}(E_{B},\mathcal{O}(C_{B}))$ vanishing on $\mathcal{P}\cap E_{B}$. Proposition 3.8.: _Suppose that $X=\mathbb{C}P^{2}$, that $B$ is a non-singular real cubic curve, and that $C_{X}=\varnothing$. If $C_{B}$ is a finite real algebraic curve and if $H^{0}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B})$ is of codimension $\|\mathcal{P}\|$ in $H^{0}(E_{B},\mathcal{O}(C_{B}))$, then there exists a finite real algebraic curve $C_{1}$ in $\mathbb{C}P^{2}$ of degree $3a$ such that_ \[\|\mathbb{R}C_{1}\|=\|\mathbb{R}C_{B}\|.\] Proof.: Recall that $E_{B}$ is a ruled surface over $B$, _i.e._, is equipped with a $\mathbb{C}P^{1}$-bundle $\pi\colon E_{B}\to B$. By [7, Lemma 2.4], we have \[H^{i}(E_{B},\mathcal{O}(C_{B}))\simeq H^{i}(B_{\infty},\pi_{}\mathcal{O}(C_{B })),\qquad i\in\{0,1,2\}.\] In particular the short exact sequence of sheaves \[0\longrightarrow\mathcal{O}(C_{B}-B_{\infty})\longrightarrow\mathcal{O}(C_{B}) \longrightarrow\mathcal{O}_{B_{\infty}}\longrightarrow 0\] gives rise to the exact sequence \[0\longrightarrow H^{0}(E_{B},\mathcal{O}(C_{B}-B_{\infty}))\longrightarrow H^{ 0}(E_{B},\mathcal{O}(C_{B}))\longrightarrow H^{0}(B_{\infty},\mathcal{O}_{B_{ \infty}})\longrightarrow\] \[\longrightarrow H^{1}(E_{B},\mathcal{O}(C_{B}-B_{\infty}))\longrightarrow H^{1 }(E_{B},\mathcal{O}(C_{B}))\xrightarrow[]{\mbox{ }\iota_{1}\mbox{ }}H^{1}(B_{ \infty},\mathcal{O}_{B_{\infty}})\longrightarrow 0.\] Furthermore, by [6, Proposition 3.1] we have $H^{1}(E_{B},\mathcal{O}(C_{B}-B_{\infty}))=0$, hence the map $\iota_{1}$ is an isomorphism. On the other hand, the short exact sequence of sheaves \[0\longrightarrow\mathcal{O}(C_{B})\otimes\mathcal{I}_{B}\longrightarrow \mathcal{O}(C_{B})\longrightarrow\mathcal{O}_{\mathcal{P}}\longrightarrow 0\] gives rise to the exact sequence \[0\longrightarrow H^{0}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B}) \longrightarrow H^{0}(E_{B},\mathcal{O}(C_{B}))\xrightarrow[]{\mbox{ }r_{1} \mbox{ }}H^{0}(\mathcal{P},\mathcal{O}_{\mathcal{P}})\longrightarrow\] \[\longrightarrow H^{1}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B}) \xrightarrow[]{\mbox{ }\iota_{2}\mbox{ }}H^{1}(E_{B},\mathcal{O}(C_{B})) \longrightarrow 0.\] By assumption, the map $r_{1}$ is surjective, so we deduce that the map $\iota_{2}$ is an isomorphism. We denote by $\widetilde{\mathcal{L}}_{0}$ the invertible sheaf on the disjoint union of $E_{B}$ and $\mathbb{C}P^{2}$ and restricting to $\mathcal{O}(C_{B})$ and $\mathcal{O}_{\mathbb{C}P^{2}}$ on $E_{B}$ and $\mathbb{C}P^{2}$ respectively. Finally, we denote by $\mathcal{L}_{0}$ the invertible sheaf on $\sigma^{-1}(0)$ for which $C_{0}$ is the zero set of a section. The natural short exact sequence \[0\longrightarrow\mathcal{L}_{0}\otimes\mathcal{I}_{B}\longrightarrow\widetilde {\mathcal{L}}_{0}\otimes\mathcal{I}_{B}\longrightarrow\mathcal{O}_{B}\longrightarrow 0\] gives rise to the long exact sequence \[0\longrightarrow H^{0}(\sigma^{-1}(0),\mathcal{L}_{0}\otimes\mathcal{I}_{B}) \longrightarrow H^{0}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B})\oplus H^ {0}(\mathbb{C}P^{2},\mathcal{O}_{\mathbb{C}P^{2}})\xrightarrow[]{\mbox{ }r_{2} \mbox{ }}H^{0}(B,\mathcal{O}_{B})\longrightarrow\] \[\longrightarrow H^{1}(\sigma^{-1}(0),\mathcal{L}_{0}\otimes\mathcal{I}_{B}) \longrightarrow H^{1}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B}) \xrightarrow[]{\mbox{ }\iota\mbox{ }}H^{1}(B,\mathcal{O}_{B})\longrightarrow H ^{2}(\sigma^{-1}(0),\mathcal{L}_{0}\otimes\mathcal{I}_{B})\longrightarrow 0.\] The restriction of the map $r_{2}$ to the second factor $H^{0}(\mathbb{C}P^{2},\mathcal{O}_{\mathbb{C}P^{2}})$ is clearly an isomorphism, hence we obtain the exact sequence \[0\longrightarrow H^{1}(\sigma^{-1}(0),\mathcal{L}_{0}\otimes\mathcal{I}_{B}) \longrightarrow H^{1}(E_{B},\mathcal{O}(C_{B})\otimes\mathcal{I}_{B}) \xrightarrow[]{\mbox{ }\iota\mbox{ }}H^{1}(B,\mathcal{O}_{B})\longrightarrow H ^{2}(\sigma^{-1}(0),\mathcal{L}_{0}\otimes\mathcal{I}_{B})\longrightarrow 0.\] Since $\iota=\iota_{1}\circ\iota_{2}$ is an isomorphism, we deduce that $H^{1}(\sigma^{-1}(0),\mathcal{L}_{0}\otimes\mathcal{I}_{B})=0$. Now the proposition follows from [20, Theorem 2.8]. ∎ ## 4. Finite curves in $\mathbb{C}P^{2}$ In the case $X=\mathbb{C}P^{2}$, Theorem 2.5 and Corollary 2.6 specialize as follows. Theorem 4.1.: _Let $C\subset\mathbb{C}P^{2}$ be a finite real algebraic curve of degree $2k$. Then,_ (4) \[\|\mathbb{R}C\|\leq k^{2}+g(C)+1,\] (5) \[\|\mathbb{R}C\|\leq\frac{3}{2}k(k-1)+1.\] In the rest of this section, we discuss the sharpness of these bounds. ### Asymptotic constructions The following asymptotic lower bound holds for any projective toric surface with the standard real structure. Theorem 4.2.: _Let $\Delta\subset\mathbb{R}^{2}$ be a convex lattice polygon, and let $X_{\Delta}$ be the associated toric surface. Then, there exists a sequence of finite real algebraic curves $C_{k}\subset X_{\Delta}$ with the Newton polygon $\Delta(C_{k})=2k\Delta$, such that_ \[\lim_{k\to\infty}\frac{1}{k^{2}}\|\mathbb{R}C_{k}\|=\frac{4}{3}{\operatorname{ Area}}(\Delta),\] _where ${\operatorname{Area}}(\Delta)$ is the lattice area of $\Delta$._ Remark 4.3.: In the settings of Theorem 4.2, assuming $X_{\Delta}$ smooth, the asymptotic upper bound for finite real algebraic curves $C\subset X_{\Delta}$ with $\Delta(C)=2k\Delta$ is given by Theorem 2.5: \[\|\mathbb{R}C\|\lesssim\frac{3}{2}{\operatorname{Area}}(\Delta).\] Proof of Theorem 4.2.: There exists a (unique) real rational cubic $C\subset(\mathbb{C}^{})^{2}$ such that * $\Delta(C)$ is the triangle with the vertices $(0,0)$, $(2,1)$, and $(1,2)$; * the coefficient of the defining polynomial $f$ of $C$ at each corner of $\Delta(C)$ equals $1$; * $\mathbb{R}C\cap\mathbb{R}_{>0}^{2}$ is a single solitary node. [FIGURE:S4.F5][ENDFIGURE] Figure 5 shows a tilling of $\mathbb{R}^{2}$ by lattice congruent copies of $\Delta(C)$. Intersecting this tilling with $k\Delta$ and making an appropriate adjustment in the vicinity of the boundary, we obtain a convex subdivision of $k\Delta$ containing $\frac{1}{3}k^{2}{\operatorname{Area}}(\Delta)+O(k)$ copies of $\Delta(C)$. Now, for each of these copies, we consider an appropriate monomial multiple of either $f(x,y)$ or $f(1/x,1/y)$. Applying Theorem 3.1, we obtain a real polynomial $f_{k}$ whose zero locus in $\mathbb{R}_{>0}^{2}$ consists of $\frac{1}{3}k^{2}{\operatorname{Area}}(\Delta)+O(k)$ solitary nodes. There remains to let $C_{k}=\{f_{k}(x^{2},y^{2})=0\}$. ∎ Corollary 4.4.: _There exists a sequence of finite real algebraic curves $C_{k}\subset\mathbb{C}P^{2}$, $\deg C_{k}=2k$, such that_ \[\lim_{k\to+\infty}\frac{1}{k^{2}}\|\mathbb{R}C_{k}\|=\frac{4}{3}.\] In the next theorem, we tweak the “adjustment in the vicinity of the boundary” in the proof of Theorem 4.2 in the case $X_{\Delta}=\mathbb{C}P^{2}$. Theorem 4.5.: _For any integer $k\geq 3$, there exists a finite real algebraic curve $C\subset\mathbb{C}P^{2}$ of degree $2k$ such that_ \[\|\mathbb{R}C\|=\begin{cases}12l^{2}-4l+2&\mbox{if }k=3l,\\ 12l^{2}+4l+3&\mbox{if }k=3l+1,\\ 12l^{2}+12l+6&\mbox{if }k=3l+2.\end{cases}\] Proof.: Following the proof of Theorem 4.2, we use the subdivision of the triangle $k\Delta$ (with the vertices $(0,0)$, $(k,0)$, and $(0,k)$) shown in Figure 6. In the $t$-axis ($t=x$ or $y$), each segment of length $1$, $2$ or $3$ bears an appropriate monomial multiple of $1$, $(t-1)^{2}$ or $(t-1)^{2}(t+1)$, respectively. Thus, each segment $\ell$ of length $2$ or $3$ gives rise to a point of tangency of the $t$-axis and the curve $\{f_{k}=0\}$, resulting in two extra solitary nodes of $C_{k}$. Similarly, each vertex of $k\Delta$ contained in a segment of length $1$ gives rise to an extra solitary node of $C_{k}$. [FIGURE:S4.F6][ENDFIGURE] ∎ Remark 4.6.: The construction of Theorem 4.5 for $k=3,4$ can easily be performed without using the patchworking technique. ### A curve of degree $12$ The construction given by Theorem 4.5 is the best known if $k\leq 5$. If $k=6$, we can improve it by $2$ more units. Proposition 4.7.: _There exists a finite real algebraic curve $C\subset\mathbb{C}P^{2}$ of degree $12$ such that_ \[\|\mathbb{R}C\|=45.\] Proof.: Let $C^{\prime}=C^{\prime}_{9}(1)$ be a finite real algebraic curve in $\Sigma_{9}(\mathcal{O}^{\prime})$ as in Lemma 3.6. Let us denote by $\mathcal{P}$ the set of nodes of $C^{\prime}$, and by $\mathcal{I}$ the sheave of ideals on $\Sigma_{9}(\mathcal{O}^{\prime})$ defining $\mathcal{P}$. Since $\mathbb{R}B\neq\varnothing$, there exists a real line bundle $\L_{0}$ of degree $3$ over $B$ such that $\mathcal{O}^{\prime}=\L_{0}^{\otimes 3}$. This bundle $\L_{0}$ embeds $B$ into $\mathbb{C}P^{2}$ as a real cubic curve for which $\L_{0}^{\otimes 3}=\mathcal{O}$ is the normal bundle. The proposition will then follow from Proposition 3.8 once we prove that $H^{0}(\Sigma_{9}(\mathcal{O}^{\prime}),\mathcal{O}(4B_{0})\otimes\mathcal{I})$ is of codimension $45$ in $H^{0}(\Sigma_{9}(\mathcal{O}^{\prime}),\mathcal{O}(4B_{0}))$. Let us show that this is indeed the case, _i.e._, let us show that given any node $p$ of $C^{\prime}$, there exists an algebraic curve in $\mathcal{O}(4B_{0})$ on $\Sigma_{9}(\mathcal{O}^{\prime})$ passing through all nodes of $C^{\prime}$ but $p$. Recall that there exists a real double covering $\rho\colon\Sigma_{9}(\mathcal{O}^{\prime})\to\Sigma_{9}(\mathcal{O}^{\prime \otimes 2})$ ramified along $B_{0}\cup B_{\infty}$ with respect to which $C^{\prime}$ is symmetric, and that $C^{\prime}$ has $18$ pairs of symmetric nodes and $9$ nodes on $B_{0}$. By Riemann-Roch Theorem, for any line bundle $\mathcal{O}$ over $B_{0}$ of degree $n\geq 1$, and given any set $\mathcal{P}$ of $n-2$ points on distinct fibers of $\Sigma_{n}(\mathcal{O})$ and any disjoint finite subset $\overline{\mathcal{P}}$ of $\Sigma_{n}(\mathcal{O})$, there exists an algebraic curve in $\mathcal{O}(B_{0})$ containing $\mathcal{P}$ and avoiding $\overline{\mathcal{P}}$. As a consequence, there exists a symmetric curve in $\mathcal{O}(2B_{0})$ on $\Sigma_{9}(\mathcal{O}^{\prime})$ passing through any $16$ pairs of symmetric nodes of $C^{\prime}$ and avoiding all other nodes of $C^{\prime}$. Altogether, we see that given any node $p$ of $C$’, there exists a reducible curve in $\mathcal{O}(4B_{0})$ on $\Sigma_{9}(\mathcal{O}^{\prime})$, consisting in the union of a symmetric curve in $\mathcal{O}(2B_{0})$ and two curves in $\mathcal{O}(B_{0})$, and passing through all nodes of $C^{\prime}$ but $p$. ∎ ### Curves of low genus Here we show that inequality $(\ref{eq:cp2 genus})$ of Theorem 4.1 is sharp when the degree is large compared to the genus. Theorem 4.8.: _Given integers $k\geq 3$ and $0\leq g\leq k-3$, there exists a finite real algebraic curve $C\subset\mathbb{C}P^{2}$ of degree $2k$ and genus $g$ such that_ \[\|\mathbb{R}C\|=k^{2}+g+1.\] Proof.: Consider a real rational curve $C_{1}\subset C^{2}$ with the following properties: * the Newton polygon of $C_{1}$ is the triangle with the vertices $(0,0)$, $(0,k-2)$ and $(2k-4,0)$, * $C_{1}$ intersects the axis $y=0$ in a single point with multiplicity $2k-4$, * $\mathbb{R}C_{1}\cap\{y>0\}$ consists of $\frac{1}{2}(k-2)(k-3)$ solitary nodes. Such a curve exists: for example, one can take a rational simple Harnack curve with the prescribed Newton polygon (see [14, 11, 1]). Shift the Newton polygon $\Delta(C_{1})$ by $2$ units up and place in the trapezoid with the vertices $(0,0)$, $(2k,0)$, $(2k-4,2)$, $(0,2)$ a defining polynomial of the curve $\widetilde{C}_{1,k-2,g+1}$ given by Lemma 3.2. Applying Theorem 3.1, we obtain a real rational curve $C_{2}\subset\mathbb{C}^{2}$ such that * $\mathbb{R}C_{2}\cap\{y>0\}$ consists of $\frac{1}{2}(k-2)(k-3)+2k+g-2$ solitary nodes, * $C_{2}$ intersects the line $y=0$ in $k-g-1$ real points of multiplicity 2, and in $g+1$ additional pairs of complex conjugated points. If $C_{2}$ is given by an equation $f(x,y)=0$ positive on $y>0$, we define $C$ as the curve $f(x,y^{2})=0$. Each node $p\in\{y>0\}$ of $C_{2}$ gives rise to two solitary real nodes of $C$, and each tangency point of $C_{2}$ and the axis $y=0$ gives rise to an extra solitary node of $C$. The genus $g(C)=g$ is given by the Riemann–Hurwitz formula applied to the double covering $C\to C_{2}$: its normalization is branched at the $2(g+1)$ points of transverse intersection of $C_{2}$ and the axis $y=0$. ∎ ## 5. Finite curves in real ruled surfaces We use the notation $B,\mathcal{O},B_{0},F,\Sigma_{n}(\mathcal{O})$ introduced in Section 3.2. A real algebraic curve $C$ in $\Sigma_{n}(\mathcal{O})$ realizing the class $u[B_{0}]+v[F]\in H_{2}(\Sigma_{n}(\mathcal{O});\mathbb{Z})$ may be finite only if both $u=2a$ and $v=2b$ are even. General results of the previous sections specialize as follows. Theorem 5.1.: _Let $C\subset\Sigma_{n}(\mathcal{O})$ be a finite real algebraic curve, $[C]=2a[B_{0}]+2b[F]\in H_{2}(\Sigma_{n}(\mathcal{O});\mathbb{Z})$, $a>0$, $b>0$. Then,_ (6) \[\|\mathbb{R}C\|\leq na^{2}+2ab+g(C)+1-2g(B),\] (7) \[\|\mathbb{R}C\|\leq\frac{1}{2}{na(3a-1)}+3ab-(a+b)+1+(a-1)g(B).\] Proof.: The statement is an immediate consequence of Theorem 2.8 and Corollary 2.6: due to Lemma 3.5, we can choose $\mathbb{R}X_{+}=\mathbb{R}X$ and $\mathbb{R}X_{-}=\varnothing$. ∎ As in the case of $\mathbb{C}P^{2}$, we do not know whether the upper bounds $(\ref{eq:hirz genus})$ and $(\ref{eq:hirz})$ are sharp in general. In the rest of the section, we discuss the special cases of small $a$ or small genus. The two next propositions easily generalize to ruled surfaces over a base of any genus (in the same sense as explained after Lemma 3.3). For simplicity, we confine ourselves to the case of a rational base. Proposition 5.2 ($a=1$).: _Given integers $b,n\geq 0$, there exists a finite real algebraic curve $C\subset\Sigma_{n}$ of bidegree $(2,2b)$ such that $\|\mathbb{R}C\|=n+2b$._ Proof.: A collection of $n+2b$ generic real points in $\Sigma_{n}$ determines a real pencil of curves of bidegree $(1,b)$, and one can take for $C$ the union of two complex conjugate members of this pencil. ∎ Proposition 5.3 ($a=2$).: _Given integers $b,n\geq 0$, and $-1\leq g\leq n+b-2$, there exists a finite real algebraic curve $C\subset\Sigma_{n}$ of bidegree $(4,2b)$ and genus $g$ such that_ \[\|\mathbb{R}C\|=4n+4b+g+1.\] _In particular, if $b+n\geq 1$, then there exists a finite real algebraic curve $C\subset\Sigma_{n}$ of bidegree $(4,2b)$ such that_ \[\|\mathbb{R}C\|=5n+5b-1.\] Proof.: We argue as in the proof of Lemma 3.6, starting from the curve $\widetilde{C}_{n,b,g+1}$ given by Lemma 3.2. The genus $g(C)$ is computed by the Riemann–Hurwitz formula. ∎ All rational ruled surfaces are toric, and Theorem 4.2 takes the following form. Theorem 5.4.: _Given integers $a>0$ and $b\geq 0$, there exists a sequence of finite real algebraic curves $C_{k}\subset\Sigma_{n}$ of bidegree $(ka,kb)$ such that_ \[\lim_{k\to+\infty}\frac{1}{k^{2}}\|\mathbb{R}C_{k}\|=\frac{4}{3}(na^{2}+2ab).\] Furthermore, the proof of Theorem 4.8 extends literally to curves in $\Sigma_{n}$. Theorem 5.5 (low genus).: _Given integers $a>0$, $b,n\geq 0$, and $-1\leq g\leq n(a-1)+b-2$, there exists a finite real algebraic curve $C\subset\Sigma_{n}$ of bidegree $(2a,2b)$ and genus $g$ such that_ \[\|\mathbb{R}C\|=na^{2}+2ab+g+1.\] ## 6. Finite curves in the ellipsoid The algebraic surface $\Sigma_{0}=\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ has two real structures with non-empty real part, namely $c_{h}(z,w)=(\bar{z},\bar{w})$ and $c_{e}(z,w)=(\bar{w},\bar{z})$. The first one was considered in Section 5. In this section, $\Sigma_{0}$ is assumed equipped with the real structure $c_{e}$, and we have $\mathbb{R}\Sigma_{0}=S^{2}$. ### General bounds Let $e_{1}$ and $e_{2}$ be the classes in $H_{2}(\Sigma_{0};\mathbb{Z})$ represented by the two rulings. The action of $c_{e}$ on $H_{2}(\Sigma_{0};\mathbb{Z})$ is given by $c_{e}(e_{i})=-e_{3-i}$, and so $\sigma^{-}_{\text{\rm inv}}(\Sigma_{0},c_{e})=1$. The classes in $H_{2}(\Sigma_{0};\mathbb{Z})$ realized by real algebraic curves are those of the form $m(e_{1}+e_{2})$. For any $m\geq 1$, a real algebraic curve of bidegree $(m,m)$ may have finite real part. Theorem 6.1.: _Let $C$ be a reduced finite real algebraic curve in $(\Sigma_{0},c_{e})$ of bidegree $(m,m)$, with $m\geq 2$. Then_ (8) \[\|\mathbb{R}C\|\leq\left\{\begin{array}[]{ll}2k^{2}+g(C)+3&\mbox{if }m=2k\\ \\ 2k^{2}+4k+g(C)&\mbox{if }m=2k+1\end{array}\right..\] _In particular we have_ (9) \[\|\mathbb{R}C\|\leq\left\{\begin{array}[]{ll}3k^{2}-2k+2&\mbox{if }m=2k\\ \\ 3k^{2}+2k&\mbox{if }m=2k+1\end{array}\right..\] Proof.: In order to apply Theorem 2.5, we note that $T_{2,1}(\Sigma_{0})=-h^{1,1}(\Sigma_{0})=-2$ and that the real locus of $(\Sigma_{0},c_{e})$ being a sphere, $\chi(\mathbb{R}\Sigma_{0})=2$. The case when $m=2k$ is then provided by Theorem 2.5 and Corollary 2.6. Indeed, in this case, $[C]=m(e_{1}+e_{2})=2k(e_{1}+e_{2})$ and letting $e=k(e_{1}+e_{2})$, we get $e^{2}=2k^{2}$ and $e\cdot c_{1}(\Sigma_{0})=2k(e_{1}+e_{2})(e_{1}+e_{2})=4k$. So suppose that $m=2k+1$ and let $p\in\mathbb{R}C$. Let $E_{1}$ and $E_{2}$ be a pair of conjugate generatrices which meet $C$ at $p$. Let $\widetilde{C}=C\cup E_{1}\cup E_{2}$ and let $\overline{C}$ be the strict transform of $\widetilde{C}$ in the blow-up $\overline{\Sigma}_{0}$ of $\Sigma_{0}$ at $p$. The class of the auxiliary curve $\widetilde{C}$ in $H_{2}(\Sigma_{0};\mathbb{Z})$ is then $[\widetilde{C}]=2(k+1)(e_{1}+e_{2})$. Let $e=(k+1)(e_{1}+e_{2})$, we get $e^{2}=2(k+1)^{2}$. Let $\overline{e}$ be half the class of $\overline{C}$ in $H_{2}(\overline{\Sigma}_{0};\mathbb{Z})$, we get $\overline{e}^{2}\leq e^{2}-4$, as the point $p$ is of multiplicity at least $4$ in $\widetilde{C}$. Furthermore, we have $g(\overline{C})=g(\widetilde{C})=g(C)-2$ and $\|\mathbb{R}C\|=\|\mathbb{R}\widetilde{C}\|=\|\mathbb{R}\overline{C}\|+1$. In order to apply Theorem 2.5 for the curve $\overline{C}$ on $\overline{\Sigma}_{0}$, it remains to note that $T_{2,1}(\overline{\Sigma}_{0})=T_{2,1}(\Sigma_{0})-1$ and $\chi(\mathbb{R}\overline{\Sigma_{0}})=\chi(\mathbb{R}\Sigma_{0})-1$. Hence we obtain $(\ref{eq:quad genus})$ from Theorem 2.5 applied to the curve $\overline{C}$ on $\overline{\Sigma}_{0}$. To get $(\ref{eq:quad})$, it suffices to remark that, $C$ being a curve of bidegree $(2k+1,2k+1)$, we have $g(C)\leq 4k^{2}-\|\mathbb{R}C\|$. ∎ Remark 6.2.: Let us consider the following problem: given a smooth real projective surface $(X,c)$ and a homology class $d\in H_{2}(X;\mathbb{Z})$, what is the maximal possible number of intersection points between $C$ and $\mathbb{R}X$ for a non-real algebraic curve $C$ in $X$ realizing the class $d$? Since any two distinct irreducible algebraic curves in $X$ intersect positively, any non-real irreducible algebraic curve $C$ in $X$ intersects $c(C)$ in $-[C]\cdot c_{}[C]$ points, and so intersects $\mathbb{R}X$ in at most $-[C]\cdot c_{}[C]$ points. It is easy to see that this upper bound is sharp in $\mathbb{C}P^{2}$. Interestingly, Theorem 6.1 shows that this trivial upper bound is not sharp in the case of the quadric ellipsoid. Any irreducible algebraic curve $C$ in $\Sigma_{0}$ realizing the class $(m-1,1)$ with $m\geq 3$ is non real and rational. Since the union of $C$ and $c_{e}(C)$ is a real algebraic curve of geometric genus $-1$ realizing the class $(m,m)$, Theorem 6.1 implies that \[\|C\cap\mathbb{R}\Sigma_{0}\|\leq\left\{\begin{array}[]{ll}2k^{2}+2&\mbox{if }m= 2k\\ \\ 2k^{2}+4k-1&\mbox{if }m=2k+1\end{array}\right.,\] whereas $(m-1,1)\cdot(1,m-1)=m^{2}-2m+2$ is at least twice as large. Next theorem is an immediate consequence of Theorem 4.2 and Proposition 3.7 Theorem 6.3.: _There exists a sequence of finite real algebraic curves $C_{m}$ of bidegree $(m,m)$ in the quadric ellipsoid such that_ \[\lim_{m\to\infty}\frac{1}{2m^{2}}\|\mathbb{R}C_{m}\|=\frac{4}{3}.\] ### Curves of low bidegree Next statement shows in particular that Theorem 6.1 is not sharp for $m=2$ and $m=5$. Proposition 6.4.: _For $m\leq 5$, the maximal possible value $\delta_{e}(m)$ of $\|\mathbb{R}C\|$ for a finite real algebraic curve of bidegree $(m,m)$ in the quadric ellipsoid is_ \[\begin{array}[]{c \| c\| c\| c\| c\| c}m&1&2&3&4&5\\ \hline\delta_{e}(m)&1&2&5&10&15\end{array}\] Proof.: We start by constructing real algebraic curves with a number of real points as stated in the proposition. For $m\leq 4$, such a curve is constructed by taking the union of two complex conjugated curves of bidegree $(m-1,1)$ and $(1,m-1)$ intersecting $\mathbb{R}\Sigma_{0}$ in $(m-1)^{2}+1$ points. For $m\leq 3$, such a curve exists since $2m-1$ points determine a pencil of curves of bidegree $(m-1,1)$. For the case $m=4$, consider $8$ points in $\mathbb{R}P^{2}$ such that there exists a non-real rational cubic $C_{0}\subset\mathbb{C}P^{2}$ passing through these 8 points (such configuration of $8$ points exist). Since $C_{0}$ has a unique nodal point, it has to be non-real. Furthermore, since $C_{0}$ intersects $\mathbb{R}P^{2}$ in an odd number of points, it has to intersect $\mathbb{R}P^{2}$ in a ninth point. Hence the union of $C_{0}$ with its complex conjugate is a real algebraic curve of degree 6 with 9 solitary points and two complex conjugate nodal points. Denote by $O$ the line passing through the two latter. Blowing up the two nodes and blowing down the strict transform of $O$, we obtain a real algebraic curve of bidegree $(4,4)$ in the quadric ellipsoid whose real part has exactly 10 points. The case $m=5$ is treated by applying the deformation to the normal cone construction to a non-singular real hyperplane section $B$, with $\mathbb{R}B\neq\varnothing$, in the quadric ellipsoid $X$. Here we use notations from Section 3.3. According to Proposition 5.3, there exists a real algebraic curve $C_{B}$ of bidegree $(4,2)$ in $E_{B}=\Sigma_{2}$ whose real part consists of 14 solitary nodes. Let $C_{X}$ be a reducible curve of bidegree $(1,1)$ in $X$ passing through $X\cap E_{B}\cap C_{B}$, and let us define $C_{0}=C_{X}\cup C_{B}$. The curve $C_{0}$ is a finite real algebraic curve with $\|\mathbb{R}C_{0}\|=15$, hence Proposition 3.7 ensures the existence of a finite real algebraic curve $C$ of bidegree $(5,5)$ in $X$ with $\|\mathbb{R}C\|=15$. We now prove that there does not exist finite real algebraic curves of bidegree $m\leq 5$ with a number of real points greater than the one stated in the proposition. By Bézout Theorem, a finite real algebraic curve of bidegree $(m,m)$ with $m=1$ or $m=2$ has at most $1$ or $2$ real points respectively. According to Theorem 6.1, a finite real algebraic curve of bidegree $(3,3)$, $(4,4)$ or $(5,5)$ in the quadric ellipsoid cannot have more that $5,10$, or $16$ real points respectively. Suppose that there exists a real algebraic curve of bidegree $(5,5)$ in the quadric ellipsoid with $16$ real points. By the genus formula, this curve is rational and its 16 real points are all ordinary nodes. By a small perturbation creating an oval for each node, we obtain a non-singular real algebraic curve of bidegree $(5,5)$ in the quadric ellipsoid whose real part consists of exactly 16 connected components, each of them bounding a disc in the sphere. This contradicts the congruence [13, Theorem 1b)]. ∎ ## References * [Bru15] E. Brugallé. Pseudoholomorphic simple Harnack curves. _Enseign. Math._, 61(3-4):483–498, 2015. * [CLR80] Man Duen Choi, Tsit Yuen Lam, and Bruce Reznick. Real zeros of positive semidefinite forms. I. _Math. Z._, 171(1):1–26, 1980. * [Deg12] Alex Degtyarev. _Topology of algebraic curves: An approach via dessins d’enfants_, volume 44 of _De Gruyter Studies in Mathematics_. Walter de Gruyter & Co., Berlin, 2012. * [DIK08] Alex Degtyarev, Ilia Itenberg, and Viatcheslav Kharlamov. On deformation types of real elliptic surfaces. _Amer. J. Math._, 130(6):1561–1627, 2008. * [Ful84] W. Fulton. _Introduction to Intersection Theory in Algebraic Geometry_, volume 54 of _BMS Regional Conf. Ser. in Math._ Amer. Math. Soc., Providence, 1984. * [GP96] F. J. Gallego and B. P. Purnaprajna. Normal presentation on elliptic ruled surfaces. _J. Algebra_, 186(2):597–625, 1996. * [Har77] R. Hartshorne. _Algebraic geometry_. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. * [Hil88] David Hilbert. Ueber die Darstellung definiter Formen als Summe von Formenquadraten. _Math. Ann._, 32(3):342–350, 1888. * [Hir86] F. Hirzebruch. Singularities of algebraic surfaces and characteristic numbers. In _The Lefschetz centennial conference, Part I (Mexico City, 1984)_, volume 58 of _Contemp. Math._, pages 141–155. Amer. Math. Soc., Providence, RI, 1986. * [IKS15] Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin. Welschinger invariants of real del Pezzo surfaces of degree $\geq 2$. _Internat. J. Math._, 26(8):1550060, 63, 2015. * [KO06] R. Kenyon and A. Okounkov. Planar dimers and Harnack curves. _Duke Math. J._, 131(3):499–524, 2006. * [Man17] Frédéric Mangolte. _Variétés algébriques réelles_, volume 24 of _Cours Spécialisés_. Société Mathématique de France, Paris, 2017. * [Mik91] G. Mikhalkin. Congruences for real algebraic curves on an ellipsoid. _Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)_, 193(Geom. i Topol. 1):90–100, 162, 1991. * [Mik00] G. Mikhalkin. Real algebraic curves, the moment map and amoebas. _Ann. of Math. (2)_, 151(1):309–326, 2000. * [Ore03] Stepan Yu. Orevkov. Riemann existence theorem and construction of real algebraic curves. _Ann. Fac. Sci. Toulouse Math. (6)_, 12(4):517–531, 2003. * [Pet38] I. Petrowsky. On the topology of real plane algebraic curves. _Ann. of Math. (2)_, 39(1):189–209, 1938. * [Shu99] E. Shustin. Lower deformations of isolated hypersurface singularities. _Algebra i Analiz_, 11(5):221–249, 1999. * [Shu05] E. Shustin. A tropical approach to enumerative geometry. _Algebra i Analiz_, 17(2):170–214, 2005. * [Shu06] Eugenii Shustin. The patchworking construction in tropical enumerative geometry. In _Singularities and computer algebra_, volume 324 of _London Math. Soc. Lecture Note Ser._, pages 273–300. Cambridge Univ. Press, Cambridge, 2006. * [ST06] E. Shustin and I. Tyomkin. Patchworking singular algebraic curves. I. _Israel J. Math._, 151:125–144, 2006. * [Vir86] O. Ya. Viro. Achievements in the topology of real algebraic varieties in the last six years. _Uspekhi Mat. Nauk_, 41(3(249)):45–67, 240, 1986. * [Wil78] G. Wilson. Hilbert’s sixteenth problem. _Topology_, 17(1):53–73, 1978.
1302.1152	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 37875, "num_imgs": 2, "llama3_tokens_count": 13864 }	[ "content_image/1302.1152/x5.png", "content_image/1302.1152/x5.png" ]	# Mutations of fake weighted projective planes Mohammad E. Akhtar Alexander M. Kasprzyk Department of Mathematics Imperial College London London, SW$7$ $2$AZ UK mohammad.akhtar03@imperial.ac.uk a.m.kasprzyk@imperial.ac.uk ###### Abstract. In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterisation of such mutations in terms of T-singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking–Prokhorov. † [FOOTNOTE:†][ENDFOOTNOTE] ## 1. Introduction In [1] we described a combinatorial notion of mutation between convex lattice polytopes. In this paper we begin to explore the geometry behind this idea. Given a convex lattice polytope $P$ containing the origin and with primitive vertices, there is a corresponding toric variety $X$ defined by the spanning fan of $P$. A mutation between polytopes $P$ and $Q$ determines a deformation between $X_{P}$ and $X_{Q}$ [5]. Our main result characterises mutations between triangles; thus we characterise certain deformations, over $\mathbb{P}^{1}$, with fibers given by fake weighted projective planes. We recover and generalise certain results of Hacking and Prokhorov [4, Theorem 4.1] connecting the fake weighted projective planes with T-singularities to solutions of Markov-type equations. We prove the following: Proposition 1.1.: _Let $X=\mathbb{P}(\lambda_{0},\lambda_{1},\lambda_{2})$ be a weighted projective plane. Up to reordering of the weights, there exists a one-step mutation to a weighted projective plane $Y$ if and only if $\frac{1}{\lambda_{0}}(\lambda_{1},\lambda_{2})$ is a T-singularity. When this is the case, $Y=\mathbb{P}\left(\lambda_{1},\lambda_{2},\frac{(\lambda_{1}+\lambda_{2})^{2}} {\lambda_{0}}\right)$. More generally, there exists a one-step mutation from the fake weighted projective plane $X/(\mathbb{Z}/n)$ to the fake weighted projective plane $Y/(\mathbb{Z}/n^{\prime})$ only if $n=n^{\prime}$ and $\frac{1}{\lambda_{0}}(\lambda_{1},\lambda_{2})$ is a T-singularity._ In Proposition 3.12 we associate to a weighted projective plane $X$ a Diophantine equation (1.1) \[mx_{0}x_{1}x_{2}=k(c_{0}x_{0}^{2}+c_{1}x_{1}^{2}+c_{2}x_{2}^{2}).\] The weights $(\lambda_{0},\lambda_{1},\lambda_{2})$ of $X$ correspond to a solution $(a_{0},a_{1},a_{2})$, where $\lambda_{i}=c_{i}a_{i}^{2}$, $i=0,1,2$, and the degree of $X$ is given by \[(-K_{X})^{2}=\frac{m^{2}}{c_{0}c_{1}c_{2}k^{2}}.\] One-step mutations of $X$ correspond to transformations of the solutions to (1.1), and all such solutions can be generated from the so-called minimal weights by mutation. When $X=\mathbb{P}^{2}$, equation (1.1) becomes the celebrated Markov equation [9]. Certain other special cases were studied by Rosenberger [10]. These cases all have finitely many minimal weights. In §4 we give an example where the corresponding Diophantine equation has infinitely many minimal weights. ## 2. Mutations of Fano polytopes Let $N\cong\mathbb{Z}^{n}$ be a lattice with dual $M:=\mathrm{Hom}\left({N,\mathbb{Z}}\right)$. A lattice polytope $P\subset N_{\mathbb{Q}}:=N\otimes_{\mathbb{Z}}\mathbb{Q}$ is called _Fano_ if it satisfies three conditions: 1. $P$ is of maximum dimension, $\dim{P}=\dim{N}$; 2. The origin is contained in the strict interior of $P$, $\mathbf{0}\in\mathrm{int}\left({P}\right)$; 3. The vertices $\mathrm{vert}\left({P}\right)$ of $P$ are primitive lattice points, i.e. for any $v\in\mathrm{vert}\left({P}\right)$ there are no other lattice points on the line segment $\overline{{\mathbf{0}}{v}}$ joining $v$ and the origin. The dual of $P$ is defined to be the polyhedron \[{P}^{\vee}:=\{u\in M_{\mathbb{Q}}\mid u(v)\geq-1\text{ for all }v\in P\} \subset M_{\mathbb{Q}}.\] By condition (2) this is a polytope with $\mathbf{0}\in\mathrm{int}\left({{P}^{\vee}}\right)$, although it need not be a lattice polytope. See [7] for an overview of Fano polytopes. We briefly recall the notation of [1, §3]. Any choice of primitive vector $w\in M$ determines a lattice height function $w:N\rightarrow\mathbb{Z}$ which naturally extends to $N_{\mathbb{Q}}\rightarrow\mathbb{Q}$. A subset $S\subset N_{\mathbb{Q}}$ is said to lie at height $h\in\mathbb{Q}$ with respect to $w$ if $w(S):=\{w(s)\mid s\in S\}=\{h\}$; we write $w(S)=h$. The set of all points of $N_{\mathbb{Q}}$ lying at height $h$ with respect to a given $w$ is an affine hyperplane $H_{w,h}:=\{v\in N_{\mathbb{Q}}\mid w(v)=h\}$. In particular, \[w_{h}(P):=\mathrm{conv}\left({H_{w,h}\cap P\cap N}\right)\subset N_{\mathbb{Q}}\] will denote the (possibly empty) convex hull of all lattice points in $P$ at height $h$. Define \[{h_{\mathrm{min}}}:=\mathrm{min}\left\{{w(v)\mid v\in P}\right\},\qquad{h_{ \mathrm{max}}}:=\mathrm{max}\left\{{w(v)\mid v\in P}\right\}.\] Since $P$ is a lattice polytope, both ${h_{\mathrm{min}}}$ and ${h_{\mathrm{max}}}$ are integers. Condition (2) guarantees that ${h_{\mathrm{min}}}<0$ and ${h_{\mathrm{max}}}>0$. Definition 2.1.: A _factor_ of $P$ with respect to $w$ is a lattice polytope $F\subset N_{\mathbb{Q}}$ satisfying: 1. $w(F)=0$; 2. For every integer $h$, ${h_{\mathrm{min}}}\leq h<0$, there exists a (possibly empty) lattice polytope $G_{h}\subset N_{\mathbb{Q}}$ at height $h$ such that \[H_{w,h}\cap\mathrm{vert}\left({P}\right)\subseteq G_{h}+(-h)F\subseteq w_{h}(P).\] Note that, for given polytope $P\subset N_{\mathbb{Q}}$ and width vector $w\in M$, a factor $F$ need not exist. When a factor does exist we make the following construction: Definition 2.2 ([1, Definition 5]).: Let $P\subset N_{\mathbb{Q}}$ be a polytope with width vector $w\in M$, factor $F$, and polytopes $\{G_{h}\}$. We define the corresponding _combinatorial mutation_ to be the convex lattice polytope \[\mathrm{mut}_{w}(P,F;\{G_{h}\}):=\mathrm{conv}\left({\bigcup_{h={h_{\mathrm{ min}}}}^{-1}G_{h}\cup\bigcup_{h=0}^{h_{\mathrm{max}}}(w_{h}(P)+hF)}\right) \subset N_{\mathbb{Q}}.\] For brevity we will often refer to a combinatorial mutation simply as a _mutation_. We summarise the key properties of mutations [1]: 1. We need only consider factors $F$ up to translation, since for any $v\in N$ such that $w(v)=0$, we have $\mathrm{mut}_{w}(P,F;\{G_{h}\})\cong\mathrm{mut}_{w}(P,v+F;\{G_{h}+hv\})$. In particular, choosing $F$ to be a point leaves $P$ unchanged. 2. If $\{G_{h}\}$ and $\{G_{h}^{\prime}\}$ are any two collections of polytopes for a factor $F$, then $\mathrm{mut}_{w}(P,F;\{G_{h}\})\cong\mathrm{mut}_{w}(P,F;\{G_{h}^{\prime}\})$. Thus the choice of collection $\{G_{h}\}$ is irrelevant and we write $\mathrm{mut}_{w}(P,F)$. 3. $P$ is a Fano polytope if and only if $\mathrm{mut}_{w}(P,F)$ is a Fano polytope. 4. Let $Q:=\mathrm{mut}_{w}(P,F)$. Then $\mathrm{mut}_{-w}(Q,F)=P$, so mutations are invertible. 5. The toric varieties defined by the spanning fans of $P$ and $\mathrm{mut}_{w}(P,F)$ have the same degree. A mutation of $P\subset N_{\mathbb{Q}}$ induces a piecewise linear transformation $\varphi$ of $M_{\mathbb{Q}}$ such that ${\left({\varphi({P}^{\vee})}\right)}^{\vee}=\mathrm{mut}_{w}(P,F)$, given by \[\varphi:u\mapsto u-{u_{\mathrm{min}}}w,\qquad u\in M_{\mathbb{Q}},\] where ${u_{\mathrm{min}}}:=\mathrm{min}\left\{{u(v_{F})\mid v_{F}\in\mathrm{vert} \left({F}\right)}\right\}$. The inner normal fan of $F\subset N_{\mathbb{Q}}$ determines a chamber decomposition of $M_{\mathbb{Q}}$, and $\varphi$ acts as a linear transformation on the interior of each maximal dimensional cone of this fan. Example 2.3.: Consider the triangle $P=\mathrm{conv}\left\{{(1,-1),(-1,2),(0,-1)}\right\}\subset N_{\mathbb{Q}}$ corresponding to the toric variety $\mathbb{P}^{2}$. Let $w=(0,1)\in M$ and set $F=\mathrm{conv}\left\{{\mathbf{0},(1,0)}\right\}\subset N_{\mathbb{Q}}$. This defines a mutation from $P$ to the triangle $Q=\mathrm{conv}\left\{{(1,2),(-1,2),(0,-1)}\right\}\subset N_{\mathbb{Q}}$, as illustrated in Figure 1. On the dual side, this corresponds to a piecewise linear map $\varphi:u\mapsto uM_{\sigma}$ for $u=(\alpha,\beta)\in M_{\mathbb{Q}}$, where \[M_{\sigma}=\left\{\begin{array}[]{ll}\small\begin{pmatrix}1&0\\ 0&1\end{pmatrix}&\text{ if }\alpha\geq 0,\\ \small\begin{pmatrix}1&-1\\ 0&1\end{pmatrix}&\text{ otherwise.}\end{array}\right.\] In particular, $\varphi({P}^{\vee})={Q}^{\vee}$. <figure><img src="content_image/1302.1152/x5.png"><figcaption>Figure 2. A one-step mutation, depicted in MQ, of the triangle conv{u0,u1,u2}to the triangle conv{u2,u3,u4}.</figcaption></figure> Mutations are particularly simple in the two-dimensional case. In this setting, a given primitive $w\in M$ defines a non-trivial mutation of $P\subset N_{\mathbb{Q}}$ if and only if $w\in\{\overline{u}\mid u\in\mathrm{vert}\left({{P}^{\vee}}\right)\}\subset M$, where $\overline{u}\in M$ is the unique primitive lattice vector on the ray passing through $u$. Nontrivial factors $F\subset N_{\mathbb{Q}}$ are just line segments, so it suffices to restrict attention to those $F$ which have vertex set $\{\mathbf{0},f\}$, for some $f\in N$ with $w(f)=0$. The inner normal fan of any factor $F$ of $P$ with respect to a given $w$ is just the linear subspace of $M_{\mathbb{Q}}$ spanned by $w$. This divides $M_{\mathbb{Q}}$ into two chambers; the piecewise linear transformation $\varphi$ acts trivially in one of the chambers, and as $u\mapsto u-u(f)w$ in the other. ## 3. One-step mutations of triangles Set $N\cong\mathbb{Z}^{2}$ and let $P:=\mathrm{conv}\left\{{v_{0},v_{1},v_{2}}\right\}\subset N_{\mathbb{Q}}$ be a Fano triangle. Since $\mathbf{0}\in\mathrm{int}\left({P}\right)$ there exists a (unique) choice of coprime positive integers $\lambda_{0},\lambda_{1},\lambda_{2}\in\mathbb{Z}_{>0}$ with $\lambda_{0}v_{0}+\lambda_{1}v_{1}+\lambda_{2}v_{2}=\mathbf{0}$. The projective toric surface $X$ given by the spanning fan of $P$ has Picard rank $1$, and is called a _fake weighted projective plane_ with weights $(\lambda_{0},\lambda_{1},\lambda_{2})$; $X$ is the quotient of $\mathbb{P}(\lambda_{0},\lambda_{1},\lambda_{2})$ by the action of a finite group of order $\mathrm{mult}\left({X}\right)$ acting free in codimension one [2, 6]. Remark 3.1.: Since the vertices of $P$ are primitive, the weights $(\lambda_{0},\lambda_{1},\lambda_{2})$ are _well-formed_: that is, $\mathrm{gcd}\left\{{\lambda_{i},\lambda_{j}}\right\}=1$, $i\neq j$. In this paper we will always require that weights are well-formed. Definition 3.2.: We say that a fake weighted projective plane $Y$ with defining Fano triangle $Q\subset N_{\mathbb{Q}}$ is obtained from $X$ by a _one-step mutation_ if $Q\cong\mathrm{mut}_{w}(P,F)$ for some choice of $w$ and factor $F$. ### One-step mutations in $M_{\mathbb{Q}}$ and weights <figure><img src="content_image/1302.1152/x5.png"><figcaption>Figure 2. A one-step mutation, depicted in MQ, of the triangle conv{u0,u1,u2}to the triangle conv{u2,u3,u4}.</figcaption></figure> Proposition 3.3.: _Let $X$ be a fake weighted projective plane with weights $(\lambda_{0},\lambda_{1},\lambda_{2})$. Suppose there exists a one-step mutation to a fake weighted projective plane $Y$. Then, up to relabelling, $\lambda_{0}\divides(\lambda_{1}+\lambda_{2})^{2}$ and $Y$ has weights_ \[\left(\lambda_{1},\lambda_{2},\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0} }\right).\] Proof.: Consider a lattice triangle $T_{1}\subset N_{\mathbb{Q}}$, $\mathbf{0}\in\mathrm{int}\left({T_{1}}\right)$, and suppose that there exists width vector $w\in M$ and factor $F\subset N_{\mathbb{Q}}$, $w(F)=0$, such that the mutation $T_{2}=\mathrm{mut}_{w}(T_{1},F)$ is also a triangle. Without loss of generality we can assume that $w=(0,1)\in M$ and $F=\mathrm{conv}\left\{{\mathbf{0},(a,0)}\right\}$ for some $a\in\mathbb{Z}_{>0}$. The mutation corresponds to a piecewise linear action on $M_{\mathbb{Q}}$ via $u\mapsto uM_{\sigma}$ given by \[M_{\sigma}=\left\{\begin{array}[]{ll}\small\begin{pmatrix}1&0\\ 0&1\end{pmatrix}&\text{ if }u\in M^{+},\\ \small\begin{pmatrix}1&-a\\ 0&1\end{pmatrix}&\text{ otherwise,}\end{array}\right.\] where $M^{+}$ is the half-space $\{(\alpha,\beta)\in M_{\mathbb{Q}}\mid\alpha>0\}$. Let ${T_{1}}^{\vee}=\mathrm{conv}\left\{{u_{0},u_{1},u_{2}}\right\}\subset M_{ \mathbb{Q}}$ be the (possibly rational) triangle dual to $T_{1}$, where $u_{2}\in M^{+}$ and so is fixed under the action of the mutation, and $u_{1}\in M^{-}:=\{(\alpha,\beta)\in M_{\mathbb{Q}}\mid\alpha<0\}$. Since ${T_{2}}^{\vee}\subset M_{\mathbb{Q}}$ is also a triangle, the only possibility is that $u_{0}$ lies on the line $\left<{w}\right>:=\{\gamma w\in M_{\mathbb{Q}}\mid\gamma\in\mathbb{Q}\}$, ${T_{2}}^{\vee}=\mathrm{conv}\left\{{u_{2},u_{3},u_{4}}\right\}$ where $u_{0}$ is contained in the line segment $\overline{{u_{2}}{u_{4}}}$ joining $u_{2}$ and $u_{4}$, and $u_{3}$ is contained in the line segment $\overline{{u_{1}}{u_{2}}}$. This situation is illustrated in Figure 2. Since $\mathbf{0}\in{T_{1}}^{\vee}$ there exist unique weights $(\lambda_{0},\lambda_{1},\lambda_{2})\in\mathbb{Z}_{>0}^{3}$, $\mathrm{gcd}\left\{{\lambda_{0},\lambda_{1},\lambda_{2}}\right\}=1$, such that (3.1) \[\lambda_{0}u_{0}+\lambda_{1}u_{1}+\lambda_{2}u_{2}=\mathbf{0}.\] Since $u_{3}=(0,\beta_{3})\in\overline{{u_{1}}{u_{2}}}$ there exists some $0<\mu<1$ such that $\mu\alpha_{1}+(1-\mu)\alpha_{2}=0$. But $\lambda_{1}\alpha_{1}+\lambda_{2}\alpha_{2}=0$, hence \[\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}\alpha_{1}+\frac{\lambda_{2}}{ \lambda_{1}+\lambda_{2}}\alpha_{2}=0.\] By uniqueness of $\mu$, (3.2) \[u_{3}=\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}u_{1}+\frac{\lambda_{2}}{ \lambda_{1}+\lambda_{2}}u_{2}.\] Similarly, since $u_{0}=(0,\beta_{0})\in\overline{{u_{2}}{u_{4}}}$ there exists some $0<\nu<1$ such that $u_{0}=\nu u_{2}+(1-\nu)u_{4}$, giving \[u_{4}=\frac{1}{1-\nu}u_{0}-\frac{\nu}{1-\nu}u_{2}.\] Comparing coefficients we see that (3.3) \[\alpha_{1}=-\frac{\nu}{1-\nu}\alpha_{2}.\] But $u_{4}=u_{1}+\kappa u_{0}$ for some $\kappa>0$. Combining this with equation (3.1) we see that \[u_{4}=\frac{\lambda_{1}\kappa-\lambda_{0}}{\lambda_{1}}u_{0}-\frac{\lambda_{2} }{\lambda_{1}}u_{2}.\] Comparing coefficients, we obtain (3.4) \[\alpha_{1}=-\frac{\lambda_{2}}{\lambda_{1}}\alpha_{2}.\] Equating equations (3.3) and (3.4) gives (3.5) \[u_{4}=\frac{\lambda_{1}+\lambda_{2}}{\lambda_{1}}u_{0}-\frac{\lambda_{2}}{ \lambda_{1}}u_{2}.\] Notice that, since both $u_{0}$ and $u_{3}$ are contained in $\left<{w}\right>$, there exists some $\gamma>0$ such that $-\gamma u_{3}=u_{0}$. Substituting into equation (3.5) we have (3.6) \[\frac{\lambda_{2}}{\lambda_{1}}u_{2}+u_{4}+\gamma^{\prime}u_{3}=\mathbf{0}\] where $\gamma^{\prime}=\gamma(\lambda_{1}+\lambda_{2})/\lambda_{1}>0$. Substituting in equation (3.2) we obtain \[\frac{\lambda_{2}}{\lambda_{1}}u_{2}+u_{4}+\frac{\gamma^{\prime}\lambda_{1}}{ \lambda_{1}+\lambda_{2}}u_{1}+\frac{\gamma^{\prime}\lambda_{2}}{\lambda_{1}+ \lambda_{2}}u_{2}=\mathbf{0}.\] Using equation (3.5) to rewrite the first two terms and clearing denominators gives: (3.7) \[(\lambda_{1}+\lambda_{2})^{2}u_{0}+\gamma^{\prime}\lambda_{1}^{2}u_{1}+\gamma^ {\prime}\lambda_{1}\lambda_{2}u_{2}=\mathbf{0}.\] Set $h:=\lambda_{0}+\lambda_{1}+\lambda_{2}$ and $\Gamma:=(\lambda_{1}+\lambda_{2})^{2}+\gamma^{\prime}\lambda_{1}^{2}+\gamma^{ \prime}\lambda_{1}\lambda_{2}$. By comparing equations (3.1) and (3.7), uniqueness of barycentric coordinates gives: \[h(\lambda_{1}+\lambda_{2})^{2} =\Gamma\lambda_{0},\] \[h\gamma^{\prime}\lambda_{1}^{2} =\Gamma\lambda_{1},\] \[h\gamma^{\prime}\lambda_{1}\lambda_{2} =\Gamma\lambda_{2}.\] In particular, \[\gamma^{\prime}=\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0}\lambda_{1}}.\] Substituting this expression for $\gamma^{\prime}$ back into equation (3.6) gives (3.8) \[\lambda_{0}\lambda_{2}u_{2}+(\lambda_{1}+\lambda_{2})^{2}u_{3}+\lambda_{0} \lambda_{1}u_{4}=\mathbf{0}.\] Finally, we consider the situation where $T_{1}\subset N_{\mathbb{Q}}$ is the triangle associated with a fake weighted projective plane with weights $(\lambda_{0},\lambda_{1},\lambda_{2})$, and assume that there exists a one-step mutation to some triangle $T_{2}\subset N_{\mathbb{Q}}$. If $\lambda_{0}$ does not divide $(\lambda_{1}+\lambda_{2})^{2}$, then by equation (3.8) the associated weights are \[\left(\lambda_{0}\lambda_{1},\lambda_{0}\lambda_{2},(\lambda_{1}+\lambda_{2})^ {2}\right),\] and these fail to be well-formed when $\lambda_{0}>1$. Therefore, we must have $\lambda_{0}\divides(\lambda_{1}+\lambda_{2})^{2}$, giving weights \[\left(\lambda_{1},\lambda_{2},\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0} }\right).\] ∎ Remark 3.4.: Let $(\lambda_{0},\lambda_{1},\lambda_{2})$ be well-formed weights such that $\lambda_{0}\divides(\lambda_{1}+\lambda_{2})^{2}$. Then \[\left(\lambda_{1},\lambda_{2},\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0} }\right)\] are also well-formed. For suppose there exists some prime $p$ such that \[p\divides\lambda_{1}\quad\text{ and }\quad p\divides\frac{(\lambda_{1}+\lambda _{2})^{2}}{\lambda_{0}}.\] Then $p\divides\lambda_{2}^{2}$ and so $p\divides\lambda_{2}$. But this contradicts $(\lambda_{0},\lambda_{1},\lambda_{2})$ being well-formed. Example 3.5.: There exists no one-step mutation from $\mathbb{P}(3,5,11)$ to any other weighted projective space, since $3\notdivides(5+11)^{2}$, $5\notdivides(3+11)^{2}$, and $11\notdivides(3+5)^{2}$. Example 3.6.: The requirement that $\lambda_{0}\divides(\lambda_{1}+\lambda_{2})^{2}$ in Proposition 3.3 is necessary but not sufficient. For example, consider the triangle $T=\mathrm{conv}\left\{{(10,-7),(-5,2),(0,1)}\right\}\subset N_{\mathbb{Q}}$. This has weights $(1,2,3)$, however there exist no one-step mutations from $T$. ### One-step mutations in $N_{\mathbb{Q}}$ and T-singularities Definition 3.7 ([8, Definition 3.7]).: A quotient surface singularity is called a _T-singularity_ if it admits a $\mathbb{Q}$-Gorenstein one-parameter smoothing. T-singularities include the du Val singularities $\frac{1}{r}(1,r-1)$, and are cyclic quotient singularities of the form $\frac{1}{nd^{2}}(1,dna-1)$, where $\mathrm{gcd}\left\{{d,a}\right\}=1$ [8, Proposition 3.10]. Lemma 3.8.: _An isolated quotient singularity $\frac{1}{r}(a,b)$ is a T-singularity if and only if $r\divides(a+b)^{2}$._ Proof.: We begin by noting that the condition that $r\divides(a+b)^{2}$ is independent of the choice of representation of $\frac{1}{r}(a,b)$. For let $c$ be any integer coprime to $r$. Then $r\divides(a+b)^{2}$ if and only if $r\divides c^{2}(a+b)^{2}=(ca+cb)^{2}$. Suppose we are given a T-singularity. Writing the singularity in the form $\frac{1}{nd^{2}}(1,dna-1)$ where $\mathrm{gcd}\left\{{d,a}\right\}=1$, we see that $nd^{2}\divides d^{2}n^{2}a^{2}$. Conversely consider the isolated quotient singularity $\frac{1}{r}(a,b)$. Since $a$ is invertible $\bmod\ r$, we can write this as $\frac{1}{r}(1,b^{\prime}-1)$, where $b^{\prime}\equiv ba^{-1}+1\ \left(\mathrm{mod}\ {r}\right)$. Write $r=nd^{2}$ where $n$ is square-free. Since $nd^{2}\divides b^{\prime 2}$ by assumption, we see that $nd\divides b^{\prime}$. In particular, we can express our singularity in the form $\frac{1}{nd^{2}}(1,dn\alpha-1)$ for some $\alpha\in\mathbb{Z}_{>0}$. Finally, we note that this really is a T-singularity: if $\mathrm{gcd}\left\{{d,\alpha}\right\}=c$ then we can absorb this factor into $n^{\prime}=nc^{2}$ whilst rescaling $d^{\prime}=d/c$ and $\alpha^{\prime}=\alpha/c$. ∎ Proposition 3.9.: _Let $X$ be a fake weighted projective plane corresponding to a triangle $T\subset N_{\mathbb{Q}}$, and suppose that the cone $C$ spanned by an edge $E$ of $T$ corresponds to a $\frac{1}{r}(a,b)$ singularity. There exists a one-step mutation to a fake weighted projective plane $Y$ given by $\mathrm{mut}_{w}(T,F)$ with $w(E)={h_{\mathrm{min}}}$ if and only if $\frac{1}{r}(a,b)$ is a T-singularity._ Proof.: Let $X$ correspond to the lattice triangle $T=\mathrm{conv}\left\{{v_{1},v_{2},v_{3}}\right\}\subset N_{\mathbb{Q}}$, where $\mathbf{0}\in\mathrm{int}\left({T}\right)$ and the vertices $\mathrm{vert}\left({T}\right)\subset N$ are all primitive. Consider the cone $C=\mathrm{cone}\left\{{v_{1},v_{2}}\right\}$ spanned by the edge $E=\overline{{v_{1}}{v_{2}}}$; this is an isolated quotient singularity (possibly smooth), so is of the form $\frac{1}{r}(a,b)$ for some $r,a,b\in\mathbb{Z}_{>0}$, $\mathrm{gcd}\left\{{r,a}\right\}=\mathrm{gcd}\left\{{r,b}\right\}=1$. Let $w\in M$ be a primitive lattice point such that $w(v_{1})=w(v_{2})=h$ for some $h<0$. Then, up to translation, there exists a factor $F\subset N_{\mathbb{Q}}$, $w(F)=0$, such that $T^{\prime}:=\mathrm{mut}_{w}(T,F)$ is a triangle if and only if $v_{1}+(-h)F=E$. Equivalently, if and only if $h\divides\left\|{E\cap N}\right\|-1$. Finally, we express the values of $h$ and $\left\|{E\cap N}\right\|-1$ in terms of the singularity $\frac{1}{r}(a,b)$. Set $k:=\mathrm{gcd}\left\{{r,a+b}\right\}$. Then the height $h=-r/k$, and the number of points on the edge $E$ is given by \[\left\|{\{m\mid m\in\{0,\ldots,r\}\text{ and }(a+b)m\equiv 0\ \left(\mathrm{mod }\ {r}\right)\}}\right\|=1+\frac{r}{h}=1+k.\] Hence $h\divides\left\|{E\cap N}\right\|-1$ if and only if $r/k\divides k$. But $r/k\divides k$ if and only if $r\divides\mathrm{gcd}\left\{{r,a+b}\right\}^{2}=\mathrm{gcd}\left\{{r^{2},(a+b )^{2}}\right\}$, and $r\divides\mathrm{gcd}\left\{{r^{2},(a+b)^{2}}\right\}$ if and only if $r\divides(a+b)^{2}$. The result follows by Lemma 3.8. ∎ Example 3.10.: Returning to Example 3.6, we see that the corresponding fake weighted projective space $X$ is a quotient of $\mathbb{P}(1,2,3)$ with $\mathrm{mult}\left({X}\right)=5$. The three singularities are $\frac{1}{5}(1,3)$, $\frac{1}{10}(1,3)$, and $\frac{1}{15}(1,11)$, none of which is a T-singularity. When $X$ is a weighted projective plane, Proposition 3.9 tells us that the condition that $\lambda_{0}\divides(\lambda_{1}+\lambda_{2})^{2}$ in Proposition 3.3 is both necessary and sufficient. ### One-step mutations and Diophantine equations Lemma 3.11.: _Let $(\lambda_{0},\lambda_{1},\lambda_{2})\in\mathbb{Z}_{>0}^{3}$ with $d=\mathrm{gcd}\left\{{\lambda_{0},\lambda_{1},\lambda_{2}}\right\}$. Write:_ 1. $\lambda_{i}=dc_{i}a_{i}^{2}$_, where_ $a_{i},c_{i}\in\mathbb{Z}_{>0}$ _and_ $c_{i}$ _is square-free;_ 2. $(\lambda_{0}+\lambda_{1}+\lambda_{2})^{2}/(\lambda_{0}\lambda_{1}\lambda_{2})= m^{2}/(rk^{2})$_, where_ $m,k,r\in\mathbb{Z}_{>0}$ _and_ $r$ _is square-free;_ 3. $c_{0}c_{1}c_{2}=gS^{2}$ _and_ $dr=hT^{2}$_, where_ $g,h,S,T\in\mathbb{Z}_{>0}$ _and both_ $g$ _and_ $h$ _are square-free._ _Then $(da_{0},da_{1},da_{2})$ is a solution to the Diophantine equation_ (3.9) \[Smx_{0}x_{1}x_{2}=Tk(c_{0}x_{0}^{2}+c_{1}x_{1}^{2}+c_{2}x_{2}^{2}).\] Proof.: By substituting expressions (1) and (3) into (2) we obtain \[gS^{2}m^{2}(da_{0})^{2}(da_{1})^{2}(da_{2})^{2}=hT^{2}k^{2}\left(c_{0}(da_{0}) ^{2}+c_{1}(da_{1})^{2}+c_{2}(da_{2})^{2}\right)^{2}.\] Comparing square-free parts, we conclude that $g=h$. Cancelling and taking square-roots on both sides establishes the result. ∎ Since the weights are assumed to be well-formed, $d=S=T=1$ and equation (3.9) becomes (3.10) \[mx_{0}x_{1}x_{2}=k(c_{0}x_{0}^{2}+c_{1}x_{1}^{2}+c_{2}x_{2}^{2}).\] Suppose that $(a_{0},a_{1},a_{2})$ is a positive integral solution to equation (3.10), so that $\lambda_{i}=c_{i}a_{i}^{2}$. The expression (3.11) \[\frac{(\lambda_{0}+\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0}\lambda_{1}\lambda _{2}}\] occurring in Lemma 3.11 is equal to the degree of $\mathbb{P}(\lambda_{0},\lambda_{1},\lambda_{2})$. More generally if $X$ is a fake weighted projective plane with weights $(\lambda_{0},\lambda_{1},\lambda_{2})$ then (3.11) is equal to $\mathrm{mult}\left({X}\right)(-K_{X})^{2}$. Proposition 3.12.: _Let $X$ be a fake weighted projective plane and suppose that there exists a one-step mutation to a fake weighted projective plane $Y$. Then the weights of $X$ and $Y$ give solutions to the same Diophantine equation (3.10). In particular, $\mathrm{mult}\left({X}\right)=\mathrm{mult}\left({Y}\right)$._ Proof.: With notation as in Lemma 3.11, we can write the weights $(\lambda_{0},\lambda_{1},\lambda_{2})$ of $X$ in the form $\lambda_{i}=c_{i}a_{i}^{2}$, where the $c_{i}$ are square-free positive integers. From Proposition 3.3 we know that $Y$ has weights \[\left(\lambda_{1},\lambda_{2},\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0} }\right)=\left(c_{1}a_{1}^{2},c_{2}a_{2}^{2},\frac{(c_{1}a_{1}^{2}+c_{2}a_{2}^ {2})^{2}}{c_{0}a_{0}^{2}}\right).\] The final weight is an integer; in particular, it has square-free part $c_{0}$. Thus the $c_{i}$ are invariant under mutation. Furthermore, \[\frac{\left(\lambda_{1}+\lambda_{2}+\frac{(\lambda_{1}+\lambda_{2 })^{2}}{\lambda_{0}}\right)^{2}}{\lambda_{1}\cdot\lambda_{2}\cdot\frac{( \lambda_{1}+\lambda_{2})^{2}}{\lambda_{0}}} =\frac{\left(\lambda_{0}\lambda_{1}+\lambda_{0}\lambda_{2}+( \lambda_{1}+\lambda_{2})^{2}\right)^{2}}{\lambda_{0}\lambda_{1}\lambda_{2}( \lambda_{1}+\lambda_{2})^{2}}\] \[=\frac{(\lambda_{0}+\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0} \lambda_{1}\lambda_{2}}\] \[=\frac{m^{2}}{rk^{2}}\] and so the ratio $m/k$ is also preserved by mutation. Hence the weights of $X$ and of $Y$ both generate solutions to the same Diophantine equation (3.10). Finally we recall that degree is fixed under mutation, hence $(-K_{X})^{2}=(-K_{Y})^{2}$. But \[\frac{m^{2}}{rk^{2}}=\mathrm{mult}\left({X}\right)(-K_{X})^{2}=\mathrm{mult} \left({Y}\right)(-K_{Y})^{2}\] and so $\mathrm{mult}\left({X}\right)=\mathrm{mult}\left({Y}\right)$. ∎ By combining Propositions 3.3, 3.9, and 3.12 we obtain Proposition 1.1. Remark 3.13.: The weights of a fake weighted projective plane correspond to a solution $(a_{0},a_{1},a_{2})$ of equation (3.10). A one-step mutation gives a second solution via the transformation: \[(a_{0},a_{1},a_{2})\mapsto\left(\frac{m}{k}\frac{a_{1}a_{2}}{c_{0}}-a_{0},a_{1 },a_{2}\right).\] Example 3.14.: Consider $\mathbb{P}^{2}$. In this case $m/k=3$, $c_{0}=c_{1}=c_{2}=1$, and $(1,1,1)\in\mathbb{Z}_{>0}^{3}$ is a solution of (3.12) \[3x_{0}x_{1}x_{2}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}.\] Up to isomorphism, there is a single one-step mutation to $\mathbb{P}(1,1,4)$, giving a solution $(1,1,2)\in\mathbb{Z}_{>0}^{3}$ of equation (3.12). Proceeding in this fashion we obtain a graph of one-step mutations corresponding to solutions of (3.12), which we illustrate to a depth of five mutations: Definition 3.15.: The _height_ of the weights $(\lambda_{0},\lambda_{1},\lambda_{2})$ is given by the sum $h:=\lambda_{0}+\lambda_{1}+\lambda_{2}\in\mathbb{Z}_{>0}$. We call the weights _minimal_ if for any sequence of one-step mutations $(\lambda_{0},\lambda_{1},\lambda_{2})\mapsto\ldots\mapsto(\lambda^{\prime}_{0} ,\lambda^{\prime}_{1},\lambda^{\prime}_{2})$ we have that $h\leq h^{\prime}$. Lemma 3.16.: _Given weights $(\lambda_{0},\lambda_{1},\lambda_{2})$ at height $h$ there exists at most one one-step mutation such that $h^{\prime}\leq h$. Moreover, if $h^{\prime}=h$ then the weights are the same._ Proof.: Without loss of generality suppose we have two one-step mutations \[\left(\lambda_{1},\lambda_{2},\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0} }\right)\qquad\text{ and }\qquad\left(\lambda_{0},\frac{(\lambda_{0}+\lambda_{ 2})^{2}}{\lambda_{1}},\lambda_{2}\right)\] with respective heights $h^{\prime}$ and $h^{\prime\prime}$ such that $h^{\prime}\leq h$ and $h^{\prime\prime}\leq h$. Since $h^{\prime}\leq h$ we obtain $(\lambda_{1}+\lambda_{2})^{2}\leq\lambda_{0}^{2}$, and so (3.13) \[\lambda_{1}^{2}+\lambda_{2}^{2}<\lambda_{0}^{2}.\] From $h^{\prime\prime}\leq h$ we obtain (3.14) \[\lambda_{0}^{2}+\lambda_{2}^{2}<\lambda_{1}^{2}.\] Combining equations (3.13) and (3.14) gives a contradiction, hence there exists at most one one-step mutation such that $h^{\prime}\leq h$. If we suppose that $h^{\prime}=h$ then \[\frac{(\lambda_{1}+\lambda_{2})^{2}}{\lambda_{0}}=\lambda_{0}\] and equality of the weights is immediate. ∎ The height imposes a natural direction on the graph of all one-step mutations generated by the weight $(\lambda_{0},\lambda_{1},\lambda_{2})$. Lemma 3.16 tells us that this directed graph is a tree, with a uniquely defined minimal weight. ## 4. Example: An infinite number of minimal weights In this section we shall focus on the Diophantine equation (4.1) \[12x_{0}x_{1}x_{2}=3x_{0}^{2}+5x_{1}^{2}+7x_{2}^{2}.\] Any solution $(a_{0},a_{1},a_{2})$ such that $(3a_{0}^{2},5a_{1}^{2},7a_{2}^{2})$ is well-formed corresponds to weighted projective space $\mathbb{P}(3a_{0}^{2},5a_{1}^{2},7a_{2}^{2})$ of degree $144/105$. One possible such solution is $(2,1,1)$ giving $\mathbb{P}(12,5,7)$. Consider the graph $\mathcal{G}$ of all such solutions. Two solutions lie in the same component if and only if there exists a sequence of one-step mutations between the corresponding weighted projective planes. Furthermore, each component is a tree with unique minimal weight. We shall show that there exists an infinite number of components, and that every component contains at most two solutions; in fact the only component with a single solution is $(2,1,1)$. ### Coprime solutions give well-formed weights Let $(a_{0},a_{1},a_{2})$ be a solution of equation (4.1) such that $\mathrm{gcd}\left\{{a_{0},a_{1},a_{2}}\right\}=1$. Clearly this is a necessary condition for the corresponding weights $(3a_{0}^{2},5a_{1}^{2},7a_{2}^{2})$ to be well-formed. We shall show that it is sufficient. For suppose that there exists some prime $p$ such that $p\divides c_{i}a_{i}^{2}$ and $p\divides c_{j}a_{j}^{2}$, $i\neq j$. Since $p$ cannot simultaneously divide both $c_{i}$ and $c_{j}$, we have that $p$ must divide either $a_{i}$ or $a_{j}$. In particular, $p\divides 12a_{0}a_{1}a_{2}$ and so, by equation (4.1), $p$ divides the remaining weight $c_{k}a_{k}^{2}$. Similarly, since $p$ can divide at most one of $3$, $5$, and $7$ we see that $p^{2}\divides 12a_{0}a_{1}a_{2}$ and so $p^{2}$ divides each of the three weights. We conclude that $p\divides\mathrm{gcd}\left\{{a_{0},a_{1},a_{2}}\right\}$, contradicting coprimality. ### A necessary and sufficient condition for rational solutions when $a_{1}$ and $a_{2}$ are fixed Fix $a_{1},a_{2}\in\mathbb{Z}_{>0}$ and consider the quadratic (4.2) \[12xa_{1}a_{2}=3x^{2}+5a_{1}^{2}+7a_{2}^{2}.\] The discriminant is given by \[12^{2}a_{1}^{2}a_{2}^{2}-12(5a_{1}^{2}+7a_{2}^{2})=12\left(5a_{1}^{2}(a_{2}^{2 }-1)+7a_{2}^{2}(a_{1}^{2}-1)\right),\] which is always non-negative. The discriminant is zero only in the case $a_{1}=a_{2}=1$, corresponding to the solution $(2,1,1)$ of equation (4.1). Furthermore, we see that a rational solution to equation (4.2) exists if and only if (4.3) \[5a_{1}^{2}(a_{2}^{2}-1)+7a_{2}^{2}(a_{1}^{2}-1)=3N^{2},\qquad\text{ for some } N\in\mathbb{Z}_{>0}.\] ### Any rational solution is an integral solution Suppose that $\alpha,\beta\in\mathbb{R}$ are the two solutions of equation (4.2). We obtain: (4.4) \[\alpha+\beta =4a_{1}a_{2},\] (4.5) \[3\alpha\beta =5a_{1}^{2}+7a_{2}^{2}.\] In particular, since the right-hand side in each case is a strictly positive integer, we see that $\alpha,\beta>0$. Furthermore, $\alpha$ is rational if and only if $\beta$ is rational. Since we are only interested in rational solutions, we can assume that both $\alpha$ and $\beta$ are rational. Let us write \[\alpha=\frac{n_{1}}{m_{1}}\qquad\text{ and }\qquad\beta=\frac{n_{2}}{m_{2}},\] where the fractions are expressed in their reduced form, i.e. $\mathrm{gcd}\left\{{n_{i},m_{i}}\right\}=1$. Then (4.6) \[m_{1}m_{2} \divides 3n_{1}n_{2},\] (4.7) \[m_{1}m_{2} \divides n_{1}m_{2}+n_{2}m_{1}.\] By (4.7), $m_{2}\divides m_{1}$ and $m_{1}\divides m_{2}$, forcing $m_{1}=m_{2}$. Without loss of generality, from (4.6) we may assume that $m_{1}\divides 3n_{2}$ and $m_{2}\divides n_{1}$. But then $m_{1}\divides n_{1}$, forcing $m_{1}=m_{2}=1$. Hence $\alpha,\beta\in\mathbb{Z}_{>0}$. ### The values $a_{1}$ and $a_{2}$ are fixed under one-step mutations We now show that, given a solution $(a_{0},a_{1},a_{2})$ such that $\mathrm{gcd}\left\{{a_{0},a_{1},a_{2}}\right\}=1$, the values of $a_{1}$ and $a_{2}$ are fixed under one-step mutation. For suppose that (4.8) \[\frac{(3a_{0}^{2}+7a_{2}^{2})^{2}}{5a_{1}^{2}}\in\mathbb{Z}.\] Without loss of generality we may take $\alpha=a_{0}$. We see that $5\divides 3a_{0}^{2}+7a_{2}^{2}=3\alpha^{2}+3\alpha\beta-5a_{1}^{2}$ by (4.5), hence $5\divides 3\alpha(\alpha+\beta)=12a_{0}a_{1}a_{2}$ by (4.4). Since the weights are pairwise coprime, the only possibility is that $5\divides a_{1}$. Returning to equation (4.8) we see that $5^{2}\divides 3a_{0}^{2}+7a_{2}^{2}$, and proceeding as before we find that $5^{2}\divides a_{1}$. Clearly we can repeat this process an arbitrary number of times, increasing the power of $5$ at each step. This is a contradiction. The case when \[\frac{(3a_{0}^{2}+5a_{1}^{2})^{2}}{7a_{2}^{2}}\in\mathbb{Z}\] is dealt with similarly. ### An infinite number of components Set $a_{1}=1$ in condition (4.3). The condition becomes $a_{2}^{2}-1=15M^{2}$, where $5M=N$. This is a Pell equation, and Emerson [3] has shown that there exists an infinite number of integer solutions given by a recurrence relation. In this case we see that $a_{2}^{(n)}$ and $M^{(n)}$ are generated by: \[a_{2}^{(0)} =1, M^{(0)} =0,\] \[a_{2}^{(1)} =4, M^{(1)} =1,\] \[a_{2}^{(n+1)} =8a_{2}^{(n)}-a_{2}^{(n-1)}, M^{(n+1)} =8M^{(n)}-M^{(n-1)}.\] Substituting these expressions back into the original quadratic (4.2) gives: \[a_{0}^{(n+1)}=2a_{2}^{(n)}\pm 5M^{(n)}.\] These solutions are coprime (since $a_{1}=1$) and so correspond to well-formed weights. We will focus on the smaller of the two solutions, corresponding to the minimum of the two weights. Substituting the expressions for $a_{2}^{(n)}$ and $M^{(n)}$ gives: \[a_{0}^{(n+1)} =2a_{2}^{(n+1)}-5M^{(n+1)}\] \[=8\left(2a_{2}^{(n)}-5M^{(n)}\right)-\left(2a_{2}^{(n-1)}-5M^{(n- 1)}\right)\] \[=8a_{0}^{(n)}-a_{0}^{(n-1)}.\] Hence we obtain the recurrence relation: \[a_{0}^{(0)} =2,\] \[a_{0}^{(1)} =3,\] \[a_{0}^{(n+1)} =8a_{0}^{(n)}-a_{0}^{(n-1)}.\] Remark 4.1.: If instead we insist that $a_{2}=1$, we obtain the Pell equation $a_{1}^{2}-1=21M^{2}$, where $7M=N$. In this case the recurrence relation is given by: \[a_{1}^{(0)} =1, M^{(0)} =0,\] \[a_{1}^{(1)} =55, M^{(1)} =12,\] \[a_{1}^{(n+1)} =110a_{1}^{(n)}-a_{1}^{(n-1)}, M^{(n+1)} =110M^{(n)}-M^{(n-1)}.\] Proceeding as above we find that \[a_{0}^{(0)} =2,\] \[a_{0}^{(1)} =26,\] \[a_{0}^{(n+1)} =110a_{0}^{(n)}-a_{0}^{(n-1)}.\] Hence we have a second infinite family of components of $\mathcal{G}$. Notice that these two families do not exhaust all the possibilities: for example, $a_{1}=5$, $a_{2}=4$ satisfies condition (4.3), giving the two solutions $(1,5,4)$ and $(79,5,4)$. ### Acknowledgments Our thanks to Tom Coates, Alessio Corti, and Song Sun for many useful conversations. The authors are supported by EPSRC grant EP/I008128/1. ## References * [ACGK12] Mohammad Akhtar, Tom Coates, Sergey Galkin, and Alexander M. Kasprzyk, _Minkowski polynomials and mutations_, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), 094, pp. 707. * [Buc08] Weronika Buczyńska, _Fake weighted projective spaces_, arXiv:0805.1211v1, May 2008. * [Eme69] Edgar I. Emerson, _Recurrent sequences in the equation $DQ^{2}=R^{2}+N$_, Fibonacci Quart. 7 (1969), no. 3, 231–242. * [HP10] Paul Hacking and Yuri Prokhorov, _Smoothable del Pezzo surfaces with quotient singularities_, Compos. Math. 146 (2010), no. 1, 169–192. * [Ilt12] Nathan Owen Ilten, _Mutations of Laurent polynomials and flat families with toric fibers_, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), 047, pp. 7. * [Kas09] Alexander M. Kasprzyk, _Bounds on fake weighted projective space_, Kodai Math. J. 32 (2009), 197–208. * [KN12] Alexander M. Kasprzyk and Benjamin Nill, _Fano polytopes_, Strings, Gauge Fields, and the Geometry Behind – the Legacy of Maximilian Kreuzer (Anton Rebhan, Ludmil Katzarkov, Johanna Knapp, Radoslav Rashkov, and Emanuel Scheidegger, eds.), World Scientific, 2012, pp. 349–364. * [KSB88] J. Kollár and N. I. Shepherd-Barron, _Threefolds and deformations of surface singularities_, Invent. Math. 91 (1988), no. 2, 299–338. * [Mar80] A. Markoff, _Sur les formes quadratiques binaires indéfinies_, Math. Ann. 17 (1880), no. 3, 379–399. * [Ros79] Gerhard Rosenberger, _Über die diophantische Gleichung $ax^{2}+by^{2}+cz^{2}=dxyz$_, J. Reine Angew. Math. 305 (1979), 122–125.
1509.03094	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31296, "num_imgs": 6, "llama3_tokens_count": 9293 }	[ "content_image/1509.03094/x1.png", "content_image/1509.03094/x2.png", "content_image/1509.03094/x3.png", "content_image/1509.03094/x5.png", "content_image/1509.03094/x8.png", "content_image/1509.03094/x9.png" ]	The Clusters AgeS Experiment (CASE). Variable stars in the field of the globular cluster M12${}^{\ast}$ J. K a l u z n y${}^{1}$†, I. B. T h o m p s o n${}^{2}$, W. Narloch${}^{1}$ W. P y c h${}^{1}$ and M. R o z y c z k a${}^{1}$${}^{1}$Nicolaus Copernicus Astronomical Center, ul. Bartycka 18, 00-716 Warsaw, Poland e-mail: (jka, wnarloch, pych, mnr)@camk.edu.pl ${}^{2}$The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA e-mail: ian@obs.carnegiescience.edu ABSTRACT The field of the globular cluster M12 (NGC 6218) was monitored between 1995 and 2009 in a search for variable stars. $BV$ light curves were obtained for 36 periodic or likely periodic variables. Thirty-four of these are new detections. Among the latter we identified 20 proper-motion members of the cluster: six detached or semi-detached eclipsing binaries, five contact binaries, five SX Phe pulsators, and three yellow stragglers. Two of the eclipsing binaries are located in the turnoff region, one on the lower main sequence and the remaining three among the blue stragglers. Two contact systems are blue stragglers, and the remaining three reside in the turnoff region. In the blue straggler region a total of 103 objects were found, of which 42 are proper motion members of M12, and another four are field stars. Forty-five of the remaining objects are located within two core radii from the center of the cluster, and as such they are likely genuine blue stragglers. We also report the discoveries of a radial color gradient of M12, and the shortest period among contact systems in globular clusters in general. _globular clusters: individual (M12) – stars: variables – stars: SX Phe – blue stragglers – binaries: eclipsing_ † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction M12 is a nearby ($(m-M)_{V}=14.01$ mag) globular cluster located at a high galactic latitude ($b=26.3$ deg) in a field of low reddening with $E(B-V)=0.19$ mag. According to Harris 1996 (2010 edition), its core radius, half-light radius, [Fe/H] index and radial velocity are equal to 0${}^{\prime}$.79, 1${}^{\prime}$.77, -1.37 and -41.4$\pm$0.2 km/s, respectively. Low reddening and relatively low concentration make it an attractive target for detailed studies with ground-based telescopes. The photometric survey presented here is a part of the CASE project (Kaluzny et al. 2005) conducted with telescopes of the Las Campanas Observatory, Chile. Early pre-CCD searches for variable stars in the field of M12 were summarized by Clement et al. (2001). These studies resulted in the detection of just one variable - a bright, long-period pulsator of W Vir type, which is overexposed in our frames. Based on CCD data, von Braun et al. (2002) found another two variables, both of W UMa type (we recovered these, and we retain their original names V1 and V2). To the best of our knowledge, no additional discoveries have been reported. In this contribution we present results of a long-term photometric survey conducted between 1995 and 2009. Section 2 contains a report on the observations and explains the methods used to calibrate the photometry. The detected variables are presented and discussed in Section 3. The paper is summarized in Section 4. ## 2 Observations Our paper is based on two sets of images. The first set was obtained using the 2.5-m du Pont telescope and the $2048\times 2048$ TEK5 CCD camera with a field of view of 8.84 arcmin on a side at a scale of 0.259 arcsec/pixel. Observations were conducted on 23 nights from April 26, 1999 to June 29, 2009. The same set of filters was used for all observations. For the analysis, we used 769 $V$-band images with seeing ranging from 0${}^{\prime\prime}$.64 to 2${}^{\prime\prime}$.07, and 216 $B$-band images with seeing ranging from 0${}^{\prime\prime}$.71 to 2${}^{\prime\prime}$.13. The median value of the seeing was 1${}^{\prime\prime}$.07 and 1${}^{\prime\prime}$.14 for $V$ and $B$, respectively. Additionally, a few $U$-band frames were taken in April/May 2001 at an average seeing of 1${}^{\prime\prime}$.25. The second set of images was obtained with the 1.0-m Swope telescope using the $2048\times 3150$ SITE3 camera. The field of view was $14.8\times 22.8$ arcmin${}^{2}$ at a scale of 0.435 arcsec/pixel. About 50% of the images were taken with a subraster providing a field of view of $14.8\times 14.8$ arcmin${}^{2}$. Observations were conducted on 57 nights from April 15, 1999 to June 08, 2008. Again, the same filters were used for all observations. For the analysis, we used 978 $V$-band images with seeing ranging from 0${}^{\prime\prime}$.95 to 2${}^{\prime\prime}$.31 and 168 $B$-band images with seeing ranging from 1${}^{\prime\prime}$.03 to 2${}^{\prime\prime}$.15. The median value of the seeing was 1${}^{\prime\prime}$.37 and 1${}^{\prime\prime}$.45 for $V$ and $B$, respectively. The photometry was performed using an image subtraction technique. The du Pont data were reduced with a modified version of the ISIS V2.1 package (Alard 2000), whereas for the frames obtained with the Swope telescope the DIAPL package¹ was used. For each set and each filter, a reference image was constructed by combining several high quality frames. Daophot, Allstar and Daogrow codes (Stetson 1987, 1990) were used to extract the profile photometry, and to derive aperture corrections for the reference images. Additionally, profile photometry was extracted for individual images from the du Pont telescope. This allowed us to obtain useful measurements for stars which were overexposed on the reference images. Also, profile photometry enabled an unambiguous identification of variable stars in crowded fields, which is sometimes problematic when image subtraction only is used. We attempted to resolve numerous blends in the Swope data for the central part of M12 by using star positions from du Pont photometry. In most cases this approach proved to be successful. The accuracy of the du Pont photometry is illustrated in Fig. 1, in which the standard deviation of the photometric measurements is plotted as a function of the average magnitude in $V$. [FOOTNOTE:1][ENDFOOTNOTE] ### Calibration The photometry collected with the du Pont telescope was transformed to the standard $UBV$ system based on observations of stars from Landolt fields (Landolt 1992). On the night of May 29, 2001 we observed 35 stars from five such fields. These data were used to find the coefficients of linear transformation from the instrumental system to the standard one. Residual differences between the standard and recovered magnitudes and colors amounted to 0.008, 0.007 and 0.009 mag for $V$, $B$ and $B-V$, respectively. The residuals did not show any systematic dependence on the color index. Transformations for the photometry obtained with the Swope telescope were based on the calibrated data from the du Pont telescope. Linear transformations proved to be entirely adequate. Fig. 2, based on reference images, shows the color-magnitude diagram (CMD) of the observed fields, illustrating the range of stellar population examined for variability (stars with formal error in $V$ larger than 0.05 mag and formal error in $B-V$ larger than 0.1 mag are not shown). The contamination of the cluster by field interlopers is rather low, but clearly increases with the larger field of view of the Swope data. Full sets of photometry can be downloaded from the CASE archive.² [FOOTNOTE:2][ENDFOOTNOTE] ### Search for variables The search for variable stars was conducted using the AOV and AOVTRANS algorithms implemented in the TATRY code (Schwarzenberg-Czerny 1996, Schwarzenberg-Czerny & Beaulieu 2006). We examined the du Pont light curves of 27586 stars with $V<21.5$ mag and the Swope light curves of 23265 stars with $V<20$ mag. The limits of detectable variability depended on the accuracy of photometric measurements, which for the du Pont data decreased from 3 mmag at $V=16$ mag to 30 mmag at $V=20$ mag and 100 mmag at $V=21.5$ mag (Fig. 1). For the Swope data the accuracy decreased from 5 mmag at $V=15$ mag to 50 mmag at $V=19$ mag and 100 mmag at $V=20$ mag. ## 3 The variables We detected 33 certain and three suspect variables, of which 20 have photometry from both telescopes. Only two of these had been known previously, the two W UMa systems discovered by von Braun et al. (2002). Fig. 3 presents finding charts for all 36 variables. Basic properties of these variables are listed in Table 1. The equatorial coordinates in Table 1 conform to the UCAC4 system (Zacharias et al. 2013), and are accurate to about 0${}^{\prime\prime}$.2. We checked that none of our variables coincides with any X-ray source from the list of Lu et al. (2009) or with the UV source discovered by Schiavon et al. (2012). The $V$-magnitudes listed in Table 1 correspond to the maximum light in the case of eclipsing binaries, while for the remaining variables average magnitudes are provided. For each variable the $B-V$ color is given, followed by the amplitude in the $V$-band. Periods of variability were found for all stars except two eclipsing binaries, for which parts of single eclipses only were observed. Some light curves show phase shifts and/or change shape from season to season. In these cases we give periods obtained for the indicated season. The last column of Table 1 gives the membership status based on proper motions (PM) taken from Zloczewski et al. (2012) and Narloch et al. (in preparation). Phased light curves of the variables from Table 1 are presented in Figs. 4a, 4b and 4c. A CMD of the cluster with the locations of the variables is shown in Fig. 5. This diagram is based on the du Pont photometry, and it only includes stars classified by Zloczewski et al (2012) as PM members of M12. Variables that are PM members of M12 are labelled in red, those with PM indicating that they are field objects in blue, and those for which the PM data are missing or ambiguous in black. The exception is V11 for which we have no PM data. We attribute to it a legitimate cluster membership based on our as yet unpublished radial velocity measurements. ### Detached eclipsing binaries We detected 11 detached eclipsing binaries, of which six are proper motion members of the cluster. Systems V10 and V11 are located at the turnoff region. With orbital periods amounting to 4.6 d and 5.2 d, respectively, they are interesting targets for a detailed follow up study aiming at the determination of their absolute parameters, and the age and distance of M12. We are presently conducting such an analysis for V11. The systemic velocity of this binary, equal to -44.4 km s${}^{-1}$, differs by less than 10% from the systemic velocity of M12 listed by Harris (1996, 2010 edition). This, together with its location at a distance of $15^{\prime\prime}$ (i.e. 0.32 core radii) from the center of the cluster leaves little doubt about its membership. It is somewhat surprising that V11 has a markedly eccentric orbit ($\epsilon\approx 0.1$) despite a relatively short period. At an age above 13 Gyr (Dotter et al. 2010), it should have been fully circularized by tidal friction (Mazeh 2008; Mathieu et al. 2004). Since we found no indication for a third body in this system, we speculate that the orbit of V11 was distorted during the last few Gyr as a result of a close encounter. The light curve of V10 shows two partial eclipses of similar depth ($\sim 0.35$ mag in $V$). As a result, the photometric solution is likely to be degenerate, allowing for a broad range of relative radii. Such a degeneracy may be overcome by the determination of the light ratio from spectra, but this will be difficult given the faintness of the system. A much deeper eclipse ($\sim 0.95$ mag in $V$) was partly observed in V19. As this binary is still fainter than V10, a really large observational effort would be required to collect enough data for an accurate analysis. The same argument concerns V20, for which an eclipse at least 0.6 mag deep in $V$ was partly observed. The PM-membership is unclear for this binary, however its location on the main sequence of the cluster and at the edge of the cluster’s core (62${}^{\prime\prime}$ from the center) both suggest that it does belong to M12. A potentially very interesting object is the eclipsing binary V16. We marked it as a nonmember in Fig. 5, however it almost meets the membership criterion employed by Zloczewski et al. (2012). It is located on the lower red giant branch of M12, 3${}^{\prime}$.8 (i.e. $\sim$2 half-light radii) away from the center of the cluster. Since we observed only three largely incomplete eclipses, the period of 2.5 d given in Table 1 is a tentative one only, and the eccentricity visible in Fig. 4b may be spurious. If this bright and well-isolated system turns out to be a member of M12, it will provide tight limits for age and distance of the cluster. V18 seems to be even more promising: based on its location on the red giant branch one might expect it to deliver more precise data than V16 which is by $\sim$1 mag fainter. Unfortunately, the proper motion of this system proves that we are dealing with a nearby binary which quickly ($\sim$14 mas yr${}^{-1}$) moves across the M12 field. The PM-membership status of V17 is ambiguous, but its location far to the right of the unevolved main sequence of M12 indicates that it is a field object - probably a pair of nearby red dwarfs. Variables V12, V13, V14, V15 are located in the blue straggler region between the turnoff and the horizontal branch. The first three of these are cluster members with periods of 1.03, 0.73 and 0.46 d, respectively, and light curves ranging from EA with proximity effect in V12 to EB-like in V14. Well defined moments of contact, together with straight ingress and egress branches of the primary minimum, suggest that they are detached rather than semi-detached. The PM-data for V15 are ambiguous: Zloczewski et al (2012) classify it as a probable member, while Narloch et al. (in preparation) as a nonmember. However, both its position on the CMD and its location in the field of view (24${}^{\prime\prime}$, i.e. 0.3 core radii from the center of M12) suggest that this system also belongs to the cluster. An additional support for the membership of V15 comes from its nature. Short period (0.54 d), moderate proximity effect and narrow total eclipses with different depths indicate a system with low mass ratio and small secondary which is much hotter than the primary (thus, in the present configuration of the system the main minimum is due to the eclipse of the secondary). Preliminary modeling with the PHOEBE facility (Prša and Zwitter 2005) yields $q\approx 0.1$, $T_{p}\approx 7500$ K, $T_{s}\approx 12000$ K, and $R_{s}/R_{p}\approx 0.15$. Apparently, the secondary has lost most of its H-rich envelope to the primary, thus conforming to one of the scenarios of the origin of blue straggler stars. The secondary minima of V12, V13 and V14 are much shallower than the primary minima, likely because of low mass ratios. We suggest that these too have experienced significant mass-transfer episodes. A detailed study of the whole quartet would certainly deliver valuable information concerning the nature and evolution of blue straggler stars. ### Contact binaries We identified nine variables with W UMa-type lightcurves, of which four are PM-members of the cluster, and another three are field interlopers. PM data for the remaining two systems are absent or not accurate enough to evaluate membership probability. Small amplitudes of V08 and V09 suggest that we may deal with elliptical variables rather than genuine contact binaries. Spectroscopical data are needed to resolve this ambiguity. Kaluzny et al. (2014) found a general paucity of contact systems on the unevolved main sequences of globular clusters. M12 perfectly conforms to this rule: PM-members V01, V03 and V06 and the suspected PM-member V05 reside in the turnoff region while another PM-member, V08, is a blue straggler. Thus we add to the growing evidence that, at least in globular clusters, the principal factor enabling contact systems to form from close but detached binaries is nuclear evolution: a contact configuration is achieved once the more massive component starts to expand quickly at the turnoff. Apparently, nuclear evolution is more important in this respect than the frequently invoked magnetic breaking; see e.g. Stepien & Gazeas (2012) and references therein. The binary V09 is a yellow straggler candidate. Based on the available data, its PM-membership cannot be firmly established. The empirical calibration of Rucinski (2000) yields a distance modulus by 0.7 mag smaller than that given by Harris (1996, 2010 edition)for M12. Given the large spread of the calibration, this is also an ambiguous result. However, V09 resides only 42${}^{\prime\prime}$ away (0.6 core radii) from the center of M12, and it may well be a cluster member. Since yellow stragglers are rather rare objects, it deserves further study aimed at clarifying its membership and evolutionary status. We note in passing that the periods of all abovementioned W UMa stars except V08 and V09 are shorter than 0.26 d, i.e. they are exceptionally short even for contact systems in globular clusters (Rucinski 2008). Among these, V03 with $P=0.210636$ d seems to be the new record holder. ### Variable stars among blue stragglers The CMD based on the du Pont data contains 103 candidate blue stragglers with $16.0<V<17.7$ and $0.23<B-V<0.70$. Of these 42 are proper motion members of M12, four are field stars, and for 57 objects no PM data are available. Of the latter 45 are located within two core radii from the center of the cluster and are likely members. In addition to the five eclipsing binaries described in Subsections 3.1 and 3.2, nine variables were found among the stragglers, seven of which are confirmed and another two suspected members of the cluster. Five of these are SX Phe pulsators with periods ranging from 0.019 to 0.049 d. The star with the longest period, V25, has the largest amplitude ($\sim$0.2 mag in $V$) and a light curve characteristic of multimodal pulsations. Unfortunately, the photometry is not good enough to enable a period analysis. Stars V30 – V33 exhibit more or less sinusoidal luminosity variations whose nature is difficult to establish. Doubling their periods makes their light curves resemble those of W UMa variables. However in that case the light curves of V30 and V31 become less regular, whereas for V32 and V33 the periods themselves grow too long for contact binaries (Rucinski 2007). ### Remaining objects V26, V29 and V36 are members of M12 located in the yellow straggler region, and as such they should be paid particular attention in future studies. V26 exhibits sinusoidal variations with $P=1.84$ d whose phase and ampliture vary from season to season. Fig. 4b shows data from our best season (2001), in which almost 650 frames were collected during 16 nights. V29 showed clear sinusoidal variation in 2001 only, with two possible periods of 0.647 and 1.847 d. V36 is a suspected variable with a very noisy light curve and amplitude of only $\sim 0.02$ mag. Another M12 member, V28, is located slightly below the asymptotic giant branch. Regular variations, perhaps the due to spots on the surface of a faint companion, were observed in 2001 only. The field object V27 is a flare star exhibiting sinusoidal variations with $P=11.4$ d whose phase amplitude and average magnitude vary from season to season. Before 2001 the star faded from 17.75 mag to 18.15 mag in $V$. On June 3, 2001 a complete flare was observed: the brightness smoothly increased by 0.2 mag in 1.2 h, and within the next 4 h it returned to the pre-flare level. Five days later regular variations were again dominating, with mean brightness 17.9 mag, amplitude $\sim 0.3$ mag, and the same period as before the flare. The flare itself is visible in Fig. 4b as a thin spike at phase 0.4. The nature of the periodic variation is unclear - spectroscopic observations are necessary to discriminate between rotation and orbital motion of V27. The membership of suspected variables V34 and V35 is unclear. Both these objects seem to exhibit roughly sinusoidal variations with $P=0.35$ d and $P=0.96$ d, respectively. ## 4 Summary We have conducted an extensive photometric survey of the globular cluster M12 in a search for variable stars. A total of 31 variables plus three suspected variables were discovered, and multiseasonal light curves were compiled for another two W UMa type eclipsing binaries that had been known before. For all variables periods accurate to 0.00001 – 0.001 d were obtained. Seven eclipsing binaries and five pulsating stars (all of them PM-members of the cluster) were found in the blue-straggler region. Four of the binaries are likely to have low mass ratios, most probably due to mass-transfer episodes. Their detailed analysis should deliver very valuable information concerning nature and evolution of blue stragglers. Yellow stragglers are represented in our sample by three confirmed and one suspected member of the cluster. The latter is a W UMa system and the nature of the former is unclear. Since yellow stragglers are even more rare and interesting than the blue stragglers, these two systems should be paid close attention during future observations. Two detached eclipsing binaries belonging to the cluster were identified in the turnoff region, and another one at the lower main sequence. A potentially very valuable discovery is the detached eclipsing binary V16 residing on the lower red giant branch. If this bright and well isolated object turns out to be a member of M12, it will provide tight limits on the age and distance of the cluster. Acknowledgements. JK, WN, WP and MR were partly supported by the grant DEC-2012/05/B/ST9/03931 from the Polish National Science Center. ## References * Alard, C. 2000, _Astron. Astrophys. Suppl. Ser._, 144, 363. * Clement, C. M., Muzzin, A., Dufton, Q., Ponnampalam, T., Wang, J. et al. 2001, _Astron. J._, 122, 2587. * Clement, C. M., Sawyer Hogg, H., and Yee, A. 1988, _Astron. J._, 96, 1642. * De Marchi, G., Pulone, L., and Paresce, F. 2007, _Astron. Astrophys._, 449, 161. * Dotter, A., Sarajedini, A., Anderson, J., Aparicio, A., Bedin, L. R. et al. 2010, _ApJ_, 708, 698. * Harris, W.E. 1996, _Astron. J._, 112, 1487. * Kaluzny, J., and Thompson, I. B. 2009, _Acta Astron._, 59, 273. * Kaluzny, J., Thompson, I. B., Krzeminski, W., Preston, G. W., Pych, W. et al. 2005, _Stellar Astrophysics with the World’s Largest Telescopes, AIP Conf. Proc._, 752, 70. * Kaluzny, J., Thompson, I. B., Krzeminski, W., and Zloczewski, K. 2010, _Acta Astron._, 60, 246. * Kaluzny, J., Thompson, I. B., Rozyczka, M., and Krzeminski, W. 2013, _Acta Astron._, 63, 181. * Kaluzny, J., Thompson, I. B., Rozyczka, M., Pych, W., and Narloch, W. 2014, _Acta Astron._, 64, 309. * Landolt, A. 1992, _Astron. J._, 104, 372. * Lu, T-N., Kong, A. K. H., Bassa, C., Verbunt, F., Lewin, W. H. G. et al. 2009, _Astrophys. J._, 705, 175. * Mathieu, R. D., Meibom, S., and Dolan, C. J. 2004, _Astrophys. J._, 602, L121. * Mazeh, T. 2008, _EAS Publ. Series_, 29, 1. * Mazur, B., Kaluzny, J., and Krzeminski, W. 1999, _MNRAS_, 306, 727. * Prša, A., and Zwitter, T. 2005, _Astrophys. J._, 628, 426. * Roediger, J. C., Courteau, S., Graves, G., and Schiavon, R. P. 2014, _Astrophys. J. Suppl. Ser._, 210, 10. * Rucinski, S. M. 2000, _Astron. J._, 120, 319. * Rucinski, S. M. 2007, _MNRAS_, 382, 393. * Rucinski, S. M., and Pribulla, T. 2008, _MNRAS_, 388, 1831. * Schiavon, R. P., Dalessandro, E., Sohn, S. T., Rood, R. T., O’Connell, R. W. et al. 2012, _Astron. J._, 143, 121. * Schwarzenberg-Czerny A. 1996, _Astrophys. J. Letters_, 460, L107. * Schwarzenberg-Czerny A., and Beaulieu, J.-Ph. 2006, _MNRAS_, 365, 165. * Stepien, K., and Gazeas, K. 2012, _Acta Astron._, 62, 153. * Stetson P. B. 1987, _P.A.S.P._, 99, 191. * Stetson P. B. 1990, _P.A.S.P._, 102, 932. * Tokovinin, A., Thomas, S., Sterzik, M., and Udry, S. 2006, _Astron. Astrophys._, 450, 681. * von Braun, K., Mateo, M., Chiboucas, K., and Athey, A. 2002, _Astron. J._, 124, 2067. * Zacharias, N., Finch, C. T., Girard, T. M., Henden, A., Bartlett, J. L. et al. 2013, _Astron. J._, 145, 44. * Zloczewski, K., Kaluzny, J., Rozyczka, M., Krzeminski, W., and Mazur, B. 2012, _Acta Astron._, 62, 357. ## Appendix: Color gradient In some GC color gradients have been observed. Roediger et al. (2014) listed six such clusters. Since that sample does not include M12, we thought it worthwhile to search for radial variations of the colors in our data. The search was limited to du Pont frames, a few of which were taken in $U$ band. Only PM-members of M12 with $V<19.5$ mag and color errors smaller than 0.015 mag were included. The field of view was divided into three concentric subfields centered on the center of the cluster, with $0\leq r\leq 2^{\prime}.36$, $2^{\prime}.36<r<3^{\prime}.74$ and $r\geq 3^{\prime}.74$, respectively. Each subfield contained almost 500 stars. Weighted means of $\overline{B-V}$ and $\overline{U-V}$ colors were calculated separately for giants and HB stars $(V<17.5$ mag), subgiants $(17.5\leq V<18.0$ mag), and dwarfs $(V\geq 18$ mag). No clear dependence of $\overline{B-V}$ of $r$ was found in any of the luminosity classes. The same negative result was obtained for $\overline{U-V}$ for giants or subgiants. A clear, albeit weak, dependence emerged only in the case of the $U-V$ color for main sequence stars, with $\overline{U-V}$ increasing from $1.1002\pm 0.0009$ mag in the central circle through $1.1142\pm 0.0009$ mag in the intermediate ring to $1.1331\pm 0.0008$ mag in the outer region. This radial trend is convincingly illustrated by the histograms shown in Fig. 6. We attribute this result at least partly to the relative underabundance of faint red main-sequence stars discovered by De Marchi et al. (2007) near the cluster center. Id \| RA \| DEC \| V \| B−V \| ΔV \| Period \| Remarksa \| Memb ---\|---\|---\|---\|---\|---\|---\|---\|--- \| [deg] \| [deg] \| [mag] \| [mag] \| [mag] \| [d] \| \| V01 \| 251.84545 \| -1.92666 \| 18.93 \| 0.72 \| 0.25 \| 0.243185 \| EW \| Y V02 \| 251.88581 \| -2.05289 \| 18.31 \| 1.18 \| 0.12 \| 0.252125 \| EW \| N V03 \| 251.80287 \| -1.95726 \| 17.86 \| 0.72 \| 0.08 \| 0.210636 \| EW \| Y V04 \| 251.90221 \| -1.91481 \| 19.45 \| 1.18 \| 0.46 \| 0.233990 \| EW \| N V05 \| 251.79329 \| -1.93622 \| 18.71 \| 0.67 \| 0.13 \| 0.225502 \| EW \| U V06 \| 251.83856 \| -1.91462 \| 18.80 \| 0.74 \| 0.34 \| 0.256196 \| EW \| Y V07 \| 251.72871 \| -1.99055 \| 16.65 \| 1.06 \| 0.07 \| 0.257078 \| EW \| N V08 \| 251.83855 \| -1.95674 \| 16.48 \| 0.35 \| 0.07 \| 0.435135 \| EW/Ell,BS \| Y V09 \| 251.82010 \| -1.94197 \| 17.12 \| 0.77 \| 0.03 \| 0.444540 \| EW/Ell \| U V10 \| 251.77274 \| -1.99331 \| 19.06 \| 0.71 \| 0.34 \| 4.595098 \| EA \| Y V11 \| 251.81257 \| -1.95160 \| 18.26 \| 0.72 \| 0.36 \| 5.218589 \| EA \| Y V12 \| 251.82194 \| -1.92852 \| 16.09 \| 0.37 \| 0.10 \| 1.025114 \| EA,BS \| Y V13 \| 251.81159 \| -1.94571 \| 17.07 \| 0.47 \| 0.26 \| 0.734141 \| EA,BS \| Y V14 \| 251.80630 \| -1.95057 \| 17.58 \| 0.54 \| 0.62 \| 0.463846 \| EA/EB,BS \| Y V15 \| 251.80844 \| -1.94113 \| 17.13 \| 0.38 \| 0.16 \| 0.541709 \| EA,BS \| U V16 \| 251.86936 \| -1.92680 \| 17.30 \| 0.83 \| 0.45 \| 2.500325 \| EA \| N V17 \| 251.83190 \| -1.86607 \| 20.00 \| 1.30 \| 0.82 \| 2.279741 \| EA \| U V18 \| 251.71477 \| -1.95946 \| 16.46 \| 0.90 \| 0.65 \| 17.47090 \| EA \| N V19 \| 251.73301 \| -2.00912 \| 19.64 \| 0.83 \| 0.97 \| - \| EA \| Y V20 \| 251.83532 \| -1.95287 \| 19.72 \| 0.75 \| 0.62 \| - \| EA \| U V21 \| 251.80641 \| -1.94114 \| 17.11 \| 0.41 \| 0.06 \| 0.019636 \| SX,BS \| Y V22 \| 251.84708 \| -1.92796 \| 17.25 \| 0.46 \| 0.05 \| 0.034364 \| SX,BS \| Y V23 \| 251.82202 \| -1.96167 \| 16.70 \| 0.41 \| 0.08 \| 0.044294 \| SX,BS \| Y V24 \| 251.80911 \| -1.89787 \| 17.20 \| 0.48 \| 0.08 \| 0.045042 \| SX,BS \| Y V25 \| 251.76203 \| -1.96051 \| 17.02 \| 0.55 \| 0.17 \| 0.049034 \| SX,BS \| Y V26 \| 251.80863 \| -1.99363 \| 16.57 \| 0.83 \| 0.12 \| 1.843644c \| Sp? \| Y V27 \| 251.84428 \| -2.01776 \| 17.89 \| 1.12 \| 0.44 \| 11.36074c \| Sp?,RG? \| N V28 \| 251.81727 \| -1.93884 \| 13.85 \| 1.00 \| 0.04 \| 0.949850c \| Sp? \| Y V29 \| 251.80863 \| -1.99363 \| 16.58 \| 0.75 \| 0.10 \| 1.842144c \| Sp? \| Y V30 \| 251.81196 \| -1.93522 \| 16.39 \| 0.50 \| 0.03 \| 0.400692 \| Ell?,BS \| Y V31 \| 251.81356 \| -1.95517 \| 17.61 \| 0.44 \| 0.08 \| 0.405218 \| Ell?,BS \| U V32 \| 251.80431 \| -1.95410 \| 17.23 \| 0.55 \| 0.07 \| 0.499084 \| Ell? \| U V33 \| 251.81356 \| -1.95518 \| 17.53 \| 0.47 \| 0.09 \| 0.682552 \| Ell?,BS \| Y V34 \| 251.80367 \| -1.96689 \| 19.55 \| 0.81 \| 0.41 \| 0.352252 \| Sp? \| U V35 \| 251.80112 \| -1.95547 \| 17.95 \| 0.63 \| 0.07 \| 0.962294 \| Ell? \| U V36 \| 251.77580 \| -1.95956 \| 16.81 \| 0.82 \| 0.02 \| 0.774844 \| Ell? \| Y ∗ We follow the naming convention of von Braun et al. (2002), who are the discoverers of the first two variables listed in the table. Their variable V1 is not the W Vir pulsator referred to as V1 by Clement et al. (1988). aEA - detached eclipsing binary, EB - close eclipsing binary, EW - contact eclipsing binary, SX - SX Phe type pulsator, Sp - spotted variable, BS- blue straggler, Ell - ellipsoidal variable, RG - red giant. bY - member, N - nonmember, U - no data or data ambiguous cFor the 2001 season. Table 1: Basic data of M12 variables identified within the present survey∗ <figure><img src="content_image/1509.03094/x1.png"><figcaption>Figure 1: Standard deviation vs. average V magnitude for light curves ofstars from the M12 field. Light curves are based on images from the du Ponttelescope.</figcaption></figure> <figure><img src="content_image/1509.03094/x2.png"><figcaption>Figure 2: CMDs of M12 based on the data from the du Pont telescope (left) andthe Swope telescope (right).</figcaption></figure> <figure><img src="content_image/1509.03094/x3.png"><figcaption>Figure 3: Finding charts for the variables. Each chart is 30 arcsec on a side;north is up and east to the left.</figcaption></figure> <figure><img src="content_image/1509.03094/x5.png"><figcaption>Figure 4a Phased V curves of variables detected in the field of M12. Insertedlabels give star ID and period in days.</figcaption></figure> <figure><img src="content_image/1509.03094/x8.png"><figcaption>Figure 5: Color-magnitude diagram for M12 with indicated locations of thevariables. Red, blue and black labels denote, respectively, members,nonmembers and objects for which the membership data is missing or ambiguous.</figcaption></figure> <figure><img src="content_image/1509.03094/x9.png"><figcaption>Figure 6: Histogram of U−V indices of M12 PM-members. Main sequence stars with18<V<19.5 mag are only shown, and the normalization is arbitrary. Black:central circle 0≤r≤2′.36; red: intermediate ring 2′.36<r<3′.74; blue: outerregion r≥3′.74.</figcaption></figure>
0910.1231	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 53906, "num_imgs": 5, "llama3_tokens_count": 17096 }	[ "content_image/0910.1231/x1.png", "content_image/0910.1231/x2.png", "content_image/0910.1231/x3.png", "content_image/0910.1231/x6.png", "content_image/0910.1231/x7.png" ]	# The phase diagram of Lévy spin glasses I. Neri, F. L. Metz and D. Bollé February 21, 2024 ###### Abstract We study the Lévy spin-glass model with the replica and the cavity method. In this model each spin interacts through a finite number of strong bonds and an infinite number of weak bonds. This hybrid behaviour of Lévy spin glasses becomes transparent in our solution: the local field contains a part propagating along a backbone of strong bonds and a Gaussian noise term due to weak bonds. Our method allows to determine the complete replica symmetric phase diagram, the replica symmetry breaking line and the entropy. The results are compared with simulations and previous calculations using a Gaussian ansatz for the distribution of fields. ## 1 Introduction The prototype mean-field model of spin glasses is the Sherrington-Kirkpatrick (SK) model [1, 2, 3]. In this fully-connected (FC) system any pair of spins is coupled through weak interactions, of order $\mathcal{O}(N^{-1/2})$ in the total number of spins $N$, whose values are drawn independently from a Gaussian distribution. Within the assumption that the free-energy landscape contains one single valley, the effective field on a given spin is a sum of a large number of uncorrelated random variables with a finite variance and the usual central limit theorem (CLT) holds. As a consequence, the effective field follows a Gaussian distribution fully characterized by its first two moments, leading to a description in terms of two observables: the magnetization and the Edwards-Anderson order parameter. The CLT reflects the independence of the macroscopic behavior of the system with respect to the details of the coupling distribution. The existence of a CLT in the SK model is technically very convenient and simplifies the replica and the cavity method at high temperatures $T$. At low temperatures extreme values become important and a more complicated description is necessary. The success of the cavity and replica method lies in the detailed and exact description they give of the intricate behavior of the SK model at low temperatures, characterized by the presence of several degenerate states separated by infinite barriers [3]. However, it was shown that materials composed of magnetic impurities, randomly distributed in a non-magnetic host and interacting through the RKKY dipolar potential, exhibit a Cauchy distribution of effective fields. This is in particular true for a small concentration of magnetic impurities [4, 5]. An analogous result was obtained in a spatially disordered system of particles with dipolar interactions [6]. These results suggest that the choice of a coupling distribution that allows a wider variation of coupling strengths would be more realistic than the traditional Gaussian assumption used in mean-field models for spin glasses. Disordered systems in which the randomness of the disorder variable $J$ is modelled by a distribution $P(J)$ that has a power-law decay $P(J)\sim\|J\|^{-1-\alpha}$ ($\alpha<2$), for large $\|J\|$, have attracted less interest. A possible reason is the technical challenge to deal with distributions that do not fulfill the classical CLT. The heavy tails of $P(J)$ give rise to the divergence of the second moment of the distribution, which invalidates the application of classical CLT. In this case, the generalized CLT of Lévy and Gnedenko holds [7, 8], and the sum of a large number of independent random variables drawn from $P(J)$ follows the same distribution as the individual summands, exhibiting only different scale factors. The role of the large tails of $J$ has proven to be crucial to the long time or large size properties of different disordered systems [9]. As examples in this context, we mention the theory of random matrices [10, 11, 12], diffusion processes [13] and the portfolio optimization problem in theoretical finance [14]. A FC model of spin glasses with interactions drawn from a distribution with power-law tails (a Lévy spin glass) was introduced by Cizeau and Bouchaud [15]. In Lévy spin glasses every spin interacts with infinitely many weak bonds of order $\mathcal{O}\left(N^{-1/\alpha}\right)$ and a finite number of strong bonds of order $\mathcal{O}\left(1\right)$. In this sense the model is a hybrid between a FC spin glass, like the SK model, and a finitely connected (FiC) spin glass, like the Viana-Bray model [16]. The authors of [15] studied the model with the cavity method under the assumption that the distribution of effective fields is Gaussian. They found a spin glass phase stable under replica symmetry breaking and it was conjectured that at zero temperature the stability of replica symmetry is restored for $\alpha<1$. Recently, this model has been studied with replica theory [17]. The effective field distribution is not Gaussian. In [17] a complete phase diagram and a discussion of replica symmetry breaking was not given. The purpose of this paper is to improve upon the foregoing studies by deriving the complete phase diagram without the Gaussian assumption, the entropy and the stability to replica symmetry breaking effects. We propose a method that consists in the insertion of a small cutoff in the distribution of the couplings $P(J)$, which gives rise to a natural distinction between “weak bonds” and “strong bonds”. This allows us to solve the problem through both the replica method and the cavity method. We obtain a solvable self-consistent equation for the distribution of effective fields. Formally this equation is similar to the self-consistent equation for the effective field distribution of a FiC spin-glass system on a random graph [18] and a straightforward implementation of the population dynamics algorithm [19] is possible. Therefore, the procedure allows us to obtain the complete phase diagram of the model for all Lévy distributions in contrast to previous works [15, 17]. We include a skewness parameter in the definition of the model, responsible for controlling the relative weight between the positive and the negative tails of the coupling distribution. The dependence of the different phases on this parameter is shown in the phase diagrams. The results are compared with simulations. We calculate the entropy of the system and the stability against replica symmetry breaking. Our results are compared with those obtained by Cizeau and Bouchaud [15]. The paper is organized as follows. In section 2, we define the model. We explain how to solve the model through replica theory in section 3 and through the cavity method in section 4. In these sections we calculate the distribution of effective fields and compare them with the Gaussian assumption on the distribution of those fields. The behavior of the magnetization is compared with simulations. In section 5 we derive the stability condition against replica symmetry breaking. The order parameter equations, derived in sections 3, 4 and 5, are solved numerically to obtain the phase diagrams and the entropy in sections 6 and 7. In section 8 we present our conclusions. The effect of the different parameters of the Lévy distributions is shown in appendix A. Some details of the replica calculations are given in appendix B. ## 2 The Lévy spin glass We study a FC system of $N$ Ising spins $\sigma_{i}=\pm 1$ ($i=1,\dots,N$) with the Hamiltonian \[H=-\sum_{i<j}J_{ij}\sigma_{i}\sigma_{j},\] (1) where the symmetric couplings $\{J_{ij}\}$ are i.d.d.r.v. drawn from a stable distribution $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$. We define the stable distributions $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ through their characteristic function $L^{J_{1},\gamma,J_{0}}_{\alpha}(q)$: \[P^{J_{1},\gamma,J_{0}}_{\alpha}(J)\equiv\int\frac{dq}{2\pi}\exp \left(-iqJ\right)L^{J_{1},\gamma,J_{0}}_{\alpha}(q).\] (2) The characterstic function is of the form \[L^{J_{1},\gamma,J_{0}}_{\alpha}(q)=\exp\left[i\frac{qJ_{0}}{N}- \left\|\frac{J_{1}q}{\sqrt{2}N^{1/\alpha}}\right\|^{\alpha}\left(1-i\gamma\Phi{ \rm sign}(q)\right)\right].\] (3) The distribution $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ is characterized by four parameters: the exponent $\alpha\in(0,1)\cup(1,2]$, the skewness $\gamma\in[-1,1]$, the scale parameter $J_{1}>0$ and the shift $J_{0}\in\mathbb{R}$. The quantity $\Phi$ is given by $\Phi=\tan{(\frac{\alpha\pi}{2})}$. The scaling with $N$ in eq. (3) ensures that the Hamiltonian (1) is of order $\mathcal{O}(N)$. Lévy distributions contain two different parameters that control the bias in the couplings: $J_{0}$ and $\gamma$. We refer the reader to appendix A for a discussion of the role of $\alpha$ and $\gamma$. For $\alpha=1$ and $\gamma\neq 0$ the quantity $\Phi$ has a different expression and we will not consider this case in the sequel. The SK model is obtained for $\alpha=2$ independent of $\gamma$: in this case the distribution $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ is Gaussian with mean $J_{0}/N$ and variance $J_{1}^{2}/N$[1]. For $\alpha<2$ and $-1<\gamma<1$, the asymptotic behavior $\rho(J)$ of $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ for $\|J\|\rightarrow\infty$ can be derived from the explicit form of $L^{J_{1},\gamma,J_{0}}_{\alpha}(q)$: \[\rho(J)\equiv N\lim_{\|J\|\rightarrow\infty}P^{J_{1},\gamma,J_{0}}_{\alpha}(J)= \left(1+\gamma{\rm sign}J\right)\frac{C_{\alpha}}{\|J\|^{\alpha+1}},\] (4) where \[C_{\alpha}=\left(\frac{J_{1}}{\sqrt{2}}\right)^{\alpha}\frac{1}{\pi}\sin\left( \frac{\alpha\pi}{2}\right)\Gamma(\alpha+1).\] (5) Accordingly, the integrals for the second and higher moments of the distribution diverge for $\alpha<2$ due to the power-law decay illustrated by eq. (4). We can define stable distributions through (3) without losing any generality, see for example [20, 21]. We remark that there are many equivalent definitions possible for the characteristic function $L$, see [21]. Loosely speaking, a random variable $x$ is stable if the sum of a given number of independent and identical copies of $x$ is characterized by the same distribution as the original variable, exhibiting only a different scale and shift. ## 3 The replica method ### The distribution of effective fields In order to study the thermodynamic behavior of the Lévy spin glass we employ the replica method [3]. The partition function of the system at inverse temperature $\beta=T^{-1}$ is defined by \[Z=\sum_{\left\{\sigma\right\}_{i=1..N}}\exp{\Big{[}-\beta H\left(\left\{\sigma \right\}_{i=1..N}\right)\Big{]}},\] (6) with $H\left(\left\{\sigma\right\}_{i=1..N}\right)$ given by eq. (1). The averaged free-energy per spin $f$ can be written as follows \[f=-\lim_{N\rightarrow\infty}\lim_{n\to 0}\frac{1}{\beta Nn}\ln{ \overline{Z^{n}}}.\] (7) The symbol $\overline{(\dots)}$ denotes the average over the quenched random couplings $\{J_{ij}\}$ with the distribution $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$. The quantity $\overline{Z^{n}}$ is computed for positive integers $n$ and the limit $n\to 0$ is taken through an analytic continuation to real values. However, the integer moments $\overline{Z^{n}}$ of the partition function diverge for real $\beta$ due to the power-law behavior of $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ for $\|J\|\rightarrow\infty$. As noted in reference [17], the introduction of an imaginary temperature $\beta=-ik$, with a real parameter $k>0$, allows a straightforward calculation of the average $\overline{Z^{n}}$ by means of the definition of the characteristic function, eq. (3). However, it is not possible to write the averaged $\overline{Z^{n}}$ in terms of the two standard order-parameters usually employed in the description of FC systems, i.e., the magnetization and the spin-glass order parameter. Therefore, it is necessary to use the replica method, as developed to deal with FiC spin glasses [22]. The macroscopic behavior is characterized in terms of a non-Gaussian effective field distribution. This procedure was followed in [17]. Following their calculations we find the equation for the free energy $f$ in the limit $N\rightarrow\infty$: \[f=f_{1}+f_{2},\] (8) with \[ikf_{1} =-\lim_{n\to 0}\frac{1}{2n}\sum_{\bm{\sigma}\bm{\tau}} \Big{[}-\Big{(}\frac{J_{1}k}{\sqrt{2}}\Big{)}^{\alpha}\|\bm{\sigma}.\bm{\tau}\|^ {\alpha}\left(1+i\gamma\rm{sign}(\bm{\sigma}\cdot\bm{\tau})\Phi\right)-ikJ_{0} (\bm{\sigma}.\bm{\tau})\Big{]}P(\bm{\sigma})P(\bm{\tau})\,\,,\] (9) \[ikf_{2} =\lim_{n\to 0}\frac{1}{n}\log{\biggl{\{}\sum_{\bm{\sigma} }\exp\left[-\sum_{\bm{\tau}}\Big{[}-\Big{(}\frac{J_{1}k}{\sqrt{2}}\Big{)}^{ \alpha}\|\bm{\sigma}.\bm{\tau}\|^{\alpha}\left(1+i\gamma\rm{sign}(\bm{\sigma} \cdot\bm{\tau})\Phi\right)-ikJ_{0}(\bm{\sigma}.\bm{\tau})\Big{]}P(\bm{\tau}) \right]\biggr{\}}}.\] The order parameter $P(\bm{\sigma})$, with $\bm{\sigma}=\left(\sigma^{1},\sigma^{2},\ldots,\sigma^{n}\right)$, fulfills the self-consistent equation \[P(\bm{\sigma}) =\frac{\exp\left(\sum_{\bm{\tau}}P(\bm{\tau})\left[-ikJ_{0}\bm{ \sigma}\cdot\bm{\tau}-\left\|\frac{J_{1}k\bm{\sigma}\cdot\bm{\tau}}{\sqrt{2}} \right\|^{\alpha}\left(1+i\gamma\rm{sign}(\bm{\sigma}\cdot\bm{\tau})\Phi\right) \right]\right)}{\sum_{\bm{\sigma}}\exp\left(\sum_{\bm{\tau}}P(\bm{\tau})\left[ -ikJ_{0}\bm{\sigma}\cdot\bm{\tau}-\left\|\frac{J_{1}k\bm{\sigma}\cdot\bm{\tau}} {\sqrt{2}}\right\|^{\alpha}\left(1+i\gamma\rm{sign}(\bm{\sigma}\cdot\bm{\tau}) \Phi\right)\right]\right)}.\] (11) We make the replica-symmetric (RS) ansatz, \[P(\bm{\sigma})=\int dhW(h)\prod^{n}_{a=1}\frac{\exp\left(-ikh \sigma^{a}\right)}{2\cosh\left(-ikh\sigma^{a}\right)},\] (12) which defines the field distribution $W(h)$. Substitution of (12) in (11) gives: \[W(h)=\int\frac{ds}{2\pi}\exp{(ish)}\exp\Bigg{\{}-\int dhW(h)\int \frac{d\hat{J}dJ}{2\pi}\] \[\times\left[\left(\frac{J_{1}}{\sqrt{2}}\right)^{\alpha}\|\hat{J}\| ^{\alpha}\Big{(}1+i\gamma\Phi{\rm sign}(\hat{J})\Big{)}+iJ_{0}\hat{J}\right] \exp\left[i\hat{J}J\right]f(J,h,s)\Bigg{\}}.\] (13) The function $f(J,h,s)$ is defined as \[f(J,h,s)\equiv\exp{\left(-\frac{is}{\beta}{\rm atanh}\Big{[}\tanh\left(\beta J \right)\tanh\left(\beta h\right)\Big{]}\right)}.\] (14) The analytic continuation of $T$ to real values has been achieved by taking $k=i\beta$ at the end of the calculation. Using eqs. (56)-(60) from appendix B, we integrate in eq. (13) over the $\hat{J}$ variable to obtain the following simplified expression for $W(h)$ \[0<\alpha < 1:\] \[W(h) = \int\frac{ds}{2\pi}\exp{\Bigg{\{}ish-isJ_{0}m+\int dhW(h)\int_{- \infty}^{\infty}dJ\rho(J)\Big{[}f(J,h,s)-1\Big{]}\Bigg{\}}},\] (15) \[1<\alpha<2:\] \[W(h)=\int\frac{ds}{2\pi}\exp{\Big{(}ish-isJ_{0}m\Big{)}}\] \[\times\exp{\Bigg{\{}\int dhW(h)\int_{-\infty}^{\infty}dJ\rho(J) \Big{[}f(J,h,s)-f^{\prime}(0,h,s)J-1\Big{]}\Bigg{\}}},\] (16) where $f^{\prime}(0,h,s)=\frac{\partial f(J,h,s)}{\partial J}\Big{\|}_{J=0}$. The distribution of couplings $\rho(J)$ is defined by eq. (4). The RS magnetization $m$ and the RS spin-glass order-parameter $q$ are determined through the averages \[m=\int dhW(h)\tanh{(\beta h)},\qquad q=\int dhW(h)\tanh^{2}{(\beta h)}.\] (17) Only the large tail behavior of the distribution $P^{J_{1},\gamma J_{0}}_{\alpha}$ appears in the equations (15) and (16). This could mean that the system exhibits a certain degree of universality: the thermodynamic behavior only depends on the large tail behavior of the coupling distribution $P(J)$. The distribution $\rho(J)$ is symmetric when $\gamma=0$, with eqs. (15) and (16) reducing to a single equation, obtained previously in [17]. ### The normalization of the coupling distribution through a cutoff The main difficulty in eqs. (15) and (16) concerns the normalization of $\rho(J)$ since the integral $\int dJ\rho(J)$ diverges for $\alpha<2$. Therefore, it is not possible to normalize the distribution. This invalidates the numerical calculation of $W(h)$ through the population dynamics algorithm [23] because it is not possible to sample random numbers from a non-normalizable distribution. In this subsection we propose a simple procedure that allows to normalize $\rho(J)$ and to derive a self-consistent equation for $W(h)$ which is similar to the order-parameter equation of FiC spin glasses on random graphs. The numerical solution of this equation can be obtained through population dynamics. The method consists of the insertion of a temperature dependent cutoff $T\epsilon>0$ in the integrals over $J$ occurring in eqs. (15) and (16), splitting each of them into an integral around zero (from $-T\epsilon$ to $T\epsilon$) plus an integral over the couplings that satisfy $\|J\|>T\epsilon$. Assuming $T\epsilon\ll 1$, the integrals around zero can be analytically performed by expanding $f(J,h,s)$ around $J=0$ up to order $\mathcal{O}(J^{2})$, resulting in the following equations \[0<\alpha<1:\] \[\int_{-\infty}^{\infty}dJ\rho(J)\Big{[}f(J,h,s)-1\Big{]}=-2is \gamma C_{\alpha}\tanh{(\beta h)}\frac{(T\epsilon)^{1-\alpha}}{1-\alpha}-s^{2} C_{\alpha}\tanh^{2}{(\beta h)}\frac{(T\epsilon)^{2-\alpha}}{2-\alpha}\] \[+\int_{-\infty}^{\infty}dJ\rho(J)\Big{[}\Theta(J-T\epsilon)+ \Theta(-J-T\epsilon)\Big{]}\Big{[}f(J,h,s)-1\Big{]},\] (18) \[1<\alpha<2:\] \[\int_{-\infty}^{\infty}dJ\rho(J)\Big{[}f(J,h,s)-f^{\prime}(0,h,s) J-1\Big{]}=-s^{2}C_{\alpha}\tanh^{2}{(\beta h)}\frac{(T\epsilon)^{2-\alpha}}{2 -\alpha}\] (19) The symbol $\Theta(J)$ denotes the Heaviside step function: $\Theta(J)=1$ if $J>0$ and $\Theta(J)=0$ otherwise. We define the normalized distribution $P_{\epsilon}(J)$ in terms of $\rho(J)$ (20) Subsituting eqs. (18) and (19) in, respectively, eqs. (15) and (16) the integrals over $s$ can be analytically calculated: \[W_{\epsilon}(h)=\sum^{\infty}_{k=0}\exp\left(-c\right)\frac{c^{k }}{k!}\int\left(\prod^{k}_{r=1}dh_{r}W_{\epsilon}(h_{r})\right)\int\left(\prod ^{k}_{r=1}dJ_{r}P_{\epsilon}(J_{r})\right)\int\mathcal{D}z\] (21) where $\mathcal{D}z=(2\pi)^{-\frac{1}{2}}\exp{(-z^{2}/2)}dz$ and: \[c=\frac{2C_{\alpha}}{\alpha(T\epsilon)^{\alpha}},\qquad\Delta= \frac{(T\epsilon)^{2-\alpha}C_{\alpha}}{2-\alpha},\] (22) \[\tilde{J}_{0}=\left(J_{0}+2\gamma C_{\alpha}\left[\frac{(T \epsilon)^{1-\alpha}}{1-\alpha}\right]\right).\] (23) To describe the thermodynamic behavior of Lévy spin glasses one has to solve the set of equations (17) and (21) for $\epsilon\to 0$. When we compare equation (21) with the order parameter equations of FiC systems [18], it describes the effective field distribution of a FiC system of Ising spins, in which the number of connections per site $k$ follows a Poissonian distribution with connectivity $c$. The values of the $k$ couplings attached to a site are drawn from the distribution $P_{\epsilon}(J)$, see eq. (20). In addition, the analytical calculation of the integrals over the couplings that satisfy $\|J\|<T\epsilon$ yields an interaction with the global magnetization with effective strength $\tilde{J}_{0}$ and an extra source of noise in eq. (21), represented by the Gaussian random variable $z$ with zero mean and variance $\Delta$. The effective strength contains the shift parameter $J_{0}$ and a term linear in $\gamma$ corresponding with the center of the distribution of the couplings. The interpretation of equation (21) is clear : the effective field contains a Poissonian term coming from a finite number of strong bonds and a Gaussian term coming from an infinite number of weak bonds it interacts with. One can perform the limit $\alpha\to 2$ to find the effective field $J_{0}m+J_{1}\sqrt{q}z$, i.e. the RS solution of the SK model. The equations (17) and (21) show explicitly how Lévy spin glasses are a hybrid between FC and FiC models. When one takes a Gaussian ansatz for the distribution $W_{\epsilon}$, equation (21) becomes in the limit $\epsilon\to 0$ equal to the result derived by Cizeau and Bouchaud [15]. The population dynamics algorithm [23] can be easily adapted to solve numerically eq. (21) and obtain $W_{\epsilon}(h)$. The idea is to obtain numerical results for sufficiently small values of $T\epsilon$ in a way that they can be extrapolated for $T\epsilon\to 0$: the first two moments of the distribution already obtain their limiting values around $T\epsilon\lesssim 0.5$. The equations become very hard to solve around $\alpha\approx 1.5$ because the mean connectivity $c$ has a maximum there. For low values of $\alpha\lesssim 0.1$ population dynamics becomes inaccurate because of numerical imprecisions due to the larger tails of the coupling distribution. In figure 1 we compare the solution of equations (17) and (21) with Monte-Carlo simulations. We simulated a Lévy spin glass using the algorithm described in [17] without the parallel tempering. The algorithm contains two update rules: single spin flip updates as usually done in Metropolis algorithms and updates of clusters of spins connected through strong bonds. For low temperatures we find a good agreement between the simulations and the theory. Around the critical temperature the magnetization obtained by the simulations is larger than the one predicted by the theory. The reason for this difference is that the simulations equilibrate very slowly. Indeed, as shown in the inset of figure 1 the magnetization decays as a power law as a function of the number of Monte-Carlo sweeps. The presence of strong bonds slows down the dynamics since the effect becomes larger for smaller values of $\alpha$. For very low temperatures the simulation results for the magnetization deviate from those of the RS result. The RS ansatz (12) is invalid for very low temperatures, see section 6. In figure 2 we plotted the solution to the self-consistent equation (21) for different values of $\alpha$. The result is compared with the Gaussian ansatz (solid lines), used in [15]. The difference between both approaches is clear. For $\alpha\to 2$ the distribution of fields becomes more and more Gaussian. For $\alpha<2$ the distribution of fields is not Lévy but leptokurtic distributions where the moments converge to a finite value as a function of the size of the population. Leptokurtic distributions have a smaller kurtosis than a Gaussian distribution with the same variance. <figure><img src="content_image/0910.1231/x1.png"><figcaption>Figure 1: The magnetization m as a function of the temperature T for severalvalues of α and J0. Simulation results (markers) are compared with resultsfrom the theory (solid lines) for J1=1, γ=0. At low temperatures theory andsimulations are in good agreement. Because of the increase in theequilibration time around the critical temperature the results from simulationoverestimate the magnetization. The inset confirms this: it shows the value ofthe magnetization as a function of the number of Monte Carlo sweeps forα=0.5,J0=0.75 and T=1.</figcaption></figure> <figure><img src="content_image/0910.1231/x2.png"><figcaption>Figure 2: The distribution of effective fields with J0=γ=0, J1=1, T=0.4 andseveral values of α. The markers give the distributions according to equation(21) while the solid lines are obtained through the theory of [15]. All themoments of the distributions are finite. Therefore we have leptokurticdistributions which are neither Gaussian nor Lévy distributions.</figcaption></figure> ## 4 Cavity method We derive the self-consistent eqs. (21) for $W(h)$ through the cavity method. In contrast with [15] we only apply the CLT to the field coming from the weak bonds, i.e. bonds smaller than the cutoff $T\epsilon$. The bonds larger than $T\epsilon$ form a backbone graph of strong bonds which is treated as a FiC system. For $\epsilon\rightarrow\infty$ we find back the results of [15]. For $\epsilon\to 0$ we expect to find the RS behavior of the spin glass. The marginal $P_{i}(\sigma_{i})\equiv\sum_{\left\{\sigma_{j}\right\}_{j=1..N}\setminus\sigma _{i}}P\left(\left\{\sigma_{j}\right\}_{j=1..N}\right)$ of the Gibbs distribution $P\left(\left\{\sigma_{j}\right\}_{j=1..N}\right)\sim\exp\left[-\beta H\left( \left\{\sigma_{j}\right\}_{j=1..N}\right)\right]$ can be written as \[P_{i}(\sigma_{i}) \sim\sum_{\left\{\sigma_{j}\right\}_{j=1..N}\setminus\sigma_{i}}P ^{(i)}\left(\left\{\sigma_{j}\right\}_{j=1..N}\setminus\sigma_{i}\right)\exp \left(\sum_{k}J_{ki}\sigma_{j}\sigma_{i}\right),\] (24) with $P^{(i)}\left(\left\{\sigma_{j}\right\}_{j=1..N}\setminus\sigma_{i}\right)$ the Gibbs distribution on the cavity graph $G^{(i)}$. The cavity graph is the subgraph of the original graph $G$ where one removed the $i$-th spin and all of its interactions with the other spins. We assume that the probability distribution on the cavity graph factorizes [3]: \[P^{(i)}\left(\left\{\sigma_{j}\right\}_{j=1..N}\setminus\sigma_{ i}\right)=\prod_{j(\neq i)}P^{(i)}_{j}(\sigma_{j}).\] (25) This factorization is valid when there is one pure phase in the system. The set $\overline{\omega}^{(i)}$ of all weak bonds and the set $\omega^{(i)}$ of all strong bonds are defined through: \[\overline{\omega}^{(i)}\equiv\left\{j\in\mathbb{N}\cap[1,N]\|J_{ij }<T\epsilon\right\},\] (26) \[\omega^{(i)}\equiv(\mathbb{N}\cap[1,N])\setminus\overline{\omega} ^{(i)}.\] (27) The cavity fields $h^{(i)}_{j}$ and $g^{(i)}_{j}$ are defined through: \[h^{(i)}_{j}\equiv\sum_{\sigma}\frac{\sigma}{2}\log\left(P^{(i)}_ {j}(\sigma)\right) \rm{if} j\in\omega^{(i)},\] (28) \[g^{(i)}_{j}\equiv\sum_{\sigma}\frac{\sigma}{2}\log\left(P^{(i)}_ {j}(\sigma)\right) \rm{if} j\in\overline{\omega}^{(i)}.\] (29) The marginal $P^{(j)}_{i}$ of the $i$-th spin on the cavity graph $G^{(j)}$ is equal to: \[P^{(j)}_{i}(\sigma_{i})\sim\prod_{k\in\overline{\omega}^{(i)} \setminus j}\sum_{\tau}\exp\left(\beta J_{ki}\sigma_{i}\tau+\beta g^{(i)}_{k} \tau\right)\prod_{k\in\omega^{(i)}\setminus j}\sum_{\tau}\exp\left(\beta J_{ki }\sigma_{i}\tau+\beta h^{(i)}_{k}\tau\right).\] (30) We used the notation $h^{(j)}_{i}$ for cavity fields where one removed a site $j$ connected with $i$ through a strong bond and the notation $g^{(j)}_{i}$ for fields where the site $j$ was connected with $i$ through a weak bond . We thus find the following set of closed equations in the cavity fields $h^{(j)}_{i}$ and $g^{(j)}_{i}$: \[g^{(j)}_{i}=z^{(j)}_{i}+\beta^{-1}\sum_{k\in\omega_{i}}{\rm{ atanh}}\left(\tanh\left(\beta h^{(i)}_{k}\right)\tanh\left(\beta J_{ki}\right) \right),\] (31) (32) where we defined a third field containing the contributions from the weak bonds \[z^{(j)}_{i}=\beta^{-1}\sum_{k\in\overline{\omega}_{i}\setminus j }{\rm{atanh}}\left(\tanh\left(\beta g^{(i)}_{k}\right)\tanh\left(\beta J_{ki} \right)\right).\] (33) In the limit $N\rightarrow\infty$ we can remove the $j$ dependency in the fields $z^{(j)}_{i}$ and $g^{(j)}_{i}$ because the sum over the weak bonds ($k\in\overline{\omega_{i}}$) contains an infinite number of terms. To take the disorder average over the couplings we define the following distributions: \[W_{\rm{w}}(g)\equiv\overline{\frac{\sum^{N}_{i=1}\delta\left(g-g _{i}\right)}{N}},\] (34) \[W_{\rm{s}}(h)\equiv\overline{\frac{\sum^{N}_{i=1}\sum_{j\in \omega_{i}}\delta\left(h-h^{(j)}_{i}\right)}{\sum^{N}_{i=1}\sum_{j\in\omega_{i }}}},\] (35) \[W_{\rm g}(z)\equiv\overline{\frac{\sum^{N}_{i=1}\delta(z-z_{i})} {N}}.\] (36) We treat the $z$-fields as a sum of infinitely many random variables on which we can apply the CLT: \[W_{\rm g}(z)=\frac{1}{\sqrt{4\pi\Delta q}}\exp\left(-\frac{(z- \tilde{J_{0}}m)^{2}}{4\Delta q}\right),\] (37) with $\Delta$ and $\tilde{J}_{0}$ as defined in eqs. (22) and (23). The parameters $m$ and $q$ determine, respectively, the mean and the variance of the Gaussian distribution $W_{\rm g}(z)$. Here is the important difference with [15] where the CLT is applied on all bonds, also on the strong ones. From eq. (33) one finds, for $N\rightarrow\infty$ and $\epsilon\ll 1$ the following expressions for the mean $m$ and the variance $q$ \[m=\left(\tilde{J}_{0}\right)^{-1}\left(N\int^{T\epsilon}_{-T \epsilon}dJP^{J_{1},\gamma,J_{0}}_{\alpha}(J)J\right)\int dg\tanh(\beta g)W_{ \rm w}(g),\] (38) \[q=\left(2\Delta\right)^{-1}\left(N\int^{T\epsilon}_{-T\epsilon} dJP^{J_{1},\gamma,J_{0}}_{\alpha}(J)J^{2}\right)\int dg\tanh^{2}(\beta g)W_{ \rm w}(g).\] (39) The integrals over the couplings in eqs. (38) and (39) can be calculated using analogous methods as used to derive the integrals in appendix B: \[N\int^{T\epsilon}_{-T\epsilon}dJP^{J_{1},\gamma,J_{0}}_{\alpha}( J)J=\tilde{J}_{0},\] (40) \[N\int^{T\epsilon}_{-T\epsilon}dJP^{J_{1},\gamma,J_{0}}_{\alpha}( J)J^{2}=2\Delta.\] (41) Using the definitions of the distributions $W_{\rm{w}}(g)$ and $W_{\rm{s}}(h)$ in eqs. (34) and (35) we get \[W_{\rm s}(h) =\sum^{\infty}_{k=0}\frac{p_{\rm poiss}(k;c)k}{c}\prod^{k-1}_{r=1 }\int dh_{r}W_{\rm s}(h_{r})\int\prod^{k-1}_{r=1}dJ_{r}P_{\epsilon}(J_{r})\int dzW _{\rm g}(z)\] (42) \[\delta\left(h-z-\beta^{-1}\sum^{k-1}_{r=1}{\rm{atanh}}\left(\tanh (\beta h_{r})\tanh(\beta J_{r})\right)\right),\] \[W_{\rm w}(g) =\sum^{\infty}_{k=0}p_{\rm poiss}(k;c)\int\prod^{k}_{r=1}dh_{r}W_ {\rm s}(h_{r})\int\prod^{k}_{r=1}dJ_{r}P_{\epsilon}(J_{r})\int dzW_{\rm g}(z)\] (43) \[\delta\left(g-z-\beta^{-1}\sum^{k}_{r=1}{\rm{atanh}}\left(\tanh( \beta h_{r})\tanh(\beta J_{r})\right)\right).\] The mean connectivity $c$ is given by \[c=\lim_{N\rightarrow\infty}\frac{\sum^{N}_{i=1}\|\omega^{(i)}( \epsilon)\|}{N}=\int^{\infty}_{T\epsilon}dJ\rho(J)+\int^{-T\epsilon}_{-\infty} dJ\rho(J)=\frac{2C_{\alpha}}{\alpha(T\epsilon)^{\alpha}},\] (44) with $\rho(J)$ the large tail behavior as defined in (4). We use the following property of Poissonian distributions: $\frac{1}{c}p_{\rm poiss}(k;c)k=p_{\rm poiss}(k-1;c)$ to find $W_{\rm w}(g)=W_{\rm s}(g)=W_{\epsilon}(g)$, i.e. the solutions to (42) and (43) are the same as the solution $W_{\epsilon}$ of (21). Indeed, eqs. (43) combined with (38) and (39) are identical to eqs. (17) and (21) derived with the replica method. From the cavity approach the importance of the CLT in Lévy spin glasses becomes clear: the couplings have a divergent variance, therefore one can not apply the CLT as done in [15]. We remark that the effective coupling $\tilde{J}_{0}$ and the parameter $2\Delta$ appearing in the replica method are the mean and the variance of the weak couplings. The distribution $W_{\epsilon}(h)$ in equations (21) is the distribution of the cavity fields propagating along the backbone graph of strong bonds. ## 5 Stability of the replica symmetric ansatz As is known [24], the RS ansatz introduced in (12) is unstable at low temperatures. It is possible to calculate the regions of stability by using the two replica method, first introduced for FiC models in [25]. This method allows us to study local and non-local replica symmetry breaking (RSB) effects. For models on graphs a relevant instability condition is proved rigorously in [26]. It determines the region where the message passing algorithms stop to converge, see for example the discussion in [27]. We start by considering two uncoupled replicas. Both replicas fulfill the equations (30)-(33). The replicas only get coupled when we take the average over the graph instance. Indeed, the effective field distribution of two sets of uncoupled spins on the same graph with the same couplings is given by: \[W_{\epsilon}(h^{1},h^{2}) =\sum^{\infty}_{k=0}\frac{p_{\rm poiss}(k;c)k}{c}\prod^{k-1}_{r=1 }\int dh^{1}_{r}dh^{2}_{r}W_{\epsilon}(h^{1}_{r},h^{2}_{r})\int\prod^{k-1}_{r= 1}dJ_{r}P_{\epsilon}(J_{r})\int dz^{1}dz^{2}W_{\rm g}(z^{1},z^{2})\] (45) \[\times\delta\left(h^{1}-z^{1}-\beta^{-1}\sum^{k-1}_{r=1}{\rm{ atanh}}\left(\tanh(\beta h^{1}_{r})\tanh(\beta J_{r})\right)\right)\] \[\times\delta\left(h^{2}-z^{2}-\beta^{-1}\sum^{k-1}_{r=1}{\rm{ atanh}}\left(\tanh(\beta h^{2}_{r})\tanh(\beta J_{r})\right)\right).\] We assume again that we can apply the CLT on the $z$-fields: \[W_{\rm g}(z^{1},z^{2})\] \[=\frac{1}{4\Delta\pi\sqrt{q^{1}q^{2}(1-\rho^{2})}}\exp\left(- \frac{1}{2(1-\rho^{2})}\left(\frac{(z^{1}-\tilde{J}_{0}m^{1})^{2}}{2\Delta q^{ 1}}+\frac{(z^{2}-\tilde{J}_{0}m^{2})^{2}}{2\Delta q^{2}}\right)\right)\] \[\times\exp\left(\frac{\rho(z^{1}-\tilde{J}_{0}m^{1})(z^{2}-\tilde {J}_{0}m^{2})}{2(1-\rho^{2})\Delta\sqrt{q^{1}q^{2}}}\right).\] (46) The order parameters become \[m^{1}=\int dg^{1}dg^{2}\tanh(\beta g^{1})W_{\epsilon}(g^{1},g^{2 }),\] (47) \[q^{1}=\int dg^{1}dg^{2}\tanh^{2}(\beta g^{1})W_{\epsilon}(g^{1}, g^{2}),\] (48) \[m^{2}=\int dg^{1}dg^{2}\tanh(\beta g^{2})W_{\epsilon}(g^{1},g^{2 }),\] (49) \[q^{2}=\int dg^{1}dg^{2}\tanh^{2}(\beta g^{2})W_{\epsilon}(g^{1}, g^{2}),\] (50) \[\rho\sqrt{q^{1}q^{2}}=\int dg^{1}dg^{2}\tanh(\beta g^{2})\tanh( \beta g^{1})W_{\epsilon}(g^{1},g^{2}).\] (51) In the limit $\alpha\to 2$ we find $W_{\epsilon}(h^{1},h^{2})=W_{\rm g}(h^{1},h^{2})$. An expansion around the RS solution $m^{1}=m^{2}=m$, $q^{1}=q^{2}=q$ and $1-\|\rho\|\sim\mathcal{O}(\delta)$, with $\delta\ll 1$, leads to the following stability condition: \[\beta^{-2}=\int^{+\infty}_{-\infty}\frac{du}{\sqrt{2\pi}}\exp \left(-\frac{u^{2}}{2}\right)\mathrm{sech}^{4}\left(\beta\sqrt{q}u+\beta J_{0} m\right).\] (52) The parameters $(q,m)$ in (52) are, respectively, the overlap parameter and the magnetization of the SK model. Equation (52) is precisely the AT line of the SK model, see [24]. ## 6 Phase Diagram The system shows three phases which depend on the values of the order parameters $m$ and $q$ defined in (17): a paramagnetic phase (P) with $m=q=0$, a ferromagnetic phase (F) with $m>0,q>0$ and a spin-glass phase (SG) with $m=0,q>0$. <figure><img src="content_image/0910.1231/x3.png"><figcaption>Figure 3: The (T/J1,J0/J1) phase diagram for several values of α and with askewness γ=0. Three phases appear: P (paramagnetic), F (ferromagnetic) and SG(spin glass). The circles present the SG-F transitions and the stars indicatewhere the F phase becomes stable against replica symmetry breaking. For α=2the phase diagram coincides with that of the SK model.</figcaption></figure> The ferromagnetic phase contains a region stable to RSB effects ($F_{\rm stable}$) and a region unstable to RSB effects $(F_{\rm unstable})$. The P-F and P-SG transitions are determined using an expansion of the self-consistent eq. (21) around the paramagnetic solution $W_{\epsilon}(h)=\delta(h)$. For $\gamma=0$ we find the same bifurcation lines as derived in [17]. To determine the SG-F transition and the $F_{\rm unstable}$ to $F_{\rm stable}$ transition one has to solve numerically, respectively, eqs. (21) and (45) with for instance a population dynamics algorithm. In figure 4 the different phases in the $(J_{0}/J_{1},T/J_{1})$ phase diagram are presented for a skewness $\gamma=0$ and several values of $\alpha$. The open circles present the SG-F transitions and the stars mark the points where the F phase becomes stable with respect to RSB. These results generalize the phase diagram obtained in the seminal paper of Sherrington and Kirkpatrick [1] to coupling distributions with a large tail. For $\gamma=0$ the P-F transition is independent of $\alpha$. When $\alpha$ increases the SG phase increases in favor of a smaller F phase. The RSB effects decrease when $\alpha$ decreases: indeed the $F_{\rm unstable}$ becomes smaller and the reentrance effect in the SG-F phase transition line diminishes and finally disappears. This is related to the decrease of frustration due to the presence of stronger bonds that dominate the systems behavior. We did not find a replica symmetric SG phase (i.e., a SG phase stable with respect to RSB), contrary to the conjecture made in [15]. Replica symmetry breaks continuously at the SG transition similar to the behavior of the SK model. We did not find any further evidence for the conjecture in [15] that replica symmetry restores at $T=0$. In figure 4 we present the $(T/J_{1},\alpha)$-phase diagram for different values of $\gamma$ and $J_{0}=0$. We consider the following regions: * $\gamma>0$ and $\alpha<1$ (left figure): the F phase increases and the SG phase decreases as a function of increasing $\gamma$. The SG phase disappears at $\gamma=1$. For values of $\gamma$ very close to one the SG phase is only present for very small values of $\alpha$. The transition temperature between the P and F phase becomes infinite for $\alpha\to 1^{-}$. * $\gamma<0$ and $\alpha>1$ (right figure): the F phase decreases and the SG phase increases as a function of increasing $\gamma$. The transition temperature between the P and F phase becomes infinite for $\alpha\to 1^{+}$. * $\gamma>0$ and $\alpha>1$ (not shown): there is no F phase but a P and SG phase. * $\gamma<0$ and $\alpha<1$ (not shown): there is no F phase but a P and SG phase. We have some additional remarks: The transition temperature becomes very large for $\alpha\to 1^{\pm}$ (for, respectively, $\gamma<0$ and $\gamma>0$) because the effective coupling $\tilde{J}_{0}\rightarrow\infty$. There is no SG phase for $\gamma=1$ and $\alpha<1$ because there are no negative couplings, only the P-F transition occurs. The P-SG transitions coincide for different values of $\gamma$. For low values of $\alpha$ the results of population dynamics become inaccurate because of numerical imprecisions when dealing with a broad range of coupling values. In this case we used the instability line of the P phase with respect to the F phase as of the location of the SG-F transition. ## 7 Entropy It is possible to calculate the free energy from the saddle point equations. We use the RS ansatz and we introduce again a cutoff $\epsilon$. The entropy is given by $s=\beta^{2}\frac{\partial f}{\partial\beta}=\lim_{\epsilon\to 0}s(\epsilon)$ with $s(\epsilon)$: \[s(\epsilon)=\beta^{2}\frac{\Delta}{2}\left(1-q^{2}\right)-\beta^ {2}\Delta\left(1-q\right)+s_{\rm site}(\epsilon)-\frac{c}{2}s_{\rm link}( \epsilon).\] (53) The quantity $s_{\rm link}$ is equal to \[s_{\rm link}(\epsilon) =-\int dhdh^{\prime}W_{\epsilon}(h)W_{\epsilon}(h^{\prime})\int dJP _{\epsilon}(J)\] (54) \[\sum_{\sigma,\tau}\frac{\exp\left(\beta J\sigma\tau+\beta h\sigma +\beta h^{\prime}\tau\right)}{\sum_{\sigma,\tau}\exp\left(\beta J\sigma\tau+ \beta h\sigma+\beta h^{\prime}\tau\right)}\log\left(\frac{\exp\left(\beta J \sigma\tau+\beta h\sigma+\beta h^{\prime}\tau\right)}{\sum_{\sigma,\tau}\exp \left(\beta J\sigma\tau+\beta h\sigma+\beta h^{\prime}\tau\right)}\right),\] and $s_{\rm site}$ reads \[s_{\rm site}(\epsilon)\] (55) \[\sum_{\sigma;(\tau_{1},\tau_{2},\cdots,\tau_{k})}\frac{\exp\left( (\beta J_{0}m+\sqrt{2q\Delta}z)\sigma\right)\prod^{k}_{\ell=1}\left(\exp\left( \beta J_{\ell}\tau_{\ell}\sigma\right)\exp\left(\beta h_{\ell}\tau_{\ell} \right)\right)}{\sum_{\sigma;(\tau_{1},\tau_{2},\cdots,\tau_{k})}\exp\left(( \beta J_{0}m+\sqrt{2q\Delta}z)\sigma\right)\prod^{k}_{\ell=1}\left(\exp\left( \beta J_{\ell}\tau_{\ell}\sigma\right)\exp\left(\beta h_{\ell}\tau_{\ell} \right)\right)}\] For $\alpha\to 2$ one gets precisely the entropy of the SK model [2]. The entropies $s_{\rm site}$ and $s_{\rm link}$ correspond with the entropy differences when performing, respectively, a site addition and a link addition on the backbone graph of strong bonds, see [19, 23]. Similar to the form of the self-consistent equation (21) for $W_{\epsilon}(h)$ we find that the entropy as given by equation (53) corresponds to the entropy of an Ising model on a Poissonian graph with mean connectivity $c$, a distribution of the bonds $P_{\epsilon}$ and an extra Gaussian noise $z$. We plotted the entropy $s$ as a function of $\alpha$ in figure 5. From this figure we see that the entropy gets less negative, for $T\to 0$. We find that for smaller values of $\alpha\lesssim 1$ the entropy becomes eventually zero for $T\to 0$. This is consistent with a decrease of RSB effects when $\alpha$ decreases. <figure><img src="content_image/0910.1231/x6.png"><figcaption>Figure 5: The entropy s as a function of the exponent α for different valuesof the temperature T, J0=0, γ=0 and J1=1. The filled markers at α=2 show theSK values. The entropy converges to the SK value for α→2.</figcaption></figure> ## 8 Conclusion In this paper we have shown how to derive the phase diagrams of Lévy spin glasses where the couplings between the spins are drawn from a distribution with power law tails characterized by an exponent $\alpha$. These models are known to have a finite number of strong bonds of order $\mathcal{O}(1)$ and an infinite amount of weak bonds of order $\mathcal{O}(N^{-1/\alpha})$. The crucial difference with previous works [15] and [17] is that we derive the phase diagrams, the entropy and the stability against replica symmetry of Lévy spin glasses without using the Gaussian assumption for the distribution of fields. We have neither found evidence for a replica symmetric spin-glass phase, nor for a restoration of the replica symmetry at zero temperature, contrary to the conjecture in [15]. We have solved the problem using the replica and the cavity method within, respectively, the replica symmetric assumption and the assumption of one pure phase. The resultant effective equations for the distribution of cavity fields show clearly the hybrid character of the model being a mixture between a finite connectivity model and a fully connected model. The phase transitions are qualitatively similar to the ones found in the SK model. Large tails do influence quantitatively the phase diagram: the Lévy spin-glass model becomes more stable with respect to replica symmetry breaking and the SG phase decreases when the tails get larger. Moreover, the reentrance effects in the replica symmetric phase diagram disappear for $\alpha\lesssim 1$. The replica symmetry breaking transitions are all continuous. The skewness $\gamma$ in the Lévy distribution can have a big influence on the size of the F phase. For $\alpha\to 2$ the effective distribution of fields becomes Gaussian and we have found back the results of the SK model. For $\alpha<2$ this distribution is neither Lévy nor Gaussian, but a distribution with finite moments and a kurtosis smaller than a Gaussian with the same variance. One of the authors (F. L. Metz) acknowledges a fellowship from CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil. ## Appendix A Stable distributions <figure><img src="content_image/0910.1231/x7.png"><figcaption>Figure 6: The Lévy distributions PJ1,γ,J0α(J) for J1=1, J0=0 and differentvalues of α and γ. The distributions with α=1.5 approach a Gaussian while theones for α=0.5 have larger tails. For γ>0, the center of the distribution goesto +∞ or −∞ for α↑1 or α↓1, respectively. The coupling distribution fulfillsPJ1,γ,J0α(J)=PJ1,−γ,J0α(−J).</figcaption></figure> The purpose of this appendix is to give some intuition on the role of the parameters $\alpha$ and $\gamma$ present in stable distributions, defined through eqs. (2) and (3). Both $\alpha$ and $\gamma$ are responsible for the shape of the distribution. The main role of the exponent $\alpha$ is to control the decay of the tails. Figure 6 shows that, for a fixed $\gamma$, a decrease in $\alpha$ gives rise to a distribution $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ with larger tails and more sharply peaked around its most probable value $J$. The center of $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ is also shifted from the negative to the positive J-axis as $\alpha$ decreases from $\alpha>1$ to $\alpha<1$. A change of $\alpha$ has no effect on the position of the center when $\gamma=0$. The skewness parameter $\gamma$ controls the relative weight between the positive and negative tails. For $\gamma>0$, the positive tail of $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ is larger than the negative one; for $\gamma<0$ vica versa. The distribution is symmetric around $J_{0}$ when $\gamma=0$. For increasing positive values of $\gamma$, see figure 6, the center of $P^{J_{1},\gamma,J_{0}}_{\alpha}(J)$ shifts to the right or left provided $\alpha<1$ or $\alpha>1$, respectively. ## Appendix B Solution of integrals In this appendix we show how to integrate over $\hat{J}$ in the following equations: \[I_{1} = \int_{-\infty}^{\infty}\frac{dJd\hat{J}}{2\pi}\|\hat{J}\|^{\alpha}e ^{i\hat{J}J}f(J),\] (56) \[I_{2} = \int_{-\infty}^{\infty}\frac{dJd\hat{J}}{2\pi}\|\hat{J}\|^{\alpha}{ \rm sign}(\hat{J})e^{i\hat{J}J}f(J),\] (57) where $f(J)$ is given by eq. (14) and $\alpha\in(0,1)\cup(1,2]$. We obtained the following results for $I_{1}$ and $I_{2}$ after integration over $\hat{J}$ \[I_{1}=-\left(\frac{\sqrt{2}}{J_{1}}\right)^{\alpha}C_{\alpha}\int_{-\infty}^{ \infty}\frac{dJ}{\|J\|^{\alpha+1}}\Big{[}f(J)-f(0)\Big{]},{\rm if}\quad 0<\alpha <2,\] (58) \[I_{2} =i\left(\frac{\sqrt{2}}{J_{1}}\right)^{\alpha}\frac{C_{\alpha}}{ \Phi}\int_{-\infty}^{\infty}\frac{dJ}{\|J\|^{\alpha+1}}{\rm sign}(J)f(J),{\rm if }\quad 0<\alpha<1,\] (59) \[=i\left(\frac{\sqrt{2}}{J_{1}}\right)^{\alpha}\frac{C_{\alpha}}{ \Phi}\int_{-\infty}^{\infty}\frac{dJ}{\|J\|^{\alpha+1}}{\rm sign}(J)\Big{[}f(J)- f^{\prime}(0)J\Big{]},{\rm if}\quad 1<\alpha\leq 2,\] (60) where the parameters $C_{\alpha}$ and $\Phi$ are defined in section 2. We have left out the dependence of $f(J)$ with respect to $h$ and $s$ since it is not important here. The aim of this appendix is to show how one can derive eqs. (58), (59) and (60) from eqs. (56) and (57). By introducing an exponential convergence factor in eqs. (56) and (57), we can rewrite them as follows \[I_{1} =\] (61) \[I_{2} = i\lim_{a\to 0^{+}}\int_{0}^{\infty}dJ\Big{[}f(J)-f(-J) \Big{]}\int_{0}^{\infty}\frac{d\hat{J}}{\pi}\hat{J}^{\alpha}\sin{(\hat{J}J)}e^ {-a\hat{J}}.\] (62) The integrals over $\hat{J}$ are calculated for $a>0$ and, afterwards, the limit $a\to 0$ is performed. Reference [28] can be used in order to integrate over $\hat{J}$ in eqs. (61) and (62), giving rise to \[I_{1} = \frac{\Gamma(\alpha+1)}{\pi}\lim_{a\to 0^{+}}\int_{0}^{ \infty}dJ\Big{[}f(J)+f(-J)\Big{]}\frac{\cos{\Big{[}(\alpha+1){\rm arctan}( \frac{J}{a})\Big{]}}}{(J^{2}+a^{2})^{\frac{\alpha+1}{2}}},\] (63) \[I_{2} = i\frac{\Gamma(\alpha+1)}{\pi}\lim_{a\to 0^{+}}\int_{0}^{ \infty}dJ\Big{[}f(J)-f(-J)\Big{]}\frac{\sin{\Big{[}(\alpha+1){\rm arctan}( \frac{J}{a})\Big{]}}}{(J^{2}+a^{2})^{\frac{\alpha+1}{2}}}.\] (64) In order to analyze the behavior of integrals (63) and (64) around $J=0$ when $a\to 0^{+}$, we insert a cutoff $\lambda>0$ and split them as follows \[I_{1} =\frac{\Gamma(\alpha+1)}{\pi}U_{1}(\lambda)+\frac{\Gamma(\alpha+1 )}{\pi}\cos{\Big{[}(\alpha+1)\frac{\pi}{2}\Big{]}}\int_{\lambda}^{\infty}\frac {dJ}{J^{\alpha+1}}\Big{[}f(J)+f(-J)\Big{]},\] (65) \[I_{2} =i\frac{\Gamma(\alpha+1)}{\pi}U_{2}(\lambda)+i\frac{\Gamma(\alpha +1)}{\pi}\sin{\Big{[}(\alpha+1)\frac{\pi}{2}\Big{]}}\int_{\lambda}^{\infty} \frac{dJ}{J^{\alpha+1}}\Big{[}f(J)-f(-J)\Big{]},\] (66) where \[U_{1}(\lambda)=\lim_{a\to 0^{+}}\int_{0}^{\lambda}\frac{ dJ}{a^{\alpha+1}}\Big{[}f(J)+f(-J)\Big{]}\frac{\cos{\Big{[}(\alpha+1){\rm arctan }(\frac{J}{a})\Big{]}}}{\Big{[}\Big{(}\frac{J}{a}\Big{)}^{2}+1\Big{]}^{\frac{ \alpha+1}{2}}},\] (67) \[U_{2}(\lambda)=\lim_{a\to 0^{+}}\int_{0}^{\lambda}\frac{ dJ}{a^{\alpha+1}}\Big{[}f(J)-f(-J)\Big{]}\frac{\sin{\Big{[}(\alpha+1){\rm arctan }(\frac{J}{a})\Big{]}}}{\Big{[}\Big{(}\frac{J}{a}\Big{)}^{2}+1\Big{]}^{\frac{ \alpha+1}{2}}}.\] (68) The limit $a\to 0^{+}$ has been performed on the right hand side of eqs. (65) and (66). The integrals present in the definition of $U_{1}(\lambda)$ and $U_{2}(\lambda)$ are computed through a power-series representation of their integrands, yielding the results \[U_{1}(\lambda) = \lim_{a\to 0^{+}}\sum_{n,l=0}^{\infty}u_{nl}\,\Big{(} \frac{\lambda}{a}\Big{)}^{2l+\alpha+1}\lambda^{2n-\alpha},\] (69) \[U_{2}(\lambda) = \lim_{a\to 0^{+}}\sum_{n,l=0}^{\infty}v_{nl}\,\Big{(} \frac{\lambda}{a}\Big{)}^{2l+\alpha+2}\lambda^{2n+1-\alpha}.\] (70) The explicit forms of the coefficients $\{u_{nl}\}$ and $\{v_{nl}\}$ are irrelevant. The analysis of eqs. (65) and (66) as $\lambda$ tends to zero, constrained to the limit $a\to 0^{+}$ in the functions $U_{1}(\lambda)$ and $U_{2}(\lambda)$, constitutes the final step of the calculation. One can notice from eq. (69) that $U_{1}(\lambda)$ diverges for $\lambda\to 0^{+}$. However, the transformation of $f(J)$ according to $f(J)\to f(J)-f(0)$ removes this divergence and makes $U_{1}(\lambda)$ go to zero for $\lambda\to 0^{+}$, provided that $\alpha<2$. This allows us to perform the limit $\lambda\to 0^{+}$ in eq. (65) which leads, after comparison with eq. (56), to the following result \[\int_{-\infty}^{\infty}\frac{dJd\hat{J}}{2\pi}\|\hat{J}\|^{\alpha}e ^{i\hat{J}J}\Big{[}f(J)-f(0)\Big{]}=-\frac{\Gamma(\alpha+1)}{\pi}\sin{\Big{(} \frac{\alpha\pi}{2}\Big{)}}\] \[\times\int_{0}^{\infty}\frac{dJ}{J^{\alpha+1}}\Big{[}f(J)+f(-J)-2 f(0)\Big{]},\,0<\alpha<2.\] (71) By integrating the term with $f(0)$ on the left hand side of the above equation we get eq. (58). The calculation of eqs. (59) and (60) proceeds in an analogous way. Depending on the value of $\alpha$, there are two different situations concerning the behavior of eq. (70) for $\lambda\to 0^{+}$. For $\alpha<1$, we obtain $\lim_{\lambda\to 0^{+}}U_{2}(\lambda)=0$, which allows to perform the limit $\lambda\to 0^{+}$ in eq. (66). For $\alpha>1$, it is necessary to transform $f(J)$ according to $f(J)\to f(J)-f^{\prime}(0)J$ in order to obtain $\lim_{\lambda\to 0^{+}}U_{2}(\lambda)=0$ and to perform the limit $\lambda\to 0^{+}$ in eq. (66). ## References * [1] D. Sherrington and S. Kirkpatrick, “Solvable model of a spin-glass,” _Phys. Rev. Lett._, vol. 35, p. 1792, 1975. * [2] S. Kirkpatrick and D. Sherrington, “Infinite-ranged models of spin-glasses,” _Phys. Rev. B_, vol. 17, p. 4384, 1978. * [3] M. Mézard, G. Parisi, and M. A. Virasoro, _Spin Glass Theory and Beyond_, vol. 9 of _World Scientific Lecture Notes in Physics_. World Scientific Pub Co Inc., 1987. * [4] M. W. Klein, “Temperature-dependent internal field distribution and magnetic susceptibility of a dilute ising spin system,” _Phys. Rev._, vol. 173, p. 552, 1968. * [5] M. W. Klein, C. Held, and E. Zuroff, “Dipole interactions among polar defects: A self-consistent theory with application to $oh^{-}$ impurities in kcl,” _Phys. Rev. B_, vol. 13, p. 3576, 1976. * [6] D. V. Berkov, “Local-field distribution in systems with dipolar interparticle interaction,” _Phys. Rev. B_, vol. 53, p. 731, 1996. * [7] P. Lévy, _Theory de l’addition de Variables Aléatoires_. Paris: Gauthier-Villars, 1937. * [8] B. V. Gnedenko and A. N. Kolmogorov, _Limit Distributions for Sums of Independent Random Variables_. Cambridge: Addison-Wesley, 1954. * [9] G. Biroli, J. P. Bouchaud, and M. Potters, “Extreme value problems in random matrix theory and other disordered systems,” _J. Stat. Mech._, p. 07019, 2007. * [10] P. Cizeau and J. P. Bouchaud, “Theory of lévy matrices,” _Phys. Rev. E_, vol. 50, p. 1810, 1994. * [11] Z. Burda, J. Jurkiewicz, M. A. Nowak, G. Papp, and I. Zahed, “Free random lévy and wigner-lévy matrices,” _Phys. Rev. E_, vol. 75, p. 051126, 2007. * [12] G. Biroli, J. P. Bouchaud, and M. Potters, “On the top eigenvalue of heavy-tailed random matrices,” _Eur. Phys. Lett._, vol. 78, p. 10001, 2007. * [13] J. P. Bouchaud and A. Georges, “Anomalous diffusion in random media: statistical mechanisms, models and physical applications,” _Phys. Rep._, vol. 195, p. 127, 1990. * [14] S. Galluccio, J. P. Bouchaud, and M. Potters, “Rational decisions, random matrices and spin glasses,” _Physica A_, vol. 259, p. 449, 1998. * [15] P. Cizeau and J.-P. Bouchaud, “Mean field theory of dilute spin-glasses with power-law interactions,” _J. Phys. A: Math. Gen._, vol. 26, p. L187, 1993. * [16] L. Viana and A. J. Bray, “Phase diagrams for dilute spin glasses,” _J. Phys. C: Solid State Phys_, vol. 18, p. 3037, 1985. * [17] K. Janzen, A. K. Hartmann, and A. Engel, “Replica theory for lévy spin glasses,” _Journal of Statistical Mechanics: Theory and Experiment_, vol. 26, p. 04006, 2008. * [18] M. Mézard and G. Parisi, “Mean-field theory of randomly frustrated systems with finite connectivity,” _Europhys. Lett._, vol. 3, p. 1067, 1987. * [19] M. Mézard and G. Parisi, “The cavity method at zero temperature,” _J. Stat. Phys_, vol. 1, p. 11, 2003. * [20] W. Paul and J. Baschnagel, _Stochastic Processes_. Springer-Verlag Berlin Heidelberg, 1999. * [21] J. P. Nolan, _Stable Distributions: Models for Heavy-Tailed Data_. Birkhauser, 2007. * [22] R. Monasson, “Optimization problems and replica symmetry breaking in finite connectivity spin glasses,” _J. Phys. A: Math. Gen._, vol. 31, p. 513, 1998. * [23] M. Mézard and G. Parisi, “The bethe lattice spin glass revisited,” _Eur. Phys. J. B_, vol. 20, p. 217, 2001. * [24] J. R. L. de Almeida and D. J. Thouless, “Stability of the sherrington-kirkpatrick solution of a spin-glass model,” _J. Phys. A: Math. Gen._, vol. 11, p. 129, 1978. * [25] C. Kwon and D. J. Thouless, “Spin glass with two replicas on a bethe lattice,” _Phys. Rev. B_, vol. 10, p. 8379, 1991. * [26] D. J. Aldous and A. Bandyopadhyay, “A survey of max-type recursive distributional equations,” _Annals of Applied Probability_, vol. 15, p. 1047, 2005. * [27] I. Neri, N. S. Skantzos, and D. Bollé, “Gallager error-correcting codes for binary asymmetric channels,” _J. Stat. Mech._, p. P10018, 2008. * [28] I. S. Gradshteyn and I. M. Ryzhik, _Table of integrals, series, and products_, p. 492 and 493. San Diego: Academic Press, 2000.
1704.05724	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 34543, "num_imgs": 6, "llama3_tokens_count": 8436 }	[ "content_image/1704.05724/x1.png", "content_image/1704.05724/x4.png", "content_image/1704.05724/x5.png", "content_image/1704.05724/x6.png", "content_image/1704.05724/x7.png", "content_image/1704.05724/x8.png" ]	# The True Destination of EGO is Multi-local Optimization Simon Wessing¹ Mike Preuss² ¹Computer Science Department Technische Universität Dortmund Germany ²European Research Center for Information Systems WWU Münster Germany simon.wessing@tu-dortmund.de mike.preuss@uni-muenster.de ###### Abstract Efficient global optimization is a popular algorithm for the optimization of expensive multimodal black-box functions. One important reason for its popularity is its theoretical foundation of global convergence. However, as the budgets in expensive optimization are very small, the asymptotic properties only play a minor role and the algorithm sometimes comes off badly in experimental comparisons. Many alternative variants have therefore been proposed over the years. In this work, we show experimentally that the algorithm instead has its strength in a setting where multiple optima are to be identified. ## 1 Introduction Efficient global optimization (EGO) is a popular algorithm for the optimization of expensive multimodal black-box functions. At its core is a Kriging metamodel, whose predictions are used to formulate a so-called infill criterion. This criterion usually is a compromise between two goals: a) to detect especially good solutions, and b) to improve the model itself in order to enable better predictions. The established model can be employed to cheaply search for potential new points because it is much faster than the original function, often by a factor of $1000$ or more. By optimizing with regard to the infill criterion, a new point can be determined for sampling the expensive function. It is clear that this works reasonably well only for functions that can be predicted from a sparse sample, namely relatively low dimensional and generally rather smooth ones. With more and more samples coming in, the model improves, so that one gets a better and better overall impression of the original function. However, the model fit also gets more and more expensive, due to necessary matrix inversions. Thus, the sample size is limited, usually to around $1000$ points. The infill criterion of choice for EGO is the so-called expected improvement (EI), which incorporates both objectives mentioned above. While EGO is conceptually elegant and its convergence rate to the global optimum can be analyzed mathematically [3], there are multiple experimental results that indicate a preference for a more greedy infill criterion in expensive global optimization. For example, Sóbester et al. [21] propose a weighted expected improvement, which can be adjusted to search more locally or globally, depending on the weights. Some more evidence has surfaced in research on “multipoint” infill criteria for parallelizing function evaluations. Bischl et al. [2] discovered that just using the model prediction was the most successful infill criterion in a comparison with expected improvement and several multipoint infill criteria for a budget of $45n$ objective function evaluations, where $n$ is the number of decision variables of the problem. Ginsbourger et al. [10] showed that an explorative variant of their _constant liar_ criterion was less competitive than the more exploitative one, when filling in four to ten points on the Branin function. Also Ursem [27] developed an _investment portfolio improvement_ function to propose three solutions per iteration, ranging from high exploitation to high exploration. Our position is, that while the rather global search strategy of expected improvement may prevent a highly accurate approximation of the global optimum with small budgets, it represents a virtue for the task of finding multiple optima. This task is also known under the names of multi-global or multi-local optimization, depending on if only global or all optima are sought. EGO has to store the sampled points and function values anyway, to build the model, and all we have to do is to add a basin identification heuristic, to decide which points correspond to distinct attraction basins, and simply select the best one of each basin. As we want to avoid the responsibility of deciding if an optimum is global or local in our algorithm, we focus on multi-local optimization here. Employing EGO as multi-local optimization algorithm may seem counterintuitive at first, but we claim that in an expensive optimization scenario, where we can afford only several hundreds of function evaluations, this makes a lot of sense. Many competing multi-local optimization algorithms rely on (multiple) local searches which turn out to be too expensive in this scenario. The focus of this paper is to experimentally analyse how well EGO is suited for expensive multi-local optimization. As expensive optimization setting, we assume budgets of $500$ evaluations or less here. With respect to the limitation concerning the number of points that can be used as basis of a Kriging model, EGO matches the expensive optimization scenario very well. We therefore run EGO against one of the most effective and robust black-box optimization methods, the (Restart-) CMA-ES [1]. The CMA-ES is a good reference as it is also part of modern multimodal optimization algorithms, e. g., it performs the local optimization step in the NEA2 method [16]. Feliot et al. [8] also compared model-based approaches with local search algorithms in a constrained optimization setting and found that the competition is quite balanced. This already shows that the situation is not as clear-cut as it seems. <figure><img src="content_image/1704.05724/x1.png"><figcaption>Figure 1: Examples of nearest-better clustering on a 2-D landscape, applied toN=100 highly uniform, random uniform, and clustered points (from left toright). Selected points representing basin centers are marked with greensquares.</figcaption></figure> Our comparison is carried out on a mix of typical global optimization benchmark functions and niching competition benchmark functions, and we look at the results from two perspectives: 1. global optimization: we are only interested in locating the global optimum (one global optimizer in case it has several preimages), 2. multi-local optimization: we want to detect as many local/global optima as possible, ideally all of them. If our reasoning from above is correct, EGO should perform better than CMA-ES and related methods under a multi-local optimization perspective, but worse if seen from a global optimization perspective. This would mean that in the expensive black-box setting, EGO is a very suitable multi-local optimization method. In order to compare it to other such methods, we need to add a basin identification method, because in the answer set of the algorithm, we only want points that are approximations of existing optima, not the full archive. According to [29], we have two methods at hand: topographical selection (TS) by [24] and nearest-better clustering (NBC) as described in [15]. Previous investigations showed that using NBC can be problematic because it produces too many clusters if the point set deviates from uniformity. An example for this effect can be found in Figure 1, where the uniformity decreases from left to right. If, e. g., the clustered points reside on a linear slope, the best point of a cluster is wrongly interpreted as representing an optimum where none exists (see rightmost subfigure, bottom left corner). The reason for this behavior is that outliers tend to be selected simply because of their large nearest-neighbor distances. We therefore rely on TS in this work. To our knowledge, EGO and related surrogate-model-based methods have never been investigated in this way. Also the whole area of expensive multi-local/multimodal optimization seems to have been sparsely visited. One of the few works in this area is [14], however, they do not employ the term _multimodal optimization_ as it has been established only years later. The next section introduces necessary problem definitions, Sect. 3 explains the employed methods, namely EGO and topographical selection. The remainder of the paper is mostly concerned with the comparison experiment (Sect. 4) and ends with the conclusions. ## 2 Problem definition In the following, we will assume to have a deterministic objective function $f:\mathcal{X}\to\mathbb{R}$, where $\mathcal{X}=[\boldsymbol{\ell},\boldsymbol{u}]\subset\mathbb{R}^{n}$ is the _search space_ or _region of interest_ (ROI) and $n\in\mathbb{N}$ is the fixed number of decision variables. The vectors $\boldsymbol{\ell}=(\ell_{1},\dots,\ell_{n})^{\top}$ and $\boldsymbol{u}=(u_{1},\dots,u_{n})^{\top}$ are called the lower and upper bounds of $\mathcal{X}$, respectively. Let $N(\boldsymbol{y})=\{\boldsymbol{x}\in\mathcal{X}\mid d(\boldsymbol{x}, \boldsymbol{y})\leq\epsilon\}$ be the neighborhood of a point $\boldsymbol{y}\in\mathcal{X}$. We say $f^{}:=f(\boldsymbol{x}^{})$ is a local minimum if $\exists\epsilon>0:\nexists\boldsymbol{x}\in N(\boldsymbol{x}^{}):f( \boldsymbol{x})<f(\boldsymbol{x}^{})$. Technically, this implies that also plateaus are considered as local optima, although they are rather not intuitively, but at least this ensures that every position of a global optimum is also one of a local optimum. A multimodal optimization problem was implicitly already defined by Törn and Žilinskas [25, pp. 2–3]. In [28, p. 6], the definition was formulated explicitly as follows: Definition 1 (Multimodal minimization problem).: _Let there be $\nu$ local minima $f_{1}^{},\dots,f_{\nu}^{}$ of $f$ in $\mathcal{X}$. If the ordering of these optima is $f_{(1)}^{}<\dots<f_{(l)}^{}<h<\dots<f_{(\nu)}^{}$, a multimodal minimization problem is given as the task to approximate the set $\bigcup_{i=1}^{l}X_{(i)}^{}$, where $X_{(i)}^{}=\{\boldsymbol{x}\in\mathcal{X}\mid f(\boldsymbol{x})=f_{(i)}^{}\}$._ The variable $h$ in this definition is simply a threshold to potentially exclude some of the worse optima. If $h=\infty$, we will be interested in all local optima of a problem. If $f_{(1)}^{}<h<f_{(2)}^{}$, we are only interested in approximating all the global optima. Let $P$ be the obtained approximation set. Additional constraints may be applied to this set, to obtain more specific problem definitions. For example, the cardinality of $P$ could be restricted by requiring $\|P\|\leq k$. If $k=1$, we have the conventional global optimization problem, where typically only one solution is sought. Another issue are diversity requirements, which could be formulated by demanding $\forall\boldsymbol{x},\boldsymbol{y}\in P,\boldsymbol{x}\neq\boldsymbol{y}:d( \boldsymbol{x},\boldsymbol{y})>\epsilon$, i. e., the distance between any two solutions may not be smaller than some threshold $\epsilon$. Alternatively, also more sophisticated diversity measures on the set $P$ may be calculated, maybe leading to a multiobjective formulation of the problem [18]. However, we will stick to the basic task of finding all optima ($h=\infty$) here. ## 3 Methods ``` 1: generate an initial design $D\subset\mathcal{X}$ 2: $\boldsymbol{y}\gets f(D)$ 3: while total evaluation budget is not exceeded do 4: fit surrogate on $D$ and obtain $\hat{f}$, $\hat{s}$ 5: get new design point $\boldsymbol{x}^{\prime}$ by optimizing the infill criterion based on $\hat{f}$, $\hat{s}$ 6: $y^{\prime}\gets f(\boldsymbol{x}^{\prime})$// evaluate new point 7: $D\leftarrow(D,\boldsymbol{x}^{\prime})$// update design 8: $\boldsymbol{y}\leftarrow(\boldsymbol{y},y^{\prime})$// update responses 9: end while 10: return $\hat{y}^{}=\min(\boldsymbol{y})$ and the associated $\hat{\boldsymbol{x}}^{}$ ``` Algorithm 1 Sequential model-based optimization Algorithm 1 illustrates the general sequential model-based optimization (MBO) framework, of which EGO is an instantiation. The main idea in model-based optimization is to approximate the expensive function $f(\boldsymbol{x})$ in every iteration by a regression model, which is much cheaper to evaluate. This is also called a meta-model or surrogate. We are using a Kriging model, which not only provides a direct estimation $\hat{f}(\boldsymbol{x})$ of the true function value $f(\boldsymbol{x})$ but also an estimation of the prediction standard error $\hat{s}(\boldsymbol{x})$, also called a local uncertainty measure. Our Kriging implementation follows the “empirical Bayes” approach with a correlation kernel and a maximum likelihood estimation of its parameters [11]. The whole MBO concept has roots in response surface methodology, which was originally applied to physical experiments (with a human in the loop) [5, pp. 7–11]. It starts by exploring the parameter space with an initial design, often constructed in a space-filling fashion. The main sequential loop can be divided into two alternating stages: 1. Fit a response surface to training data (including estimation of the model’s parameters). 2. Use the surface to compute new search points under the assumption that the parameters are correct. In [12], the now standard expected improvement criterion was proposed. It is defined as \[\text{EI}(\boldsymbol{x}) =\mathbb{E}[\max\{0,\hat{y}^{}-\hat{f}\}]=\left(\hat{y}^{}-\hat {f}(\boldsymbol{x})\right)\leavevmode\nobreak\ \Phi\left(\dfrac{\hat{y}^{}- \hat{f}(\boldsymbol{x})}{\hat{s}(\boldsymbol{x})}\right)+\hat{s}(\boldsymbol{x })\leavevmode\nobreak\ \phi\left(\dfrac{\hat{y}^{}-\hat{f}(\boldsymbol{x})}{ \hat{s}(\boldsymbol{x})}\right)\,,\] where $\phi$ and $\Phi$ are the density and cumulative distribution function of the standard normal distribution, respectively. Hence, the sought point is $\boldsymbol{x}^{}=\arg\max_{\boldsymbol{x}\in\mathcal{X}}\text{EI}( \boldsymbol{x})$. ``` 0: points $\mathcal{P}=\{\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{N}\}$, number $k$ of nearest neighbors 0: nodes of the topograph with no outgoing edges 1: create a directed graph $G=(V,E)$ with $V=\{v_{1},\dots,v_{N}\}$ and $E=\emptyset$ 2: for all $i\in\{1,\dots,N\}$ do 3: $J\leftarrow$ indices of the $k$ nearest neighbors of $\boldsymbol{x}_{i}$ 4: for all $j\in J$ do 5: if $f(\boldsymbol{x}_{j})<f(\boldsymbol{x}_{i})$ then 6: $E\gets E\cup\{(v_{i},v_{j})\}$// add edge to graph 7: else if $f(\boldsymbol{x}_{i})<f(\boldsymbol{x}_{j})$ then 8: $E\gets E\cup\{(v_{j},v_{i})\}$// add edge to graph 9: end if 10: end for 11: end for 12: return $\{v\in V\mid\operatorname{deg}^{+}(v)=0\}$// select nodes with no outgoing edges ``` Algorithm 2* Topographical selection Topographical selection is provided as pseudo-code in algorithm 2. It has similarities to nearest-better clustering because the basic idea is to compare points regarding their quality to their closest neighbors. While NBC argues with relative distances, TS relies on fixed-size $k$-neighborhoods, such that $k$ is a parameter of the algorithm. We start with an empty graph and for every point, and detect the $k$ nearest neighbors. For each neighbor, we add an edge that points from the worse to the better point. After finishing the loop, we return all points that have only incoming edges as basin representatives. As performance measure for an approximation set $P$ in the case of global optimization, we use the deviation from the global optimum $f_{\Delta}=\hat{y}^{}-f_{1}^{}$. In the multi-local case the number of found optima $o=\|\{\boldsymbol{x}^{}\in X^{}\mid d_{\mathrm{nn}}(\boldsymbol{x}^{},P)\leq r\}\|$ divided by the total number of optima $\|X^{}\|=\nu$ as peak ratio $\operatorname{PR}(P)=o/\nu$ is employed [26, 23]. Another measure is the averaged Hausdorff distance (AHD) [20] by using $X^{}$ as a reference set. The function $d_{\mathrm{nn}}(\boldsymbol{x},X)$ denotes the Euclidean distance of a point $\boldsymbol{x}$ to its nearest neighbor in a set of points $X$. Magnitude \| Application ---\|--- n⋅101 \| Initial designs in model-based optimization [12] n⋅102 \| Expensive optimization n⋅103 \| n⋅104 \| Budget of the CEC 2005 competition [22] n⋅105 \| Budget of the CEC 2013 niching competition [13] n⋅106 \| Budget of the black-box optimization benchmark (BBOB) [9] Table 1: Different magnitudes for the number of function evaluations Nf. Problem name \| Dim. n \| #Local optima \| #Global optima \| Ref. ---\|---\|---\|---\|--- Shekel5 \| 4 \| 5 \| 1 \| [6] Shekel7 \| 4 \| 7 \| 1 \| [6] Shekel10 \| 4 \| 10 \| 1 \| [6] Hartman3 \| 3 \| 3 \| 1 \| [6] Hartman6 \| 6 \| 2 \| 1 \| [6] Goldstein-Price \| 2 \| 5 \| 1 \| [6] Branin \| 2 \| 3 \| 3 \| [6] Vincent \| 2 \| 36 \| 36 \| [13] Vincent \| 3 \| 216 \| 216 \| [13] Modified Rastrigin \| 4 \| 48 \| 1 \| [4] Modified Rastrigin \| 8 \| 48 \| 1 \| [4] Six-hump camelback \| 2 \| 6 \| 2 \| [13] Table 2: Test problems used in the experiment. <figure><img src="content_image/1704.05724/x4.png"><figcaption>Figure 2: Convergence graphs for approximating the global optimum. Note thelogarithmic scale.</figcaption></figure> ## 4 Experiment _Research question:_ How does the assessment of optimization algorithms depend on the performance measurement in expensive optimization, i. e., are the results in global optimization different from multi-local optimization? _Pre-experimental planning:_ Table 1 shows how some budgets are associated with research areas and benchmarks. Measuring consumed resources simply as the number of objective function evaluations is generally deemed admissible if this number is small, because then the assumption of expensive function evaluations in relation to the overhead of an optimization algorithm holds. For extremely large budgets, this is rather unlikely [7]. However, in expensive optimization, often very computationally demanding algorithms are used, so it is also an interesting question where the actual break-even point between two optimization algorithms is in terms of wall clock time. <figure><img src="content_image/1704.05724/x5.png"><figcaption>Figure 3: Averaged Hausdorff distances (AHD) over the course of optimization.</figcaption></figure> _Task:_ We assume an anytime scenario for assessment, that is, the algorithms could be stopped at any time. We record three different performance measures over the course of optimization, namely the deviation from the global optimum, the peak ratio (PR), and the averaged Hausdorff distance (AHD). For PR, the position of an optimum is considered as approximated if a point is within a Euclidean distance of $0.01$ in the normalized search space (see setup below). AHD is used with an exponent of one. For PR and AHD, the solutions up to the measuring point are filtered by topographical selection (TS), to stay close to a real-world scenario. Topographical selection, originally proposed by [24], contains a parameter $k$, specifying a number of neighbors. To determine this parameter, we use the model \[k(n,N)=0.215n+0.74N^{1/2}\,,\] depending on the dimension $n$ and the number of points $N$. It was developed in [29] for random uniform samples. Although the solution sets produced by the optimization algorithms are not uniformly distributed (except for MmLHS, see below), we feel certain that this is not a severe problem, as TS proved quite robust to changes in the distribution in previous experiments [29]. <figure><img src="content_image/1704.05724/x6.png"><figcaption>Figure 4: Peak ratio (PR, r=0.01) over the course of optimization.</figcaption></figure> _Setup:_ Table 2 contains the test problems used in this experiment. They consist of the classic test set for global optimization by Dixon and Szegö [6], and some problems taken from the 2013 niching competition [13]. The former problems mostly contain fewer minima than the latter ones. However, the latter ones in part have other properties that make them easier, i. e., separability (modified Rastrigin) or no local optima (Vincent). All problems have in common the rather low dimension, bound constraints, and the fact that positions of _all_ local and global minima are known. The last aspect is crucial for carrying out the assessment in the multi-local case with PR and AHD. The search spaces are always normalized to the unit hypercube. <figure><img src="content_image/1704.05724/x7.png"><figcaption>Figure 5: The number of basins identified by topographical selection aftereach evaluation.</figcaption></figure> We compare three different algorithms, namely CMA-ES, EGO, and a maximin Latin hypercube sampling (MmLHS). MmLHS acts as a representative of random search here. Our MmLHS designs are produced on-the-fly by a greedy construction heuristic. Thus, they are not exactly optimal according to the maximin-distance criterion, but possess a significantly higher uniformity than random uniform sampling (see [28, p. 58] for details). For EGO, we try two variants, which only differ in the amount of function evaluations invested into the intial sample. The sample size is determined as $cn$, with $c=2$ or $c=10$, in accordance with Tab. 1. The initial sample for EGO is also drawn by MmLHS, thus MmLHS alone can be seen as a limiting case for EGO where the whole budget is spent on the initial sample. EGO’s Kriging model uses the power-exponential kernel \[\operatorname{corr}(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=\exp\left(-\sum_{ \ell=1}^{n}\theta_{\ell}\|x_{i,\ell}-x_{j,\ell}\|^{p}\right)\,.\] For the kernel parameters, we require $-2\leq\log_{10}(\theta_{\ell})\leq 2$ and $0.5\leq p\leq 2$. As virtually all contemporary EGO implementations, our code differs from the algorithm in [12] in the way the infill criterion is optimized. Instead of a branch-and-bound approach, which consumes a lot of memory and restricts the kernel choice, we simply use CMA-ES, started once from the best of $100n$ uniformly distributed points. Also the likelihood function for fitting the model is optimized with CMA-ES, based on recommendations in [17]. CMA-ES is the candidate in this test set with a strong focus on local search. We use version 1.1.7 of the Python implementation¹. Its “tolfun” and “tolfunhist” stopping criteria are set to $10^{-3}$ and $10^{-5}$, respectively, to stop really early and thus potentially have some budget left for starting another search. The starting points for CMA-ES are drawn by the maximin reconstruction algorithm, as recommended in [28]. The initial step size is set to $0.15$. [FOOTNOTE:1][ENDFOOTNOTE] In total, this experimental setup is chosen deliberately rather in favor of EGO than of CMA-ES, by including the test problems EGO was originally proposed for [12]. Additionally, CMA-ES is geared to being a very _robust_ black-box optimizer, so it is not necessarily the most efficient one on these low-dimensional, continuously differentiable problems [19]. _Results:_ Figures 2, 3, and 4 illustrate the development of the three indicators over the course of optimization. Additionally, we report the number of selected solutions in Fig. 5. Figure 6 shows the wall clock times of the algorithms. The curves contain the time for running the algorithm for the number of evaluations specified on the x-axis, plus the time for executing topographical selection once. In all figures, the curves represent mean values over 75 stochastic replications, with a 95% confidence interval for the mean under normality assumption. _Observations:_ Figure 2 shows that even under the quite restricted budget of 500 evaluations, CMA-ES significantly beats EGO on some problems EGO was developed for, if only the deviation from one global optimum counts. The variance is larger for CMA-ES, because some runs naturally only converge to local optima, but the average performance is clearly better. However, EGO is always better than CMA-ES in terms of AHD and PR. On problems with a large number of optima, MmLHS obtains a still better AHD than EGO, but the optima are not approximated very well. Thus, EGO always has the better peak ratio. With two exceptions, its peak ratio is also always better than that of CMA-ES. Only few significant differences can be found between the choices $c=2$ and $c=10$ for EGO, but the results seem to be slightly in favor of $c=2$. Figure 5 shows that the number of selected solutions is mostly nicely correlated with the problems’ actual number of optima. On Branin and Shekel5, even the correct number of optima is reliably identified towards the end of the optimization. However, the approximation quality does not satisfy the PR criterion for all optima. The running times of the algorithms in wall clock time naturally differ by orders of magnitude (see Fig. 6). The curve for MmLHS is slightly misleading, as the whole 500-point sample was computed in advance. So the curve begins with this cost and subsequently adds the cumulative cost of the test functions plus the cost for one topographical selection. In reality, MmLHS is of course always the cheapest algorithm in this setting, because one would not sample more points than one wants to evaluate. <figure><img src="content_image/1704.05724/x8.png"><figcaption>Figure 6: The wall clock time required by the algorithms.</figcaption></figure> _Discussion:_ The results show that EGO represents an intermediate strategy between CMA-ES and MmLHS regarding the exploration-exploitation trade-off. It has the potential to detect several attraction basins with quite small budgets, but lacks the ability to approximate the corresponding optima with high precision. The $f_{\Delta}$ may stagnate for several hundred function evaluations. On the other hand, performance measures from multimodal optimization do often keep improving during this time. The bends in the curves of CMA-ES in Fig. 2 are probably due to the aggressive stopping criteria, which prevent the algorithm from approximating the global optimum to a higher precision. This is the only explanation on problems as Branin or Vincent, which only contain global optima, and where we would expect a linear convergence behavior otherwise. However, this is not to be seen as a drawback, as we deliberately chose this setting to obtain a better global search, and the restarts do clearly improve other measures (see Figs. 3 and 4). Tuning the initial step size might improve the CMA-ES performance slightly more. However, note that any improvement of CMA-ES would further strengthen the support for our hypothesis, so the omission is not critical. ## 5 Conclusions We showed that _efficient global optimization_ (EGO) is in fact not always the best algorithm for global optimization, i. e., the application it was originally designed for, except for extremely small budgets of approximately up to $200$ function evaluations. By our experimental setup, this statement is restricted to rather low-dimensional ($2\leq n\leq 8$), smooth, and generally well-behaved objective functions. However, as higher-dimensional, more multimodal, and/or non-continuous functions would pose even more difficulties to the meta-modelling, and other optimization algorithms as, e. g., CMA-ES are inherently more robust to such difficulties, because they use less assumptions about the problem, the statement might be extended to broader settings in the future. Of course more sophisticated sequential model-based optimization algorithms do already exist, and may not share some of the basic EGO’s weaknesses in global optimization, but our point is that EGO is actually fairly well-suited for the slightly different problem definition of finding multiple optima, when the optimization problem is expensive. Thus, combining it with a suitable basin identification heuristic makes it a strong competitor in this domain. In future work, we shall look again at the basin identification mechanism and find better values or heuristics for the $k$ parameter, or even another algorithm altogether. Also, additional model-based optimization algorithms may be tested to find out if there exists an approach that is competitive in both the global and multi-local optimization case. Finally, the methods shall be thoroughly benchmarked on a larger set of problems in order to make stronger claims on their strengths and weaknesses. ## References [1] Anne Auger and Nikolaus Hansen. Performance evaluation of an advanced local search evolutionary algorithm. In _IEEE Congress on Evolutionary Computation (CEC)_, volume 2, pages 1777–1784, 2005. * [2] Bernd Bischl, Simon Wessing, Nadja Bauer, Klaus Friedrichs, and Claus Weihs. Moi-mbo: Multiobjective infill for parallel model-based optimization. In _Learning and Intelligent Optimization, Lion 8_, pages 173–186. Springer, 2014. * [3] Adam D. Bull. Convergence rates of efficient global optimization algorithms. _Journal of Machine Learning Research_, 12:2879–2904, 2011. * [4] Kalyanmoy Deb and Amit Saha. Multimodal optimization using a bi-objective evolutionary algorithm. _Evolutionary Computation_, 20(1):27–62, 2012. * [5] Enrique del Castillo. _Process Optimization_, volume 105 of _International Series in Operations Research & Management Science_. Springer, 2007. * [6] L.C.W. Dixon and G.P. Szegö. The global optimization problem: an introduction. In L.C.W. Dixon and G.P. Szegö, editors, _Towards Global Optimisation 2_, pages 1–15. North-Holland, Amsterdam, 1978. * [7] Agoston E. Eiben and Mark Jelasity. A critical note on experimental research methodology in EC. In _IEEE Congress on Evolutionary Computation (CEC)_, volume 1, pages 582–587, 2002. * [8] Paul Feliot, Julien Bect, and Emmanuel Vazquez. A Bayesian approach to constrained single- and multi-objective optimization. _Journal of Global Optimization_, 67(1):97–133, 2017. * [9] Steffen Finck, Nikolaus Hansen, Raymond Ros, and Anne Auger. Real-parameter black-box optimization benchmarking 2009: Presentation of the noiseless functions. Technical Report 2009/20, Research Center PPE, 2009. Updated February 2010. * [10] David Ginsbourger, Rodolphe Le Riche, and Laurent Carraro. Kriging is well-suited to parallelize optimization. In Yoel Tenne and Chi-Keong Goh, editors, _Computational Intelligence in Expensive Optimization Problems_, pages 131–162. Springer, 2010. * [11] Donald R. Jones. A taxonomy of global optimization methods based on response surfaces. _Journal of Global Optimization_, 21(4):345–383, 2001. * [12] Donald R. Jones, Matthias Schonlau, and William J. Welch. Efficient global optimization of expensive black-box functions. _Journal of Global Optimization_, 13(4):455–492, 1998. * [13] Xiadong Li, Andries Engelbrecht, and Michael G. Epitropakis. Benchmark functions for CEC’2013 special session and competition on niching methods for multimodal function optimization. Technical report, RMIT University, Evolutionary Computation and Machine Learning Group, Australia, 2013. * [14] Ian C. Parmee. The maintenance of search diversity for effective design space decomposition using cluster-oriented genetic algorithms (COGA) and multi-agent strategies (GAANT). In _Proceedings of 2nd International Conference on Adaptive Computing in Engineering Design and Control, ACEDC ’96_, pages 128–138. University of Plymouth, 1996. * [15] Mike Preuss. Improved topological niching for real-valued global optimization. In _Applications of Evolutionary Computation_, volume 7248 of _Lecture Notes in Computer Science_, pages 386–395. Springer, 2012. * [16] Mike Preuss. _Multimodal Optimization by Means of Evolutionary Algorithms_. Springer, 2015. * [17] Mike Preuss, Günter Rudolph, and Simon Wessing. Tuning optimization algorithms for real-world problems by means of surrogate modeling. In _Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation_, GECCO ’10, pages 401–408. ACM, 2010. * [18] Mike Preuss and Simon Wessing. Measuring multimodal optimization solution sets with a view to multiobjective techniques. In _EVOLVE – A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV_, volume 227 of _Advances in Intelligent Systems and Computing_, pages 123–137. Springer, 2013. * [19] Luis Miguel Rios and Nikolaos V. Sahinidis. Derivative-free optimization: a review of algorithms and comparison of software implementations. _Journal of Global Optimization_, 56(3):1247–1293, 2013. * [20] Oliver Schütze, Xavier Esquivel, Adriana Lara, and Carlos A. Coello Coello. Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization. _IEEE Transactions on Evolutionary Computation_, 16(4):504–522, 2012. * [21] András Sóbester, Stephen J. Leary, and Andy J. Keane. On the design of optimization strategies based on global response surface approximation models. _Journal of Global Optimization_, 33(1):31–59, 2005. * [22] Ponnuthurai N. Suganthan, Nikolaus Hansen, Jing Jane Liang, Kalyanmoy Deb, Ying-Ping Chen, Anne Auger, and Santosh Tiwari. Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. Technical report, Nanyang Technological University, Singapore, May 2005. * [23] René Thomsen. Multimodal optimization using crowding-based differential evolution. In _IEEE Congress on Evolutionary Computation_, volume 2, pages 1382–1389, 2004. * [24] Aimo Törn and Sami Viitanen. Topographical global optimization. In Christodoulos A. Floudas and Panos M. Pardalos, editors, _Recent Advances in Global Optimization_, Princeton Series in Computer Sciences, pages 384–398. Princeton University Press, 1992. * [25] Aimo Törn and Antanas Žilinskas. _Global Optimization_, volume 350 of _Lecture Notes in Computer Science_. Springer, 1989. * [26] Rasmus K. Ursem. Multinational evolutionary algorithms. In Peter J. Angeline, editor, _Proceedings of the Congress of Evolutionary Computation (CEC 99)_, volume 3, pages 1633–1640. IEEE Press, 1999. * [27] Rasmus K. Ursem. From expected improvement to investment portfolio improvement: Spreading the risk in kriging-based optimization. In _Parallel Problem Solving from Nature – PPSN XIII: 13th International Conference, Proceedings_, pages 362–372. Springer, 2014. * [28] Simon Wessing. _Two-stage methods for multimodal optimization_. PhD thesis, Technische Universität Dortmund, 2015. URL http://hdl.handle.net/2003/34148. * [29] Simon Wessing, Günter Rudolph, and Mike Preuss. Assessing basin identification methods for locating multiple optima. In Panos M. Pardalos, Anatoly Zhigljavsky, and Julius Žilinskas, editors, _Advances in Stochastic and Deterministic Global Optimization_, pages 53–70. Springer, 2016.
0910.2812	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 5016, "num_imgs": 0, "llama3_tokens_count": 1781 }	[]	# The Algol type eclipsing binary X Tri: BVRI modelling and O-C diagram analysis Liakos, A.${}^{1}$, Zasche, P.${}^{2,3}$ and Niarchos, P.${}^{1}$ ###### Abstract CCD photometric observations of the Algol-type eclipsing binary X Tri have been obtained. The light curves are analyzed with the Wilson-Devinney (WD) code and new geometric and photometric elements are derived. A new O-C analysis of the system, based on the most reliable timings of minima found in the literature, is presented and apparent period changes are discussed with respect to possible and multiple Light-Time Effect (LITE) in the system. Moreover, the results for the existence of additional bodies around the eclipsing pair, derived from the period study, are compared with those for extra luminosity, derived from the light curve analysis. ${}^{1}$Department of Astrophysics, Astronomy and Mechanics, National and Kapodistrian University of Athens, GR 157 84 Zografos, Athens, Greece ${}^{2}$Astronomical Institute, Faculty of Mathematics and Physics, Charles University Prague, CZ-180 00 Praha 8, V Holešovičkách 2, Czech Republic ${}^{3}$Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 70-264, México, DF 04510, Mexico ## 1 Observations and analyses The system was observed during 5 nights from October 2008 to January 2009 at the Athens University Observatory, using a 40-cm Cassegrain telescope equipped with the CCD camera ST-8XMEI and B, V, R, I Bessell filters. The light curves (hereafter LCs) were analysed with the $PHOEBE~{}0.29d$ software (Prša & Zwitter, 2005) which uses the 2003 version of the WD code (Wilson & Devinney, 1971; Wilson, 1979). Due to the lack of spectroscopic mass ratio of the system, the q-search method was applied in Mode 2, 4 and 5 in order to find the most probable value of the (photometric) mass ratio (q). The least squares method with statistical weights has been used for the analysis of the O-C diagram. The current O-C diagram of X Tri includes 572 times of minima taken from the literature. The following ephemeris: $Min.I=HJD~{}2422722.285+0.9715341^{d}\times E$(Kreiner et al., 2001) was used as the initial one for the O-C analysis of the compiled times of minima. ## 2 Discussion and Conclusions The results of the LCs solution (see figure 1 and table 1) show that X Tri is a semi-detached system with the secondary star filling its Roche Lobe. The periodic variations of the orbital period of the system could be explained by adopting the existence of three additional components, which were found to have minimal masses 0.18, 0.24 and 0.22, respectively (see figure 1 and table 2). An extra light contribution to the luminosity of the EB was taken into account in the LCs solution but it was found to be less than 1%. Such a small extra luminosity could be explained by the small values of the minimal masses of the possible additional components found. The steady decrease rate of its period is probably due to angular momentum loss, since the direction of the flow (from the more massive to the less massive component), revealed from the O-C diagram analysis, comes in disagreement with the one derived from the LCs analysis. Parameter \| value \| Parameter \| value ---\|---\|---\|--- q (m2/m1) \| 0.599 (2) \| \| B \| V \| R \| I i (deg) \| 87.9 (1) \| x1∗∗∗ \| 0.551 \| 0.478 \| 0.402 \| 0.322 T1∗∗(K), T2 (K) \| 8600, 5188 (4) \| x2∗∗∗ \| 0.835 \| 0.692 \| 0.597 \| 0.503 A1∗, A2∗ \| 1, 0.5 \| L1/LT \| 0.893 (2) \| 0.839 (2) \| 0.795 (2) \| 0.739 (1) g1∗, g2∗ \| 1, 0.32 \| L2/LT \| 0.107 (1) \| 0.160 (2) \| 0.201 (2) \| 0.246 (3) Ω1, Ω2 \| 4.27 (1), 3.06 \| L3/LT \| 0.000 (1) \| 0.000 (1) \| 0.004 (1) \| 0.015 (2) χ2 \| 1.278 \| \| \| \| \| ∗assumed, \| ∗∗ Giuricin et al. (1983), \| ∗∗∗Van Hamme (1993), \| LT=L1+L2+L3 \| \| \| Table 1: The parameters of X Tri derived from the LCs solutions Parameters of the EB \| value \| Parameters of the LITEs \| value ---\|---\|---\|--- M∗1+M2 (M⊙) \| 2.1 + 1.26 \| \| 3rdbody \| 4thbody \| 5thbody Min.I (HJD) \| 2442502.731 (2) \| P (yrs) \| 36.9 (5) \| 22.4 (3) \| 16.8 (4) P (days) \| 0.9715318 (2) \| T0 (HJD), ω (deg) \| 2452916 (373), 220 (98) \| 2455069 (335), 34 (13) \| –, – c2 (×10−10) \| -2.0308 (2) \| A (days), e \| 0.0052 (3), 0.2 (2) \| 0.0040 (4), 0.5 (1) \| 0.003 (2), 0.0 ˙P (×10−10) \| -1.5269 (2) \| Mmin (M⊙) \| 0.18 (1) \| 0.24 (1) \| 0.22 (1) ∗assumed \| \| \| \| \| Table 2: The results of the O-C diagram analysis for X Tri [FIGURE:S2.F1][ENDFIGURE] ## References * Giuricin et al. (1983) Giuricin, G., Mardirossian, F. & Mezzetti, M. 1983, ApJSS, 52, 35 * Kreiner et al. (2001) Kreiner, J. M., Kim, C.-H., & Nha, I.-S. 2001, $An~{}Atlas~{}of~{}O-C~{}diagrams~{}of~{}Eclipsing~{}Binary~{}Stars$, Cracow Pedagogical University Press * Prša & Zwitter (2005) Prša, A., & Zwitter, T. 2005, ApJ, 628, 426 * Van Hamme (1993) van Hamme, W. 1993, AJ, 106, 2096 * Wilson & Devinney (1971) Wilson, R. E., & Devinney, E. J. 1971, ApJ, 166, 605 * Wilson (1979) Wilson, R. E. 1979, ApJ, 234, 1054
1306.6647	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 56825, "num_imgs": 8, "llama3_tokens_count": 18737 }	[ "content_image/1306.6647/x1.png", "content_image/1306.6647/x2.png", "content_image/1306.6647/x3.png", "content_image/1306.6647/x4.png", "content_image/1306.6647/x5.png", "content_image/1306.6647/x6.png", "content_image/1306.6647/x9.png", "content_image/1306.6647/x10.png" ]	# Photometric and spectroscopic variability of the FUor star V582 Aurigae E. H. Semkov 1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose blvd., BG-1784 Sofia, Bulgaria, 1esemkov@astro.bas.bg S. P. Peneva 1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose blvd., BG-1784 Sofia, Bulgaria, 1esemkov@astro.bas.bg U. Munari 2INAF Osservatorio Astronomico di Padova, Sede di Asiago, I-36032 Asiago (VI), Italy 2 M. Dennefeld 3Institut d’Astrophysique de Paris, CNRS, and Université Pierre et Marie Curie, 98 bis Boulevard Arago, 75014 Paris, France 3 H. Mito 4Kiso Observatory, Institute of Astronomy, School of Science, the University of Tokyo, Mitake-mura, Kiso-gun, Nagano-ken 397-0101, Japan 4 D. P. Dimitrov 1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose blvd., BG-1784 Sofia, Bulgaria, 1esemkov@astro.bas.bg S. Ibryamov 1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose blvd., BG-1784 Sofia, Bulgaria, 1esemkov@astro.bas.bg K. A. Stoyanov 1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose blvd., BG-1784 Sofia, Bulgaria, 1esemkov@astro.bas.bg Received ; accepted Key Words.:stars: pre-main sequence – stars: variables: T Tauri, Herbig Ae/Be – stars: individual: V582 Aur† [FOOTNOTE:†][ENDFOOTNOTE] ###### Abstract Context: Aims:We present results from optical photometric and spectroscopic observations of the eruptive pre-main sequence star V582 Aur. Variability of the star was reported a few years ago when it was suspected as a possible FU Orionis object. Due to the small number of currently known FUors, a new object of this type is ideal target for follow-up photometric and spectroscopic observations. Methods:We carried out $BVRI$ CCD photometric observations in the field of V582 Aur from 2009 August to 2013 February. We acquired high-, medium-, and low-resolution spectroscopy of V582 Aur during this period. To study the pre-outburst variability of the target and construct its historical light curve, we searched for archival observations in photographic plate collections. Both CCD and photographic observations were analyzed using a sequence of 14 stars in the field of V582 Aur calibrated in $BVRI$. Results:The pre-outburst photographic observations of V582 Aur show low-amplitude light variations typical of T Tauri stars. Archival photographic observations indicate that the increase in brightness began in late 1984 or early 1985 and the star reached the maximum level of brightness at 1986 January. The spectral type of V582 Aur can be defined as G0I with strong P Cyg profiles of H$\alpha$ and Na I D lines, which are typical of FU Orionis objects. Our $BVRI$ photometric observations show large amplitude variations $\Delta V\sim 2\aas@@fstack{m}8$) during the 3.5 year period of observations. Most of the time, however, the star remains in a state close to the maximum brightness. The deepest drop in brightness was observed in the spring of 2012, when the brightness of the star fell to a level close to the pre-outburst. The multicolor photometric data show a color reversal during the minimum in brightness, which is typical of UX Ori variables. The corresponding spectral observations show strong variability in the profiles and intensities of the spectral lines (especially H$\alpha$), which indicate significant changes in the accretion rate. On the basis of photometric monitoring performed over the past three years, the spectral properties of the maximal light, and the shape of the long-term light curve, we confirm the affiliation of V582 Aur to the group of FU Orionis objects. Conclusions: ## 1Introduction Photometric and spectral variability is the most common characteristic of pre-main sequence (PMS) stars. Both classes of PMS stars, the wide-spread low mass ($\it M$**$\leq$$2M_{\sun}$) T Tauri stars and the more massive Herbig Ae/Be stars, show various types of variability. The main physical mechanisms causing the brightness variation of PMS stars are defined by Herbst et al. (1994, 2007) and Bouvier et al. (1995). The study of the large amplitude brightness variations of PMS stars is of great importance in understanding stellar evolution. These variations comprise transient increases in brightness (outbursts), temporary drops in brightness (eclipses), and large amplitude irregular or regular variations for a short or long time scales. In many cases large amplitude variations in brightness are accompanied by changes of the spectral type or by variability in the profiles and the presence of individual spectral lines (Herbig et al. 2003, Grinin et al. 2001, Rodgers et al. 2002, Kurosawa & Romanova 2013). This is especially true for the H$\alpha$ emission line, which is the most prominent feature in the PMS spectra. The large amplitude outbursts of PMS stars can be grouped into two main types, named after their respective prototypes: FU Orionis (FUor; Ambartsumian 1971) and EX Lupi (EXor; Herbig 1989). The flare-up of FU Orionis itself occurred in 1936 and for several decades it was the only known object of that type. Herbig (1977) defined FUors as a class of young variables after the discovery of two new FUor objects, V1057 Cyg and V1515 Cyg. Several more objects were assigned to this class of young variables over the next four decades (see Reipurth & Aspin 2010 and references therein). Because only a small number of FUor stars have been detected to date, photometric and spectral studies of every new object are of great interest. Due to the large-scale optical and infrared monitoring programs carried out in several observatories and the contributions of amateur astronomers, some new objects have been observed to undergo large amplitude outbursts, V733 Cep (Reipurth et al. 2007, Peneva et al. 2010), V2493 Cyg (Semkov et al. 2010, 2012; Miller et al. 2011, Kóspál et al. 2011), V2492 Cyg (Aspin 2011, Hillenbrand et al. 2013, Kóspál et al. 2013), V2494 Cyg (Aspin et al. 2009a), V2495 Cyg (Movsessian et al. 2006), V900 Mon (Reipurth et al. 2012), V2775 Ori (Fischer et al. 2012). A typical outburst of FUor objects can last for several decades, and the rise time is shorter than that of the decline. All known FUors share the same defining characteristics: a $\Delta$$V$$\approx$4-6 mag. outburst amplitude, association with reflection nebulae, location in star-forming regions, an F-G supergiant spectrum during outbursts, a strong LiI 6707 Å line in absorption, and CO bands in near-infrared spectra (Herbig 1977, Reipurth & Aspin 2010). Typically the decrease in brightness goes smoothly, but several events of temporary drops in brightness have been registered in the cases of V1515 Cyg (Kenyon et al. 1991, Clark et al. 2005), V733 Cep (Peneva et al. 2010) and V1735 Cyg (Peneva et al. 2009). An important feature of FUors is the massive supersonic wind observed as a P Cyg profile most commonly for both H$\alpha$ and Na I D lines. The EXor objects undergo frequent, irregular, and relatively brief (a few weeks to a few months or one year) outbursts with amplitudes $\Delta$$V$$\approx$2-5 mag. During such events, the cool spectrum of the quiescence is veiled, and strong emission lines from single ionized metals are observed together with appearance of reversed P-Cyg absorption components (Herbig 2007). Both types of eruptive stars, FUors and EXors, seem to be related to the low-mass PMS objects (T Tauri stars), which have massive circumstellar disks. These objects have been classified in terms of their wide range of available photometric and spectral properties, but their outbursts are thought to have the same cause: an enhanced accretion rate from the circumstellar disk onto the central star (Hartmann & Kenyon 1996, Herbig 2007). At the time of their outbursts, FUor objects undergo an increase in their accretion rate from $\sim$10${}^{-7}$$M_{\sun}$$/$yr up to $\sim$10${}^{-4}$$M_{\sun}$$/$yr. The periods of enhanced accretion are thought to be triggered by thermal or gravitational instability in the circumstellar disk (Hartmann & Kenyon 1996, Zhu et al. 2009). Another possible triggering mechanism could be the interactions of the circumstellar disk with a planet or nearby stellar companion on an eccentric orbit (Lodato & Clarke 2004, Reipurth & Aspin 2004, Pfalzner 2008). For a period of $\sim$ 100 years, the circumstellar disk adds $\sim 10^{-2}$ M$\sun$ onto the central star and ejects $\sim$10% of the accreting material in a high-velocity stellar wind. Some FUor objects were found to exhibit periodic spectroscopic (Herbig et al. 2003, Powell et al. 2013) or low amplitude photometric (Kenyon et al. 2000, Green et al. 2013, Siwak et al. 2013) variability in short time-scale (days). Approximately 25$\%$ of Herbig Ae/Be stars and some T Tauri stars of F-G type show strong photometric variability with sudden quasi-Algol drops of brightness and amplitudes up to 3 mag. ($V$) (Natta et al. 1997). During the deep minima of brightness, an increase of polarization and specific color variability are observed. The prototype of this group of PMS objects with intermediate mass named UXors is UX Ori. The widespread explanation of its variability are variable obscurations from orbiting circumstellar clumps of dust or edge-on circumstellar disks (Grinin et al. 1991). The discovery of a new FUor candidate in Auriga was reported by Anton Khruslov (Samus 2009). The star was cataloged as USNO A2.0 1200-03303169 and according to the General Catalog of Variable Stars as V582 Aur. V582 Aur is located in a region of active star formation near Auriga OB2 association (Fig. 1). Samus (2009) examined the brightness of the star on photographic plates from the Moscow collection (1965-1992) and on the images from the Digitized Sky Survey (1954-1993), which suggests that the increase in brightness started between 1982 and 1986. Munari et al. (2009) obtained the first low-resolution spectrum of V582 Aur on 2009 August 6 and registered the presence of absorption lines of the Balmer series, Na I D and Ba II ($\lambda$ 6496) and the absence of the Li I ($\lambda$ 6707) line in the spectrum. The photometric observations of V582 Aur reported by Munari et al. (2009) show the star near to maximal brightness on 2009 August 5 and 6. Using the mid-infrared part of the AKARI All-Sky Survey, Takita et al. (2010) identified V582 Aur as a T Tauri star candidate. Our first results from photometric and spectral observations of the star were reported in Semkov et al. (2011). In the paper we come to the conclusion that V582 Aur has all observational characteristics of FUor objects. <figure><img src="content_image/1306.6647/x1.png"><figcaption>Figure 1: Color image of V582 Aur obtained with the 2 m RCC telescope in NAORozhen. A faint cometary nebula arising from the star is clearly visible.</figcaption></figure> Recent data from photometric and spectral observations and results from archival photographic plate measurements are reported in the present paper. We try to collect regular observations (spectroscopy and multicolor photometry) of V582 Aur in order to clarify the nature of variability of the object. ## 2Observations ### Photometric CCD observations The CCD photometric observations of V582 Aur were performed with the 2 m RCC, the 50/70 cm Schmidt, and the 60 cm Cassegrain telescopes of the National Astronomical Observatory (NAO) Rozhen (Bulgaria) and with the 1.3 m RC telescope of the Skinakas Observatory¹ of the Institute of Astronomy, University of Crete (Greece). Observations were performed with four types of the CCD camera Vers Array 1300B at the 2 m RCC telescope, ANDOR DZ436-BV at the 1.3 m RC telescope, FLI PL16803 at the 50/70 cm Schmidt telescope, and FLI PL9000 at the 60 cm Cassegrain telescope. All frames were exposed through a set of standard Johnson-Cousins filters. All the data were analyzed using the same aperture, which was chosen as 4″in radius (while the background annulus was from 9″to 14″) in order to minimize the light from the surrounding nebula. [FOOTNOTE:1][ENDFOOTNOTE] In order to facilitate transformation from instrumental measurement to the standard Johnson-Cousins system, fourteen stars in the field of V582 Aur were calibrated in $BVRI$ bands. Calibration was made during seven clear nights in 2010 and 2011 with the 1.3 m RC telescope of the Skinakas Observatory. Standard stars from Landolt (1992) were used as a reference. Table 1 contains photometric data for the $BVRI$ comparison sequence. The corresponding mean errors in the mean are also listed. The stars are labeled from A to N in order of their V-band magnitude. In regions of star formation a great percentage of stars can be photometric variables. Therefore, there is a possibility that some of our standard stars are low amplitude variables, and we advise observers to use our photometric sequence with care. The finding chart of the comparison sequence is presented in Fig. 2. The field is $8\arcmin\times 8\arcmin$, north is at the top and east is to the left. The chart is retrieved from the STScI Digitized Sky Survey Second Generation Red. Star \| B \| σB \| V \| σV \| Rc \| σR \| Ic \| σI ---\|---\|---\|---\|---\|---\|---\|---\|--- A \| 15.635 \| 0.037 \| 14.153 \| 0.023 \| 13.303 \| 0.055 \| 12.486 \| 0.013 B \| 16.120 \| 0.053 \| 14.303 \| 0.026 \| 13.242 \| 0.063 \| 12.144 \| 0.013 C \| 16.271 \| 0.028 \| 15.324 \| 0.023 \| 14.766 \| 0.041 \| 14.156 \| 0.017 D \| 16.708 \| 0.044 \| 15.774 \| 0.024 \| 15.193 \| 0.039 \| 14.529 \| 0.020 E \| 17.254 \| 0.049 \| 16.206 \| 0.028 \| 15.590 \| 0.046 \| 14.982 \| 0.018 F \| 17.148 \| 0.073 \| 16.250 \| 0.021 \| 15.727 \| 0.036 \| 15.161 \| 0.029 G \| 17.382 \| 0.063 \| 16.267 \| 0.026 \| 15.620 \| 0.046 \| 14.920 \| 0.024 H \| 17.298 \| 0.061 \| 16.282 \| 0.022 \| 15.682 \| 0.045 \| 14.930 \| 0.020 I \| 18.055 \| 0.153 \| 16.584 \| 0.044 \| 15.672 \| 0.060 \| 14.748 \| 0.014 J \| 17.706 \| 0.051 \| 16.779 \| 0.034 \| 16.190 \| 0.049 \| 15.599 \| 0.048 K \| 18.642 \| 0.099 \| 17.004 \| 0.029 \| 15.990 \| 0.064 \| 14.839 \| 0.030 L \| 17.948 \| 0.110 \| 17.016 \| 0.042 \| 16.531 \| 0.030 \| 15.816 \| 0.037 M \| 18.276 \| 0.156 \| 17.154 \| 0.054 \| 16.469 \| 0.046 \| 15.714 \| 0.034 N \| 18.418 \| 0.131 \| 17.301 \| 0.029 \| 16.656 \| 0.047 \| 15.944 \| 0.049 Table 1: Photometric data for the BVRI comparison sequence. <figure><img src="content_image/1306.6647/x2.png"><figcaption>Figure 2: Finding chart for the BVRI comparison sequence around V582 Aur.</figcaption></figure> The results from our photometric CCD observations of V582 Aur are summarized in Table 2. The measured magnitudes from the photometric observations of V582 Aur made with the Asiago Schmidt telescope on 2009 August are also included in the table. All CCD frames are measured with the same parameters and standard stars reported in the present paper. The columns of Table 2 provide the date and Julian date (J.D.) of observation, $\it IRVB$ magnitudes of V582 Aur, and the telescope and CCD camera used. The typical errors in the reported magnitudes are $0\aas@@fstack{m}01$-$0\aas@@fstack{m}02$ for $I$ and $R$-band data, $0\aas@@fstack{m}01$-$0\aas@@fstack{m}03$ for $V$, and $0\aas@@fstack{m}02-0\aas@@fstack{m}04$ for $B$-band. 2 Date \| J.D. (24…) \| I \| R \| V \| B \| Telescope \| CCD ---\|---\|---\|---\|---\|---\|---\|--- 2009 Aug 05 \| 55048.581 \| 12.26 \| 13.60 \| 14.85 \| 16.72 \| Asiago Sch \| 2009 Aug 06 \| 55049.592 \| 12.21 \| 13.59 \| 14.91 \| 16.62 \| Asiago Sch \| 2009 Oct 07 \| 55111.519 \| 12.70 \| - \| - \| - \| Sch \| FLI 2009 Oct 07 \| 55112.491 \| 12.70 \| 14.29 \| - \| - \| Sch \| FLI 2009 Oct 08 \| 55113.491 \| 12.67 \| 14.24 \| - \| - \| Sch \| FLI 2009 Oct 09 \| 55114.491 \| 12.66 \| 14.24 \| 15.53 \| - \| Sch \| FLI 2009 Oct 28 \| 55133.397 \| 12.49 \| 14.03 \| 15.27 \| - \| Sch \| FLI 2009 Nov 19 \| 55155.352 \| 12.34 \| 13.84 \| 15.06 \| 16.83 \| Sch \| FLI 2009 Nov 20 \| 55156.418 \| 12.33 \| 13.83 \| 15.05 \| 16.80 \| Sch \| FLI 2009 Nov 21 \| 55157.449 \| 12.34 \| 13.83 \| 15.06 \| 16.81 \| Sch \| FLI 2009 Nov 25 \| 55161.476 \| 12.28 \| 13.72 \| 14.94 \| 16.73 \| 2m \| VA 2010 Mar 11 \| 55267.327 \| 12.34 \| 13.80 \| 15.01 \| - \| 2m \| VA 2010 Mar 12 \| 55268.247 \| 12.32 \| 13.79 \| 15.00 \| 16.70 \| 2m \| VA 2010 Apr 13 \| 55300.294 \| 12.11 \| - \| - \| - \| Sch \| FLI 2010 May 14 \| 55331.281 \| 12.53 \| 14.02 \| - \| - \| Sch \| FLI 2010 May 16 \| 55333.286 \| 12.43 \| 13.96 \| 15.30 \| - \| Sch \| FLI 2010 Aug 06 \| 55415.577 \| 12.60 \| 14.13 \| 15.38 \| 17.04 \| Sch \| FLI 2010 Aug 07 \| 55416.549 \| 12.60 \| 14.14 \| 15.40 \| 17.17 \| Sch \| FLI 2010 Aug 12 \| 55420.572 \| 12.63 \| 14.17 \| 15.45 \| 17.20 \| 1.3m \| ANDOR 2010 Aug 13 \| 55421.566 \| 12.65 \| 14.19 \| 15.47 \| 17.22 \| 1.3m \| ANDOR 2010 Aug 15 \| 55423.598 \| 12.71 \| 14.29 \| 15.60 \| - \| 1.3m \| ANDOR 2010 Aug 24 \| 55432.538 \| 12.11 \| 13.50 \| 14.66 \| 16.29 \| 1.3m \| ANDOR 2010 Aug 25 \| 55432.554 \| 12.06 \| 13.46 \| 14.60 \| 16.23 \| 1.3m \| ANDOR 2010 Aug 26 \| 55434.535 \| 12.03 \| 13.41 \| 14.54 \| 16.18 \| 1.3m \| ANDOR 2010 Aug 27 \| 55435.551 \| 12.00 \| 13.37 \| 14.50 \| 16.13 \| 1.3m \| ANDOR 2010 Sep 08 \| 55447.609 \| 11.92 \| 13.28 \| 14.38 \| - \| Sch \| FLI 2010 Sep 09 \| 55448.561 \| 11.89 \| 13.25 \| 14.34 \| - \| Sch \| FLI 2010 Sep 10 \| 55449.589 \| 11.91 \| 13.26 \| 14.37 \| - \| Sch \| FLI 2010 Sep 20 \| 55459.581 \| 11.87 \| 13.21 \| 14.32 \| 15.93 \| 1.3m \| ANDOR 2010 Oct 12 \| 55481.517 \| 11.70 \| 13.02 \| 14.11 \| - \| 1.3m \| ANDOR 2010 Oct 29 \| 55499.455 \| 11.84 \| 13.09 \| 14.17 \| 15.79 \| 2m \| VA 2010 Oct 30 \| 55500.483 \| 11.82 \| 13.08 \| 14.16 \| 15.78 \| 2m \| VA 2010 Oct 31 \| 55501.476 \| 11.82 \| 13.07 \| 14.16 \| 15.78 \| 2m \| VA 2010 Oct 31 \| 55501.493 \| 11.79 \| 13.09 \| 14.18 \| 15.78 \| Sch \| FLI 2010 Nov 01 \| 55502.476 \| 11.81 \| 13.06 \| 14.13 \| 15.75 \| 2m \| VA 2010 Nov 02 \| 55502.522 \| 11.78 \| 13.09 \| 14.17 \| 15.73 \| Sch \| FLI 2010 Nov 02 \| 55503.485 \| 11.80 \| 13.12 \| 14.19 \| 15.80 \| Sch \| FLI 2010 Nov 03 \| 55504.458 \| 11.80 \| 13.11 \| 14.20 \| 15.75 \| Sch \| FLI 2010 Nov 04 \| 55505.430 \| 11.81 \| 13.11 \| 14.19 \| 15.77 \| Sch \| FLI 2010 Nov 05 \| 55506.313 \| 11.82 \| 13.13 \| 14.20 \| 15.77 \| Sch \| FLI 2010 Nov 06 \| 55507.492 \| 11.83 \| 13.16 \| 14.25 \| 15.84 \| Sch \| FLI 2011 Jan 01 \| 55563.377 \| 11.80 \| 13.11 \| 14.17 \| 15.76 \| Sch \| FLI 2011 Jan 06 \| 55568.360 \| 11.91 \| 13.21 \| 14.31 \| 15.96 \| 2m \| VA 2011 Jan 08 \| 55570.257 \| 11.94 \| 13.23 \| 14.34 \| 15.99 \| 2m \| VA 2011 Jan 09 \| 55571.364 \| 11.92 \| 13.24 \| 14.35 \| 16.02 \| 2m \| VA 2011 Jan 11 \| 55573.289 \| 11.94 \| 13.25 \| 14.37 \| 16.05 \| 2m \| VA 2011 Feb 06 \| 55599.305 \| 12.05 \| 13.43 \| 14.59 \| 16.21 \| Sch \| FLI 2011 Feb 07 \| 55600.324 \| 12.06 \| 13.45 \| 14.60 \| 16.27 \| Sch \| FLI 2011 Apr 04 \| 55656.296 \| 11.72 \| 13.01 \| 14.06 \| 15.61 \| Sch \| FLI 2011 Apr 09 \| 55661.280 \| 11.71 \| 12.99 \| - \| - \| 2m \| VA 2011 Aug 17 \| 55790.579 \| 11.84 \| 13.11 \| 14.17 \| 15.70 \| 1.3m \| ANDOR 2011 Aug 18 \| 55791.577 \| 11.83 \| 13.10 \| 14.15 \| 15.68 \| 1.3m \| ANDOR 2011 Aug 24 \| 55797.524 \| 11.81 \| 13.09 \| 14.12 \| 15.67 \| Sch \| FLI 2011 Aug 25 \| 55798.559 \| 11.80 \| 13.07 \| 14.10 \| 15.63 \| Sch \| FLI 2011 Aug 26 \| 55799.535 \| 11.82 \| 13.08 \| 14.11 \| 15.64 \| Sch \| FLI 2011 Sep 11 \| 55815.538 \| 11.84 \| 13.10 \| 14.14 \| 15.66 \| 1.3m \| ANDOR 2011 Sep 12 \| 55816.550 \| 11.82 \| 13.08 \| 14.12 \| 15.64 \| 1.3m \| ANDOR 2011 Sep 19 \| 55824.491 \| 11.83 \| 13.09 \| 14.14 \| 15.64 \| 1.3m \| ANDOR 2011 Oct 07 \| 55842.444 \| 11.92 \| 13.21 \| 14.27 \| 15.81 \| 1.3m \| ANDOR 2011 Oct 13 \| 55848.442 \| 11.97 \| 13.28 \| 14.36 \| 15.92 \| 1.3m \| ANDOR 2011 Oct 30 \| 55865.451 \| 12.06 \| 13.44 \| 14.53 \| 16.16 \| 2m \| VA 2011 Oct 31 \| 55866.448 \| 12.03 \| 13.41 \| 14.53 \| 16.16 \| 2m \| VA 2011 Nov 26 \| 55892.390 \| 12.24 \| 13.64 \| 14.76 \| 16.45 \| 2m \| VA 2011 Nov 27 \| 55893.362 \| 12.18 \| 13.53 \| 14.63 \| 16.26 \| Sch \| FLI 2011 Nov 28 \| 55894.329 \| 12.15 \| 13.51 \| 14.60 \| 16.23 \| Sch \| FLI 2011 Nov 29 \| 55895.433 \| 12.14 \| 13.48 \| 14.58 \| 16.21 \| Sch \| FLI 2011 Nov 30 \| 55896.395 \| 12.15 \| 13.51 \| 14.59 \| 16.22 \| Sch \| FLI 2011 Dec 29 \| 55925.433 \| 12.10 \| 13.45 \| 14.52 \| 16.15 \| Sch \| FLI 2012 Jan 01 \| 55928.303 \| 12.07 \| 13.40 \| 14.47 \| 16.07 \| Sch \| FLI 2012 Jan 29 \| 55956.251 \| 12.48 \| 13.90 \| 15.05 \| - \| 2m \| VA 2012 Mar 16 \| 56003.346 \| 14.22 \| 15.70 \| 16.73 \| 18.30 \| Sch \| FLI 2012 Mar 20 \| 56007.379 \| 14.21 \| 15.65 \| 16.75 \| 18.20 \| 60cm \| FLI 2012 Mar 22 \| 56009.268 \| 14.25 \| 15.69 \| 16.78 \| 18.28 \| 60cm \| FLI 2012 Mar 23 \| 56010.275 \| 14.25 \| 15.68 \| 16.80 \| 18.30 \| 60cm \| FLI 2012 Mar 28 \| 56016.346 \| 14.29 \| 15.74 \| 16.82 \| 18.23 \| 2m \| VA 2012 Apr 03 \| 56021.398 \| 14.32 \| 15.75 \| 16.76 \| - \| 60cm \| FLI 2012 Apr 10 \| 56028.305 \| 14.30 \| 15.73 \| 16.85 \| - \| Sch \| FLI 2012 Apr 12 \| 56030.293 \| 14.29 \| 15.73 \| 16.83 \| 18.21 \| Sch \| FLI 2012 Jul 31 \| 56139.590 \| 12.61 \| 14.13 \| 15.40 \| 17.17 \| 1.3m \| ANDOR 2012 Aug 02 \| 56141.568 \| 12.50 \| 13.99 \| 15.25 \| 17.01 \| 1.3m \| ANDOR 2012 Aug 03 \| 56142.569 \| 12.47 \| 13.95 \| 15.20 \| 16.96 \| 1.3m \| ANDOR 2012 Aug 04 \| 56143.590 \| 12.44 \| 13.93 \| 15.20 \| - \| 1.3m \| ANDOR 2012 Aug 11 \| 56150.601 \| 12.37 \| 13.84 \| 15.06 \| 16.72 \| 1.3m \| ANDOR 2012 Aug 13 \| 56152.609 \| 12.38 \| 13.84 \| 15.06 \| 16.73 \| 1.3m \| ANDOR 2012 Aug 14 \| 56153.598 \| 12.36 \| 13.83 \| 15.06 \| 16.71 \| 1.3m \| ANDOR 2012 Aug 17 \| 56156.619 \| 12.40 \| 13.87 \| 15.09 \| 16.81 \| 1.3m \| ANDOR 2012 Aug 18 \| 56157.604 \| 12.39 \| 13.85 \| 15.07 \| 16.76 \| 1.3m \| ANDOR 2012 Aug 20 \| 56159.545 \| 12.42 \| 13.88 \| 15.08 \| 16.79 \| Sch \| FLI 2012 Aug 21 \| 56160.556 \| 12.41 \| 13.87 \| 15.10 \| 16.81 \| 1.3m \| ANDOR 2012 Aug 21 \| 56160.557 \| 12.42 \| 13.90 \| 15.11 \| 16.85 \| Sch \| FLI 2012 Aug 22 \| 56161.564 \| 12.42 \| 13.89 \| 15.10 \| 16.79 \| Sch \| FLI 2012 Sep 03 \| 56173.529 \| 12.45 \| 13.92 \| 15.11 \| 16.81 \| 1.3m \| ANDOR 2012 Sep 04 \| 56174.531 \| 12.45 \| 13.93 \| 15.13 \| 16.82 \| 1.3m \| ANDOR 2012 Sep 05 \| 56175.483 \| 12.49 \| 13.97 \| 15.15 \| 16.76 \| Sch \| FLI 2012 Sep 06 \| 56176.513 \| 12.50 \| 13.99 \| 15.17 \| 16.87 \| Sch \| FLI 2012 Sep 09 \| 56179.502 \| 12.53 \| 14.05 \| 15.26 \| 16.94 \| 1.3m \| ANDOR 2012 Sep 09 \| 56180.387 \| 12.61 \| 14.11 \| 15.32 \| - \| 60cm \| FLI 2012 Sep 10 \| 56180.515 \| 12.59 \| 14.13 \| 15.37 \| 17.07 \| 1.3m \| ANDOR 2012 Sep 11 \| 56182.476 \| 12.65 \| 14.20 \| 15.45 \| 17.16 \| 1.3m \| ANDOR 2012 Sep 13 \| 56183.520 \| 12.67 \| 14.25 \| 15.50 \| 17.22 \| 1.3m \| ANDOR 2012 Sep 23 \| 56193.515 \| 12.68 \| 14.29 \| 15.58 \| 17.32 \| 1.3m \| ANDOR 2012 Sep 23 \| 56193.523 \| 12.70 \| 14.30 \| 15.58 \| 17.31 \| Sch \| FLI 2012 Sep 24 \| 56194.504 \| 12.71 \| 14.30 \| 15.57 \| 17.28 \| Sch \| FLI 2012 Oct 09 \| 56209.526 \| 12.16 \| 13.54 \| 14.68 \| 16.31 \| Sch \| FLI 2012 Oct 10 \| 56211.496 \| 12.13 \| 13.49 \| 14.60 \| 16.23 \| Sch \| FLI 2012 Oct 11 \| 56212.405 \| 12.09 \| 13.43 \| 14.54 \| 16.17 \| 60cm \| FLI 2012 Oct 13 \| 56214.357 \| 12.08 \| 13.39 \| 14.52 \| 16.13 \| 2m \| VA 2012 Oct 25 \| 56226.402 \| 12.03 \| 13.37 \| 14.45 \| 16.04 \| Sch \| FLI 2012 Oct 26 \| 56227.480 \| 12.04 \| 13.40 \| 14.50 \| 16.12 \| Sch \| FLI 2012 Nov 17 \| 56249.408 \| 12.07 \| 13.42 \| 14.55 \| 16.21 \| Sch \| FLI 2012 Nov 18 \| 56250.435 \| 12.06 \| 13.42 \| 14.53 \| 16.17 \| Sch \| FLI 2012 Nov 27 \| 56258.648 \| 12.04 \| 13.38 \| 14.47 \| 16.07 \| Sch \| FLI 2012 Dec 12 \| 56274.270 \| 12.08 \| 13.45 \| 14.51 \| 16.11 \| 2m \| VA 2012 Dec 14 \| 56275.532 \| 12.06 \| 13.44 \| 14.49 \| 16.11 \| 2m \| VA 2012 Dec 14 \| 56276.393 \| 12.08 \| 13.44 \| 14.49 \| 16.10 \| 2m \| VA 2012 Dec 30 \| 56292.345 \| 12.04 \| 13.38 \| 14.42 \| 16.17 \| 60cm \| FLI 2013 Jan 01 \| 56294.359 \| 12.03 \| 13.34 \| 14.41 \| 16.07 \| 60cm \| FLI 2013 Jan 03 \| 56296.382 \| 12.02 \| 13.34 \| 14.42 \| 16.03 \| 60cm \| FLI 2013 Jan 16 \| 56309.306 \| 12.03 \| 13.33 \| 14.41 \| 16.03 \| Sch \| FLI 2013 Jan 19 \| 56312.308 \| 11.99 \| 13.25 \| 14.35 \| 15.97 \| 2m \| VA 2013 Feb 02 \| 56326.247 \| 11.91 \| 13.20 \| 14.23 \| 15.81 \| Sch \| FLI 2013 Feb 04 \| 56328.416 \| 11.90 \| 13.18 \| 14.21 \| 15.75 \| Sch \| FLI 2013 Feb 05 \| 56329.401 \| 11.91 \| 13.19 \| 14.24 \| 15.80 \| Sch \| FLI 2013 Mar 04 \| 56356.461 \| - \| 13.16 \| 14.24 \| - \| 60cm \| FLI 2013 Mar 05 \| 56357.441 \| 11.91 \| 13.19 \| 14.24 \| 15.85 \| 60cm \| FLI 2013 Mar 17 \| 56369.361 \| 11.92 \| 13.23 \| 14.24 \| 15.79 \| 2m \| VA 2013 Mar 19 \| 56371.361 \| 11.97 \| 13.27 \| 14.28 \| 15.80 \| 2m \| VA 2013 Apr 10 \| 56393.316 \| 11.96 \| 13.26 \| 14.30 \| 15.85 \| Sch \| FLI 2013 Apr 11 \| 56394.305 \| 11.96 \| 13.27 \| 14.31 \| 15.87 \| Sch \| FLI 2013 Apr 12 \| 56395.289 \| 11.96 \| 13.26 \| - \| - \| Sch \| FLI Table 2: Continued. ### Spectral observations At the time of our photometric monitoring of V582 Aur, a total of sixteen optical spectra of the star were obtained. High-, medium-, and low-resolution spectroscopy of V582 Aur was performed using spectral equipment in three observatories: Asiago (Italy), Haute-Provence (France), and Skinakas (Greece). All data reduction was performed within IRAF, Table 3 provides a log of spectral observations. Low-dispersion, absolutely fluxed spectra of V582 Aur were obtained with the Asiago 1.22 m + B&C telescope operated by the Department of Physics and Astronomy of the University of Padova. A 300 ln/mm grating blazed at 5000 Å provided a dispersion of 2.31 Å/pix and a FWHM (PSF) $\sim$3.0 pix. The detector was an Andor iDus DU440 CCD camera with a 2048$\times$512 pixel array, which is of high UV efficiency as the whole spectrograph optical train. The primary spectrophotometric standard star was HR 1729, which was only a few degrees away and observed immediately before or after V582 Aur. High-resolution spectra of V582 Aur were secured with the Asiago 1.82 m telescope equipped with an REOSC echelle spectrograph and an Andor DW436BV CCD camera, housing a back-illuminated E2V CCD4240 AIMO detector with a 2048$\times$2048 pixel array. A binning of 2$\times$2 provided a resolving power of 12000. At both telescopes the slit was oriented along the parallactic angle and widened to 2.0 arcsec. One high-resolution and one low-resolution spectrum of V582 Aur were obtained with the 1.93 m telescope at the Haute-Provence Observatory. The high-resolution spectrum was obtained with the cross-dispersed echelle spectrograph (Sophie) on 2010 January 15 and the low-resolution spectrum with the long-slit Cassegrain spectrograph (Carelec) on 2012 January 18. The CCD chips used are EEV 42-20 CCD ($2048\times 1024$ pixels) for Carelec and EEV ($4096\times 2048$) for Sophie. Observations in the Skinakas Observatory were carried out with the focal reducer of the 1.3 m RC telescope and ISA 608 spectral CCD camera ($2000\times 800$ pixels, 15$\times 15$$\mu$m) on 2012 August 31, September 1, and September 23. Two gratings (1300 and 2400 lines per mm) and 160$\mu$m slit were used. The first grating yield a resolving power $\lambda/\Delta\lambda$$\sim$ 1300, while the second grating yielded $\lambda/\Delta\lambda$$\sim$ 2500 at H$\alpha$ line. The exposures of V582 Aur were followed immediately by an exposure of an FeHeNeAr comparison lamp and exposure of a spectrophotometric standard star. Date \| UT \| Exp. t. \| Dispersion or \| λ range \| Tel. ---\|---\|---\|---\|---\|--- \| \| (sec) \| resolving power \| (Å) \| 2009 Aug 06 \| 02:34 \| 3600 \| disp. 2.31 Å/pix \| 4000−6950 \| 1.22m+B&C 2010 Jan 15 \| 23:26 \| 3000 \| res. pow. 45 000 \| 3872−6943 \| 1.93m+SOPHIE 2011 Dec 21 \| 22:04 \| 2700 \| disp. 2.31 Å/pix \| 3700−7550 \| 1.22m+B&C 2012 Jan 13 \| 19:27 \| 3600 \| res. pow. 12 000 \| 4400−7335 \| 1.82m+REOSC 2012 Jan 18 \| 20:22 \| 2400 \| res. pow. 900 \| 3600−7000 \| 1.93m+CARELEC 2012 Feb 08 \| 21:38 \| 900 \| res. pow. 12 000 \| 4400−7335 \| 1.82m+REOSC 2012 Feb 18 \| 20:02 \| 900 \| disp. 2.31 Å/pix \| 3700−7550 \| 1.22m+B&C 2012 Mar 02 \| 20:48 \| 1800 \| res. pow. 12 000 \| 4400−7335 \| 1.82m+REOSC 2012 Mar 30 \| 19:01 \| 1800 \| disp. 2.31 Å/pix \| 3700−7550 \| 1.22m+B&C 2012 Aug 31 \| 01:41 \| 3600 \| disp. 0.48 Å/pix \| 5750−6720 \| 1.30m+focal red. 2012 Sep 01 \| 23:50 \| 1800 \| disp. 1.05 Å/pix \| 5490−7590 \| 1.30m+focal red. 2012 Sep 23 \| 00:22 \| 3600 \| disp. 1.05 Å/pix \| 5490−7590 \| 1.30m+focal red. 2012 Oct 21 \| 22:55 \| 900 \| disp. 2.31 Å/pix \| 3700−7550 \| 1.22m+B&C 2012 Oct 29 \| 23:08 \| 900 \| res. pow. 12 000 \| 4400−7335 \| 1.82m+REOSC 2012 Nov 15 \| 22:02 \| 1200 \| disp. 0.61 Å/pix \| 5680−6910 \| 1.22m+B&C 2012 Dec 28 \| 20:15 \| 1500 \| res. pow. 12 000 \| 4400−7335 \| 1.82m+REOSC Table 3: Journal of spectroscopic observations ### Archival photographic observations The construction of the historical light curves of FUors could be very important for determining the exact moment of the beginning of the outburst and the time to reach the maximum light. Another important option is to study the pre-outburst variability of the star. The only possibility for such a study is a search in the photographic plate archives at the astronomical observatories around the world. Most suitable for this purpose are the plate archives of the big Schmidt telescopes that have a large field of view. Unfortunately, the collection and analysis of old photographic observations requires a very long and laborious amount of work. In this paper, we present our first result of exploring the whole photographic plate stack of the 105/150 cm Schmidt telescope at Kiso Observatory (Japan), and several plates around the expected time of the outburst obtained with the 67/92 cm Schmidt telescope at Asiago Observatory (Italy). We also used the digitized plates from the Palomar Schmidt telescope, available via the website of the Space Telescope Science Institute. In addition we checked for archival observations of V582 Aur in the photographic plate collection of the 2 m RCC and the 50/70 cm Schmidt telescopes at NAO Rozhen (Bulgaria), but found none. The plates from Asiago Schmidt telescope are inspected visually through a high-quality Carl Zeiss microscope offering a variety of magnifications (Munari et al. 2001). The magnitude is then derived by comparing the stars in the photometric sequence with the variable, identifying those that are more closely bracketing the variable. If “a” and “b” are such two stars of the sequence, visual inspection estimate the quantities n1, n2, which represent the fraction of the total a-b magnitude difference by which the variable V is fainter than “a” and brighter than “b”: a - n1 - V - n2 - b. The magnitude of V follows from a simple proportion. If more than one such pair is available, more estimates are derived and weighted according to the “a-b”, “c-d” etc. mag interval. Typical estimated errors are of the order of 0.10 mag. The plates from Kiso Schmidt telescope were scanned with Canon CanoScan LiDE 600F portable scanner, which has 1200 dpi resolution. Each photographic plate was put on the scanner glass plate, and three sheets of white paper were stacked on the photographic plate. A fluorescent tube was used to light the photographic plates. Aperture photometry of the digitized plates was performed with DAOPHOT routines using the same aperture radius and the background annulus as for CCD photometry. The results of the measured magnitudes of V582 Aur from the archival photographic plates are given in Table 4. The columns provide the name of the observatory, the plate number, date and Julian date (J.D.) of observation, photographic emulsions and filters used, the magnitude estimated or plate limit, and the corresponding errors. Observatory \| Plate No. \| Date \| J.D. \| Emulsion \| Filter \| Magnitude \| Er. ---\|---\|---\|---\|---\|---\|---\|--- Palomar \| 001315E \| 1954 Dec 29 \| 2435105.778 \| 103aE \| Plexi \| R=16.87 \| ±0.06 Palomar \| 001315O \| 1954 Dec 29 \| 2435105.809 \| 103aO \| none \| pg=19.5 \| ±0.1 Kiso \| 000901 \| 1977 Oct 10 \| 2443427.258 \| IN \| RG695 \| I>15.0 \| Kiso \| 001521 \| 1978 Mar 20 \| 2443587.927 \| IN \| RG695 \| I=15.8 \| ±0.2 Kiso \| 001965 \| 1979 Jan 02 \| 2443875.957 \| 103aE \| RG645 \| R=16.6 \| ±0.2 Kiso \| 001967 \| 1979 Jan 02 \| 2443876.159 \| IIaD \| GG495 \| V=18.0 \| ±0.3 Kiso \| 001973 \| 1979 Jan 04 \| 2443878.072 \| 103aE \| RG645 \| R=17.1 \| ±0.2 Kiso \| 002409 \| 1979 Nov 15 \| 2444193.160 \| IIaD \| GG495 \| V>17.6 \| Kiso \| 002413 \| 1979 Nov 15 \| 2444193.287 \| IN \| RG695 \| I>15.0 \| Kiso \| 002492 \| 1979 Dec 14 \| 2444222.201 \| 103aE \| RG645 \| R=17.2 \| ±0.2 Kiso \| 002529 \| 1979 Dec 22 \| 2444230.004 \| 103aE \| RG645 \| R>16.7 \| Kiso \| 003010 \| 1980 Nov 14 \| 2444558.256 \| IIaD \| GG495 \| V>17.5 \| Kiso \| 003057 \| 1980 Dec 10 \| 2444584.059 \| IIaD \| GG495 \| V>17.5 \| Palomar \| 000310V \| 1982 Oct 21 \| 2445263.955 \| IIaD \| W12 \| V=18.9 \| ±0.1 Asiago \| 011872 \| 1983 Jan 12 \| 2445347.360 \| 103aO \| GG13 \| B>18.5 \| Asiago \| 011873 \| 1983 Jan 12 \| 2445583.380 \| IN \| RG5 \| I>15.9 \| Asiago \| 012690 \| 1984 Nov 29 \| 2446036.444 \| IN \| RG5 \| I>15.9 \| Asiago \| 013276 \| 1986 Jan 17 \| 2446448.436 \| IN \| RG5 \| I=15.25 \| ±0.08 Palomar \| 001013 \| 1986 Dec 29 \| 2446793.756 \| IIIaJ \| GG385 \| B=15.74 \| ±0.05 Asiago \| 013706 \| 1987 Feb 01 \| 2446828.377 \| 103aD \| GG14 \| V=14.25 \| ±0.08 Palomar \| 002236 \| 1988 Dec 01 \| 2447496.817 \| IIIaF \| RG610 \| R=13.85 \| ±0.05 Palomar \| 002812 \| 1989 Oct 04 \| 2447803.958 \| IIIaF \| RG610 \| R=13.86 \| ±0.06 Palomar \| 002935 \| 1989 Nov 19 \| 2447849.846 \| IVN \| RG9 \| I=12.21 \| ±0.03 Palomar \| 005536 \| 1993 Oct 23 \| 2449283.920 \| IIIaJ \| GG385 \| B=15.86 \| ±0.06 Palomar \| 007538 \| 1997 Oct 30 \| 2450751.851 \| IVN \| RG9 \| I=12.06 \| ±0.03 Table 4: Photometric data from the photographic observations of V582 Aur ## 3Results ### Photometric monitoring The historical $BVRI$ light curves of V582 Aur from all available photometric observations are plotted in Fig. 3. On the figure, the filled diamonds represent the CCD observations from the present paper, the filled circles photographic data from the 67/92 cm Asiago Schmidt telescope, the filled triangles photographic data from the 105/150 cm Kiso Schmidt telescope, and the filled squares photographic data from the Oschin Schmidt Telescope on Palomar. <figure><img src="content_image/1306.6647/x3.png"><figcaption>Figure 3: Historical BVRI light curves of V582 Aur for the period 1954December − 2013 February.</figcaption></figure> The light curve of V582 Aur allows us to conclude that its photometric behavior is similar to that of classical FUor stars. The $R$ and $V$ magnitudes measured from Kiso and Palomar photographic plates suggest that before the outburst the star was variable with an amplitude at about 1 mag. Archival photographic observations indicate that the increase in brightness began in late 1984 or early 1985 and the star brightness reached its maximum value in 1986 January. Hence, the rise in brightness is relatively fast at about one year and the registered amplitude is $\sim$3.6 mag. ($V$). The photographic data from Palomar plates indicate that during the first decade after the outburst (1986-1997) the star keeps its maximum brightness with approximately constant $B$, $R$, and $I$ magnitudes (Table 4). The CCD photometric data reported in the present paper show a very strong and fast variability in brightness, while the star remains in a state close to the maximum brightness most of the time. Consequently, the outburst of V582 Aur continued for approximately 28 years. Figure 4 shows the $R$-light curve of V582 Aur for the period of our CCD observations and the dates of spectroscopic observations (marked by blue arrows). The present photometric data show large amplitude variations ($\Delta R\sim 2\aas@@fstack{m}8$) during the 3.5 year period of observations. The photometric variability does not show periodicity, as the brightness variations sometimes occur very rapidly in the time scale of days and weeks. The fastest changes in brightness were registered in 2010 August, when for a twelve-day period the stellar magnitude increased by $1\aas@@fstack{m}10$ ($V$) and in 2012 September-October, when for a seventeen-day period the stellar magnitude increased by $1\aas@@fstack{m}05$ ($V$). But the most remarkable event in the light curve of V582 Aur is the large drop in brightness during the spring of 2012. A strong decline in brightness with $2\aas@@fstack{m}26$ ($V$) from January 1 to March 16 was observed and the star remained in a state of very low brightness for at least one month. The observed $BVRI$ magnitudes during this period (2012 March-April) are only by 1-1.5 mag. higher than the pre-outburst magnitudes measured from the archival photographic plates. <figure><img src="content_image/1306.6647/x4.png"><figcaption>Figure 4: R-light curve of V582 Aur for the period 2009 August − 2013February. Dates of spectroscopic observations are marked by arrows.</figcaption></figure> Another important result from our photometric study is the variation of color indices with stellar brightness. Figure 5 shows the measured color indices $V-I$ versus stellar magnitude $V$ during the period of our observations. A clear dependence can be seen over almost the entire period: the star becomes redder as it fades. Such color variations are typical for FUor stars, which have a relatively fast set in brightness, as for example, V1057 Cyg (Hartmann & Kenyon 1996). In the case of V582 Aur, the changes of color indices occur very rapidly within days and weeks as we observe drops in brightness accompanied by reddening and increases in brightness accompanied by blue color indices. But the multicolor photometric data obtained through the deep minimum in brightness (spring 2012) indicated a different relationship. From a certain turning point, V582 Aur gets bluer, fading further to $V$$\sim 16\aas@@fstack{m}8$ on 2012 March. Such color variations are observed as well for the $V-R$ and $B-V$ indices. The observed change of color indices suggests the existence of a color reversal (or so-called “blueing”) in the minimum light, a typical feature of PMS stars from the UXor type. <figure><img src="content_image/1306.6647/x5.png"><figcaption>Figure 5: V−I vs. V color-magnitude diagram for V582 Aur during the outburst(2009 August - 2013 February).</figcaption></figure> ### Spectroscopic monitoring Besides the strong photometric variability, relatively rapid changes of the profiles and intensities at different spectral lines have been registered. The first low-dispersion spectrum obtained on 2009 August 6 is dominated by absorption lines of the Balmer series (H$\alpha$ and H$\beta$), Na I D, and Ba II 6497 $\AA$, and emission lines are not noticed. The spectrum appears like a supergiant of temperature about 5500 K, but the Li I 6707 $\AA$ line is weak, not emerging above the noise (Munari et al. 2009). On the high-resolution spectrum obtained on 2010 January 15, the H$\alpha$ line and Na I doublet show the P Cyg profiles, which are typical of FUor stars (Fig. 6). The broad blueshifted H$\alpha$ absorption seems to be saturated, extending to about -800 km/s due to powerful, rapidly expanding wind. All high- and medium-resolution spectra, except those obtained in 2012 February-March, show the same spectral features, but with changes in the P Cyg profiles of H$\alpha$ line (Fig. 6). Moreover, at increased brightness of the star, the absorption component becomes deeper and wider and the emission component becomes less intensive. Therefore, there is a noticeable correlation between the variations in the optical depth of H$\alpha$ line and the photometric properties of the star. <figure><img src="content_image/1306.6647/x6.png"><figcaption>Figure 6: Profiles of Hα and Na I D lines from different periods ofobservations. Left: high-resolution profiles of Hα obtained with the SOPHIEand REOSC echelle spectrograph. Center: medium-resolution profiles of Hα fromthe Skinakas and Asiago telescopes. Right: high-resolution profile of Na I Dobtained with the SOPHIE echelle spectrograph.</figcaption></figure> Figure 7 presents a comparison between the low-resolution spectra of V582 Aur, which are obtained before, during, and after the large drop in brightness in spring 2012. During this event, the spectroscopic properties of the star changed dramatically. The absorption lines and P Cyg profiles disappeared from the spectrum, and only the H$\alpha$ emission line without an absorption component remained. Therefore, during the decline in brightness with $\sim$$2\aas@@fstack{m}26$ ($V$), the spectral features of the stellar spectrum change from a typical FUor to typical T Tauri star spectrum. With the ending of the deep decline in brightness of V582 Aur, the spectral characteristics change significantly again. The absorption lines of H$\alpha$, H$\beta$, Na I D, and Ba II 6497 $\AA$, along with the P Cyg profiles at H$\alpha$ and Na I D, appear again in 2012 August - September. A significant change in the SED of V582 Aur relative to spectra obtained at periods of high and low brightness is also observed. <figure><img src="content_image/1306.6647/x9.png"><figcaption>Figure 7: Low-resolution spectra of V582 Aur.</figcaption></figure> To classify the absorption spectrum of V582 Aur, we compared our spectra with the Asiago atlas of MKK spectral types observed with exactly the same instrumental configuration of V582 Aur (Munari 2013, in preparation). We took the absolutely fluxed spectra of V582 Aur and the MKK atlas and continuum normalized them using the same function (a Legendre polynomial of fifth order limited to the range of wavelength covered in Figure 8, which corresponds to those recommended for the classification within the MKK system). As a first classification pass, we applied a simple $\chi^{2}$ matching to determine the area of the Herzsprung-Russell diagram on which our deeper analysis was focused. The match found by the $\chi^{2}$ is not perfect, since the stellar spectra originates in stationary atmospheres where a three-dimensional treatment is generally unnecessary, while the absorption lines in V582 Aur instead form in a moving medium, the wind. We then proceeded to refine the classification by using an eye inspection of the spectra and found that the closest (even though imperfect) match was for a G0I type star. Figure 8 shows how the properties of the absorption spectrum of V582 Aur are similar to those of supergiants of the G0 types. <figure><img src="content_image/1306.6647/x10.png"><figcaption>Figure 8: Low-resolution spectrum of V582 Aur obtained on 2012 December 21 iscompared with spectra of the G supergiant stars HD 29645, HD 6903, and HD26630 (from the Asiago spectral database).</figcaption></figure> ## 4Discussion Any case in which a large amplitude outburst of PMS star is observed gives rise to the question: FUor or EXor? The main differences between these two types are the spectral appearance (presence or absence of certain spectral lines, their profiles, and intensity) and the photometric properties (duration and amplitude of the outburst and the shape of light curves). The outburst of V582 Aur differs from EXor eruptions both in duration and spectral appearance. Photometric and spectroscopic data so far accumulated imply that V582 Aur is indeed proceeding through a _bona fide_ FUor outburst. The presence of a 3.6 mag. amplitude outburst in optic and a rise in brightness in the period 1984-1986 are well documented. Despite the sparse and random photometry available during 1986 to 2009, all available data suggest that the star keeps its maximal brightness at this period. Therefore, a large amplitude outburst with duration of several decades is registered. The spectrum dominated with prominent absorption lines of Balmer series, Na I and Ba II, the strong P Cyg profiles at H$\alpha$ and Na I D, and the presence of a reflection nebula around the star are all well-established characteristics of the classical FUor objects. The only discrepancies are the relatively weak absorption line of LiI 6707 Å and the very strong and fast photometric variability, which are not typical for classical FUor stars. The observed profile of the H$\alpha$ line is highly variable, with a deep and high-velocity blueshifted absorption component. This feature can be interpreted as evidence of a strong and time-variable outflow driven by the central FUor object. The strong photometric variability observed over the 3.5 year period can be explained by 1) time-variable extinction or 2) changes in accretion rate from the circumstellar disk onto the stellar surface. The variable accretion affects the speed and intensity of the stellar wind and is manifested as changes in the P Cyg profile of the H$\alpha$ line. Similar transformation of several spectral lines from absorption to emission have been observed in the deep minima of some UXor stars (Rodgers et al. 2002, Eaton & Herbst 1995). The spectral changes were explained by the different sizes of the star and the circumstellar envelope: the occulting screen is larger than the star but the circumstellar envelope is significantly larger than the screen. In contrast to FUor stars the absorption and emission components in this case have symmetrical profiles and P Cyg profiles are not observed. During the large drop in brightness in the spring of 2012, an appearance of dust particles in the immediate circumstellar environment of the star was registered. An evidence of this statement is the observed effect of color reversal on the color/magnitude diagram (Fig. 5). The widely accepted explanation of the color reversal effect is variations of the column density of dust in the line of sight to the star. Normally the star becomes redder when its light is covered by dust clumps or filaments on the line of sight. But when the obscuration rises sufficiently, the part of the scattered light in the total observed light become significant and the star color gets bluer. Then, the decline in brightness was caused by either the reformation of circumstellar dust or motion of dust clumps into the line of sight toward the star. In the fall of 2012, the dust particles were sublimated by the stellar wind or removed from the line of sight and the star returned to its previous photometric condition. The strong photometric variability at maximum light is typical for EXors, but not for FUors. However, for several FUor objects similar short-time drops in brightness were registered. The well-known one is the 1980 minimum in the light curve of V1515 Cyg, a strong decrease in brightness by about $1\aas@@fstack{m}5$ (B) in few months (Kolotilov & Petrov 1983). This minimum in brightness was explained by an obscuration from a dust material ejected from the star (Kenyon et al. 1991). Evidence for strong light variability in the time of set in brightness ($\Delta$$V$=1$\aas@@fstack{m}$2) during the period from 1986 to 1992 was reported in the photometric study of another FUor object, V1735 Cyg (Peneva et al. 2009). A short drop in brightness was observed in 2009 (decrease by 0$\aas@@fstack{m}$4 ($I$) and return to its previous level) in the light curve of V733 Cep, which is also recognized as a FUor object (Peneva et al. 2010). Unfortunately, spectral observations during minimum light are not available for these objects, and the cause of such short-time drops in brightness is not known. The large amplitude variability may result from the superposition of both phenomena, the variable accretion rate and time variable extinction, and it is very difficult to distinguish the two phenomena using only photometric data (Semkov & Peneva 2012). In recent studies, such a scenario is used to explain the light variability of two PMS objects with characteristics similar to those of FUor and EXor – V1647 Ori (Aspin et al. 2009b, Aspin 2011) and V2492 Cyg (Hillenbrand et al. 2013, Kóspál et al. 2013). It seems that the time variable extinction is characteristic not only of some Herbig Ae/Be stars (UXor variables) but is also a common phenomenon during the evolution of all types of PMS stars. In the case of V582 Aur, we have direct evidence from multicolor photometry, which suggests the presence of dust around the star during the decline in brightness. Even though many PMS stars indicate evidence of time variable accretion, the physical cause of this phenomenon is still under discussion. One of the possible reasons for the variable accretion rate could be fragmentation of the circumstellar disk. Because the FUor phenomenon is probably repeatable, up to 50% of the protostellar mass can be accumulated as a result of such episodes of strong accretion burst. Stamatellos et al. (2011, 2012) suggest that episodic accretion may initially promote disc fragmentation. In the early stages of PMS evolution, fragmentation does not happen and disk accretion is assumed to be constant. After several episodic accretion bursts, the circumstellar disk is gradually fragmented and thus prevents new FUor events. Therefore, it can be supposed that FUor outbursts during different periods of stellar evolution may vary in amplitude, duration, and shape of the light curve due to the different state of disk fragmentation. Strong accretion bursts may also be the triggering mechanism of planet formation with different masses inside the circumstellar disk. If the above suggestions are correct, the studies of FUor and EXor objects would be useful not only for understanding stellar evolution but also for understanding the formation of planets and asteroids and the frequency of planetary systems. ## 5Conclusions Photometric data presented in this paper show the usefulness of systematically spectral and photometric monitoring of the regions of star formation. These data can be used to detect new FUor or EXor events and to determine the type of the registered outbursts. On the basis of our photometric monitoring over the past 3.5 years and the spectral properties at maximal light (a G0I supergiant spectrum with strong P Cyg profiles of H$\alpha$ and Na I D lines), we have confirmed that the observed outburst of V582 Aur is of the FUor type. On the other hand, the observed effect of color reversal at the minimum light is evidence of the possible symbiotic nature of V582 Aur (FUor + UXor). Therefore, the collection of new photometric data (from photographic plate archives and ongoing photometric monitoring) will be of great importance for a precise determination of the type of variability. At the same time, according to existing observations the light curve of V582 Aur remains unique, confirming the hypothesis that each known FUor has a different rate of increase and decrease in brightness and a different light curve shape. We plan to continue our spectroscopic and photometric monitoring of the star during the next few months and years and strongly encourage similar follow-up observations. ###### Acknowledgements. This work was partly supported by grants DO 02-85 and DO 02-362 of the National Science Fund of the Ministry of Education, Youth and Science, Bulgaria. The authors thank the Director of Skinakas Observatory Prof. I. Papamastorakis and Prof. I. Papadakis for granting telescope time. We also thank Prof. R. Zamanov for the fruitful discussion on the obtained results. The Digitized Sky Survey was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. This research has made use of the NASA Astrophysics Data System.** ## References * (1971) Ambartsumian, V. A. 1971, Astrophysics, 7, 331 * (2011) Aspin, C. 2011, AJ, 142, 135 * (2009) Aspin, C., Beck, T. L., Pyo, T.-S., et al. 2009a, AJ, 137, 431 * (2009) Aspin, C., Reipurth, B., Beck, T. L. et al. 2009b, ApJ, 692, L67 * (1995) Bouvier, J., Covino, E., Kovo, O. et al. 1995, A&A, 299, 89 * (2005) Clarke, C., Lodato, G., Melnikov, S. Y., & Ibrahimov, M. A. 2005, MNRAS, 361, 942 * (2005) Eaton, N. L., & Herbst, W. 1995, AJ, 110, 2369 * (2012) Fischer, W. J., Megeath, S. T., Tobin, J. J., et al. 2012, ApJ, 756, 99 * (2013) Green, J. D., Robertson, P., Baek, G. et al. 2013, ApJ, 764, 22 * (1991) Grinin, V. P., Kiselev, N. N., Minikulov, N. KH., Chernova, G. P., & Voshchinnikov, N. V. 1991, Ap&SS, 186, 283 * (2001) Grinin, V. P., Kozlova, O. V., Natta, A., Ilyin, I., Tuominen, I., Rostopchina, A. N., & Shakhovskoy, D. N. 2001, A&A, 379, 482 * (1996) Hartmann, L., & Kenyon, S. J. 1996, ARA&A, 34, 207 * (1977) Herbig, G. H. 1977, ApJ, 217, 693 * (1989) Herbig, G. H. 1989, in ESO Workshop on Low-Mass Star Formation and Pre-Main-Sequence Objects, ed. B. Reipurth (Garching: ESO), 233 * (2009) Herbig, G. H. 2007, AJ, 133, 2679 * (2003) Herbig, G. H., Petrov, P. P., & Duemmler, R. 2003, ApJ, 595, 384 * (1994) Herbst, W., Herbst, D. K., Grossman, E. J., & Weinstein, D. 1994, AJ, 108, 1906 * (2007) Herbst, W., Eisloffel, J., Mundt, R., Scholz, A., 2007, in Protostars and Planets V, eds. B. Reipurth, D. Jewitt, K. Keil, University of Arizona Press, Tucson, 297 * (2013) Hillenbrand, L. A., Miller, A. A., Covey, K. R. et al. 2013, AJ, 145, 59 * (1991) Kenyon, S. J., Hartmann, L. W., & Kolotilov, E. A. 1991, PASP, 103, 1069 * (2000) Kenyon, S. J., Kolotilov, E. A., Ibragimov, M. A., & Mattei, J. A. 2000, ApJ, 531, 1028 * (1983) Kolotilov, E. A., & Petrov, P. P. 1983, Pis’ma Astron. Zh., 9, 171 * (2011) Kóspál, Á., Ábrahám, P., Acosta-Pulido, J. A. et al. 2011, A&A, 527, A133 * (2013) Kóspál, Á., Ábrahám, P., Acosta-Pulido, J. A. et al. 2013, A&A, 551, A62 * (2013) Kurosawa, R., & Romanova, M. M. 2013, MNRAS, 431, 2673 * (1992) Landolt, A. U. 1992, AJ, 104, 340 * (2004) Lodato, G., & Clarke, C. J. 2004, MNRAS, 353, 841 * (2011) Miller, A, A., Hillenbrand, L. A., Covey, K. R. et al. 2011, ApJ, 730, 80 * (2006) Movsessian, T. A., Khanzadyan, T., Aspin, C., et al. 2006, A&A, 455, 1001 * (2009) Munari, U., Siviero, A., Ochner, P., Fiorucci, M. & Dallaporta, S. 2009, CBET, 1898, 1 * (2012) Munari, U. 2013, in preparation * (1997) Natta, A., Grinin, V. P., Mannings, V., & Ungerechts, H. 1997, ApJ, 491, 885 * (2009) Peneva, S. P., Semkov, E. H., & Stavrev, K. Y. 2009, Ap&SS, 323, 329 * (2010) Peneva, S. P., Semkov, E. H., Munari, U., & Birkle, K. 2010, A&A, 515, A24 * (2008) Pfalzner, S. 2008, A&A, 492, 735 * (2013) Powell, S. L., Irwin, M., Bouvier, J., & Clarke, C. J. 2013, MNRAS, 426, 3315 * (2004) Reipurth, B., & Aspin, C. 2004, ApJ, 608, L65 * (2007) Reipurth, B., Aspin, C., Beck, T. et al. 2007, AJ, 133, 1000 * (2010) Reipurth, B., & Aspin, C. 2010, in Evolution of Cosmic Objects through their Physical Activity, eds. H. A. Harutyunian, A. M. Mickaelian, Y. Terzian (Yerevan: Gitutyun), p. 19 * (2012) Reipurth, B., Aspin, C., & Herbig, G. H. 2012, ApJ, 748, L5 * (2002) Rodgers, B., Wooden, D. H., Grinin, V., Shakhovsky, D., & Natta, A. 2002, ApJ, 564, 405 * (2009) Samus, N. 2009, CBET, 1896, 1 * (2010) Semkov, E. H., Peneva, S. P., Munari, U., Milani, A., & Valisa, P. 2010, A&A, 523, L3 * (2011) Semkov, E. H., Peneva, S. P., & Dennefeld, M. 2011, BlgAJ, 15, 65 * (2012) Semkov, E. H., Peneva, S. P., Munari, U. et al. 2012, A&A, 542, A43 * (2012) Semkov, E. H., & Peneva, S. P. 2012, Ap&SS, 338, 95 * (2013) Siwak, M., Rucinski, S. M., Matthews, J. M. et al. 2013, MNRAS, 432, 194 * (2011) Stamatellos, D., Whitworth, A. P., & Hubber, D. A. 2011, ApJ, 730, 32 * (2012) Stamatellos, D., Whitworth, A. P., & Hubber, D. A. 2012, MNRAS, 427, 1182 * (2010) Takita, S., Kataza, H., Kitamura, Y. et al. 2010, A&A, 519, A83 * (2009) Zhu, Z., Hartmann, L., Gammie, C., & McKinney, J. C. 2009, ApJ, 701, 620
1409.7773	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35566, "num_imgs": 0, "llama3_tokens_count": 12679 }	[]	# Operator-Valued Frames for the Heisenberg Group† [FOOTNOTE:†][ENDFOOTNOTE] Benjamin Robinson Benjamin Robinson and Douglas CochranSchool of Mathematical & Statistical Sciences Arizona State University Tempe, AZ 85287-1809 USA ¹William Moran RMIT University Melbourne, VIC 3000 Australia ²Stephen D. Howard Defence Science & Technology Organisation Edinburgh, SA 5111 Australia ³ William Moran Benjamin Robinson and Douglas CochranSchool of Mathematical & Statistical Sciences Arizona State University Tempe, AZ 85287-1809 USA ¹William Moran RMIT University Melbourne, VIC 3000 Australia ²Stephen D. Howard Defence Science & Technology Organisation Edinburgh, SA 5111 Australia ³ Douglas Cochran Benjamin Robinson and Douglas CochranSchool of Mathematical & Statistical Sciences Arizona State University Tempe, AZ 85287-1809 USA ¹William Moran RMIT University Melbourne, VIC 3000 Australia ²Stephen D. Howard Defence Science & Technology Organisation Edinburgh, SA 5111 Australia ³ Stephen D. Howard Benjamin Robinson and Douglas CochranSchool of Mathematical & Statistical Sciences Arizona State University Tempe, AZ 85287-1809 USA ¹William Moran RMIT University Melbourne, VIC 3000 Australia ²Stephen D. Howard Defence Science & Technology Organisation Edinburgh, SA 5111 Australia ³ [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:6][ENDFOOTNOTE] ###### Abstract A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on $\mathbb{R}$, restricted to $E=(-1/2,1/2)$, forms a Hilbert-space frame for $L^{2}(E)$. For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group $H_{n}$, where frames are replaced by _operator-valued frames_. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of $L^{2}(E)$ for suitable subsets $E$ of $H_{n}$ in terms of representations of $H_{n}$. Keywords Operator-valued frames, G-frames, Representations, Heisenberg Group, Sampling AMS Subject Classification 42C15 ## 1 Introduction A frame for a Hilbert space $\mathcal{H}$ is a sequence $\{\xi_{j}\,\|\,j\in\mathbb{N}\}\subset\mathcal{H}$ such that \[A\left\\|\xi\right\\|_{\mathcal{H}}^{2}\leq\sum_{j\in\mathbb{Z}}\left\|\left<\xi_ {j},\xi\right>_{\mathcal{H}}\right\|^{2}\leq B\left\\|\xi\right\\|_{\mathcal{H}}^ {2}\] for positive real numbers $A$ and $B$ and all $\xi\in\mathcal{H}$. In the case that $\{\xi_{j}\}$ is an orthonormal basis for $\mathcal{H}$, this sequence is a Parseval frame for $\mathcal{H}$, meaning $A=B=1$. In particular, when $\mathcal{H}=L^{2}(-1/2,1/2)$, the sequence $\{\xi_{j}\}$ may be taken to be the standard Fourier basis $\{e^{2\pi ij{\,\cdot\,}}\,\|\,j\in\mathbb{Z}\}$. In their seminal 1952 paper [8], Duffin and Schaeffer proved that it is possible to perturb the numbers $j\in\mathbb{Z}$ and still preserve the frame condition: Theorem 1.1: _Denote by $E$ the interval $(-1/2,1/2)\subset\mathbb{R}$. Let $M>0$ and $\delta>0$, and suppose $\{\chi_{j}=e^{2\pi i\omega_{j}\cdot}\,\|\,j\in\mathbb{Z}\}$ is a sequence of characters of $\mathbb{R}$ with_ 1. $\|\omega_{j}-j\|<M$_, for all_ $j\in\mathbb{Z}$_, and_ 2. $\|\omega_{j}-\omega_{k}\|\geq\delta$ _for all_ $j\neq k\in\mathbb{Z}$_._ _Then there exist positive real numbers $B\geq A>0$ such that, for any $f\in L^{2}(E)$,_ \[A\left\\|f\right\\|_{L^{2}(E)}^{2}\leq\sum_{j\in\mathbb{Z}}\left\|\left<\chi_{j}, f\right>_{L^{2}(E)}\right\|^{2}\leq B\left\\|f\right\\|_{L^{2}(E)}^{2}.\] Modern literature in frame theory widely acknowledges that the subject was initiated by this paper of Duffin and Schaeffer, although it received relatively little attention until the late 1980s when Daubechies, Grossmann, and Meyer connected this important idea with the rapidly expanding area of wavelet analysis [6]. Theorem 1.1 has inspired numerous generalizations and extensions, including the highly significant work of Kadets (Kadec) [13] and Avdonin [2] on Riesz bases of exponentials. These and other results are described in [15]. The objective of this paper is to extend these perturbation results to the real Heisenberg group $H_{n}$ in a natural way. In this context, the appropriate notion of a frame is a “g-frame” or “operator-valued frame.” This concept is found in [17] and also appears in [4; 5; 14; 12]. In what follows, the term _operator-valued frame_ (OVF) will be used to mean a countable sequence of (arbitrary-rank) linear operators $T_{j}:\mathcal{H}\rightarrow\mathcal{K}_{j}$ mapping a separable, complex Hilbert space $\mathcal{H}$ into separable, complex Hilbert spaces $\mathcal{K}_{j}$ with the property that there are positive real numbers $B\geq A>0$ such that \[A\left\\|\xi\right\\|^{2}\leq\sum_{j}\left\\|T_{j}\xi\right\\|_{\mathcal{K}_{j}}^{ 2}\leq B\left\\|\xi\right\\|^{2}\] for all $\xi\in\mathcal{H}$. In some papers, the definition includes the additional assumption that the maps $T_{j}$ all have the same rank, but that condition will not be required. A frame in the usual sense is a special case of this definition with the additional requirement that $\dim\mathcal{K}_{j}=1$ for all $j$. Much of the study of OVFs thus far has been motivated by multiwavelets [14] and by certain applications in distributed processing [5]. One straightforward way to construct OVFs [14; 12] is to fix a locally compact group $G$, a unitary representation $\pi$ of $G$ on a Hilbert space $\mathcal{H}$, and a sequence of points $\{x_{j}\}\subset G$, and set $T_{j}=T\pi(x_{j})$ where $T$ is some fixed operator from $\mathcal{H}$ into a separable, complex Hilbert space $\mathcal{K}$. This recipe is also used to generate many rank-one frames of interest [6]. However, as Theorem 1.1 suggests, there are other constructions for frames. Among them are the so-called _frames of exponentials_, as described in Theorem 1.1. Others have extended Theorem 1.1 to more general locally compact abelian groups (see [10]). The general goal of this paper is to extend the circle of results of Duffin and Schaeffer to the setting of a non-trivial subspace $\mathcal{H}$ of $L^{2}(G)$ where $G$ is a unimodular locally compact Lie group. In the case of a noncommutative group, such as $H_{n}$, there will of course be representations of dimension greater than one. This means that frames will have to be replaced by OVFs. Some conditions are presented in Section 2 for $\{\pi_{j}\}$ corresponding, in a fairly general context, to the orthogonality of harmonic frames of exponentials in the classical case studied by Duffin and Schaeffer. Section 3 sets forth results analogous to Theorem 1.1 for the special case where $G$ is $H_{n}$. ## 2 Harmonic OVFs of representations This section begins with a synopsis of a few aspects of representation theory needed in subsequent discussion. It proceeds to describe some Parseval OVFs. The setting is a unimodular locally compact Lie group $G$ with a discrete co-compact closed subgroup $\Gamma$. Let $\mu$ be a finite invariant measure on right cosets $\Gamma\backslash G$(11, Theorem 2.49) that is normalized so that $\mu(\Gamma\backslash G)=1$. Then the _quasi-regular_ representation $R$ of $(G,\Gamma)$ is defined on $L^{2}(\Gamma\backslash G):=L^{2}(\Gamma\backslash G,d\mu)$ by \[\left(R(y)\phi\right)(x)=\phi(xy)\] with $y\in G$, $x\in\Gamma\backslash G$, and $\phi\in L^{2}(\Gamma\backslash G)$. The symbol $C_{c}(G)$ ($C_{c}^{\infty}(G)$) will denote continuous (smooth) compactly supported functions from $G$ into $\mathbb{C}$ and, when $E$ is open in $G$, the symbol $C_{E}^{\infty}(G)$ will denote the subset of $C_{c}^{\infty}(G)$ with support contained in $E$. The archetypal example of this setting is when $G=\mathbb{R}$ and $\Gamma=\mathbb{Z}$, in which case $\Gamma\backslash G$ can be identified with $[-1/2,1/2)$. For these particular groups, $R$ decomposes as \[R=\bigoplus_{j\in\mathbb{Z}}\pi_{j}\] where $\pi_{j}$ is the character on $G=\mathbb{R}$ defined by $\pi_{j}(x)=e^{2\pi ijx}$. By the Poisson summation formula, for $f\in C_{c}^{\infty}(G)$, \[\left\\|\sum_{k\in\mathbb{Z}}f({\,\cdot\,}+k)\right\\|_{L^{2}(-1/2,1/2)}^{2}= \sum_{j\in\mathbb{Z}}\left\|\left\langle\pi_{j},f\right\rangle\right\|^{2}.\] Restricting to $f$ supported on $E=(-1/2,1/2)$ gives \[\left\\|f\right\\|_{L^{2}(E)}^{2}=\sum_{j\in\mathbb{Z}}\left\|\left\langle\pi_{j} ,f\right\rangle\right\|^{2}\] for all $f\in C_{E}^{\infty}(G)$. This equality can be extended to all of $L^{2}(E)$ by density, showing that the harmonic exponentials $\{\pi_{j}\}$ form a Parseval frame for $L^{2}(E)$. While this is just the Plancherel theorem for Fourier series, the derivation here serves to illustrate the general case. Let $G$ be a locally compact group, let $dx=dm(x)$ be Haar measure on $G$, let $E\subset G$ be open with compact closure and $m(E)>0$, and let $\{\pi_{j}\,\|\,j\in\mathbb{N}\}$ be a set of representations of $G$ on separable Hilbert spaces $\{\mathcal{H}_{j}\,\|\,j\in\mathbb{N}\}$. Further, assume that $\{\pi_{j}\,\|\,j\in\mathbb{N}\}$ has the property that, for each $j$ and all $f\in L^{2}(E)$, the operator defined by $\pi_{j}(f)=\int_{G}f(x)\pi_{j}(x)\,dx$ is a Hilbert-Schmidt class operator on $\mathcal{H}_{j}$. Then $\{\pi_{j}\}$ will be called an _OVF of representations_ for $L^{2}(E)$ provided there exist $B\geq A>0$ such that \[A\left\\|f\right\\|_{L^{2}(E)}^{2}\leq\sum\left\\|\pi_{j}(f)\right\\|_{\text{HS}}^ {2}\leq B\left\\|f\right\\|_{L^{2}(E)}^{2}\] (1) for all $f\in L^{2}(E)$. In this expression and subsequently, $\left\\|{\,\cdot\,}\right\\|_{\text{HS}}$ denotes the Hilbert-Schmidt norm. In the case that each $\mathcal{H}_{j}=\mathbb{C}$, each $\pi_{j}(x)$ for $x\in G$ and $j\in\mathbb{N}$ can be viewed either as a scalar or as an operator on $\mathbb{C}$, and for all $f\in L^{2}(E)$. In particular, when each $\pi_{j}$ in the preceding paragraph is regarded as a representation on $\mathbb{C}$, the inequality (1) holds with $A=B=1$. In the more general setting, the appropriate replacement for $E=(-1/2,1/2)$ as it occurs in the $(\mathbb{R},\mathbb{Z})$ case, will be called a $(G,\Gamma)$_reproducing set_; i.e., a non-empty open set $E$ with compact closure having the property that $EE^{-1}$ is disjoint from every conjugate of $\Gamma-\{\mathbbm{1}_{G}\}$. Existence of such an $E$ is equivalent to existence of a non-empty open set $U\subset G$ such that $\cup_{g\in G}\,g^{-1}Ug$ intersects $\Gamma$ only in the point $\mathbbm{1}_{G}$. Further, if $R$ is as above, then $R$ decomposes discretely as \[R=\bigoplus\pi_{j},\qquad(\text{listed with multiplicities})\] (2) where each $\pi_{j}$ is represented on a Hilbert space $\mathcal{H}_{j}$ and has finite multiplicity (7, Lemma 9.2.7). Now and henceforth it is assumed that $dx$ is chosen so that for $f\in C_{c}(G)$ \[\int_{G}f(x)\,dx=\int_{\Gamma\backslash G}\sum_{\gamma}f(\gamma x)\,d\mu(x)\] for $\mu$ as described above. This is possible by (11, Theorem 2.49), and in this case, the above equality also holds for all $f\in L^{1}(G)$[3]. Further, $R(f)$ will denote the operator on $\tilde{\mathcal{H}}=L^{2}(\Gamma\backslash G)$ obtained by integrating the representation $R$ against a function $f\in L^{1}(G)$; i.e., \[R(f)=\int_{G}f(y)R(y)\,dy.\] (3) In what follows, the algebra of trace-class operators on a separable Hilbert space $\mathcal{K}$ will be denoted by $L^{1}(\mathcal{K})$ and the Hilbert space of Hilbert-Schmidt class operators on $\mathcal{K}$ will denoted by $L^{2}(\mathcal{K})$. Further, the trace of an operator $T\in L^{1}(\mathcal{K})$ will be denoted $\operatorname{Tr}(T)$, the Hilbert-Schmidt inner product of $S$ and $T\in L^{2}(\mathcal{K})$ will be denoted $\left\langle S,T\right\rangle_{L^{2}(\mathcal{K})}=\operatorname{Tr}(T^{}S)$. With the necessary background and terminology established, the objective in the remainder of this section is to prove the following: Theorem 2.1: _Let $E$ be a $(G,\Gamma)$ reproducing set. Then the decomposition of $R$ into irreducible representations $\{\pi_{j}\}$, listed with multiplicities, implies that $\{\pi_{j}\}$ forms a Parseval OVF for $L^{2}(E)$._ This begins with a preliminary result: Lemma 1: _Let $E\subset G$ be non-empty and open with compact closure, and let $\mathcal{H}=L^{2}(E)$. Then $\check{R}:f\mapsto R(f)$ is a bounded linear map from $\mathcal{H}$ into $L^{2}(\tilde{\mathcal{H}})$._ Proof: It is shown in [1] that, for $f\in C_{c}(G)$, the Hilbert-Schmidt norm of $R(f)$ is given by \[\left\\|R(f)\right\\|_{\mathrm{HS}}^{2}=\int_{\Gamma\backslash G}\int_{\Gamma \backslash G}\left\|\sum_{\gamma\in\Gamma}f(x^{-1}\gamma y)\right\|^{2}\,d\mu(x) \,d\mu(y).\] (4) Because $E$ has compact closure, this formula carries over _mutatis mutandis_ to the case $f\in L^{2}(E)$. If $q:G\rightarrow\Gamma\backslash G$ is the canonical quotient map and $F$ is a measurable subset of $G$, then \[\int_{G}\chi_{F}(x)\,dx=\int_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}\chi_{F }(\gamma x)\,d\mu(x)\geq\int_{\Gamma\backslash G}\chi_{q(F)}\,d\mu.\] By (11, Lemma 2.46), there is a compact set $K\subset G$ such that $q(K)=F$. Thus, given $S\subset\Gamma\backslash G$ and taking $F=q^{-1}(S)\cap K$ in the above yields $\int_{K}\chi_{S}\circ q(x)\,dx\geq\int_{\Gamma\backslash G}\chi_{S}\,d\mu$. That is, $\int_{K}g\circ q(x)\,dx\geq\int_{\Gamma\backslash G}g\,d\mu$ for all characteristic functions $g$ on $\Gamma\backslash G$, and thus all non-negative measurable functions on $\Gamma\backslash G$. The right-hand side of (4) then becomes bounded by \[\int_{\Gamma\backslash G}\int_{K}\left\|\sum_{\gamma\in\Gamma}f(x^{-1}\gamma y) \right\|^{2}\,dx\,d\mu(y)\leq\int_{K}\int_{K}\left\|\sum_{\gamma\in\Gamma}f(x^{- 1}\gamma y)\right\|^{2}\,dx\,dy.\] The sum in the integrand vanishes off the set $\Gamma_{0}=\Gamma\cap KEK^{-1}$, which is compact and discrete, hence finite. An application of the Cauchy-Schwarz inequality yields the following upper bound for $\left\\|R(f)\right\\|_{L^{2}(\tilde{\mathcal{H}})}^{2}$, \[\left\\|R(f)\right\\|_{\mathrm{HS}}^{2} \leq\|\Gamma_{0}\|\int_{K}\int_{K}\sum_{\gamma\in\Gamma_{0}}\left\|f (x^{-1}\gamma y)\right\|^{2}\,dx\,dy\] \[\leq\|\Gamma_{0}\|\sum_{\gamma\in\Gamma_{0}}\int_{K}\int_{G}\left\|f (x^{-1}\gamma y)\right\|^{2}\,dx\,dy\] \[\leq\|\Gamma_{0}\|^{2}m(K)\|\|f\|\|_{L^{2}(E)}^{2}\] as desired. Now (2) and (3) yield a unitary $V:\bigoplus\mathcal{H}_{j}\rightarrow\tilde{\mathcal{H}}$ for which, as an operator on $\bigoplus\mathcal{H}_{j}$, \[V^{}R(f)V=\bigoplus\pi_{j}(f).\] It follows that each $\pi_{j}(f)$ is a Hilbert-Schmidt class operator on $\mathcal{H}_{j}$ and that \[\left\\|R(f)\right\\|_{\mathrm{HS}}^{2}=\sum\left\\|\pi_{j}(f)\right\\|_{\mathrm{ HS}}^{2}.\] (5) The condition for the operators $\{\pi_{j}\}$, which respectively map into the Hilbert spaces $L^{2}(\mathcal{H}_{j})$, to form a Parseval OVF for $L^{2}(E)$ is \[\left\\|f\right\\|_{L^{2}(E)}^{2}=\sum\left\\|\pi_{j}(f)\right\\|_{\mathrm{HS}}^{2}.\] (6) In view of (5), this inequality follows from $\left\\|f\right\\|_{L^{2}(E)}^{2}=\left\\|R(f)\right\\|_{\mathrm{HS}}^{2}$, a sufficient condition for which is that $E$ is a $(G,\Gamma)$ reproducing set. Verification of this sufficiency is achieved in the following lemmas. Lemma 2: _Let $M\in L^{1}(\tilde{\mathcal{H}})$ and $E$ be an open subset of $G$ with compact closure and positive Haar measure. Then the function $f^{M}:E\rightarrow\mathbb{C}$ defined by $f^{M}(x)=\operatorname{Tr}(R(x^{-1})M)$ is bounded, and $\check{R}^{}M=f^{M}$._ Proof: First it will be shown that $f^{M}$ is well-defined. If $M$ has eigenvalues $\{\lambda_{j}\,\|\,j\in\mathbb{N}\}$ and corresponding eigenbasis $\{e_{j}\}\subset\tilde{\mathcal{H}}$, and $U$ is any unitary operator on $\tilde{\mathcal{H}}$, then \[\left\|\operatorname{Tr}(UM)\right\| \leq\sum_{j}\left\|\left\langle UMe_{j},e_{j}\right\rangle_{\tilde {\mathcal{H}}}\right\|\] \[\leq\sum_{j}\left\\|UMe_{j}\right\\|_{\tilde{\mathcal{H}}}\] \[=\sum_{j}\left\\|Me_{j}\right\\|_{\tilde{\mathcal{H}}}=\sum_{j}\| \lambda_{j}\|.\] Thus, $f^{M}(x)=\sum_{j}\left\langle R\left(x^{-1}\right)Me_{j},e_{j}\right\rangle$ converges absolutely to a bounded function on $E$. It will now be shown that \[\left<R(f),M\right>_{L^{2}(\tilde{\mathcal{H}})}=\left<f,f^{M}\right>_{L^{2}(E )}.\] The right-hand side is equal to \[\int_{E}f(x)\operatorname{Tr}(M^{}R(x))\,dx.\] As implied by the above estimates, the series $\operatorname{Tr}(M^{}R(x))=\overline{\operatorname{Tr}(R(x^{-1})M)}$, expanded using $\{e_{j}\}$, converges absolutely to a bounded function. This means the integrand is dominated by a multiple of $\|f(x)\|$ and, since $f\in L^{2}(E)\subset L^{1}(E)$, it follows from the dominated convergence theorem that \[\left<f,f^{M}\right>_{L^{2}(E)} =\operatorname{Tr}\left(\int_{E}f(x)M^{}R(x)\,dx\right)\] which is just \[\operatorname{Tr}\left(M^{}\int_{E}f(x)R(x)\,dx\right).\] The latter is equal to $\left\langle R(f),M\right\rangle_{L^{2}(\tilde{\mathcal{H}})}$, as desired. Lemma 3: _Let $E$ be a $(G,\Gamma)$ reproducing set and $f\in L^{2}(E)$. Then (6) holds._ Proof: Suppose $f\in C_{E}^{\infty}(G)$. By [1], $R(f)$ is trace-class. Thus, with the notation $f_{x}(y)=f(yx)$, Lemma 2 implies that the function $\check{R}^{}R(f)$ has the following very specific form: \[(\check{R}^{}R(f))(x) =\operatorname{Tr}\left(R\left(x^{-1}\right)R(f)\right)\] \[=\operatorname{Tr}\left(R\left(x^{-1}\right)\int_{G}f(y)R(y)\,dy\right)\] \[=\operatorname{Tr}\left(\int_{G}f(y)R(y)\,dyR\left(x^{-1}\right)\right)\] \[=\operatorname{Tr}\left(\int_{G}f_{x}(y)R(y)\,dy\right)\] \[=\operatorname{Tr}\left(R(f_{x})\right)\] \[=\int_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}f_{x}(y^{-1} \gamma y)\,d\mu(y)\] (7) \[=f_{x}(\mathbbm{1}_{G})\mu(\Gamma\backslash G)+\int_{\Gamma \backslash G}\sum_{\mathbbm{1}_{G}\neq\gamma\in\Gamma}f_{x}(y^{-1}\gamma y)\,d \mu(y)\] where (7) follows from the Selberg Trace Formula applied to the function $f_{x}$ (see [1]). If $x$ is such that $\operatorname{supp}f_{x}$ is disjoint from all conjugates of $\Gamma-\{\mathbbm{1}_{G}\}$, then the integral term vanishes and the right-hand side becomes $f_{x}\left(\mathbbm{1}_{G}\right)$, which is just $f(x)$. But this will happen if $x\in E$, since $\operatorname{supp}f_{x}=Ex^{-1}\subset EE^{-1}$, which has the desired disjointness property. Hence, for $x\in E$ and $f\in C_{E}^{\infty}(G)$, $(\check{R}^{}R(f))(x)=f(x)$. Consequently, for all $f$ in a dense subspace of $\mathcal{H}=L^{2}(E)$, and hence for all of $\mathcal{H}$. As noted above, the desired Parseval frame condition (6) follows from this equality. This section has established that, if $G$ is a locally compact, unimodular Lie group, $\Gamma$ is a discrete, co-compact, closed subgroup, $\{\pi_{j}\,\|\,j\in\mathbb{N}\}$ is a list (with multiplicities) of the subrepresentations of the quasi-regular representation of $(G,\Gamma)$, and $E$ is a $(G,\Gamma)$ reproducing set, then $\{\pi_{j}\}$ is an OVF of representations for $L^{2}(E)$ with $A=B=1$. Such entities will be called _harmonic_ OVFs. ## 3 OVFs of representations for the real Heisenberg group With the context established in Section 2, this section returns to the matter of finding a generalization of the Duffin-Schaeffer theorem on non-harmonic Fourier series in which the pair $(\mathbb{R},\mathbb{Z})$ is replaced by a more general $(G,\Gamma)$. In this setting, when one is given a reproducing neighborhood $E$ and a harmonic OVF of representations $\{\pi_{j}\,\|\,j\in\mathbb{N}\}$ for $L^{2}(E)$, one may ask whether there is a “neighborhood” of $\{\pi_{j}\}$ consisting only of OVFs of representations for $L^{2}(E)$; i.e., consisting only of sequences of representations $\{\tilde{\pi}_{j}\,\|\,j\in\mathbb{N}\}$ for which there are $B\geq A>0$ such that \[A\left\\|f\right\\|_{L^{2}(E)}^{2}\leq\sum\left\\|\tilde{\pi}_{j}(f)\right\\|_{ \text{HS}}^{2}\leq B\left\\|f\right\\|_{L^{2}(E)}^{2}\] for all $f\in L^{2}(E)$. This section takes up this question for $H_{n}$, the real Heisenberg group, defined as ordered triples $(x,\xi,t)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}$ with the operation \[(x,\xi,t)(x^{\prime},\xi^{\prime},t^{\prime})=\left(x+x^{\prime},\xi+\xi^{ \prime},t+t^{\prime}+\frac{1}{2}(x\cdot\xi^{\prime}-x^{\prime}\cdot\xi)\right).\] The discrete subgroup $\Gamma$ consists of ordered triples in $\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\frac{1}{2}\mathbb{Z}$ and the reproducing neighborhood $E$ will be $D\times(-1/4,1/4)$, where $D=(-1/2,1/2)^{n}\times(-1/2,1/2)^{n}$. It is necessary to verify that $E$ really is a $(H_{n},\Gamma)$ reproducing set. To see this, first observe that $\Gamma-\{0\}^{2n+1}=\Gamma_{1}\cup\Gamma_{2}$ with $\Gamma_{1}=\left(\mathbb{Z}^{2n}-\{0\}^{2n}\right)\times\frac{1}{2}\mathbb{Z}$ and $\Gamma_{2}=\{0\}^{2n}\times\left(\frac{1}{2}\mathbb{Z}-\{0\}\right)$. Since the first $2n$ scalar components of $EE^{-1}$ lie in $(-1,1)$ and since the orbit of $\Gamma_{1}$ under conjugation in $G$ consists only of members of $(\mathbb{Z}^{2n}-\{0\}^{2n})\times\mathbb{R}$, $EE^{-1}$ is disjoint from this orbit. On the other hand, $\Gamma_{2}$ is in the center of $H_{n}$, so it is equal to its orbit under conjugation. If $(x,\xi,t)\in H_{n}$, then $(x,\xi,t)^{-1}=(-x,-\xi,-t)$, so if $(x,\xi,t),(x^{\prime},\xi^{\prime},t^{\prime})\in E$ and if $(x,\xi,t)(x^{\prime},\xi^{\prime},t^{\prime})^{-1}\in\Gamma_{2}$, then $x=x^{\prime}$, $\xi=\xi^{\prime}$, and $t-t^{\prime}\in\frac{1}{2}\mathbb{Z}-\{0\}$, which is impossible since $t,t^{\prime}\in(-1/4,1/4)$. Thus, $EE^{-1}$ does not intersect $\Gamma_{2}$. It remains to explicitly describe the subrepresentations of $R$ and their corresponding multiplicities. Up to equivalence, the representations of $H_{n}$ are of two types. The infinite-dimensional representations of $H_{n}$ have the form (see [18]) $\rho_{\omega}:H_{n}\times L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$ \[\left(\rho_{\omega}(x,\xi,t)\phi\right)(y)=e^{-2\pi i\omega(t+x\cdot y+\frac{1 }{2}x\cdot\xi)}\phi(y+x)\] with $\omega\in\mathbb{R}^{}=\mathbb{R}-\{0\}$ and $\phi\in L^{2}(\mathbb{R}^{n})$. The others are (one-dimensional) characters, given by $\chi_{b,\beta}(x,\xi,t)=e^{-2\pi i(b\cdot x+\beta\cdot\xi)}$ for $b,\beta\in\mathbb{R}^{n}$. To decompose $L^{2}(\Gamma\backslash H_{n})$ into $R$-invariant subspaces, observe first that $g\in L^{2}(\Gamma\backslash H_{n})$ may be viewed as a function on $H_{n}$ that is invariant under left translations in $\Gamma$. Such a function satisfies, in particular, $g(x,\xi,t)=g(x,\xi,t+1/2)$. Thus, $L^{2}(\Gamma\backslash H_{n})=\bigoplus_{k\in\mathbb{Z}}\mathcal{K}_{2k}$, where $\mathcal{K}_{2k}$ is the $R$-invariant space $\{h\in L^{2}(\Gamma\backslash H_{n}):h(x,\xi,t)=e^{4\pi ikt}h(x,\xi,0)\}$. The action of $R$ on $\mathcal{K}_{0}$ factors through the action of the right regular representation of $\mathbb{R}^{2n}$ on $L^{2}(\mathbb{T}^{2n})$, and can therefore be shown to decompose into the sum \[\bigoplus_{a,\alpha\in\mathbb{Z}^{n}}\chi_{a,\alpha}.\] Further, it is shown in [18] that the action of $R$ on $\mathcal{K}_{2k}$, $k\neq 0$, splits into $\|2k\|^{n}$ irreducible actions, each of which is equivalent by a Weil-Brezin-Zak transform to the action of $\rho_{2k}$ on $L^{2}(\mathbb{R}^{n})$. Thus, \[R\cong\bigoplus_{a,\alpha\in\mathbb{Z}^{n}}\chi_{a,\alpha}\oplus\bigoplus_{k \in\mathbb{Z}^{}}{\|2k\|}^{n}{\rho_{2k}}\] where $\mathbb{Z}^{}=\mathbb{Z}-\{0\}$. From this it follows that the frame condition (6) becomes \[\left\\|f\right\\|_{L^{2}(E)}^{2}=\sum_{a,\alpha\in\mathbb{Z}^{n}}\left\|\chi_{a, \alpha}(f)\right\|^{2}+\sum_{k\neq 0}{\|2k\|}^{n}\left\\|{\rho_{2k}}(f)\right\\|_{ \mathrm{HS}}^{2}\] for all $f\in L^{2}(E)$. The goal of this paper can now be rephrased as the following result about perturbing the values of the equispaced parameters $a$, $\alpha$, and ${2k}$ to vectors $\{b_{a}\,\|\,a\in\mathbb{Z}^{n}\}\subset\mathbb{R}^{n}$ and $\{\beta_{\alpha}\,\|\,\alpha\in\mathbb{Z}^{n}\}\subset\mathbb{R}^{n}$ and real numbers $\{\omega_{k}\,\|\,k\in\mathbb{Z}^{}\}$. Theorem 3.1: _Suppose $\{b_{a}\,\|\,a\in\mathbb{Z}^{n}\}$ and $\{\beta_{\alpha}\,\|\,\alpha\in\mathbb{Z}^{n}\}$ are sequences of real $n$-vectors and $\{\omega_{k}\,\|\,k\in\mathbb{Z}^{}\}$ is a sequence of real numbers. Define_ \[M=\max\left\{\sup_{a\in\mathbb{Z}^{n}}\left\\|b_{a}-a\right\\|_{\infty},\sup_{ \alpha\in\mathbb{Z}^{n}}\left\\|\beta_{\alpha}-\alpha\right\\|_{\infty},\sup_{k \neq 0}\left\|\omega_{k}-{2k}\right\|\right\}.\] _If $M>0$ is sufficiently small, then there exist $A=A(M)>0$ and $B=B(M)$ such that_ \[A\left\\|f\right\\|_{L^{2}(E)}^{2}\leq\sum_{a,\alpha}\left\|\chi_{b_{a},\beta_{ \alpha}}(f)\right\|^{2}+\sum_{k\neq 0}{\|2k\|}^{n}\left\\|\rho_{\omega_{k}}(f) \right\\|_{\mathrm{HS}}^{2}\leq B\left\\|f\right\\|_{L^{2}(E)}^{2}\] _holds for all $f\in L^{2}(E)$._ Proof: Let $f\in L^{2}(E)$. For $b,\beta,\omega\in\mathbb{R}^{n}$, the (Euclidean) Fourier transform of $f$ at $(b,\beta,\omega)$ is defined to be \[\hat{f}(b,\beta,\omega)=\int\int\int f(x,\xi,t)e^{-2\pi i(b\cdot x+\beta\cdot \xi+\omega t)}\,dx\,d\xi\,dt.\] Let $\mathcal{F}_{1}$, $\mathcal{F}_{2}$, and $\mathcal{F}_{3}$ denote the corresponding Fourier transforms with respect to the first, second, and third variables, respectively. Further, the symbols $p$, $q$, and $r$ will denote the quadratic forms \[q(f)=\sum_{a,\alpha}\left\|\chi_{b_{a},\beta_{\alpha}}(f)\right\|^{2}\] and \[r(f)=\sum_{k\neq 0}{\|2k\|}^{n}\left\\|\rho_{\omega_{k}}(f)\right\\|_{\mathrm{HS}} ^{2}\] and \[p(f)=q(f)+r(f).\] The result to be proven, in effect, is that for $M>0$ sufficiently small, the seminorm $p^{1/2}$ is equivalent to $\left\\|{\,\cdot\,}\right\\|_{L^{2}(E)}$. The key step in this proof will be a simple extension of Duffin and Schaeffer’s (8, Lemma II) for the domain $E$. Specifically, given $J={\mathbb{Z}^{2n}\times 2\mathbb{Z}}$ and given $\tilde{\ }:J\rightarrow\mathbb{R}^{2n+1}$ and given that the number \[M^{\prime}=\sup_{z\in J}\left\\|\tilde{z}-z\right\\|_{\infty}\] is sufficiently small, there is $T=T(M^{\prime})$ such that \[\sum_{z\in J}\left\|\hat{f}(\tilde{z})-\hat{f}(z)\right\|^{2}\leq T(M^{\prime}) \sum_{z\in J}\left\|\hat{f}(z)\right\|^{2}\] for every $f\in L^{2}(E)$. By the triangle inequality, this means that the quantity \[\sum_{z\in J}\left\|\hat{f}(\tilde{z})\right\|^{2}\] (8) is bounded above and below by the quantity \[\left(1\pm T(M^{\prime})^{1/2}\right)^{2}\sum_{z\in J}\left\|\hat{f}(z)\right\|^ {2}=\left(1\pm T(M^{\prime})^{1/2}\right)^{2}\left\\|f\right\\|_{L^{2}(E)}^{2}.\] Thus, it suffices to show that $p(f)$ is bounded above and below by positive multiples of (8) for some $\tilde{z}$’s for which $M^{\prime}=M$. For $\omega\neq 0$ and $f\in L^{2}(E)$, it will be useful to obtain a formula for $\left\\|\rho_{\omega}(f)\right\\|_{\mathrm{HS}}^{2}$. By an argument in Chapter 7 of [11], the operator $\rho_{\omega}(f):L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$ has Hilbert-Schmidt norm \[\left\\|\rho_{\omega}(f)\right\\|_{\mathrm{HS}}^{2}=\frac{1}{\|\omega\|^{n}}\int \int\left\|\mathcal{F}_{3}f(u,v,\omega)\right\|^{2}\,du\,dv\] for Haar measure on $H_{n}$ normalized to coincide with Lebesgue measure on $\mathbb{R}^{2n+1}$. Further, the facts that $g=\mathcal{F}_{3}f({\,\cdot\,},{\,\cdot\,},\omega)$ is supported on $D$ and is square-integrable imply that $\left\\|\rho_{\omega}(f)\right\\|_{\mathrm{HS}}^{2}$ may be written using the $2n$-dimensional Fourier series expansion of $g$ as \[\left\\|\rho_{\omega}(f)\right\\|_{\mathrm{HS}}^{2}=\frac{1}{\|\omega\|^{n}}\sum_{ a,\alpha\in\mathbb{Z}^{n}}\left\|\mathcal{F}_{1}\mathcal{F}_{2}\mathcal{F}_{3}f (a,\alpha,\omega)\right\|^{2}=\frac{1}{\|\omega\|^{n}}\sum_{a,\alpha}\left\|\hat{f }(a,\alpha,\omega)\right\|^{2}\] for any $\omega\neq 0$. Consider $\|r(f)-\phi(f)\|$, where \[\phi(f) =\sum_{k\neq 0}\sum_{a,\alpha}\left\|\hat{f}(a,\alpha,\omega_{k}) \right\|^{2}\] (9) The quantity has the following upper bound: \[\left\|r(f)-\phi(f)\right\| \leq\sum_{k\neq 0}\left\|\left\|\frac{{2k}}{\omega_{k}}\right\|^{n}- 1\right\|\|\omega_{k}\|^{n}\left\\|\rho_{\omega_{k}}(f)\right\\|_{\mathrm{HS}}^{2}\] \[\leq\sup_{k\neq 0}\left\|\left\|\frac{{2k}}{\omega_{k}}\right\|^{n}- 1\right\|\sum_{k\neq 0}\|\omega_{k}\|^{n}\left\\|\rho_{\omega_{k}}(f)\right\\|_{ \mathrm{HS}}^{2}\] \[=\sup_{k\neq 0}\left\|\left\|\frac{{2k}}{\omega_{k}}\right\|^{n}-1 \right\|\phi(f).\] For $M\ll 1$, a bound may be obtained by replacing $\left\|\omega_{k}\right\|^{n}$ by ${\|2k\|}^{n}-nM{\|2k\|}^{n-1}$. A corresponding bound on the supremum terms is $nM/({\|2k\|}-nM)$, which is decreasing in $\left\|k\right\|$. Thus, the supremum term is less than $C(M)=nM/({2}-nM)$, which goes to zero as $M$ goes to zero. In other words, \[(1-C(M))\phi(f)\leq r(f)\leq(1+C(M))\phi(f).\] (10) The inequality \[(1-C(M))(\phi(f)+q(f))\leq p(f)\leq(1+C(M))(\phi(f)+q(f))\] (11) results from adding $(1-C(M))q(f)\leq q(f)\leq(1+C(M))q(f)$ to (10). For each $b,\beta\in\mathbb{R}^{n}$, the quantity $\chi_{b,\beta}(f)$ is equal to $\hat{f}(b,\beta,0)$, so \[q(f)=\sum_{a,\alpha\in\mathbb{Z}^{n}}\left\|\hat{f}(b_{a},\beta_{\alpha},0) \right\|^{2}\] for Haar measure as above. Thus, combining the above with (11) and (9) gives \[(1-C(M))\sum_{z\in J}\left\|\hat{f}(\tilde{z})\right\|^{2}\leq p(f)\leq(1+C(M)) \sum_{z\in J}\left\|\hat{f}(\tilde{z})\right\|^{2}\] where, when $k=0$, $(a,\alpha,{2}k)\tilde{\ }=(b_{a},\beta_{\alpha},0)$ and, when $k\neq 0$, $(a,\alpha,{2}k)\tilde{\ }=(a,\alpha,\omega_{k})$. For these values of $\tilde{z}$, the number $M^{\prime}$ is equal to $M$, and \[(1-C(M))(1-T(M)^{1/2})^{2}\left\\|f\right\\|_{L^{2}(E)}^{2}\] \[\leq p(f)\] \[\leq(1+C(M))(1+T(M)^{1/2})^{2}\left\\|f\right\\|_{L^{2}(E)}^{2}\] as desired. Observe that by making the perturbations small, $A$ and $B$ can be made as close to one as desired, resulting in a “nearly Parseval” OVF of representations. Thus, viewing the list of representations $\{\chi_{b_{a},\beta_{\alpha}}\}\cup\{\rho_{\omega_{k}}\}$ with the appropriate number of repetitions, the desired result about OVFs of representations on $H_{n}$ is obtained: all that is needed to specify one is a sequence of numbers satisfying a Duffin-Schaeffer type stability condition. In particular, since an element $f$ in a Hilbert space $\mathcal{H}$ is uniquely specified by $\{T_{j}f\}$ when $\{T_{j}\}$ is a OVF for $\mathcal{H}$, the preceding shows that $f\in L^{2}(E)$ is uniquely specified by $\{\pi_{j}(f)\}$. ## 4 Conclusion The preceding sections have described what it means for a OVF of representations on a locally compact Lie group $G$ to be harmonic. For $G=H_{n}$, $\Gamma=\mathbb{Z}^{2n}\times\frac{1}{2}\mathbb{Z}$, and $E=D\times(-1/4,1/4)$, a family of OVFs that are are “almost harmonic” was constructed by perturbing a harmonic OVF of representations in a particular way. This construction is analogous to the development of frames of non-harmonic exponentials in $L^{2}(E)$ starting with an orthonormal basis of harmonic exponentials given by Duffin and Schaeffer. The OVFs constructed here appear to stand in contrast those found in current literature, which are typically generated as the unitary orbit of a single fixed operator [12; 14]. The nature of the OVFs introduced here is more similar to the non-harmonic Fourier frames of [8] than to wavelet or Gabor systems, and they are more closely related to the problem in representation theory described above. The development in this paper is restricted to OVFs for $L^{2}(E)$, where $E$ is a proper subset of $G$. A possible extension of interest is the case where $E=G$, seeking a theory that provides an analysis of $L^{2}(G)$ that provides features akin to Gabor analysis for $L^{2}(\mathbb{R})$. As noted, the condition set forth by Duffin and Schaeffer to get a Fourier frame is quite general, whereas the condition given in this paper is less so. It would be interesting to quantify the deviation from harmonic OVFs that is possible while still remaining within the set of OVFs. ## References (1) Arthur, J.: An introduction to the trace formula. In: J. Arthur, D. Ellwood, R. Kottwitz (eds.) Harmonic Analysis, the Trace Formula, and Shimura Varieties, vol. 4, pp. 1–263. Amer. Math. Soc. and Clay Math. Inst. (2005) * (2) Avdonin, S.A.: On the question of Riesz bases of exponential functions in ${L}_{2}$. Vestnik Leningrad Univ. Ser. Mat. 13, 5–12 (1974) * (3) Bourbaki, N.: Integration (1963) * (4) Casazza, P., Kutyniok, G.: Frames of subspaces. In: C. Heil, P.E. Jorgensen, D.R. Larson (eds.) Wavelets, Frames, and Operator Theory, vol. 345, pp. 87–113. Amer. Math. Soc. (2004) * (5) Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 114–132 (2008) * (6) Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271 (1986) * (7) Dietmar, A., Echterhoff, S.: Principles of Harmonic Analysis, vol. 1. Springer (2009) * (8) Duffin, R., Schaeffer, A.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. pp. 341–366 (1952) * (9) Feichtinger, H.G., Gröchenig, K.: Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In: C.K. Chui (ed.) Wavelets: A tutorial in theory and applications, vol. 2, pp. 359–398. Academic Press, Boston (1992) * (10) Feichtinger, H.G., Gröchenig, K.: Irregular sampling theorems and series expansions of band-limited functions. J. Math. Anal. Appl. 167(2), 530–556 (1992) * (11) Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press (1995) * (12) Han, D., Larson, D.R., Liu, B., Liu, R.: Operator-Valued Measures, Dilations, and the Theory of Frames. Mem. Amer. Math. Soc. (2014) * (13) Kadec, M.I.: The exact value of the Paley-Wiener constant. In: Dokl. Akad. Nauk SSSR, vol. 155, pp. 1253–1254 (1964) * (14) Kaftal, V., Larson, D., Zhang, S.: Operator-valued frames. Trans. Amer. Math. Soc. 361(12), 6349–6385 (2009) * (15) Lindner, A.: On lower bounds of exponential frames. J. Fourier Anal. Appl. 5(2), 187–194 (1999) * (16) Moran, B., Howard, S.D., Cochran, D.: Positive-operator-valued measures: A general setting for frames. In: T. Andrews, R. Balan, W. Czaja, K. Okoudjou, J. Benedetto (eds.) Excursions in Harmonic Analysis, vol. 2, pp. 49–64. Springer (2013) * (17) Sun, W.: G-frames and g-Riesz bases. J. Math. Anal. Appl. 322(1), 437–452 (2006) * (18) Thangavelu, S.: Harmonic analysis on Heisenberg nilmanifolds. Rev. Un. Mat. Argentina 50(2), 75–93 (2009)

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.